Classical field theory

A global symmetry has transformation parameters constant across spacetime and preserves the action. It can relate physically distinct states. Continuous global symmetries give Noether currents and conserved charges when boundary fluxes vanish; discrete symmetries do not receive currents from the same infinitesimal theorem. A gauge redundancy instead uses local parameters to relate descriptions of the same physical state, except that nontrivial boundary transformations may carry physical charges. The global phase symmetry of a complex scalar field is an important internal example.

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.

An antisymmetric tensor $A_{mn}$ with $F=dA$ obeys ${\partial }^m{F}_{mnp}={\mu }^2{A}_{np}$ for $\mu \ne 0$. Taking a divergence yields ${\partial }^m{A}_{mn}=0$, and the remaining equation is the Klein-Gordon equation for each component. The physical particle polarizations form the exterior square of the massive little group vector representation, of dimension $(D-1)(D-2)/2$. Unlike a two-form gauge field, it has physical longitudinal polarizations.

If ${\partial }_{-}$ is invertible, the divergence constraint of a massive two-form field makes $A_{-i}$ and $A_{ij}$ independent. It gives $A_{+-}=-{\partial }_{-}^{-1}{\partial }_i{A}_{-i}$ and $A_{+i}={\partial }_{-}^{-1}(-{\partial }_{+}{A}_{-i}+{\partial }_j{A}_{ji})$. All independent components obey the Klein-Gordon equation. Their count is $(D-2)+\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$.

A Hamiltonian density is a local integrand for the field-theory Hamiltonian. For canonical fields with invertible velocity-momentum relation it is the Legendre transform in mechanics of the Lagrangian density, summing over all conjugate field components. Gauge theories require constraints and multiplier terms as well.

For a real scalar field with Lagrangian density $\mathcal{L}={\dot{\varphi }}^2/2-|\nabla \varphi {|}^2/2-V(\varphi )$, the canonical momentum is $\pi =\dot{\varphi }$. Its Hamiltonian density is the sum of momentum energy, gradient energy and potential energy. Hamilton's equations recover the Euler-Lagrange equation.

A classical field theory is a gauge theory when its description has a local redundancy represented by gauge transformations. A gauge field defines the covariant derivative that compares matter fields at different points, and its gauge field strength measures the curvature of that comparison. Electromagnetism is the familiar Abelian example; Yang-Mills theory generalizes it to a non-Abelian internal group. Both classical equations and their quantum field theory can have this structure. Physically equivalent gauge descriptions must give the same observables, even though the gauge potential changes.

Localizing a parameter of a global symmetry produces its Noether current times a derivative of that parameter. Introduce a gauge field with an inhomogeneous transformation and couple it to the Noether current to cancel this variation. Recompute the remaining variation and add the required interaction and transformation terms. Closure of the local symmetry algebra constrains the completion. For supersymmetry, its supercurrent couples to the gravitino, and the translation in the supersymmetry algebra requires a vierbein.

An internal symmetry changes field values while leaving spacetime coordinates fixed, preserving the action. A constant phase rotation of a complex scalar field is an example. This distinguishes internal charges from the spacetime translation charges in the canonical stress-energy tensor. Localizing the parameter generally requires gauge fields; a global internal symmetry is not automatically a gauge symmetry.

A differentiable infinitesimal transformation is a variational symmetry when its change of the Lagrangian density is a divergence, $\delta \mathcal{L}=\epsilon {\partial }_{\mu }{K}^{\mu }$. Under suitable boundary conditions the action is invariant, so exact pointwise invariance is unnecessary. The off-shell variation identity gives the Noether current for a first-derivative scalar field. This definition is for a constant transformation parameter; a spacetime-dependent parameter is a different, gauge-symmetry problem.

A scalar field assigns one scalar to each point of a specified space, such as Euclidean space or spacetime. For example, $F(x,y)=x^2+2y^2$ is a real-valued function on a plane, and its change along a moving observation is given by the derivative along a curve. Under a change of coordinates, a scalar field's value at the same physical point is unchanged.

A real scalar field is minimally coupled to a metric tensor when its kinetic term uses the metric contraction of first derivatives and its scalar potential depends only on the field, without an explicit scalar curvature coupling. For metric signature $(-,+,+,+)$ the displayed Lagrangian and volume density $\sqrt{-g}$ give $\square \varphi -V^{\prime }(\varphi )=0$. Its stress-energy tensor is $T_{ab}={\nabla }_a\varphi {\nabla }_b\varphi -g_{ab}[\frac{1}{2}(\nabla \varphi {)}^2+V]$. The scalar stress-energy divergence identity explains its conservation on the field equation.

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.

