Relativistic Quantum Fields

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Relativistic Quantum Fields

¯b

W⁺

¯t

¯ s

g

¯s s

¯t

Lecture notes

Based on the course given by Riccardo Rattazzi

Johnny Espin

2012

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Foundations and Matter

5

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Chapter 1

Introduction

Quantum field theory is the theoretical “apparatus” that is needed to describe how nature works at its most fundamental level, that is to say at the shortest distances. Fundamental also means simplicity as opposed to complexity. We shall see towards the end of this course, after having digested the necessary set of mathematical and physical concepts (in a simple word: formalism), that the laws that govern particle physics can be written in a few lines with absolute precision and great empirical adequacy. In a sense particle physics is orthogonal to the sciences that deal with complexity.

1.1 What is quantum field theory?

Technically, quantum field theory (QFT) is the application of quantum mechanics to dynamical systems of fields, very much like basic quantum mechanics concerns the quantisation of dynamical systems of particles. If quantum mechanics deals with mechanical systems with a finite number of degrees of freedom, quantum field theory describes the quantisation of systems with infinitely many degrees of freedom. Specifically in this course, we are more interested in relativistic QFT. Relativistic QFTs explain and describe the existence of particles and their mutual interactions. Indeed, the fact that nature at its most basic level consists of particles can be viewed merely as a consequence of relativistic QFT. The domains of application of the latter in modern physics are quite broad: from the study of collisions among elementary particles in high energy accelerators, to early Universe cosmology. For instance, the primordial density fluctuations that later gave rise to structures like galaxies, the origin of dark matter or black-hole radiation are all described by relativistic QFT. Nevertheless there are also applications of quantum field theory to non-relativistic systems, in particular in condensed-matter physics: superfluidity, superconductivity, quantum Hall effect,. . .

1.2 Why quantum field theory?

We saw what QFT describes, but why do we need a field theory to describe particles? Is this a necessity? It basically is: relativistic QFT is the only way to reconcile ordinary quantum mechanics with special relativity. We can understand this necessity with several qualitative arguments:

• Special relativity postulates the existence of a limiting speed. This implies that instantaneous interactions between two particles separated in space are not possible. One needs a medium (a field) to “propagate”

interactions. For example, two boats on the Lac Léman affect each other through the waves they generate at the surface of water. The first example is the electromagnetic field. So basically, interactions require field dynamics. However, in classical electrodynamics, the fields’ sources, the charged particles, are ordinary mechanical point-like objects and are thus not associated to fields. With the addition of quantum mechanics, the only way things can be made compatible (consistent) is if matter as well is described at its basic level by fields.

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The need to associate matter particles as well to fields, or in other words, the failure of the point-like picture in relativistic quantum mechanics is hinted by the following heuristic arguments:

• If one tries to generalise the Schrödinger equation to make it relativistically invariant in the most obvious way, one is forced with a Hamiltonian which is unbounded from below, i.e. negative energy states appear.

The stability of the system is jeopardised!

+i~∂t+~²∇² 2m

ψ(t, x) = 0⇒ −~²∂_t²+c²~²∇²+m²c⁴

ψ(t, x) = 0 (1.1) E= ~²k²

2m ⇒E=±cp

mc²+~²k² (1.2)

• More indirectly, if one somehow disposes of the extra negative energy states, then one finds that with the single particle picture there is trouble implementing causality¹.

+i~∂t+~²∇² 2m

ψ(t, x) = 0⇒

+i~∂t−p

c²~²∇²+m²c⁴

ψ(t, x) = 0 (1.3) E=~²k²

2m ⇒E= +cp

mc²+~²k²>0 (1.4)

but:

hy|e⁻ⁱ^Ht/^ˆ ^~|xi=A(x→y, t) (1.5) is non-zero whenxandy are not causally related!

There is however a very basic reason why at the fundamental level a single particle description cannot work. Indeed the famous Einstein relationE = mc² tells us that energy and mass are equivalent. Therefore, in a relativistic process, we have no right to assume that part of the energy cannot be used to create extra particles (e.g.electron- positron pairs). Quantum mechanics makes this possibility more urgent and relates to the fundamental properties of matter. Consider indeed the uncertainty principle with relativity for a particle of massm:

Relativity: ⇒ E=cp

p²+m²c² (1.6)

Quantum mechanics: ⇒ ∆x∆p&~ (1.7)

so that if:

∆x < ~

mc=λ_Compton (1.8)

then:

∆E > mc² (1.9)

In other words, if we try to localise a particle to a distance shorter than its characteristic Compton wavelength, it becomes relativistic and the excess energy can lead to the production of extra particles! The production of these extra particles implies that the one-particle states of ordinary quantum mechanics stop making sense for localisations smaller than ∆x'λCompton. Empirically, it is a fact that particles are indeed created and destroyed in fundamental processes.

1Causality: information cannot travel faster than light.

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1.3 Units of measure

There are three dimensionful quantities in mechanics (CGS):

Length: [L] → cm (1.10)

Time: [t] → s (1.11)

Mass: [m] → g (1.12)

Energy and momentum are not fundamental quantities and are expressed in terms of the above three:

Energy: [E] → erg=g×cm s

²

(1.13)

Momentum: [p] → g×cm

s (1.14)

In relativistic QFT we have fundamental equations coming from both special relativity and quantum mechanics.

