BASICS OFQUANTUMFIELDTHEORY 5Massless particles : In the massless case, we look for Lorentz transformations Σµ that leaveqν = ω, 0, 0, ω invariant.. GELIS– A STROLLTHROUGHQUANTUMFIELDS1.
Trang 2A Stroll Through QUANTUM FIELDS
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Trang 31.1 Special relativity 1
1.2 Free scalar fields, Mode decomposition 6
1.3 Interacting scalar fields 11
1.4 LSZ reduction formulas 14
1.5 From transition amplitudes to reaction rates 17
1.6 Generating functional 22
1.7 Perturbative expansion and Feynman rules 27
1.8 Calculation of loop integrals 33
1.9 K¨allen-Lehmann spectral representation 36
1.10 Ultraviolet divergences and renormalization 38
1.11 Spin 1/2 fields 47
1.12 Spin 1 fields 52
1.13 Abelian gauge invariance, QED 57
1.14 Charge conservation, Ward-Takahashi identities 60
1.15 Spontaneous symmetry breaking 63
1.16 Perturbative unitarity 71
2 Functional quantization 85 2.1 Path integral in quantum mechanics 85
2.2 Classical limit, Least action principle 89
2.3 More functional machinery 89
2.4 Path integral in scalar field theory 96
2.5 Functional determinants 98
2.6 Quantum effective action 101
2.7 Two-particle irreducible effective action 107
2.8 Euclidean path integral and Statistical mechanics 114
i
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3 Path integrals for fermions and photons 119
3.1 Grassmann variables 119
3.2 Path integral for fermions 125
3.3 Path integral for photons 127
3.4 Schwinger-Dyson equations 130
3.5 Quantum anomalies 133
4 Non-Abelian gauge symmetry 143 4.1 Non-abelian Lie groups and algebras 144
4.2 Yang-Mills Lagrangian 152
4.3 Non-Abelian gauge theories 157
4.4 Spontaneous gauge symmetry breaking 162
4.5 θ-term and strong-CP problem 168
4.6 Non-local gauge invariant operators 176
5 Quantization of Yang-Mills theory 187 5.1 Introduction 187
5.2 Gauge fixing 189
5.3 Fadeev-Popov quantization and Ghost fields 191
5.4 Feynman rules for non-abelian gauge theories 193
5.5 On-shell non-Abelian Ward identities 197
5.6 Ghosts and unitarity 199
6 Renormalization of gauge theories 211 6.1 Ultraviolet power counting 211
6.2 Symmetries of the quantum effective action 212
6.3 Renormalizability 218
6.4 Background field method 223
7 Renormalization group 231 7.1 Callan-Symanzik equations 231
7.2 Correlators containing composite operators 234
7.3 Operator product expansion 237
7.4 Example: QCD corrections to weak decays 241
7.5 Non-perturbative renormalization group 248
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Trang 5CONTENTS iii
8.1 General principles of effective theories 260
8.2 Example: Fermi theory of weak decays 264
8.3 Standard model as an effective field theory 267
8.4 Effective theories in QCD 274
8.5 EFT of spontaneous symmetry breaking 284
9 Quantum anomalies 295 9.1 Axial anomalies in a gauge background 295
9.2 Generalizations 307
9.3 Wess-Zumino consistency conditions 314
9.4 ’t Hooft anomaly matching 318
9.5 Scale anomalies 320
10 Localized field configurations 327 10.1 Domain walls 328
10.2 Skyrmions 331
10.3 Monopoles 333
10.4 Instantons 343
11 Modern tools for tree level amplitudes 357 11.1 Shortcomings of the usual approach 357
11.2 Colour ordering of gluonic amplitudes 358
11.3 Spinor-helicity formalism 364
11.4 Britto-Cachazo-Feng-Witten on-shell recursion 374
11.5 Tree-level gravitational amplitudes 385
11.6 Cachazo-Svrcek-Witten rules 395
12 Worldline formalism 407 12.1 Worldline representation 407
12.2 Quantum electrodynamics 413
12.3 Schwinger mechanism 417
12.4 Calculation of one-loop amplitudes 420
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13.1 Discretization of bosonic actions 432
13.2 Fermions 437
13.3 Hadron mass determination on the lattice 441
13.4 Wilson loops and confinement 442
13.5 Gauge fixing on the lattice 446
13.6 Lattice worldline formalism 450
14 Quantum field theory at finite temperature 457 14.1 Canonical thermal ensemble 457
14.2 Finite-T perturbation theory 458
14.3 Large distance effective theories 477
14.4 Out-of-equilibrium systems 492
15 Strong fields and semi-classical methods 501 15.1 Introduction 501
15.2 Expectation values in a coherent state 503
15.3 Quantum field theory with external sources 509
15.4 Observables at LO and NLO 510
15.5 Green’s formulas 516
15.6 Mode functions 527
15.7 Multi-point correlation functions at tree level 531
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Trang 7Let us consider two frames F and F′, in which the coordinates of a given eventare respectivelyxµandx′µ A Lorentz transformation is a linear transformation such
that the intervalds2≡ dt2− dx2is the same in the two frames2 If we denote thecoordinate transformation by
that achieves this is called the conformal group In four space-time dimensions, the conformal group is 15
dimensional, and in addition to the 6 orthochronous Lorentz transformations it contains dilatations as well
as non-linear transformations called special conformal transformations.
