Lecture Notes on Quantum Field Theory

Transcription

1 Lecture Notes on Quantum Field Theory Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM January 25, 2016

2 Contents I Effective Action Approach in Quantum Field Theory V 1 Introduction Fundamental Physics Relation of Physical Theories Problems of the Fundamental Physics Classical Mechanics Classical Field Theory Special Relativity Quantum Mechanics Quantum Field Theory Classical Gravity and General Relativity Quantum Gravity Fields and Particles Relativistic Invariance Spacetime Poincarè and Lorentz Groups Poincaré Group Group of Translations Lorentz Group Poincarè and Lorentz Algebras Tensor Fields Representations of the Lorentz Group Tensor Representations Reflections and Pseudo-tensors Spinor Fields Dirac Matrices Spinor Representation I

3 II CONTENTS Covariant Spinor Representation Reflections Spinors of Higher Rank Action Functional Action in Classical Mechanics Action in Field Theory Noether Theorem Energy and Momentum Angular Momentum and Spin Current and Charge Models in field theory Path Integrals Action in Quantum Mechanics Gaussian Path Integrals Functional Integration Stationary Phase Method Path Integrals with Fermions Background Field Method Scattering Matrix Generating Functional Path Integral for Generating Functional Chronological Mean Values Effective Action Feynmann Diagrams Loop Expansion Effective Action in Scalar Field Theory Gauge Theories Dynamical Configuration Subspace Invariant Measure Ward Identities Physical Field Variables Propagators De Witt Gauge Conditions Effective Action in Gauge Theories Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 1

4 CONTENTS III 6.8 Yang-Mills Theory and Quantum Gravity Yang-Mills Theory General Relativity II Mathematical Supplement Lie Groups and Lie Algebras Abstract Group Continuous Groups Invariant Subgroups Homomorphisms Direct and Semi-direct Products Group Representations Multiple-valued Representations. Universal Covering Group Matrix Lie Groups The Structure Constants of a Lie Group Exponential Mapping Algebra of S U(2) Algebra of S O(3) Representations of S U(2) and S O(3) Algebra so(3) Representations of S O(3) Group S U(2) Algebra su(2) Representations of S U(2) Double Covering Homomorphism S U(2) S O(3) Heisenberg Algebra, Fock Space and Harmonic Oscillator Operators on Finite-Dimensional Inner Product Spaces III Physical Supplement Classical Field Theory Introduction Superclassical fields Field configurations Field functionals Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 2

5 IV Contents Dynamics Models in field theory Small disturbances and Green functions Wronskian Retarded and advanced Green functions Cauchy problem for Jacobi fields Feynman propagator Classical perturbation theory Quantum Mechanics Mathematical Foundations of Quantum Mechanics Kinematics Dynamics Classical Mechanics Quantum Mechanics Semiclassical Approximation Path Integrals Quantum Field Theory at Finite Temperature 209 Notation 215 Bibliography 217 Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 3

6 Part I Effective Action Approach in Quantum Field Theory V

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8 Chapter 1 Introduction These are the lecture notes for the Seminar in Quantum Field Theory (QFT). They are not supposed to replace a regular course on QFT but rather give a conceptual introduction to the subject for motivated students of physics and mathematics. For a more detailed exposition the reader is referred to the literature at the end of these notes, the books I recommend are [5, 20, 21, 23, 24, 26, 27, 28, 32, 33, 35, 37]. The book [26] is the closest in the spirit and the methodology to these lectures. 1.1 Fundamental Physics Relation of Physical Theories Like any other physical theory Quantum Field Theory is a theory designed to describe physical phenomena at a certain length scale. Contrary to mathematics physics does not know infinities. All observations are measured by some apparatus that returns some finite values. That is why, physics does not really deal with the infinite lengths and zero lengths. These are idealization of reality. In reality physicists deal with the scales from the largest possible ones of about cm, which is roughly equal to the size of the observable Universe, to cm, which is believed to be the smallest theoretically possible scale. The typical length scales of various physical objects are given in the table 1

9 2 CHAPTER 1. INTRODUCTION Objects Scale (cm) observable Universe clusters of galaxies Solar system Earth molecules 10 6 atoms 10 8 nuclei proton LHC scale GUT scale Planck scale Here the LHC stands for the Large Hadron Collider and GUT stands for the Grand Unification Theory. The physics at length scales in the range cm is well established. The cosmological and the astrophysical length scales are well described by General Relativity with few problems like the dark energy (cosmological scale) and the dark matter (galactical scale). The standard macroscopic scales are very well described by the Newtonian Classical Mechanics. The molecular and the atomic scales are well described by the non-relativistic Quantum Mechanics. The scales from the atomic ones to the LHC ones are the scales of the elementary particle physics (or high energy physics). These are the scales that require relativistic Quantum Mechanics or Quantum Field Theory for the adequate description. The current theory of the elementary particles is called the Standard Model. This model was finalized in the 1970 s and is still working pretty good. However, it starts to show some cracks in its foundation. One of the facts that does not fit in the model are the non-zero masses of neutrino. Also, there are some recent indications from the LHC that there is some new physics beyond the Standard Model. The higher energy scales (or the smaller length scales) are not understood very well and are not studied experimentally. These energy scales cannot be reach at the particle accelerators and the only data come from astrophysics (cosmic rays etc). Quantum Field Theory is still applied at those energy scales but at this stage these theories are speculative. Finally, to understand the physics at the Planckian scales one requires a consistent theory of quantum gravity which is currently absent. The most popular candidates for the theory of quantum gravity are String Theory, Loop Quantum Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 4