The first unifies space and time through the speed of light:

c[t] = [L], c= 2.99×10¹⁰ cm

s (1.15)

and the second associates a (wave-)length to a momentum through the Planck constant:

[L] = ~

[p], ~= 1.05×10⁻²⁷ erg·s (1.16)

Since these two theories form the basic pillar of QFT, it is natural to choose a system of units in which:

c= 1 =~ (1.17)

so that:

[t] = [L] = 1 [p] = 1

[E] (1.18)

In particle physics, it is customary to measure all quantities in units of an energy unit. We shall use as unit the GeV = 10⁹ eV. Using 1eV = 1.6×10¹² erg, we can compute as an exercise the following relations:

Unit QFT system to CGS Length GeV⁻¹= 1.97×10⁻¹⁴ cm

Time GeV⁻¹= 6.58×10⁻²⁵ s Mass GeV = 1.78×10⁻²⁴g Action GeV⁰ (~= 1) Velocity GeV⁰ (c= 1)

(1.19)

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We understand why it is equivalent to talk either about high-energy physics or about short-distance physics.

High-energy accelerators are truly huge microscopes testing the short distance behaviour of matter and forces!

1.4 Overview of particle physics

The goal of the course is obviously to learn the basics of QFT, nonetheless before that it would be helpful to receive an overview of present understanding of the micro world.

There are four well known fundamental forces:

1. The electromagnetic force which is responsible for the stability of atoms, namely for binding nuclei and electrons into atoms. This is a long range force.

2. The weak force: a short-range force (L ≈ 10⁻¹⁶ cm) which is responsible for the beta-decay of nuclei, neutrons and for muon-decay:

n → p + e⁻ + ¯ν_e (1.20)

µ⁻ → e⁻ + ¯ν_e + ν_µ (1.21)

3. The strong force is responsible for the existence of nuclei (for their stability). Also, for instance, more than 99% of the mass of nuclei is due to the strong force alone. The latter explains most of our mass. The strong force is also short-ranged (L≈10⁻¹³cm)

4. The gravitational force: it is very weak, however it is long ranged and universally attractive. For these reasons, gravity plays nonetheless an essential role in our life. In order to understand how weak gravity is compared to the other forces, we can calculate the ratio between the gravitational and electromagnetic forces’ amplitude between two protons:

FG

F_E ' G_Nm²_p

e² ∼10⁻⁴⁰ (1.22)

We see immediately that the difference is huge!

As we shall learn, in QFT forces are associated to fields and thus to particles. The electromagnetic force is mediated by the photonγ, the weak force by theW^± andZ⁰and the strong force by eight gluonsg. These particles are all helicity±1 vector bosons. On the other hand, gravity is mediated by the graviton which is a helicity±2 boson.

This is what makes gravity so different from the other forces. In addition, there is a mysterious “Higgs” force that remains to be studied. It might be mediated by one (or more) scalar particle (spin 0).

We saw which forces are present in Nature as far as we know. They all are mediated by bosonic particles. All other fundamental particles known so far arefermions with spin 1/2 and come into two classes:

1. The leptons: they only feel the electromagnetic and weak forces. They come into three distinct generations.

1^st 2^nd 3^rd

Q=−1 electron e⁻ me= 0.511 M eV

muon µ⁻ mµ= 105 M eV

tau τ⁻ mτ = 1.777 GeV

Q= 0 νe νµ ντ

(1.23)

The masses of the neutrinos are non-zero but very small (mν .1 eV). In addition, every particle in this table comes in pairs with its antiparticle.

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2. The quarks: they also feel the strong force. Because of this, quarks are strongly bound intohadrons and are always confined into these ones. We have basically two types of hadrons: the baryons (like the proton) made up of three quarks and themesons (like the pions) made up of a quark and an antiquark. They are also organised in three families.

1^st 2^nd 3^rd

Q= 2/3 up u

mu'1.5−3 M eV

charm c mc'1.25±0.1GeV

top t mt'171.4±2 GeV Q=−1/3 down d

md'3−7M eV

strange s ms'95±25M eV

bottom b mb'4.20±0.07GeV

(1.24)

We should understand how these observational facts fit into a coherent theoretical picture. More pragmatically, how well we describe the observed particles and their natural interactions. Particle physics, to a beginner, often gives the impression of being all about “finding particles and classifying them” much like botanics. Nothing could be more wrong than that! What is being sought for are indeed the fundamental laws of Nature. Underlying the above empirical fact, there is a well developed theoretical understanding; in certain aspects, our understanding is truly remarkable, while open problems and some mystery remain. What we understand well is the structure of the electromagnetic, weak and strong forces, and quarks and leptons are successfully described by one QFT called the Standard Model (SM). What do we mean by understanding something? Understanding means also being able to describe and predict: for example, some of the existing particles (the weak bosons the charm and the top quarks) have been predicted from logical (or mathematical) consistency of the theory before their experimental discovery.

Also, the prediction of some observables is made and verified to a level of accuracy which is unmatched in any other field. One of the best results of our QFT formalism is the derivation of quantum effects in electrodynamics.

