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A Non-Abelian gauge theory describes fields with a local redundancy, not an independent physical degree of freedom for each component of its gauge field. Take a compact gauge group with Hermitian generators $T^a$ and Lie algebra structure constants defined by $[T^a,T^b]=i{f}^{abc}{T}^c$. In a mostly-plus metric signature, the pure Yang-Mills theory hasWith the matter gauge covariant derivative $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$, a finite gauge transformation acts as $A_{\mu }\mapsto U{A}_{\mu }{U}^{-1}-(i/g)({\partial }_{\mu }U)U^{-1}$. Expanding $U=e^{ig{\omega }^a{T}^a}$ givesThe Yang-Mills action is constant along each gauge orbit. Consequently a naive functional integral $\int \mathcal{D}A{e}^{iS[A]}$ repeatedly integrates the same physical configuration and contains the infinite formal volume of the local gauge group. At quadratic order the same problem appears as noninvertibility: the free kinetic operator annihilates $A_{\mu }={\partial }_{\mu }\omega$, so no unique gauge-boson propagator exists. In canonical language the Gauss law constraint in gauge theory and the absence of an independent momentum for the time component show that the field components cannot all be independent oscillators. In covariant quantization, time-like and longitudinal polarizations must not be counted as additional physical states.

The Faddeev-Popov determinant implements a local quotient by the gauge orbit. Choose $G^a[A]={\partial }^{\mu }{A}_{\mu }^a$ and first impose $G^a[A]=f^a(x)$ for a specified function $f$. This is the Lorenz gauge condition when $f=0$. Its infinitesimal change isThe direct Faddeev-Popov operator is therefore $M_0^{ab}={\partial }^{\mu }(D_{\mu }{)}^{ab}$. Locally, if the condition cuts each orbit once and residual zero modes have been removed by boundary conditions, changing variables from $\omega$ to $G[A^{\omega }]-f$ proves the Faddeev-Popov gauge-orbit identityThe determinant is the functional Jacobian of this change of variables. A finite-dimensional version is $\int d\omega \delta (G(\omega )-f)=1/|\det (\partial G/\partial \omega )|$ near a unique root; the perturbative determinant has a fixed sign or phase, absorbed into normalization. This is the same change-of-variables mechanism used in Bastianelli's derivation of the Faddeev–Popov construction.

Insert the identity into the normalized functional integral for a gauge-invariant observable $\mathcal{O}[A]$:For each fixed $\omega$, change the integration variable from $A$ to $A^{\omega }$. The action, observable and measure are invariant, and the determinant is the orbit Jacobian at the resulting gauge slice. The integrand is now independent of the separately integrated orbit parameter. Its integral is $\mathrm{Vol}\mathcal{G}$, canceling the denominator. Thus, up to normalization,This assumes an anomaly-free measure and a local perturbative gauge slice; it is not an assertion of a globally unique representative on every gauge orbit.

Average over $f$ with weight $\exp [-i\int d^{4}x{f}^a{f}^a/(2\xi )]$. The delta functional sets $f=G[A]$, so the average produces the covariant gauge termNext represent the Faddeev-Popov determinant by a Grassmann Gaussian integral. Choose $M=-M_0=-{\partial }^{\mu }{D}_{\mu }$; changing $M_0$ to $-M_0$ changes the regulated determinant by a field-independent factor that is absorbed in normalization. For a Faddeev-Popov ghost field $c$ and independent antighost field $\bar{c}$,To see why it is a determinant rather than its reciprocal, diagonalize a finite-dimensional quadratic form formally: for each Grassmann pair only the term proportional to ${\bar{c}}_j{c}_j$ survives integration, supplying one eigenvalue. Multiplying the surviving eigenvalues gives $\det M$, with an overall factor of $i$ per pair. The functional version is defined with the same regulator as the rest of the theory.

