Classical field theory

A fixed-coordinate field variation compares two fields at the same spacetime point. If a transformation also changes coordinates by $\delta x^{\mu }$, its total variation and fixed-coordinate variation differ by $\delta x^{\mu }{\partial }_{\mu }\varphi$. Keeping these separate is essential when applying the Noether current for a spacetime symmetry.

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.

Adding a free real scalar field $\varphi$ to covariantly gauge-fixed QED gives an infinitesimal global invariance $\delta A_{\mu }=\epsilon {\partial }_{\mu }\varphi$, $\delta \psi =-ie\epsilon \varphi \psi$, $\delta \varphi =-\epsilon (\partial \cdot A)/\xi$. The matter terms transform as an ordinary gauge transformation. The scalar kinetic variation combines with the gauge fixing variation into $\delta \mathcal{L}=-(\epsilon /\xi ){\partial }_{\mu }[(\partial \cdot A){\partial }^{\mu }\varphi ]$. Consequently the action is invariant under boundary conditions removing this total derivative, without using field equations.

An internal circle group action with spacetime-independent parameter $\alpha$. For canonical complex scalar fields, Noether's theorem gives $j^{\mu }=i{\sum }_j{q}_j{\phi }_j^{†}\overleftrightarrow{{\partial }^{\mu }}{\phi }_j$. The Noether charge of the particle annihilated by ${\phi }_j$ is $q_j$, and its antiparticle has charge $-q_j$.

The interaction $-\frac{1}{2}(h{\varphi }^2\Phi +h^{\ast }{\varphi }^{†2}{\Phi }^{†})$ is invariant under a global phase symmetry precisely when the displayed charge relation holds. With $q_{\varphi }=1$, a heavy particle annihilated by $\Phi$ has charge $-2$, so its allowed decay channel is $\Phi \to \bar{\varphi }\bar{\varphi }$. The reverse heavy antiparticle decays into $\varphi \varphi$. Crossing only one heavy external state changes the exchange channel of mixed-species quantum scattering.

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.

An interaction whose gauge boson couples to a matter current fixed by a gauge symmetry. In the Standard Model, the photon, gluons, W boson and Z boson have different flavour, charge and chirality selection rules.

A theory embedding the Standard Model gauge factors in a larger gauge group. Particular embeddings can relate couplings and matter multiplets or predict proton decay. The unification hypothesis alone does not specify a unique mass spectrum or an experimentally established extension.

The SU(6) representations $15={\Lambda }^{2}6$ and $\overline{6}$ branch under SU(5) as $10\oplus 5$ and $\overline{5}\oplus 1$. Their cubic gauge anomaly coefficients are $2$ and $-1$ in fundamental normalization. Thus $15\oplus 2\overline{6}$ is anomaly-free and gives one Standard Model family $10\oplus \overline{5}$, a vectorlike gauge spectrum $5\oplus \overline{5}$, and two singlets. With these representations and their conjugates, precisely one net chiral family requires $m_{15}-m_{\overline{15}}=1$ and $m_{\bar{6}}-m_6=2$, so this is the smallest such choice.

A quark-lepton unification framework with conventional minimal gauge group $SU(4{)}_C\times SU(2{)}_L\times SU(2{)}_R$. leptons supply a fourth color charge, and a right-handed neutrino completes each family. fermion masses additionally depend on symmetry breaking and Yukawa couplings.

The charge relation combines left and right weak isospin with the generator distinguishing quarks from leptons inside the extended color charge group.

For the right-handed neutrino, $T_{3R}=1/2$ and $B-L=-1$, so $Y=T_{3R}+(B-L)/2=0$. Its colour-singlet and left-weak-singlet assignments make it sterile under the Standard Model after symmetry breaking.

A gauge theory whose left- and right-chiral fermions transform differently under its gauge group. The weak charged current gives a physical example. A chiral representation assignment must satisfy the relevant gauge anomaly cancellation conditions to define a consistent quantum theory.