One example is the study of the magnetic moment of a particle. The latter is proportional to the spin and is given by:

~ µ=g e

2m~s (1.25)

where g is the gyromagnetic ratio. Classically, we expect g = 1; for a relativistic electron in Dirac’s theory, we haveg= 2. In QED, however, we can compute the digression from the value of 2:

g−2

2 = α

2π+c2

α 2π

² +c3

α 2π

³

+. . . (1.26)

The coefficients ci are predicted by QED and are of order one. Alpha (α) is the fine structure constant of the theory and is given by:

α= e²

4π₀~c ∼ 1

137 (1.27)

The digression ong is measured experimentally with very high accuracy:

g−2 2

exp

= 1159652180.85(76)×10⁻¹² (1.28)

Once this quantity is measure, we can use it to computeαwith precision and then use this value to test the theory with other observable:

g−2 2

exp

→ α⁻¹= 137.035999705(36) (1.29)

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and for example, from atomic clocks (Rb and Cs):

α⁻¹(Rb) = 137.03599878(91) (1.30)

These results are a great confirmation that QFT is the correct description of the subatomic world. The success of QFT and of the SM make the open problems in the theory only more exciting! Here is a list if what we understand less well in decreasing order of understanding:

1. The SM necessarily predicts the existence of a fifth force to explain the origin of the mass of the weak bosons as well as of the quarks and leptons. However, there are various possibilities in nature for this necessary force. The simplest is that it is mediated by a boson of spin zero, a scalar particle called the Higgs boson.

However the presence of this boson has not been experimentally confirmed² and moreover many physicists question this simple picture proposing alternative theories. This is a very exciting time because before the end of this course, the Large Hadron Collider (LHC) at CERN should have had announced its discovery or it dismissal.

2. Why are there three families of fermions?

3. What is the reason for the big ratios of masses?

4. What is the role of gravity? As far as we understand, gravity makes sense as a quantum field theory only at sufficiently low energy. A full quantum description is however missing. Possibly gravity will require to go beyond QFT (e.g.String theory or other alternative Quantum Gravity theories). The problem of gravity is not necessarily urgent. Indeed, the needed energy cannot be reached in any imaginable laboratory. It is however crucial to develop a theory of the very early Universe to answer question such as “How did the Universe start?” or “Was it inevitable?”. During our course, we shall neglect gravitational perturbations and work with a static Minkowski metric:

gµν →ηµν = diag(+1,−1,−1,−1) (1.31)

After this short introduction (for a very broad subject), we shall start learning its technicalities.

2As of when these notes were typed.

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Chapter 2

Classical Field Theory

2.1 Lagrangian mechanics

Non-relativistic quantum mechanics, in Schrödinger’s equation formulation, is based on the Hamiltonian. In relativistic quantum field theory one instead normally uses the Lagrangian formulation. One reason for this, which will become clear below, is that in the Lagrangian approach relativistic invariance is manifest, while in the Hamiltonian it is not. In other words, the Hamiltonian depends on the reference frame while the Lagrangian does not.

According to the Lagrangian formulation of mechanics, the dynamics of a system ofN degrees of freedom (generalised coordinates)q1, . . . , qN is characterised by a function:

L({qi},{q˙i}) (2.1)

When dealing with a system which evolves from timet_i to timet_f one defines the action:

S[{qi}]≡ Z tf

ti

L({qi},{q˙i})dt (2.2)

The action is afunctional: a real valued function of the trajectory. By the least action principle, the time evolution of the system is given by the trajectory(-ries) ¯q_i that extremises the action:

δS[{q¯_i}] = 0 (2.3)

Let us derive what the equations of motion look like in the Lagrangian formulation. Eq.2.3implies that if ones considers a tiny deviation around the trajectory, the variation of the action will be zero:

δS[{q¯i}] =S[{¯qi+δqi}]−S[{q¯i}] = 0 (2.4) Using the integral form of the action, this variation is given by (at first order):

δS[{q¯i}] = Z t_f

t_i

dt ∂L

∂qi

δqi+ ∂L

∂q˙i

δq˙i

= Z tf

ti

dt ∂L

∂q_i − d dt

∂L

∂q˙_i

δq_i+ ∂L

∂q˙_iδq_i

t_f

t_i

(2.5)

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where the second term vanishes when considering the most general variations with fixed boundariesq(ti) andq(tf).

Extremality is then equivalent to:

∂L

∂qi

− d dt

∂L

∂q˙i

= 0 (2.6)

This set of N second order differential equations for the generalised coordinates is called the set of Lagrange equations. Equivalently, the dynamics of the system can be described in the Hamiltonian formulation where one ends up with 2N first order differential equations for the generalised coordinates and momenta. The Hamiltonian is defined as the Legendre transform of the Lagrangian with respect to the generalised momenta:

pi≡ ∂L

∂q˙i

(2.7) This relation is then inverted and the velocities are expressed in terms of the coordinates and momenta:

˙

qi→q˙i(qi, pi) (2.8)

The Hamiltonian is finally given by:

H≡H(q_i, p_i, t) =p_iq˙_i−L(q_i,q˙_i) (2.9) Notice that we use throughout the whole course Einstein’s summation convention. The Lagrangian equations of motion read in the Hamiltonian formalism:

˙ qi= ∂H

∂p_i

˙

pi=−∂H

∂qi

(2.10)

Another important mathematical notion in Hamiltonian mechanics is thePoisson bracket of the functions of the coordinates and momenta:

{A, B} ≡ ∂A

∂p_i

∂B

∂q_i −∂A

∂q_i

∂B

∂p_i (2.11)

The Poisson brackets allow us to write Hamiltonian mechanics inalgebraicform,e.g.:

˙

q_i={H, q_i} (2.12)

˙

pi={H, pi} (2.13)

while for any observableO({qi},{pi}), the time dependence is:

d

dtO= ∂O

∂pi

˙ pi+∂O

∂qi

˙ qi

=−∂O

∂pi

∂H

∂qi

+∂O

∂qi

∂H

∂pi

={H, O}

(2.14)

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Similarly, the Poisson brackets encapsulate the group structure of the symmetries. This algebraic property of Hamiltonian mechanics is unchanged when going to the quantum description of the system:

Classical Quantum {·,·} [·,·]

O({qi},{pi}) Oˆ({qˆi},{pˆi})

