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For fields ${\Phi }_a$ with a first-derivative Lagrangian density, suppose an infinitesimal global transformation at fixed coordinates is $\delta {\Phi }_a=\alpha {\Delta }_a$ and $\delta \mathcal{L}=\alpha {\partial }_{\mu }{K}^{\mu }$, with constant infinitesimal $\alpha$. Noether theorem gives the on shell conserved currentThis follows by varying the Lagrangian, integrating the terms with ${\partial }_{\mu }{\Delta }_a$ by parts, and setting the Euler-Lagrange expressions to zero. If spatial boundary flux vanishes, $\int d^{3}x{J}^0$ is a conserved Noether charge. Spacetime symmetries are covered by including the induced field variation at fixed coordinates and the corresponding total derivative $K^{\mu }$.

An example of a theory with a Lorentz four-vector is the source-free Maxwell field with electromagnetic four-potential $A_{\mu }$. Under an Abelian gauge transformation,The change in $F_{\mu \nu }$ is ${\partial }_{\mu }{\partial }_{\nu }\chi -{\partial }_{\nu }{\partial }_{\mu }\chi =0$, so the Maxwell Lagrangian is gauge-invariant. For $A_{\mu }=(\Phi ,-\mathbf{A})$ define $E_i=F_{0i}$ and $F_{ij}=-{ϵ}_{ijk}{B}_k$. With signature $+---$,The minus sign in $-F^2/4$ gives the positive electric kinetic term and positive physical field energy; reversing it would reverse the Hamiltonian's sign for the physical modes. The normalization $1/4$ accounts for antisymmetry and yields the conventional equations. Varying $A_{\nu }$ gives ${\partial }_{\mu }{F}^{\mu \nu }=0$.

For time translation at fixed coordinates take ${\Delta }_{\nu }={\partial }_0{A}_{\nu }$. Since $\alpha$ is constant, $\delta F_{\mu \nu }=\alpha {\partial }_0{F}_{\mu \nu }$, and hence $\delta \mathcal{L}=\alpha {\partial }_0\mathcal{L}$. Thus $K^{\mu }={\delta }^{\mu }{}_0\mathcal{L}$. The momentum derivative isNoether theorem therefore gives the canonical stress-energy tensor's time-translation currentHere ${\dot{A}}_i=E_i+{\partial }_i{A}_0$ was used. The canonical momentum of $A_0$ vanishes, while that of the lower spatial component $A_i$ is $E_i$: $A_0$ acts as a multiplier for Gauss law, not a propagating degree of freedom. The density above is not manifestly gauge-invariant, but its charge agrees with the physical energy on the constraint surface and with appropriate boundary conditions.

The gauge-covariant time translation instead has ${\Delta }_{\nu }={\partial }_0{A}_{\nu }-{\partial }_{\nu }{A}_0=F_{0\nu }$. It is time translation combined with a gauge transformation of parameter $\chi =-\alpha A_0$. Directly,using the definition of $F$, equivalently its Bianchi identity. The total derivative in $\delta \mathcal{L}$ is unchanged. The improved current isIt is the gauge-invariant Maxwell stress-energy tensor. In particular,The spatial current is the Poynting vector $\mathbf{E}\times \mathbf{B}$, so conservation gives ${\partial }_{t}u+\nabla \cdot (\mathbf{E}\times \mathbf{B})=0$. This is the familiar positive local electromagnetic field energy density, expressed entirely in measurable fields.

To compare the currents,The last term vanishes on the source-free Maxwell equations, leaving a stress-energy tensor improvement generated by a stress-energy superpotential. At the charge level,when the surface term vanishes and Gauss law holds. Thus the symmetries yield the same conserved energy under these conditions while the second supplies a gauge-invariant local density. Boundary terms would have to be retained if those boundary conditions were not imposed.

