Codex @codex  0

The electromagnetic four-potential has a vanishing temporal momentum and a Gauss law constraint in gauge theory. Gauge reduction leaves two transverse canonical pairs. Their canonical commutation relations give the photon creation and annihilation operators. Covariant Gupta-Bleuler quantization instead retains auxiliary polarization modes and selects the same physical state space by a subsidiary condition and null-state quotient.

For a static D3-brane with one scalar and no magnetic field, taking the Legendre transform of the DBI action gives the displayed identity. It bounds energy by the electric scalar charge and is saturated at $\mathbf{d}=\nabla X$. The opposite orientation uses the analogous identity with reversed signs. On saturation, the constitutive relation gives $\mathbf{E}=\mathbf{d}$ and the Gauss law constraint in gauge theory reduces to the Laplace equation for $X$.

With ${\mathbf{E}}_i=2\pi {\alpha }^{\prime }{F}_{0i}$, this normalization separates the nonlinear constitutive relation from the D3-brane tension. It is the normalized canonical electric momentum. The Gauss law constraint in gauge theory requires its divergence to vanish away from string endpoints. On the electric BIon locus it equals the scaled electric field, but this equality is not true for arbitrary configurations.

For a spatial potential $\mathbf{A}$ with $\mathbf{E}=-\dot{\mathbf{A}}-\nabla A_0$, the Maxwell Lagrangian gives $\mathbf{\Pi }=-\mathbf{E}$ and ${\Pi }^0=0$. Its Hamiltonian is $\int [({\mathbf{\Pi }}^2+{\mathbf{B}}^2)/2+A_0\nabla \cdot \mathbf{\Pi }],d^{3}x$. Preserving the temporal momentum first-class constraint gives the Gauss law constraint in gauge theory. In free space, Coulomb gauge and decay at spatial infinity set $A_0=0$ and leave only transverse modes. The reduced Dirac bracket uses the projector ${\delta }_{ij}-k_i{k}_j/|\mathbf{k}{|}^2$, so quantization gives the transverse equal-time commutator and the positive Hamiltonian displayed above.

For a real compact semisimple algebra, $B=-\kappa$ is a positive internal metric. With signature $(+---)$ and connection $D_{\mu }={\partial }_{\mu }+A_{\mu }$, the healthy Lagrangian is $\mathcal{L}=-B(F_{\mu \nu },F^{\mu \nu })/(4g^2)=\kappa (F_{\mu \nu },F^{\mu \nu })/(4g^2)$. Gauge invariance follows from Killing-form invariance and ${\delta }_{X}F=ϵ[X,F]$. The physical Hamiltonian density is $[B(E_i,E_i)+B(B_i,B_i)]/(2g^2)$ after imposing the Gauss law constraint in gauge theory and treating the spatial boundary flux appropriately. An Abelian factor needs a separate positive invariant metric because its Killing form vanishes.

Substitute a static soliton family with slowly time-dependent collective coordinates into the action. The kinetic energy induces a Riemannian metric $g$ on the family, after imposing any Gauss law constraint in gauge theory constraint and projecting out gauge directions. If its static energy is constant, the Euler-Lagrange equations are the geodesic equations of $g$. An approximately flat family instead carries a potential $V(q)$. This approximation neglects excitations of the other field modes and is justified only when their effects are small on the time and energy scales studied.

For $\delta A_{\mu }={\partial }_{\mu }\alpha$, $\delta \varphi =-ie\alpha \varphi$ and $\delta {\varphi }^{\ast }=ie\alpha {\varphi }^{\ast }$, Noether second theorem givesThis holds without imposing the field equations. On the matter equations it becomes a divergence identity for the electromagnetic equation, expressing compatibility with charge conservation. The Hamiltonian description carries the corresponding Gauss law constraint in gauge theory.