(2.15)

For example for the angular momentum operators’ algebraic structure:

{Li, Lj}=ijkLk → [Li, Lj] =i~ijkLk (2.16)

2.2 From mechanics to field theory

The notion of afield arises when we consider a coordinate spaceX (e.g.Rⁿ,Sⁿ,. . . ) and we associate a number ofdynamical variablesφ_α(x, t) (α= 1, . . . , N) to each pointx∈ X. Thetimecoordinatetencodes the dynamics of the system. From a mechanical perspective, the total number of degrees of freedom is given by N ×dimX, where by dimX we indicate the number of points inX. Of course if X is a continuum, likeRⁿ orSⁿ, then dimX is infinite. For example, consider the electric and magnetic fields:

E(~~ x, t) (2.17)

B(~~ x, t) (2.18)

The coordinate space is hereR³3~x. This is infinite dimensional, both because the points form a continuum and because~xextends to infinity. Thus each of these two fields has an infinite number of degrees of freedom! Ordinary mechanics concerns a finite number of dynamical variablesq_i(t) withi= 1, . . . , M. One can view field theory as a limit of mechanics where the discrete labeliis replaced by (α, x), which is continuous, as long asxis a continuous coordinate. As we talk about quantum field theory, the reader should expect to be dealing with infinitely many degrees of freedom throughout this course. In particular, most of the time the coordinate space will be given by R³ unless specifically indicated. At this stage, the reader should have already dealt with fields other than the electromagnetic ones, in particular, in fluid mechanics:

ρ(~x, t) (2.19)

P(~x, t) (2.20)

~v(~x, t) (2.21)

From now on we will drop the indexαand consider R³ as our coordinate space. The generalisation of the action to the field theory framework is trivial:

S[φ] = Z

dtL φ,φ˙

(2.22)

We want the action of a field configuration to be defined as a global quantity that does not depend on the coordinate space points. Indeed, it should describe the whole “trajectory” regardless of its details. Much like

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the whole electric charge of a system of particles does not depend on where the particles are and how they move through it. Therefore, in field theory, we can define the Lagrangian such that, when evaluated on a specific field configuration, it is only a function of time:

L(t)≡ Z

d³xL

φ(~x, t),φ(~˙ x, t), ~∇φ(~x, t)

(2.23) This involves a new quantity called theLagrangian density L. Using the four-vector notation:

L ≡ L(φ(x), ∂µφ(x)) (2.24)

where we denote the four-vectorx^µ byxfor simplicity as it is clear from context that it is not a one-dimensional coordinate. Finally:

S[φ] = Z

d⁴xL(φ(x), ∂µφ(x)) (2.25)

whered⁴x≡dtd³x. Notice that the Lagrangian density was taken to depend only on fields and their derivatives at each coordinate space point. This property is called locality. It is intuitively obvious that, if a field theory description is to help us to get rid of instantaneous interactions between spacelike separated objects, then its Lagrangian density must be local. Notice also that we stick to Lagrangian densities that depend only on one derivative with respect to the coordinates. This is analogous to the Lagrangian in ordinary mechanics. The systems we will be interested in have this property but it is straightforward to generalise the action principle to higher derivative Lagrangians. Finally, as it is clear from the context of field theory that we will always be dealing with local Lagrangian densities, we will simply call them “Lagrangians” from here onwards.

2.3 Least action principle in field theory

Now that we have written the action in the field theory framework, the least action principle is immediately generalised as well. We consider variations of the action induced by a small variation of the field:

φ(x)¯ →φ(x) +¯ δφ(x), δφ(~x, ti) =δφ(~x, tf) = 0 and δφ(x)^|x|→−∞−→ 0 fast enough (2.26) Then:

0≡δSφ¯

= Z

d⁴x ∂L

∂φδφ(x) + ∂L

∂(∂_µφ)δ(∂µφ)(x)

= Z

d⁴x ∂L

∂φ−∂µ

∂L

∂(∂µφ)

δφ(x) + Z

d³x∂L

∂φ˙δφ(x)

t_f

t_i

+ Z t_f

ti

dt Z

d²Σ~n· ∂L

∂(∇φ)~ δφ(x)

(2.27)

where Σ is the boundary of our coordinate space. The last two terms vanish since the variation on the time boundary is zero and we assume that for Σ → ∞ the variation goes to zero fast enough (i.e., the variation is local). We can therefore neglect the boundary terms and write the variation of the action in terms of its functional derivativein analogy with ordinary derivatives of functions of many variables. If we consider a function of many-variablesf(qi), then:

δf=X

i

∂f

∂qi

δq_i+O(δqi)² (2.28)

In analogy:

δS= Z

d⁴x δS

δφ(x)δφ(x) + O(δφ(x))² (2.29)

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When we evaluate this variation on a “trajectory” it must vanish for any variation of the field and thus:

δS δφ(x)

_φ_¯

≡ ∂L

∂φ −∂µ

∂L

∂(∂_µφ)

_φ_¯

= 0 (2.30)

These are the so-calledEuler-Lagrange equations.