Combining these steps gives the gauge-fixed Yang–Mills LagrangianThis is the gauge-fixed Yang-Mills Lagrangian in a covariant gauge. Integrating its ghost term by parts givesThe second term is the ghost-gluon vertex in the antighost convention chosen here. The Faddeev-Popov ghosts are anticommuting Lorentz scalars introduced to represent a determinant; they are not extra asymptotic particles. A closed ghost loop has a minus sign from their Grassmann parity. In a Non-Abelian gauge theory the determinant depends on $A$, so these loops are essential. In an Abelian gauge theory $D_{\mu }$ in the adjoint sector reduces to ${\partial }_{\mu }$, the determinant is field independent, and its ghost sector decouples.

The quadratic gauge field kernel at non-null momentum isDefine the mixed-index longitudinal projector of a vector field $P_L{}^{\mu }{}_{\nu }=k^{\mu }{k}_{\nu }/k^2$ and transverse projector of a vector field $P_T=1-P_L$. The kernel is $-k^2(P_T+{\xi }^{-1}{P}_L)$ and hence has inverse $-k^{-2}(P_T+\xi P_L)$. Multiplying by $i$ gives the gauge-boson propagator, off its poles,The vacuum Feynman i-epsilon prescription completes this off-pole formula; with the mostly-plus convention its physical transverse denominator is $k^2-i0$. The choice $\xi =1$ is Feynman gauge; $\xi \to 0$ is covariant Landau gauge. The invertible quadratic kernel supplies perturbation theory, while the ghost determinant ensures the gauge-orbit measure has been accounted for.

The consistency of the unphysical sector is organized by BRST symmetry. Introduce a bosonic Nakanishi-Lautrup field $B^a$ and a left-acting BRST differentialThe graded Leibniz rule and the Jacobi identity give $s^2=0$: for example $s^{2}A=D(sc)+g[Dc,c]=0$ since $D[c,c]=2[Dc,c]$, and $s^{2}c$ vanishes by the graded Jacobi identity. Nilpotence on $\bar{c}$ and $B$ is immediate. Using the odd antighost field, the same graded Leibniz rule givesEliminating $B^a$ by its algebraic field equation $B^a=-G^a/\xi$ recovers the displayed gauge-fixing and ghost terms. Thus they are BRST-exact together, and the total action preserves BRST symmetry. At the level of states, the BRST charge $Q$ implements physical states as the ghost-number-zero BRST cohomology $\ker Q/\mathrm{im}Q$. This prescription removes longitudinal, time-like and ghost excitations from physical amplitudes while allowing them on internal lines. For a BRST-closed observable, changing $\xi$ inserts $s({\bar{c}}^a{B}^a/2)$; its expectation vanishes by the BRST change-of-variables identity when the measure preserves the symmetry. This explains the independence of physical quantities from the arbitrary gauge parameter.

Finally, a local condition such as Lorenz gauge can have several solutions on the same gauge orbit, the Gribov ambiguity. The preceding Faddeev-Popov gauge-orbit identity and perturbative gauge fixing construction apply in a neighborhood where the linearized operator is invertible after residual modes are excluded. They resolve perturbative gauge overcounting and supply consistent covariant Feynman rules; they do not by themselves prove a globally unique nonperturbative gauge fixing.

A symmetry preserves the action, possibly up to a boundary term. In a classical field theory, this connects transformations of fields to conserved quantities. In a quantum field theory, it also constrains states, observables and scattering amplitudes. Continuous symmetries have Noether currents; discrete symmetries can be equally important without supplying a current through the same theorem.

For a local density depending on fields ${\Phi }_a$ and their first derivatives, define ${\Pi }_a^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }{\Phi }_a)$ and ${\mathcal{E}}_a=\partial \mathcal{L}/\partial {\Phi }_a-{\partial }_{\mu }{\Pi }_a^{\mu }$. Suppose a one-parameter transformation has fixed-coordinate variation $\delta {\Phi }_a=\epsilon R_a$ and satisfies $\delta \mathcal{L}=\epsilon {\partial }_{\mu }{K}^{\mu }$. The Noether's first theorem asserts thatFor Grassmann-valued fields the derivatives and ordering are chosen consistently; the same integration-by-parts argument applies. To prove the theorem, expand the density variation and integrate its derivative term once algebraically:Comparison with the assumed divergence gives ${\partial }_{\mu }{j}^{\mu }=-{\sum }_a{\mathcal{E}}_a{R}_a$. On shell every Euler-Lagrange expression vanishes. Integrating this continuity equation over space gives the Noether charge $Q=\int j^0{d}^{3}x$ and $dQ/dt=-{\int }_{\partial {\mathbb{R}}^3}\mathbf{j}\cdot d\mathbf{S}$. Thus $Q$ is conserved whenever the boundary flux vanishes. Boundary conditions are part of the conservation statement; the local identity alone does not discard nonzero flux.