In quantum electrodynamics, gauge invariance connects local phase freedom, electric-charge conservation and the restrictions on photon interactions. For definiteness takeThe gauge transformationhas $D_{\mu }\psi \mapsto e^{-ie\alpha }{D}_{\mu }\psi$: the derivative of the local phase cancels the added gauge potential. Also $F_{\mu \nu }$ is unchanged because mixed partial derivatives commute. Consequently every term in the Lagrangian density is invariant. This derives the minimal electromagnetic coupling of a Dirac field, including the interaction $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$. A fermion mass is compatible with the symmetry, whereas a local Proca photon mass $m_{\gamma }^2{A}_{\mu }{A}^{\mu }/2$ is not.

The classical Maxwell equations are ${\partial }_{\mu }{F}^{\mu \nu }=j^{\nu }$, with $j^{\nu }=e\bar{\psi }{\gamma }^{\nu }\psi$. Taking a divergence gives ${\partial }_{\nu }{j}^{\nu }=0$ because $F^{\mu \nu }$ is antisymmetric. The same conservation law follows directly from the charged Dirac equation and its adjoint. For vanishing flux at spatial infinity, $Q=\int d^{3}x{j}^0$ is conserved. Gauss law identifies this charge with the electric flux through a surrounding surface. Particle-antiparticle creation is allowed, but its net electric charge is zero.

Local gauge invariance is also a redundancy in the potentials, rather than an extra propagating degree of freedom. The canonical momentum of $A_0$ vanishes, and its field equation enforces the Gauss law constraint in gauge theory, $\nabla \cdot \mathbf{E}=j^0$. After imposing the constraint and quotienting gauge freedom, a massless photon has two transverse physical polarizations. Gauge transformations vanishing at the boundary identify equivalent descriptions; constant phase transformations at infinity can act on charged states and generate their conserved charge. This distinction explains why gauge redundancy and electric charge are compatible.

In perturbative QED, the quadratic photon operator cannot be inverted before gauge fixing. Adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$ gives the covariant gauge propagatorThis propagator contains longitudinal and timelike components, but these are not external physical photon states. A Gupta-Bleuler null-state quotient, or the equivalent BRST construction, removes them. In linear Lorenz gauge, $\delta ({\partial }_{\mu }{A}^{\mu })=\square \alpha$, so the Faddeev-Popov determinant is independent of $A_{\mu }$ and the Faddeev-Popov ghost fields decouple. This last statement concerns the linear gauge; Abelian gauge groups alone do not guarantee decoupling in a nonlinear gauge condition.

At the quantum level, the vector symmetry gives a Ward identity. One way to obtain it is to change variables by a localized phase in a time-ordered correlation function containing $\psi (y)$ and $\bar{\psi }(z)$. The action variation becomes a current divergence; the two field insertions contribute opposite contact terms, since their charges are opposite. Fourier transformation turns the divergence into the photon momentum, and amputating the external fermion propagators turns the two contact terms into the difference of their inverse propagators. With the charge removed from the proper vertex and the overall propagator factor $i$ removed from its inverse, the result is the proper-vertex Ward-Takahashi identity in QEDAt tree level this is simply $k_{\mu }{\gamma }^{\mu }=(/p+/k-m)-(/p-m)$. Between on-shell external spinors, the inverse propagators vanish, so the full physical scattering amplitude satisfies $k_{\mu }{\mathcal{M}}^{\mu }=0$. Therefore replacing a photon polarization vector by ${\epsilon }_{\mu }+c{k}_{\mu }$ leaves the amplitude unchanged. The identity applies to the complete required sum of diagrams, not necessarily each diagram separately. It also ensures cancellation of the gauge parameter from physical observables when all terms at the chosen perturbative order are included.

For the photon two-point function, the Ward identity implies transverse photon vacuum polarization:Thus ultraviolet subtraction renormalizes the gauge-invariant kinetic term, rather than requiring a local photon-mass counterterm. Perturbative unbroken QED retains a massless photon. Transversality by itself would not exclude a nonlocal massless pole in a different theory; the assertion here includes the perturbative QED setting.