A formulation of a gauge theory with discretized spacetime. Matter fields are assigned to sites and gauge-group elements to links between nearby sites. Products of link variables around closed paths provide gauge-invariant quantities. This structure preserves the local gauge symmetry while making nonperturbative numerical calculations possible. Physical predictions require control of finite spacing, finite volume and statistical errors rather than treating one finite grid as the continuum theory.

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.

For $N$ Dirac fields with common mass, $\mathcal{L}=\bar{\Psi }(i/\partial -m{1}_N)\Psi$ is invariant under constant complex-linear rotations $\Psi \mapsto U\Psi$ with $U\in U(N)$. A Hermitian flavour generator $T^A$ has Noether current $j_A^{\mu }=\bar{\Psi }{\gamma }^{\mu }{T}^A\Psi$. A nondegenerate fermion mass matrix restricts the symmetry to unitary matrices commuting with it. If the theory is entirely free and rotations into charge-conjugate fields are allowed, its enhanced internal symmetry of free Dirac fields can be larger than this subgroup.

Without fixed gauge charges or interactions distinguishing particles from antiparticles, $N$ free Dirac fields give $2N$ left-handed Weyl fields after charge conjugation of the right-handed fields. Their massless kinetic term has unitary group $U(2N)$. Equal nonzero Dirac masses have equal positive singular values and can be written as $2N$ equal Majorana mass terms; their preserving group is the orthogonal group $O(2N)$. Explicitly, in a basis with symmetric mass matrix $m1$, invariance requires $V^{†}V=V^{T}V=1$, hence real $V$. Restricting to transformations preserving a chosen common fermion number gives the usual $U(N{)}_L\times U(N{)}_R$ at zero mass and $U(N)$ at nonzero mass.

The free kinetic terms of left- and right-handed Weyl fields are separately invariant under unitary group rotations in flavour space. For $N$ massless Dirac fields, the subgroup preserving common fermion number is $U(N{)}_L\times U(N{)}_R$. A common nonzero Dirac mass term couples the two chirality sectors and leaves its diagonal $U(N)$ subgroup. Gauge interactions can obstruct some of the classical axial currents through a chiral anomaly.

Two canonically normalized complex scalar fields with potential $V=m^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2$ are four real fields with a radial potential. Consequently the isolated action is invariant under the orthogonal group $O(4)$, not only under complex-linear $U(2)$ transformations. Writing ${\varphi }_1=(q_1+i{q}_2)/\sqrt{2}$ and ${\varphi }_2=(q_3+i{q}_4)/\sqrt{2}$ makes the kinetic term ${\sum }_A(\partial q_A{)}^2/2$ and the potential a function of ${\sum }_A{q}_A^2$. The six real-plane rotations give currents $j_{AB}^{\mu }=q_B{\partial }^{\mu }{q}_A-q_A{\partial }^{\mu }{q}_B$. Gauge or Yukawa couplings can break this larger symmetry, so it should not be asserted for an arbitrary interacting doublet.

For $\mathcal{L}={\varphi }_{1,t}{\varphi }_{2,t}-c^2{\varphi }_{1,x}{\varphi }_{2,x}$, the real linear internal transformations preserving the density are $\mathrm{diag}(r,r^{-1})$ and $\left(\begin{array}{cc}0& r\\ r^{-1}& 0\end{array}\right)$ with $r\ne 0$. There are four connected components; the identity component is the displayed reciprocal rescaling with $r=e^s>0$. In the sum-and-difference basis it preserves a bilinear form of signature $(1,1)$, with the identity component isomorphic to that of the One-dimensional Lorentz group. Its Noether current in coordinates $(t,x)$ is $j^0={\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t}$ and $j^x=c^2({\varphi }_2{\varphi }_{1,x}-{\varphi }_1{\varphi }_{2,x})$. The two wave equations give ${\partial }_t{j}^0+{\partial }_x{j}^x=0$. The energy quadratic form is indefinite: the fields ${\varphi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ have equal and opposite kinetic signs.

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 ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$.

The densitycontains the term linear in the potential from minimal coupling, but omits $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\square +m^2)\varphi +ie(\partial \cdot A)\varphi +2ieA\cdot \partial \varphi =0$, together with its complex conjugate. The omitted quadratic interaction must not be added silently when solving this specified theory.

