Quantum Field Theory ( QFT ) / Quantum Optics ( QED ) Mukul Agrawal April19, 2004

Contents I II 1

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Conventions/Notations

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Background/Introduction

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Quantum Field Theory ( QFT ) - Why We Need It? 1.1 Two “Flavors” of QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 QFT as a Special Application of Qunatum Mechanics . . . . . . . . . . . . . . . 1.3 QFT as a Fundamentally New Physics – Relativistic Multiparticle Quantum Physics 1.3.1 QFT as convenient alternative language for non-relativistic multiparticle quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 In Summary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Comments on Philosophy of QM 2.1 Brief Comment on Linearity of Physical Systems in Quantum Mechanics 2.2 Brief Comment on Waves and Particles . . . . . . . . . . . . . . . . . . 2.2.1 Usage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Usage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Usage 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Brief Comments on Canonical Quantization . . . . . . . . . . . . . . . . 2.4 Brief Comments on Identical Particles . . . . . . . . . . . . . . . . . . .

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III A First Introduction to QFT- Non Relativistic Canonical Quantization of Classical Wave Phenomenon - Quantization of Displacement (Elastic) Fields 14

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Single Harmonic Oscillator 3.1 Quick Classical Recap . . . . . . . . . . . 3.1.1 Complex Variable Transformation . 3.2 Quantization of Single Harmonic Oscillator 3.2.1 Normal Ordered Hamiltonian . . . 3.3 Alternative Interpretation . . . . . . . . . . 3.4 Commutation and Bosonic Behavior . . . .

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Chain of N Coupled Harmonic Oscillators - Quantization in Real Space

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Chain of N Coupled Harmonic Oscillators - Quantization in k - Space 22 5.1 Alternative Interpretation – Phonons . . . . . . . . . . . . . . . . . . . . . . . . 26

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Continuum – Quantization of Elastic Waves in Continuous Media

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QFT As A Non-Relativistic Multi-Particle Quantum Theory Identical Particle Hilbert Space Representations 7.1 Fock Space and Fock States . . . . . . . . . . . . . . 7.1.1 Basis Conversion . . . . . . . . . . . . . . . 7.2 Operators in Fock Space . . . . . . . . . . . . . . . 7.3 Ground State or Vacuum . . . . . . . . . . . . . . . 7.4 Creation and Annihilation Operators . . . . . . . . . 7.5 The Wavefunction Operators or The Field Operators . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . .

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Canonical Quantization of Schrodinger’s Field (Second Quantization)

Non-Relativistic Quantum Field Theory ( QFT )

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Non-relativistic, Massive, Fermionic Fields 38 9.1 First Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.2 Second Scheme – Second Quantization of Schrodinger’s Field . . . . . . . . . . 42

10 Non-relativistic, Massive, Bosonic Fields

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Relativistic Quantum Field Theory ( QFT )

11 Klein Gordon (KG) Field 44 11.1 Derivation of ’Classical’ KG Field . . . . . . . . . . . . . . . . . . . . . . . . . 44 11.2 Classical Hamiltonian Theory of Real Valued Scalar KG Field . . . . . . . . . . 45

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11.2.1 Classical Hamiltonian and Lagrangian . . . . . . . . . . . . . . 11.2.2 Ψ(x,t) Field in Momentum and Energy Basis . . . . . . . . . . 11.2.3 π(x,t) Field in Momentum and Energy Basis . . . . . . . . . . . 11.2.4 Classical Hamiltonian in Momentum and Energy Basis . . . . . 11.2.5 Field Expansions in Normalized Variables . . . . . . . . . . . . 11.2.6 Inverse Transforms . . . . . . . . . . . . . . . . . . . . . . . . Quantization of Real-Valued Scalar ’Classical’ KG Field . . . . . . . . 11.3.1 Quantized Field Operators . . . . . . . . . . . . . . . . . . . . 11.3.2 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Number State basis . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Coherent and Squeezed States . . . . . . . . . . . . . . . . . . 11.3.5 Heisenberg Representation . . . . . . . . . . . . . . . . . . . . 11.3.6 Two Point Correlation Functions/Propagators . . . . . . . . . . Classical Hamiltonian Theory of Complex-Valued Scalar KG Field . . . 11.4.1 Classical Hamiltonian and Lagrangian . . . . . . . . . . . . . 11.4.2 Classical φ(x,t) and φ? (x,t) Fields in Energy/Momentum Basis 11.4.3 Classical π(x,t) and π? (x,t) Fields in Momentum/Energy Basis 11.4.4 Classical Hamiltonian in Energy Basis . . . . . . . . . . . . . . 11.4.5 Classical Field Expansions in Normalized Variables . . . . . . . 11.4.6 Classical Inverse Transforms . . . . . . . . . . . . . . . . . . . Quantization of Complex-Valued Scalar KG Field . . . . . . . . . . . . 11.5.1 Quantization Field Operators . . . . . . . . . . . . . . . . . . . 11.5.2 Quantized Field Operators in Energy/Momentum Basis . . . . . 11.5.3 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 11.5.4 Number State Basis . . . . . . . . . . . . . . . . . . . . . . . . 11.5.5 Associated Charge . . . . . . . . . . . . . . . . . . . . . . . . Quantization of Complex Valued Vector Klein Gordon (KG) Field . . . 11.6.1 Quantized Field Operators and Hamiltonian . . . . . . . . . . . 11.6.2 Associated Charge . . . . . . . . . . . . . . . . . . . . . . . .

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12 Dirac Field

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Quantum Electrodynamics ( QED )

13 Non-relativistic Electromagnetic Field Quantization ( Quantum Optics ) 65 13.1 Coherent States / Poisonian Distribution . . . . . . . . . . . . . . . . . . . . . . 68 14 Relativistic Electromagnetic Field Quantization ( QED )

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Interaction Between Free Fields

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15 Different Pictures of Time Evolution 15.1 Exponential of Operators . . . . . . . . . 15.2 Schrodinger’s Picture of Time Evolution . 15.3 Heisenberg Picture of Time Evolution . . 15.4 Interaction Picture . . . . . . . . . . . . . 15.5 Inter-Relation . . . . . . . . . . . . . . . 15.6 Diagrammatic Perturbation Theory . . . . 15.7 Expectation Values . . . . . . . . . . . . 15.7.1 Quantum Ensemble Expectations

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Statistical Quantum Field Theory (Condensed Matter Physics) References

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References

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Further Resources

Part I

Conventions/Notations • Some books use a “hat” or “caret” on top of a symbol to represent an operator as opposed to a classical variable. I prefer not to use these “hats” but they sometimes become essential. Usually, the distinction between a classical variable and a quantum operator is generally clear from the context. At the same time, I believe, text is more readable if simple symbols with least numbers of subscripts, superscripts, hats, bars, bold faces etc. are used! In this article, “hat” on a symbol is used explicitly only in places where there are chances of confusion. Sometimes in the initial development of the subject we would use these hats to explicitly differentiate between classical and quantum analogues but would then gradually get rid of it. • Creation and annihilation operators are always written in Schrodinger picture. Other field operators would have subscripts to explicitly differentiate between the Schrodinger, Heisenberg or interaction picture. • I would start with SI units initially. But I would also try to gradually move toward natural units (~ = 1, c = 1) rather than SI units as we develop the subject.

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Part II

Background/Introduction 1 1.1

Quantum Field Theory ( QFT ) - Why We Need It? Two “Flavors” of QFT

QFT has two flavors. On one hand its a simple application of standard non-relativistic quantum mechanics to describes the classical wave phenomena in quantum world. For example quantum physics of elastic waves is described by QFT which in this problem is not a fundamentally new science. Its a very simple application of standard non-relativistic quantum mechanics in systems with infinite degrees of freedom. On the other hand QFT also describes quantum physics of classical particles. In non relativistic limit, physics involved in QFT is same as physics involved in standard quantum mechanics. This is a special case. More generally, quantum physics of classical particles in relativistic limit is a fundamentally new science known as QFT. This is can be called a non-trivial extension of quantum mechanics (this extension would need new postulates) but it is certainly not a mere application of standard quantum mechanics.

1.2

QFT as a Special Application of Qunatum Mechanics

In its one of the incarnation, QFT is simply a quantum theory of systems that can have infinite degrees of freedom. QFT is simply a quantum theory of classical problems which has many coupled dynamical variables. Notice that a field is nothing but a bunch of coupled variables. For example a scalar field A(x,t) can simply be represented as a bunch of variables Ax (t). Notice that the time evolution of one of the variable Ax1 (t) might depend on the time evolution of other variable Ax2 (t). This second notation, I guess, makes it amply clear what we mean by multiple coupled variables. Canonical quantization of classical field would now be done in exactly same fashion as the canonocal quantization of other many variable problems like multi-particle problems. We would work out the details in sections below.

1.3

QFT as a Fundamentally New Physics – Relativistic Multiparticle Quantum Physics

Fundamentally, Quantum Field Theory (QFT) is required to combine ideas from special relativity and quantum mechanics. In relativistic regime, quantum physics can not be described in any way that would closely resemble standard non-relativistic quantum mechanics. As we would discuss below, there is no consistent theory of single particle relativistic quantum mechanics (similar to Schrodinger’s equation that represents mechanics of single non-relativistic particle in sub-atomic world). In literature, there exist many attempts at it (like Pauli’s equation, Klein Gordon equation or Dirac equation etc – each of which we would discuss below, but in more modern formalism). 5

But, as we would see, all of them suffer from one or the other difficulties1 . Fundamentally, these difficulties arise from the fact that Einstein mass-energy equivalence removes the possibility of single particles. Since, particles can dynamically break into smaller mass particles and smaller mass particles can recombine to give larger mass particles, any relativistic quantum theory needs to be fundamentally a multi-particle theory. Hence there is no single particle relativistic quantum mechanical equation. Simplest we can have is a multi-particle equation which describes the motion of potentially infinitely many particles with possibility of annihilation and creation. What follows is called quantum field theory. The reason this subject goes by the name of “field theory” is because we start from field equations and the concept of particles (infinitely many of them) emerges as we develop the formalism. At this point, let us remind ourselves how the subject of non-relativistic quantum mechanics is usually developed. We start with classical single particle equation of motion. Perform canonical quantization and obtain single particle quantum equation of motion (which goes by the name of Schrodinger’s equation). It turn out that this euqation of motion is actually a “wave equation” and it correctly represents the dynamics of single non-relativistic particle in sub-atomic world. Now suppose I want to develop a non-relativistic multi-particle quantum theory. It is a very simple extension of single particle quatum mechanics. All we need to do is to use direct-product spaces and direct-product states (see below). Just for fun, we can try another alternative route (we know it works even though there is no reason for it to work – and that’s why its fun!). We can take the Schrodinger’s single particle quantum wave equation and “quantize” it treating it as a classical “field”2 (details of how classical fields are quantized would be discussed below). Why are we trying this alternative route? This would become clear when we talk about relativistic physcis in a bit. Now, if we try this second approach, what emerges is multi-particle quantum theory! Yeah – a multi-particle theory out of a single particle theory3 ! Exactly same physics as what we get from previous route. Srprised? Don’t worry, it is very surpring at first sight. First approach usually seems to be more direct approach but remember that the canonical quantization itself is a postulation — so saying that this second approach is more arbitrary than the first approach is, strictly speaking, not correct. Both methods are postulations which we know work. Now, people tried copying this exact same reciepe for relativistic physics. There was some bad news and some good news. Bad news first. If we start with single particle relativistic classical equation of motion and quantize it, we don’t get a self consistent theory of single particle relativistic quatum physics. We get a wave equation (or a field) which has many problems. This euqation can not be treated as a valid “single particle” equation of motion. Now the good news. If we treat this equation of motion as a classical field and quantize it, we do get a valid multi-particle relativistic quantum physics. This whole procedure has no justification. It should, strictly speaking, be treated as postulation. As a side note – non-relativistic physics was just a special case. In general, a multi-particle theory can not fall out of a single particle theory without further postulations. Philosophically, this a bit comforting. Secondly, we were very lucky that canonical quan1 Most

perturbing problems were existence of negative energy states, non-positive-definite probability and nonunitary Lorentz transformation. 2 Note that term “field” is more apropriate than the term “wave”. Entity that we are trying to quatize is simply a function of space and time – it might not necessarily represent anything “propagating” or “moving”. 3 Non-relativistc physics is a special case – this does not happen in more general relativistic case.

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tization of calssical particles physics lead us to Schrodinger/Heisenberg/Dirac/Von-Neumann etc type of quantum particle physics which worked. Again this is a special case. Canonical quantization of relativistic classical “particle” physics does not lead to any valid quantum physics. So, in conclusion, fundamentally, particle theories actually emerges from field theories with non-relativistic situation being a special case where field theory does not take a fundamentally important stand (and actually seems very uncomfortable, unnecessary and bizarre). Second point that I want to stress again before closing this section is that this whole part of science which deals with canonical quantization, is not usually taught in very accurate fashion. These are just “tricks” that give us physics that experiments have confirmed to be true. These tricks can not even be called postulates in mathematically rigourous sense. Just like in standard non-relativistic quantum mechanics, there is no way to “prove” or “derive” expressions for quantum Hamiltonians (canonical quantization is just a trick that works for a few important problem), simmilarly there is no way to derive Hamiltonians in terms of field operators (that we use in QFT). These are supposed to be fundamental postulates. But there exist a simmilar canonical quantization scheme, as we would study latter, using which we can “derive” quantum field theory operators starting from a “wrong” supposedly-single-particle relativistic quantum wave equation (like Dirac equation of KG equation). 1.3.1

QFT as convenient alternative language for non-relativistic multiparticle quantum physics

As I said above, it is not necessary to quantize a field to obtain multi-particle non-relativistic quantum mechanics. One can obtain the same physics using the concept of direct product spaces in seemingly more direct approach. So, QFT is not essential for multi-particle non-relativistic systems. As the methods and tools of QFT gained popularity, we have learned that they are very useful even in non-relativistic multi-particle situations. Almost all problems in condensed matter physics (a big class of these problems are non-relativistic) are now a days conveniently described in terminologies and formalism of QFT.

1.4

In Summary ...

As is clear from above discussion, QFT as two utilities. First, it is a quantum theory of classical waves. Second, it is a quantum relativistic/non-relativistic theory of multi-particle (varying number of particle or infinite number of particles – both can be handled) physics. As far as first usage is concerned, QFT is not a completely new theory. At least if we restrict ourself to non-relativistic systems, this formalism has nothing fundamentally new physics in it (just a few mathematical tools required to handle infinite degrees). In this case it is a very simple extension of elementary quantum mechanics. Now as for the second usage, it becomes a bit tricky. As we would see latter, we treat single particle quantum mechanics (like Schrodinger’s equation or the Dirac equation etc) as “classical” wave equation and quantize it using same rules as those used for quantizing classical waves. We would see that this leads to multi-particle quantum theory (relativistically consistent). This new story is not mathematically as stright forward, though. As we move to relativistic 7

systems, mathematical formulation becomes very intricate. The reason is that it was not very simple to enforce Lorentz invariance on classical wave equations. Moreover representations of Lorentz transformation in Hilbert space has it’s own nitty-gritties. So mathematically speaking, relativistic QFT 4 is far more difficult than elementary quantum mechanics (although very little completely new physics as such). To summarize, here are a couple of fundamental reason why we need QFT :• QFT is needed to explain quantization of classical wave phenomenon - with no quantum description available. One can argue that as far as non-relativistic Schrodinger’s equation is considered QFT only gives a new interpretation of same physics included in Schrodinger’s wave equation5 . But when we quantize fields like electromagnetic fields (which has no quantum description besides QFT) we get fundamentally new concepts using QFT. QFT makes various predictions of non-classical “states of fields” – like squeezed light etc. For example if you pick one normal mode of EM wave then classically all you can change is its amplitude or phase. In classical light, if you pick one normal mode, there is a Poissonian distribution of probability of number of photon – that is its a linear superposition of states having different number of photons in that EM mode with expansion coefficients being Poissonian distributed. But QFT predicts that this mode can be in many other states like a superposition of having one and two photons in one normal mode of EM wave — such a state of light would not give Poisonian distribution in photon arrival time. There are many more such phenomena that can not be explained without QFT. • In relativistic QM, QFT removes many problems associated with negative energies, infinite energies and negative probability densities by coherently bringing the concepts of antiparticles. For example, if we try to solve free-space Klein-Gordon equation (which is just a simple relativistic version of free-space Schrodinger’s equation and is known to be correct description of a family of particles despite some earlier historical doubts), ’each’ state of the system is associated with two energies - one positive and one negative. One does not know how to interpret it physically staying within non-quantized wave equation description. QFT resolves such difficulties by predicting the presence of anti-particles each particle has its anti-particle that sits in the negative energy state (somewhat like holes in semiconductor theory). QFT predicts existence of many such particles. Besides, QFT also provides a very neat and straight forward conceptual formulation under varied circumstances (that can strictly be studied in conventional QM as well), like :4 Fundamental

physicists might insist here that “relativistic” qualifier is not required as QFT in fundamentally not required for non-relativistic systems – for them QFT is relativistic QFT! 5 Technically speaking, second quantization involves no new physics. But field quantization do. Second quantization can be studied in two ways. One can either use concepts of field quantization and quantize the Schrodinger’s field – in such a case it does appear that we are postulating new physics. Alternatively, one can simply use the anti-symmetry (Pauli’s Exclusion) postulate of Fermi particles to obtain exactly same physics as that by field quantization. Such alternative routes are not available if a quantum description (something like Schrodinger’s equation) does not already exist - such as in EM waves case. In such case field quantization is a postulation of new physics.

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• It removes the different notions of single-particle equation or multi-particle equation and brings in a more complete picture. Its makes it much easier to describe real physical situation where number of particles is not constant and keeps on changing as a result of interaction with surroundings. All QFTs are multi-particle theories. One starts from so called single-particle Schrodinger’s equation and gets a multi-particle QFT. This is no miracle. If we look at conventional quantum mechanics, even with single particle Schrodinger equation one can study multi-particle systems if one ignores particle-particle interaction. If you build a QFT of electrons what you get is a non-interacting multi-electron theory6 . In QFT inter-particle interaction is treated as inter-field interaction. Main benefit of QFT is its conceptually simple and straight forward formulation. • It removes the notion of interacting or non-interacting multi-particle theories. Interparticle interaction comes from ’inter-field interaction’. For example quantization of Schrodinger’s equation leads to multi-particle QFT. For including electron-electron interaction one needs to include the photons in the picture. So one type of field only brings out one character of a type of particle. One needs to study inter-field interaction to get a complete picture of an electron, for example. This kind of set up is much more handy - at least conceptually.

2

General Comments on Philosophy of QM

2.1

Brief Comment on Linearity of Physical Systems in Quantum Mechanics

• One would notice the equations governing the motion (like Schrodinger’s equation) in quantum physics are linear. So when we claim that we understand all the fundamental laws of how universe operate, how are we proposing to handle non-linear processes? Please check the other tutorial article on linearity of dynamical equations in quantum physics. Concepts explained there hold even for QFT.

2.2

Brief Comment on Waves and Particles

In quantum physics waves/fields and particles are very closely interleaved – much more than what we have seen in Schrodinger’s type quantum mechanics. One can describe each and every phenomena through particles only (including static forces like electrostatic attraction as well as interference/diffraction – we don’t need any field concept) or using waves/fields only. In fact all particles are made up of fields themselves in QFT. However, while building these two exactly equivalent formalism we would define very clearly what we actually mean by ’particle’ 6 Description

remains incomplete unless all other fields are included as well. For example in QFT electronelectron Coulomb interaction comes from the interaction of electron QFT with photons QFT. So one may want to call the Schrodinger’s quantized field (without the presence of other fields) as ’Quasi-Electrons’ – they are strictly eigen energy state as long as other fields are not present. See the following points to make things clear.