Translations illustrate the theorem in every relativistic field theory. For $R_a=a^{\nu }{\partial }_{\nu }{\Phi }_a$ and $K^{\mu }=a^{\mu }\mathcal{L}$, the current is $a^{\nu }{T}^{\mu }{}_{\nu }$, whereThe four charges are energy and momentum. For the Klein-Gordon field they reduce to the expressions derived in Question 1. Lorentz invariance gives angular momentum and boost currents. With a symmetric improved stress-energy tensor, these are $M^{\lambda \mu \nu }=x^{\mu }{T}^{\lambda \nu }-x^{\nu }{T}^{\lambda \mu }$. The canonical Dirac expression also carries an intrinsic spin current; improvement incorporates it into the symmetric tensor. Together translations and Lorentz transformations form the Poincaré group.

Internal symmetries act on field components without moving spacetime points. The global phase symmetry of a complex scalar field, $\delta \varphi =-i\epsilon \varphi$, hasIts charge distinguishes particles and antiparticles. A Dirac phase transformation $\delta \psi =-i\epsilon \psi$ gives $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$. This is preserved by the real-scalar Yukawa interaction, because the phases of $\bar{\psi }$ and $\psi$ cancel; consequently fermion number is conserved even while scalar and fermion particles interact. A multiplet of real scalars with a potential depending only on ${\sum }_a{\varphi }_a^2$ has orthogonal internal rotations. Their antisymmetric currents can be written $j_{ab}^{\mu }={\varphi }_a{\partial }^{\mu }{\varphi }_b-{\varphi }_b{\partial }^{\mu }{\varphi }_a$.

Discrete examples include $\varphi \mapsto -\varphi$ for an even real-scalar potential, parity symmetry in quantum field theory, and charge conjugation, which interchanges a complex scalar with its conjugate and reverses the electromagnetic potential. Discrete transformations are not generated by an infinitesimal continuous parameter, so Noether theorem does not attach a local conserved current to each of them. Nevertheless they forbid interaction terms and relate physical processes.

A global symmetry in field theory uses parameters constant across spacetime and can act nontrivially on physical states. A gauge redundancy allows arbitrary spacetime-dependent parameters and relates descriptions of the same physical configuration. In scalar electrodynamics, replacing the scalar's global phase by a local phase requires the compensating gauge transformation of $A_{\mu }$ and the gauge covariant derivative; the invariant contraction uses the opposite representation for ${\varphi }^{\ast }$. In Yang-Mills theory, matter transforms in a representation of a non-Abelian gauge group and the gauge potential transforms so that covariant derivatives and field strengths transform covariantly. Gauge fixing, such as Coulomb gauge, chooses representatives of these descriptions rather than changing physical predictions. The current-contracted propagator identity of Question 3 illustrates why a noncovariant gauge choice leaves scattering Lorentz invariant.

The arbitrary local parameter produces a stronger statement than one independent conserved charge for each function. For the abelian scalar transformation, vary the action with an arbitrary compactly supported $\alpha (x)$ and integrate its derivative term:ThereforeThis consequence of Noether second theorem, the Noether identity for abelian scalar gauge symmetry, holds off shell: gauge invariance makes the field equations dependent. In Hamiltonian language the same redundancy is reflected by the Gauss law constraint in gauge theory and the elimination of unphysical gauge degrees of freedom. Gauge transformations that vanish at the boundary are redundancies; transformations with nontrivial boundary behaviour can instead carry physical surface charges. This is why identifying every gauge transformation with a zero-charge operation would be too strong.