For $\delta \varphi =-iϵ\varphi$, the conserved current is $J^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2e{A}^{\mu }|\varphi {|}^2$. The conjugate field equations implywhich directly cancels the divergence of the explicit background term. Conservation requires no Lorenz gauge condition; constancy of total charge additionally requires vanishing boundary flux.

For the linear-in-potential charged-scalar Lagrangian, $\pi ={\dot{\varphi }}^{\ast }-ie{A}_0{\varphi }^{\ast }$ and ${\pi }^{\ast }=\dot{\varphi }+ie{A}_0\varphi$. With $A_{\mu }=(A_0,-\mathbf{A})$, the Hamiltonian density isThe first product includes $+e^2{A}_0^2|\varphi {|}^2$ after the Legendre transform; that term can arise in canonical variables even though the original Lagrangian has no interaction quadratic in $A$.

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.

The special unitary group $SU(2)$ acts transitively on every sphere in its defining ${\mathbb{C}}^2$: for $|u{|}^2+|v{|}^2=1$, the matrix with columns $(u,v{)}^T$ and $(-\bar{v},\bar{u}{)}^T$ is in $SU(2)$. Thus any invariant real polynomial in a single commuting doublet and its complex conjugate depends only on ${\varphi }^{†}\varphi$. Through total degree four it is $V_0+m^2{\varphi }^{†}\varphi +\lambda ({\varphi }^{†}\varphi {)}^2$. A holomorphic polynomial invariant of positive degree does not exist for one doublet: ${\mathrm{Sym}}^{d}2$ is a nontrivial irreducible representation for every $d>0$.

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.

The parameter multiplying a scalar quadratic mass term. A negative value describes an instability of the expansion point, rather than a negative value of the physical mass of a stable excitation. Nonnegative quartic terms do not stabilize a negative quadratic term along an exactly flat quartic ray.

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

For a field $\Phi$,

An off shell field configuration is not required to satisfy the Euler-Lagrange field equation. An external four-momentum is off shell if it is not constrained to its physical mass shell. On shell identities can therefore fail for general untransformed correlation functions, even when the associated S-matrix is unchanged by a field redefinition.

Adding $({\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t})/(2\tau )$ to the cross-coupled scalar Lagrangian density produces independent equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}+{\varphi }_{1,t}/\tau =0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}-{\varphi }_{2,t}/\tau =0$. For the convention $e^{i(kx-\omega t)}$, their frequencies are respectively $-i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$ and $+i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$. Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies $(A+Bt)e^{\mp t/(2\tau )}$. At $k=0$ each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.

For an action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,{\partial }^2\varphi )d^{d+1}x$, vary the field with compact support, or fix both $\delta \varphi$ and its first derivatives on the boundary. One integration by parts on the first-derivative term and two on the second-derivative term give the displayed Euler-Lagrange field equation. With ordered repeated indices, define the second-derivative coefficient symmetrically so that $\delta \mathcal{L}={\mathcal{L}}_{\varphi }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu }}{\partial }_{\mu }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu \nu }}{\partial }_{\mu }{\partial }_{\nu }\delta \varphi$. This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.

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.

Rotating the two transverse displacement components around the string axis gives density ${\ell }_x=\rho ({\psi }_y{\dot{\psi }}_z-{\psi }_z{\dot{\psi }}_y)$ and flux ${\mathcal{J}}_x=-F({\psi }_y{\psi }_z^{\prime }-{\psi }_z{\psi }_y^{\prime })$. The pair of wave equations yields ${\partial }_t{\ell }_x+{\partial }_x{\mathcal{J}}_x=0$. The total charge is physical angular momentum about the axis; it is conserved only when the net endpoint torque flux vanishes.

For $\mathcal{L}=\rho ({\dot{\psi }}_y^2+{\dot{\psi }}_z^2)/2-F(({\psi }_y^{\prime }{)}^2+({\psi }_z^{\prime }{)}^2)/2$, constant displacement symmetry in $y$ gives $j^0=\rho {\dot{\psi }}_y$ and $j^x=-F{\psi }_y^{\prime }$. The wave equation implies ${\partial }_t{j}^0+{\partial }_x{j}^x=0$. Its integrated charge is physical transverse momentum; endpoint forces can change that charge on a finite segment.