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and what we actually mean by a ’wave’. We would see that these are not exactly same as our naive understanding of particle and wave concepts. In the prevalent literature of modern physics people speak of a ’particle’ or a ’wave’ in several very distinct senses:2.2.1

Usage 1

• ’Elementary particles’ are often simply called ’particles’. We would understand these better while studying QFT. These results from the studies of free fields (no interaction). ’Particles’ are treated as a ’quanta’ of something like spin angular momentum or mass or electric charge etc (each elementary particle has a set of associated quantum numbers). So we claim (this actually falls of naturally from the theory that we would discuss below, its not treated as a separate postulate) that these quantities like charge or mass (or it may be something else) can only be incremented or reduced in steps and not continuously. This is the correct meaning of particle behaviour. For, example when electron gets absorbed from a quantum well by a physically “localized” absorber, “entire electron” gets absorbed “at once” from entire quantum well irrespective of the shape of wavefunction. The process is not something like water flow going into a drain. How relativistic causality is preserved in this process would be explored in the theory of QFT. Note that localization in physical space should not be considered as a proof or guarantee of particle behaviour. On the other hand, interference and diffraction should be associated with correct meaning of wave behaviour. To allow both the effects in one physical entity, say an electron, under different experimental setup, the way we explain the physics is to propose that there is ’something’ (like wavefunctions in Schrodinger’s mechanics) associated with every system that is more fundamental than the properties we can measure. So the kind of language we would be using while studying quantum physics is that – certain “mode” has so many numbers of “quanta” of certain quantity. The “mode” is usually a state vector in state space of continuous functions. Various “modes” are allowed to go through interference which finally leads to interference effects in observable quantities like spatial distribution etc. Note that simply associating a spatial function is not a proof or guarantee of wave like behaviour. We need to allow these spatial wavefunction to take negative or complex values (mathematical it should form a vector space) so that we can add them to get interference effects. A positive-definite spatial function can not show interference effects. Also, very roughly speaking, the amplitude of “mode” can only change in discrete fashion. And this would finally lead to particle behaviour. And this is how wave-particle duality is inbuilt in quantum mechanics. So this state vector or mode itself can be a function of spatial coordinates that describes the distribution of quantity concerned in space in some particular fashion. This space function may or may not accept a straight Born probability interpretation that we have seen in Schrodinger’s quantum mechanics. The mapping between state vector and actual amplitude/probability distribution (Born like interpretation) depends on the quantity we are talking about. This interpretation again would fall of naturally as we develop the theory. • A comment should be added here about widespread misconception about energy and mo10

mentum of elementary particles. It is widely taught that photons are supposed to be monochromatic (state of definite energy and state of definite momentum in freespace). Remember E = ~ω and p = ~k? This statement is completely incorrect7 . We can take a photon and place it in a linear superposition of two energy eigen state (in free space for example). So we can have a single photon whose energy and momentum are not well defined. 2.2.2

Usage 2

• In condensed matter physics, when we quantize complicated interacting system, we would be lead to many other types of ’quanta’ that are not elementary particles. Examples of these are electron-polarons (quanta of coupled electron phonon system) or phonon-polariton (quanta of coupled phonon-photon system) etc. Basically these are elementary excitation of ground state (lowest energy state) of such systems. By ’elementary excitation’ we mean that ’minimal’ excitation of system beyond ground state would have one of these elementary excitation. Some people refer to quanta defined like this either as normal ’particles’ or ’quasi-particles’ depending upon the “lifetime”. Quasi-particles are something that represent energy eigen state as long as interactions with other particles/fields are neglected. Interactions give finite lifetime. In this sense, actually everything known as ’particles’ are just ’quasi-particles’! • We need to clarify the meaning of last statement a bit further. Let us assume that we have a single molecule with one extra electron. What happens is that the molecule goes into a different state of vibrations when extra electron is added. This is what we call as electron-polaron. Let us assume that there are no other interactions. So if we place electron-polaron in an energy eigen state of this multi-particle system, it would remain in that state for ever (life time of the state is infinite). We can also place this electron-polaron, if we want, in a linear superposition of two energy eigen states of this multi-particle system. Then system would keep on oscillating between two energy eigen state for ever. But the electron-polaron would never ’decompose’. In realistic cases, lifetime is never infinite but can be long compared to other processes (electron-polaron would decompose after long time but not within the time scale we maybe interested in). In such a case people simply refer to this elementary excitation as a ’particle’. In case this elementary excitation decays faster than the time scale we are studying, people usually refer to it as ’quasi-particle’. 2.2.3

Usage 3

• The other common usage of the term ’particle’ is more intended for fixing analogies between (loose) classical and quantum interpretations. In a very loose classical sense, a particle is supposed to have a definite position and velocity whereas waves never have both of them in definite quantities at same time. Note that this is not a real indication of particle behaviour even in classical mechanics - lots and lots of people misunderstand the concept 7 Two

expressions, known as Einstein and De Broglie relations respectively, are correct depending upon how you interpret and how you use them. Abuses of these in both teaching and practice are common!

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of particles. For example you can take a completely classical acoustic wave and create wave-packets out of it – and it might seem that you got the wave-particle duality explained with a theory that is fundamentally a wave theory! Certainly not, because you would never be able to explain instantaneous and localized absorption of phonons from everywhere in one go (process would always be like water flowing into drain as long as you stay in classical mechanics). So the way people very loosely put it is that — ’wave-particle duality can be resolved/institutionalized by accepting that actual world obeys field/wave mechanics but one can obtain ’particle’ like behaviour by creating wave-packets by superposition of field/waves’. This is interpretation is not correct. It gives an impression that quantum mechanics is just a simple classical wave-mechanics. I have even seen books on quantum mechanics with titles like “Wave Mechanics”. Complete resolution of wave/particle duality comes only when one understands that wave-like modes or quantum states (which obeys linear algebra) contains quantized amount of matter. For example example when electron would get absorbed, electron would disappear from ’every-where’ unlike classical wave theories. Even in Schrodinger’s quantum mechanics, you would appreciate the real strength of quantum mechanics only when you start building multi-particle theory and start studying things like emission and absorption of particles. • In conclusion, one should understand that if you associate interference/diffraction phenomenon as defining properties of waves then the wave-like behavior is built-in in quantum mechanics since it associates a more fundamental ’entity’ to all system and that entity has a linear algebraic structure. Phrases like ’wave like’ or ’field like’ are often used to indicate that quantity obeys linear algebra and its the ’amplitude’ associated with the quantity that adds up and not the probabilities which are interpreted as square of amplitudes. So this can be taken as another usage/interpretation of word ’wave’. Phrases like - ’particle-like’ actually means that when it is absorbed it is absorbed as a whole from everywhere – physical localization is not real indicator of particle behaviour. Physical localization is easily achievable constructing wave-packets out-off completely classical wave mechanics.

2.3

Brief Comments on Canonical Quantization

Over time, many schemes of quantizing a classical Hamiltonian to obtain a quantum Hamiltonian have been developed. All of them are ultimately just “rules”. For any given problem, these “rules” needs to be verified against empirical data. QFT, as a theoretical framework assumes that Hamiltonian is given. Most popular of these quantization rules is the canonical quantization due to Dirac. In field quantization setup, canonical quantization also goes by the name of second quantization. • Readers might also want to check other articles on Quantum Mechanics and Symmetries in Physical World. • There are many ways of obtaining quantum Hamiltonian and other operators from classical analogues. These are known as quantization rules. Most common of these is the so called 12

canonical quantization due to Dirac. In field theory, canonical quantization is sometimes also called ’second quantization’. • It is instructive, and probably interesting as well, to stress that the Dirac’s canonical quantization rule of ’inventing’ quantum Hamiltonians from the known classical Hamiltonians by converting canonical co-ordinate and momenta into operators and enforcing a commutation relation between them, is in essence, a claim that one can use the same force/interaction laws in subatomic world as in classical macroscopic world. Only the system description (the kinematical part of the story) is different because of Nature’s reluctance to allow us to specify the state of a system beyond certain limit. This statement is true in many common situations but not in general (for examples spins have no classical counterparts and can not be invented from classical Hamiltonians). But it is at the heart of most of the conventional quantum mechanics that we do in day to day semiconductor physics. When experimentally one come to know that it does not work then one look around for a new quantum mechanical Hamiltonian that would describe the new physics. One such famous example is spin. • Also note that time t has a special and distinct status in non-relativistic quantum mechanics and is different from the status of spatial position. We don’t replace t with an operator when using Dirac’s canonical quantization rule. t remains a functional parameter whereas spatial co-ordinates and conjugate momenta are promoted (or may be demoted whatever you like!) to the status of operators. This, intuitively speaking, might be related to the fact that we never observe quantized time (or rather we never explain it that way). • Since all classical physical quantities can always be represented as a function of Cartesian co-ordinates and conjugate momenta, Dirac’s canonical momentum, in effect, tells us that all we need to do is to get new definitions of co-ordinate and momentum such that those definitions somehow include our inability to specify co-ordinate and momentum simultaneously. The rest of all the physical quantities would still hold identical meaning in quantum mechanics and we can use exactly identical definitions. These new definitions would be postulates. Or in other words we postulate a commutation relation between two variables which obey the same classical physics. • Note that Dirac’s quantization rule and the enforced (postulated) commutation works in Cartesian co-ordinates only. If you change the co-ordinates you need to obtain new commutation rules using the co-ordinate transformation equations. One example is the angular momentum. In Hamiltonian formulation angular position and angular momenta can simply be treated as generalized co-ordinates and conjugate momenta but they would not obey Dirac’s simple commutation rule. One can obtain commutation relation between any set of generalized co-ordinates and conjugate momenta starting from the postulated commutation relation of Cartesian co-ordinates and Cartesian momenta. • These would also work only as long as quantities involved have integer spins. This we would explore in more details below. 13

2.4

Brief Comments on Identical Particles

• QFT would result in different families of identical particles. So its important to understand what happens if particles are identical or in other words indistinguishable from each other. • Postulate of symmetrization of basic quantum mechanics states that there are only two types (classes) of identical particles – bosons or fermions. This means that:– All states need to be either symmetric or anti-symmetric. For wavefunctions of two identical particles:φ(r1 , r2 ) = ±φ(r2 , r1 ) – All operators corresponding to physically measurable quantities need to preserve the Bosonic/Fermionic properties of states. This means that operators corresponding to physical properties need to commute with so called exchange operator. Hence operators need to be completely symmetric with respect to exchange of particles. For any operator that operates on two identical particle Hilbert space:A(r1 , r2 ) = A(r2 , r1 ) • Further, one can prove that Bosonic and Fermionic behaviour is associated with particles and not with states (this is not a postulate, this statement can be proved). This means that – Suppose we have a system of two identical particles. Suppose we know that system of particles is in an anti-symmetric state. When we add a third identical particle to the system, it would remain anti-symmetric. One can prove this from physical arguments. – If we have system of three identical particles, there does not exist any state of the system so that it is symmetric with respect to exchange of two of these and antisymmetric with respect to exchange of another two. This statement can also be proved from group theoretic arguments.

14

Part III

A First Introduction to QFT- Non Relativistic Canonical Quantization of Classical Wave Phenomenon Quantization of Displacement (Elastic) Fields As mentioned above, QFT has two usages. For physical phenomenon that are classically wave motions (like electromagnetic waves or continuum elastic waves etc), QFT is a simple extension of quantum mechanics into systems with infinite degrees of freedom. As far as quantization of fields is concerned, we do not need any new physics or any new postulates. Standard quantum mechanics is sufficient. On the other hand QFT also provides us multi-particle theories for those particles which obey single particle quantum mechanical wave equations. For example quantization of Schrodinger’s wave and the Dirac waves gives us non relativistic and relativistic physics of multi-electron systems respectively. This part of QFT is a bit tricky. We treat Schrodinger’s equation or the Dirac equation as a “classical” wave equation and quantize it as usual. As it turns out, this quantization (also known as second quantization) leads to multi-particle physics. In this part we would only discuss the quantization classical wave motion. For this part we would not need any new physics. We would discuss how to apply standard quantum mechanics in systems with infinite degrees and we would discuss how to do canonical quantization of classical waves. In next part we would discuss the second flavor of QFT and we would try to rationalize why we are justified in treating single particle quantum mechanical equation of motions as “classical” wave equations.

3 3.1

Single Harmonic Oscillator Quick Classical Recap

Classical Hamiltonian for a harmonic oscillator with natural frequency of oscillation ω and mass m is written as p2 1 H(q, p) = mω2 q2 + 2 2m q Natural frequency is related to the spring-constant and mass as ω = mk . q and p are Cartesian co-ordinate and momentum respectively. Hamilton’s equation of motion (or equivalently

15

Newtonian equation of motion) can be written as dp = −mω2 q dt

(1)

and

dq =p dt We know that the most general solution for these equations is m

q(t) = A cos(ωt − φ) and

p(t) = Amω sin(ωt − φ)

Here A and φ are real valued variables. Any other solution or superpositions of solutions can always be written in above form 8 . 3.1.1

Complex Variable Transformation

One should notice that A and φ are the real valued variables in the above solution. From above form of the solution for q and p one can immediately see that p is 90o out of phase from q. Hence we can, if we want to, write the solutions as real and imaginary parts of a complex number. In other words, if B is a complex valued number, then we must be able to write such that q ∝ B + B? and p ∝ i(B − B? ). We can write the above written most general solution in a more convenient and completely alternative form as q(t) = C exp(−iωt) +C? exp(iωt) ≡ B(t) + B? (t) and p(t) = −imωC exp(−iωt) + imωC? exp(iωt) ≡ −imω(B(t) − B? (t)) C can now be a complex valued variable and includes the phase information. Let, in most general case, A C = exp(iφ) 2 where A and φ are again real valued, then we retrieve our previous form of the solution q(t) = A cos(ωt − φ). One can now immediately check that q2 = CC exp(−i2ωt) +C?C? exp(i2ωt) + 2CC? 8 Here comes a very important point. One should notice that the reason q(t) and p(t) are real valued functions is because the classical equation of motion ( 1 ) has second order time derivative of q(t). If this derivative had been a first order then complex solution would be acceptable. As we would see in the next section, quantum mechanical equation only involves a first order time derivative (Schrodinger’s equation). Hence the solutions are complex exponentials. So we would have to add two solutions to get a real valued solution. This a very generic result. A very important difference between non-relativistic and relativistic QFT of elementary particles. It also creates anomalies within non-relativistic QFT. For example, one would also notice that the wavefunction operator for a single phonon would include two terms (so called positive energy and negative energy terms) whereas non-relativistic single particle electron wavefunction operator contains only one positive energy term.

16

and

p2 = −m2 ω2CC exp(−i2ωt) − m2 ω2C?C? exp(i2ωt) + 2m2 ω2CC?

Hence, the Hamiltonian can now be written as 1 p2 H(q, p) = mω2 q2 + = 2mω2CC? ≡ mω2 (BB? + B? B) 2 2m From here onward we would use this complex exponential form of the solution. We would call the B ≡ C exp(−iωt) as the “positive energy” part and B? ≡ C? exp(iωt) as the “negative energy” part of the solution. It sometimes help to remember that C = A2 exp(iφ). And the solution is q(t) = B + B? and p(t) = −imω(B − B? ). In general, its not necessary that q and p be sinusoidal functions of time and be 90o out of phase. Since a complex valued function of time is simply a combination of two arbitrary real valued of functions of time, we can always write q and p as real and imaginary parts of a complex valued function of time. Reversing the transformation of variables, one can write B=

p q +i 2 2mω

and

p q −i 2 2mω This transformation can be used to obtain the complex variable B when the classical solution is not a sinusoidal function. We would also use normalized variables very often. These normalizations give much simpler expressions. Suppose we want to write the Hamiltonian as B? =

H = bb? This can easily be done by defining “normalized field variables” as:√ b = 2mω2 B Here are the conclusions • Most general solution is made up of “positive energy” (B = C exp(−iωt)) and “negative energy” (B? = C? exp( jωt)) parts. Coefficients are complex valued and conjugate of each other. We would often represent these two parts as B and B? . We would often refer to these parts as “field variables”. These includes phase as well as time information. • b and b? are just the normalized B and B? . • When we would quantize the problem in next section, we would find that its very helpful in finding the eigen energy spectrum of them Hamiltonian to try to factorize the classical Hamiltonian as above. General procedure would be try and write the real-valued classical canonical variables q and p in terms of single complex variable b such that q ∝ b + b? and p ∝ −i(b − b? ). 17

3.2

Quantization of Single Harmonic Oscillator

One can easily get a quantum mechanical Hamiltonian through Dirac’s quantization rule by converting time dependent canonical variables q and p into Hermitian time independent Schrodinger’s picture operators qˆ and pˆ obeying the canonical commutation relation [q, ˆ p] ˆ = i~ (we experimentally know that it works) pˆ2 1 ˆ H= + mω2 qˆ2 2m 2 As we know from elementary quantum mechanics, energy spectrum (eigen values and eigen vectors) of this Hamiltonian can be obtained using standard differential equations methods. Instead following that straight attack, we would develop a more handy method here. We know that the classical q and p were given by q = B + B? and p = −imω(B − B? ) Now when q and p turn into Hermitian time independent operators then B and B? also into time independent operators but non-Hermitian ones – we will call them Bˆ and Bˆ † (as would become obvious in a minute, these two operators are indeed Hermitian conjugate of each other). Transformation of B and B? into operators can easily be seen from classical inverse transformations. B(t) =

p(t) q(t) +i 2 2mω

B? (t) =

q(t) p(t) −i 2 2mω

and So we get

qˆ pˆ Bˆ = + i 2 2mω and

qˆ pˆ Bˆ † = − i 2 2mω If we want, we can call these expression just as their definitions in terms of operators qˆ and p. ˆ All we are doing here is to motivate these definitions. We want to justify the reader why I want to define Bˆ and Bˆ † the way I did. Alternatively, one can see above expressions as natural consequence of Dirac’s quantization of q and p. One can just consider above quantities as quantized versions of associated classical quantities. Incidentally note that qˆ and pˆ are Hermitian by definition and hence Bˆ and Bˆ † as defined above are Hermitian conjugate of each other. Inversing these definitions, one can write qˆ and pˆ in terms of Bˆ and Bˆ † qˆ = Bˆ + Bˆ † 18

and pˆ = −imω(Bˆ − Bˆ † ) Now, the defined commutation relation between qˆ and pˆ would also enforce a commutation relation between Bˆ and Bˆ † . We can see that ˆ Bˆ † ] = − [B,

i ~ [q, ˆ p] ˆ = 2mω 2mω

And after some explicit calculations we would see that ˆ Hˆ = mω2 (Bˆ Bˆ † + Bˆ † B) Now suppose I want to write this as ~ω ˆ ˆ † ˆ † ˆ Hˆ = (bb + b b) 2 This can easily be done by defining normalized operator bˆ as follows: r 2mω ˆ B bˆ ≡ ~ and hence in terms of these new variables ˆ bˆ † ] = 1 [b, So if we want, using the commutator, we also can write the Hamiltonian as 1 Hˆ = ~ω(bˆ † bˆ + ) 2 After so much of formalism and variable transformations, you may ask what’s the point? We started with a Hamiltonian and ended with Hamiltonian, we haven’t solved anything yet? Well, problem is actually solved by this point. The above “factorized” form of Hamiltonian is easily diagonalizable. Lets explore this. The good thing about this scheme is that if |0i is the normalized lowest energy eigen state then bˆ † |0i is also an energy eigen state (may not be normalized) with energy of (1 + 21 )~ω which can easily be checked by applying Hamiltonian on bˆ † |0i. And this chain keeps on going on and on toward higher energy side. Similarly if |ni is the normalized energy eigen state with ˆ is also an energy eigen state (may not be normalized) with energy of energy (n + 12 )~ω then b|ni 1 ˆ (n − 1 + 2 )~ω which can easily be checked by applying Hamiltonian on b|ni. Specifically note † that bˆ |0i, for example might not turn out to be normalized even though |0i is normalized. So if we separate out the prefactors √ bˆ † |ni = n + 1|n + 1i and ˆ = b|ni

√

n|n − 1i

19

So knowing ground state we can form all the states without explicitly solving the rather difficult energy eigen value problem. Even finding the ground state is pretty simple. One can show that d ˆ b|0i = 0. And hence dq + q = 0. Using the integration factor of exp(q2 /2) one can easily see that 1 |0i = 4 exp(−q2 /2) π in terms of normalized q. Hence the ground state wavefunction is Gaussian distributed over real space with a normalization factor in front of it. 3.2.1

Normal Ordered Hamiltonian

Second part of the Hamiltonian written above give the zero point energy. Very often we would shift our zero reference of the energy scale to throw out this part in the energy spectrum. If we decide to do this, we can simply write the Hamiltonian as ˆ Hˆ = ~ω(bˆ † b) In this simple problem it was obvious how to remove the constant shift from the Hamiltonian. In more complex problems it might not be as obvious. Following is the formal method of getting rid off constant shift. Remember that initially Hamiltonian was written as ~ω ˆ † ˆ ˆ ˆ † (b b + bb ) Hˆ = 2 Now we define normal ordering N() of the operators by requiring all creation operators to go to the left hand and all annihilation operators to go to the right. We claim that doing this removes the constant zero-point energy. In standard text books normal ordering is also represented by :: symbol. So ˆ = ~ω(bˆ † b) ˆ ˆ = ~ω N(bˆ † bˆ + bˆ bˆ † ) = ~ω (2bˆ † b) N(H) 2 2

3.3

Alternative Interpretation

We can interpret it (or talk about it) in a completely equivalent and an alternate language. This alternate language for the time being would look unnecessary but its usefulness would become more clear once entire subject of QFT is developed. An oscillator of natural frequency ω is treated as one single “field-mode” which can take different quanta of energies in steps of ~ω. Depending upon the number of quanta in this single field mode the “wave-function of the field” would take different shape. Instead of saying that an oscillating particle absorbs a photon and goes to higher energy eigen state we say that the photon is absorbed and a quanta of energy is emitted (or created) in the “field mode ω”. Note that there is still a “field-wavefunction” associated. “Field” in this extremely trivial case has no spatial degree of freedom (in the sense that we don’t really have a phonon like wave in this example). The “field-wavefunction” represents the uncertainty associated with the “field amplitude”. Again, in this simple case “field amplitude” is simply the displacement of the oscillating particle from the equilibrium position. 20

3.4

Commutation and Bosonic Behavior

Another important point is that above commutation relation between annihilation and creation operator “indicates Bosonic identical particles” as we would explore below in more details. What this means is that the quanta of energy discussed above would follow a thermal equilibrium statistics similar to photons and phonons.