Finally, symmetry of the action need not imply symmetry of a chosen vacuum. For a complex scalar with a symmetry-preserving potential whose minima occur at nonzero $|\varphi |$, different constant phases label degenerate vacua. Choosing one breaks the global phase symmetry of the state; the Goldstone theorem supplies a massless mode under its usual relativistic assumptions. When the same phase is gauged, the Higgs mechanism describes the phase degree of freedom becoming the longitudinal polarization of a massive gauge boson; the local redundancy is still present in the underlying description. At the quantum level a classical symmetry can also fail through a quantum anomaly, so a classical Noether derivation alone does not guarantee an exact quantum Ward identity. Conserved charges, physical global transformations and local gauge redundancy are related but distinct consequences of symmetry.

Fix the current sign convention by defining the localized variation as ${\delta }_{ϵ}S=-\int d^{d}x({\partial }_{\mu }{ϵ}^a)j_a^{\mu }$. For a first-derivative Lagrangian invariant without a boundary term under constant parameters, this means $j_a^{\mu }=-\partial \mathcal{L}/\partial ({\partial }_{\mu }\varphi )t_a\varphi$; include the usual improvement term if the constant variation is a total derivative. On solutions, arbitrary compactly supported parameters imply ${\partial }_{\mu }{j}_a^{\mu }=0$. The Noether charge is $Q_a=\int d^{d-1}x{j}_a^0$ and is conserved if the spatial flux vanishes. This sign convention matches the printed Ward identity; reversing the current also reverses the corresponding generator convention.

Assume an invariant regulated functional measure, invariant vacuum boundary conditions and no quantum anomaly. Changing variables in the normalized path integral gives $0=⟨{\delta }_{ϵ}X⟩+i⟨X{\delta }_{ϵ}S⟩$. Integration by parts then yieldsFor a product of scalar fields, the local Ward identity contact terms areAway from the insertions this is current conservation. Time-ordering or the distributional functional identity supplies the contact terms.

For the gauge theory write $[X,Y{]}^a=f^{abc}{X}^b{Y}^c$ and use the left-acting BRST differential $s$, with ${\delta }_{ϵ}=ϵs$. It obeys the graded Leibniz rule andAssume the structure constants satisfy the Jacobi identity and the dot product is invariant; antisymmetry alone would not be enough. Since $c$ and $D_{\mu }c$ are odd, their Lie brackets are symmetric in these two arguments. ConsequentlyAlso $s{F}_{\mu \nu }=D_{\mu }{D}_{\nu }c-D_{\nu }{D}_{\mu }c=[F_{\mu \nu },c]$. The graded Jacobi identity gives $s[c,c]=0$, so $s^{2}c=0$; the other three fields have zero second variation immediately. For independent odd parameters, ${\delta }_{{ϵ}^{\prime }}{\delta }_{ϵ}=-{ϵ}^{\prime }ϵs^2=0$. These are off-shell BRST nilpotence identities because the Nakanishi-Lautrup field $b$ is retained.

The Yang-Mills theory variation is proportional to $F^{\mu \nu }\cdot [F_{\mu \nu },c]=0$. The gauge-fixing variation is $({\partial }^{\mu }b)\cdot D_{\mu }c$, while the ghost variation is its negative; the $b^2$ term does not vary. Equivalently these terms are $s\Psi$ for the gauge-fixing fermion $\Psi =({\partial }^{\mu }\bar{c})\cdot A_{\mu }+\xi \bar{c}\cdot b/2$. BRST nilpotence makes this expression invariant. Ghost-number scaling also leaves every term invariant: the ghost and antighost factors carry opposite weights.

Here are the two explicit currents in the same sign convention as the Ward identity. Localizing the even ghost parameter gives coefficient $({\partial }_{\mu }\theta )[\bar{c}\cdot D^{\mu }c-({\partial }^{\mu }\bar{c})\cdot c]$. Localizing the odd parameter, keeping it on the left, gives coefficientThe last sign comes from moving the odd parameter through the odd antighost derivative. Since the current was defined as minus this coefficient,These are the ghost-number Noether current and the BRST current in derivative-b gauge fixing. If the opposite Noether sign is used, both displayed currents acquire an overall minus sign. Integrating the gauge-fixing term by parts changes the Noether representative by the associated boundary improvement; mixing the two Lagrangian conventions without that improvement gives incorrect signs.