The Maxwell equations with vacuum permittivity and permeability give $\rho ={ϵ}_0\nabla \cdot E$ and $J={\mu }_0^{-1}\nabla \times B-{ϵ}_0{\partial }_{t}E$. The current identity is the Ampère-Maxwell equation, and Faraday's law gives ${\partial }_{t}B=-\nabla \times E$. With $g={ϵ}_{0}E\times B$,The identity $(\nabla \times B)\times B=(B\cdot \nabla )B-\frac{1}{2}\nabla B^2$, and its electric analogue, turn this into the divergence of the Maxwell stress tensor $T$. Thus the tensor in the question is its negative:The vector $g$ is electromagnetic momentum density, and ${\sigma }_{ij}$ is the outward flux of its $i$th component across a surface normal in the $j$ direction. In static conditions the total force on matter is $F_i=-\oint {\sigma }_{ij}{n}_{j}dS$; generally subtract the time derivative of the enclosed electromagnetic field momentum as well.

For the two infinite sheets, superposition and the electromagnetic field jumps give $E=(\sigma /{ϵ}_0)e_x$, $B={\mu }_{0}K{e}_y$ between them and zero outside. Substitution yields exactlyA thin pillbox enclosing the $x=0$ sheet has only its $+x$ face in the interior electromagnetic field. ThereforeDirectly, omit the sheet's self-electromagnetic field: the other sheet supplies $E_{\mathrm{other}}=\sigma e_x/(2{ϵ}_0)$ and $B_{\mathrm{other}}={\mu }_{0}K{e}_y/2$. The Lorentz force per area is $\sigma E_{\mathrm{other}}+K{e}_z\times B_{\mathrm{other}}$, giving the same result. The electric contribution is attractive and the magnetic contribution repulsive.

If $K=\sigma v$ for a single moving charge population, use ${\mu }_0{ϵ}_0=c^{-2}$ to obtain $F/A={\sigma }^2(1-v^2/c^2)e_x/(2{ϵ}_0)$. For physical $|v|<c$ this is attractive, so motion of these charges cannot make the force repulsive. Independently imposed currents need not satisfy that restriction on $K/\sigma$.

A commitment to testing claims against experience is a defensible scientific norm. A strict account according to which every meaningful scientific statement must be conclusively verified, or translated into a fixed vocabulary of theory-free observations, is much harder to sustain. Logical empiricism should also be understood as a changing family of projects, not reduced without qualification to its strongest early formulations.

An unrestricted universal scientific law concerns cases beyond a finite observation record. No finite list of favourable cases deductively entails all of them. Requiring conclusive verification would therefore exclude much ordinary theoretical science. Replacing that demand with scientific confirmation retains a relation to evidence while accepting that support is fallible. This is a substantive revision, rather than proof that universal laws were conclusively verified after all.^[ref]

A second difficulty concerns observations. A detector count relies on assumptions about sensitivity, background and the production of a signal. Theory-ladenness makes a sharp division between completely neutral observations and theoretical claims implausible. But dependence on some assumptions does not imply dependence on the truth of the target theory. A calibration can be supported independently, and agreement across instruments using different mechanisms can make a putative effect less vulnerable to one shared error.

Nor should unobservable vocabulary be dismissed merely because its referents cannot be seen directly. A theory of electrons and electromagnetic fields can organize measured currents, spectra and scattering through a common model. What matters scientifically is how the theoretical commitments constrain observations and survive attempts to discriminate them from alternatives. Insisting that each term independently translate into an observation report can miss the role of a whole theoretical structure.

An empiricist can accept much of this. Constructive empiricism permits literal theoretical discourse and commitment to using a theory while restricting belief to empirical adequacy. It separates withholding a stronger ontological belief from refusing to reason with theoretical models.^[ref] Whether this restraint is preferable to selective scientific realism remains an argument about what successful testing warrants.