10 1.1. FUNDAMENTAL PHYSICS 3 Gravity and some others. These theories should be able to describe physical phenomena in the early Universe (Big Bang singularity, the nature of time, quantum cosmology) as well as the physics of black holes (quantum explosion of black holes, collisions of black holes, structure of the event horizon of black holes, information paradox, firewall, entropy of black holes etc). In 1832 Gauss realized that in fundamental physics one needs only three fundamental units of length (cm), mass (g) and time (s), which led to the so-called CGS unit system. Among many physical constants only three are really fundamental. These are the speed of light, c, the Planck constant,, and the gravitational constant, G. they have the following dimensions [c] = LT 1, [ ] = L 2 MT 1, [G] = L 3 M 1 T 2, where L, T and M are units of length, time and mass. To put it in a prospective let us recall the history of fundamental physical theories. The classical mechanics and gravity theory was created by Newton in The theory of classical electrodynamics was finalized by Maxwell in The classical mechanics was superseded by Einstein s Special Theory of Relativity in 1905 and the Newtonian gravity theory was replaced by Einstein s General Relativity in Quantum Mechanics was created by Bohr, Schrödinger and Heisenberg in The first Quantum Field Theory, Quantum Electrodynamics, was put on a solid ground in The current theory of elementary particles, the Standard Model, which unified the electromagnetic, weak and strong interactions was finalized in The search for the theory of Quantum Gravity is still ongoing with various success, it is not completed yet. The relations between different theories can be illustrated on Fig Therefore, the Quantum Field Theory and the Quantum Gravity can be viewed on as deformations of the classical theories corresponding to the fundamental constants c,, G. In 1899 Planck proposed so called natural units by combining these three fundamental physical constants. The Planck length, mass and time are defined by L p = M p = T p = ( ) 1/2 G cm, (1.1) {?} c 3 ( ) 1/2 c 10 5 g, (1.2) {?} G ( ) 1/2 G s, (1.3) {?} c 5 Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 5

11 4 CHAPTER 1. INTRODUCTION fig1 Quantum Gravity G 0 0 Quantum Field Theory General Relativity c 0 G 0 c 7 Quantum Mechanics Special Relativity Newtonian Gravity 0 c G 0 Classical Mechanics Figure 1.1: Relations of physical theories Therefore, one can set = c = G = 1, which makes all quantities dimensionless, and measure everything in Planck units. This is the natural unit system in quantum gravity. In quantum field theory one usually sets = c = 1, which leaves only one unit, mass (or energy), and measure everything in that unit. I this system the unit of length and time is the same and is the reciprocal of the mass unit L = T = 1 M, that is, [G] = 1 M 2. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 6

12 1.1. FUNDAMENTAL PHYSICS 5 In the following we always put = c = G = 1, except for some explicit expressions where they are left for convenience Problems of the Fundamental Physics Four of the most important problems of the fundamental physics, which are probably related to each other, are 1. Quantization of gravity: find a way to combine the basic principles of general relativity with the basic principles of quantum mechanics. 2. Foundations of quantum mechanics: develop a consistent interpretation of quantum mechanics. 3. Unified field theory: combine all known interactions and particles in a single theory. 4. Cosmology: explain the origin of the dark matter and the dark energy Classical Mechanics Any physical theory consists of two parts: kinematics and dynamics. Kinematics describes the states of a system and dynamics predicts the evolution of the state in time, so given an initial state at an initial time we should be able to predict the state of the system at any time in the future. This is exactly what happens in classical mechanics. The configuration of a mechanical system can be described by a finite number of independent parameters called the coordinates. Each such coordinate is called a degree of freedom and the total number of them is called the number of degrees of freedom. That is why, a classical mechanical system is a system with finitely many degrees of freedom. However, the knowledge of all coordinates at an initial time is not enough to predict the dynamical evolution of the system since prescribing the coordinates only does not describe the state of the system. Two systems could have the same coordinates but be in completely different states since one system could be at rest and the other could be moving. That is why, a complete description of the state of a classical mechanical system requires prescribing not only the coordinates but also the velocities (or momenta). Then, given the coordinates and momenta at an initial time uniquely determines the evolution of the system in the future. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 7