The concept of functional derivatives is obviously straightforwardly generalised to arbitrary space dimensions and to densities that depend on coordinates as well. For example, define a functionalF:

F[φ] = Z

dⁿxf(φ, ∂iφ, x) (2.31)

Then:

δF δφ(x)≡ ∂f

∂φ−∂i

∂f

∂(∂iφ) (2.32)

In particular, we can choose as density:

f(φ, ∂iφ, x) =φ(x)δ⁽ⁿ⁾(x−x0) (2.33)

So thatF[φ] =φ(x0). This yields on one hand:

δF

δφ(x)= δφ(x₀)

δφ(x) (2.34)

and on the other hand:

δF

δφ(x) = ∂f

δφ(x)=δ⁽ⁿ⁾(x−x₀) (2.35)

Yielding finally an important generalisation of _∂q^∂qⁱ

j =δⁱ_j: δφ(x0)

δφ(x) ≡δ⁽ⁿ⁾(x−x0) (2.36)

Notice further than in the Lagrangian description of field theory, time and space coordinates are treated on equal grounds. This is an advantage of the Lagrangian formalism when considering relativistic field theory!

2.4 Hamiltonian formalism

The Hamiltonian formalism can be developed along the same lines as in ordinary mechanics. One defines the canonically conjugated momentaπ(x) to the fieldφ(x):

π(x)≡∂L

∂φ˙ (2.37)

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and then inverts this relation to express the time derivative of the field as a function of the momenta and fields¹:

φ(x)˙ →φ(φ(x), π(x))˙ (2.38)

The Hamiltonian is defined in analogy with mechanics by a Legendre transform:

H(π, φ) = Z

d³x π(x) ˙φ(x)− L(φ, π)

(2.39) where:

H=≡π(x) ˙φ(x)− L(φ, π) (2.40)

is the Hamiltonian density. The Hamiltonian is a local functional of the momenta and fields over the three dimensional space! As expected, in the Hamiltonian formalism the time coordinate has been singled out so that in this formalism relativistic invariance is no longer manifest. Also, as we work with functionals, the formalism we developed in the previous section about functional derivatives becomes handy. “Equal time” Poisson brackets (we make the time dependence implicit from here on) can be generalised to functional by employing the latter. Given two functionals:

A[π, φ] = Z

d³x a(π, φ) (2.41)

B[π, φ] = Z

d³x b(π, φ) (2.42)

we have:

{A, B} ≡ Z

d³x δA

δπ δB

δφ −δA δφ

δB δπ

(2.43) In particular, using Eq.2.36, we have:

{π(~x, t), φ(~y, t)}=δ⁽³⁾(~x−~y) (2.44) {π(~x, t), π(~y, t)}= 0 ={φ(~x, t), φ(~y, t)} (2.45) Similarly to ordinary mechanics, the equations of motion are:

˙

π={H, π}=−δH

δφ =∇ ·~ ∂H

∂(∇φ)~ −∂H

∂φ φ˙ ={H, φ} =δH

δπ = ∂H

∂π

(2.46)

1Note that∇φ(x) is to be considered as a function of~ φ(x)

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Chapter 3

Symmetries

The study of symmetries is of capital importance in theoretical physics as they greatly simplify any problem we have to deal with. For example, in ordinary mechanics the study of balistics, in the presence of gravity only, becomes a two dimensional problem since the system is symmetric under any translation along the direction perpendicular to both the gravitational field and the direction of movement. There are many other possible symmetries in physics (e.g. rotational invariance) and their study falls into the name group theory. Very often (but not always) the symmetry operation involves alinear transformation and this naturally leads us to the idea of finding a set of matrices that realises it. This procedure falls into the topic ofrepresentation theory. Below, we will put these ideas into a mathematical language and study one of the most important groups in relativistic physics as well as the most beautiful consequence of the presence of symmetries in a system.

3.1 Group theory

3.1.1 Groups

Let us start with a short summary of basic group theory. It is highly suggested to any reader, familiar or not with the topic, to have a look at one among any reference book on group theory (e.g.Lie Algebras in Particle Physics by H. Georgi Chap. 1-3).

Definition. A groupGis a set of elements{g_i},i= 1, . . . ,|G|, where|G|is called the order of the group, endowed with a binary operation that assigns to each ordered pair of elements a third element of the group:

◦ :G×G7→G

(g₁, g₂)7→g₁◦g₂=g₃ (3.1)

This binary operation is usually written with a multiplicative notation:

g1◦g2 →g1g2 (3.2)

ForG to be a group, the binary operation must satisfy the following axioms:

1. Associativity:

g1(g2g3) = (g1g2)g3, ∀g1, g2, g3∈G (3.3) 2. Existence of a (left-)identity:

∃e∈G s.t. ∀ g∈G eg=g (3.4)

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3. Existence of a (left-)inverse:

∀ g∈G, ∃g⁻¹∈G s.t. g⁻¹g=e (3.5) In many books, it is required both the identity and the inverse to act also from the right. It is actually enough to have a left-identity/inverse. Indeed, from these three axioms it is easy to prove the following corollaries:

Corollary. The group axioms have the following consequences:

1. The left-inverse is also a right-inverse:

gg⁻¹=e (3.6)

2. The left-identity acts in the same way from the right:

ge=g (3.7)

3. Uniqueness of the identity:

∃!e∈G s.t. ∀ g∈G eg=g=ge (3.8) 4. Uniqueness of the inverse :

∀ g∈G, ∃! g⁻¹∈G s.t. gg⁻¹=e=g⁻¹g (3.9) The proofs are left as an exercise.

Two elementsg₁ andg₂ are said tocommuteifg₁g₂=g₂g₁. Moreover ifg₁g₂=g₂g₁∀g₁, g₂∈G, thenGis called anabelian group.

If|G|is finite, thenGis called afinitegroup. Examples of finite groups are the group of permutations ofnobjects Sn which has ordern! or the cyclic groupZn of order ndefined as¹ the group generated by one single elementg such thatgⁿ =e. The cyclic group is an finite abelian group.