4

Chain of N Coupled Harmonic Oscillators - Quantization in Real Space

Let us now look at a multi coupled dynamical variable problem. This would shed some light on the QFT. Suppose there are N atoms connected together with identical linear springs. Note that we can choose the N Cartesian displacements as independent co-ordinates and corresponding Cartesian momenta to write a simple classical Hamiltonian as:p2l 1 1 H(ql , pl ) = ∑ + mω2 (ql+1 − ql )2 + mω2 (ql − ql−1 )2 2 2 l 2m Where l is the atomic site index. The equation assumes that the spring constants between all the identical nearby atoms are exactly same (for simplicity) and assumes only nearest neighbour coupling9 . Both ql and pl are functions of time. The Hamiltonian leads to a set of 2N separate Hamilton’s equations of motion which would be equivalent to N Newton’s equation of motion and N definitions of momenta in terms of co-ordinates d pl = mω2 (ql+1 − ql ) − mω2 (ql − ql−1 ) dt dql dt Note that the equation of motion for any one Cartesian displacement (or Cartesian momentum) would depend upon the other Cartesian displacements as they are coupled. We can go ahead and do the Dirac’s quantization and obtain a quantum mechanical Hamiltonian. Please note that the commutation relations enforced would be pl = m

[qˆn , pˆl ] = i~δn,l [qˆn , qˆl ] = 0 [ pˆn , pˆl ] = 0 9 Note

that as far as order of coupling is finite, field theory developed would be a “local field theory” and hence conceptually they are all same – nearest neighbour coupling makes it easier to illustrate the physics. When order of coupling becomes infinite, then this becomes non-local field theory – something that we would not discuss in this article.

21

One should notice how one quantization rule automatically and logically gets replaced by three in coupled oscillator case – simply because now we have a multi-particle problem at hand. Note that these are local Cartesian co-ordinates and momenta. Such a quantization is scheme is usually known as “quantization in real space” (real space in this particular case is local Cartesian coordinate and momenta but in general it can be local generalized co-ordinate and momenta as in case of EM field quantization in real space as we would see below). The quantum mechanical Hamiltonian can easily be written as :1 pˆl 2 1 ∑ 2m + 2 mω2(qˆl+1 − qˆl )2 + 2 mω2(qˆl − qˆl−1)2 l The state can be written as

Ψ(t) = Πl Ψl (ql ,t)

Note that Ψl depends only on one co-ordinate and Πl represents the direct product over all l. This would lead to a Schrodinger’s equation:pˆl 2 1 1 ∂Ψ(t) ∑ 2m + 2 mω2(qˆl+1 − qˆl )2 + 2 mω2(qˆl − qˆl−1)2Ψ(t) = i~ ∂t l Which is equivalent to set of N Schrodinger’s equations each associated with a single l:pˆl 2 Ψl (ql ,t) 1 1 + mω2 (qˆl+1 − qˆl )2 Ψl+1 (ql ,t)Ψl (ql ,t) + mω2 (qˆl − qˆl−1 )2 Ψl−1 (ql ,t)Ψl (ql ,t) 2m 2 2 ∂ = i~ Ψl+1 (ql ,t)Ψl (ql ,t)Ψl−1 (ql ,t) ∂t Similarly one can write Heisenberg equations as well which would be exactly similar to the classical Hamilton’s equations. As a result of direct correspondence we can easily see that all the N Schrodinger’s equations are also coupled. Note that at each atomic site there is an associated wavefunction of its own Cartesian displacement. So we have N attached wavefunctions. And the wavefunction evolution at each site is dependent on the wavefunction evolution attached with other sites. Solving such a equation directly would not be simple.

5

Chain of N Coupled Harmonic Oscillators - Quantization in k - Space

There exist a linear transformation of co-ordinates in which the classical Hamilton’s equation, when written as matrix, becomes diagonal. These are simply the classical Normal mode decomposition. An important effect of this change of co-ordinates is that even the resulting Schrodinger’s equations would be decoupled. Mathematically, what we are doing is that we are taking linear combinations of different Schrodinger’s equations and then defining new differential operators so that a bunch of new differential equations in terms of these new operators are 22

linearly independent. Which straightaway tells us that these would lead to the quantum mechanical eigen energy states. Another strange thing is that, as it turns out, even the Hamiltonian in new co-ordinates have the same basic form of Harmonic oscillator. Which tells us that the amplitude of each classical normal mode obeys a decoupled harmonic oscillator Hamiltonian when quantized. We would now explore in details how this transformation actually works. I believe reader is well aware of methods to diagonalize matrix equations and find eigen solutions and classical dispersion relations from classical mechanics. If not, one can easily check by substitution that following are indeed the linearly independent solutions for the simultaneous equations of motion (for this back substitution check one would still have to assume that dispersion relation is given). For a most general pattern of time evolution one simply takes a linear superposition over all eigen functions. For unbounded medium the Cartesian displacement (field amplitude) and momentum of any atomic site written in terms of classical normal modes amplitudes are:ql (t) = ∑ Ak cos(kla − ωkt − φk ) k

and

pl (t) = ∑ mωk Ak sin(kla − ωkt − φk ) k

Inverse transformations are given at the bottom of this section just for reference as they are not used much. By substituting such a plane wave solution back into equations of motion one can find the phonon dispersion relation. We would again find it very useful to write these two variables as as real and imaginary part of some complex variable. We would again break the solution into “positive energy part” and “negative energy part”. 1 ql (t) = √ ∑ Ck exp(ikla) exp(−iωkt) +Ck∗ exp(−ikla) exp(iωkt) N k and

1 pl (t) = √ ∑ −imωkCk exp(ikla) exp(−iωkt) + imωkCk∗ exp(−ikla) exp(iωkt) N k

√1 N

is normalization for exponentials. For periodic boundary conditions one can see that k = 2πn Na where n would be an integer between −N/2 and N/2. Note that we need two terms in the above equation to make the displacements real. Ck and Ck∗ contains the phase and the amplitudes information of the normal modes of oscillations. If we want we can relate them explicitly to the amplitude and phase as Ck = A2k exp(iφk ). Usually one defines Ck exp(−iωkt) ≡ Bk and Ck∗ exp(iωkt) ≡ B∗k . Hence, time, phase and mode oscillation amplitude information is buried into the field co-ordinates defined in this way whereas spatial dependence is taken out. And hence 1 ql (t) = √ ∑ Bk exp(ikla) + B∗k exp(−ikla) N k and

1 pl (t) = √ ∑ −imωk Bk exp(ikla) + imωk B∗k exp(−ikla) N k 23

By direct substitution of above into the classical Hamiltonian and then using the orthonormal conditions and dispersion relations to enforce relationship between ω and k, after simple but tedious algebraic manipulations one can show that classical Hamiltonian can be written as H = ∑ mω2k {B∗k Bk + Bk B∗k } k

Suppose we want to write this as (it makes things much more handy) H =∑ k

~ωk ∗ {bk bk + bk b∗k } 2

This can be done by defining normalized field variables as s ~ Bk = bk 2mωk Remember that spatial parts are orthonormalized and field variables include initial phase, time evolution and mode oscillation amplitude information and the amplitudes are further made dimensionless through prefactors. Now let us do our standard quantization (I want stress here that we are not doing any new postulation) by enforcing the three commutation relations between ql and pl mentioned in the previous section. This converts the time dependent variables ql and pl into time independent operators in Schrodinger’s picture. Now simply from the co-ordinate transformation equation one can see that when classical Cartesian co-ordinates (actually field/displacement at one atomic site) that is ql and corresponding momenta that is pl , ∀l, are lifted to the status of operators, the classical time dependent normalized field oscillation amplitudes (generalized co-ordinates and concommutationjugate momenta) that is Bk and B∗k become time independent field operators in Schrodinger’s picture and time dependent operator in Heisenberg picture. Note that Bk are complex numbers and, hence, have two degrees of freedom. Here we chose to select the Bk and B∗k as independent variables instead of real and imaginary parts. The reasons for this would be explained below. Using the Dirac’s commutation relation between Cartesian co-ordinates (actually field/displacement at one atomic site) and momenta (as given above) and using the inverse transformations given at the bottom of this section, after simple but tedious algebraic manipulations, one can see that these field operators would obey the set of Bosonic commutation relations defined previously :[bˆn , bˆ †k ] = δn,k †

[bˆn , bˆ †k ] = 0 [bˆn , bˆ k ] = 0 These are called “equal time commutation relation (ETCR)” (this is because when working in Heisenberg representation value of time variable needs to be kept same in these relations). This 24

second scheme of quantization that we followed in this section can be called “quantization in k space” or simply the normal-mode-quantization. Note that in this special case it turns out that local real space commutation relation between Cartesian co-ordinates and Cartesian momenta leads to the same commutation relation between generalized co-ordinate and generalized momenta (normalized field oscillation amplitudes). This is not a generic rule10 . The quantum mechanical Hamiltonian can simply be obtained by promoting field amplitudes to be operators 1 H = ∑ ~ω{bˆ †k bˆ k + } 2 k And hence one can see that field amplitude of each classical normal mode obeys a single decoupled harmonic oscillator Schrodinger’s equation. So in these transformed co-ordinates the ground state would be simply be the properly symmetrized direct product of various kets representing the ground states on various classical normal modes. The ket corresponding to the ground state of any one of the classical normal mode would be Gaussian function in field amplitudes |Ψk (bk )i = π14 exp(−b2k /2). Note that this is a wavefunction associated with a field mode. The next higher states for any of the classical normal modes can be build up using raising and lowering operators and would turn out to be Hermite polynomials. Any general state would be properly symmetrized direct product of such Hermite polynomials attached with all the classical normal modes. The field operators operate on such a state. This is a good place for us to discuss why we chose Bk and B∗k as independent variables and not real and imaginary parts of Bk , for example. Any choice is a valid quantization scheme. But the one we chose here is very convenient as it leads to simply harmonic oscillator Hamiltonian and hence, as we would see below, one can develop a QFT of phonons. Any other choice, although valid, would not lead to a concept of identical particles. One can write the Cartesian displacement operator at any site in terms of these field operators by replacing operators in classical co-ordinate transformation equation. One can also see that these field operators are still related to co-ordinate operators in similar fashion :r r mω 1 k ) pˆl + i( )qˆl ] exp(−ikla) bˆk = ∑[( 2~N 2~mω N k l and

r r mω 1 k bˆk = ∑[( ) pˆl − i( )qˆl ] exp(ikla) 2~N 2~mωk N l †

10 As we would see in QED (where there is no obvious choice for analogous “real space” conjugate variables), depending upon the k-space conjugate variables chosen and enforced Bosonic commutation relation (or Fermionic or any other if one wants to try different choices) in k-space quantization would not lead to same (Bosonic, Fermionic or any other) commutation in real space. One should note that in EM waves case, as stressed previously, even in real space, conjugate variables would be generalized co-ordinates and momenta that obeys Hamilton’s equations (which should be equivalent to Maxwell’s equations). Vice-versa if one choose a set of generalized real space conjugate variables (this choice is not unique) and enforce Dirac’s commutation (which I simply call Bosonic commutation) one can develop a similar theory. One would end up with some commutation between the conjugate variables in k-space. Such a theory in general would not be correct. Experimental proofs tell us that only that choice of real space conjugate variables that finally leads to a Bosonic commutation relation. We would discuss this in latter sections.

25

Note that field amplitude operators do not have any spatial dependence (exponential goes off when you sum it over l.) It is similar to Fourier transform. ql and pl operators would be functions of l and when you sum them up all l dependence goes off. The entire quantum theory of single harmonic oscillator works now for each of the normal † modes. bˆk and bˆk can easily be shown to be creation and annihilation operators respectively just as we did for a single harmonic oscillator in previous section.

5.1

Alternative Interpretation – Phonons

One should now be able to see how QFT language starts making more and more sense. There are many normal modes. And each mode can go in many different quantum states. We call that the energy of each of normal field modes can be increased in quanta named as phonons. Each quantum state of any normal mode is represented by a field-wavefunction which in this case are simply the Hermite polynomials just like for a single harmonic oscillator. These fieldwavefunctions are functions of field oscillation amplitude and represents the uncertainty in field oscillation amplitudes. One should also notice that QFT (and in this case even standard QM) predicts completely non-classical states of coherent elastic waves. Closest quantum mechanics can goto coherent classical elastic wave is only when the field oscillation amplitude uncertainty is small and remains constant as system evolves in time. This would happen only if we put each normal mode in a quantum state which is linear superposition of states with different number of quanta in one normal mode such that the expansion coefficients are Poisonian distributed. Any other distribution represents a highly non-classical wave. Second important point is about co-ordinate operators. Note specifically that qˆl does NOT represent the Cartesian co-ordinate operator of a phonon. That probably would be represented by some hypothetical operator corresponding to index number l! Such operator does not exist in QFT! So, I strongly recommend that one should try to come out of this way of looking at things. One should forget that Cartesian co-ordinates of QFT particles (that is something equivalent to site index l) can be operators. In QFT Cartesian co-ordinates of elementary excitations (QFT particles) are treated as parameters just like time. In QFT its not conventional to ask a question like – “what is the Cartesian co-ordinate operator for a phonon?” – since QFT is not written in this language. In fact this is completely in-sinc with relativity as it is required that both time and co-ordinate needs to be treated on equal footings. Another important point one should notice is that concept of field quantization or concept of “quasi-particles” can not be generalized to any random problem. Looking at single harmonic oscillator problem, one might get an impression (this would become more clear when we quantize Schrodinger’s field in any random quantum well, say a non-parabolic one) that one can simply calculate the energy eigen states using Schrodinger’s equation and then start filling it up one by one and hence once can define creation and annihilation operators. Remember that entire QFT stands on the concept of identical particles. For example if our simple harmonic oscillator had a non-linear restoring force then the energy eigen states would not be equally spaced and one would not be able to set up a QFT in such a simple form (in terms of identical phonons) –

26

one would need to introduce the concept of phonon-phonon interaction and even the concept of phonon would be practically useful only when phonon-phonon interaction is not two big (so that a concept of quasi-particle has some sort of experimental validity).

6

Continuum – Quantization of Elastic Waves in Continuous Media

Suppose the atomic sites come infinitesimally close and become continuum. Results have intuitive extensions. Probability amplitudes and probabilities both become density functions. Equation of motion simply use Lagrangian and Hamiltonian density functions. So basically, instead of having one classical Hamilton equation or one Heisenberg equation for each spatial coordinate we would have operators which are functions of spatial co-ordinates. Similarly all other operators would actually become operator density. Now again its much more easier to solve the problem by co-ordinate transformation so that equation at one generalized coordinate is independent of equations at other points. Even the eigen mode/normal modes would be infinite and infinitesimally closely spaced and hence field operators would also be operator densities. Secondly the commutators would become double integrals and Kronecker deltas would become Dirac-delta functions. One important point change is that the dynamical variables like local Lagrangian density L would now be the function of local co-ordinates, time-derivatives of local co-ordinates, time, as well as local space-derivative of all orders of local co-ordinates. This is needed to develop the notion of local Lagrange densities and local field theories. Note that the Lagrange’s equation then becomes :∂L ∂ ∂L ∂ ∂L ( )= − ∑ ( ∂Ψ ) ˙ ∂t ∂Ψi ∂Ψi x,y,z ∂x ∂ i ∂x

and one of the two Hamilton equations (for each field) also get modified according to ∂Πi ∂H ∂ ∂H =− + ∑ ( ∂Ψ ) ∂t ∂Ψi x,y,z ∂x ∂ i ∂x

Everything else follows exactly as above.

27

Part IV

QFT As A Non-Relativistic Multi-Particle Quantum Theory 7

Identical Particle Hilbert Space Representations

Non-relativistic QFT of identical massive particles can be “completely”11 understood if we try to understand the properties and representation of identical particle Hilbert spaces without any further postulates. Relativistic case gives some troubles because a particle can not exist without it’s anti-particle12 . Moreover ground state consists of infinite many of anti-particles. Once these things are understood, it’s actually possible to build the Hilbert space representations in relativistic and to understand the relativistic QFT “completely” from there on. One should keep in mind that this is no magic! QFT is in fact developed to understand these complications with anti-particles, ground states, spin statistics theorem etc. Once you take these as postulates, obviously everything else is just symbols and notations and no new physics. So its obvious that with these things taken as proved one can understand the structure of QFT simply by understanding the structure of Hilbert space. With this motivation, lets explore the identical particle state and operator representations in some depth. We would only look into non-relativistic case. Relativistic case would be discussed latter.

7.1

Fock Space and Fock States

In basic quantum mechanics, multi-particle Hilbert space is postulated to be the direct product of single particle Hilbert Spaces. Further, the fact, that identical multi-particle state space is either a Bosonic subspace (for integer spin particles) or Fermionic subspace (for half-integer spin particles) of the entire direct product of all single particle spaces, is treated as one of the basic postulates of non-relativistic quantum theory (see the article on Quantum Mechanics). The multi-particle state space can be generated/spanned from infinitely many choices of basis sets. One of the obvious choice is the direct product of the basis states that generate/span single particle state spaces for each of these particles. Most common one is the set of direct products of single particle energy eigen states. Please note that this works irrespective of whether the particles interact with each other or not. If they do then, for example, the direct product of energy eigen states of single particles would not be an energy eigen state of the multiparticle system. But it would still be a valid candidate as a basis state. This postulation of 11 “Completely”

does not mean that we can understand the spin statistics theorem – why half integer spin particles are Fermions and integer spin particles are Bosons needs to be taken as a postulate in the arguments in this section. 12 This is related to the fact that relativistic quantum mechanical single particle equations involve second order time derivatives unlike single particle Schrodinger’s equation.

28

multi-particle state space as direct product space would directly imply the additivity of linear momentum, for example. And additivity of energy in case of non-interacting particles. In case of identical particles, a somewhat simpler way of choosing the basis state might be to choose occupation number (number of particles in single particle energy states or maybe any other single particle basis state) as a quantum index. We call it occupation number state representation. These multi-particle basis states are simply the states with definite number of particles in any of the single particle basis state. We would show in a minute that this latter scheme works only for identical particles – both for Bosonic or Fermionic identical particles. These basis states are known as Fock states. Fock states are the so called natural basis of Hilbert space of identical particles. The Hilbert space of identical particles (technically unknown number of them) that is the symmetric or anti-symmetric subspace of the entire direct product Hilbert space is known as Fock space. For the time being assume that particles are not forced to be Bosons or Fermions. We would show that this scheme does not work unless particles are Fermions or Bosons or unless particles are forced to be in some other kind of symmetric subspace of direct-product space. Let us choose the single particle energy eigen states as the basis of single particle Hilbert space. We can decide to symbolically write a generic single particle state |Ψi = ∑ cm |ΨEm i m

as

∑ cm|01, 02, ...., 1m, 0m+1, ....0∞i m

Similarly, we might try to write a generic two particle state (which is not required to be symmetric or anti-symmetric state) |Ψi = ∑ ∑ ci j (|ΨEi i ∗ |ΨE j i) i

as

j

∑ ∑ ci, j |01, 02, ....1i, 0i+1, .....1 j , 0 j+1, ......0∞i i

j

Here symbol ∗ stands for direct product13 . Similarly we might develop notations for manyparticle states. If identical particles are not forced to be either bosons or fermions then what we are claiming here is that symbolically |01 , 02 , ....1i , 0i+1 , .....1 j , 0 j+1 , ......0∞ i ≡ |ΨEi (r1 )i ∗ |ΨE j (r2 )i where r1 is a variable that keeps track of the co-ordinate of the distinguishable particle labelled as ’1’ and r2 keeps track of the co-ordinate of another distinguishable particle labelled as ’2’. Note that the number state representation is not complete and is confusing when particles are not forced to be either boson or fermion. For example how would you distinguish between|ΨEi (r1 )i ∗ |ΨE j (r2 )i and |ΨEi (r2 )i ∗ |ΨE j (r1 )i? One might try to introduce another 13 When

it is not imperative to stress the special presence of a direct product we sometime also use the notation |ΨEi i|ΨE j i to represent direct product.