The mass of a canonically normalized scalar field is determined by its quadratic fluctuation potential at a stationary vacuum. With several real fields, physical squared masses are the eigenvalues of the scalar mass matrix. A negative squared mass signals an unstable direction rather than a stable massive particle.

A scalar field transforming in the Adjoint representation of a Lie algebra transforms by a Lie bracket with the infinitesimal gauge parameter.

A scalar field with no spacetime coordinates reduces its path integral to an ordinary integral. It preserves Wick contractions and Feynman-diagram symmetry factors, making it a useful model of perturbative quantum field theory.

For $m^2>0$, the zero-dimensional scalar field theory with Euclidean action $m^2{x}^2/2+\lambda x^4/4!$ has a Gaussian integral normalization $I(0)=\sqrt{2\pi /m^2}$. Its Feynman rules assign $1/m^2$ to each propagator, $-\lambda$ to each quartic vertex, and the reciprocal Feynman-diagram symmetry factor to each vacuum diagram. The integral itself includes disconnected vacuum diagrams; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with Feynman-diagram symmetry factors $128,16,48$ respectively. Their weights add to $35{\lambda }^2/(384m^8)$.

The Wick contractions give the general coefficient $I(0)(-1{)}^v(4v-1)!!/[v!(4!{)}^v{m}^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an asymptotic expansion as $\lambda \downarrow 0$, rather than a convergent power series. For every fixed truncation the Taylor theorem remainder estimate for $e^{-u}$, $u\ge 0$, can be integrated against the Gaussian distribution to justify the corresponding remainder bound.

A charged scalar field is a complex scalar field transforming nontrivially under a gauge symmetry. For an electromagnetic potential $A_a$, its gauge-covariant derivative is ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$.

For electric charge $q$ and electromagnetic four-potential $A_a$, the derivative ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$ transforms covariantly under an electromagnetic gauge transformation.

The massless gauge-covariant wave equation is ${\mathcal{D}}_a{\mathcal{D}}^a\Phi =0$. In a stationary charged background, the electrostatic potential shifts the local frequency from $\omega$ to $\omega +q{A}_t$.

A scalar potential is the derivative-free energy density $V(\varphi )$ of a scalar field. Its derivative supplies the force term in the field equation, and its local minima describe candidate vacuum field values.

In vacuum analysis, an exact flat direction of a scalar potential is a nonconstant continuous family of degenerate stationary vacua. The scalar potential stays constant along this family and its gradient vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing eigenvalue of the quadratic Hessian matrix also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical scalar potential.

A scalar-field shift symmetry acts by $\varphi (x)\mapsto \varphi (x)+c$ for constant $c$. Its Noether charge is the spatial integral of the canonical momentum, up to the normalization of $c$, and interactions invariant under the symmetry contain derivatives of the scalar field.

Lagrangian field theory specifies an action by integrating a Lagrangian density $\mathcal{L}(\varphi ,\partial \varphi ,x)$ over spacetime. Requiring its first variation to vanish for compactly supported field variations yields the field Euler-Lagrange equations. The variational Lagrangian is the integrand, whereas this field-theoretic formalism specifies the continuum fields and their dynamics.

A Lagrangian density is integrated over spacetime to give the action $S=\int d^{d}x\mathcal{L}$ in Lagrangian.

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

A gauge-covariant kinetic term replaces ordinary derivatives by covariant derivatives. Expanding it generates interactions between the matter field and one or more gauge fields, including contact vertices with two gauge bosons.

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

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

For a field $\Phi$,

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

A continuous variational symmetry gives a current satisfying ${\partial }_{\mu }{j}^{\mu }=0$ on shell. Integrating $j^0$ over space gives a conserved charge when boundary flux vanishes.

For a symmetric conserved stress-energy tensor $\Theta$ and a Conformal Killing vector field $\omega$ in dimension $d$, the displayed conformal current has divergence $(\partial \cdot \omega ){\Theta }^{\mu }{}_{\mu }/d$. This follows by contracting the Conformal Killing equation with $\Theta$. A traceless tensor therefore gives a conserved current for every conformal generator.

A virial current $V^{\mu }$ represents the trace of a conserved stress-energy tensor as a divergence. Then the dilation Noether current $D^{\mu }=x_{\nu }{T}^{\mu \nu }-V^{\mu }$ is conserved, since its divergence is $T^{\mu }{}_{\mu }-{\partial }_{\mu }{V}^{\mu }$. This establishes scale invariance; a traceless stress-energy tensor improvement is the extra condition needed for conformal currents.

