Quantum Field Theory I

Transcription

1 October 12, 2011 Quantum Field Theory I Ulrich Haisch Rudolf Peierls Centre for Theoretical Physics, University of Oxford OX1 3PN Oxford, United Kingdom

2 Abstract This course deals with modern applications of quantum field theory with emphasize on the quantization of theories involving scalar and spinor fields.

3 Recommended Books and Resources There is a vast array of quantum field theory texts, many of them with redeeming features. Here I mention a few of them, mostly the ones that I used or looked at when preparing this course. To a large extent, I will follow the first section of M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory This is a very clear and comprehensive book, covering essentially everything in this course as well as many advanced aspects of quantum field theory that go (far) beyond the scope of this lecture. S. Weinberg, The Quantum Theory of Fields: Volume 1, Foundations This is the first in a three volume series by one of the masters of quantum field theory. It takes a unique route through the subject, focussing initially on particles rather than fields. Since it has a very particular viewpoint, it is difficult to digest, but certainly worth reading. L. Ryder, Quantum Field Theory This elementary text has a nice discussion of much of the material in this course. It is good for a first reading. A. Zee, Quantum Field Theory in a Nutshell This is a charming book, where emphasis is placed on physical understanding and the author isn t afraid to hide the ugly truth when necessary. It contains many gems. By browsing the web, I also found interesting material. Nice introductions to quantum field theory (of different length and viewpoint) have been written by C. Anastasiou and D. Tong. The corresponding scripts can be found at: babis/teaching/qfti/qft1.pdf Other links to useful resources can be found on the web page of D. Tong: For completeness, I will also give relevant references at the end of each section of this script. The interested reader can consult them for further details on the discussed topics. 1

4 Contents 1 Introduction Why QFT? Scales and Units Elements of Classical Field Theory Dynamics of Fields Noether s Theorem Example: Electrodynamics Space-Time Symmetries Problems Klein-Gordon Theory Klein-Gordon Field as Harmonic Oscillators Structure of Vacuum Particle States Two Real Klein-Gordon Fields Complex Klein-Gordon Field Heisenberg Picture Klein-Gordon Correlators Non-Relativistic Limit Problems Interacting Fields Classification of Interactions Interaction Picture First Look at Scattering Processes Wick s Theorem Second Look at Scattering Processes Feynman Diagrams Third Look at Scattering Processes Yukawa Potential Connected and Amputated Feynman Diagrams From Correlation Functions to Scattering Matrix Elements Decay Widths and Cross Sections Problems Dirac Theory Spinor Representation Discrete Symmetries of Dirac Theory Continuous Symmetries of Dirac Theory Solutions to Dirac Equation Quantization of Dirac Theory Problems

5 1 Introduction As the term quantum field theory (QFT) suggests, QFT is the application of quantum mechanics (QM) to dynamical systems of fields, in the same sense that QM is concerned mainly with the quantization of dynamical systems of particles. QFT is not only a subject that is absolutely essential to understand the current state of elementary particle physics as well as modern aspects of cosmology, but also plays a crucial role in many active areas of research, ranging from atomic over nuclear and condensed-matter physics to pure mathematics. Since the ultimate goal of this course is to gain a basic understanding of the fundamental laws of nature, we will in the following focus mainly on the physics of elementary particles and hence deal mostly with relativistic fields. 1.1 Why QFT? The primary reason for introducing the concept of fields in classical physics is to construct laws of nature that are local. The old laws of Newton (Coulomb) involve action at a distance. This means that the force felt by a planet (an electron) changes immediately if a distant star (proton) moves. The laws of Newton and Coulomb thus feature non-local interactions. The field theories of Einstein (general relativity) and Maxwell (electrodynamics) remedied the situation, with all interactions mediated in a local fashion by fields. The requirement of locality remains a strong motivation for studying QFTs. However, there are further good reasons to treat the quantum field (and not the particle) as fundamental (or as Steven Weinberg puts it in [1]: Quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum made out of them. ). QM and Special Relativity A first reason is that the combination of QM and special relativity implies that particle number is not conserved. Consider a particle of mass m trapped in a box of size L. Heisenberg s uncertainty principle tells us that the uncertainty in the momentum of our particle is p /L. In the relativistic limit, momentum and energy can be treated on equivalent footing, and one has an uncertainty in the energy of order E c/l. Yet, if E = 2mc 2, there is enough energy available to create a virtual particle-antiparticle pair from the vacuum (Dirac sea). This little exercise shows that when a particle with mass m is localized within a distance λ Compton = /(mc), talking about a single particle loses its sense. For distances smaller than this Compton wavelength there is a high probability that we will detect particle-antiparticle pairs swarming around the single particle that we initially put into the box. Notice that λ Compton is always smaller than the de Broglie wavelength given by λ de Broglie = / p. 1 If you like, the de Broglie wavelength is the distance at which the wavelike nature of particles becomes apparent, while the Compton wavelength is the distance at which the concept of a single pointlike particle breaks down and one has to start thinking about how to describe multiparticle states. 1 Throughout this course we will use boldface type (ordinary italic type) to denote 3-vectors (4-vectors). 3