The strongest sustainable empiricism is consequently methodological and fallible: demand explicit empirical consequences, scrutinize calibration and preserve the possibility of correction. The stricter verificationist reduction does not adequately describe how laws, theoretical terms and measurement actually work. Rejecting that reduction need not weaken the central demand that scientific claims remain accountable to evidence.

There are two different issues: whether a law follows logically from the other laws, and whether an experiment can test it without assuming anything else. Newton's laws of motion have empirical content when combined with rules for identifying inertial frames, measuring mass and specifying forces. An experiment tests predictions of that combination. Calling every observed acceleration an otherwise unspecified force would remove the empirical constraint rather than verify Newton's second law. The distinction is central to the empirical testing of Newton's laws of motion.

Newton's first law contributes the claim that a common family of inertial frames exists: bodies with zero net external force have uniform straight motion in those frames. It is not enough to choose coordinates which straighten the trajectory of just one body; an admissible frame must accommodate many independent bodies and interactions. Once that frame and the ordinary constant-positive-mass form of Newton's second law are assumed,This conditional deduction does not establish the existence of the required frame, or independently identify the physical circumstances in which the net force vanishes. Defining an inertial frame and then counting a straight trajectory as proof of the definition would not be a test.

Nevertheless, the law of inertia has testable implications in a network of assumptions. Consider low-friction motion along an air track: determine positions at measured times, reduce the known residual influences, and ask whether the remaining acceleration approaches zero in the same laboratory frame for different bodies. Rotating the frame changes the description through Coriolis forces and centrifugal forces; this dependence must be accounted for. No finite experiment establishes that every unmeasured influence is absent, but varying the apparatus can discriminate an interaction from a frame effect. A failed prediction is evidence against some part of the package, not automatically the isolated verbal statement of Newton's first law.

Newton's second law is more than a definition if the force law and mass assignments constrain predictions before the new acceleration is measured. For instance, independently characterize a spring's reproducible extension response, specify the same interaction in several configurations, and compare the resulting accelerations for different bodies. If its strength and every mass are freely refitted to each outcome, agreement is empty. If they are fixed by an independently checked model and calibration, the law predicts held-out trajectories. Such a test remains conditional on the spring model, initial conditions and frame identification; this is an instance of the Duhem problem, not a reason to deny all empirical content.

Newton's third law supplies a different constraint. For two bodies exchanging forces, Newton's second law giveswhen there is no net external force. Equal-and-opposite interaction forces imply conservation of momentum for the pair. The two second-law equations alone do not require that cancellation: assigning ${\mathbf{F}}_{12}=\mathbf{f}$ and ${\mathbf{F}}_{21}=-2\mathbf{f}$ is algebraically consistent with both second-law equations while violating the third law. This is a logical countermodel, not a proposed isolated physical interaction. Measuring paired impulses or checking the pair's total momentum therefore tests an additional constraint, conditional on the assumed absence of momentum exchange with anything else.

The system boundary matters. In electromagnetism, matter can exchange momentum with the electromagnetic field. Instantaneous mechanical forces between two particles need not satisfy the simple pairwise statement, although total matter-plus-field momentum is conserved under the appropriate closed-system conditions. Likewise, Noether theorem can derive conservation of momentum from a translationally invariant action; that uses an extra symmetry assumption and does not derive the third law from the second alone.

Thus the first law is not an additional differential consequence once the full second-law framework is fixed, and the third law is not a consequence of that framework. All three can contribute to empirical tests, but none is tested in total isolation from frame, interaction and measurement assumptions. This preserves the difference between logical dependence and experimental testability.^[ref]

Historically, electromagnetism was central to Albert Einstein's 1905 argument. His opening example compares a moving magnet and a moving conductor. The same relative motion produces the observed induction, but a description using electric fields in one frame and the magnetic part of the Lorentz force in another seemed to divide the explanation according to which object was called stationary. The unsuccessful search for an observable preferred state of rest also motivated treating the laws of mechanics, optics and electromagnetism on a common footing. That establishes a strong historical connection, not a deduction that relativity applies only to electromagnetic systems.^[ref]