If|G|is infinite obviously the group is said to be infinite. One example the reader should be familiar with is the group of rotations in two dimensionsSO(2). It is trivially infinite as we can label the group elements by gθ and for eachθ∈[0,2π) we have a different rotation. This is an example of aLie group. We will define below what a Lie group is more specifically.

Let us now be more precise about why group theory is relevant in physics. Besides the rotations group in n dimensions, there are many other groups which have great importance. Take for exampleN coordinates{qi}, i= 1, . . . , N. The space of configurations {q_i} is amanifold and there existsfreedom in choosing coordinates (e.g.

cartesian, spherical,. . . ). For example if {q_i} ≡ θ ∈ [0,2π), the manifold is a circle. A change of coordinates corresponds to a change in the “point of view” on the system. Mathematically:

q_i 7→q_i⁰=f_i({q_i}) (3.10)

such that for any configuration there is one and only one{q_i⁰}. Thenf is a bijection and the set of all possible changes in coordinates is a group with respect to the binary operation which is the composition of functions!

Obviously iff andgare two changes of coordinates, thenf◦gis also a change of coordinates. Therefore, to prove that we deal with a group, we need only to check the three axioms that a group is required to obey:

1. Associativity. Letf, g, h be three changes of coordinates and let us work, without loss of generality, with one coordinate ({qi} ≡q). Then:

(f◦g)◦h(q) =f(g(h(q))) =f◦(g◦h(q)) (3.11)

1The cyclic groupZnis isomorphic to the addition of integers modulonand thus can also be defined this way

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2. The identity is simply:

q7→q (3.12)

3. Inverse. As we consider the set of bijective functions, by definition, for all changes of coordinatesf there existsf⁻¹, its inverse, such that:

f⁻¹◦f(q) =q=f◦f⁻¹(q) (3.13)

Hence, the set of changes of coordinates endowed with the operation of functions’ composition is indeed a group.

An important subgroup is the group of changes of coordinates that leave the laws of motion invariant. If we consider Newton’s second law:

d~p

dt =XF~ext (3.14)

The group that leaves the equations of motion invariant is called the Galileo group. Sets of transformations that do not change the form of the laws are called symmetries and in this case, the Galileo group is the group of symmetries which leave classical mechanics invariant! Galileo group consists of three dimensional rotations, uniform translations of spacetime as well as linear uniform boosts:

~x 7→ ~x⁰=R(α, β, γ)~x+~v₀t+~x₀ (3.15)

t 7→ t+t0 (3.16)

Transformations are generates by 10 parameters : α, β, γ, ~v₀, ~x₀ and t₀. Any of them has an infinite number of possible values, for example,~x₀∈R³. It is therefore an infinite group. As we shall see shortly, Galileo group is a Lie group. In the language of Lie groups, it has dimension 10 as it is generated by 10 parameters. Be particularly aware of the fact that the dimension of the Lie group can be finite even if the group itself is infinite, “infinite”

meaning here that it contains an infinite number of elements! From now on we shall always work with a group Gthat describes a symmetry of the system. Moreover, in relativistic QFT the most important groups are all Lie groups. Thus, unless specifically stated, we shall only work with infinite groups that are Lie groups.

Let us start defining concretely what a Lie group is:

Definition. A Lie groupGis a group and at the same time a smooth manifold. It is a manifold because its group elements are labelled by continuous variables which, regarded as coordinates, span the latter. IfGis spanned byN variables then it is anN-dimensional Lie group. It is a smooth manifold because its group operations are smooth in the sense that (without loss a generality we consider a one-dimensional Lie group):

g(α)g(β) =g(γ), γ=f(α, β) (3.17)

g⁻¹(α) =g(i(α)) (3.18)

and bothf andiare smooth functions of their parameters.

The parameters labelling a Lie group can be put into a vector form:

~

α= (α1, . . . , αN) (3.19)

It is a convention to call the identity the group element which parameters correspond to the null vector:

g(~0) =e (3.20)

In general, we work with an explicitrealisation of the group (realisation: how the group acts) and not with the abstract notion of a group (defined through Eq.3.17). A realisation is a concrete way of writing group elements

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in terms of transformations on the same space. For example, consider the one-dimensional colinear group on the real line. A realisation of the group is:

g(α₁, α₂)x=α₁x+α₂, x∈R, α₁>0 (3.21) It is a two-dimensional Lie group which manifold is the upper half-plane ofR².

An important realisation of Lie groups is thematrix representation (we drop the term “matrix” from here on).

Definition. Ann-dimensional representation of a group Gis formally defined to be a homomorphism from Gto GL(V), whereV isn-dimensional vector space:

D : G → GL(V)

g 7→ D(g) (3.22)

such that:

D(g1)D(g2) =D(g1g2), ∀ g1, g2∈G and D(e) =1 (3.23) Then, the realisation of the Lie group on the vector spaceV is as follows:

∀ g∈G, ~v7→D(g)~v∈V, ∀~v∈V (3.24) Let us give a list of important definitions attached to representations:

Definition. Consider a representationD of a group Gon the vector spaceV.

1. D isreducible if there exists on non-trivial invariant subspaceU 6=V,{0} of V. Formally:

∃ U ⊂V, s.t. ∀ ~u∈U, D(g)~u∈U, ∀g∈G (3.25) 2. D isirreducible if it is not reducible. That is, the only invariant subspaces areV itself and{0}.

3. D is completely reducible if and only if we can decompose V into a non-trivial partition of invariant subspaces:

V =U₁⊕U₂⊕. . . (3.26)

In other words, there exists a basis of V such that:

∀ g∈G, D(g) =







D1(g) 0 0 0 D₂(g) 0

0 0 . ..