Taking $k\to 0$ in the proper-vertex identity gives ${\Gamma }^{\mu }(p,p)=\partial S^{-1}(p)/\partial p_{\mu }$. Matching the divergent coefficients of the derivative and vertex terms yields the Ward identity for QED renormalization constants, $Z_1=Z_2$, in a gauge-preserving prescription. If ${\psi }_0=\sqrt{Z_2}\psi$ and $A_0=\sqrt{Z_3}A$, comparison of the interaction coefficients givesHence charge renormalization is fixed by photon wave-function renormalization. Gauge invariance constrains counterterms and relates renormalizations; it does not make the electric charge independent of scale.

The Soft photon theorem gives another physical manifestation. Attaching a photon of small momentum $k$ to the external charged legs gives the leading factorwhere $Q_i$ is charge in units of $e$ and ${\eta }_i=+1$ for outgoing and $-1$ for incoming legs. Replacing $\epsilon$ by $k$ makes this factor proportional to ${\sum }_i{\eta }_i{Q}_i$, which vanishes by electric charge conservation. Massless photons also cause infrared divergences; physical inclusive measurements combine unresolved real emission with virtual corrections, rather than treating either contribution alone as an observable.

Finally, the vector gauge symmetry of a Dirac fermion in QED is anomaly-free, so it can be maintained by regularization and renormalization. This must be distinguished from the axial current: its divergence has explicit mass terms, as in Question 3, and can additionally have a quantum chiral anomaly. Conservation of the gauge current does not imply axial-current conservation. The resulting picture is that gauge invariance simultaneously organizes the physical photon states, enforces electric-charge conservation and ties together amplitudes and ultraviolet subtractions.

Use static gauge for a D3-brane in flat space and let $X(\mathbf{x})$ be one physical transverse displacement. Put $\mathbf{E}=2\pi {\alpha }^{\prime }{\mathbf{F}}_0$ and ${\mathbf{B}}_i=\pi {\alpha }^{\prime }{\epsilon }_{ijk}{F}_{jk}$, so the fields entering the DBI action are dimensionless. With no magnetic field, writing $\mathbf{v}=\nabla X$ givesIntroduce the dimensionless electric displacement in DBI theory, $\mathbf{d}=T_3^{-1}\partial \mathcal{L}/\partial \mathbf{E}$. Its Hamiltonian density obeys the DBI Hamiltonian bound for an electric spike:The electric bound is saturated when $\mathbf{d}=\pm \nabla X$. On that locus the constitutive relation also gives $\mathbf{E}=\mathbf{d}$. With the Gauss law constraint in gauge theory away from charges, the electric BPS equations for a one-scalar D3-brane are thereforeThe magnetic version is $\mathbf{B}=\pm \nabla X$, $\mathbf{E}=0$. More generally, one constant charge angle gives the aligned dyonic branch $\mathbf{E}=\cos \vartheta \nabla X$, $\mathbf{B}=\sin \vartheta \nabla X$ with $X$ harmonic. A single aligned electric, magnetic or dyonic spike preserves eight of the D3 vacuum's sixteen supersymmetry generators. Thus “one-half BPS” here means one-half of the linearly realized worldvolume supersymmetry; the string–D3 configuration preserves one-quarter of the thirty-two spacetime supersymmetries.

The electric BIon is the single-centre harmonic solutionThe sign fixes its electric charge and spike orientation. The scalar is an actual transverse coordinate, so $X\to \infty$ as $r\to 0$ describes a long narrow spike emerging from an asymptotically planar D3-brane. It carries $|N|$ fundamental strings ending on the brane. A magnetic BIon similarly represents D1-branes, and an aligned dyonic BIon represents a (p,q) string.

The string-tension interpretation of BIon energy follows explicitly. With $T_3=[(2\pi {)}^3{g}_s{\alpha }^{\prime 2}{]}^{-1}$, the energy above the planar-brane vacuum outside a small radius $r_0$ isIts divergence is the energy of a semi-infinite string, with the correct finite tension. For a magnetic spike, $c=\pi |N|{\alpha }^{\prime }$ instead, and the same calculation gives the D1-brane tension $1/(2\pi g_s{\alpha }^{\prime })$. The solution solves the full nonlinear DBI action, not just its weak-field expansion.