6 The presence of a multitude of particles and antiparticles at short distances (or high energies) tells us that any attempt to write down a relativistic version of the one-particle Schrödinger equation is doomed to fail. There is no mechanism in standard non-relativistic QM to deal with changes in the particle number. Indeed, any attempt to naively write down a relativistic version of the one-particle Schrödinger equation meets serious problems: negative probabilities, infinite towers of negative energy states, or a breakdown of causality are the common issues that arise. QM and Causality Let us have a closer look at the issue of breakdown of causality. Consider the amplitude A(t) = y e iet/ x, (1.1) that describes the propagation of a free particle from the point x to y. In non-relativistic QM one has E = p 2 /(2m) and hence 2 A(t) = y [ ( exp i p 2 /(2m) ) t/ ] x d 3 p [ ( = y exp i p 2 /(2m) ) t/ ] p p x (2π ) 3 d 3 p = (2π ) exp [ i ( p 2 /(2m) ) t/ ] exp [ ip (y x)/ ] 3 ( m ) 3/2 [ = exp im (y x) 2 /(2 2 t) ]. 2πi t Here we have made use of the completeness d 3 p/(2π ) 3 p p = 1 of p and a little bit of algebra. The expression (1.2) is non-zero for all y and t, indicating that a particle can propagate between any two points in an arbitrarily short time. In a relativistic theory, this conclusion would signal a violation of causality. One might hope that using the relativistic expression E = p 2 c 2 + m 2 c 4 for the energy would cure the problem, but it does not. In fact, in the relativistic case one has A(t) = y [ exp it/ p2 c 2 + m 2 c ] 4 x d 3 p = (2π ) exp [ it/ p 2 c 2 + m 2 c ] 4 exp [ ip (y x)/ ] 3 (1.3) = 1 2π 2 2 y x 0 dp p sin (p y x / ) exp [ it/ p 2 c 2 + m 2 c 4 ]. (1.2) This integral can be evaluated explicitly in terms of Bessel functions, but for our purposes it is sufficient to consider its asymptotic behavior for L 2 = y x 2 c 2 t 2, i.e., separations well outside the light-cone. We use the method of stationary phase. The relevant phase function 2 The symbol p denotes the momentum operator, which in many QM books is indicated by a. To avoid clutter, I will not use the latter notation, but simply write p. 4

7 pl t p 2 c 2 + m 2 c 4 has a stationary point p = imcl/ L 2 c 2 t 2. Plugging this value into (1.3), we find that (up to a rational function of L and t), A(t) exp [ m/ ] L 2 c 2 t 2. (1.4) This expression is small but non-zero outside the light-cone and causality is again violated. In both cases, the observed failure is telling us that we need a new formalism to preserve causality. This formalism is QFT. It solves the causality problem in a miraculous way. We will see later that in QFT the propagation of a particle across a space-like interval is indistinguishable from the propagation of an antiparticle in the opposite direction. When we ask whether an observation made at point x can affect an observation made at point y, we will find that the amplitudes for particle and antiparticle propagation cancel in such a way that causality is preserved. What else is QFT good for? Besides solving the causality problem, QFT also provides an elegant framework to describe transitions between states of different particle number and type. An example physical processes, exhaustivelly studied (from 1989 until 2000) at the Large Electron Positron (LEP) collider in Geneva, is the production of a muon (µ ) and its antiparticle (µ + ) out of the annihilation of an electron (e ) and its antiparticle (the positron e + ): e + e + µ + µ +. (1.5) The experimental confirmation of the QFT predictions for processes such as (1.5), often to an unprecedented level of accuracy, is our real reason for studying QFT. But the power of QFT does not end here. In traditional QM the relation between spin and statistics has to be put in by hand. To agree with experiment, one should choose Bose statistics (no minus sign if one exchanges two identical particles) for integer spin particles, and Fermi statistics (minus sign if one exchanges two identical particles) for half-integer spin particles. On the other hand, in QFT the relationship between spin and statistics is a consequence of the framework, following from the commutation quantization conditions for boson fields and anticommutation quantization conditions for fermion fields. 1.2 Scales and Units There are three fundamental dimensionful constants in nature: the speed of light c, Planck s constant (divided by 2π), and Newton s constant G N. Their dimensions are [c] = length time 1, [ ] = length 2 mass time 1, (1.6) [G N ] = length 3 mass 1 time 2. 5

8 In order to avoid unnecessary clutter, we will work throughout this course in natural units, defined by c = = 1. 3 This allows us to express all dimensionful quantities in terms of a single scale which we choose to be mass or, equivalently, energy (since E = mc 2 has become E = m). Energies will be given in units of ev (the electron volt) or more often GeV = 10 9 ev or TeV = ev, since we are typically dealing with high energies. To convert the unit of energy back to units of length or time, we have to insert the relevant powers of c and. E.g., the length scale λ associated to a mass m is λ = h/(mc). Remembering that hc ev m, (1.7) one finds that the length scale corresponding to the electron with mass m e 511 kev is λ e m. Throughout this course we will refer to the dimension of a quantity, meaning the mass dimension. Newton s constant, e.g., has [G N ] = 2 and defines a mass scale G N = M 2 P, (1.8) where M P GeV is the Planck scale. This energy corresponds to a length scale L P m the Planck length. The Planck length is believed to be the smallest length scale that makes sense: beyond this scale quantum gravity effects are likely to become important and its no longer clear that the concept of space-time can be applied. The largest length scale we can talk of is the size of the cosmological horizon, roughly L P. A number for particle physics and cosmology relevant masses and the corresponding length scales are shown in Table 1. Let me go through the list and spend some words on the most important quantities. After the size of the observable universe, the first scale we encounter is the cosmological constant (Λ) measured to be around 10 3 ev. Since nobody can really explain why the cosmological constant has this particular value, let s forget about it real quick and turn our attention to the masses of the known elementary particles. These range from less than 1 ev for the neutrinos (ν s) to around 175 GeV for the top quark (t). The (in)famous Higgs boson (h), which is the only not yet observed ingredient of the standard model (SM) of elementary particle physics, is believed to weigh in at about 100 to 200 GeV. For scales around 1 TeV, i.e., the terascale, the predictive power of the SM is expected to break down. This is precisely the energy regime that the Large Hadron Collider (LHC) at CERN in Geneva has started to explore, having a design center-of-mass energy of 14 TeV. Beyond the electroweak scale (v) of around 250 GeV, again nobody knows with certainty what is going on. One could find a plethora of new (elementary) particles or a great desert. There are experimental hints that the coupling constants of electromagnetism, and the weak and strong forces unify at around M GUT = GeV, i.e., the grand unification scale (GUT). Everything is topped off at the Planck scale where a QFT description might no longer be possible and a quantum theory including the effects of gravity is needed to describe the physics of fundamental interactions. The most likely possibility for such a theory seems to be some kind of string theory. But also many other ideas such as loop quantum gravity, Hoŕava-Lifshitz gravity, etc. exist. In fact, the theory of everything (TOE) could also be a QFT, but one in which the finite or 3 The whole point of units is that you can choose whatever units are most convenient! 6