The logical starting points of the kinematical argument are Einstein's principle of relativity and an invariant speed of light in vacuum, together with the assumptions needed to compare homogeneous inertial frames and physical measuring standards. Einstein synchronisation assigns distant clock readings by light exchange. In standard coordinates, the resulting Lorentz transformation for a boost along $x$ isNeither the derivation of this transformation nor the deductions of relativity of simultaneity and time dilation require solving Maxwell equations for every particular material system. They constrain the relation between measurements made in different inertial frames.^[ref]

There is nevertheless an empirical commitment in identifying $c$ with the speed of vacuum light. It is not a freely chosen numerical label: the theory says different inertial observers, using the prescribed standards, obtain the same measured value. More generally, $c$ specifies the causal structure of Minkowski spacetime. A theory could contain a massive scalar field with a Lorentz invariant equation and no electromagnetic field at all. Its particles, proper times and causal propagation would still be governed by relativistic geometry. This gives a direct sense in which the framework is logically more general than electromagnetism.

Conversely, the presence of Maxwell equations alone would not establish the full relativity principle for all matter. One could append a law involving a physically preferred timelike vector field, giving a nonelectromagnetic sector a preferred frame while retaining the usual electromagnetic equations. The covariant equation with a preferred timelike vector is an explicit model of this distinction. Requiring every relevant interaction to respect Lorentz invariance adds content; writing its components in four-dimensional notation does not supply that content automatically.

The magnet–conductor puzzle is resolved because electric fields and magnetic fields are related parts of one electromagnetic field, with the Lorentz force transforming consistently. The measured current must be compared for the same physical arrangement, including the conductor's state of motion, rather than by demanding the same electric and magnetic components separately. This is an application of the general frame transformations to a particular interaction, not the definition of every possible relativistic interaction.

Albert Einstein's later distinction between a principle theory and a constructive theory clarifies what the 1905 argument provides. It places universal restrictions on admissible laws without constructing every rod and clock from a particular microscopic model. A constructive explanation can additionally investigate why actual material standards approximate the required behavior. Harvey R. Brown and Oliver Pooley emphasize this dynamical question; it does not make the kinematical transformations meaningless or confine them to electromagnetic phenomena.^[ref][ref]

The strongest account therefore distinguishes historical motivation, the operational use of light, and the theory's universal scope. The 1905 paper addresses an electromagnetic problem through a framework whose restrictions extend to every physical process claimed to obey special relativity.

On the curved wire surface, the outward normal is $\hat{\mathbf{r}}$ and the Poynting vector has the constant radial value $N_r(a)=-I^2/(2\sigma {\pi }^2{a}^3)$. Its outward flux isThe Poynting vector is radial, so it has zero normal component on the two end faces in this approximation. This is therefore also the outward flux through the complete wire surface. Equivalently, positive electromagnetic power $P_{\mathrm{loss}}$ flows inward.

The fields are steady, so the stored electromagnetic energy does not change. Poynting's theorem consequently requires the incoming power to equal the Ohmic heating in part (e), exactly as found. The energy reaches the wire through the surrounding electromagnetic field. A local check is $\nabla \cdot \mathbf{N}=r^{-1}{\partial }_r(r{N}_r)=-I^2/(\sigma {\pi }^2{a}^4)=-\mathbf{E}\cdot \mathbf{J}$.