29

labeling scheme to keep track of particle labels. For example one might be inclined to try |01 , 02 , ....11i , 0i+1 , .....12j , 0 j+1 , ......0∞ i for above mentioned state. But, as we are going to see, even this can not be done. For example a valid two particle state (c1 |ΨE1 (r1 )i + c2 |ΨE2 (r1 )i) ∗ (c3 |ΨE1 (r2 )i + c4 |ΨE2 (r2 )i) can not be written in terms of number states unless coefficients are inter-related. Here particle ’1’ is in some superposition and particle ’2’ is also in some superposition. Since in both Fermionic and Bosonic sub-spaces of multi-particle Hilbert space such a co-relation between coefficients exist, we can use such a representation for identical particle systems. I hope that by this time reader is convinced that Fock states can only be used as basis for identical particles. There are several of visualization benefits with number state representation :• Firstly, common processes that we want to study (like optical absorption emission etc) are non-particle-number-conserving. Number state representation handles this pretty nicely as we would see below. • When we would write down the operators in this representation, we would see that it hides a lot of details about the actual force laws that particles are obeying (for example irrespective of whether a charged fermion is in electrostatic potential or a fermion with permanent magnetic dipole is in magnetic field, the operators and their operations on states look exactly same). The actual details are dumped into the single particle states over which number state representation is built. So even if you don’t know the exact states you can build an overall theoretical structure. 7.1.1

Basis Conversion

Suppose number state representation of certain generic identical multi-particle state is given. What we mean is that the state is given as a linear superposition of Fock states. Fock states are built on top of certain basis of single particle Hilbert space. For example we might have chosen single particle energy eigen states as the basis for the single particle Hilbert space and build our number state representation on top of it. What if I want to change this underlying basis of the single particle Hilbert space. This would also change the Fock states and hence the number state representation of the given multi-particle state. We want to figure out a formal way of doing this basis change. After a bit of formalism, it would turn out to be very simple and very useful. If the particles are Fermions then two identical particles can only occupy an anti-symmetric subspace of the entire direct product of single particle Hilbert spaces. In this two particle subspace, any generic state would can simply be written as a so called Slater determinant with multiplication of state vectors being interpreted as a direct product of state vectors : |1, ΨEi i |2, ΨEi i (2) ∑ ∑ ci j |1, ΨE j i |2, ΨE j i i j where |1, ΨEi i means particle 1 is in single particle state |ΨEi i. And hence a set of following states |1, ΨEi i |2, ΨEi i |01 , 02 , ....1i , 0i+1 , .....1 j , 0 j+1 , ......0∞ i ≡ |1, ΨE j i |2, ΨE j i 30

can be taken as the two particle Fermionic basis states. Statement can easily be generalized for multiple particles. Similarly a generic two particle Boson state would be written as 1

ˆ ΨEi i|2, ΨE j i] ∑ ∑ ci j √2 ∑ P[|1, i

j

(3)

P

√ where Pˆ is a linear operator that generates 2! permutations by exchanging particles among fixed single particle states. i and j can take √ all values from 1 to N where N is the number of single particle states available. Note that 2! is correct only when i and j are different. One can easily figure out a normalization constant when i = j. And hence 1 ˆ ΨEi i|2, ΨE j i] |01 , 02 , ....1i , 0i+1 , .....1 j , 0 j+1 , ......0N i = √ ∑ P[|1, 2 P are the two particle Bosonic basis states. Statement can easily be generalized for multiple particles. 7.1.1.1 Co-ordinate Representation to Number State Representation Our first aim is to obtain a number state representation if a completely antisymmetric or symmetric |ri representation of a multi-particle state is given. Suppose φi ’s are the wavefunctions of the single particle basis states. Let us be specific and work out the details for fermions case first. Analogous results can be worked out for Bosons case. If the wavefunction of a two identical fermion particle state is given as Ψ = c1 (φ1 (r1 )φ2 (r2 ) − φ2 (r1 )φ1 (r2 )} + c2 (φ3 (r1 )φ4 (r2 ) − φ4 (r1 )φ3 (r2 )) then one can see that this state can be equivalently represented as c1 |11 , 12 , ....0i , 0i+1 , .....0 j , 0 j+1 , ......0N i + c2 |01 , 02 , 13 , 14 , ....0i , 0i+1 , .....0 j , 0 j+1 , ......0N i But how do we do it more formally so that this is possible for more difficult cases which are not that easy to see? The ket representation of the above state in |r1 i ∗ |r2 i ≡ |r1 i|r2 i basis would be |Ψi =

∑ (c1(φ1(r1)φ2(r2) − φ2(r1)φ1(r2)) + c2(φ3(r1)φ4(r2) − φ4(r1)φ3(r2)))|r1i|r2i

r1 ,r2

Even though this basis is capable of spanning entire direct product space, the coefficients are such that the particles stay only in Fermionic subspace. We now use the normal linear algebra trick to change the basis. Following operator when operated over the wavefunction of the state gives the ket representation of the state in new basis (change from |r1 i|r2 i → |φm (r1 )i|φn (r2 )i basis):Z

∑ |φm(r1)i|φn(r2)i

dr1 dr2 (φ∗m (r1 )φ∗n (r2 )

m,n

31

This gives us the representation in terms of some new basis states |φm (r1 )i|φn (r2 )i. But we want the representation in terms of number states? This is straight forward now. Following is the required operator :-

∑

Z

|....0i , 0i+1 , 1m ...., 1n .....0N i

dr1 dr2 (φ∗m (r1 )φ∗n (r2 ) − φ∗n (r1 )φ∗m (r2 ))

(4)

m,n>m

Note that it works properly when you know apriori that particles are for sure Fermions and wavefunction is properly antisymmetric. Otherwise we have seen that in the complete directproduct space this number state representation does not make any sense and these can not span the entire space. We can also write the same operator as :-

∑

Z

|....0i , 0i+1 , 1m ...., 1n .....0N i

dr1 dr2 φ∗m (r1 )φ∗n (r2 )

(5)

m,n>m

7.1.1.2 Number State Representation to Co-ordinate Representation Using similar linear algebra concepts we can also do the inverse transformation. The inverse transformation is obtained as :-

∑

(φm (r1 )φn (r2 ) − φn (r1 )φm (r2 ))h....0i , 0i+1 , 1m ...., 1n .....0N |

(6)

m,n>m

7.1.1.3 General Basis Transformation Using these one can easily do a general basis transformations of underlying single particle basis set. Using 6 to go to r representation then do usual basis conversion and then use 5 to get new number state representation.

7.2

Operators in Fock Space

Suppose we are given some operator in r representation A(r1 , r2 ) that operates on two particle Hilbert space. Let this operator correspond to some physically measurable property of two identical particle system. Hence, as argued before this operator needs to be symmetrical with respect to exchange of particles. A(r1 , r2 ) = A(r2 , r1 ) We want to know how this operator operates on a state given in number state representation. Effectively we want to find the equivalent representation of this operator in number state representation. From above two transformation equations, we can also build a number state representation of an operator A(r1 , r2 ) given in |r1 i|r2 i basis as R

∑m,n>m |....1m ...., 1n .....i ∗ ∗ ∗ dr1 dr2 (φm (r1 )φn (r2 ) − φn (r1 )φ∗m (r2 )) A(r1 , r2 ) ∑i, j>i (φi (r1 )φ j (r2 ) − φi (r1 )φ j (r2 )) h...., 1i ...., 1 j .....|

32

Now we know that the operation of any symmetric linear operator A(r1 , r2 ) on any antisymmetric state leaves the final sate antisymmetric. So we can remove the second antisymmetric parts from the above expression :-

∑

Z

|....1m ...., 1n .....i

dr1 dr2 φ∗m (r1 )φ∗n (r2 )A(r1 , r2 )

m,n>m

∑ φi(r1)φ j (r2)h...., 1i...., 1 j .....|

i, j>i

The same operator can also be written as Z

1 1 dr1 dr2 ( √ ) ∑ |....1m ...., 1n .....iφ∗m (r1 )φ∗n (r2 )A(r1 , r2 )( √ ) ∑ φi (r1 )φ j (r2 )h...., 1i ...., 1 j .....| 2 m,n 2 i, j (7) in order to remove the double counting. Note that we do not need factor of 4 as someone might think without looking too deeply. We have divided the factor of two into two parts just for symmetry purposes. We have also changed the order of integration and summation. The process works (except factor of 2 needs to be replaced by factor of N!) for any number of particles. These result for are exactly same for both bosons and fermions. The difference stays in the meaning of number states and latter we would bury the difference into creation and annihilations operators that would be used to generate number states.

7.3

Ground State or Vacuum

Definition, identification and proof of existence of ground state is actually a very tricky part of the QFT. Things are somewhat simpler at least in free particle non relativistic theories. We claim that there exist a unique and non-degenerate lowest energy state. We further claim that this is the state in which there are no particles in the system. We would represent this state as |0i. As we build the theory more rigorously, we would discuss in details the issues with the existence of a ground state in QFT.

7.4

Creation and Annihilation Operators

One can first intuitively define annihilation a†j and creation a j operators giving them the properties that their names suggest. One can then easily prove the anticommutation relations among them utilizing their pre-assumed Fermionic behaviour:{a†j , a†k } = a†j a†k + a†k a†j = 0 {a j , ak } = a j ak + ak a j = 0 and

{a†j , ak } = a†j ak + ak a†j = δ jk

Note that these anticommutation rules completely and uniquely define the Fermionic subspace of the direct product of single identical particle spaces. Hence one can treat these as

33

completely equivalent statement of anti-symmetrization postulate of Quantum Mechanics. One can also prove that Nk = a†k ak is a number operator. Similarly for bosons, one can first intuitively define annihilation b†j and creation b j operators giving them the properties that their names suggest. One can then easily prove the commutation relations among them utilizing their pre-assumed Bosonic behaviour:[b†j , b†k ] = b†j b†k − b†k b†j = 0 [b j , bk ] = b j bk − bk b j = 0 and

[bk , b†j ] = bk b†j − b†j bk = δ jk

Note that these commutation rules completely and uniquely define the Bosonic subspace of the direct product of single identical particle spaces. Hence one can treat these as completely equivalent statement of symmetrization postulate of Quantum Mechanics. One can also prove that Nk = b†k bk is a number operator.

7.5

The Wavefunction Operators or The Field Operators

As we would see below, the above derivation (7.2) is the logic behind the definition and usage of something called “wavefunction operators” or the “field operator”. One should first note that |0, 0, 1m , 0, 1n , 0, 0ih0, 0, 1i , 0, 1 j , 0, 0| = a†m a†n ai a j This is self obvious. Readers can convince themselves of the equivalence by just looking at the operation of both sides on some state given in number state representation. Let us define so ˆ and creation (Ψ ˆ † ) wavefunction operators respectively as :called annihilation (Ψ) ˆ 1 , r2 ) ≡ ( √1 ) ∑ ai a j φi (r1 )φ j (r2 ) Ψ(r 2 i, j

(8)

ˆ † (r1 , r2 ) ≡ ( √1 ) ∑ a†m a†n φ∗m (r1 )φ∗n (r2 ) Ψ 2 m,n

(9)

Exactly analogous definitions can be given for boson wavefunction operators by just replacing fermion creation and annihilation operators inside summation by boson counterparts. These definitions of wavefunction operators can easily be generalized for more numbers of particles. We simply include more number of operators and wavefunctions inside the summation, increase

34

the dimension of sums and replace 2 by N!. By exchanging the order of multiplication and summation, multi-particle wavefunction operator can also be written as

where

ˆ 1 , r2 , r3 , ....) = √1 Ψ(r ˆ 1 )Ψ(r ˆ 2 )Ψ(r ˆ 3 )..... Ψ(r N!

(10)

ˆ n ) = ∑ ai φi (r1 ) Ψ(r

(11)

i

Annihilation wavefunction operator is simply the Hermitian adjoint of this operator. With this we can now write down the symmetric multi-particle operators (generalized version of 7) in number state representation for both fermions and bosons in much simpler fashion Z

ˆ † (r1 , r2 , ...)A(r1 , r2 , ...)Ψ(r ˆ 1 , r2 , ...) dr1 dr2 ....Ψ

(12)

One should notice that we haven’t postulated anything new. The definitions of these wavefunction operators and their importance have just fallen out from the different way (number state representation) we started to represent the Fock space.

7.6

Summary

Lessons we want to learn from this whole exercise are:• Field quantization (concept of particles, creation/annihilation operators etc) can be understood without invoking any postulates provided we know before hand what kind of particles are we dealing with. When we don’t know this – we have to properly quantize a wave equation (second quantization). • Process outlined here is correct for non-relativistic fermions and bosons that satisfy Schrodinger’s equation. It would not work for relativistic elementary particles (since they coexist with anti-particles). But a similar derivation can be given once spin statistics theorem as well co-existence of anti-particles are assumed. I would not try to give such a derivation. Reason is that the formal procedure of field quantization leads to both the concept of coexistence of anti-particles as well as all the expressions for wavefunction operators, creation/annihilation operators etc. So a similar derivation would not provide any new insight. • Process outlined here would not work even for non-elementary particles (elementary excitation on condensed matter physics for example) like phonons. Even for these we need to include the negative energy solutions as we would see latter (equivalent to inclusion of anti-particles). • This exercise is useful in pointing out what is new and what is old physics in QFT. We would see that all this language of creation/annihilation operators etc, which although seems new, falls out naturally from all the physics that we already know. So this part 35

does not contain any new physics in it. What is new in QFT are things like existence of anti-particles, spin-statistics theorem etc. These would fall of naturally when we take Schrodinger equation or Dirac equation as a “classical” equation and quantize it. This so canonical quantization of field equations (or so called second quantization) is the new physics. And this is what would lead us to the concepts of anti-particles, spin-statistics etc.

8

Canonical Quantization of Schrodinger’s Field (Second Quantization)

Now in this section, we would see that the exactly same physics can be obtained if we treat Schrodinger’s field as a classical field and quantize it. We know that Ψ is a complex field. Hence we might be tempted to treat it as two independent fields (or may be vector field having two components) corresponding to its real and imaginary parts. Or alternatively we might want to treat Ψ and Ψ∗ as two independent fields. Having two fields is similar to having two harmonic oscillators at each spatial point. So the Hamiltonian or the Lagrangian would contain variables corresponding to two independently moving particles. Specifically, in continuous variable case, for the latter choice the Lagrangian ’denisty’ would be written as : ˙ Ψ˙∗ ) L = L(Ψ, Ψ∗ , ∇Ψ, ∇Ψ∗ , Ψ, But we know that in classical mechanics of two ’unconstrained’ particles it is necessary to provide initial co-ordinates and initial velocities of both the particles to know its future evolution in time with known forces. This is why one claims that q1 and q2 , and q˙1 and q˙2 are independent variables and one needs to know the Lagrangian as a function of all these variables in order to know the time evolution of the system. Such is not the case with Schrodinger’s equation. If we know real and imaginary part of Ψand ∇Ψat t = 0 (at any single spatial point and its ’neighbourhood’), its time evolution is completely known. Hence all four variables are not independent variables. In other words if write Hamiltonian we do not need all four variables Ψ, Ψ∗ and their associated conjugate momenta. Secondly, one should note that Lagrangian function (and Hence Hamiltonian function as well) is never unique. One can add total time derivative of any function of co-ordinates and time. It simply adds a constant to action and hence its minima still remains same. Similarly for Lagrangian density one can add a total time derivative (of any function of filed amplitudes or space or time) at each spatial location and one can also add divergence of any vector field (which could also be any function of filed amplitudes or space or time). Assuming this new vector field goes to zero at infinity. When we integrate (over infinite space) this to calculate total Lagrangian the divergence integrate to zero as there is no field at surfaces. Hence, its no wonder that there are different choices of Lagrangian density available in literature. Sometimes people choose Ψr as co-ordinate and it turns out that the imaginary part Ψi can be taken as conjugated momenta associated with it. On the other hand one can also take Ψ as co-ordinate and Ψ∗ as its conjugate momenta. It depends a lot on how you write your Lagrangian density which is not a uniquely identified function. 36

Now we know that the equation of motion for the two independent fields are ∂Ψ ~ 2 ∇2 Ψ i~ =− +V Ψ ∂t 2m and its complex conjugate. But note that what is more important for us is to know Lagrangian density corresponding to a system whose Euler-Lagrange equation is as given above. As noted above, the choice of Lagrangian density is not unique. Some choices are preferred over others but for the time being let us choose to have 2 ˙ Ψ∗ , ∇Ψ∗ , Ψ˙∗ ) = i~Ψ∗ Ψ ˙ − ~ ∇Ψ∗ ∇Ψ −V Ψ∗ Ψ L(Ψ, ∇Ψ, Ψ, 2m

This Lagrangian takes redundant variables and treats Ψand Ψ∗ as independent fields. Note that such a Lagrangian is not Hermitian and hence a better choice is one in which the first term is ˙ − i~Ψ˙∗ Ψ)/2. One can easily verify that both choices of chosen symmetrically as well i~(Ψ∗ Ψ Lagrangian densities lead to Euler-Lagrange equations which are same as Schrodinger’s Equations for Ψ and Ψ∗ . Let us choose the non-Hermitian choice since at the end we simply need a Hermitian Hamiltonian and we would not care about Lagrangian any more. One can see that the conjugate momenta associated with Ψis i~Ψ∗ and conjugate momenta associated with Ψ∗ is identically zero. This makes our life much simpler. Suppose we want to write Hamiltonian. We can immediately see that Hamiltonian is only going to be a function of Ψ and it conjugate momenta Π (which is just same as Ψ∗ ). Which automatically removes redundancy from Hamiltonian. Hence Hamiltonian density would be ~2 ∂Ψ ∗˙ − (i~Ψ Ψ − ∇Ψ∗ ∇Ψ −V Ψ∗ Ψ) H = (i~Ψ∗) ∂t 2m Which we can write as

~2 ∇Ψ∗ ∇Ψ +V Ψ∗ Ψ 2m This is completely Hermitian. This is the reason why we didn’t care much that we are starting with redundant and non-Hermitian Lagrangian. Note that occurrence of gradient of conjugate momenta is still a painful experience. But I believe it can be written as linear superposition of gradient of Ψand Π. Note that both Ψand Πare non-Hermitian. Now once we have identified Ψand Π as “canonical variables” we can lift them to the status of “operators” representing “generalized” co-ordinate and momentum in QFT picture. The commutation relation to be enforced would either be Bosonic or Fermionic – both leads to physically correct results. One of them represents Fermionic fields and other represents Bosonic fields. Now wavefunctions can be expanded in linear superposition of energy eigen wavefunctions of the Schrodinger’s equation (single particle). Hence when wavefunction is lifted to the status of an operator the coefficient of expansions also become operators – these are exactly same as the creation and annihilation operators discussed in previous section. Wave function operators are simply recognized as creation and annihilation operators at one spatial location. Hence they obey the same commutation relation as creation and annihilation operators discussed in previous section. H=

37

Part V

Non-Relativistic Quantum Field Theory ( QFT ) QM of non-relativistic massive particles is given by Schrodinger’s equation. QFT for nonrelativistic massive particles can be very easily understood if we simply try to represent states and operators in number state representation pretending that they don’t interact with each other. The reason for this is that inter-particle interaction in QFT comes from inter-field interaction. One can use either Bosonic or Fermionic symmetrization of multi-particle space.

9

Non-relativistic, Massive, Fermionic Fields

Our aim is to build a non-relativistic version of QFT for fermions. This is basically a multiparticle QFT equivalent of Schrodinger’s equation. A non-relativistic fermion QFT does not need any more postulations beyond those included in the Schrodinger’s formulation. Al wee need to do is to learn how to represent states and operators in number state representation. This part of the job we have already done in a previous section (7). We just need to learn how to use that scheme now. But for a moment, let me give alternative historical routes to this problem. As we discuss this we would also learn how we can use these schemes. Note that the Fermionic Hilbert space is a sub-space of full direct product of single particle Hilbert spaces. And the number states can be used as basis to generate the Fermionic Hilbert space. So when operators are written explicitly in number state representations then they would only operate on Fermionic subspace unlike the |r > representation of the operator that can mathematically operate on entire direct product. So in a sense the scheme for writing the operators in number state representation is essentially equivalent to take the projection of operators on Fermionic subspace. These schemes are not simple basis change or mathematically speaking these are not similarity transformation of operators. The main idea is to obtain a representation of various operators in terms of number states because that allows us to think in terms of quanta. So we can pretend as if we are working with classical physics like particles and entire wave-like physics is dumped into the inter relationship between operators and hence selection rules etc. There are two major schemes available in the literature to obtain the operators that operate on multi-particle Fermionic Hilbert space. More common scheme available in most text books is sometimes known as second quantization. What has been proposed is to quantize the Schrodinger’s equation exactly similar to the procedure followed in quantizing Maxwell’s wave equations as we would see below. This scheme sounds more like a postulate. Whereas it is not supposed to be one. Since all the postulates are already included in the Schrodinger’s equation. The second scheme, which was first taught to me by Prof. D. A. B. Miller in EE-223, is more enlightening as it clearly shows us that no more postulates are needed in order to obtain 38

the Fermionic operators in state space representation. Prof Miller had introduce the concept of wavefunction-operators in semi-ad hoc fashion. On the other hand I have already shown how to derive the concept of wavefunction operators rigorously in section (7). In this section I would attempt to further justify the operators by showing the complete equivalence of the operators in |r > and number state representations in the sense that the give exactly same states when they operate on exactly same states from within the Fermionic subspace of direct product space. Another important point is that, from the very general derivation given in section (7), we can see that it works for Bosonic operators as well. The reason is pretty simple. As we discussed in the previous section, the number space representation is simply a projection of operators into smaller subspaces. The difference between Bosons and Fermions is simply buried into the commutation relationship between creation and annihilation operators. On the other hand second quantization picture has its own benefits. What if the quantum mechanical description of single particle physics is not known? For the case of massive nonrelativistic particles physics in known in terms of Schrodinger’s equation – so in this case there is no problem. For example time evolution of a single photon is not known in quantum mechanics so there is no point of being able to built a multi-particle number state representation of operators. In such case second quantization proceeds by identifying canonical variables and quantizing a classical wave equation. In such cases this procedure is known as field quantization rather than second quantization but the procedure would be exactly same.