For $x^{\prime \mu }=x^{\mu }+\epsilon {\xi }^{\mu }$ and ${\varphi }_a^{\prime }(x^{\prime })={\varphi }_a(x)+\epsilon {\Delta }_a$, the fixed-coordinate field variation is $Q_a={\Delta }_a-{\xi }^{\nu }{\partial }_{\nu }{\varphi }_a$. Suppose action invariance is expressed as ${\delta }_0\mathcal{L}/\epsilon +{\partial }_{\mu }(\mathcal{L}{\xi }^{\mu })={\partial }_{\mu }{K}^{\mu }$. Writing ${\Pi }_a^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }{\varphi }_a)$ and $E_a={\mathcal{L}}_{{\varphi }_a}-{\partial }_{\mu }{\Pi }_a^{\mu }$, the product rule gives ${\partial }_{\mu }{j}^{\mu }=-{\sum }_a{E}_a{Q}_a$. Thus $j^{\mu }$ is a conserved current on shell. A vanishing spatial boundary flux is additionally required for its Noether charge to be conserved. This formula includes internal transformations, translations and Lorentz transformations.

For $\delta \varphi =\epsilon X$ and a variational symmetry of a Lagrangian density, write ${\Pi }^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }\varphi )$ and $\mathcal{E}={\mathcal{L}}_{\varphi }-{\partial }_{\mu }{\Pi }^{\mu }$. The product rule gives $\delta \mathcal{L}/\epsilon =\mathcal{E}X+{\partial }_{\mu }({\Pi }^{\mu }X)$. Therefore $j^{\mu }={\Pi }^{\mu }X-K^{\mu }$ obeys ${\partial }_{\mu }{j}^{\mu }=-\mathcal{E}X$, so it is a conserved current on shell. Spatial integration gives a conserved Noether charge only when the boundary flux vanishes. The formula extends to several components by summation.

A conserved current is a vector field $j^{\mu }$ satisfying the continuity equation ${\partial }_{\mu }{j}^{\mu }=0$. Its spatial integral is constant in time when no current crosses the boundary.

An integral of the time component of a conserved current is a conserved charge when the spatial boundary flux vanishes. Charges need not come from an immediately apparent elementary symmetry; a field theory can have an infinite higher-derivative current hierarchy. A Noether charge is obtained from the symmetry current supplied by the Noether theorem.

In a massless four-dimensional theory, current conservation gives a transverse tensor structure, $\int d^{4}x{e}^{-ikx}⟨J_{\mu }(x)J_{\nu }(0)⟩=(k_{\mu }{k}_{\nu }-{\delta }_{\mu \nu }{k}^2)G(k^2)$. With current mass dimension three, the scalar coefficient has dimension zero.

A Noether charge is the spatial integral $Q=\int j^0{d}^{3}x$ of a Noether current. It is time-independent when the current has vanishing flux through the spatial boundary, and in a Hamiltonian formulation it generates the associated continuous symmetry.

The dilatation current associated with scale transformations is $D^{\mu }=x_{\nu }{T}^{\mu \nu }$ up to improvement terms. Its divergence is the trace of the stress-energy tensor, so classical scale invariance requires a traceless improved tensor on shell.

Spacetime translations give the canonical stress-energy tensor

For a real scalar field in one spatial dimension with metric signature $(+,-)$ and potential $V$, the stress-energy conservation law is ${\partial }_t{T}^{01}+{\partial }_x{T}^{11}=0$. The force on the field to the left of a fixed cut is $-T^{11}$ at the cut when the far-left boundary flux vanishes. Finite-energy field configurations also permit this conclusion by spatial cutoffs, without assuming pointwise decay of all derivatives.

For the free real scalar field with signature $(+,-,-,-)$,The canonical stress-energy tensor is $T^{\mu \nu }={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi -{\eta }^{\mu \nu }\mathcal{L}$. Its divergence vanishes by the Klein-Gordon equation, giving constant integrated charges when integrals and boundary flux are well behaved. The minus sign in contravariant spatial momentum is from ${\partial }^i=-{\partial }_i$; lower-index spatial charges have the opposite sign. A plane wave $\cos (Et-\mathbf{k}\cdot \mathbf{x})$ has momentum density pointing along $\mathbf{k}$.

For a canonically quantized real scalar field, the quadratic Hamiltonian contains $\frac{1}{2}{\sum }_{\mathbf{p}}{E}_{\mathbf{p}}$ in a regulated box. Normal ordering subtracts this zero-point energy, leavingThe vacuum momentum is zero with an inversion-symmetric regulator. The canonical commutation relation then gives $[P^{\mu },a_{\mathbf{p}}^{†}]=p^{\mu }{a}_{\mathbf{p}}^{†}$, identifying the momentum and energy added by one particle. The un-subtracted energy differs by a constant and is not literally the same operator.