13 6 CHAPTER 1. INTRODUCTION To develop the mathematical formalism one introduces the set of all states of a system called the phase space, which is a finite dimensional symplectic manifold. Then the state is a point in the phase space and the dynamics is a curve in the phase space, which is governed by the Hamiltonian equations (a system of first order ordinary differential equations). The equations of classical mechanics are invariant under the Galilean transformations, which include translations of time, translations of space, rotations of space and the uniform motions of space (Galilean boosts). It is worth stressing one very important point here. Suppose that our system consists of two subsystems. If these subsystems interact then, strictly speaking, the whole system should be treated as a whole. In other words the equations of motion above do not decouple. However, if the interaction of these subsystems becomes negligible, for example, they move apart at large distance, then the equations of motion decouple and we have two subsystems whose dynamics is completely independent. Then the knowledge of the evolution of each subsystem determines the dynamics of the whole system. This is an intuitively obvious property that holds for classical systems but does not hold in quantum systems (see below) Classical Field Theory Besides classical mechanical systems classical physics also deals with the classical field theory, in particular, electromagnetic fields. A field can be viewed as a mechanical system with infinitely many degrees of freedom, one for each point in space. That is why, the state of a classical field is described by specifying the field and its time derivative at each point in space. Thus, the phase space is now the set of all smooth functions on space and the dynamical evolution is a curve in this space governed by the field equations (a system of partial differential equations), in particular, Maxwell equations. Maxwell equations are experimentally verified with a very good precision and there is no doubt that they are valid in all macroscopic phenomena including electromagnetism, in particular, light propagation. The problem is that they are not invariant under the Galilean transformations of classical mechanics but rather under the Lorentz transformations, which include translations of time, translations of space, rotations of space and the pseudo-rotations of space-time (Lorentz boosts). Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 8

14 1.1. FUNDAMENTAL PHYSICS Special Relativity It is an experimental fact that light propagates with the same speed c in all inertial reference frames. Einstein analyzed the physics of the classical mechanics and showed that classical mechanics is inconsistent with the properties of the light propagation. He modified the laws of the classical mechanics and formulated a new theory called Special Relativity. In this theory all physical laws are the same in all inertial reference frames, that is, all equations of physics are invariant under the Lorentz transformation of the space-time coordinates (ct, x, y, z). The parameter that measures the tendency of an object to stay at rest is called the inertial mass. According to special relativity when the speed of an object increases then the inertia increases and becomes infinite when the speed reaches the speed of light. That is why, nothing can travel faster than light and the new relativistic effects become important for objects moving close to the speed of light. The only such objects are elementary particles. If the mass of an elementary particle is m then it is relativistic if its speed is large enough so that its kinetic energy is comparable to mc 2. Of course, for massless particles this means that they are always relativistic, so they always travel with the speed of light. Beside the photon, there are some other massless particles like neutrino (according to recent data it may have a small mass), graviton (not discovered yet) and others. There was a controversy recently about new experimental data that neutrino could propagate faster than light, but now almost everybody agrees that that was an experimental error. If that were true, then the whole body of modern physics should have been reevaluated and modified somehow. Mathematically, one could say that special relativity is a deformation of classical mechanics, which is recovered in the limit c. Note that classical Maxwell field theory is already relativistic Quantum Mechanics It is an experimental fact that microscopic objects exhibit wave phenomena (like diffraction and interference) that cannot be explained by the classical mechanics. One of the most important features of a quantum system is that it cannot be reduced to its parts but should be treated as a whole; one says, that different parts of a quantum system are entangled. This is a very new feature that is absent in classical mechanics. If a quantum system consists of two subsystems then even if their interaction vanishes (let say, they move apart at large distance) they are still not independent. The dynamics of the whole system is not determined by the Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 9

15 8 CHAPTER 1. INTRODUCTION dynamics of its (even non interacting) subsystems. Of course, one could say that this means that these subsystems do interact, but the interaction is not the usual local one but rather some kind of a non-local interaction. This is what is meant when one says that the subsystems are entangled. This leads to a number of surprising apparent paradoxes, like the Schrödinger cat paradox, which lead some authors to conclude that quantum mechanics is an incomplete theory with some deep foundational problems which need to be resolved before it can be united with gravity. The state of a quantum system is described by specifying a complex-valued function on the physical space called the wave function. The wave function is determined only up to a constant normalizing factor which can be chosen so that it has a unit norm, and, additionally, up to a constant phase factor. Given an initial state, that is, an initial wave function, at an initial time quantum mechanics predicts the wave function at all future times. The standard approach to quantum mechanics (called the Copenhagen interpretation) is somewhat minimalistic. It is assumed that the only way to know the state of a quantum system is by providing a set of measurements. A measurement is an interaction of a quantum system with a classical system governed by the classical mechanics. That is why, quantum mechanics requires classical mechanics for its logical formulation. The main problem of quantum mechanics is therefore reduced to predicting the outcomes of measurements. Another very important feature of quantum mechanics is that it cannot predict the outcome of a measurement with complete certainty; the most it can do is to predict the probabilities of various outcomes of a measurement. There two main physical principles in quantum mechanics: the superposition principle and the uncertainty principle. The uncertainty principle says that in general two physical observables cannot have specific values in a given state, there is a limit at which they can be determined. The product of uncertainties in these two physical variables is bounded from below, that is, if one approaches zero then the other must go to infinity. Suppose that we have two initial states ψ 1 (0) and ψ 2 (0) of a system. Then quantum mechanics determines the evolution of each of these states with time, say ψ 1 (t) and ψ 2 (t). The superposition principle says that the time evolution of the initial state ψ(0) = a 1 ψ 1 (0) + a 2 ψ(0), (1.4) {?} (where a 1 and a 2 are complex numbers such that a a 2 2 = 1) is determined Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 10