Consider a conventional spin-singlet, weak-coupling, three-dimensional superconductor. Set $\hslash =k_B=1$ in the microscopic action and restore electromagnetic units below. An attraction $\lambda >0$ in an energy shell about the Fermi surface gives a Grassmann actionThe attractive sign permits pairing even when the single-particle Fermi gas is otherwise stable. A complex Hubbard–Stratonovich transformation introduces a pair field $\Delta$ proportional at the saddle to $\lambda ⟨{\psi }_{\downarrow }{\psi }_{\uparrow }⟩$. The action is then quadratic in fermions. Integrating them out gives the BCS pairing-field effective actionHere the trace uses one independent Nambu spinor pairing block; a fully doubled particle-hole representation instead carries its corresponding one-half factor. At a uniform saddle the Bogoliubov--de Gennes Hamiltonian can be written asThe negative branch describes the paired vacuum; excitations are Bogoliubov quasiparticles mixing particles and holes. Extremizing the effective action givesThe physical cutoff or a renormalized interaction is part of the microscopic model; an unregulated contact integral must not be used indiscriminately. With constant density of states $N(0)$ per spin and shell half-width ${\Omega }_c$, the zero-temperature weak-coupling solution is ${\Delta }_0\simeq 2{\Omega }_c{e}^{-1/[N(0)\lambda ]}$. The exponentially small scale follows from the logarithmic pairing susceptibility of states near the Fermi surface. For this ideal weak-coupling model, $2{\Delta }_0/(k_B{T}_c)\simeq 3.53$. The microscopic theory relates the gap, pairing and thermodynamic transition.^[ref]

Close to a continuous transition, for fields varying slowly compared with microscopic scales, expand the effective action in powers and gradients of $\Delta$. Its static part is the Ginzburg-Landau theory of superconductivity:The normalization of $\Delta$ fixes the coefficients; here $a(T)$ changes sign at $T_c$. The uniform minimum has $|{\Delta }_0{|}^2=-a/b$ for $a<0$, condensation free-energy density $-a^2/(2b)$, and characteristic coherence-length scale $\xi =\sqrt{K/|a|}$. Above the transition the minimum is at zero. The quartic stabilization and gradient cost are precisely the ingredients of Landau theory for a complex order parameter. This controlled near-transition expansion supplies a bridge between microscopic BCS theory and macroscopic magnetic response; its coefficients and validity depend on the material and fluctuation regime.^[ref]

Write $\Delta =(|{\Delta }_0|+\eta )e^{i\varphi }$. The radial fluctuation has positive local restoring curvature, whereas the phase has no local potential in a neutral theory with global $U(1)$ symmetry. Freezing the amplitude givesThe neutral case supports a gapless collective phase mode. In the charged case the electromagnetic field couples to this phase rigidity. Under $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ and $\varphi \mapsto \varphi +q\chi /\hslash$, the displayed combination is invariant. Locally choosing a gauge with constant phase leaves a quadratic term in $\mathbf{A}$. Varying the free energy gives the transverse London current and magnetic screening scale,Combining this relation with static Maxwell equations gives exponential field penetration, the Meissner effect. Single-valuedness of the pair field around a loop with negligible current gives flux in units ${\Phi }_0=2\pi \hslash /|q|=\pi \hslash /e$. These are gauge-invariant physical consequences of the condensate stiffness.

For the dynamics, the neutral phase and density mode is reorganized by the Coulomb interaction and the dynamical electromagnetic field. In a bulk three-dimensional system the long-wave longitudinal mode has a plasma gap, rather than remaining the neutral sound mode. This is the Anderson-Higgs mechanism in a superconductor: a phase coordinate becomes part of the electromagnetic collective response instead of surviving as an independent massless physical particle. Local gauge transformations are redundancies of the description, so a literal breaking of a local redundancy is not an observable. Phase coherence, current response and magnetic screening provide the physical statements.^[ref]

A static Landau functional alone cannot determine that full dynamical spectrum. A two-dimensional film, for example, need not have the same long-wave plasma dispersion as a bulk charged system. Nor does a positive radial curvature automatically guarantee a sharp Higgs amplitude resonance: coupling to quasiparticles and other modes can damp it. The chosen mean-field phase is also a thermodynamic symmetry-breaking description; an isolated finite system can be formulated with definite particle number. The microscopic pairing saddle, the near-transition Landau expansion and the electromagnetic Anderson-Higgs response describe successive scales of the same superconducting state, with distinct assumptions at each step.