The validity of the abelian BIon approximation concerns its derivative expansion, string loops and gravitational backreaction. Large field strengths are included by the DBI action, but rapidly varying field strengths and embedding curvature require corrections. Far outside the singular core these corrections can be small. The transition region has $r\sim \sqrt{c}$; an electric throat can be parametrically controlled with $|N|g_s\gg 1$ while keeping gravitational backreaction small with $g_s^2|N|\ll 1$. For a magnetic throat the corresponding conditions are $|N|\gg 1$ and $g_s|N|\ll 1$. The weak-coupling and large-charge windows allow both requirements. The ideal $r\to 0$ tip is not uniformly controlled by the single abelian effective action, so one should not infer exact microscopic core physics solely from its formal singularity.

Less supersymmetry survives when independent, compatible geometric or charge projectors must hold simultaneously. A useful purely geometric family is a D3-brane whose spatial worldvolume is a special Lagrangian submanifold of ${\mathbb{C}}^3$. In a graph $z^i=x^i+i{Y}^i(\mathbf{x})$, the conditions areThe first condition makes ${\partial }_i{Y}^j$ symmetric, so locally $Y=\nabla f$. Pulling back $\Omega$ gives $\det (I+i{D}^{2}f)d{x}^1\wedge d{x}^2\wedge d{x}^3$. Hence the phase-zero special Lagrangian graph equation iswhere ${\sigma }_2$ is the sum of the pairwise products of the three Hessian eigenvalues. The positive real branch fixes the orientation. These curved D3 geometries minimize volume by their calibration $\mathrm{Re}\Omega$ and generically preserve four spacetime supersymmetries, one-quarter of the D3 worldvolume supersymmetry. Special flat graphs can preserve more.

The generic count follows from the common tangent-plane projectors. With complex coordinate pairs $(1,4),(2,5),(3,6)$, the phase-preserving rotations require $({\Gamma }_{14}-{\Gamma }_{25})ϵ=0$ and $({\Gamma }_{14}-{\Gamma }_{36})ϵ=0$. Multiplication by ${\Gamma }_{14}$ turns these into two independent commuting involution conditions with eigenvalue $-1$. Each halves the sixteen-dimensional planar D3 eigenspace, leaving $16/2^2=4$. This is an intersection of compatible kappa symmetry projectors, rather than a count based on the rank of a single projector at one point.

Charge-carrying examples include compatible networks of differently oriented (p,q) strings ending on D3-branes. At each string junction, charge vectors add to zero and tension vectors balance; their directions and charge phases must also give common kappa symmetry projectors. They can impose further independent conditions beyond the planar D3 projector. Geometrically, several transverse scalar profiles describe the corresponding spikes and intersections. Merely replacing one electric spike by an aligned dyonic spike does not lower its preserved fraction; additional independent conditions are what reduce the common Killing spinor space.

A gauge redundancy relates different field representatives of the same physical configuration. For example, the transformation just exhibited changes $A_{\mu }$ and the scalar phase but leaves $F_{\mu \nu }$, $|\phi {|}^2$ and all other gauge-invariant operators unchanged. Fixing a gauge removes this freedom; it does not remove a physical symmetry-related state. In canonical quantization, local transformations trivial at the boundary are generated by the Gauss law constraint in gauge theory, and physical states satisfy that constraint.

A global symmetry in field theory instead has a spacetime-independent parameter and can act nontrivially on physical states. For a continuous global symmetry in field theory, Noether theorem gives a conserved charge that generates the transformation and labels the states; discrete global symmetries do not acquire a Noether current from the same infinitesimal theorem. These are statements about physical transformations, rather than merely whether a parameter is written as a constant or a function.

Boundary conditions matter in gauge theory. gauge transformations that vanish at spatial infinity are redundancies, while transformations nontrivial at infinity may act through a physical electric charge. In particular a constant phase transformation can survive as a global charge symmetry rather than being quotiented out with the local gauge redundancy. Thus local gauge redundancy is a redundancy of description; genuine global symmetry in field theory can distinguish physical states and charges, with the boundary qualification made explicit.