16 1.1. FUNDAMENTAL PHYSICS 9 by the same linear combination ψ(t) = a 1 ψ 1 (t) + a 2 ψ(t). (1.5) {?} Moreover, if the measurement of a physical observable A in the state ψ 1 always returns the value A 1 and the measurement of A in the state ψ 2 always returns the value A 2, then the measurement of A in the state ψ = a 1 ψ 1 + a 2 ψ can only give either A 1 and A 2 with probabilities a 1 2 and a 2 2. The mathematical formalism of quantum mechanics is provided by the theory of self-adjoint operators in a Hilbert space. The states of a quantum system (the wave function) are described by unit vectors in a Hilbert space. The physical observables are described by self-adjoint operators in the Hilbert space. The possible values of a physical observable A are given by the eigenvalues A n of the operator A. The probability that a measurement of a physical quantity in a state ψ returns A n is determined by P n = (ψ n, ψ) 2, (1.6) {?} where ψ n are the eigenfunctions of the operator A corresponding to the eigenvalue A n, so that the expectation value of the observable A in a state ψ is given by A = (ψ, Aψ) = P n A n. (1.7) {?} The dynamical evolution of a state of a quantum system is described by the Schrödinger equation, which is a linear partial differential equation. One could say that quantum mechanics is a deformation of classical mechanics, which is recovered in the limit Quantum Field Theory Of course, quantum mechanics is not a relativistic theory. It does not apply to physical phenomena involving relativistic quantum objects like elementary particles. The most important property of elementary particles is that the number particles is not conserved; the particles are being created and annihilated all the time. This is what one usually sees in the experiments. There are certain ways to register an elementary particle and to measure its mass, its spin, its charge, its momentum and other characteristics. One collides say a beam of electrons with a beam of positrons (or something else) coming from large distances at certain angles. The states of the particles in both beams are measured in advance and known, like their momentum, the polarization, etc. The interaction occurs in a n Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 11

17 10 CHAPTER 1. INTRODUCTION compact space region in a very small amount of time. Then one registers the products of this collision going in all possible directions with all possible momenta. What one sees is not just the two particles that collided (such a collision is called an elastic scattering; it happens usually at low energies) but a lot of new particles that were created during the collision. The original particles may have disappeared totally. One could get a pretty good visual picture of what is going on by drawing so called Feynman diagrams. Quite often the results of the collision further decay in some other particles which are detected. So, if one knows in what it could decay one can judge of the existence of some new particles. This is how usually the discoveries of new elementary particles are made. The mathematical theory of the relativistic quantum mechanics is called Quantum Field Theory. The most common problems in quantum theory are scattering problems. One prepares particles in a certain specified states and collides them either with each other or with a target and then observes the outcome, that is, registers the states of the outgoing particles. If ψ in is the initial state and ψ out is the final state then (ψ in, ψ out ) is called the amplitude of this process, its square (ψ in, ψ out ) 2 gives the probability of the initial state ψ in becoming the final state ψ out. The set of all such amplitudes is called the scattering matrix, or, simply, the S -matrix. The primary goal of quantum field theory is the calculation of the scattering matrix, that is, such amplitudes. The quantum field theories that describe the interaction of elementary particles are non-linear, and, therefore, cannot be solved exactly. The only reasonable way to carry out calculations is the perturbation theory. In this approach one splits the field in a free non-interacting part and small quantum perturbations that interact with each other. Then one expands in powers of the perturbation and gets an infinite series in powers of a parameter called a coupling constant that describes the strength of the interaction. Such a series can be represented graphically by the famous Feynmann diagrams. It turns out that, roughly speaking, the order of the perturbation theory is related to the number of loops in Feynman diagrams. The tree diagrams are purely classical, they do not take into account the quantum nature of the elementary particles. All loop diagrams represent quantum corrections to the classical processes. And then one realizes that there are two main problems with such an expansion. First, the whole series is only an asymptotic series and it diverges. Second, each term in this series in all orders of the perturbation theory, if computed formally, also diverges. The reason for these divergences, called the ultra-violet divergences, is the local nature of the interaction. A reasonable thing to do is to regularize the integrals representing the Feynmann diagrams to make them finite and then to take off the regularization at the Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 12