1
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Trang 82 F GELIS– A STROLLTHROUGHQUANTUMFIELDS
the matrixΛ of the transformation must obey
transfor-proper transformations), and do not change the direction of the time axis since
Λ0 = 1≥ 0 (they are called orthochronous) Any combination of such infinitesimal
transformations shares the same properties, and their set forms a subgroup of the fullgroup of transformations that preserve the Minkowski metric.c
1.1.2 Representations of the Lorentz group
More generally, a Lorentz transformation acts on a quantum system via a tionU(Λ), that forms a representation of the Lorentz group, i.e
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(The prefactori/2 in the second term of the right hand side is conventional.) Sincetheωµνare antisymmetric, the generatorsMµνcan also be chosen antisymmetric
By using eq (1.7) for the Lorentz transformationΛ−1Λ′Λ, we arrive at
U−1(Λ)MµνU(Λ) = ΛµρΛν Mρσ, (1.9)indicating thatMµνtransforms as a rank-2 tensor When used with an infinitesimaltransformationΛ = 1 + ω, this identity leads to the commutation relation that definesthe Lie algebra of the Lorentz group
Mµν, Mρσ
= i(gµρMνσ− gνρMµσ) − i(gµσMνρ− gνσMµρ) (1.10)When necessary, it is possible to divide the six generatorsMµνinto three generators
Jifor ordinary spatial rotations, and three generatorsKifor the Lorentz boosts alongeach of the spatial directions:
Pµ
p, σ = pµ
p, σ , withp0≡pp2+ m2 (1.14)Consider now the stateU(Λ)
p, σ is an eigenstate of momentum with eigenvalue (Λp)µ, and
we may write it as a linear combination of all the states with momentumΛp,U(Λ)
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1.1.4 Little group
Any positive energy on-shell momentumpµ can be obtained by applying an thochronous Lorentz transformation to some reference momentumqµthat lives onthe same mass-shell,
The choice of the reference4-vector is not important, but depends on whether theparticle under consideration is massive or not Convenient choices are the following:
• m > 0 : qµ≡ (m, 0, 0, 0), the 4-momentum of a massive particle at rest,
• m = 0 : qµ≡ (ω, 0, 0, ω), the 4-momentum of a massless particle moving inthe third direction of space
Then, we may define a generic one-particle state from those corresponding to thereference momentum as follows
is the group of all rotations in3-dimensional space The additional quantum number
σ is therefore a label that enumerates the possible states in a given representation ofSO(3) These representations correspond to the angular momentum, but since we are
in the rest frame of the particle, this is in fact the spin of the particle For a spins, thedimension of the representation is2s + 1, and σ takes the values −s, 1 − s,· · · , +s
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Massless particles : In the massless case, we look for Lorentz transformations
Σµ that leaveqν = (ω, 0, 0, ω) invariant For an infinitesimal transformation,
Σµ ≈ δµ + ωµ , this gives the following general form
Thus, the little group for massless particles is three dimensional, with generatorsJ3
(the projection of the angular momentum in the direction of the momentum) and4
B1,2 Using eq (1.10), we have
The numberσ is called the helicity of the particle After a rotation of angle θ = 2π,
the state must return to itself (bosons) or its opposite (fermions), implying that thehelicity must be a half integer:
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1.1.5 Scalar field
A scalar fieldφ(x) is a (number or operator valued) object that depends on a spacetimecoordinatex and is invariant under a Lorentz transformation, except for the change ofcoordinate induced by the transformation:
1.2 Free scalar fields, Mode decomposition
1.2.1 Quantum harmonic oscillators
Let us consider a continuous collection of quantum harmonic oscillators, each of themcorresponding to particles with a given momentump These harmonic oscillatorscan be defined by a pair of creation and annihilation operatorsa†p, ap, wherep is a3-momentum that labels the corresponding mode Note that the energy of the particles
is fixed from their 3-momentum by the relativistic dispersion relation,
If we denote by H the Hamiltonian operator of such a system, the property that
a†pcreates a particle of momentump (and therefore of energy Ep) implies that
H, a†p
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Trang 131 BASICS OFQUANTUMFIELDTHEORY 7
