Quantum field theory

Integrating out a field means performing its path integral while retaining the other fields as backgrounds. The result is generally a nonlocal quantum effective action; for a field much heavier than all external momenta, a derivative expansion turns it into local effective interactions.

Derivative expansion

A derivative expansion expresses a nonlocal effective interaction as a series of local operators with successively more derivatives. For a field of mass

M

, each additional pair of derivatives is typically suppressed by

M^{- 2}

Spontaneous symmetry breaking (SSB)

Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold.

Vacuum manifold

The vacuum manifold is the space of field configurations that minimize the energy. If a group

G

acts transitively and a chosen vacuum has stabilizer

H

, its connected component is the homogeneous space

G / H

Goldstone theorem

In a relativistic quantum field theory, each spontaneously broken generator of a continuous internal symmetry produces a massless scalar excitation. For

G \to H

, the standard setting therefore has

dim G - dim H

Goldstone bosons.

Nonlinear sigma model

A nonlinear sigma model has fields valued in a curved target space such as a vacuum manifold

G / H

. Its leading action is quadratic in spacetime derivatives but nonlinear in any unconstrained coordinates used for the target.

Discrete symmetry in quantum field theory

Charge conjugation ( $C$ )

Charge conjugation exchanges particles with antiparticles. For a matrix-valued gauge potential it acts as

A_{μ} \mapsto - A_{μ}^{T}

Time-reversal symmetry ( $T$ )

Time-reversal symmetry reverses the time coordinate and is represented antiunitarily in quantum theory.

Standard Model (SM)

The Standard Model is a gauge theory with gauge group

S U (3)_{c} \times S U (2)_{L} \times U (1)_{Y}

. Its Higgs field breaks the electroweak subgroup to the electromagnetic

U (1)_{Q}

Electroweak interaction

The electroweak interaction is the

S U (2)_{L} \times U (1)_{Y}

gauge theory that unifies the weak and electromagnetic interactions.

Hypercharge ( $Y$ )

Hypercharge is the generator of the electroweak

U (1)_{Y}

factor. With the convention

D_{μ} = \partial_{μ} - i g W_{μ}^{a} T^{a} - i g^{'} Y B_{μ}

, electric charge is

Q = T^{3} + Y

Higgs field

The Standard Model Higgs field is a complex

S U (2)_{L}

doublet of hypercharge

1/2

. Its nonzero vacuum expectation value triggers the Higgs mechanism.

Higgs mechanism

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

Unitary gauge

Unitary gauge uses a gauge transformation to remove the Goldstone fields from the Higgs multiplet, leaving the physical radial Higgs excitation and massive vector fields.

Higgs boson

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

Electroweak gauge boson

W boson ( $W^{\pm}$ )

The charged electroweak gauge fields are

W_{μ}^{\pm} = (W_{μ}^{1} \mp i W_{μ}^{2}) / 2

. After electroweak symmetry breaking they have electric charges

\pm 1

and mass

gv /2

Z boson ( $Z^{0}$ )

The neutral massive electroweak gauge field is proportional to

g W_{μ}^{3} - g^{'} B_{μ}

and has mass

v g^{2} + g^{'2} /2

Standard Model fermion

A Standard Model generation contains left-handed quark and lepton doublets and right-handed up-quark, down-quark and charged-lepton singlets. A right-handed neutrino may be added as a gauge singlet.

Right-handed neutrino

A right-handed neutrino transforms as

(1, 1)_{0}

under

S U (3)_{c} \times S U (2)_{L} \times U (1)_{Y}

, so it is neutral under every Standard Model gauge interaction.

Yukawa interaction

A Yukawa interaction couples left- and right-handed fermions through a scalar field. Replacing the Higgs field by its vacuum expectation value turns a gauge-invariant Standard Model Yukawa interaction into a fermion mass term.

Classical field theory

Lagrangian density ( $L$ )

A Lagrangian density is integrated over spacetime to give the action

S = \int d^{d} x L

Kinetic term

A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.

Mass term

A mass term is a quadratic term without derivatives whose coefficient sets the mass scale of a free field.

Field interaction term

A field interaction term is nonlinear in the fields and produces the interaction vertices of perturbation theory.

Euler-Lagrange field equation

For a field

Φ

\frac{\partial L}{\partial Φ} - \partial_{μ} \frac{\partial L}{\partial ( \partial _{μ} Φ )} = 0.

On shell

A field configuration or identity is on shell when the equations of motion hold.

Noether current ( $j^{μ}$ )

A continuous variational symmetry gives a current satisfying

\partial_{μ} j^{μ} = 0

on shell. Integrating

j^{0}

over space gives a conserved charge when boundary flux vanishes.

Canonical stress-energy tensor ( $T_{can}^{μν}$ )

Spacetime translations give the canonical stress-energy tensor

T^{μ}_{ν} = \sum_{a} \frac{\partial L}{\partial ( \partial _{μ} Φ _{a} )} \partial_{ν} Φ_{a} - δ_{ν}^{μ} L .