(3.27)

whereDi is ani×ni matrix, n1+n2+. . .=nand all Di are irreducible. D is said to be adirect sumof irreducible representations:

D=D₁⊕D₂⊕. . . (3.28)

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4. Two representations D1 andD2 areequivalent if:

∃ S∈GL(n,C), s.t. S⁻¹D1(g)S=D2(g), ∀g∈G (3.29) S is said to be a change of basis.

5. D isunitaryif it is equivalent to a representation in which :

D⁻¹(g) =D^†(g), ∀ g∈G (3.30)

The most important representations are those which are irreducible. It is however not a straightforward task to tell, just by looking at it, if a representation is irreducible or not. In order to do this, we shall develop now some tools allowing us to easily classify representations by their reducibility.

3.1.2 Schur’s lemma

The most useful result about the reducibility of representations comes from Schur’s lemma and two corollaries:

Schur’s lemma. Let D₁ andD₂ be two irreducible representations ofGacting onV₁ andV₂ respectively and an intertwining operatorΛ : V₁→V₂ such that:

ΛD₁=D₂Λ (3.31)

Then:

1. Either Λ≡0 orΛ is invertible, in which caseV1 andV2 have the same dimension andD1= Λ⁻¹D2Λ.

2. If V1=V2=V andD1=D2=D thenΛ =λ1.

Now letD be a completely reducible representation andΛan hermitian operator such that:

ΛD=DΛ (3.32)

3. Then Λ is block-diagonal in the same basisD is block-diagonal:

D=







D₁ 0 0

0 D2 0

0 0 . ..







, Λ =







λ₁1 0 0 0 λ21 0 0 0 . ..







(3.33)

We give a short proof of the first and second Schur’s lemmas:

Proof. To prove the first lemma, consider the two following objects:

[Ker(Λ)≡subspace ofV₁annihilated byΛ] and [Im(Λ)≡ΛV₁] (3.34)

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Eq.3.31 shows that Ker(Λ) ⊆V1 and Im(Λ) ⊆ V2 are invariant subspaces for the representations D1 and D2

respectively:

∀~v1∈Ker(Λ) : Λ(D1~v1) =D2(Λ~v1) = 0 ⇒ ∀~v1∈Ker(Λ), D1~v1∈Ker(Λ) (3.35) and

∀ ~v2∈Im(Λ), ∃~v1∈V1, s.t. ~v2= Λ~v1: D2~v2=D2Λ~v1

D2~v2= Λ(D1~v1) ⇒ ∀~v2∈Im(Λ), D2~v2∈Im(Λ) (3.36) This proves that the kernel and the image of the intertwining operator are invariant subspaces underD₁ andD₂ respectively. Now, since we assume these two representations to be irreducible (meaning that there is no non-trivial invariant subspace for any of them), eitherΛ≡0 or these subspaces are trivial:

Ker(Λ) ={0}, Im(Λ) =V₂ (3.37)

In the latter case,Λ is injective and surjective, i.e.bijective and invertible.

The second lemma can be seen as a corollary since it uses the first one to be proven. Eq.3.31becomes in this case:

ΛD=DΛ (3.38)

This equation remains obviously true ifΛ is replaced byΛλ= Λ−λ1:

ΛλD=DΛλ (3.39)

Now, by Schur’s first lemma we have that eitherΛ_λ≡0orΛ_λ is invertible. However, by the fundamental theorem of algebra,

det(Λ_λ) =det(Λ−λ1) (3.40)

is a polynomial of order n and has at least one root. If we choose λto be this root, the polynomial is zero and det(Λλ) = 0, meaning it is not invertible. Therefore, by Schur’s first lemma:

Λλ≡0 ⇒Λ =λ1 (3.41)

The third lemma is proven in the same spirit using the first two lemmas. The proof is left as an exercise.

The proof relies strongly on the fact that the representations we consider are irreducible. Therefore, Schur’s lemma can be seen as a tool to prove representations are reducible by finding a non-zero, non-invertible intertwiner satisfying Eq.3.31.

Notice that, physically, Schur’s third lemma means that, if we find an operator which commutes with all group symmetries, the eigenvalues of this operator can be used to label the different representations of our theory! For example, in quantum mechanics, we know that we label the spin of the particles with their total spin operator J²=JiJieigenvalue (0,1/2, . . .). Formally, it is becauseJ²commutes with all generators of the algebra describing spin, theSU(2) algebra.

This closes this section devoted to the irreducibility of representations and we will now focus more specifically on the mathematics of Lie groups.

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3.1.3 Lie algebras

We shall start this section with an important result from the theory of Lie groups:

Ado’s theorem. Every finite dimensional Lie group can be represented faithfully², at least locally, by finite dimensional matrices.

This implies that we do not lose any information by studying the matrix representation of the group. As we saw earlier, we choose to parametrise the elements in such a way that α = 0 corresponds to the identity element.

Therefore if we find a representation of the group, it will be parametrised in the same way:

D(g(0))≡D(0) =1 (3.42)

where from here on we forget the abstract notation and will write simplyD(α) for all elements. By Ado’s theorem, the tangent space at the identity fully describes the behaviour of the group, up to global effects. This leads to the concept ofLie algebra:

Definition. A Lie algebra A is an n-dimensional vector space endowed with a second mulitplication, the Lie bracket or Lie product:

[·,·] : A × A → A

(X, Y) 7→ [X, Y] (3.43)

satisfying:

Antisymmetry

[X, Y] =−[Y, X] (3.44)

Jacobi identity

[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0 (3.45) Linearity

[aX+bY, Z] =a[X, Z] +b[Y, Z] (3.46) Since it is a vector space, we can find a basisTiand, as for the standard product, it is enough to know the product rule for this basis vectors:

[Ti, Tj] =ifijkTk (3.47)

wherefijk are called thestructure constants. It is a set of number defining completely the Lie algebra, much like the metricgij =e^T_i ·ej defines completely the geometry in standard Riemannian spaces.