With no BRST anomaly, the conserved odd BRST charge has $Q_B^2=0$. The BRST cohomology identifies closed states $Q_B|\psi ⟩=0$ modulo exact states $Q_B|\chi ⟩$. Nilpotence puts every exact state in the closed space. In the usual indefinite gauge-fixed state space, a Hermitian BRST charge makes exact states orthogonal to closed states; the standard no-ghost/positivity assumptions then give a physical inner product on the quotient. The physical sector is its ghost-number-zero component,The ghost number assigns $+1$ to $c$, $-1$ to $\bar{c}$ and zero to gauge and auxiliary fields. Gauge-invariant observables and the chosen vacuum have ghost number zero; unphysical ghost excitations are removed in BRST pairs. Thus physical representatives are expected to satisfy $Q_G|\psi ⟩=0$. This zero-grading selection is part of the physical-state prescription, not a consequence of nilpotence alone.

A Non-Abelian gauge theory promotes an internal group representation symmetry to a spacetime-dependent transformation and introduces a gauge field so that differentiation transforms covariantly. The symmetry is a gauge redundancy: physically meaningful quantities must be invariant under local changes of field representative. We work in four-dimensional Minkowski spacetime with signature $(+---)$, and use skew-Hermitian matrix generators throughout.

For the compact Simple Lie group $G$, its real Lie algebra has a negative-definite Killing form. Hence $B=-\kappa$ is a positive invariant bilinear form on a Lie algebra. Choose a $B$-orthonormal basis $T_a$, $a=1,\dots ,\dim G$, with $[T_a,T_b]=f_{ab}{}^c{T}_c$. Invariance makes $f_{abc}=B([T_a,T_b],T_c)$ fully antisymmetric. This construction uses the Adjoint representation and works independently of any chosen scalar representation.

Let ${\mathcal{A}}_{\mu }={\mathcal{A}}_{\mu }^a{T}_a$ be a Lie-algebra-valued gauge potential, with the coupling absorbed into this connection. If $R$ is the given finite-dimensional irreducible representation of $G$, its Lie algebra representation $\rho =dR$ definesFor a finite local transformation $U(x)\in G$, require ${\Phi }^{\prime }=R(U)\Phi$ and $D_{\mu }^{\prime }{\Phi }^{\prime }=R(U)D_{\mu }\Phi$. Differentiating the first equation forces the gauge-field transformation lawThus the gauge potential transforms inhomogeneously; an ordinary derivative alone would leave an uncancelled derivative of $U$.

The commutator of gauge covariant derivatives determines the gauge field strength:The Lie algebra representation preserves the bracket, so this formula holds for any $R$, including a trivial scalar representation. The gauge field strength itself is defined in the gauge algebra. The transformation of the connection impliesIt is therefore covariant, rather than invariant as a Lie-algebra-valued quantity.

For infinitesimal $U=1+ϵ+O({ϵ}^2)$, the same equations give the infinitesimal gauge transformation with an anti-Hermitian connection:Substitution into $\delta (D_{\mu }\Phi )$ cancels the terms containing ${\partial }_{\mu }ϵ$ and gives $\delta (D_{\mu }\Phi )=\rho (ϵ)D_{\mu }\Phi$, the gauge covariance of a scalar covariant derivative.

The Yang-Mills theory has Lagrangian densityThis is the Killing-form Yang-Mills Lagrangian. Its sign uses a positive internal form $B$ and the stated spacetime signature. In a $B$-orthonormal basis it is $-{\mathcal{F}}_{\mu \nu }^a{\mathcal{F}}^{a\mu \nu }/(4g_{\mathrm{YM}}^2)$. Gauge invariance follows because $B$ is invariant under the adjoint group action. Infinitesimally, the variation is proportional toAfter rescaling ${\mathcal{A}}_{\mu }=g_{\mathrm{YM}}{A}_{\mu }$, the gauge coupling appears in $D_{\mu }={\partial }_{\mu }+g_{\mathrm{YM}}\rho (A_{\mu })$ and in the non-Abelian part of the canonically normalized field strength. Squaring that field strength gives cubic and quartic gauge-field self-interactions. An ordinary local quadratic mass term in the gauge potential is not invariant because its transformation contains ${\partial }_{\mu }ϵ$.