Belinfante-Rosenfeld stress-energy tensor

The Belinfante-Rosenfeld construction adds the divergence of a spin-current superpotential to the canonical stress-energy tensor, producing a symmetric conserved tensor with the same integrated four-momentum under suitable boundary conditions.

Dirac current ( $j^{μ} = \overset{ˉ}{ψ} γ^{μ} ψ$ )

The global phase symmetry of a Dirac field gives the conserved Dirac current

j^{μ} = \overset{ˉ}{ψ} γ^{μ} ψ

Relativistic quantum field

Real scalar field

A free real scalar field obeys the Klein-Gordon equation and creates neutral spin-zero particles.

Complex scalar field

A complex scalar field has distinct particle and antiparticle excitations and a global phase symmetry.

Dirac field

A free Dirac field has Lagrangian density

L = i \overset{ˉ}{ψ} γ^{μ} \partial_{μ} ψ - m \overset{ˉ}{ψ} ψ

and obeys the Dirac equation.

Dirac equation ( $(i γ^{μ} \partial_{μ} - m) ψ = 0$ )

The Dirac equation is the relativistic spin-one-half field equation

(i γ^{μ} \partial_{μ} - m) ψ = 0

Weyl spinor

A left- or right-handed Weyl spinor transforms in the Lorentz representation

(1/2, 0)

(0, 1/2)

. Parity exchanges the two chiralities.

Dirac spinor

A Dirac spinor transforms as

(1/2, 0) \oplus (0, 1/2)

, so it combines the two Weyl chiralities into a parity-invariant representation.

Gauge field

A gauge field is a connection associated with a local symmetry. An Abelian gauge potential

A_{μ}

has field strength

F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

Yang-Mills theory

Yang-Mills theory is the non-Abelian gauge theory with field strength

F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i g [A_{μ}, A_{ν}] .

Under an infinitesimal gauge transformation it transforms covariantly as

δ F_{μν} = - i g [F_{μν}, α]

Yang-Mills theta term ( $θ$ term)

In four dimensions the Yang-Mills theta term is proportional to

θ Tr (F_{μν}^{⋆} F^{μν})

. Its density is a total derivative, but nontrivial gauge-field topology can make its spacetime integral physically relevant in the quantum theory.

Chern-Simons current ( $K^{μ}$ )

The Chern-Simons current satisfies

\partial_{μ} K^{μ} = Tr (F_{μν}^{⋆} F^{μν})

. It is not gauge invariant even though its divergence is.

Quantum anomaly

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.

Gauge anomaly

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Chiral anomaly

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

Axial current ( $J_{A}^{μ}$ )

For massless Dirac fermions, the classical axial current is

J_{A}^{μ} = \overset{ˉ}{ψ} γ^{μ} γ^{5} ψ

. The chiral anomaly makes its divergence proportional to

F_{μν}^{⋆} F^{μν}

't Hooft anomaly

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

't Hooft anomaly matching

't Hooft anomaly matching requires the massless infrared degrees of freedom, topological sector, or symmetry-breaking pattern to reproduce every anomaly of an unbroken global symmetry measured in the ultraviolet theory.

Gauge fixing

Gauge fixing removes the degeneracy among gauge-equivalent field configurations so that the kinetic operator has an inverse propagator.

Covariant gauge

Adding

- (\partial_{μ} A^{μ})^{2} / (2 α)

gives a family of Lorentz-covariant gauges;

α = 1

is Feynman gauge.

Feynman gauge

Faddeev-Popov determinant ( $Δ_{FP}$ )

The Faddeev-Popov determinant is the functional Jacobian

Δ_{FP} [A] = det (\frac{δ G ( A ^{α} )}{δ α})_{G = 0}

that compensates for the change from integration along a gauge orbit to a gauge-fixing condition

G (A) = 0

. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

Faddeev-Popov ghost field

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

Axial gauge

An axial gauge imposes

n^{μ} A_{μ} = 0

for a fixed vector

n^{μ}

. Its Faddeev-Popov operator is

n^{μ} D_{μ}

; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

BRST symmetry

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential

s

replaces an infinitesimal gauge parameter by the ghost field and satisfies

s^{2} = 0

Gauge-fixing fermion ( $Ψ$ )

A gauge-fixing fermion is a Grassmann-odd functional whose BRST transformation

s Ψ

supplies the gauge-fixing and ghost terms. Nilpotence gives

s (s Ψ) = 0

immediately.

BRST cohomology

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

Quantized field

Mode expansion of a free field

A free field is expanded in positive- and negative-frequency plane waves multiplied by annihilation and creation operators.

Fock state

A Fock state has specified occupation numbers. Applying creation operators to the vacuum constructs multiparticle momentum eigenstates.

Intrinsic parity ( $γ_{p}$ )

Intrinsic parity is the phase by which a field transforms under spatial inversion after its spacetime argument is reflected. For a Hermitian scalar field it is

+ 1

- 1

Quantum field theory propagator

A propagator is a time-ordered two-point function and a Green function for the quadratic kinetic operator with an

i ϵ

boundary prescription.

Photon propagator

In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

Perturbative quantum field theory

Feynman diagram

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

Feynman rule

Feynman rules translate each diagram into momentum-space factors for propagators, vertices, external states, momentum conservation and loop integration.