9.1

First Scheme

• One can first intuitively define annihilation a†j and creation a j operators giving them the properties that their names suggest. One can then easily prove the anticommutation relations among them utilizing their pre-assumed Fermionic behaviour:{a†j , a†k } = a†j a†k + a†k a†j = 0 {a j , ak } = a j ak + ak a j = 0 and

{a†j , ak } = a†j ak + ak a†j = δ jk

Note that these anticommutation rules completely and uniquely define the Fermionic subspace of the direct product of single identical particle spaces. Hence one can treat these as completely equivalent statement of anti-symmetrization postulate of Quantum Mechanics. One can also prove that Nk = a†k ak is a number operator. • Secondly we define a (annihilation) wavefunction operator from annihilation operators as:ˆ = √1 Ψ ∑ aaabac......Ψa(r1)Ψb(r2)Ψc(r3).... N! a,b,c.... 39

where N is the number of particles and a, b, c... are all the single particle energy eigen states (if one chose the number state representation using single particle energy eigen states as the base). Note that we are multiplying the wavefunctions and not the kets. One can easily prove that for any operator Aˆ that operates on entire direct product space of single particle spaces, following operator: Z

ˆ † Aˆ Ψdr ˆ 1 dr2 ... Ψ

would be the “projection” of A on N identical Fermionic Hilbert space. By projection what I mean is that this operator only operates on the Fermionic subspace and gives exactly the same result as that given by A operating on the same state. The rigorous prove follows from the previous section’s (7) arguments used to build operators in number state representation. One can easily verify this by testing it on any generic state given as ∑basis cbasis (SlaterDeterminat o f order N)basis and with noninteracting multi-particle Hamiltonian operator. Its easily verifiable for two particles. The most generic state of two non interacting fermions can be written as : |1, ΨE (r1 )i |2, ΨE (r2 )i m n |Ψi = ∑ cmn |1, Ψ (r )i |2, Ψ (r )i E E 1 2 m n m,n =

∑ cmn(|ΨEm (r1)i|ΨEn (r2)i − |ΨEn (r1)i|ΨEm (r2)i)

m,n

Where multiplication should be read as direct product of kets. Also, if one so wishes, one can also write it as

∑ cmn|01, 02, ....1m, 0m+1, ....1n, 0n+1...... >

m,n

Also using Dirac quantization rule for non-interacting multi-particles we would have the |r > representation as :~2 2 ~2 2 ˆ H= ∇ + ∇ +V (r1 ) +V (r2 ) 2m r1 2m r1 So the operation of Hamiltonian on most generic state can be written as :-

∑ cmn(En + Em)[|ΨEm (r1)i|ΨEn (r2)i − |ΨEm (r1)i|ΨEn (r2)i]

m,n

On the other hand the wavefunction operator for two particle fermions is given as :ˆ = √1 ∑ am an Ψa (r1 )Ψb (r2 ) Ψ 2 m,n So we are proposing that the projection of the H in r representation onto the two particle Fermionic sub space in number state representation is :1 ∑ ∑ a†ma†naia j 2 m,n i, j

Z

Ψ?a (r1 )Ψ?b (r2 )H(r1 , r2 )Ψi (r1 )Ψ j (r2 )dr1 dr2 40

Which is same as 1 ∑ ∑ a†ma†naia j (Ei + E j ) 2 m,n i, j

Z

Ψ?a (r1 )Ψ?b (r2 )Ψi (r1 )Ψ j (r2 )dr1 dr2

And hence can be written as 1 ∑ a†ma†naman(Em + En) 2 m,n One can easily satisfy oneself that this is same as ∑ j E j N j . So the operation of this Hamiltonian on a most generic two particle Fermionic state

∑ cmn|01, 02, ....1m, 0m+1, ....1n, 0n+1...... >

m,n

gives us

∑ cmn(En + Em)|01, 02, ....1m, 0m+1, ....1n, 0n+1...... >

m,n

which is exactly same as what we obtained earlier but using more conventional Hamiltonian and r representation of the states. • Hence one can easily get the Hamiltonian of any number of non interacting particles in number state representation can be written as Hˆ = ∑ E j N j j

• Similarly one can obtain other operators as well. This rule of generating projection of operators in number state representation is true for other operators as well. Hence one can easily guess (and obviously prove with little bit of algebra) that Aˆ = ∑ N j (∑ |ci j |2 αi ) j

i

where ci j is the projection of ith eigen state of a single particle operator A onto a jth single particle basis state and αi is the eigen value. Note that additivity of single particle quantity Aˆ of particles are assumed (that is two particle operator A has a property so that ˆ α1 > |Φα2 >= (α1 + α2 )|Φα1 > |Φα2 >) and direct product of single particle energy A|Φ states are always taken as basis (whether interacting or not). The rule generalizes to energy as well since for non-interacting particles, energy is additive. ˆ † = √1 ∑i, j,k.... ai † a j † ak † ......Ψ∗ (r1 )Ψ∗ (r2 )Ψ∗ (r3 )...., Ψ ˆΨ ˆ † = −Ψ ˆ †Ψ ˆ and • Noting that Ψ i j k N! hence multi-particle wavefunction operator satisfy the anticommutation relation.

41

9.2

Second Scheme – Second Quantization of Schrodinger’s Field

We know that Ψ is a complex field. Hence we might be tempted to treat it as two independent fields (or may be vector field having two components) corresponding to its real and imaginary parts. Or alternatively we might want to treat Ψ and Ψ∗ as two independent fields. Having two fields is similar to having two harmonic oscillators at each spatial point. So the Hamiltonian or the Lagrangian would contain variables corresponding to two independently moving particles. Specifically, in continuous variable case, for the latter choice the Lagrangian ’denisty’ would be written as : ˙ Ψ˙∗ ) L = L(Ψ, Ψ∗ , ∇Ψ, ∇Ψ∗ , Ψ, But we know that in classical mechanics of two ’unconstrained’ particles it is necessary to provide initial co-ordinates and initial velocities of both the particles to know its future evolution in time with known forces. This is why one claims that q1 and q2 , and q˙1 and q˙2 are independent variables and one needs to know the Lagrangian as a function of all these variables in order to know the time evolution of the system. Such is not the case with Schrodinger’s equation. If we know real and imaginary part of Ψand ∇Ψat t = 0 (at any single spatial point and its ’neighbourhood’), its time evolution is completely known. Hence all four variables are not independent variables. In other words if write Hamiltonian we do not need all four variables Ψ, Ψ∗ and their associated conjugate momenta. Secondly, one should note that Lagrangian function (and Hence Hamiltonian function as well) is never unique. One can add total time derivative of any function of co-ordinates and time. It simply adds a constant to action and hence its minima still remains same. Similarly for Lagrangian density one can add a total time derivative (of any function of filed amplitudes or space or time) at each spatial location and one can also add divergence of any vector field (which could also be any function of filed amplitudes or space or time). Assuming this new vector field goes to zero at infinity. When we integrate (over infinite space) this to calculate total Lagrangian the divergence integrate to zero as there is no field at surfaces. Hence, its no wonder that there are different choices of Lagrangian density available in literature. Sometimes people choose Ψr as co-ordinate and it turns out that the imaginary part Ψi can be taken as conjugated momenta associated with it. On the other hand one can also take Ψ as co-ordinate and Ψ∗ as its conjugate momenta. It depends a lot on how you write your Lagrangian density which is not a uniquely identified function. Now we know that the equation of motion for the two independent fields are i~

~ 2 ∇2 Ψ ∂Ψ =− +V Ψ ∂t 2m

and its complex conjugate. But note that what is more important for us is to know Lagrangian density corresponding to a system whose Euler-Lagrange equation is as given above. As noted above, the choice of Lagrangian density is not unique. Some choices are preferred over others but for the time being let us choose to have ˙ Ψ∗ , ∇Ψ∗ , Ψ˙∗ ) = i~Ψ∗ Ψ ˙− L(Ψ, ∇Ψ, Ψ, 42

~2 ∇Ψ∗ ∇Ψ −V Ψ∗ Ψ 2m

This Lagrangian takes redundant variables and treats Ψand Ψ∗ as independent fields. Note that such a Lagrangian is not Hermitian and hence a better choice is one in which the first term is ˙ − i~Ψ˙∗ Ψ)/2. One can easily verify that both choices of chosen symmetrically as well i~(Ψ∗ Ψ Lagrangian densities lead to Euler-Lagrange equations which are same as Schrodinger’s Equations for Ψ and Ψ∗ . Let us choose the non-Hermitian choice since at the end we simply need a Hermitian Hamiltonian and we would not care about Lagrangian any more. One can see that the conjugate momenta associated with Ψis i~Ψ∗ and conjugate momenta associated with Ψ∗ is identically zero. This makes our life much simpler. Suppose we want to write Hamiltonian. We can immediately see that Hamiltonian is only going to be a function of Ψ and it conjugate momenta Π (which is just same as Ψ∗ ). Which automatically removes redundancy from Hamiltonian. Hence Hamiltonian density would be H = (i~Ψ∗)

2 ∂Ψ ˙ − ~ ∇Ψ∗ ∇Ψ −V Ψ∗ Ψ) − (i~Ψ∗ Ψ ∂t 2m

Which we can write as

~2 ∇Ψ∗ ∇Ψ +V Ψ∗ Ψ H= 2m This is completely Hermitian. This is the reason why we didn’t care much that we are starting with redundant and non-Hermitian Lagrangian. Note that occurrence of gradient of conjugate momenta is still a painful experience. But I believe it can be written as linear superposition of gradient of Ψand Π. Note that both Ψand Πare non-Hermitian. Now once we have identified Ψand Π as “canonical variables” we can lift them to the status of “operators” representing “generalized” co-ordinate and momentum in QFT picture. The commutation relation to be enforced would either be Bosonic or Fermionic – both leads to physically correct results. One of them represents Fermionic fields and other represents Bosonic fields. Now wavefunctions can be expanded in linear superposition of energy eigen wavefunctions of the Schrodinger’s equation (single particle). Hence when wavefunction is lifted to the status of an operator the coefficient of expansions also become operators – these are exactly same as the creation and annihilation operators discussed in previous section. Wave function operators are simply recognized as creation and annihilation operators at one spatial location. Hence they obey the same commutation relation as creation and annihilation operators discussed in previous section.

10

Non-relativistic, Massive, Bosonic Fields

Follows exactly same two alternative schemes given above. Only difference is in the commutation relations among field operators or creation/annihilation operators.

43

Part VI

Relativistic Quantum Field Theory ( QFT ) 11 11.1

Klein Gordon (KG) Field Derivation of ’Classical’ KG Field

Klein-Gordon equation is the most simple relativistic quantum equation. Let us see how one can obtain KG equation following standard quantization rule. We know that p2 + m2 = E 2 . In 4-vector notations pµ pµ = m2 where momentum 4-vector is defined as pµ ≡ {E, p1 , p2 , p3 , p4 } One then do the standard quantization (need the energy operator as well, see below) and obtains −~2 ∂µ ∂µ = m2 In natural units (~ = 1,c = 1) above equation is written as −∂µ ∂µ = m2 This is the KG equation. Writing it in standard form ∂2 ∂t 2 Alternative way of obtaining the above equation (which follows the standard quantization more closely) is as follows. ∂ H = i~ ∂t Hence ∂2 H 2 = −~2 2 ∂t 2 2 Now in relativistic quantum mechanics H = p + m2 . Converting p to operator and plugging in the above equation one can again get the KG equation. Only difference from standard quantization technique is that one needs square of energy as there is square root involved. This is the starting points of all troubles in relativistic quantum mechanics. There were lots of problems with the KG equation. Thats the reason for which initially it was discarded. We now know that it is correct equation for spin zero bosons like pions. We now know how to interpret the peculiarities of this equation. Major problem with this equation is that it can accept both positive and negative energies. We would see that this can be interpreted as anti-particle states. If one interprets Ψ2 as probabilities then one gets into the trouble of negative probabilities. We will discuss the Noether’s theorem latter. Basically its a continuity equation. From the continuity equation one obtains the correct interpretation of probabilities and densities associated with the Ψ. We would then interpret Ψ2 as charge density. −∇2 + m2 = −

44

11.2

Classical Hamiltonian Theory of Real Valued Scalar KG Field

11.2.1

Classical Hamiltonian and Lagrangian

A free-space KG equation is written as (−∇2 + m2 )Ψ(x,t) = −

∂2 Ψ(x,t) ∂t 2

˙ ∂Ψ(x),t) so that we First of all, we want to find the Lagrangian density function L(Ψ(x), Ψ(x), can find the conjugate momentum variable π(x) which is conjugate to Ψ(x). Note that we are already working in generalized co-ordinate system. I would sometime drop the explicit time dependence from Ψ(x,t) for smaller notations, spatial dependence is usually kept to specifically indicate that we dealing with continuum version of infinite set of generalized variables. We start by selecting Ψ(x) as infinite set of generalized co-ordinate variable. This choice is never unique. The choice of Lagrangian density is never unique as well. We should be able to find a function so that the associated Lagrange’s equation is the above KG field equation. Let us try (we would justify that this is one of the many possible valid choices) :1 1 ˙2 1 ˙ − (∇Ψ)2 − m2 Ψ2 L(Ψ(x), Ψ(x), ∂Ψ(x),t) = Ψ 2 2 2

(13)

The Euler-Lagrange’s equation of motion would be :d ∂L ∂L ∂L ∂2 ) + ∇( ( )− = ( 2 − ∇2 + m2 )Ψ(x) = 0 ˙ dt ∂(Ψ(x)) ∂(∇Ψ(x))) ∂Ψ(x) ∂t Which is exactly same as the KG equation. Hence, above expression for Lagrange density is one of the many possible acceptable expressions. From L we can easily obtain π(x) (variable conjugate to Ψ(x)) as follows :π(x) ≡

∂L ˙ = Ψ(x) ˙ ∂(Ψ(x))

(14)

And we can now construct the Hamiltonian density :˙ − L = 1 π2 + 1 (∇Ψ)2 + 1 m2 Ψ2 H(Ψ(x), π(x), ∇(Ψ(x))t) ≡ π(x)Ψ(x) 2 2 2

(15)

If we move to energy eigen basis, we can find a more convenient expression for Hamiltonian density. So let try that first. 11.2.2

Ψ(x,t) Field in Momentum and Energy Basis

One can easily check that the energy eigen states of free-space KG equation are :ΨE (x,t) = A exp(ipx − iEt) 45

where, p E = ± m2 + p2 = ±E p Here E p is always a positive number. Note that for one value of p, there are two values of E possible - one positive and one negative (classical KG field theory does not have any acceptable justification for negative energies). Hence we note that classical field can be written as a general superposition of all possible modes :1 Ψ(x,t) = 2π

Z

1 d p{Ψ+ (p) exp(ipx) exp(−iE pt)} + 2π

Z

d p{Ψ− (p) exp(ipx) exp(iE pt)} (16)

The factors of 2π have been included just because of conventions. These can be dumped into Ψ+ and Ψ− which we anyway would-re normalize latter on. Above can also be written as: Ψ(x,t) = where,

1 2π

Z

d p{ f (p,t) exp(ipx)}

f (p,t) = Ψ+ (p) exp(−iE pt) + Ψ− (p) exp(iE pt)

(17)

(18)

Note that 17 is exactly a spatial Fourier transform. Which is also same as expanding Ψ(x,t) in momentum basis. Note that energy basis expansion needs two expansion coefficient for one |E| (or for one p) but momentum space expansion needs only one coefficient. Ψ+ (p) and Ψ− (p) would be recognized as energy mode amplitudes and f (p,t) as momentum mode amplitude of field Ψ(x). Let us restrict our analysis for the time being to only those solutions which are real-valued. More generic complex valued solutions and their interpretations would be seen latter. Hence:

=

1 2π

1 2π Z

Z

1 2π Z

Z

d p{Ψ+ (p) exp(ipx) exp(−iE pt)} +

d p{Ψ− (p) exp(ipx) exp(iE pt)}

1 2π

d p{Ψ∗− (p) exp(−ipx) exp(−iE pt)}

d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} +

This can also be re-written as Z

Z

1 1 d p{Ψ+ (p) exp(ipx) exp(−iE pt)} + d p{Ψ− (−p) exp(−ipx) exp(iE pt)} 2π Z 2π Z 1 1 = d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} + d p{Ψ∗− (−p) exp(ipx) exp(−iE pt)} 2π 2π Hence: Z

1 d p{Ψ+ (p) − Ψ∗− (−p)} exp(ipx) exp(−iE pt) 2π Z 1 = d p{Ψ∗+ (p) − Ψ− (−p)} exp(−ipx) exp(iE pt) 2π This would be true only if

Ψ+ (p) − Ψ∗− (−p) = 0 46

or (and)

Ψ∗+ (p) − Ψ− (−p) = 0

(19)

This is an important result. It shows that the restriction that the field Ψ(x,t) is real ensures that the mode amplitude of negative energy state is always related to the mode amplitude of positive energy state. With this, one can write 16 as: 1 Ψ(x,t) = 2π

11.2.3

Z

1 d p{Ψ+ (p) exp(ipx) exp(−iE pt)} + 2π

Z

d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} (20)

π(x,t) Field in Momentum and Energy Basis

Differentiating 20 with respect to time and using 14: −iE p π(x,t) = { 2π

Z

Z

d p{Ψ+ (p) exp(ipx) exp(−iE pt)} −

d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} (21)

Which can also be written as: −iE p { π(x,t) = 2π

Z

Z

d p{Ψ+ (p) exp(ipx) exp(−iE pt)} −

d p{Ψ− (p) exp(ipx) exp(iE pt)} (22)

And analogy to 17, we can also write π(x,t) = where,

−iE p 2π

Z

d p{g(p,t) exp(ipx)}

g(p,t) = Ψ+ (p) exp(−iE pt) − Ψ− (p) exp(iE pt)

(23)

(24)

With this, we have been able to write the generalized co-ordinate and its conjugate momentum as B + B? and B − B? just as in the case of single Harmonic oscillator case. We would see latter how these two parts of solution gets converted to creation and annihilation operators and how Hamiltonian gets ’factorized’ and subsequently diagonalized when we do the quantization in the next section. 11.2.4

Classical Hamiltonian in Momentum and Energy Basis

One should notice that the classical Hamiltonian (not density) can be written as : Z

H=

1 1 1 dx{ π2 + (∇Ψ)2 + m2 Ψ2 } = 2 2 2

Or,

Z

H=

dp

Z

dp

1 {E 2 + p2 +m2 }{Ψ+ (p)Ψ∗+ (p)+Ψ∗+ (p)Ψ+ (p)} 4π (25)

1 2 E {Ψ+ (p)Ψ∗+ (p) + Ψ∗+ (p)Ψ+ (p)} 2π

47

(26)

Suppose we want to write the classical Hamiltonian as Z

H=

dp

1 Ep {Ψ+ (p)Ψ∗+ (p) + Ψ∗+ (p)Ψ+ (p)} 2π 2

(27)

This can easily be done by defining new normalized: Ψ+ (p)|old → 11.2.5

Ψ+ (p)|new p 2E p

(28)

Field Expansions in Normalized Variables Z

Z

1 p Ψ(x,t) = { d p{Ψ+ (p) exp(ipx) exp(−iE pt)} + d p{Ψ− (p) exp(ipx) exp(iE pt)} 2π 2E p Z Z 1 p = { d p{Ψ+ (p) exp(ipx) exp(−iE pt)} + d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} 2π 2E p Z 1 = d p{ f (p,t) exp(ipx)} (29) 2π where,

1 {Ψ+ (p) exp(−iE pt) + Ψ− (p) exp(iE pt)} 2E p

f (p,t) = p −i

p

Z

(30)

Z

E p /2 π(x,t) = { d p{Ψ+ (p) exp(ipx) exp(−iE pt)} − d p{Ψ− (p) exp(ipx) exp(iE pt)} 2π p Z Z −i E p /2 { d p{Ψ+ (p) exp(ipx) exp(−iE pt)} − d p{Ψ∗+ (p) exp(−ipx) exp(iE pt)} = 2π Z 1 = d p{g(p,t) exp(ipx)} (31) 2π where, q g(p,t) = −i E p /2{Ψ+ (p) exp(−iE pt) − Ψ− (p) exp(iE pt)} 11.2.6

(32)

Inverse Transforms

One can also try to find the inverse expansions by exploiting the orthonormality of exponentials: Z

Z

1 dxΨ(x,t) exp(−iqx) = p {Ψ+ (q) exp(−iEqt) + Ψ∗+ (−q) exp(iEqt)} 2Eq r

Eq {Ψ+ (q) exp(−iEqt) − Ψ∗+ (−q) exp(iEqt)} 2 Adding and subtracting above two expressions: dxπ(x,t) exp(−iqx) = −i

48

Ψ+ (q) exp(−iEqt) = Ψ∗+ (−q) exp(iEqt) =

Z

r dx{

Z

r dx{

This last expression can also be written as: r Z Ψ∗+ (q) exp(iEqt) =

dx{

Eq i Ψ(x,t) + p π(x,t)} exp(−iqx) 2 2Eq

(33)

Eq i Ψ(x,t) − p π(x,t)} exp(−iqx) 2 2Eq

Eq i Ψ(x,t) − p π(x,t)} exp(iqx) 2 2Eq

11.3

Quantization of Real-Valued Scalar ’Classical’ KG Field

11.3.1

Quantized Field Operators

(34)

With this much of background preparations we are now ready to do the standard quantization. One can do the standard quantization in real space (obviously with generalized co-ordinates though) by converting the infinite set of time dependent real space co-ordinates and corresponding conjugate momenta – Ψ(x,t) and π(x,t) – into time independent (in Schrodinger’s picture) ˆ ˆ Hermitian operators Ψ(x) and π(x) and enforcing standard commutation relation between these: ˆ ˆ [Ψ(x), Ψ(y)] = 0 ˆ ˆ [π(x), π(y)] = 0 ˆ ˆ [Ψ(x), π(y)] = iδ(x − y)

(35)

Note that the right hand side on the last expression a delta function (not a Kronecker delta). Delta R function is defined such that δ(x − y) = 0 ∀x 6= y and such that δ(x − y)dx = 1. After doing this, we notice from 33 and 34 that even the of energy mode expansion coefficients Ψ+ (p) exp(−iE pt) and Ψ∗+ (p) exp(iE pt) get converted into operators : r Z Eq ˆ i ˆ Ψ(x) + p π(x)} exp(−ipx) (36) a(p) ˆ = dx{ 2 2Eq Z †

aˆ (p) =

r dx{

Eq ˆ i ˆ Ψ(x) − p π(x)} exp(ipx) 2 2Eq

(37)

ˆ One can easily check that indeed 37 is the adjoint of 36 as direct consequence of ˆΨ(x) and ˆ π(x) being the Hermitian operators by definition. We would see below that these new operators actually behave as creation and annihilation operators and the Hamiltonian can easily be written in diagonal forms with their help. Using 36 and 37 one can easily check that