9 Quantity Mass Length Observable universe ev m ly Cosmological constant (Λ) 10 3 ev 10 3 m Neutrinos (ν s) 1 ev 10 6 m Electron (e ) 511 kev m Muon (µ ) 106 MeV m Charm quark (c) 1.3 GeV m Tau (τ ) 1.78 GeV m Bottom quark (b) 4.6 GeV m Top quark (t) 175 GeV m Higgs boson (h) [100, 200] GeV [6, 12] m Electroweak scale (v) 250 GeV m LHC energy 14 TeV m GUT scale (M GUT ) GeV m Planck scale (M P ) GeV m Table 1: An assortment of masses and corresponding lengths scales that appear in the context of particle physics and cosmology. infinite number of renormalized couplings do not run off to infinity with increasing energy, but hit a fixed point of the renormalization group equation. This possibility goes by the name of asymptotic safety. Don t worry if haven t understood a single word of what I have mumbled about possible TOEs. All this is way too advanced to be covered in this course. I only mentioned it, to make propaganda for the research of Joe Conlon (string theory), Andre Lukas (string theory), and John Wheater (quantum gravity), which work on such theories here in Oxford. Ask them if you want to know more about it. References [1] S. Weinberg, What is quantum field theory, and what did we think it was?, arxiv:hepth/ [2] S. Weinberg, The Search for Unity: Notes for a History of Quantum Field Theory, Daedalus, Vol. 106, No. 4, Discoveries and Interpretations: Studies in Contemporary Scholarship, Volume II (1977), 17 p. [3] Chapter 1 of S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge, UK, Univ. Pr. (1995), 609 p. [4] F. Wilczek, Rev. Mod. Phys. 71, S85 (1999) [arxiv:hep-th/ ]. 7

10 2 Elements of Classical Field Theory In this second section we will discuss various aspects of classical fields. We will cover only the bare minimum ground necessary before turning to the quantum theory, and will return to classical field theory at several later stages in this course when we need to introduce new concepts or ideas. 2.1 Dynamics of Fields A field is a quantity defined at every space-time point x = (t, x). While classical particle mechanics deals with a finite number of generalized coordinates q a (t), indexed by a label a, in field theory we are interested in the dynamics of fields φ a (t, x), (2.1) where both a and x are considered as labels. We are hence dealing with an infinite number of degrees of freedom (dofs), at least one for each point x in space. Notice that the concept of position has been relegated from a dynamical variable in particle mechanics to a mere label in field theory. Lagrangian and Action The dynamics of the fields is governed by the Lagrangian. In all the systems we will study in this course, the Lagrangian is a function of the fields φ a and their derivatives µ φ a, 4 and given by L(t) = d 3 x L(φ a, µ φ a ), (2.2) where the official name for L is Lagrangian density. Like everybody else we will, however, simply call it Lagrangian from now on. For any time interval t [t 1, t 2 ], the action corresponding to (2.2) reads t2 S = dt d 3 x L = d 4 x L. (2.3) t 1 Recall that in classical mechanics L depends only on q a and q a, but not on the second time derivatives of the generalized coordinates. In field theory we similarly restrict to Lagrangians L depending on φ a and φ a. Furthermore, with an eye on Lorentz invariance, we will only consider Lagrangians depending on φ a and not higher derivatives. Notice that since we have set = 1, using the convention described in Section 1.2, the dimension of the action is [S] = 0. With (2.3) and [d 4 x] = 4, it follows that the Lagrangian must necessarily have [L] = 4. Other objects that we will use frequently to construct Lagrangians are derivatives, masses, couplings, and most importantly fields. The dimensions of the former two objects are [ µ ] = 1 and [m] = 1, while the dimensions of the latter two quantities depend on the specific type of coupling and field one considers. We therefore postpone 4 If there is no (or only little) room for confusion, we will often drop the arguments of functions and write φ a = φ a (x) etc. to keep the notation short. 8

11 the discussion of the mass dimension of couplings and fields to the point when we meet the relevant building blocks. Principle of Least Action The dynamical behavior of fields can be determined by the principle of least action. This principle states that when a system evolves from one given configuration to another between times t 1 and t 2 it does so along the path in cofiguration space for which the action is an extremum (usually a minimum) and hence satisfies δs = 0. This condition can be rewritten, using partial integration, as follows { L δs = d 4 x L } = d 4 x δφ a + φ a ( µ φ a ) δ( µφ a ) {[ ( L L µ φ a ( µ φ a ) )] δφ a + µ ( )} L ( µ φ a ) δφ a = 0. The last term is a total derivative and vanishes for any δφ a that decays at spatial infinity and obeys δφ a (t 1, x) = δφ a (t 2, x) = 0. For all such paths, we obtain the Euler-Lagrange equations of motion (EOMs) for the fields φ a, namely ( ) L µ L = 0. (2.5) ( µ φ a ) φ a Hamiltonian Formalism (2.4) The link between the Lagrangian formalism and the quantum theory goes via the path integral. While this is a powerful formalism, we will for the time being use canonical quantization, since it makes the transition to QM easier. For this we need the Hamiltonian formalism of field theory. We start by defining the momentum density π a (x) conjugate to φ a (x), In terms of π a, φa, and L the Hamiltonian density is given by π a = L φ a. (2.6) H = π a φa L, (2.7) where, as in classical mechanics, we have eliminated φ a in favor of π a everywhere in H. The Hamiltonian then simply takes the form H = d 3 x H. (2.8) 2.2 Noether s Theorem The role of symmetries in field theory is possibly even more important than in particle mechanics. There are Lorentz symmetry, internal symmetries, gauge symmetries, supersymmetries, etc. We start here by recasting Noether s theorem in a field theoretic framework. 9