An RF cavity stores an oscillating electromagnetic field. Its longitudinal electric field does work on a correctly phased bunch, with energy gain $\Delta E=q{V}_{\mathrm{RF}}\mathcal{T}\sin {\varphi }_s$ for a chosen phase convention and transit-time factor $\mathcal{T}$. The bunch arrival time is synchronized so successive gaps accelerate rather than decelerate it. The magnetic force guides and focuses the beam but does no work.

A linear accelerator uses new accelerating structures along its path; cell lengths and timing accommodate the particle speed, and each bunch normally traverses a given section once. A circular synchrotron reuses the same RF cavities each revolution. Its synchronous phase supplies both acceleration and radiation replacement; RF frequency or harmonic synchronization and bending-field strength follow the required orbit and particle speed. Longitudinal phase stability captures nearby particles into RF buckets.

For very high-energy electron and positron colliders, a linear geometry avoids repeated synchrotron losses: at fixed radius, loss per turn scales as $E^4/m^4$. This is not a statement that all high-energy colliders are preferably linear. The LHC was already a circular proton collider in the 2015 context; the proton's much larger mass greatly suppresses radiation at the same energy. Circular electron machines can also be competitive at lower energies. Magnet limits, luminosity, cost and beamstrahlung enter any real comparison.

A neutral superfluid has a complex order parameter whose phase describes coherent particle transport. For a Bose condensate it is represented by a field amplitude; for a paired Fermi superfluid it is a pair amplitude. In a symmetry-breaking description, selecting its phase breaks the global particle-number $U(1)$ symmetry. A common phase shift leaves the energy unchanged, while a phase gradient costs energy and produces superfluid velocity. Long-wavelength phase oscillations give a gapless Goldstone mode, generally a sound wave when density fluctuations provide the conjugate dynamics. A finite isolated state with exact particle number need not have a nonzero field expectation: the phase-selected language describes the ordered limit and relative-phase coherence.

In a superconductor, the paired order parameter carries charge of magnitude $2e$ and couples to the electromagnetic field. The meaningful spatial phase combination is $\hslash \nabla \theta -q\mathbf{A}$, with signed pair charge $q$. Gauge symmetry is a redundancy, so “breaking local gauge symmetry” is not a literal breaking of a measurable physical symmetry. The gauge-invariant electromagnetic response includes the Meissner effect and a finite London penetration depth. Long-range Coulomb forces move the would-be neutral phase mode to a plasma scale through the Anderson-Higgs mechanism in a superconductor.

Additional symmetries can be broken by particular pairing states. Conventional isotropic spin-singlet pairing need not break spatial rotations or time reversal. A d-wave superconductor has a gap transforming nontrivially under crystal symmetries, and some triplet or multicomponent states can also break spin-rotation or time-reversal symmetries. These additional choices are not required by superconductivity itself. A nonzero pairing amplitude, phase coherence and electromagnetic screening describe related but distinct aspects of the ordered state.

Write $\Gamma =(1-v^2/c^2{)}^{-1/2}$ for the Lorentz factor of the inertial frame $S^{\prime }$, distinguishing it from the particles' changing Lorentz factors. Apply the Lorentz transformationto each world line in a. Both particles initially have velocity $-v$ in $S^{\prime }$. The switching events have coordinatesThus the Positron starts accelerating earlier in $S^{\prime }$. Its velocity first becomes less negative, vanishes when its laboratory velocity reaches $v$, and then becomes positive. The electron continues at $-v$ until $t^{\prime }=0$, then accelerates toward $-c$. The relativistic velocity-addition formula makes this precise:The electric field parallel to the Lorentz boost has unchanged magnitude, but its switching surface is $t^{\prime }+v{x}^{\prime }/c^2=0$, rather than $t^{\prime }=0$ everywhere.