The canonical commutation relation gives $[P^{\mu },a_p^{†}]=p^{\mu }{a}_p^{†}$ for a free real scalar field. Differentiate the exponential conjugation: its derivative is a scalar multiple of the same operator, yielding the displayed phase. The annihilation operator receives the opposite phase, so at equal time $e^{-i\mathbf{P}\cdot \mathbf{y}}\varphi (\mathbf{x})e^{i\mathbf{P}\cdot \mathbf{y}}=\varphi (\mathbf{x}+\mathbf{y})$. The spatial translation operator translates a state's position wavefunction in the opposite argument direction.

For signature $(+,-,-,-)$, translation invariance of the Maxwell Lagrangian gives this tensor. It is generally neither symmetric nor gauge invariant. The antisymmetric superpotential improvement $-{\partial }_{\rho }(F^{\rho \mu }{A}^{\nu })$ produces, on the source-free equations of motion, the symmetric electromagnetic stress-energy tensor ${\Theta }^{\mu \nu }=-F^{\mu \rho }{F}^{\nu }{}_{\rho }+{\eta }^{\mu \nu }{F}^2/4$, which is traceless in four spacetime dimensions.

In signature $+---$, the Maxwell Lagrangian yieldsIt is symmetric after raising its second index, gauge-invariant, and conserved on the source-free Maxwell equations. Its energy density is $\frac{1}{2}({\mathbf{E}}^2+{\mathbf{B}}^2)$ and its energy flux is the Poynting vector. Its difference from the canonical stress-energy tensor is an antisymmetric superpotential divergence plus an equation-of-motion term, so integrated charges coincide when the constraints and vanishing boundary terms are imposed.

If $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$, commuting partial derivatives gives ${\partial }_{\mu }{\partial }_{\rho }{B}^{\rho \mu \nu }=0$. Such an improvement preserves conservation of a stress-energy tensor and changes integrated charges only by a spatial boundary term. The Belinfante-Rosenfeld stress-energy tensor is obtained through this type of improvement.

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.

For a symmetric conserved stress-energy tensor, the Lorentz current is $J^{\mu \rho \sigma }=x^{\rho }{T}^{\mu \sigma }-x^{\sigma }{T}^{\mu \rho }$. Its spatial integrals generate rotations and boosts.

The global phase symmetry of a Dirac field gives the conserved Dirac current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

For a commuting spinor in mostly-plus signature, orient its Dirac current to the future and write $V^i={ϵ}^{†}{\alpha }^iϵ$, with Hermitian ${\alpha }^i=-{\gamma }^0{\gamma }^i$. For every spatial unit vector $u$, $(u\cdot \alpha {)}^2=1$ implies $|u\cdot V|\le {ϵ}^{†}ϵ=V^0$. Choosing $u$ along the spatial current proves $V^2\le 0$. Thus spinor bilinears supply causal vectors suitable for energy-condition estimates. A Grassmann-valued spinor does not have this ordinary positivity interpretation.

In four dimensions a single commuting Majorana spinor is specified by a two-component complex spinor $\xi$. Its future Dirac current is proportional to $({\xi }^{†}\xi ,{\xi }^{†}\mathbf{\sigma }\xi )$, and the Pauli matrix multiplication law implies $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. Thus it is null. A Dirac spinor or a sum of independent Majorana currents can be timelike. This distinction matters when applying a timelike-current classification to minimal real supersymmetry.

The Gordon identity rewrites a vector-current matrix element between equal-mass on-shell Dirac spinors using a momentum term and an antisymmetric spin term. In the mostly-minus convention $\{{\Gamma }^a,{\Gamma }^b\}=2{\eta }^{ab}$, $(/p-m)u=0$, and ${\Gamma }^{ab}=[{\Gamma }^a,{\Gamma }^b]/2$, it reads $2m{\bar{u}}^{\prime }{\Gamma }^{a}u={\bar{u}}^{\prime }[(p^{\prime }+p{)}^a-{\Gamma }^{ab}(p_b^{\prime }-p_b)]u$. Derive it by adding the two external Dirac equation identities. The equivalent formula in the mostly-plus Dirac convention is ${\bar{u}}^{\prime }[{\gamma }^{ab}(p_b^{\prime }-p_b)+2m{\gamma }^a+p^a+p^{\prime a}]u=0$. The identity before division by $m$ remains valid for massless equal-mass states.

The Dirac electromagnetic current is the electric charge times the Dirac current. Its coupling $-e{j}^{\mu }{A}_{\mu }$ to the electromagnetic four-potential is the matter interaction of quantum electrodynamics.

An interaction $-\lambda j^{\mu }{\partial }_{\mu }\varphi$ with an exactly conserved current is redundant: integration by parts gives $\lambda \varphi {\partial }_{\mu }{j}^{\mu }$ up to a boundary term. For the Dirac current, the local phase redefinition $\psi =e^{-i\lambda \varphi }\chi$ removes the coupling, so its on-shell scattering amplitudes vanish.