18 1.1. FUNDAMENTAL PHYSICS 11 end of the calculations. For example, one could just cut off the integrals at some large momenta (or small distances) or by introducing a smooth cutoff function. It turns out that in some important cases of quantum field theories (quantum electrodynamics, Yang-Mills theory, Dirac theory, Higgs theory, etc) only finitely many types of singularities occur, and, therefore, it is possible to isolate all singularities and then absorb them in the redefinition (or the renormalization) of the classical parameters of the fields (like mass, coupling constant, the field itself etc.). This means that one classifies all Feynmann diagrams according to their potential for the divergences. The high-order diagrams will usually converge but all of them have some diverging subdiagrams. The problem is complicated even further by the overlapping of the divergences from one subdiagram with another. All this can be done and has been done. It is not easy. People like Feynmann and Schwinger got Nobel prizes for doing that in quantum electrodynamics and t Hooft got his Nobel prize for doing the same thing in Yang-Mills theory. If this is possible, then the theory is called renormalizable. In such theories one can compute all quantities of interest and compare them with the experiments, which makes them consistent quantum field theories. The meaning of the renormalizability is that the physics at low energies does not depend on the unknown physics at high energies. All other theories are called non-renormalizable. In such theories the number of different types of divergences is infinite and it is impossible to get rid of all of them by redefining only finitely many parameters of the classical fields. Unfortunately, this is a generic case, and General Relativity is a perfect example of a non-renormalizable theory. In such theories the details of the physics at high energies impact the physical phenomena at low energies. Now, one could think of quantum field theory as a deformation of the quantum mechanics by relativizing it or the deformation of special relativity by quantizing it. In either way, in the limits 0 and c one should recover the classical mechanics Classical Gravity and General Relativity Classical Newtonian Gravity is a theory of gravitational interaction of massive objects. In the classical theory the gravitational phenomena are described by a scalar gravitational field. Every massive body creates a gravitational field whose gradient determines the force exerted by the gravitational field on another massive body. This interaction is instantaneous, in other words, it propagates with infinite speed. The main equation of Newtonian gravity is an elliptic partial differential equation, that is, there is no time, it is static. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 13

19 12 CHAPTER 1. INTRODUCTION Einstein showed that the classical theory of gravitation contradicts special relativity and found a way to modify the gravity theory so that it becomes relativistic. One of the most important physical principles of general relativity is the Equivalence Principle that asserts that locally the effects of the gravitational field is equivalent to the acceleration of the reference frame. That is why, he required that the equation of physics should be invariant not only under linear Lorentz transformations but also under general coordinate transformations (or diffeomorphisms). This lead him to introduce the pseudo-riemannian metric, the connection and the curvature of the space-time. Therefore, according to Einstein gravity is described not by a single scalar field but by a metric, which is a symmetric 2-tensor, and all gravitational phenomena are the manifestations of the curvature of the spacetime. In particular, there is no instantaneous gravitational interaction; instead, all massive bodies move along the geodesics of the curved space-time. One could think of general relativity as a deformation of the special relativity such that the special relativity is recovered as G 0 (no gravity). In the limit as c general relativity turns to a non-relativistic classical gravity theory Quantum Gravity General relativity is constructed by using the following fundamental objects and concepts: 1. events, 2. spacetime, 3. topology of spacetime, 4. manifold structure of spacetime, 5. smooth differentiable structure of spacetime, 6. diffeomorphism group invariance, 7. causal structure (global hyperbolicity), 8. dimension of spacetime, 9. (pseudo)-riemannian metric with the signature ( + +), 10. canonical connections on spin-tensor bundles over the spacetime. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 14

20 1.1. FUNDAMENTAL PHYSICS 13 Every attempt to quantize general relativity immediately encounters many problems such as: 1. what is the space-time? 2. how relevant are the space-time concepts of general relativity? 3. what exactly should be quantized (metric, connection, topology)? 4. should one consider connection an independent quantum object? 5. is it possible to use the standard interpretation of quantum mechanics? 6. is it possible to quantize gravity separately from all other interactions? 7. how much of the space-time structure should remain fixed: topology, smooth structure, causal structure, etc? 8. are the continuum concepts of standard theory valid at ultra-small scales? 9. is space-time discrete? 10. does the continuum structure appear only in coarse-grained sense? 11. does the topology change? 12. what exactly is quantum (does fluctuate)? 13. what is the dimension of the space-time? 14. can the signature of the metric change? There are many approaches to quantum gravity based on how they answer the above questions. The very first one is the attempt to quantize general relativity in the same way as an ordinary quantum field theory. We fix the topology and decompose the metric in the Minkowski metric and a fluctuation, which describes the propagation of gravitons, massless particles of spin 2. This is the minimalistic approach since we do not change anything else, including the interpretation of quantum mechanics. The main problem with this approach is that general relativity is non-renormalizable and, therefore, this approach in its simplest form just does not work. One of the reasons for non-renormalizability of general relativity is its non-polynomial behavior. There are infinitely many types of interaction of Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 15