For the scalar field kinetic term, compactness supplies a positive invariant Hermitian form $h$ on its representation space. One can obtain it by averaging any positive form with normalized Haar measure, the unitarization of a compact-group representation. Then $R(U)$ is unitary for $h$, and $\rho (ϵ)$ is anti-Hermitian. The Yang-Mills theory coupled to an arbitrary scalar representation hasHere $V$ is any real gauge-invariant scalar potential, so $V(R(U)\Phi )=V(\Phi )$. For example,is available for every unitary representation, with $\lambda \ge 0$ giving a bounded quartic contribution. Other invariant interactions may exist for particular representations; the potential is not required to depend only on $h(\Phi ,\Phi )$.

The scalar kinetic term is gauge-invariant because both covariant derivatives transform by the same unitary $R(U)$. Infinitesimally its variation vanishes byThe potential is invariant by its defining condition. For a real irreducible representation, use the averaged positive symmetric form instead and write the kinetic term as $\frac{1}{2}h(D_{\mu }\Phi ,D^{\mu }\Phi )$; all transformation and invariance arguments remain valid. Expansion of the covariant kinetic term produces both the linear gauge-scalar interaction and the quadratic term needed by local gauge invariance.

The invariant inner products, gauge covariant derivatives and gauge field strength together construct the full theory for every compact Simple Lie group and every supplied finite-dimensional irreducible representation of the scalar fields. Both finite and infinitesimal transformation laws have been included, and every sign follows the same anti-Hermitian convention.

Use Lie algebra generators with $[T_b,T_c]=f^a{}_{bc}{T}_a$, and define the adjoint gauge covariant derivative by $(D_{\mu }X{)}^a={\partial }_{\mu }{X}^a+f^a{}_{bc}{A}_{\mu }^b{X}^c$. Varying the gauge field strength gives $\delta F_{\mu \nu }=D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$. Antisymmetry followed by integration by parts therefore givesFor variations vanishing at the boundary, the Yang-Mills equations areThe Jacobi identity for the commutators of gauge covariant derivatives, using $[D_{\mu },D_{\nu }]X=[F_{\mu \nu },X]$, gives the gauge-theory Bianchi identity:In the second form of the gauge-theory Bianchi identity ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. The identity is a geometric consequence of the definition of curvature, not a second dynamical field equation.

For the usual non-Abelian Yang-Mills theory, the quantum running coupling becomes strong at low energies. Dimensional transmutation generates a scale absent from the classically scale-invariant equations. The expected confining dynamics and massive colour-singlet spectrum involve confinement and a mass gap, rather than freely propagating weakly coupled coloured waves. These nonperturbative effects are not captured by simply solving the classical equations; this is not a claim of a mathematical proof of the Yang-Mills mass gap.

Write the gauge condition as ${\chi }^a[A]=f^a[A]$ to distinguish it from the structure constants. Under an infinitesimal gauge transformation, ${\delta }_{\omega }{A}_{\mu }=D_{\mu }\omega$. Define the Faddeev-Popov operator by ${\mathcal{M}}^{ab}=\delta {\chi }^a[A+D\omega ]/\delta {\omega }^b{|}_{\omega =0}$. In a Euclidean convention, the gauge-fixing and ghost additions can be chosen asA nonlocal kernel would require a double integral; a local functional $\chi$ instead makes $\mathcal{M}$ a local differential operator. Choosing a local gauge functional thus retains a local gauge-fixed action and the ordinary local interaction structure of perturbative quantum field theory.

The fields $c^a$ and ${\overline{c}}^a$ are independent Grassmann-valued fields in the Adjoint representation, Lorentz scalars with ghost numbers $+1$ and $-1$. They are the Faddeev-Popov ghost fields: their functional integral produces $\det \mathcal{M}$, the Faddeev-Popov determinant correcting the volume element along a gauge orbit. They are internal fields, not physical asymptotic particles. Their statistics supply a minus sign for every closed ghost loop. No auxiliary field is needed in the displayed gauge-fixing representation.