In the laboratory, the total three-momentum is zero for all $t$: the two Lorentz forces cancel at equal laboratory times. In $S^{\prime }$, each active force is still $qE$, since this is one-dimensional motion with a parallel electric field. Define $t_p^{\prime }=-\Gamma vd/c^2$. For $v>0$ the total three-momentum, evaluated at equal $t^{\prime }$, isThis follows by integrating $d{p}_p^{\prime }/d{t}^{\prime }=F$ from its switching event and $d{p}_e^{\prime }/d{t}^{\prime }=-F$ from its later switching event. During the interval between them only the Positron feels the prescribed electric field, so momentum conservation for the particle pair fails. Once both forces act, the total three-momentum is constant again.

The equal-time momentum of an externally driven particle pair illustrates a distinction. The sum of particle four-momenta at a fixed pair of events is always a four-vector. However, a fresh sum at equal time in each inertial frame generally uses different events. Here transforming an equal-$t$ laboratory sum would give $-2\Gamma v\sqrt{m^2{c}^4+F^2{c}^2{t}^2}/c^2$, which varies with $t$; this differs from the equal-$t^{\prime }$ answer above. For example, with $c=m=F=d=1$ and $v=0.6$, the late-time equal-$t^{\prime }$ sum is $-0.75$, whereas transforming the equal-$t=0$ sum gives $-1.5$. This is a concrete failure of treating the pair's equal-time sums as components of a single frame-independent four-vector.

A closed system's total four-momentum, including the electromagnetic field and the apparatus supplying it, is conserved and independent of the choice of spacelike integration surface under the usual isolated-system boundary conditions. The particles alone exchange energy and momentum with an external system, so neither that independence nor universal particle-only momentum conservation is guaranteed.

In the same Minkowski metric and electromagnetic units, $T^{\mu \nu }=-F^{\mu }{}_{\lambda }{F}^{\nu \lambda }+\frac{1}{4}{g}^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }$. Differentiating givesTo simplify the last two terms, contract ${\partial }_{\mu }{F}_{\nu \lambda }+{\partial }_{\nu }{F}_{\lambda \mu }+{\partial }_{\lambda }{F}_{\mu \nu }=0$ with $F^{\mu \lambda }$. Its first and third terms are equal, and its middle term is $-\frac{1}{2}{\partial }_{\nu }(F_{\alpha \beta }{F}^{\alpha \beta })$. Raising $\nu$ therefore gives $F^{\mu }{}_{\lambda }{\partial }_{\mu }{F}^{\nu \lambda }=\frac{1}{4}{\partial }^{\nu }(F_{\alpha \beta }{F}^{\alpha \beta })$. The cancellation leavesThis is the electromagnetic stress-energy divergence identity in rationalized units. As a sign check, $T^{00}=(E^2+B^2)/2$ is positive and $T^{0i}=(\mathbf{E}\times \mathbf{B}{)}_i$ is the Poynting vector. The integrated four-momentum $P^{\nu }=\int T^{0\nu }{d}^{3}x$ is conserved if the boundary flux vanishes. With matter sources, the electromagnetic field alone instead obeys ${\partial }_{\mu }{T}^{\mu \nu }=-F^{\nu \lambda }{J}_{\lambda }$: energy and momentum transfer to matter restore conservation of the total stress-energy tensor.

The Pauli term couples a fermion bilinear to electromagnetic field strength through $\bar{\psi }{\sigma }^{\mu \nu }{F}_{\mu \nu }\psi$. In four dimensions it is a dimension-five interaction.

A photon is one quantum of the electromagnetic field. In vacuum it has zero rest mass, energy $E=h\nu =\hslash \omega$, and momentum magnitude $p=E/c$. Its two physical polarization degrees of freedom are transverse to its propagation direction. A monochromatic field mode can contain zero, one or several photons; its photon number distinguishes these quantum states.