21 14 CHAPTER 1. INTRODUCTION gravitons with each other. The simplest way to see why general relativity is nonrenormalizable is to look at the dimension of its coupling constant (which is G, the Newton constant), [G] = 1, (1.8) {?} M2 where M is the unit of energy (or mass). This means that the expansion in powers of G produces more and more divergent diagrams. One can modify the theory by including some other fields and some extra dimensions and require very specific interactions, eventually, this approach led to the string theory. This is the approach of high-energy physicists to the gravitational problem. Other approaches to quantum gravity include: loop quantum gravity, dynamical triangulations, non-commutative geometry and others. 1.2 Fields and Particles Quantum field theory (QFT) is a theory of relativistic fields for describing the properties of elementary particles and their interaction. Quantum fields enable to describe such physical phenomena as creation and annihilation of elementary particles, and their transformation in each other. The relativistic field is, in fact, a typical example of a continuum system. However, it can be described by a discrete mechanical system with infinite many degrees of freedom, so called field oscillators. This enables one to quantize the classical fields by associativing to the field some discrete quanta of energy, which corresponds to different energetic states of the field oscillators. The elementary particles have some specific spins, internal angular momentum, electric charge and other characteristics. They are identified with the quanta of corresponding relativistic fields. The kinematic properties of elementary particles are described by quantum theory of free (noninteracting) fields. The quantum theory of interacting fields is the theory of the interaction of elementary particles. The QFT is a quantum relativistic theory. It contains two fundamental physical constants, reflecting these properties: the speed of light c and the Plank constant. The more ambitious quantum gravity is a further generalization (not completed yet) and contains in addition the Newton gravitational constant G. QFT is the theory of elementary particles Properties of elementary particles Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 16

22 1.2. FIELDS AND PARTICLES 15 Mass Spin Electric charge Lifetime, stable and unstable particles Transmutation of particles Electromagnetic, weak, strong, gravitational interactions Conservation laws and symmetries Noether theorem External conservation laws: energy, momentum, angular momentum Internal conservation laws: electric charge, barion charge, color charge, etc Particle-field dualism Fields have infinitely many degrees of freedom Dynamics and interactions of particles are described by relativistic quantum fields Quantum field enables one to describe all possible states of (infinitely) many particles by one object Quantum fields are obtained from classical fields by quantization Whereas a classical field is a function, a quantum field is an operator acting on a Hilbert space This allows to describe the transmutation of particles To each degree of freedom of a quantum field corresponds a harmonic oscillator Quantum field is interpreted as an infinite collection of oscillators (quants, particles) corresponding to various energetic states of the oscillators. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 17

23 16 CHAPTER 1. INTRODUCTION Electromagnetic field is described by a real vector field. They have zero mass, do not electric charge and have spin 1. Particles with zero mass move with the speed of light Massive particles move with speeds less than the speed of light. Neutral particles are described by real fields Particles with electric charge are described by complex fields Particles without spin are described by scalar fields Particles with integer spin are described by vector and tensor fields Particles with half-integer spin are described by spinor and spin-tensor fields Free particles are described by linear wave equations Interacting particles are described by nonlinear wave equations. Lorentz transformations Lorentz group and its subgroups Poincare group Representations of groups Representations of Lorentz group Tensor and spinor representations Examples: scalars, vectors, covectors, spinors, 2-tensors Space reflections Pseudo-tensors: pseudo-scalars, pseudo-vectors, Parity Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 18

24 Chapter 2 Relativistic Invariance 2.1 Spacetime. Classical mechanical systems have finite degrees of freedom, i.e. they are described by a finite set of configuration variables (e.g. coordinates of the particles in the space), q i (i = 1,, N), which are functions of time: q i = q i (t). The field, in contrary, is described by a finite set of functions over the space ϕx A, (A = 1,, D), x denoting a point in the space, which are functions of time ϕx A = ϕ A (t, x). The number of degrees of freedom of a field is, therefore, proportional to the number of the points x of the space, which is, of course, infinite. Thus, the field has inifinitely many degrees of freedom and is an example of an infinitely-dimensional system. It is helpful to treat the space argument of the field just as on a continuous label, i.e. to replace i = (A, x) and q i (t) = ϕ A (t, x). So, the fields are just functions of the time and space coordinates. The collection of the time t together with space coordinates x = (x, y, z) define an event with coordinates x = (x 0, x 1, x 2, x 3 ), where x 0 = t, x 1 = x, x 2 = y, x 3 = z. The set of all events determines the spacetime M, one of the basic object of any physical theory. Although there are only three physical space coordinates, sometimes there is a need to consider physical models in spaces of lower or higher dimension. That is why we will assume that there are d 1 space coordinates so that x = (x 1,..., x d 1 ) and x = (x 0, x 1,..., x d 1 ). To enumerate the space coordinates we will always use the small Latin index, x i, (i = 1, 2,..., d 1) and for the spacetime coordinates we use the small Greek one, x µ, (µ = 0, 1, 2,..., d 1). The spacetime should be endowed with a metric, i.e. a rule for calculating the 17