There is a set of three theorems due to Sophus Lie linking Lie algebras and Lie groups. We do not intend to be rigorous here and will only give a small overview of their content.

Lie theorems I,II,III. The first of the theorems states that we can associate a Lie algebra to any Lie group.

Consider an n-dimensional Lie group andD a representation of the latter. The tangent space atα¯ can be obtained by Taylor expanding any matrix of the group:

2For all element in the group, there is a different matrix

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D(α) =D( ¯α) + ∂

∂α_iD(α) _{α= ¯}_α

(αi−α¯i) +. . . (3.48) In particular around the identity:

D(α) =1+ ∂

∂αi

D(α) α=0

α_i+. . .≡1+iα_iX_i+. . . (3.49)

where we put the complex unityi in evidence by (physicists’) convention. The quantitiesTi,i= 1, . . . N (whereN is the dimension of the group) are called thegeneratorsof the group and are given by:

Xi≡ −i ∂

∂αi

D(α) _α=0

(3.50) The factor of i is removed from the generators because if the representation is unitary, the generators will be hermitean (something physicists appreciate!).

Lie theorem II states that the generatorsX_i form the basis of a Lie algebra:

[Xi, Xj] =ifijkXk (3.51)

The structure constants follow from the group multiplication law and are independent of the representation. In particular if we consider the group structure:

D(α)D(β) =D(f(α, β)) (3.52)

the functionf can only depend on the structure constants.

Finally, Lie theorem III states that the tangent spaceT_g at any pointg( ¯α) is also a Lie algebra and:

iX_i( ¯α)≡D⁻¹( ¯α) ∂

∂αi

D(α) _{α= ¯}_α

=iX_k(0)V_i^k( ¯α) (3.53)

where V is an invertible matrix. The last equality means that the generators at any point are simply a linear combination of the generators at the origin: anyA(g)is isomorphic toA(e). This is not surprising since with the group action we can translate from the origin to any point!

Notice that Lie theorems imply that for any groupGthere is one Lie algebra. However, a Lie algebraAcan give rise to different groups (e.g. SU(2) and SO(3)). However, a theorem states that given any Lie algebraAthere is one and only one connected and simply connected Lie group associated to it (e.g. SU(2) is simply connected and connected,SO(3) is only connected).

We will focus from this point on the Lie algebra around the origin. Consider the following quantities:

D(α)D(β) =1+i(αi+βi)Xi+. . . (3.54) D(λα) =1+i(λα_i)X_i, ∀ λ∈C (3.55) We see that locally, the product of two group elements is linearised. This leads to the concept ofexponential map.

The exponential map is a particular parametrisation of the Lie groupG. As we saw, close enough from the origin, the group product structure becomes linear and therefore we can reach any group element by proceeding step by step in the following way:

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D(α) =D()ⁿ, ≡ α

n (3.56)

Taking smaller and smaller steps, we obtain:

D(α) = lim

n→∞D()ⁿ= lim

n→∞

1 +iαi

nXi+O(1/n²)n

≡e^iαⁱ^Xⁱ (3.57)

Theα_i become a set of exponential coordinates, furthermore, this implies a particular functionf describing the product:

D(α)D(β) =eîαⁱ^Xⁱeîβ^j^X^j =eîf^k^(α,β)X^k =D(f(α, β)) (3.58) We can prove this equation using theHaussdorf-Campbell-Baker formula:

exp(A) exp(B) = exp(A+B+1

2[A, B] + 1

12([A,[A, B]] + [B,[B, A]]) +. . .) (3.59) whenA=αiXi andB=βjXj each term will be linear inXk with complicated factors coming from the structure constants. We recover the previous statement that the group product structure depends only on thefijk! Now, this parametrisation is valid close to the origin, but more precisely, how much of G can we recover with this exponential map?

Theorem. If Gis a compact and connected Lie group, then the exponential map covers the whole group.

Compact means that the coordinatesαitake all values on a compact set, in other words, the Lie group’s manifold is compact. Connected refers as well to the manifold describing the Lie group. Intuitively, the theorem can be understood in the following way: compactedness implies that we do not risk to diverge from the expontial map as we try to reach distant points; connectedness implies that all points of the manifold can be reached without

“jumps” from one side to another.

It is sometimes the case in physics that the groups we have to deal with are connected but not compact. In this case another theorem becomes useful.

Theorem. If Gis a connected but non-compact Lie group, then every element of Gcan be written as Y

n

e^iα⁽ⁿ⁾ⁱ ^Xⁱ (3.60)

where, generally, the number of group elements needed to span the whole group is finite.

To summarise, when dealing with Lie groups, all we have to do is to find a representation of the Lie algebra, and to exponentiate! We will now put all this machinery to work in one of the most important groups in high-energy physics.

3.2 Lorentz and Poincaré groups

3.2.1 Construction

Let us start with a bit of history. Before the 20^th century, the physics of mechanical systems was governed by Newton’s laws of motion. These laws are left invariant by the action of a symmetry group: Galileo group. As we saw before, its realisation on space and time is as follows:

Relativistic Quantum Fields