The photoelectric effect illustrates a photon's discrete energy exchange: increasing incident frequency increases the available electron kinetic energy, whereas increasing light intensity primarily increases the rate of events in the weak-illumination regime. Compton scattering additionally tests the associated momentum transfer. Interference still depends on probability amplitudes, so describing light in terms of photons does not discard its wave behavior.

For a single electromagnetic field mode, the eigenvalues of the number operator count its photons.

The Planck constant $h$ connects the frequency $\nu$ of an electromagnetic field mode with a single photon's energy, $E=h\nu$. Its physical dimension is action, or energy multiplied by time. It sets the scale of quantum mechanics: the de Broglie wavelength of a particle is $\lambda =h/p$, and discrete energy levels often have spacings involving $h$.

In the International System of Units, $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ exactly. Fixing this value defines the kilogram through already defined units of length and time; it does not remove experimental uncertainty from a practical mass measurement.^[ref] The reduced Planck constant $\hslash =h/(2\pi )$ is convenient when angular frequency replaces ordinary frequency.

Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

In geometrical optics, suppose radiation has a nonnegative radiance $L(\mathbf{x},\mathbf{s})$ whose directional contributions add as power. Here $\mathbf{s}$ is the actual propagation unit vector, $d\Omega$ is a solid angle element, and the collecting oriented surface $A$ has chosen unit normal $\mathbf{n}(\mathbf{x})$. Assume the following integrals exist. The outward and inward radiant fluxes areBoth quantities are nonnegative; their difference is the signed net radiant flux. The dot product supplies the cosine of the angle with the unit normal, hence the projected area. For incoming radiation the outward unit normal gives a negative dot product; the minus sign makes the collected power positive. Measuring the incoming angle from a source-facing unit normal instead gives the familiar positive $\cos \theta$. The units check is $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\times {\mathrm{m}}^2\times \mathrm{sr}=\mathrm{W}$.

Equivalently, the incoming surface integral is ${\Phi }_{-}={\int }_{A}EdA$ with local irradiance $E$; the emitted part of ${\Phi }_{+}$ is ${\int }_{A}MdA$ with radiant exitance $M$. For a small flat patch and a narrow accepted cone, nearly constant radiance and angle give $\Phi \simeq LA\cos \theta \Delta \Omega$. This approximation requires uniformity across the patch and cone. Direction-independent outgoing radiance over a whole hemisphere gives the different exact result ${\Phi }_{+}=\pi LA$, by Lambert's cosine law and the spherical coordinates calculation in radiant exitance. Replacing the cosine-weighted integral by the hemisphere's $2\pi$ solid angle would double the answer.^[ref]

These radiance formulas presume a description in additive directional power; they do not reconstruct arbitrary coherent wave interference from separate ray contributions. For general electromagnetic fields, the signed instantaneous power crossing an oriented surface is ${\int }_A\mathbf{S}\cdot \mathbf{n}dA$, using the Poynting vector $\mathbf{S}$; optical measurements commonly average over many wave periods. A zero net radiant flux can conceal equal nonzero incoming and outgoing radiant fluxes.

For a free electromagnetic field with suitable boundary conditions, combine Coulomb gauge ${\partial }_i{A}_i=0$ with $A_0=0$. Gauss's law makes the second condition consistent in the source-free problem. The remaining field operator is spatially transverse and describes the two radiative photon polarizations. A constrained scalar potential must be restored when charges are present.

Radiation reaction is the influence on a source's motion of its own electromagnetic field, including the balance of radiated energy and momentum with changes of the near field. For sufficiently weak radiation loss, an averaged power balance can give the leading damping law without solving the full self-field dynamics. Replacing a constant-frequency radiation formula by its instantaneous slowly varying value requires an additional time-scale assumption; it is not an exact treatment of arbitrarily rapid motion changes.

In the radiation zone of a localized source, the leading electromagnetic fields decay as $1/R$, are transverse to the observation direction, and satisfy $B=\hat{R}\times E/c$. Terms decaying faster than $1/R$ do not contribute to the limiting radiated power.