25 18 CHAPTER 2. RELATIVISTIC INVARIANCE spacetime interval, i.e. the distance, between two close points x and x + dx, ds 2 = η µν dx µ dx ν (2.1) 1.2 where η = (η µν ) is a symmetric d d matrix. Here and everywhere below we use the usual convention that one should perform a summation over repeated (dummy) indices. In the special theory of relativity it is postulated to be the Minkowski metric. If the coordinates x are Cartesian, then it has the form η =.... (2.2)?1.1? or, in short, η = diag ( 1, 1,, 1). The metric of this kind is called pseudo- Euclidean metric. The spacetime interval can be positive, negative or zero. The corresponding vector dx = (dt, dx) is called space-like, time-like or light (also null or isotropic) vector. The light vectors form the light cone, so that the timelike vectors lie inside and the spacelike outside it. 2.2 Poincarè and Lorentz Groups The principle of relativistic invariance states that all systems of coordinates are physically equivalent. This means that all physical observables should be invariant under the Poincaré group Poincaré Group Let us remind the definition of the Poincaré group. The set of inhomogeneous linear transformations g of the coordinates or in matrix form x µ = Λ µ αx α + a µ (2.3)?1.3? x = Λx + a leaving the interval (2.1) invariant is called the general Poincaré group (or the general inhomogeneous Lorentz group) and will be denoted by P. The invariance of the spacetime interval means (2.4)?1.3a? Λ µ αη µν Λ ν β = η αβ or Λ T ηλ = η, (2.5) 1.5 Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 19

26 2.2. POINCARÈ AND LORENTZ GROUPS 19 where T denote the transposition. So, an element of the Poincaré group g is determined by a matrix Λ = (Λ µ ν), satisfying the condition (2.5), and a vector a = (a µ ) The identity transformation is given by g = (Λ, a). (2.6)?1.47? e = (I, 0), (2.7)?1.48? where I is the identity matrix. It is easy to compute the product of two transformations and the inverse g 1 g 2 = (Λ 1 Λ 2, a 1 + Λ 1 a 2 ) (2.8) 1.49 g 1 = (Λ 1, Λ 1 a). (2.9)?1.50? Group of Translations It is obvious that the transformations of the coordinates i.e. the elements of the Poincaré group of the form x = x + a, (2.10) {?} τ = (I, a) (2.11) {?} form a subgroup of the Poincaré group called the group of translations and denoted by T. From (2.8) it is also clear that this group is commutative, or Abelian, τ 1 τ 2 = τ 2 τ 1 = (1, a 1 + a 2 ). (2.12) {?} Lorentz Group The set of homogeneous linear transformations x = Λx, (2.13) 1.6 Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 20

27 20 CHAPTER 2. RELATIVISTIC INVARIANCE with Λ satisfying the condition (2.5), consists of the elements of the Poincaré group of the form l = (Λ, 0) (2.14) {?} and forms another subgroup of the Poincaré group, called the general Lorentz group L. It is isomorphic to the group O(1, d 1) of matrices satisfying the condition (2.5): L O(1, d 1). This group is non-commutative (or non-abelian). From the condition (2.13) we have and, therefore, Besides Hence ( det Λ) 2 = 1 (2.15)?1.7? det Λ = ±1. (2.16)?1.8? (Λ 0 0) 2 + δ ik Λ i 0Λ k 0 = 1. (2.17)?1.9? Λ 0 0 = ± 1 + δ ik Λ i 0Λ k 0. (2.18)?1.10? An connected component of a Lie group is a subset (not necessarily a subgroup) such that all transformations from this subset can be transformed into each other by a continuous transformation. A connected component of a Lie group that contains the identity element is a subgroup called its proper subgroup. The general Lorentz group has four connected components I. Proper Lorentz group L I. L = {L I, L II, L III, L IV }. (2.19) {?} det Λ = +1, Λ 0 0 > 0. (2.20)?1.11? It contains obviously the identity transformation and all the pseudo-orthogonal rotations. Λ = I (2.21)?1.12? Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 21

28 2.2. POINCARÈ AND LORENTZ GROUPS 21 II. Products of proper Lorentz transformations and the time reflection L II. det Λ = 1, Λ 0 0 < 0. (2.22) 1.18 This component contains the products of proper Lorentz transformations and the reflection of the time coordinate T : x 0 x 0 = x 0. (2.23) 1.19 given by the matrix Λ(T) = (2.24) 1.20 III. Products of the proper Lorentz transformations and the space reflection L III, det Λ = 1, Λ 0 0 > 0. (2.25)?1.22? These transformations are the products of the pseudoorthogonal rotations from the proper Lorentz group and the reflection of one space coordinate, say x 1, P : x 1 x 1 = x 1 (2.26) {?} given by the matrix Λ(P) = (2.27)?1.24? IV. Products of the proper Lorentz transformations with the time reflection and the space reflection L IV, det Λ = +1, Λ 0 0 < 0. (2.28)?1.25? Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 22