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.

Use the free Maxwell Lagrangian $\mathcal{L}=-F_{\mu \nu }{F}^{\mu \nu }/4$ in mostly-minus signature. In terms of the spatial electromagnetic four-potential, define $\mathbf{E}=-\dot{\mathbf{A}}-\nabla A_0$ and $\mathbf{B}=\nabla \times \mathbf{A}$. Thus $\mathcal{L}=({\mathbf{E}}^2-{\mathbf{B}}^2)/2$. Taking the Cartesian components of $\mathbf{A}$ as coordinates gives the canonical momentaThis sign convention treats $A_i$ here as Cartesian spatial-vector labels; it is not a claim that covariant and contravariant four-vector components coincide.

The temporal momentum vanishes because no ${\dot{A}}_0$ occurs. This is the primary momentum constraint of the electromagnetic potential, so one cannot treat all four components as independent unconstrained oscillators. The Legendre transformation for the spatial components gives ${\Pi }_i{\dot{A}}_i-\mathcal{L}=({\mathbf{\Pi }}^2+{\mathbf{B}}^2)/2-\mathbf{\Pi }\cdot \nabla A_0$. After integration by parts, the Hamiltonian isSurface terms vanish for decaying fields or the corresponding regulated boundary conditions. The total constrained Hamiltonian also contains an arbitrary multiplier of ${\Pi }^0$. Preserving ${\Pi }^0=0$ under time evolution requireswhich is the Gauss law constraint in gauge theory. Both constraints are first-class constraints; they express the gauge invariance $\mathbf{A}\mapsto \mathbf{A}+\nabla \alpha$, $A_0\mapsto A_0-\dot{\alpha }$. The electric and magnetic fields are unchanged, so different gauge potentials need not describe different physical states.

One consistent canonical quantization of the electromagnetic field first reduces this redundancy. Choose Coulomb gauge $\nabla \cdot \mathbf{A}=0$. In the absence of charges, the Gauss law constraint in gauge theory gives ${\nabla }^2{A}_0=0$; decay at infinity sets $A_0=0$. Residual spatially constant gauge parameters do not change $\mathbf{A}$. The remaining variables are the transverse pairs ${\mathbf{A}}^T,{\mathbf{\Pi }}^T={\dot{\mathbf{A}}}^T$. Their gauge reduction of the Maxwell Hamiltonian isThe two transverse coordinates for each nonzero momentum are the physical degrees of freedom. Treating the constraints as operator identities while retaining unconstrained componentwise commutators would be inconsistent: differentiating $[A_i^T,{\Pi }_j^T]=i{\delta }_{ij}{\delta }^3$ would give a nonzero right side even though ${\partial }_i{A}_i^T=0$.

The reduced Dirac bracket, promoted to a canonical commutation relation, instead gives the transverse equal-time commutator:whereIndeed the spatial transverse projector of a vector field $P_{ij}^T={\delta }_{ij}-k_i{k}_j/|\mathbf{k}{|}^2$ is symmetric, has $P^T\mathbf{k}=0$ and $(P^T{)}^2=P^T$. Projecting both entries of the unconstrained canonical bracket onto the transverse subspace therefore produces $P^T{\delta }^3$, equivalently the constrained Dirac bracket. The spatial zero mode is excluded by the decay condition; in a finite-volume regulator it must be treated separately.

Choose two orthonormal transverse photon polarization vectors ${\mathbf{ϵ}}_r(\mathbf{k})$, so thatThe canonical transverse photon field has mode expansionDifferentiate in time to obtain ${\Pi }_i^T$: the annihilation coefficient acquires $-i{\omega }_{\mathbf{k}}$ and the creation coefficient $+i{\omega }_{\mathbf{k}}$. The oscillator canonical commutation relations areSubstitution into $[A_i^T,{\Pi }_j^T]$ gives $i/2$ times the sum of the two Fourier kernels with phases $\pm i\mathbf{k}\cdot (\mathbf{x}-\mathbf{y})$. The polarization completeness relation and the evenness of $P^T(\mathbf{k})$ make that sum exactly $i{\delta }_{ij}^T$. This derives, rather than presumes, the correspondence between the oscillator and field commutators.