12 Currents and Charges Noether s theorem states that every continuous symmetry of the Lagrangian gives rise to a conserved current J µ (x), so that the EOMs (2.5) imply µ J µ = 0, (2.9) or in components dj 0 /dt + J = 0. To every conserved current there exists also a conserved (global) charge Q, i.e., a physical quantity which stays the same value at all times, defined as Q = d 3 x J 0. (2.10) R 3 The latter statement is readily shown by taking the time derivative of Q, dq R3 dt = d 3 x dj 0 = d 3 x J, (2.11) dt R 3 which is zero, if one assumes that J falls off sufficiently fast as x. Notice, however, that the existence of the conserved current J is much stronger than the existence of the (global) charge Q, because it implies that charge is in fact conserved locally. To see this, we define the charge in a finite volume V by Q V = d 3 x J 0. (2.12) Repeating the above analysis, we find dq V = dt V V d 3 x J = ds J, (2.13) S where S denotes the area bounding V, ds is a shorthand for n ds with n being the outward pointing unit normal vector of the boundary S, and we have used Gauss theorem. In physical terms the result means that any charge leaving V must be accounted for by a flow of the current 3-vector J out of the volume. This kind of local conservation law of charge holds in any local field theory. Proof of Theorem In order to prove Noether s theorem, we ll consider infinitesimal transformations. This is always possible in the case of a continuous symmetry. We say that δφ a is a symmetry of the theory, if the Lagrangian changes by a total derivative δl(φ a ) = µ J µ (φ a ), (2.14) for a set of functions J µ. We then consider the transformation of L under an arbitrary change of field δφ a. Glancing at (2.4) tells us that in this case [ ( )] ( ) L L L δl = µ δφ a + µ φ a ( µ φ a ) ( µ φ a ) δφ a. (2.15) 10

13 When the EOMs are satisfied than the term in square bracket vanishes so that we are simply left with the total derivative term. For a symmetry transformation satisfying (2.13) and (2.14), the relation (2.15) hence takes the form ( ) L µ J µ = δl = µ ( µ φ a ) δφ a, (2.16) or simply µ J µ = 0 with J µ = L ( µ φ a ) δφ a J µ, (2.17) which completes the proof. Notice that if the Lagrangian is invariant under the infinitesimal transformation δφ a, i.e., δl = 0, then J µ = 0 and J µ contains only the first term on the right-hand side of (2.17). We stress that that our proof only goes through for continuous transformations for which there exists a choice of the transformation parameters resulting in a unit transformation, i.e., no transformation. An example is a Lorentz boost with some velocity v, where for v = 0 the coordinates x remain unchanged. There are examples of symmetry transformations where this does not occur. E.g., a parity transformation does not have this property, and Noether s theorem is not applicable then. Energy-Momentum Tensor Recall that in classic particle mechanics, spatial translation invariance gives rise to the conservation of momentum, while invariance under time translations is responsible for the conservation of energy. What happens in classical field theory? To figure it out, let s have a look at infinitesimal translations x ν x ν ɛ ν = φ a (x) φ a (x + ɛ) = φ a (x) + ɛ ν ν φ a (x), (2.18) where the sign in the field transformation is plus, instead of minus, because we are doing an active, as opposed to passive, transformation. If the Lagrangian does not explicitly depend on x but only through φ a (x) (which will always be the case in the Lagrangians discussed in this course), the Lagrangian transforms under the infinitesimal translation as L L + ɛ ν ν L. (2.19) Since the change in L is a total derivative, we can invoke Noether s theorem which gives us four conserved currents T µ ν = (J µ ) ν one for each of the translations ɛ ν (ν = 0, 1, 2, 3). From (2.18) we readily read off the explicit expressions for T µ ν, T µ ν = L ( µ φ a ) νφ a δ µ νl. (2.20) This quantity is called the energy-momentum (or stress-energy) tensor. It has dimension [T µ ν] = 4 and satisfies µ T µ ν = 0. (2.21) 11

14 The four conserved charges are (µ = 0, 1, 2, 3) P µ = d 3 x T 0µ, (2.22) Specifically, the time component of P µ is P 0 = d 3 x T 00 = ( ) d 3 x π a φa L, (2.23) which (looking at (2.7) and (2.8)) is nothing but the Hamiltonian H. We thus conclude that the charge P 0 is the total energy of the field configuration, and it is conserved. In fields theory, energy conservation is thus a pure consequence of time translation symmetry, like it was in particle mechanics. Similarly, we can identify the charges P i (i = 1, 2, 3), P i = d 3 x T 0i = d 3 x π a i φ a, (2.24) as the momentum components of the field configuration in the three space directions, and they are of course also conserved. 2.3 Example: Electrodynamics As a simple application of the formalism we have developed so far in this section, let us try to derive Maxwell s equations using the field theory formulation. In terms of the electric and magnetic fields E and B and the charge density ρ and 3-vector current j, these equations take the well-known form B = 0, (2.25) E + B t = 0, (2.26) E = ρ, (2.27) B E t = j. (2.28) The E and B fields are spatial 3-vectors and can be expressed in terms of the components of the 4-vector field A µ = (φ, A) by E = φ A t, B = A. (2.29) This definition ensures that the first two homogeneous Maxwell equations (2.25) and (2.26) are automatically satisfied, ( A) = ɛ ijk i j A k = 0, (2.30) ( φ A ) + t t ( A) = ( φ) = ɛ ijk j k φ = 0. (2.31) 12