29 22 CHAPTER 2. RELATIVISTIC INVARIANCE These are the products of the proper Lorentz transformations with the time reflection T and one reflection P of the space coordinate given by the matrix Λ(TP) = (2.29)?1.27? The proper Lorentz subgroup L 1 together with the component L IV form a subgroup of the general Lorentz group S O(1, d 1) = {L 1, L IV } (2.30)?1.26a? with the property det Λ = 1. (2.31) {?} The proper Lorentz subgroup L 1 together with the component L III form another subgroup of the general Lorentz group L + = {L 1, L III } (2.32)?1.26b? with the property Λ 0 0 > 0. (2.33) {?} This subgroup is called the full (complete) orthochronous Lorentz group. It is clear that the proper component of all subgroups of the Lorentz group is the proper Lorentz group. The Lorenz group has also an Abelian discrete subgroup of all reflections Γ = {1, T, P, T P} ; (2.34) {?} obviously, T P = PT, T 2 = P 2 = (T P) 2 = I. (2.35)?1.45? The relation between different connected components of the Lorenz group is illustrated in Fig. 2.1 Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 23

30 2.3. POINCARÈ AND LORENTZ ALGEBRAS 23 P L II L IV T T T P T P L 1 P L III fig2b Figure 2.1: General Lorentz group 2.3 Poincarè and Lorentz Algebras The infinitesimal Poincaré transformations read where g = e + ω, (2.36)?1.56? ω = (ɛ, a), (2.37)?1.57? a is an infinitesimal vector and ɛ = (ɛ µ ν) is an infinitesimal matrix satisfying the condition or meaning where ɛ µ αη µβ + η αµ ɛ µ β = 0, (2.38)?1.14? ɛ τ η + ηɛ = 0, (2.39) 1.58 ɛ µν = ɛ νµ, (2.40) 1.16 ɛ µν = η µα ɛ α ν. (2.41)?1.17? Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 24

31 24 CHAPTER 2. RELATIVISTIC INVARIANCE The number of independent components of an infitisemal vector a is, obviously, equal d. Thus the dimension of the group of translation is equal to d dim T = d. (2.42) {?} The Lorentz transformations are determined by the antisymmetric matrix ɛ µν which has d(d 1)/2 independent parameters. Therefore, the dimension of the Lorentz group is d(d 1) dim L =. (2.43) {?} 2 Therefore, the dimension of the Poincaré group is dim P = d + d(d 1) 2 = d(d + 1). (2.44) {?} 2 Any infinitesimal Poincaré transformation can be presented, hence, in the form ω = 1 2 ɛαβ M αβ + a γ P γ = λ a X a, (2.45)?1.59? where λ a = (ɛ αβ, a γ ) (2.46) {?} are the infinitesimal parameters and X a = (M αβ, P γ ) (2.47)?1.61? are the generators, which have the form M αβ = M βα = (M µ ναβ, 0) (2.48)?1.62? with M µ ναβ = δ µ αη βν δ µ β η αν = 2δ µ [α η β]ν. (2.49)?1.63? and P γ = (0, P µ γ) (2.50)?1.64? with P µ γ = δ µ γ. (2.51)?1.65? Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 25

32 2.4. TENSOR FIELDS 25 The generators M αβ and P α of the Poincaré group satisfy the commutation relations [M αβ, M γδ ] = η αγ M βδ η βδ M αγ + η αδ M βγ + η βγ M αδ, (2.52) 1.66 [M αβ, P µ ] = η µα P β η µβ P α, (2.53)?1.67? [P α, P β ] = 0, (2.54) 1.68 and form the Lie algebra of the Poincaré group, called the Poincaré algebra. It is immediately seen that the generators of translations P α form an Abelian subalgebra (2.54) and the generators of Lorentz transformations form the non- Abelian Lorentz algebra (2.52). The infinitesimal form of pure Lorentz transformations is Λ = I + ɛ = I ɛαβ M αβ. (2.55)?1.13? Exercise. Obtain the structure constants of the Lorentz group. [M αβ, M γδ ] = 1 2 Cµν αβγδm µν. (2.56)?1.69? Answer: C µν αβγδ = 8δ [µ [δ δµ] [α η β]γ] = δ ν δ δµ αη βγ + (2.57)?1.70? 2.4 Tensor Fields Representations of the Lorentz Group Let us consider now a set of some smooth functions over the spacetime ϕ(x) = ϕ A (x) (2.58)?1.71? This set of functions defines a field if it transforms according to some specific rule under transformation of coordinates from the Poincaré group x µ x µ = Λ µ νx ν + a µ. (2.59) 1.72x Namely, to the transformation of the coordinates (2.59) it is assigned a homogeneous linear transformation of the field components ϕ(x) ϕ (x ) = D(Λ)ϕ(x) (2.60)?1.73? where the operator D(Λ) is completely determined by the matrix Λ. Ivan G. Avramidi: Lectures on Quantum Field Theory (2016), NMT intro-qft2.tex; May 4, 2016; 18:30; p. 26