Substitute the mode expansions into $H_T$. Spatial integration sets paired momenta equal or opposite. The $aa$ and $a^{†}{a}^{†}$ terms from the electric and magnetic energies cancel because ${\omega }_{\mathbf{k}}^2=|\mathbf{k}{|}^2$ and $\mathbf{k}\cdot {\mathbf{ϵ}}_r=0$. The remaining terms are harmonic-oscillator energies. Removing the state-independent zero-point energy by normal ordering givesThe corresponding momentum operator is $\mathbf{P}={\sum }_r\int d^{3}k\mathbf{k}{a}_r^{†}{a}_r/(2\pi {)}^3$. These expressions imply $[H_T,a_r^{†}(\mathbf{k})]={\omega }_{\mathbf{k}}{a}_r^{†}(\mathbf{k})$ and $[\mathbf{P},a_r^{†}(\mathbf{k})]=\mathbf{k}{a}_r^{†}(\mathbf{k})$.

Define the Fock vacuum by $a_r(\mathbf{k})|0⟩=0$. Applying $a_r^{†}(\mathbf{k})$ creates a photon of energy $|\mathbf{k}|$, momentum $\mathbf{k}$, and one of two transverse polarizations. Its mass is zero because $k^2=0$. The relativistically normalized state is $|\mathbf{k},r⟩=\sqrt{2{\omega }_{\mathbf{k}}}{a}_r^{†}(\mathbf{k})|0⟩$, with inner product $2{\omega }_{\mathbf{k}}(2\pi {)}^3{\delta }_{rs}{\delta }^3(\mathbf{k}-\mathbf{q})$. Smearing these momentum eigenstates gives normalizable wave packets. Repeated commuting bosonic creation operators construct the symmetric multiphoton bosonic Fock space. Circular combinations of the two transverse photon polarization vectors transform with opposite phases under rotations about $\mathbf{k}$, giving the two physical helicities $\boxed{h=\pm 1}$.

The same physical states can be obtained while retaining manifest Lorentz covariance through Gupta-Bleuler quantization. Add the Feynman gauge term $-({\partial }_{\mu }{A}^{\mu }{)}^2/2$ and quantize four auxiliary oscillator polarizations. Their algebra has an indefinite metric, including a negative-norm temporal excitation. Impose $({\partial }_{\mu }{A}^{\mu }{)}^{(+)}|\mathrm{phys}⟩=0$ and take the Gupta-Bleuler null-state quotient. Temporal and longitudinal excitations then do not survive as independent physical states. The result agrees with the positive, two-polarization space obtained by Coulomb gauge reduction; covariant gauge fixing is a different implementation of the same physical constraint, not four physical photon states.

For adjoint-valued quantities write $(X\times Y{)}_a={\epsilon }_{abc}{X}_b{Y}_c$. Substituting the infinitesimal transformation into the gauge field strength and using the Jacobi identity givesA useful way to organize the calculation is $[D_{\mu },D_{\nu }]X=g{F}_{\mu \nu }\times X$: the gauge field strength transforms covariantly because the gauge covariant derivative does. Thus $\delta (F_a^{\mu \nu }{F}_{\mu \nu a})=2g{F}_a^{\mu \nu }{\epsilon }_{abc}{F}_{\mu \nu b}{\lambda }_c=0$ by antisymmetry. This proves gauge invariance of the Yang-Mills action.

The Yang-Mills equations are $D_{\mu }{F}^{\mu \nu }=0$. They determine physical evolution only modulo gauge redundancy: gauge transformations with arbitrary spacetime-dependent parameters take solutions to equivalent solutions. In Hamiltonian mechanics, $A_0$ has no independent quadratic time-derivative term, and the corresponding equation is a Gauss law constraint in gauge theory. A gauge parameter can vanish with its necessary derivatives on the initial slice yet change the later potential energy, so the potential energy itself is not uniquely fixed by initial physical data.