49

[a(p), ˆ a(q)] ˆ = 0 † [aˆ (p), aˆ (q)] = 0 [a(p), ˆ aˆ† (q)] = δ(p − q) †

(38)

Note that the right hand side on the last expression a Kronecker delta function (not a delta function). Kronecker delta function is defined such that δ(x − y) = 0 ∀x 6= y and δ(x − y) = 1 for x = y. Also, from 30 and 32, the momentum mode expansion coefficients also get converted to operators: 1 fˆ(p) = p {a(p) + a† (−p)} (39) 2E p q g(p) ˆ = −i E p /2{a(p) − a† (−p)} (40) Please note that these operators are not Hermitian. The commutation relation between these operators can also be easily evaluated: −i −i ˆ aˆ† (−q)] + [aˆ† (p), a(−q)]} ˆ = {−(δ(p + q)) + (−δ(p + q))} [ fˆ(p), g(q)] ˆ = {−[a(p), 2 2 ˆ [ f (p), g(q)] ˆ = iδ(p + q) Also inverse transformations can readily be calculated as well: Z

1 1 ˆ Ψ(x) = d p{ p }{a(p) ˆ exp(ipx) + aˆ† (p) exp(−ipx)} (41) 2π 2E p Z q 1 ˆ d p{−i E p /2}{a(p) ˆ exp(ipx) − aˆ† (p) exp(−ipx)} (42) π(x) = 2π Its good to see that these operators are certainly Hermitian as we initially defined them to be. 11.3.2

Quantum Hamiltonian

Now the quantum version of Hamiltonian can be found starting from 66: Z 1 Ep ˆ H = dp {a(p) ˆ aˆ† (p) + aˆ† (p)a(p)} ˆ 2π 2 Z 1 1 = d p E p {aˆ† (p)a(p) ˆ + [a(p), ˆ aˆ† (p)]} (43) 2π 2 Z 1 1 = d p E p {aˆ† (p)a(p) ˆ + δ(0)} (44) 2π 2 In the last expression we have used the 38 derived above. Note that the second term on the right hand sign sums to infinity. Since its a constant energy term, we would throw out this term. Note that in most physical problems only energy difference is of concern. There are some places where infinite energy of ground state can lead to problems but we would not encounter any such situation in this article. Hence we would write: Z 1 Hˆ = d p E p {aˆ† (p)a(p)} ˆ (45) 2π 50

11.3.3 11.3.3.1

Number State basis Ground State

11.3.3.2 Higher States of Definite Particle Number Now let us try to explore some properties of aˆ† (p) and a(p) ˆ operators. Suppose |φi is a known eigen state of Hamiltonian defined in ˆ 93. So that H|φi = E|φi. aˆ† (q)|φi would be some state in the same Hilbert space. Let us operate Hamiltonian on this state. So Z

Z

dp

1 1 E p aˆ† (p)a(p) ˆ aˆ† (q)|φi = d p E p aˆ† (p)(δ(p − q) + aˆ† (q)a(p))|φi ˆ 2π 2π Z 1 = d p E p (δ(p − q)aˆ† (p) + aˆ† (p)aˆ† (q)a(p))|φi ˆ 2π Z 1 ˆ = d p E p (δ(p − q)aˆ† (p) + aˆ† (q)aˆ† (p)a(p))|φi 2π Z 1 = d p{ E p (δ(p − q)aˆ† (p)|φi} + aˆ† (q)E|φi 2π † = Eq aˆ (q)|φi + aˆ† (q)E|φi = (E + Eq )aˆ† (q)|φi

Hence, aˆ† (q)|φi is also an energy eigen state (with energy of E + Eq ) of the Hamiltonian. Similarly we can show that a(q)|φi ˆ is also an energy eigen state (with energy of E − Eq ) of the Hamiltonian. More concise mathematical prove is to show that: ˆ aˆ† (p)] = E p aˆ† (p) [H,

(46)

ˆ a(p)] [H, ˆ = −E p a(p) ˆ

(47)

and Above two properties guarantee that proposed aˆ† (p) and a(p) ˆ indeed behave like creation and annihilation operators respectively. 11.3.3.3 11.3.4

Orthonoarmality and Completeness Coherent and Squeezed States

Let us look at following state (this actuallyPoisonian is a coherent state as we would show) : ∞

|ψc i = exp(caˆ† (p))|0i = ∑ { i=0

ci (aˆ† (p))i }|0i i!

Here c is a complex number. Now, we also know that |n(p)i = p

1 (aˆ† (p))n(p) |0i n(p)! 51

Hence state |Ψc i can be written as ∞ cn(p) |Ψc i = ∑ p |n(p)i n(p)! i=0

We notice that in one single classical momentum mode we have an indefinite number of particles. The probability amplitude of finding certain number of particles has Poisonian distribution. We would then like to study U(t ← 0)|Ψc i. This would justify that, indeed, the above constructed state is a coherent state. 11.3.5

Heisenberg Representation

Using 94, one can generalize that relation and obtain Hˆ n aˆ† (p) = aˆ† (p)(Hˆ − E p )n Note that the creation and annihilation operators encountered in previous sections are all in Schrodinger pictures. Now using the Taylor expansion of exponential operators, one can easily check that the Heisenberg representation of these operators is given by aˆ†h (p) = Us† (t ← 0)aˆ†s (p)Us (t ← 0) = aˆ†s (p) exp(−iE pt) and aˆh (p) = Us† (t ← 0)aˆs (p)Us (t ← 0) = aˆs (p) exp(iE pt) Hence, field operators in Heisenberg representation in relativistic notations (4-vectors) can be written as Z

1 d p{ p }{aˆs (p) exp(−ip.x) + aˆ†s (p) exp(ip.x)}| p0 =E p (48) 2E p Z q 1 πˆ h (x) = d p{−i E p /2}{aˆs (p) exp(−ip.x) − aˆ†s (p) exp(ip.x)}| p0 =E p (49) 2π Notice that the signs in the exponentials have reversed. One should notice that p.x = pµ xµ = pµ xµ = E pt − p1 x1 − p2 x2 − p3 x3 . This justifies the sign reversal. Spatial part automatically get the previous sign. These are very important and very generic results. Time and space would always have opposite signs. Time exponential with negative sign is called “positive energy part of the solution”. Time exponential with positive sign is called “negative energy part of the solution”. Creation operator always goes with negative energy part of the solution and annihilation operator always goes with the positive energy solution part of the solution. ˆ h (x) = 1 Ψ 2π

11.3.6

Two Point Correlation Functions/Propagators

Propagators are related to two point correlation functions. Sometime names might be used interchangeably. Two point correlation functions are also related to Green’s functions of KG operator. Many different types of Green’s functions can be defined. First two we would are not 52

really Green’s function. But they are usually referred to as Green’s function because these two are the building blocks for the 4 different types of Green’s functions of the KG operator. The most basic one is Particle Propagator or a Greater Green’s Function. In relativistic notations this is defined as ˆ h (x)Ψ ˆ † (y)|0h i G> (x − y) ≡ G> (x ← y) ≡ h0h |Ψ h Note that what we are doing here is that we are creating a single particle at y (a 4-vector) and finding the amplitude of probability that system propagates to become particle at x (again a 4ˆ h (x) = Ψ ˆ † (x) as the field operator vector). Also note that for real valued scalar KG field Ψ h is Hermitian. But above expression can be further generalized by creating a single particle in, say, single particle classical momentum mode p (that is state |pi) and then looking an amplitude probability that state evolves into a state |qi. Using this Greater Green’s function one can study the causality of the theory. Note that it is not necessary that y0 < x0 . Another fundamentally important Green’s function is what is known as Anti-Particle (or Hole) Propagator or a Lesser Green’s Function. This is defined as follows ˆ † (x)Ψ ˆ h (y)|0h i G< (x − y) ≡ G< (x ← y) ≡ h0h |Ψ h So basically, G> (x − y) = G< (y − x) Also for real valued scalar KG field G> (x − y) = G< (x − y). Note that x and y are the 4-vectors in above equation. So we are not only flipping the time but also the space co-ordinates. One ˆ and Ψ ˆ † are the equal very important point is that the commutation relation between Ψ time commutation relation. There is no defined relationship between field operators at different times. So there would be no obvious correlation between lesser and greater Green’s function. Let us try to obtain the Fourier transform of these two correlation function (these are not really the Green’s functions of KG operator). Substituting 48 in the definition of Greater Green’s function we can immediately write Z >

G (x − y) =

Similarly,

Z >

G (x − y) =

d3 p 1 exp(−ip.(x − y))| p0 =E p (2π)3 2E p

d3 p 1 exp(−ip.(y − x))| p0 =−E p (2π)3 2E p

We would find these expressions useful in the following discussion. One should notice that the coefficients of exponentials in right hand side are NOT really 4D Fourier transform. One can consider this coefficient multiplied by time exponential as spatial Fourier transform which is a function of time. Above two functions are the basic building blocks for other types of correlation functions which have netter physical and mathematical properties. Using these two functions we define 53

following four other correlation functions which are more commonly encountered in text books. A set of functions that are sometime useful are time ordered and anti time ordered Green’s functions. Time ordered Green’s function is also sometime (in the case of free/non-interacting fields) called the Feynman propagator. ˆ h (x)Ψ ˆ † (y)|0h i Gt (x − y) ≡ Gt (x ← y) ≡ h0h |T Ψ h ˆ h (x)Ψ ˆ † (y)|0h i Gt¯(x − y) ≡ Gt (x ← y) ≡ h0h |T¯ Ψ h Note that t



G> i f x0 > y0 G< i f x0 < y0





G< i f x0 > y0 G> i f x0 < y0



G (x − y) = t¯

G (x − y) =

Two other functions are retarded and advanced Green’s functions. Retarded Green’s function is defined as ˆ † (x), Ψ ˆ h (y)]|0h i) = −iΘ(x0 − y0 )(G> (x − y) − G< (x − y) GR (x − y) ≡ −iΘ(x0 − y0 )(h[0h |Ψ h Note that Θ is a Heaviside step function. Note that some books leave out the −i factor. Similarly Advanced green’s Function is defined as ˆ † (x), Ψ ˆ h (y)]|0h i) = −iΘ(y0 − x0 )(G> (x − y) − G< (x − y) GA (x − y) ≡ −iΘ(y0 − x0 )(h[0h |Ψ h These four correlation functions are actually very closely related to each other. We would show that Gt , Gt¯, GR and GA are all Green’s function of KG equation. That is, if we write a inhomogeneous KG equation with a 4-dimensional delta function on right hand side then above Green’s functions satisfy the equation. This gives a prescription of actually calculating these functions. One can Fourier transform the whole equation. Which then get converted to a simple algebraic equation. From this one can calculate the Fourier transform of the Green’s function which can then be inverse transformed to calculated GF in real space. Suppose G(x − y) is a Green’s func˜ ˜ tion of KG operator. Let its 4D Fourier transform be G(p). By definition, then, G(p) would satisfy the following algebraic equation ˜ G(p) =

i p2 − m2

Now taking its 4D Fourier inverse Z

d4 p i exp(−ip.(x − y)) 4 2 (2π) p − m2 Z i/2E p d 4 p i/2E p = ( 0 − 0 ) exp(−ip.(x − y)) 4 (2π) p − E p p + E p

G(x − y) =

Note that p2 − m2 = (p0 )2 − E p2 = E 2 − E p2 . Now, if we look at the integral over energy scale more closely, we would realize that the integrand has two poles at p0 = ±E p . First thing is 54

that we need to choose a contour along the real axis so that poles do not fall exactly on the contour. There are four ways of selecting this part of contour. Depending upon how we decide to integrate we get above defined four different GFs. Secondly, we need to close the contour either in upper or lower plane depending upon x0 > y0 or y0 > x0 . Let us do the integration on energy scale. If we choose to go “above” both the poles then for x0 < y0 G(x − y) = 0. For x0 > y0 G(x − y) = G> (x − y) − G< (x − y). Hence this is retarded Green’s function. Similarly if we go “below” both poles we would get advanced GF. If we below negative energy below and above positive energy pole (this is known as Feynman prescription) we would get time ordered GF. If go above negative energy pole but below positive energy pole then we would get anti-time ordered GF.

11.4

Classical Hamiltonian Theory of Complex-Valued Scalar KG Field

11.4.1

Classical Hamiltonian and Lagrangian

Action for the classical complex valued field is given as Z

S≡

Z

Ldt =

d 4 x(∂µ φ? ∂µ φ − m2 φ? φ)

Hence, Lagrangian density function can be written as L(φ(x), φ? (x), ∂µ φ(x), ∂µ φ? (x),t) = ∂µ φ? ∂µ φ − m2 φ? φ ∂φ? ∂φ ~ ? ~ − ∇φ .∇φ − m2 φ? φ = ∂t ∂t

(50)

Note that we don’t have explicit time dependence in this problem. We know from Lagrange Principle of least action that the extremum would give the Euler-Lagrange equation of motion. The Euler-Lagrange’s equation of motions would be :∂L ∂L ∂L ∂2 d ( ) + ∇( )− = ( 2 − ∇2 + m2 )φ? (x) = 0 ˙ dt ∂(φ(x)) ∂(∇φ(x))) ∂φ(x) ∂t and

d ∂L ∂L ∂L ∂2 ( ) + ∇( ) − = ( − ∇2 + m2 )φ(x) = 0 ˙ ? (x)) dt ∂(φ ∂(∇φ? (x))) ∂φ? (x) ∂t 2

Both of these are exactly same as the KG equation. Hence both φ(x) and φ? (x) satisfies KG equation. From the expression for Lagrangian density 50, the conjugate momentum density function (conjugate to φ(x)) would be:∂L ∂φ? π1 (x) ≡ = ˙ ∂t ∂φ(x) and that conjugate to φ? (x) is π2 (x) ≡

∂L ∂φ = ? ˙ ∂t ∂φ (x) 55

We notice that

π1 (x) = π?2 (x)

So we can use more simple notations: ∂L ∂φ? π (x) = π1 (x) ≡ = ˙ ∂t ∂φ(x)

(51)

∂φ ∂L = ? ˙ ∂t ∂φ (x)

(52)

?

π(x) = π2 (x) ≡

Now, once we know the conjugate momenta density, one can easily obtain the Hamiltonian density function H(φ(x), φ? (x), π? (x), π(x), ~∇φ(x)~∇φ? (x),t) as ˙ + π(x)φ˙ ? (x) − L H ≡ π? (x)φ(x) ? ˙ + π(x)φ˙ ? (x) − ∂φ ∂φ + ~∇φ? .~∇φ + m2 φ? φ = π? (x)φ(x) ∂t ∂t Now using 51 and 52 we can write the classical Hamiltonian density function as: H(φ(x), φ? (x), π? (x), π(x), ~∇φ(x)~∇φ? (x),t) = π(x)π? (x) + ~∇φ? .~∇φ + m2 φ? φ

(53)

Note that the order of π? π and another similar term is not determined of classical mechanR ics. Applying integration operator d 3 x one can then get the Classical Hamiltonian function. 11.4.2

Classical φ(x,t) and φ? (x,t) Fields in Energy/Momentum Basis

A free-space KG equation is written as (−∇2 + m2 )Ψ(x,t) = −

∂2 Ψ(x,t) ∂t 2

One can easily check that the classical energy eigen states of free-space KG equation are :ΨE (x,t) = A exp(ipx − iEt) where, p E = ± m2 + p2 = ±E p Here E p is always a positive number. Note that for one value of p, there are two values of E possible - one positive and one negative . Hence we note that arbitrary classical field can be written as a general superposition of all possible solutions :φ(x,t) =

1 2π

Z

d p{φ+ (p) exp(ipx) exp(−iE pt)} +

1 2π

Z

d p{φ− (p) exp(ipx) exp(iE pt)} (54)

The factors of 2π have been included just because of conventions. These can be dumped into Ψ+ and Ψ− which we anyway would-re normalize latter on. One should note that above expansion 56

is simply spatial Fourier expansion. This can be understood by noting that above can also be written as: Z 1 φ(x,t) = d p{ f (p,t) exp(ipx)} (55) 2π where, f (p,t) = φ+ (p) exp(−iE pt) + φ− (p) exp(iE pt) (56) Note that 55 is exactly a spatial Fourier transform. Which is also same as expanding Ψ(x,t) in momentum basis. Note that energy basis expansion needs two expansion coefficient for one |E| (or for one p) but momentum space expansion needs only one coefficient. Ψ+ (p) and Ψ− (p) would be recognized energy mode amplitudes and f (p,t) as momentum mode amplitude of field Ψ(x). Now 54 can also be written as 1 φ(x,t) = 2π

Z

Z

1 d p{φ+ (p) exp(ipx) exp(−iE pt)} + 2π

d p{φ− (−p) exp(−ipx) exp(iE pt)} (57)

Now taking the complex conjugate of 57: 1 φ (x,t) = 2π

Z

?

11.4.3

d p{φ?− (−p) exp(ipx) exp(−iE pt)} +

1 2π

Z

d p{φ?+ (p) exp(−ipx) exp(iE pt)} (58)

Classical π(x,t) and π? (x,t) Fields in Momentum/Energy Basis

Differentiating 57 with respect to time and using 52: −iE p π(x,t) = { 2π

Z

Z

d p{φ+ (p) exp(ipx) exp(−iE pt)} −

d p{φ− (−p) exp(−ipx) exp(iE pt)} (59)

And analogy to 55, we can also write −iE p π(x,t) = 2π where,

Z

d p{g(p,t) exp(ipx)}

g(p,t) = φ+ (p) exp(−iE pt) − φ− (p) exp(iE pt)

(60)

(61)

Now taking the complex conjugate of 59: π? (x,t) =

Z

−iE p [ 2π

Z

d p{φ?− (−p) exp(ipx) exp(−iE pt)} − {

d p{φ?+ (p) exp(−ipx) exp(iE pt)}] (62)

Note that the condition 51 is automatically enforced.

57

11.4.4

Classical Hamiltonian in Energy Basis

One should notice that the classical Hamiltonian (not density) can be written as : Z

H =

dx{π(x)π? (x) + ~∇φ? .~∇φ + m2 φ? φ}

Z

=

dp

Or,

(63)

1 {E 2 + p2 + m2 }{φ+ (p)φ∗+ (p) + φ− (−p)φ?− (−p)} 2π p

(64)

Z

2 2 E p {φ+ (p)φ∗+ (p) + φ− (−p)φ?− (−p)} 2π Suppose we want to write the classical Hamiltonian as H=

dp

Z

H=

dp

1 Ep {φ+ (p)φ∗+ (p) + φ− (−p)φ?− (−p)} 2π 2

(65)

(66)

This can easily be done by defining new normalized field variables such that: Ψ+ (p)|old → 11.4.5

Ψ+ (p)|new p 2 Ep

(67)

Classical Field Expansions in Normalized Variables Z

1 φ(x,t) = p { 4π E p 1 p

φ? (x,t) =

Z

d p{φ+ (p) exp(ipx) exp(−iE pt) +

Z

d pφ− (−p) exp(−ipx) exp(iE pt)} Z

(68)

d p{φ?− (−p) exp(ipx) exp(−iE pt) + d pφ?+ (p) exp(−ipx) exp(iE pt)} 4π E p (69) p Z −i E p Z π(x,t) = { d p{φ+ (p) exp(ipx) exp(−iE pt)} − d p{φ− (−p) exp(−ipx) exp(iE pt)} 4π (70) p Z Z −i Ep ? ? π (x,t) = [ d p{φ− (−p) exp(ipx) exp(−iE pt)} − { d p{φ?+ (p) exp(−ipx) exp(iE pt)}] 4π (71) 11.4.6

Classical Inverse Transforms

One can also try to find the inverse expansions by exploiting the orthonormality of exponentials: Z

1 dxφ(x,t) exp(−iqx) = p {φ+ (q) exp(−iEqt) + φ− (q) exp(iEqt)} 2 Eq p Z −i Eq dxπ(x,t) exp(−iqx) = {φ+ (q) exp(−iEqt) − φ− (q) exp(iEqt)} 2 58

(72)

(73)

Z

1 dxφ? (x,t) exp(−iqx) = p {φ?− (−q) exp(−iEqt) + φ?+ (−q) exp(iEqt)} 2 Eq p Z Eq ? −i {φ− (−q) exp(−iEqt) − φ?+ (−q) exp(iEqt)} dxπ? (x,t) exp(−iqx) = 2 Adding and subtracting 72 and 73 one gets: φ+ (q) exp(−iEqt) = φ− (q) exp(iEqt) =

Z

dx{

Z

p

i Eq φ(x,t) + p π(x,t)} exp(−iqx) Eq

(74) (75)

(76)

p i dx{ Eq φ(x,t) − p π(x,t)} exp(−iqx) Eq

This last expression can also be written as: φ− (−q) exp(iEqt) =

Z

p i dx{ Eq φ(x,t) − p π(x,t)} exp(iqx) Eq

(77)

Similarly, adding and subtracting 74 and 75, one gets: φ?− (−q) exp(−iEqt) = φ?+ (−q) exp(iEqt) =

Z

Z

dx{

i Eq φ? (x,t) + p π? (x,t)} exp(−iqx) Eq

p

(78)

p i dx{ Eq φ? (x,t) − p π? (x,t)} exp(−iqx) Eq

This last expression can also be written as: φ?+ (q) exp(iEqt) =

Z

p i dx{ Eq φ? (x,t) − p π? (x,t)} exp(iqx) Eq

11.5

Quantization of Complex-Valued Scalar KG Field

11.5.1

Quantization Field Operators

(79)