For a complex scalar field multiplet with canonical kinetic term and scalar potential invariant under $\Phi \mapsto e^{i\epsilon T}\Phi$, a constant Hermitian matrix $T$ gives the displayed Noether current. Vary the field and its conjugate independently in the Noether current for a first-derivative scalar field. A common alternative convention reverses both the parameter and current sign. Its divergence vanishes on shell.

For a longitudinal displacement field $\psi (z,t)$ and Lagrangian density $\mathcal{L}=A[\rho {\dot{\psi }}^2-K{\psi }^{\prime 2}]/2$, uniform field shifts give Noether current $j^0=A\rho \dot{\psi }$, $j^z=-AK{\psi }^{\prime }$. Its spatial integral is the rod's longitudinal mechanical momentum. A prescribed position-dependent mass density changes $j^0$ to $A\rho (z)\dot{\psi }$ but does not break this internal shift symmetry. Boundary traction can carry momentum into or out of a finite rod, so action symmetry alone does not guarantee conservation of the integrated charge.

If a Lagrangian density depending on first and second derivatives has no undifferentiated field, the scalar-field shift symmetry $\delta \varphi =ϵ$ gives the displayed Noether current. Its divergence is minus the Euler-Lagrange field equation with second derivatives, so it is conserved on shell. The spatial integral of $j^0$ is constant only if the boundary current vanishes. For $\mathcal{L}=\rho {\varphi }_t^2/2+\alpha {\varphi }_x^2/2-\beta {\varphi }_{xx}^2/2$, it gives $j^0=\rho {\varphi }_t$, $j^x=\alpha {\varphi }_x+\beta {\varphi }_{xxx}$. Fixed transverse endpoint positions break the global shift symmetry of the boundary problem; support forces can then change the total transverse momentum even though the bulk continuity equation still holds.

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.

For $\mathcal{L}=(\partial \varphi {)}^2/2-\lambda {\varphi }^4$ and the displayed-weight dilation variation $\delta \varphi =\varphi +x\cdot \partial \varphi$, the full divergence identity is $(4-n)\mathcal{L}+\delta \varphi (\square \varphi +4\lambda {\varphi }^3)$. The field equation removes the second term. In four spacetime dimensions the density changes by a total divergence, so the Noether current includes the subtraction $-x^{\mu }\mathcal{L}$. This is a classical statement about generic solutions and this particular current.

For a first-derivative real scalar field in one space and one time dimension, put $j_D^{\mu }={\mathcal{L}}_{{\partial }_{\mu }\psi }(z{\psi }^{\prime }+t\dot{\psi })-x^{\mu }\mathcal{L}$. Its on-shell divergence isA quadratic derivative density with arbitrary field-dependent coefficients has zero derivative-homogeneity term. An extra term $-\alpha {\psi }^{\prime 4}$ instead gives $-2\alpha {\psi }^{\prime 4}$. A position-dependent kinetic coefficient $A\rho (z)/2$ gives $-Az{\rho }^{\prime }(z){\dot{\psi }}^2/2$. These source terms distinguish scale invariance from internal field-shift symmetry.

For constant $A,\rho ,K$ on all space and time, the active waveform transformation ${\psi }_b(z,t)=\psi (bz,bt)$ leaves the action of $\mathcal{L}=A(\rho {\dot{\psi }}^2-K{\psi }^{\prime 2})/2$ invariant: the density gains $b^2$ and the integration measure gains $b^{-2}$. With canonical momentum $\Pi =A\rho \dot{\psi }$, its Noether current can be chosen asThe integrated charge $D=\int j_D^{0}dz$ is conserved if its boundary flux vanishes and the weighted integral exists. For a rightgoing wave packet $\psi =f(z-ct)$, $c=\sqrt{K/\rho }$, it becomes $D=-A\rho c\int \xi [f^{\prime }(\xi ){]}^{2}d\xi$. A fixed finite interval generally breaks this scale symmetry unless its boundaries and boundary data are transformed too.

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.