Likewise, sinceapdestroys a particle with the same energy, we have
†
pap+ V Ep
where V is the volume of the system To make contact with the usual treatment6of a
harmonic oscillator in quantum mechanics, it is useful to introduce the occupation numberfpdefined by,
2
The expectation value offphas the interpretation of the number of particles par unit
of phase-space (i.e per unit of volume in coordinate space and per unit of volume
in momentum space), and the1/2 in fp+1
2is the ground state occupation of eachoscillator7 Of course, this additive constant is to a large extent irrelevant since onlyenergy differences have a physical meaning Given eq (1.34), the commutationrelations (1.32) and (1.33) are fulfilled provided that
constrained by the positive energy mass-shell condition 2π θ(p 0 ) δ(p 2 − m 2 ).
6 In relativistic quantum field theory, it is customary to use a system of units in which ¯ h = 1, c = 1 (and also kB= 1 when the Boltzmann constant is needed to relate energies and temperature) In this system of units, the action S is dimensionless Mass, energy, momentum and temperature have the same dimension, which is the inverse of the dimension of length and duration:
= temperature
= length−1
= duration−1
Moreover, in four dimensions, the creation and annihilation operators introduced in eq (1.34) have the dimension of an inverse energy:
7 This is reminiscent of the fact that the energy of the level n in a quantized harmonic oscillator of base energy ω is E n = (n +12)ω.
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1.2.2 Scalar field operator, Canonical commutation relations
Note that in quantum mechanics, a particle with a well defined momentump is notlocalized at a specific point in space, due to the uncertainty principle Thus, when wesay thata†pcreates a particle of momentump, this production process may happenanywhere in space and at any time since the energy is also well defined Instead ofusing the momentum basis, one may introduce an operator that depends on space-time
in order to give preeminence to the time and position at which a particle is created ordestroyed It is possible to encapsulate all theap, a†pinto the following Hermiteanoperator8
φ(x)≡
Z
d3p(2π)32Ep
a†pe+ip·x− ape−ip·x
(1.39)
Given the commutation relation (1.37), we obtain the following equal-time
commuta-tion relacommuta-tions forφ and Π,
These are called the canonical field commutation relations In this approach (known
as canonical quantization), the quantization of a field theory corresponds to
promot-ing the classical Poisson bracket between a dynamical variable and its conjugatemomentum to a commutator:
It is possible to invert eqs (1.38) and (1.39) in order to obtain the creation and
8 In four space-time dimensions, this field has the same dimension as energy:
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annihilation operators given the operatorsφ and Π These inversion formulas read
H=
Z
d3x 12Π2(x) +12(∇φ(x))2+12m2φ2(x) (1.44)From this Hamiltonian, one may obtain equations of motion in the form of Hamilton-Jacobi equations Formally, they read
One may also obtain a Lagrangian L(φ, ∂0φ) that leads to the Hamiltonian (1.44)
by the usual manipulations Firstly, the momentum canonically conjugated toφ(x)should be given by
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This gives the following Lagrangian,
x+ m2
which is known as the Klein-Gordon equation This equation is of course equivalent
to the pair of Hamilton-Jacobi equations derived earlier.c
1.2.4 Noether’s theorem
Conservation laws in a physical theory are intimately related to the continuoussymmetry of the system This is well known in Lagrangian mechanics, and can beextended to quantum field theory Consider a generic Lagrangian L(φ, ∂µφ) thatdepends on fields and their derivatives with respect to the spacetime coordinates, andassume that the theory is invariant under the following variation of the field,
Such an invariance is said to be continuous when it is valid for any value of theinfinitesimal parameterε If the Lagrangian is unchanged by this transformation, wecan write
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that the variation of the Lagrangian is zero implies the following continuity equationfor this current
This is the simplest case of Noether’s theorem, where the Lagrangian itself is invariant.