In axial gauge, ${\chi }^a=n^{\mu }{A}_{\mu }^a$, soWith the Fourier transform convention $e^{ik\cdot x}$, the ghost Feynman rules in the original normalization of $A$ areMomentum conservation accompanies the vertex; each ghost loop has the extra minus sign. With a canonically normalized field $A=g{A}_{\mathrm{can}}$, the vertex is instead $-g{f}^a{}_{bc}{n}_{\mu }$. The gauge fixing term is quadratic and introduces no further interaction vertex.

The Euclidean quadratic kernel of $A$ is $g^{-2}[k^2{\delta }_{\mu \nu }-k_{\mu }{k}_{\nu }+{\xi }^{-1}{n}_{\mu }{n}_{\nu }]$. For $n\cdot k\ne 0$, its inverse isThis follows by multiplication with the kernel: the $k^2\delta -kk$ piece gives $I-n{k}^T/(n\cdot k)$, and the ${\xi }^{-1}nn$ piece supplies the missing $n{k}^T/(n\cdot k)$. The strict axial-gauge propagator is thereforeIt obeys $n^{\mu }{D}_{\mu \nu }=0$. Every ghost attachment to an internal gauge propagator therefore vanishes in the strict limit; equivalently, $\mathcal{M}=n\cdot \partial$ on the gauge slice. Thus the interacting Faddeev-Popov ghosts decouple. Residual gauge transformations and the poles at $n\cdot k=0$ require compatible boundary conditions and an axial-gauge pole prescription.

After Wick rotation, the corresponding Minkowski propagator has ${\delta }_{\mu \nu }$ replaced by ${\eta }_{\mu \nu }$ and the usual overall factor $-i$, with a compatible causal prescription. The fixed $n$ makes the gauge-fixed propagator nonmanifestly Lorentz covariant. Nevertheless, gauge invariance makes physical observables independent of this gauge-choice vector, so their perturbative predictions are Lorentz invariant. Already at tree level, contraction with conserved external currents removes all terms containing $k_{\mu }$ or $k_{\nu }$, leaving $g^{2}j\cdot j^{\prime }/k^2$. The quantum Ward identities give the analogous cancellation in complete physical amplitudes; arbitrary gauge-dependent Green functions need not be independent of $n$.

The three real components form the Adjoint representation of $SU(2)$. Around a nonzero vacuum, a local gauge transformation rotates the direction of $\varphi (x)$ to the third axis. In this unitary gauge, the two angular fields are removed from the scalar parametrization and become longitudinal polarizations of two gauge bosons, through the Higgs mechanism. Thus $\varphi =(0,0,v+\eta {)}^T$ describes the perturbative physical fluctuations, with $v^2=-m^2/\lambda$.

The subgroup generated by $t^3$ fixes $v{e}_3$, giving $SU(2)\to U(1)$. Define the physical vector fieldsThe gauge covariant derivative on the radial configuration isso the scalar kinetic term becomes $\frac{1}{2}(\partial \eta {)}^2+g^2(v+\eta {)}^2{W}_{\mu }^{+}{W}^{-\mu }$.

Here is an exact form of the Lagrangian in these physical fields, up to its additive vacuum constant. DefineandThe minus sign in the printed non-Abelian gauge field strength gives precisely these signs. ThenIts quadratic terms give $m_A=0$, $m_{W^{+}}=m_{W^{-}}=gv$, and $m_{\eta }=\sqrt{2\lambda }v$. The two massive vectors have six real polarization degrees of freedom in total; the massless vector has two and the scalar one. Their total of nine equals the original six vector plus three scalar degrees of freedom.

Under the surviving $U(1)$, $W^{+}$ and $W^{-}$ have opposite charges and $\eta$ is neutral. The Yang-Mills theory supplies $A{W}^{+}{W}^{-}$ and $AA{W}^{+}{W}^{-}$ couplings and quartic charged-vector self-interactions. The scalar term supplies $2g^{2}v\eta W^{+}{W}^{-}$ and $g^2{\eta }^2{W}^{+}{W}^{-}$, together with the cubic and quartic scalar self-interactions. There is no direct $\eta AA$ term.