Feynman parameter

A Feynman parameter combines propagator denominators; for example,

\frac{1}{A B} = \int_{0}^{1} \frac{d x}{[ x A + ( 1 - x ) B ] ^{2}} .

After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

Tree-level Feynman diagram

A tree-level diagram has no loops and gives the leading classical contribution allowed by the interaction vertices.

Loop order

The loop order of a connected Feynman diagram is the number

L = I - V + 1

of independent momentum cycles, where

I

is the number of internal lines and

V

the number of vertices. Each loop introduces an unconstrained momentum integral.

Connected Feynman diagram

A connected Feynman diagram has a path between every pair of its vertices. Normalized correlation functions discard disconnected vacuum bubbles, while full correlators can still factor into disconnected components carrying external insertions.

One-particle-irreducible Feynman diagram (1PI Feynman diagram)

A one-particle-irreducible Feynman diagram is a connected Feynman diagram that remains connected after any one internal line is cut. The vertices of the quantum effective action generate precisely these diagrams.

Schwinger-Dyson equation

A Schwinger-Dyson equation follows from invariance of a path integral under a change of integration variable and relates correlation functions through the field equations and contact terms.

Quantum effective action ( $Γ$ )

The quantum effective action is the Legendre transform of the generating functional of connected correlation functions. Its functional derivatives are the one-particle-irreducible vertex functions.

Renormalization

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

Counterterm

A counterterm is a local term added to a regulated Lagrangian density to cancel ultraviolet divergences and impose chosen renormalization conditions. Field-strength, mass and coupling counterterms respectively adjust propagator normalization, pole position and interaction strength.

Renormalization condition

A renormalization condition defines a renormalized field or parameter by prescribing a correlation function at a chosen kinematic point. Changing that point changes the renormalized parameters while leaving physical predictions invariant.

Renormalization scale ( $μ$ )

The renormalization scale is the auxiliary momentum or energy scale at which renormalization conditions define the parameters of a quantum field theory.

Running coupling

A running coupling is a renormalized coupling regarded as a function of the renormalization scale. Its scale derivative is its beta function.

Beta function (physics, $β (g) = μ d g / d μ$ )

The beta function of a coupling

g

β (g) = μ \frac{d g}{d μ} = \frac{d g}{d l o g μ} .

It describes how the running coupling changes when the renormalization scale changes.

Regularization in quantum field theory

Regularization modifies divergent loop integrals by introducing an auxiliary parameter. The regulator is removed after its dependence has been absorbed into counterterms.

Cutoff regularization

Cutoff regularization restricts loop momenta to

∣ ℓ ∣ < Λ

. The ultraviolet cutoff

Λ

makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

Dimensional regularization

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to

d = d_{0} - ϵ

. Ultraviolet logarithms then appear as poles in

1/ ϵ

Minimal subtraction scheme ( $MS$ scheme)

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator

ϵ

, without additional finite terms.

Quantum electrodynamics ( $QED$ )

Quantum electrodynamics couples a Dirac field to an Abelian gauge field through

- e \overset{ˉ}{ψ} γ^{μ} ψ A_{μ}

Photon vacuum polarization ( $Π^{μν} (k)$ )

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

Scalar quantum electrodynamics (Scalar QED)

Scalar quantum electrodynamics minimally couples a complex scalar field to the electromagnetic gauge field through

D_{μ} = \partial_{μ} + i e A_{μ}

Seagull vertex

The scalar-QED term

e^{2} A_{μ} A^{μ} ϕ^{*} ϕ

produces a local two-photon two-scalar vertex, conventionally called the seagull vertex.

Ward identity

For an on-shell amplitude

M = ϵ_{μ} (k) M^{μ}

with an external photon, gauge invariance implies

k_{μ} M^{μ} = 0

Power counting in quantum field theory

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

Superficial degree of divergence ( $D$ )

The superficial degree of divergence

D

is the ultraviolet power predicted by rescaling all independent loop momenta together. For a scalar graph with

L

loops and

I

propagators behaving as

p^{- 2}

d

dimensions,

D = d L - 2 I

before cancellations and subdivergences are considered.

Marginal coupling

A marginal coupling is dimensionless under classical power counting.

Relevant coupling

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

Irrelevant coupling

An irrelevant coupling has negative mass dimension and is suppressed toward long distances.

Quantum field theory in curved spacetime

Quantum field theory in curved spacetime quantizes matter fields on a prescribed classical spacetime geometry.

Klein-Gordon inner product

For complex solutions of the Klein-Gordon equation,

(ϕ_{1}, ϕ_{2})_{K G} = i \int_{Σ} d Σ^{a} (ϕ_{1}^{*} \nabla_{a} ϕ_{2} - ϕ_{2} \nabla_{a} ϕ_{1}^{*}) .

The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

Positive-frequency solution

Relative to a timelike Killing field

K

, a positive-frequency solution obeys

i L_{K} u = ω u

with

ω > 0

. Such solutions define the one-particle space and annihilation operators of a stationary vacuum.

Vacuum state in a stationary spacetime