Once we have the classical conjugate variables and classical Hamiltonian, we are now ready to do the standard quantization. One can do the standard quantization in real space (obviously with generalized co-ordinates though) by converting the infinite set of time dependent real space coordinates and corresponding conjugate momenta – φ(x,t), π? (x,t) and φ? (x,t), π(x,t) – into time ˆ ˆ independent (in Schroedinger’s picture) operators φ(x) , πˆ † (x) and φˆ † (x) , π(x) respectively14 and enforcing standard commutation relation between these: ˆ x), φ(~ ˆ y)] = 0 [φ(~ [πˆ † (~x), πˆ † (~y)] = 0 ˆ x), πˆ † (~y)] = iδ(~x −~y) [φ(~

(80)

we convert real and imaginary part of φ into Hermitian operators, that is Re{φ} → φˆ r and Im{φ} → φˆ i ˆ From this one can at once see that φ? → φ† . so that φ → φ. 14 Suppose

59

[φˆ † (~x), φˆ † (~y)] = 0 ˆ x), π(~ ˆ y)] = 0 [π(~ † ˆ y)] = iδ(~x −~y) [φˆ (~x), π(~

(81)

ˆ x) and φˆ † (~x) as well as π(~ ˆ x) and πˆ † (~x) are independent variables, we postulate Also since φ(~ that all other pairs commute with each other. Note that the right hand side of above two expression are Dirac delta functions (not the Kronecker delta). In Heisenberg representation these all are equal time commutation relations. Now the corresponding quantized Hamiltonian density operator can be written as: Z

Hˆ =

ˆ ˆˆ ˆ x)} ˆ x)πˆ † (~x) + ~∇φˆ † (~x).~∇φ(~ d 3 x{π(~ x) + m2 φˆ † (~x)φ(~

(82)

We notice that due to postulated commutation relations, there is no confusion about ordering of the operators while going from classical to quantum representation. Now, we can convert these operators into time depending operators by moving to Heisenberg representation. Noting that the Hamiltonian is explicitly time independent, we can write, for φˆ 1 (z) field operator for example, in Heisenberg representation ˆ x,t) = exp(iHt) ˆ x) exp(−iHt) ˆ φ(~ ˆ φ(~ Similarly other operators are also converted into Heisenberg representation. Note that in this representation all above commutation relations still hold but they are equal time relations. Now ˆ x) is:Heisenberg equation of motion for field φ(~ Z

ˆ x,t) ∂φ(~ ˆ x,t), d 3 xH] ˆ i = [φ(~ ∂t Z ˆ ˆˆ ˆ ˆ y,t)}] ˆ y,t)πˆ † (~y,t) + ~∇φˆ † (~y,t).~∇φ(~ = [φ(~x,t), d 3 y{π(~ y,t) + m2 φˆ † (~y,t)φ(~ Z

= Z

=

ˆ ˆ y,t)[φ(x,t), d 3 y{π(~ πˆ † (~y,t)]} ˆ y,t)iδ(~x −~y) = iπ(~ ˆ x,t) d 3 yπ(~

Also, Z

i

ˆ x,t) ∂π(~ ˆ ˆ x,t), d 3 xH] = [π(~ ∂t Z ˆ x,t), = [π(~ Z

=

ˆ y,t)[π(x,t), ˆ y,t)[π(x,t), ˆ ˆ d 3 y{−∇2 φ(~ φˆ † (~y,t)] + m2 φ(~ φˆ † (~y,t)]}

Z

= −

ˆ ˆˆ ˆ y,t)}] ˆ y,t)πˆ † (~y,t) + ~∇φˆ † (~y,t).~∇φ(~ d 3 y{π(~ y,t) + m2 φˆ † (~y,t)φ(~

ˆ y,t) + m2 φ(~ ˆ y,t)}iδ(~x −~y) = −i{−∇2 φ(x,t) ˆ ˆ d 3 y{−∇2 φ(~ + m2 φ(x,t)}

60

Combining the two equations: ∂2ˆφ(x,t) ˆ ˆ = ∇2 φ(x,t) − m2 φ(x,t) 2 ∂t Repeating exactly same problem one can also show that: ∂2 φˆ † (x,t) = ∇2 φˆ † (x,t) − m2 φˆ † (x,t) 2 ∂t This proves that the Heisenberg equation of motion for the quantized field operators is just the KG equation written for field operators. 11.5.2

Quantized Field Operators in Energy/Momentum Basis

Once we do the standard quantization by converting φ(x,t), π? (x,t) and φ? (x,t), π(x,t) into ˆ ˆ time independent (in Schroedinger’s picture) operators φ(x) , πˆ † (x) and φˆ † (x) , π(x) respectively and by enforcing standard commutation relation 80 and 81 (and all other pairs commuting) between these operators we immediately notice from 76, 77 , 79 and 78 that even the of energy mode expansion coefficients φ+ (p) exp(−iE pt), φ− (−q) exp(iEqt), φ?− (−q) exp(−iEqt) and φ∗+ (p) exp(iE pt) get converted respectively into operators : φ+ (p) exp(−iE pt) → a(p) ˆ =

Z

φ− (−q) exp(iEqt) → bˆ † (p) = ˆ φ?− (−q) exp(−iEqt) → b(p) = φ∗+ (p) exp(iE pt) → aˆ† (p) =

p ˆ + pi π(x)} ˆ exp(−ipx) dx{ E p φ(x) Ep

(83)

p ˆ − pi π(x)} ˆ exp(ipx) E p φ(x) Ep

(84)

Z

dx{

Z

dx{ Z

dx{

p

i E p φˆ † (x) + p πˆ † (x)} exp(−ipx) Ep

p i E p φˆ † (x) − p πˆ † (x)} exp(ipx) Ep

(85) (86)

One can easily check that indeed 86 and 84 are the adjoint of 83 and 85 respectively and hence the choice of symbols is well justified. We would see below that these new operators actually behave as creation and annihilation operators and the Hamiltonian can easily be written in diagonal forms with their help. Using 83 and 86 one can easily check that:

[a(p), ˆ a(q)] ˆ = 0 † [aˆ (p), aˆ (q)] = 0 [a(p), ˆ aˆ† (q)] = δ(p − q) †

Similarly, using 84 and 85 one can easily check that:

61

(87)

ˆ ˆ [b(p), b(q)] = 0 † † [bˆ (p), bˆ (q)] = 0 ˆ [b(p), bˆ † (q)] = δ(p − q)

(88)

And all other pairs commute. Note that the right hand sides on the last expressions in above two sets of commutation relations are Kronecker deltas (not a delta functions). Kronecker deltas defined such that δ(x − y) = 0 ∀x 6= y and δ(x − y) = 1 for x = y. Also inverse transformations can readily be calculated as well: ˆ φ(x) =

Z

1 p { 4π E p

Z

d pbˆ † (p) exp(−ipx)}

d p{a(p) ˆ exp(ipx) + Z

Z

1 ˆ p d p{b(p) exp(ipx) + d paˆ† exp(−ipx)} 4π E p p Z −i E p Z ˆ π(x) = { d p{a(p) ˆ exp(ipx)} − d p{bˆ † (p) exp(−ipx)} 4π p Z −i E p Z † ˆ πˆ (x) = { d p{b(p) exp(ipx) − d p{aˆ† (p) exp(−ipx)} 4π φˆ † (x) =

11.5.3

(89) (90) (91)

(92)

Quantum Hamiltonian

Now the quantum version of Hamiltonian can be found starting from 66 and using 83-79: Z

Hˆ =

dp

1 Ep † ˆ {aˆ (p)a(p) ˆ + bˆ † (p)b(p)} 2π 2

(93)

Factor 1/2 is there because of the normalization of the field variables I chose. I encounter the problem of operator ordering which basically originates from the ordering of π? and π in classical Lagrangian. I have straighten up the order and threw away the infinite term. 11.5.4

Number State Basis

Now let us try to explore some properties of aˆ† (p) and a(p) ˆ operators. From 93 we can immediately see that: ˆ aˆ† (p)] = E p aˆ† (p) [H, (94) and ˆ a(p)] [H, ˆ = −E p a(p) ˆ

(95)

Exactly same relations hold for other set of creation and annihilation operators as well. Above ˆ indeed behave like creation two properties guarantee that proposed aˆ† (p) and a(p) ˆ (and bˆ † and b) ˆ and annihilation operators respectively. Also note that the state operator φ(x) simultaneously creates one and destroys other so they do behave like particle and antiparticle. 62

11.5.5

Associated Charge

Z

Q = −i Hence,

Z

Qˆ = −i Hence,

d 3 x(π? φ − φ? π) ˆ d 3 x(πˆ † φˆ − φˆ † π)

Z

−1 ˆ d 3 p(aˆ† (p)a(p) ˆ − bˆ † (p)b(p)) 4(2π)3 Note that factor of 1/4 is result of the definition normalized field variables. This we can get rid of by properly normalizing variables. Also I have again straighten up the ordering of the operators. Now, let us evaluate the following commutation relation: Qˆ =

ˆ ˆ [aˆ† (p)a(p) ˆ + bˆ † (p)b(p), aˆ† (q)a(q) ˆ − bˆ † (q)b(q)] = − + − =

[aˆ† (p)a(p), ˆ aˆ† (q)a(q)] ˆ † † ˆ [aˆ (p)a(p), ˆ bˆ (q)b(q)] ˆ [bˆ † (p)b(p), aˆ† (q)a(q)] ˆ † † ˆ ˆ [bˆ (p)b(p), bˆ (q)b(q)] 0

Since H and Q commute, the eigen states of Q are also the eigen states of H and vice versa. More explicitly, let a single eigen state of H be |1 p ; 0iwhere first number represents that there is one particle of type 1 in state p and zero particle of type 2. Hence H|1 p ; 0i = |1 p ; 0i. Whereas Q|1 p ; 0i = |1 p ; 0i. On the other hand H|0; 1 p i = |0; 1 p i and Q|0; 1 p i = −|0; 1 p i. Hence Q can be interpreted as some sort of quantized “charge” carried my massive particles. Antiparticle carries negative and equal amount of charge as does a particle.

11.6

Quantization of Complex Valued Vector Klein Gordon (KG) Field

11.6.1

Quantized Field Operators and Hamiltonian

The Lagrangian density function generalizes to L(φi (x), φ?i (x), ∂µ φi (x), ∂µ φ?i (x),t) = ∂µ φ?i ∂µ φi − m2 φ?i φi ∂φ?i ∂φi ~ ? ~ − ∇φi .∇φi − m2 φ?i φi = ∂t ∂t

(96)

Where in this problem i ∈ {1, 2} and sum convention is assumed. From the expression for Lagrangian density 96, the conjugate momentum density function (conjugate to φ1 (x)) would be:∂φ? ∂L π?1 (x) ≡ = 1 ˙ ∂t ∂φ(x) and that conjugate to φ?1 (x) is π?1 (x) ≡

∂L ∂φ1 = ? ˙ ∂t ∂φ (x) 63

And exactly analogous results exist for i = 2. Now, once we know the conjugate momenta density, one can easily obtain the Hamiltonian density function H(φi (x), φ?i (x), π?i (x), πi (x), ~∇φi (x)~∇φ?i (x),t) as ? H ≡ π?i (x)φ˙i (x) + πi (x)φ˙i (x) − L ∂φ?i ∂φi ~ ? ~ ? ? ˙ ˙ = πi (x)φi (x) + πi (x)φi (x) − + ∇φi .∇φi + m2 φ?i φi ∂t ∂t

So we can write the classical Hamiltonian density function as: H(φi (x), φ?i (x), π?i (x), πi (x), ~∇φi (x)~∇φ?i (x),t) = πi (x)π?i (x) + ~∇φ?i .~∇φi + m2 φ?i φi

(97)

Again note that the order of π? π and another similar term is not determined of classical R 3 mechanics. Applying integration operator d x one can then get the Classical Hamiltonian function. Commutation relations generalizes in a straight forward manner. Considering two fields to be “independent” we first of all postulate exactly same set of commutation rules as 80 , 81 and other pairs independently for both i = 1 and i = 2. Taking clue from the theory of quantization of multi-particle systems we can postulate that all other pairs with i 6= j commutes. Since both the fields behave independently and due to the postulated commutation relations the proof that φˆ1 and φˆ2 has the Heisenberg equation of motions as Klein Gordon equation is exactly same as the that in problem 1. I don’t feel like repeating the derivation un-necessary. Proof is step by step exactly same.We move energy basis and denies the operators φˆ i , φˆ †i and their conjugate momenta in terms of two independent set of creation and annihilation operators of particles and antiparticles (hence we would have 8 such operators in this problem). Let us call these aˆi , aˆ†i , bˆ i and bˆ †i with i ∈ {1, 2}. Derivations again are exactly same. I am not repeating those. Hamiltonian generalizes to : Z

Hˆ = 11.6.2

dp

1 Ep † {aˆi (p)aˆi (p) + bˆ †i (p)bˆ i (p)} 2π 2

(98)

Associated Charge

Expression for Qˆ also generalizes in straight forward fashion as well : Z

Qˆ = −i

d 3 x(πˆ †i φˆi − φˆ †i πˆ i )

−1 = 4(2π)3

Z

d 3 p(aˆ†i (p)aˆi (p) − bˆ †i (p)bˆ i (p))

This commutes with H and, again, the proof is exactly same as before Z

i Qˆp = − d 3 x(πˆ †i σˆ p i j φˆ i − φˆ †i σˆ p i j πˆ j ) 2 I could not work out the commutations. Somehow above expression does not look right to me. 64

12

Dirac Field

Part VII

Quantum Electrodynamics ( QED ) 13

Non-relativistic Electromagnetic Field Quantization ( Quantum Optics )

Non-relativistic electromagnetic field quantization is better known as Quantum Optics. The reason is because non-relativistic quantization Maxwell equations are mostly useful for explaining the properties of light and its interaction with materials. A more elaborate relativistic version of the subject is more commonly known as Quantum Electrodynamics (QED) and comes under the bigger subject area of QFT of Gauge Fields. QED is mostly useful for describing electromagnetic interaction between elementary particles. Hence the name QED is popular among particle physicists. Laser and device people mostly need non-relativistic version and in this community subject is better known as Quantum Optics (or sometimes term cavity-QED is also used). The idea is to get a classical wave equation. Identify the generalized co-ordinate and conjugate momentum. Form a Lagrangian density function so that Lagrange’s equation of motion gives the classical wave equation. Form the classical Hamiltonian density function. Elevate the generalized co-ordinate and conjugate momentum to the status of operators and force a commutation relationship between them. It turns out that only if you force Bosonic commutation relation you get the correct physics. This equivalent to postulate that EM fields are build of Bosons. Obtain quantum mechanical Hamiltonian. Hamilton equation of motion would turn into a Heisenberg equation of motion. Note that in classical mechanics few components of electric and magnetic field at each spatial point can be treated as generalized co-ordinates and conjugate momenta. Now same as what we discussed in phonons case this scheme theoretically works but has a major practical problem in the sense that these different Heisenberg equations at different spatial point would be coupled together. Operator equation for one spatial point contain operators at other spatial points as well. This can easily be resolved by doing normal mode expansion of classical field from there to obtain a co-ordinate transformation. Note that its easy to visualize that same co-ordinate transformation would decouple the quantized Heisenberg equations as well. So at the end we would have one Heisenberg equation of motion for each normal mode of classical field. The transformed operators are recognized as the mode amplitude operator or field operator. Note again that the field variables include phase and time dependence (field operators become time independent after quantization in Schroedinger’s picture) and are normalized to be dimensionless. The rest of the part of the complete field profile in a mode is supposed to be orthonormal. Since the Hamiltonian with respect to each mode variables has exactly same form as that of a classical harmonic oscillator, with only difference being that now

65

the variable is the amplitude of normal mode field oscillation, the solutions are also exactly same. Hence each normal mode gets quantized as a separate decoupled harmonic oscillator. But co-ordinate that is oscillating is the mode amplitude. So we have one ’wavefunction’ associates with each classical normal mode. The wavefunction would be Gaussian representing uncertainty in field amplitude within each mode. From the postulated commutation relation between the generalized co-ordinates and conjugate momenta (field amplitudes at each spatial point) one can obtain the commutation relation between the mode amplitude operators by simple use of co-ordinate transformation equations. They would also be Bosonic. One can show that the field amplitude operators are actually creation and annihilation operators. Using the co-ordinate transformation equations one can relate it to the field oscillation operators at each spatial points. Let us say that the classical normal mode expansion gives us :E(r,t) =

1 A j η j (t) f j (r) + A∗j η∗j (t) f j∗ (r) 2∑ j

A∗j ∗ 1 1 Aj H(r,t) = ∑ κ j (t)∇x f j (r) + κ j (t)∇x f j∗ (r) 2 j µ0 ω j ωj where

η j (t) = exp(−iω j t) exp(−ω j t/2Q j ) κ(t) =

1 dη j (t) ω j dt

Hence approximately, κ j (t) = −i exp(−iω j t) exp(−ω j t/2Q j ) = −iη j (t) Note that the mode orthonormalization is enforced as:Z

ε(r) f j (r) f j∗0 (r)dr = δ j j0

g

Usually f j = √jε . Now if we define mode oscillation amplitude variable B j = A j η j (t) and B∗j = A∗j η j (t) one can write the classical Hamiltonian as :1 W = ∑ B∗j B j + B j B∗j 4 j But, if want to write it as :W =∑ j

~ω j ∗ b b j + b j b∗j 2 j

66

then we normalize the mode oscillation amplitudes to define normalized mode oscillation amplitude variable as s 1 b j = −i Bj 2~ω j and we get r

r ~ω j ~ω j ∗ ∗ E(r,t) = ∑ i b j f j (r) + b f (r) 2 2 j j j s s ~ ~ 1 b j ∇x f j (r) + b∗j ∇x f j∗ (r) H(r,t) = ∑ 2µ0 ω j 2µ0 ω j ω j j The classical Hamiltonian can now be written as W =∑ j

~ω j ∗ b b j + b j b∗j 2 j

If we write b j = q j + ip j and b∗j = q j − ip j (q and p being real numbers) then we can see that q j and p j turns out to be generalized co-ordinates and conjugate momentum in the sense that they satisfy the Hamilton’s equation of motion. Note that q j turns out to be the normalized real mode oscillation electric field amplitude. Whereas p j turns out to be normalized real mode oscillation magnetic field amplitude. When we quantize it and elevate q j and p j to the status of operators satisfying Bosonic commutation relation then one can prove that b†j and b j behaves exactly as the creation and annihilation operators of a decoupled harmonic oscillator. One can easily show that these operators obey the Bosonic commutation relations as well. The fact that this scheme works in which we treat fields as conjugate variables should be treated as a postulate of quantum optics. • H = ∑ j ~ω j (b†j b j + 12 ), N j = b†j b j • [b†i , b†j ] = 0, [bi , b j ] = 0 and [bi , b†j ] = δi j • bj →

p √ n j , b†j → n j + 1

67

13.1

14

Coherent States / Poisonian Distribution

Relativistic Electromagnetic Field Quantization ( QED )

Part VIII

Interaction Between Free Fields 15 15.1

Different Pictures of Time Evolution Exponential of Operators

Suppose we have an equation which we want to solve: i~

∂Ψs (t) = H0,s Ψs (t) ∂t

where H0,s is time independent. Let us try to solve it in following fashion. Let us assume first that Ψs (t) = Ψs (0). This is our zeroth order approximation. First order correction term can be found by substituting this into the right hand side of above equation. That gives us (1) Ψs (t) = (−

i ~

Z t 0

it dtH0,s )Ψs (0) = (− H0,s )Ψs (0) ~

Substituting this again in the right hand side of original equation: 1 −itH0,s 2 (2) ) Ψ(0) Ψs (t) = ( 2 ~ Similarly, after calculating the nth order correction term and adding them all we can write ∞

1 −itH0,s n ( ) }Ψ(0) ~ n=0 n!

Ψs (t) = { ∑

Please note that H0,s is an operator and not a simple number. Still, we note that the terms under summation sign resembles a Taylor expansion. We define a “symbolic notation” for such a series as ∞ −iH0,st 1 −itH0,s n Ψs (t) = { ∑ ( ) }Ψ(0) ≡ exp( )Ψ(0) ~ ~ n=0 n! Note that this notation is valid only when H0,s is time independent. When operator becomes time dependent we can follow a similar route but would have to be more careful. We do this in the following section 15.6.