With suitable fermions, this spectrum offers a massless photon and massive charged mediators of short-range weak charged currents, so it captures part of the weak/electromagnetic pattern. It differs crucially from the electroweak interaction of the Standard Model: the latter gauges $SU(2{)}_L\times U(1{)}_Y$, uses a complex Higgs doublet, and breaks to $U(1{)}_{\mathrm{em}}$. Three rather than two generators are broken, leaving a massive neutral Z boson as well as $W^{\pm }$ and a photon. Its charge operator $Q=T^3+Y$ and independent hypercharge coupling give a nontrivial weak mixing angle. The present theory has no massive neutral mediator, no such mixing angle, and does not reproduce the observed chiral fermion charge assignments or their usual Yukawa couplings from the standard doublet/singlet assignments.

Use anti-Hermitian representation matrices for the unitary representation, so $R(X{)}^{†}=-R(X)$. Absorb the gauge coupling into the connection, consistently with the transformation law in the PDF. The appropriate gauge covariant derivative isThis has a plus sign because the inhomogeneous part of the connection transformation is $-ϵ{\partial }_{\mu }X$. Its variation isThus gauge covariance of a scalar covariant derivative givesThe cancellation uses that $R$ is a Lie algebra representation, including its commutator identity.

The gauge field strength isIt is the curvature appearing in $[D_{\mu },D_{\nu }]\varphi =R(F_{\mu \nu })\varphi$. The index positions in the PDF are related by the constant Minkowski metric; lowering the index in its transformation law gives the convention used here.

Take signature $(+,-,-,-)$, let $B$ be a real symmetric gauge-invariant internal form, and let $⟨\cdot ,\cdot {⟩}_V$ be the positive Hermitian form preserved by the unitary matter representation. One consistent Lorentz transformation scalar Lagrangian density isHere $g_{\mathrm{YM}}>0$, $m^2\ge 0$, and $\lambda \ge 0$ give a simple stable potential. The coupling is in the kinetic normalization because it was absorbed into $A_{\mu }$; it is not inserted again into $D_{\mu }$. Taking a zero or another gauge-invariant potential is also possible. A positive nondegenerate $B$ gives the usual physical gauge kinetic energy, as for a compact gauge algebra. More generally the invariance calculation below applies to any symmetric invariant bilinear form on a Lie algebra, including the Killing form.

Under the infinitesimal gauge-field transformation law,Invariance and symmetry of $B$ give the cancellation. For matter, both $\varphi$ and $D_{\mu }\varphi$ transform by $R(X)$, so anti-Hermiticity givesfor either pairing used in the Lagrangian. Its potential is a function of the invariant scalar norm. Therefore $\boxed{{\delta }_X\mathcal{L}=0}$, establishing gauge invariance of this Yang-Mills theory coupled to a scalar. Contraction of spacetime indices with the Minkowski metric gives Lorentz invariance.

For the Adjoint representation, define the Lie algebra structure constants byInternal superscripts here are basis labels, as in the paper; spacetime indices alone are raised with the Minkowski metric. The adjoint covariant derivative and curvature have componentsThese formulas involve no extra factor of $i$, because the generators use the Lie-algebra convention rather than Hermitian physics generators.

To make the Killing-form sign explicit, write $B=s\kappa$ and $⟨\varphi ,\psi {⟩}_V=s{\kappa }^{ab}({\varphi }^a{)}^{\ast }{\psi }^b$. The literal contraction requested in the paper corresponds to $s=1$. For a compact real form of a complex semisimple gauge algebra, its actual positive inner product has $s=-1$, since the Killing form is negative definite. With a complex adjoint scalar, the fully expanded Killing-form Lagrangian for an adjoint scalar isAll repeated internal labels are summed; $s^2=1$ explains the absence of $s$ from the quartic term. Invariance is equivalently the component identity $f^{ca}{}_d{\kappa }^{db}+f^{cb}{}_d{\kappa }^{ad}=0$. For a real adjoint scalar, omit complex conjugation and insert $1/2$ in the scalar kinetic and quadratic mass terms; the same invariance proof applies.