15 Here ɛ ijk is the fully antisymmetric Levi-Civita tensor with ɛ 123 = ɛ 123 = +1 and the indices i, j, k = 1, 2, 3 are summed over if they appear twice. The remaining two inhomogeneous Maxwell equations (2.27) and (2.28) follow from the Lagrangian L = 1 2 ( µa ν ) ( µ A ν ) ( µa µ ) 2 A µ J µ, (2.32) with J µ = (ρ, j). From the rules presented in Section 2.1, we gather that the dimension of the vector field and current is [A µ ] = 1 and [J µ ] = 3, respectively. The funny minus sign of the first term on the right-hand side is required to ensure that the kinetic term 1/2A 2 i is positive using the Minkowski metric. Notice also that the Lagrangian (2.32) has no kinetic term 1/2A 2 0 and hence A 0 is not dynamical. Why this is and necessarily has to be the case will only become fully clear if you attend the advanced QFT course. Yet, we can already get an idea what is going on by remembering that the photon (the quantum of electrodynamics) has only two polarization states, i.e., two physical dofs, while the massless vector field A µ has obviously four dofs. The fact that time component A 0 is not dynamical reduces the number of independent dofs in A µ from four to three. But this is still one too many. The last unwanted dof can be gauged away using the gauge symmetry of the quantum version of electromagnetism aka quantum electrodynamics (QED). Enough said, let s do serious business and compute something. To see that the statement made before (2.32) is indeed correct, we first evaluate L A ν = J ν, L ( µ A ν ) = µ A ν + ( ρ A ρ ) η µν, (2.33) from which we derive the EOMs, ( ) L 0 = µ L = [ 2 A ν + ν ( ρ A ρ ) ] + J ν = µ ( µ A ν ν A µ ) + J ν. (2.34) ( µ A ν ) A ν Introducing now the field-strength tensor we can write (2.32) and (2.34) quite compact, F µν = µ A ν ν A µ, (2.35) L = 1 4 F µνf µν J µ A µ. (2.36) µ F µν = J ν, (2.37) Does this look familiar? I hope so. Notice that [F µν ] = 2. In order to see that (2.37) indeed captures the physics of (2.27) and (2.28), we compute the components of F µν. We find F 0i = F i0 = 0 A i i A 0 = F ij = F ji = i A j j A i = ɛ ijk B k, 13 ( φ + A ) i = E i, t (2.38)

16 while all other components are zero. With this in hand, we then obtain from µ F µ0 = ρ and µ F µ1 = j 1, µ F µ0 = 0 F 00 + i F i0 = E = ρ, µ F µ1 = 0 F 01 + i F i1 = E1 t + B3 x 2 B2 x 3 = ( B E ) 1 = j 1. t (2.39) Similar relations hold for the remaining components i = 2, 3. Taken together this proves the second inhomogeneous Maxwell equation (2.28). Let me also derive the energy-momentum tensor T µν of electrodynamics, ignoring for the moment the source term A µ J µ. Using (2.33) one finds T µν = ( ν A µ )( ρ A ρ ) ( µ A ρ )( ν A ρ ) ηµν F ρσ F ρσ. (2.40) Notice that the first term in (2.40) is not symmetric, which implies that T µν T νµ. In fact, this is not really surprising since the definition of the energy-momentum tensor (2.20) does not exhibit an explicit symmetry in the indices µ and ν. Nevertheless, there is typically a way to massage the energy-momentum tensor of any theory into a symmetric form. 5 To learn how this can be done in the case under consideration is the objective of a homework problem. 2.4 Space-Time Symmetries One of the main motivations to develop QFT is to reconcile QM with special relativity. We thus want to construct field theories in which space and time are placed on an equal footing and the theory is invariant under Lorentz transformations, with x µ (x ) µ = Λ µ νx ν, (2.41) η µν = η ρσ Λ µ ρλ ν σ, (2.42) so that the distance ds 2 = η µν dx µ dx ν is preserved. Here η µν = η µν = diag (1, 1, 1, 1) denotes the Minkowski metric. E.g., a rotation by the angle θ about the z-axis, and a boost by v < 1 along the x-axis are respectively described by the following Lorentz transformations γ γv 0 0 Λ µ 0 cos θ sin θ 0 ν = 0 sin θ cos θ 0, γv γ 0 0 Λµ ν = , (2.43) with γ = 1/ 1 v 2. The Lorentz transformations form a Lie group under matrix multiplication. You can learn more about this if you attend the lecture course on group theory held 5 One (but not the only) reason that you might want to have a symmetric energy-momentum tensor T µν is to make contact with general relativity, since such an object sits on the right-hand side of Einstein s field equations. 14

17 by Andre Lukas. Alternatively, you can study the group theory crash course written by Martin Bauer (a PhD student at Mainz University). It can be found on my Oxford homepage. The various fields belong to different representations of the Lorentz group. The simplest example is the scalar field φ, which under the Lorentz transformation x Λx, 6 transforms as φ(x) φ (x) = φ(λ 1 x). (2.44) The inverse Λ 1 appears in the argument because we are dealing with an active transformation, in which the field is truly shifted. To see why this means that the inverse appears, it will suffice to consider a non-relativistic example such as a temperature field. Suppose we start with an initial field φ(x) which has a hotspot at, say, x = (1, 0, 0). Let s now make a rotation x Rx about the z-axis so that the hotspot ends up at x = (0, 1, 0). If we want to express the new field φ (x) in terms of the old field φ(x), we have to place ourselves at x = (0, 1, 0) and ask what the old field looked like at the point R 1 x = (1, 0, 0) we came from. This R 1 is the origin of the Λ 1 factor in the argument of the transformed field in (2.44). The Lagrangian formulation of field theory makes it especially easy to discuss Lorentz invariance, since an EOM is automatically Lorentz invariant if it follows from a Lagrangian that is a Lorentz scalar. This is an immediate consequence of the principle of least action. If a Lorentz transformation leaves the Lagrangian unchanged, the transformation of an extremum in the action will be another extremum. To give an example, let s look at the following Lagrangian L = 1 2 ( µφ) m2 φ 2. (2.45) where φ is a real scalar and, as we will see later, m is the mass of φ (for now on just think about m as a parameter). Obviously, the dimension of the field is [φ] = 1. You will show in a homework assignment that the EOM corresponding to (2.45) takes the form ( µ µ + m 2) φ = 0. (2.46) This equation is the famous Klein-Gordon equation. The Laplacian in Minkowski space is sometimes denoted by. In this notation, the Klein-Gordon equation reads ( + m 2 )φ = 0. Let us first check that a Lorentz transformation Λ leaves the Lagrangian (2.45) and its action invariant. According to (2.44), the mass term transforms as 1/2 m 2 φ 2 (x) 1/2 m 2 φ 2 (x ) with x = Λ 1 x. The transformation of µ φ is µ φ(x) µ (φ(x )) = (Λ 1 ) ν µ ( νφ)(x ). (2.47) Using (2.43) we thus find that the derivative term in the Klein-Gordon Lagrangian behaves as 1 2 ( µφ(x)) (Λ 1 ) ρ µ ( ρφ)(x )(Λ 1 ) σ ν ( σφ)(x ) η µν = 1 2 ( ρφ)(x )( σ φ)(x ) η ρσ (2.48) = 1 2 ( µφ(x )) 2, 6 To shorten the notation we will often use matrix notation and drop the indices µ, etc. 15