But for the theory to be unmodified by the transformation of eq (1.52), it is onlynecessary that the action be invariant, which is also realized if the Lagrangian ismodified by a total derivative, i.e
(The proportionality toε follows from the fact that the variation must vanish when
ǫ → 0.) When the variation of the Lagrangian is a total derivative instead of zero, thecontinuity equation is modified into:
∂µ Jµ− Kµ
whereJµis the same current as before As we shall see later, there are situationswhere a conservation equation such as (1.54) is violated by quantum effects, due to adelicate interplay between the symmetry responsible for the conservation law and theultraviolet structure of the theory.c
1.3 Interacting scalar fields
1.3.1 Interaction term
Until now, we have only considered non-interacting particles, which is of course ofvery limited use in practice That the Hamiltonian (1.34) does not contain interactionsfollows from the fact that the only non-trivial term it contains is of the forma†pap, thatdestroys a particle of momentump and then creates a particle of momentum p (hencenothing changes in the state of the system under consideration) By momentumconservation, this is the only allowed Hermitean operator which is quadratic inthe creation and annihilation operators Therefore, in order to include interactions,
we must include in the Hamiltonian terms of higher degree in the creation andannihilation operators The additional term must be Hermitean, since H generates thetime evolution, which must be unitary
The simplest Hermitean addition to the Hamiltonian is a term of the form
wheren is a power larger than 2 The real constant λ is called a coupling constant
and controls the strength of the interactions, while the denominatorn! is a symmetry
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factor that will prove convenient later on At this point, it seems that any degreenmay provide a reasonable interaction term However, theories with an oddn have anunstable vacuum, and theories withn > 4 are non-renormalizable in four space-timedimensions, as we shall see later For these reasons,n = 4 is the only case which iswidely studied in practice, and we will stick to this value in the rest of this chapter.With this choice, the Hamiltonian and Lagrangian read
lim
x 0
→±∞
What we have in mind here is thatλ goes to zero adiabatically at asymptotic times,
i.e much slower than all the physically relevant timescales of the theory underconsideration Therefore, atx0=±∞, the theory is a free theory whose spectrum ismade of the eigenstates of the free Hamiltonian Likewise, the fieldφ(x) should be
in a certain sense “close to a free field” in these limits In the case of thex0
→ −∞limit, let us denote this by9
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Eq (1.61) can be made more explicit by writing
φ(x) = U(−∞, x0) φin(x) U(x0, −∞) , (1.63)whereU is a unitary time evolution operator defined as a time ordered exponential of
the interaction term in the Lagrangian, evaluated with theφinfield:
U(t2, t1)≡ T exp i
Zt 2
t 1
dx0d3x LI(φin(x)) , (1.64)where
U that connects φ and φin
1.3.3 In and Out states
The in creation and annihilation operators can be used to define a space of eigenstates
of the free Hamiltonian, starting from a ground state (vacuum) denoted
0in
Forinstance, one particle states would be defined as
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free field that coincides with the interacting fieldφ(x) in the limit x0
→ +∞ (withthe same caveat about field renormalization) Starting from a vacuum state ... functions of the field operators thatare calculable in quantum field theory The missing link to connect this to experi-mental measurements is an explicit formula relating reaction rates to these transitionamplitudes... scalar fields, Mode decomposition
1.2.1 Quantum harmonic oscillators
Let us consider a continuous collection of quantum harmonic oscillators, each of themcorresponding to. .. conjugatemomentum to a commutator:
It is possible to invert eqs (1.38) and (1.39) in order to obtain the creation and
8 In four space-time dimensions, this field has the same