68

15.2

Schrodinger’s Picture of Time Evolution

Suppose we break the Hamiltonian involved in the time dependent Schrodinger’s equation (from here on we would call it Hamiltonian operator in Schrodinger’s picture) in two parts – Hs (t) = H0,s + Vs (t). Here, H0,s is time independent part of the Hamiltonian. Whereas Vs (t) may or may not be time independent. In cases where Vs is time independent, separation of Hamiltonian into two time independent parts is simply for mathematical convenience. Its helpful to take out that part of Hamiltonian whose eigen states are trivial. Then one can build a time-independent perturbation theory to calculate the eigen states of the total Hamiltonian. In any case, we would keep our discussions very general and would keep the possibility of Vs being time dependent. We would consider many different situations involving equilibrium and non-equilibrium. Usually in equilibrium Vs would be time independent. We would discuss these cases in details latter on. The actual time evolution of system would be given by the Schrodinger’s equation of motion: i~

∂Ψs (t) = (H0,s +Vs (t))Ψs (t) ∂t

(99)

Let us assume, for the time being, that Vs (t) = 0. We need to consider this case to define of some of the terms that we would use latter. In this case the Schrodinger’s equation (99) simplifies to: ∂Ψs (t) = H0,s Ψs (t) i~ ∂t Now, since, H0,s is time independent, one can symbolically write the solution of the above equation as: Ψs (t) = exp(−iH0,st/~)Ψs (0) Or equivalently, by defining unperturbed time-propagation operator in Schrodinger’s picture as: U0,s (t) ≡ exp(−iH0,st/~) (100) the same equation can also be written as i~

∂U0,s (t) = H0,sU0,s (t) ∂t

(101)

Please note two specific points. First of all, if you have seen Heisenberg equation of motion before, its instructive to make it clear that we are not writing the time evolution of the system in Heisenberg picture (equation is strikingly similar). Moreover, U0,s (t) is not Hermitian (U0,s (t) 6= † † −1 U0,s (t)), rather it is Unitary (U0,s (t) = U0,s (t)). That is, its eigen values are complex. Moving back to actual case where Vs (t) 6= 0, we define time propagator in Schrodinger’s picture as: Ψs (t) ≡ Us (t)Ψs (0) (102) And with this definition, the Schrodinger equation (99) becomes: i~

∂Us (t) = Hs (t)Us (t) ∂t 69

Note that only in a very special case when Hs 6= Hs (t) (usually equilibrium), one can write: Us (t) = exp((−iH0,st − iVst)/~) Moreover, even in this special case, its not correct to break exponentials into two parts as Vs and H0,s might not commute. We would try to avoid such exponentials as much as possible and develop a general theory. Key point to remember is that in Schrodinger picture states have a very complicated time dependence. Operators can have a time dependence but this dependence is preknown. So the whole problem is to solve the time dependence of the states.

15.3

Heisenberg Picture of Time Evolution

In Heisenberg representation, states have no time dependence whereas operators have all the time dependence. One defines the state of the system in Heisenberg representation as: Ψh = Us† (t)Ψs (t)

(103)

Hence, comparing with (102), one can easily conclude that: Ψh = Ψs (0)

(104)

One similarly defines operator in Heisenberg picture15 as: Ah (t) = Us† (t)As (t)Us (t)

(105)

I would stress again that only in the special case of time independent total Hamiltonian (Hs 6= Hs (t)), one can write: Us (t) = exp(−iHst/~) We would try to avoid such exponentials as much as possible and develop a general theory. Key point to remember is that in Heisenberg picture, states have absolutely no time dependence whereas operators have all the time dependence. We take out time dependence from Schrodinger states by multiplying it by Us† from left side. And we deposit this time dependence into operators by multiplying Schrodinger operators by Us† from left side and by Us from right side. Language of Heisenberg representation sometimes become very confusing. One needs a little bit of experience to make some sense out of it. State left to themselves have no time dependence. But since operators are time dependent, the eigen states of operators calculated at different times 15 Schrodinger’s equation gets converted into time evolution equation for operators in Heisenberg representation. One can easily show that any Hermitian operator would obey following equation

i~

∂Ah (t) = [Ah (t), H] ∂t

70

would be different. Let us take a trivial example to make things absolutely clear. Let us consider the problem of a single free particle. We would choose our basis as a co-ordinate basis. In strict sense they are eigen states of co-ordinate operator calculated at certain time say t = 0. We call these to be in Heisenberg representation, which means they don’t have any time dependence. To make things absolutely clear xˆh (t = 0)|xih = x|xih . Hence whenever you see the symbol |xih – one thing should be clear that they are eigen states of position operator constructed at t = 0. In this ∂2 basis our Hamiltonian can be written as Hs = H0,s = −~ ∂x 2 . Momentum operator in Schrodinger ∂ picture would be Pˆs = Pˆ0,s = −i~ ∂x . Eigen states of this operator in Schrodinger picture would be time dependent. States left to themselves evolve in time:- |p(t)is = exp(−ip2t/2m~) exp(ipx). Momentum operator itself is time independent. In more complicated systems states would evolve in much more complicated fashion. A momentum eigen state t = 0 might not remain a momentum eigen state at latter time. In Heisenberg representation momentum operator become time ∂ ) exp(−iH0,st/~). Eigen states constructed at dependent: Pˆh (t) = Pˆ0,h (t) = exp(iH0,st/~)(−i~ ∂x different times would be different. After that, states left to themselves would not evolve with time.

15.4

Interaction Picture

We define the state of the system in interaction picture as: Ψs (t) = exp(−iH0,st/~)Ψi (t) = U0,s (t)Ψi (t)

(106)

Basically what we are doing here is that we are removing the free-system time evolution as this part of the evolution is pre-known and we want to concentrate on the evolution due to time dependent perturbation part of Hamiltonian. One can invert above equation and write: † Ψi (t) = U0,s (t)Ψs (t)

(107)

Substituting t = 0 one can easily see that: Ψi (0) = Ψs (0)

(108)

Substituting (106) into the complete time evolution equation in Schrodinger’s picture (99): i~ exp(−iH0,st/~)

∂Ψi (t) + H0,s Ψs (t) = H0,s Ψs (t) + (Vs (t) exp(−iH0,st/~)Ψi (t)) ∂t

∂Ψi (t) = (exp(iH0,st/~)Vs (t) exp(−iH0,st/~))Ψi (t) ∂t Now if we define any general Hermitian (not Unitary like Ui ) operators in interaction picture as: Vi (t) = exp(iH0,st/~)Vs (t) exp(−iH0,st/~) (109) which can also be written in notational format as: † Vi (t) = U0,s (t)Vs (t)U0,s (t)

71

(110)

we can then write the time evolution of wavefuncton in interaction picture as (this is basically Schrodinger’s equation in interaction picture): ∂Ψi (t) = Vi (t)Ψi (t) ∂t

(111)

One can also define the time propagation operator in interaction picture as Ψi (t) = Ui (t)Ψi (0)

(112)

Hence one can write

∂Ui (t) = Vi (t)Ui (t) (113) ∂t This is a very important equation that we would use for generating perturbation sequence16 . Key point to remember is that in Interaction picture, states have only non-trivial time dependence whereas operators have only trivial time dependence (interaction picture is closer to Schrodinger picture then Heisenberg picture). We take out trivial time depen† dence from Schrodinger states by multiplying it by U0,s from left side. And we deposit this i~

† trivial time dependence into operators by multiplying Schrodinger operators by U0,s from left side and by U0,s from right side.

15.5

Inter-Relation

Using definition of state (106) and (??) propagator in interaction picture one can see that: Ψs (t) = U0,s (t)Ψi (t) = U0,s (t)Ui (t)Ψi (0) Moreover, from the boundary condition (108) and the time propagator in Schrodinger’s picture (102) : Us (t) = U0,s (t)Ui (t)

(115)

16 Note

that we can write down the time evolution of any general operator in interaction picture as well (in interaction picture operators might become explicit functions of time). Let us assume that an operator in Schrodinger’s picture As does not have any explicit time dependence. Following argument can be generalized if needed. Using the definition of operators in interaction picture (109) and using the equation describing the equation of motion of U0,s (101): † ∂U0,s (t) ∂U0,s (t) ∂Ai (t) † = i~ AsU0,s (t) + i~U0,s As (t) i~ ∂t ∂t ∂t ∂Ai (t) † † † i~ = −H0,s U0,s (t)AsU0,s (t) +U0,s As (t)H0,sU0,s (t) ∂t now, simply by looking at the definition, one can figure out that U0,s (t) and H0,s commute (using this same property one can show that H0,s = H0,i ). Hence: i~

∂Ai (t) † = U0,s (t)[As , H0,s ]U0,s (t) = [Ai , H0,i ] ∂t

72

(114)

This is a pretty important relationship. After developing the perturbation in interaction picture, this equation is used to move back to Schrodinger’s picture. Secondly, using the definition of operator representation in Heisenberg picture (105) and above expression (115): † Ah (t) = Ui† (t)U0,s (t)As (t)U0,s (t)Ui (t)

Which in turn means: Ah (t) = Ui† (t)Ai (t)Ui (t)

(116)

This another, very important relationship that we would use time and again.

15.6

Diagrammatic Perturbation Theory

The time evolution equation of Ui (t) derived above (113) can be used to obtain a perturbation solution for Ui (t). Only important point to remember is that the Hamiltonian at different times do not necessarily commute. So one needs to include the time ordering operator when converting equation into integral equation in terms of exponentials. Let us explore this explicitly. Integrating (113), symbolically one can write: Z t

Ui (t) −Ui (0) = −i

0

Vi (t1 )Ui (t1 )dt1

Noting that Ψi (0) = Ψs (0) one can see that Ui (0) = 1 (an identity operator). Hence Z t

Ui (t) − 1 = −i

0

Vi (t1 )Ui (t1 )dt1

Now a zeroth order approximation term would obtained if we assume Vi (t) = 0. So (0)

Ui (t) = 1 Substituting this back into above integral, the first order correction term would be: (1) Ui (t) = −i

Z t 0

Vi (t1 )dt1

(117)

Similarly, second order correction term would be: (2) Ui (t)

2

Z t

= (−i)

0

2

Z t1

Vi (t1 ){

Z t

= (−i)

0

Z t1

dt1

0

0

Vi (t2 )dt2 }dt1

dt2Vi (t1 )Vi (t2 )

(118) (119)

First thing to be stressed is to point out that this is an integral of time dependent operators – which might not commute at different times. Hence, in general, Vi (t1 )Vi (t2 ) 6= Vi (t2 )Vi (t1 ). So one should not try to shuffle the operators in a product like this. Secondly, one should notice that 73

one has to keep the order of integrations. For example, in above integration scheme, one needs to evaluate the integral in curly braces before doing the second integral. Another way of saying the same thing is that for any given value of the variable t1 , the variable t2 can only takes values from 0 up to t1 . Hence, t2 ≤ t1 . Combining above two peculiarities, one should make sure that in the second form of the above equation (118), operators with latter times as variables should go to the left. Or in other words time variable of any operator should never be allowed to become greater than the time variable of an operator sitting to its left. Now these different time limits on two integral make things very difficult. If we can find a way of making the two limits on two integrals independent of each other it would make things much simpler. Let us define a new operator which shuffles the operators in a product so that latter always time sits on the left. We call it a time ordering operator. That is   Vi (t1 )Vi (t2 ) i f t1 > t2 T [Vi (t1 )Vi (t2 )] ≡ (120) Vi (t2 )Vi (t1 ) i f t2 > t1 One would notice that W (t1 ,t2 ) = T [Vi (t1 )Vi (t2 )] is a symmetric operator in t1 and t2 plane about the line t1 = t2 . In other words W (1, 2) = W (2, 1). Looking at the equation 118, one would (2) realize that Ui (t) is proportional to the upper triangular integration of a two variable function W (t1 ,t2 ) on t1 and t2 plane. Moreover since, W (t1 ,t2 ) is symmetrical about t1 = t2 , we can also write above expression as (2) Ui (t) =

(−i)2 2

Z t 0

Z t

dt1

dt2 T [Vi (t1 )Vi (t2 )]

0

Hence we were able to get rid of the different limits on the integrals by introducing a concept of time ordering operator. Same way, one can write an nth order correction term as well. Summing all terms, an infinite order estimate of Ui (t) can be written as: ∞

(−i)n Ui (t) = I + ∑ n=1 n!

Z

Z

dt1

Z

dt2

Z

dt3 .....

dtn T [Vi (t1 )Vi (t2 )Vi (t3 ).....Vi (tn )]

(121)

With T being the time-ordering operator such that left most operator has biggest time. That is (tn < tn−1 .... < t1 ) if above written order is already time-ordered. All integrals go from ti = 0 to ti = t. We now define a symbolic representation for the above series. We would write in short: Z t

Ui (t) = T [exp(−i

0

dt 0Vi (t 0 ))]

One should compare this with the time-independent version defined in previous section 15.1.

15.7

Expectation Values

15.7.1

Quantum Ensemble Expectations

15.7.1.1 Gell-Mann and Low Theorem First of all, suppose instead of propagating a state from t = 0 to t = t, I want to propagate from t = t1 to t = t2 . This can easily be achieved as: Ui (t2 ← t1 ) = Ui (t2 )Ui† (t1 ) 74

(122)

Similarly, Us (t2 ← t1 ) = Us (t2 )Us† (t1 )

(123)

We can also generalize the relation between above two propagators (111) as follows. We first note that: Ψi (t2 ) = Ui (t2 ← t1 )Ψi (t1 ) † Now using (111) we get Ψs (t) = U0,s (t)Ψi (t) and its inverse Ψi (t) = U0,s (t)Ψs (t). Hence † Ψs (t2 ) = U0,s (t2 )Ui (t2 ← t1 )Ψi (t1 ) = U0,s (t2 )Ui (t2 ← t1 )U0,s (t1 )Ψs (t1 )

Hence † Us (t2 ← t1 ) = U0,s (t2 )Ui (t2 ← t1 )U0,s (t1 )

(124)

Similarly other relations can also be generalized to two-time operators. Now let us first define the zero reference level of our energy spectrum. Let |0i0,h be the lowest energy eigen state of the unperturbed Hamiltonian. We write it in Heisenberg representation to explicitly specify that this state does not has any time dependence whatsoever. Or in other words |0i0,h represents |0(t = 0)i0,s or any other time that is taken as a reference time. We define the reference level of energies to be such that Hs,0 |0i0,h = 0 Now, let |nih be the energy eigen state at (t = 0) of perturbed system. Again Heisenberg subscript has been added to explicitly say that these states do not have any time dependence. Hs |nih = En |nih One should specifically note that Hs in above equation can certainly be time dependent. This is just an eigen value equation (which certainly can have time dependent operators). To make things clear, in Schrodinger’s picture we would have |nis = exp(−iEnt/~)|nih We are specifically interested in |0ih . This state is unknown and we hope that this can be calculated by evolving |0i0,h from −∞ to present time. In general one can always write ∞

Us (t2 ← t1 )|0i0,h =

∑ αn exp(−iEn(t2 − t1)/~)|nih

n=0

where αn = h hn|0i0,h . One of the important assumptions that we have to make here is that α0 6= 0. As far as concept of “perturbation” is valid, we claim that this has to be true. There are formal problems in this assertion. It can be shown in QFT that, in infinite volumes, any two states are orthogonal. We can get rid of this problem by working with finite volume and taking limits at the end. Moreover we would assume that the state |0ih is non-degenerate. This can not be proved mathematically. In fact there are interesting counter examples like those of spontaneously 75

broken symmetry. But for the time being we would exclude those exotic problems and assume that perturbed ground state is unique. Additionally, we would make another claim without proving it. We would assume that we are allowed to send the time variable to complex values in the above expression. This can be rigorously justified but I am skipping that part. Procedure we would follow is known as analytic continuation. This type of QFT is known as “Euclidean QFT”. Basically, if real time QFT is Lorentz invariant then analytic continued theory would be O(4) invariant. We would keep the inner product independent of rotation in complex t plane. So let us send t1 → −∞(1 − iε) where ε is a small real number. Also choose t2 = 0. Hence lowest energy term would be slowest decaying term. Hence, |0ih =

1 Us (0 ← t1 )|0i0,h t1 →−∞(1−iε) α0 exp(iE0t1 ) lim

† † Now from 124 we know that Us (0 ← t1 ) = Ui (t2 ← t1 )U0,s (t1 ). And moreover U0,s (t1 )|0i0,h = exp(i0(t1 )/~)|0i0,h = |0i0,h from the definition of the zero level of energy scale. Hence, we can write the above expression as

|0ih =

1 Ui (0 ← t1 )|0i0,h t1 →−∞(1−iε) α0 exp(iE0t1 ) lim

(125)

From here onward, we would simply write this expression in shorthand notation as |0ih = βUi (0 ← −∞)|0i0,h

(126)

Here β is a complex number. Limit is understood. On similar lines, one can also derive h0|h = βUi (+∞ ← 0)h0|0,h For the bra state one can get an alternative expression simply by taking adjoint of 126. h0|h = β?Ui (−∞ ← 0)h0|0,h This is known as Gell-Mann and Low Theorem. 15.7.1.2 Quantum Ensemble Expectation One interesting point, and that is the most important reason why we are studying it in statistical mechanics context, is that even evaluation of expectation values of “instantaneous time operators” can be included in the perturbation expansion. What I mean by instantaneous time operators is that operators that perturb the state of the system instantaneously. Something like a creation or annihilation operators, for example. Let cλ,i (t1 ) and cλ,h (t1 )be the interaction and Heisenberg representation of one such operator. Suppose, I want to evaluate something like hΨh |cλ,h (t1 )|Ψh i Where |Ψh i is the Heisenberg representation of an exact energy eigen state of the system (proper of visualizing such an expression is discussed in Green’s function section). We are simply evaluating a quantum mechanical expectation value. Note that, usually, even Ψs (0) = Ψi (0) is unknown since this state includes the effects of perturbation (interaction among particles of system). 76

Now, from Gel-Mann and Low Theorem discussed above, one can write above expression as hΨh,0 |Ui (−∞ ← 0)Ui (0 ← t1 )cλ,i (t1 )Ui (t1 ← 0)Ui (0 ← −∞)|Ψh,0 i

(127)

Where I have used the fact that Ui (0 ← t1 ) = Ui† (t1 ← 0). Here states are now non-interacting ground states. In certain cases above expression can also be written as: hΨh,0 |Ui (∞ ← 0)Ui (0 ← t1 )cλ,i (t1 )Ui (t1 ← 0)Ui (0 ← −∞)|Ψh,0 i hΨh,0 |Ui (+∞ ← −∞)|Ψh,0 i

(128)

This second expression can be justified provided hΨs,0 |Ui (+∞ ← 0) is a well defined state. In equilibrium this can be justified. But not in non-equilibrium. Now if we include time ordering operator, above can be written as: hΨh,0 |T Ui (−∞ ← −∞)cλ,i (t1 )|Ψh,0 i

(129)

hΨh,0 |T Ui (+∞ ← −∞)cλ,i (t1 )|Ψh,0 i hΨh,0 |Ui (+∞ ← −∞)|Ψh,0 i

(130)

or, in some cases:

After expanding Ui in perturbation, cλ,i is properly placed in time sequence and above expectation can easily be evaluated. This is the way one extract equilibrium information about complicated interacting systems using perturbation.

Part IX

Statistical Quantum Field Theory (Condensed Matter Physics) Please refer to a separate tutorial article on Non-Equilibrium Green’s Functions (Statistical QFT).

Part X

References • Quantum Field Theory – Peskin’s book (Peskin [9]) on quantum field theory is nowadays becoming a standard text book. This book is mostly written from elementary particle physics point of view. It’s mostly useful in learning how to solve real world field theory problems. It’s fairly mathematically sound but gives very little insight into the physical aspects of the subject itself. 77

– Weinberg’s book (Weinberg [11]) is another modern standard text book. As far as physical insight into the subject is concerned, it’s better than Peskin’s book, but still not very satisfying. – Three related texts books by Greiner on – QFT (Bromley and Greiner [2]), QED (Greiner and Reinhardt [6]), and relativistic QM (Bromley and Greiner [3]) are also useful. They are not very readable. Organization of the books is also very erratic. • Quantum Field Theory for Solid State Physics – Haken’s book (Haken [7]) is a nice elementary introduction to field theory as used in solid state physics (it’s mostly QFT of quasi-particles). – David Pines’s book (Pines [10]) is another elementary readable book in this subject. • Statistical Quantum Field Theory – Abrikosov’s (Abrikosov [1]) book is a masterly written classic on this subject. – Fetter’s book (Fetter and Walecka [5]) is somewhat more readable than Abrikosov’s book. – Mahan’s book (Mahan [8]) is another popular text book in this subject though I personally don’t like it much. The latest edition is actually much much better than older versions. – Doniach’s book (Doniach [4]) on Green’s functions as used in solid state physics can also be useful in learning how to solve practical problems.

References [1] A. A. Abrikosov. Methods of Quantum Field Theory in Statistical Physics (Selected Russian Publications in the Mathematical Sciences.). Dover Publications, 1977. [2] D.A. Bromley and Walter Greiner. Physics). Springer, 1998.

Classical Electrodynamics (Classical Theoretical

[3] D.A. Bromley and Walter Greiner. Relativistic Quantum Mechanics. Wave Equations. Springer, 2000. [4] S Doniach. Green’s functions for solid state physicists (Frontiers in physics, 44). W. A. Benjamin, 1974. [5] Alexander L. Fetter and John Dirk Walecka. Quantum Theory of Many-Particle Systems. Dover Publications, 2003. [6] Walter Greiner and Joachim Reinhardt. Quantum Electrodynamics. Springer, 2002. 78

[7] H. Haken. Quantum Field Theory of Solids: An Introduction. Elsevier Science Publishing Company, 1983. [8] Gerald D. Mahan. Many Particle Physics (Physics of Solids and Liquids). Springer, 2000. [9] Michael E. Peskin. An Introduction to Quantum Field Theory. HarperCollins Publishers, 1995. [10] David Pines. Elementary Excitations in Solids : Lectures on Phonons, Electrons, and Plasmons (Advanced Book Classics). Perseus Books Group, 1999. [11] Steven Weinberg. The Quantum Theory of Fields, Vol. 1: Foundations. Cambridge University Press, 1995. ISBN 0521550017.

Part XI

Further Resources Here are some other tutorial articles: • Mathematical background • Article on Quantum Mechanics • Authors Homepage

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