18 under the Lorentz transformation Λ. Putting things together, we find that the action of the Klein-Gordon theory is indeed Lorentz invariant, S = d 4 x L(x) d 4 x L(x ) = d 4 x L(x ) = S. (2.49) Notice that changing the integration variables from d 4 x to d 4 x, in principle introduces an Jacobian factor det (Λ). This factor is, however, equal to 1 for Lorentz transformation connected to the identity, that we are dealing with. A similar calculation also shows that, as promised, also the EOM of the Klein-Gordon field φ is invariant, ( 2 + m 2) φ(x) ( 2 + m 2) φ(x ) = [(Λ 1 ) ν µ ν(λ 1 ) ρµ ρ + m 2 ] φ(x ) (2.50) = ( η νρ ν ρ + m 2) φ(x ) = 0. In the case of the Klein-Gordon theory, we hence conclude that the statements made before (2.45) are indeed correct. Representations of Lorentz Group The transformation law (2.44) is the simplest possible transformation law for a field. In fact, it is the only possibility for a one-component field aka a real scalar. Yet, it is also clear that in order to describe nature (think only about electromagnetism) we need multicomponent fields, which have more complicated transformation properties. The most familiar case is that of a vector field, such as the vector potential A µ, which we have already met in Section 2.3. In this case the quantity that is distributed in space-time also carries an orientation which must be rotated and/or boosted. In fact, we will learn in this course that the Lorentz group has a variety of representations, corresponding to particles with integer (bosons) and half-integer spins (fermions) in QFT. These representations are normally constructed out of spinors. To start this general (and somewhat formal) discussion, let me examine the allowed possibilities for linear field transformations φ a (x) φ a(x ) = M(Λ) ab φ b (x), (2.51) under (2.41). The first important point to notice is that the Lorentz transformations form a group. This means that two successive Lorentz transformations, can also be described in terms of a single one x x = Λx, x x = Λ x, (2.52) x x = Λ x, (2.53) with Λ = Λ Λ. (2.54) 16

19 What happens to (2.51) under this set of Lorentz transformations? For x x = Λ x, we have (in matrix notation) φ(x) φ (x ) = M(Λ )φ(x). (2.55) On the other hand, for x x = Λx x = Λ Λx, we get φ(x) φ (x ) = M(Λ )φ (x ) = M(Λ )M(Λ)φ(x). (2.56) In order for the last two equations to be consistent with each other, the field transformations M must obviously fulfill M(Λ Λ) = M(Λ )M(Λ). (2.57) In group theory terminology, this means that the matrices M furnish a representation of the Lorentz group. Field Lorentz transformations are therefore not random, but they can be found if we find all (finite dimensional) representations of the Lorentz group. So how do the common representation of the Lorentz group look like and how do we get all of them? While both questions will be answered in this lecture, I believe it is best to do it case-by-case whenever we will meet a new type of (quantum) field. Since we already talked about the real scalar φ (Klein-Gordon field) and the vector A µ (potential in electrodynamics), it makes nevertheless sense to give the representations for these two types of fields already at this point. Since a scalar field by definition does not change under Lorentz transformations, φ(x) φ (x ) = φ(x), the scalar representation of the Lorentz group is simply M(Λ) = 1. (2.58) This was easy! The representation of the vector A µ is also not difficult to figure out. Let me for the time being only state the result. One finds M(Λ) = Λ, (2.59) which means that a vector field A µ transforms under a Lorentz transformation as (restoring indices) A µ (x) (A ) µ (x ) = Λ µ νa ν (x ). (2.60) It is important to notice that the latter transformation property implies that any term build out of A µ and µ, where all Lorentz indices are contracted is invariant under Lorentz transformations. As an exercise you are supposed to show this explicitly for terms like µ A µ, etc. Angular Momentum In classical particle mechanics, rotational invariance gives rise to conservation of angular momentum. What is the analogy in field theory? Moreover, we now have further Lorentz transformations, namely boosts. What conserved quantity do they correspond to? In order to address these questions, we first need the infinitesimal form of the Lorentz transformations Λ µ ν = δ µ ν + ω µ ν, (2.61) 17