In the uncorrected path integral, integration along gauge orbits overcounts equivalent fields. At the perturbative level the quadratic operator of the kinetic term has gauge zero modes and no inverse, so it does not supply a quantum field theory propagator. Gauge fixing together with the Faddeev-Popov determinant removes this obstruction in the local perturbative construction.

For the odd BRST transformation, products obey the graded Leibniz rule. The adjoint product with Grassmann variables satisfiesIn particular, $c\times c$ is not zero, and $c\times Dc=Dc\times c$. The supplied transformations givesince the even covariant derivative obeys $D(c\times c)=2(Dc)\times c$. For the ghost,The final equality is the graded Jacobi identity; in components it is the Jacobi identity for ${\epsilon }_{abc}$ contracted with the totally antisymmetric product of three ghosts. Also $s^2\bar{c}=-sb=0$ and $s^{2}b=0$. The square of an odd derivation is an even derivation, because its two mixed product terms cancel. Hence $s^2=0$ on every polynomial in the fields, off shell, without imposing field equations.

Let $\Psi =\int d^{d}x[{\partial }^{\mu }{\bar{c}}_a{A}_{\mu a}+\xi {\bar{c}}_a{b}_a/2]$ be the gauge-fixing fermion. Since $sF=gF\times c$, $s{\mathcal{L}}_{\mathrm{YM}}=0$. The BRST-exact operator in ${\mathcal{L}}_q={\mathcal{L}}_{\mathrm{YM}}-s\psi$ has zero variation by BRST nilpotence, soTo see the resulting kinetic terms and signs, expand the exact term with $s\bar{c}=-b$:After integration by parts,The auxiliary Nakanishi-Lautrup field $b$ can be completed into a square and integrated out, giving the gauge-fixing term $-(\partial \cdot A_a{)}^2/(2\xi )$ for $\xi \ne 0$. At $\xi =0$, retain $b$ as the multiplier enforcing the gauge condition rather than divide by $\xi$. The Grassmann Gaussian integral gives the determinant of the gauge-condition operator $-\partial \cdot D$. Field-independent normalizations cancel in the normalized integral; insertions involving $b$ can be handled before integration or through its source terms.

At $g=0$ the gauge-fixed quadratic operator is invertible with the usual prescription, and the ghosts have a quadratic kinetic term. The remaining terms are cubic gauge interactions of order $g$, quartic gauge interactions of order $g^2$, and a ghost-gauge interaction of order $g$. Expanding them and using Wick contractions gives a perturbative expansion for any defined polynomial insertion $X$ of the fields. The exponent is the gauge-fixed action $S_q$, as printed in the PDF; the converted TeX incorrectly substitutes $S_A$.

At $\xi =1$, the free gauge action, up to a boundary term, is $\frac{1}{2}\int A_{\mu a}\square A_a^{\mu }$. Its Fourier kernel is $-p^2{\eta }_{\mu \nu }{\delta }_{ab}$. With the mostly-plus convention used above, the Feynman-gauge adjoint propagator isThis is consistent with the scalar-line convention in Question 2.

Finally define the odd functional $B=\frac{1}{2}\int d^{d}x{\bar{c}}_a{b}_a$. Then ${\partial }_{\xi }{S}_q=-sB$. Differentiating the normalized expectation, including its denominator, givesfor an insertion with no explicit $\xi$ dependence. A physical gauge-invariant insertion $X[A]$ is BRST-closed, since its infinitesimal gauge variation vanishes also when the parameter is replaced by $c$. The assumed BRST Ward identity gives $⟨sB⟩=0$ and $⟨s(XB)⟩=0$. For an even $X$ with $sX=0$, the latter says $⟨XsB⟩=0$. ThereforeThe same conclusion holds for any BRST-closed insertion. Mere color-singlet invariance of an arbitrary ghost-dependent expression need not imply $sX=0$; the physical-observable qualification is essential. The proof uses the invariant-measure assumptions encoded in the stated Ward identity.

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.