20 where ω µ ν is infinitesimal. The condition (2.42) for Λ to be a Lorentz transformation becomes in infinitesimal form η µν = η ρσ (δ µ ρ + ω µ ρ) (δ ν σ + ω ν σ) = η µν + ω µν + ω νµ + O(ω 2 ), (2.62) which implies that ω µν must be an antisymmetric matrix, ω µν = ω νµ. (2.63) Notice that an antisymmetric 4 4 matrix has six independent parameters, which agrees with the number of different Lorentz transformations, i.e., three rotations and three boosts. Applying the infinitesimal Lorentz transformation to our real scalar field φ, we have φ(x) φ(x ωx) = φ(x) ω µ νx ν µ φ(x), (2.64) where the minus sign arises from the factor Λ 1 in (2.43). The variation of the field φ under an infinitesimal Lorentz transformation is hence given by δφ = ω µ ν x ν µ φ. (2.65) By the same line of reasoning, one shows that the variation of the Lagrangian is δl = ω µ ν x ν µ L = µ (ω µ ν x ν L), (2.66) where in the last step we used the fact that ω µ µ = 0 due to its antisymmetry. The Lagrangian changes by a total derivative, so we can apply Noether s theorem (2.17) with J µ = ω µ ν x ν L to find the conserved current, J µ = L ( µ φ) ωρ ν x ν ρ φ + ω µ ν x ν L [ ] L = ω ρ ν ( µ φ) ρφ δ µ ρl x ν = ω ρ νt µ ρx ν. (2.67) Stripping off ω ρ ν, we obtain six different currents, which we write as These currents satisfy (J λ ) µν = x µ T λν x ν T λµ. (2.68) λ (J λ ) µν = 0, (2.69) and give (as usual) rise to six conserved charges. For µ, ν 0, the Lorentz transformation is a rotation and the three conserved charges give the total angular momentum of the field (i, j = 1, 2, 3): Q ij = d 3 x ( x i T 0j x j T 0i). (2.70) What s about the boosts? In this case, the conserved charges are Q 0i = d 3 x ( x 0 T 0i x i T 00). (2.71) 18

21 The fact that these are conserved tells us that 0 = dq0i = d 3 x T 0i + t d 3 0i dt x d dt dt dt = P i + t dp i dt d d 3 x x i T 00. dt d 3 x x i T 00 (2.72) Yet, also the momentum P i is conserved, i.e., dp i /dt = 0, and we conclude that d d 3 x x i T 00 = const.. (2.73) dt This is the statement that the center of energy of the field travels with a constant velocity. In a sense it s a field theoretical version of Newton s first law but, rather surprisingly, appearing here as a conservation law. Notice that after restoring the label a our results for (J λ ) µν etc. also apply in the case of multicomponent fields. Poincaré Invariance We now require that a physical system possesses both space-time translation (2.18) and Lorentz transformation symmetry (2.41). The symmetry group that includes both transformations is called the Poincaré group. Notice that for any Poincaré-invariant theory the two charge conservation equations (2.21) and (2.69) should hold. This is only possible if the energymomentum tensor T µν is symmetric. Indeed, 0 = λ (J λ ) µν = λ ( x µ T λν x ν T λµ) = x µ λ T λν + T λν λ x µ x ν λ T λµ T λµ λ x ν (2.74) = T λν δ λ µ T λµ δ λ ν = T µν T νµ. Since Maxwell s theory is Poincaré invariant, this general result tells us that the expression of the energy-momentum tensor in (2.40) can be made symmetric without changing physics. The key to actually do it, lies in making use of the conservation law (2.21) in an appropriate way. 2.5 Problems i) Suppose that a no further specified Lagrangian L depends not only on φ and µ φ but also on the second derivatives of the fields: 7 L = L(φ, µ φ, µ ν φ). (2.75) For the case that the variations δφ vanish at the endpoints and that δ( µ1... µn φ) = µ1... µn (δφ) holds, derive the Euler-Lagrange EOMs for such a theory. 7 For the sake of brevity, we have omitted the subscript a labelling the different fields. 19

22 Apply your result to obtain the EOMs for the field φ with Lagrangian L = 1 2 ( tφ) ( x φ) + α 6 ( xφ) 3 ν 2 ( ν µ φ) 2. (2.76) ii) Let us study the dynamics of acoustic waves in an elastic medium (e.g. air), as described by the Lagrangian L = 1 ( ) 2 y 2 ρ 1 t 2 ρv2 sound ( y) 2, (2.77) with ρ the density of the medium and v sound the speed of sound. Find the Euler-Lagrange EOMs for the system and their solutions. What do they describe? Calculate the Hamiltonian H. iii) Consider the Klein-Gordon Lagrangian (2.45). Derive the kinetic and potential energy (T and V with L = T V ) as well as the Euler-Lagrange EOMs for the field φ. Write down the energy-momentum tensor T µν and show that it indeed satisfies µ T µν = 0. Give the expressions for the conserved energy E and momentum P i. iv) Using (2.60) show that the terms µ A µ, ( µ A ν ) 2, and ( µ A ν )( ν A µ ) are Lorentz invariant. What are the dimensions of these terms? v) We saw that in the case of electrodynamics in vacuum using (2.20) leads to an energymomentum tensor T µν that is not symmetric. To remedy that, one can add to T µν a term of the form λ Γ λµν, where Γ λµν is antisymmetric in its first two indices, i.e., Γ λµν = Γ µλν. Show that such an object is automatically divergenceless, i.e., it obeys µ λ Γ λµν = 0. This feature implies that instead of T µν one can also use Θ µν = T µν + λ Γ λµν, (2.78) without changing the physics, since Θ µν momentum as T µν. Show that this construction, with has the same globally conserved energy and Γ λµν = F µλ A ν, (2.79) leads to an energy-momentum tensor Θ µν that is symmetric and yields the standard formulas for the electromagnetic energy and momentum densities: E = 1 ( E 2 + B 2), S = E B. (2.80) 2 20

23 References [1] Chapter 4 of L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Fourth Edition: Vol. 2 (Course of Theoretical Physics Series), Butterworth-Heinemann (1975), 481 p. [2] Chapter 5 of B. Thidé, Electromagnetic Field Theory, revised and extended 2nd edition, 21

24 3 Klein-Gordon Theory In QM, canonical quantization is a recipe that takes us from the Hamiltonian formalism of classical dynamics to the quantum theory. The recipe tells us to take the generalized coordinates q a and their conjugate momenta p a = L/ q a and promote them to operators. The Poisson bracket structure of classical mechanics descends to the structure of commutation relations between operators, namely [q a, q b ] = [p a, p b ] = 0, [q a, p b ] = iδ a b, (3.1) where [a, b] = ab ba is the usual commutator. If one wants to construct a QFT, one can proceed in a similar fashion. The idea is to start with the classical field theory and then to quantize it, i.e., reinterpret the dynamical variables as operators that obey canonical commutation relations, 8 [φ a (x), φ b (y)] = [π a (x), π b (y)] = 0, [φ a (x), π b (y)] = iδ (3) (x y)δ a b. (3.2) Here φ a (x) are field operators and the Kronecker delta in (3.1) has been replaced by a delta function since the momentum conjugates π a (x) are densities. Notice that for now, we are working in the Schrödinger picture which means that the operators φ a (x) and π a (x) do only depend on the spatial coordinates but not on time. The time dependence sits in the states ψ which obey the usual Schödinger equation i d ψ = H ψ. (3.3) dt While all this looks pretty much the same as good old QM there is an important difference. The wavefunction ψ in QFT, is a functional, i.e., a function of every possible configuration of the field φ a, and not a simple function. 9 So things are more complicated in QFT than in QM after all. The Hamiltonian H, being a function of φ a and π a, also becomes an operator in QFT. In order to solve the theory, one task is to find the spectrum, i.e., the eigenvalues and eigenstates of H. This is usually very difficult, since there is an infinite number of dofs within QFT, at least one for each point x in space. However, for certain theories, called free theories, one can find a way to write the dynamics such that each dof evolves independently from all the others. Free field theories typically have Lagrangians which are quadratic in the fields, so that the EOMs are linear. 8 This procedure is sometimes referred to as second quantization. We will not use this terminology here. 9 In functional analysis, a functional is a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a vector as its argument or input and returns a scalar. Commonly, the vector space is a space of functions, so the functional takes a function as its argument, and so it is sometimes referred to as a function of a function. The use of functionals goes back to the calculus of variations where one searches for a function which minimizes a certain functional. A particularly important application in physics is to search for a state of a system which minimizes the energy functional. 22

25 3.1 Klein-Gordon Field as Harmonic Oscillators So far the discussion in this section was rather general. Let us be more specific and consider the simplest relativistic free theory as a practical example. It is provided by the classical Klein-Gordon equation (2.45). To exhibit the coordinates in which the dofs decouple from each other, we only have to Fourier transform the field φ, d 3 p φ(t, x) = (2π) 3 ei p x φ(t, p). (3.4) In momentum space (2.45) simply reads [ 2 t + ( p 2 + m 2) ] φ(t, p) = 0, (3.5) 2 which tells us that for each value of p, the Fourier transform φ(t, p) solves the equation of a harmonic oscillator with frequency ω p = p 2 + m 2. (3.6) We see that the most general solution of the classical Klein-Gordon equation is a linear superposition of simple harmonic oscillators, each vibrating at a different frequency with a different amplitude. In order to quantize the field φ, we must hence only quantize this infinite number of harmonic oscillators (as Sidney Coleman once said [1]: The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. ). Let s recall how to do it in QM. Harmonic Oscillator in QM Consider the QM Hamiltonian H = 1 2 p ω2 q 2, (3.7) with the canonical commutation relations [q, p] = i. In order to find the spectrum of the system, we define annihilation and creation operators (also known as lowering and raising or ladder operators) ω a = 2 q + i ω p, a = 2ω 2 q i p. (3.8) 2ω Expressing q and p through a and a gives q = 1 2ω (a + a ), ω p = i 2 (a a ). (3.9) The commutator of the operators introduced in (3.8) is readily computed. One finds [a, a ] = 1. Expressing the Hamiltonian (3.7) through a and a gives H = ω ( aa + a a ) ( = ω a a + 1 ). (3.10)

26 It is also easy to show that the commutator of H with a and a takes the form [H, a] = ωa, [H, a ] = ωa. (3.11) These relations imply that if ψ is an eigenstate of H with energy E, i.e., H ψ = E ψ, then we can construct other eigenstates by acting with the operators a and a on ψ : Ha ψ = (E ω) a ψ, Ha ψ = (E + ω) a ψ, (3.12) This feature explains why a (a ) is called annihilation (creation) operator. From the latter equation it is also clear that the spectrum of (3.7) has a ladder structure,..., E 2ω, E ω, E, E +ω, E +2ω,.... If the energy is bounded from below, there must be a ground state 0, which satisfies a 0 = 0. This state has the ground state or zero-point energy H 0 = ω/2 0. Excited states n are then created by the repeated action of a, and satisfy (a ) n 0 = n (n 1)... 1 n = n! n, (3.13) ( H n = ω N + 1 ) ( n = ω n + 1 ) n, (3.14) 2 2 where N = a a is the number operator with N n = n n. The prefactor on the right-hand side of (3.13) is needed to guarantee that the states n are normalized to 1, i.e., n n = 1. Quantization of Real Klein-Gordon Field If we treat each Fourier mode of the field φ as an independent harmonic oscillator, we can apply canonical quantization to the real Klein-Gordon theory, and in this way find the spectrum of the corresponding Hamiltonian. In analogy to (3.9), we write φ and π as a linear sum of an infinite number of operators a p and a p, labelled by the 3-momentum p, d 3 p 1 [ ] φ(x) = a (2π) 3 p e i p x + a p e i p x, 2ωp π(x) = d 3 p (2π) 3 ( i) The commutation relations (3.2) become ωp 2 [a p, a q ] = [a p, a q] = 0, [a p, a q] = (2π) 3 δ (3) (p q). [a p e i p x a p e i p x ]. Let us assume that the latter equations hold, it then follows that d 3 p d 3 q i ωq ( ) [φ(x), π(y)] = [a (2π) 6 p, a 2 ω q] e i p x i q y + [a p, a q ] e i p x+i q y p d 3 p d 3 q i ωq ( ) = (2π) 3 δ (3) (p q) e i p x i q y e i p x+i q y (2π) 6 2 ω p d 3 p i ( ) = e i p (x y) e i p (x y) = iδ (3) (x y), (2π) (3.15) (3.16) (3.17)