Angular Momentum and Spin Current in Fields

Six parameters of the Lorentz group remain unharvested. Translations gave the stress-energy tensor; internal rotations gave charge; now rotations and boosts — the transformations that mix the coordinates among themselves — produce the chapter's most structurally rich conservation law, and inside it, something genuinely new. Run Noether's machine on a scalar field and the angular-momentum current is exactly what particle mechanics would guess: position wedged with momentum, $x \wedge T$ , the continuum version of $\mathbf{r}\times\mathbf{p}$ . Run it on a field that carries indices — a vector field, a spinor — and an extra term appears that has no orbital reading at all: the spin current, angular momentum stored in how the field's components rotate among themselves, present even in a plane wave with no spatial structure whatsoever. Spin, which quantum mechanics is usually blamed for, is sitting in classical field theory as a Noether output. The article's second theme is a precise identity with consequences across three articles: demanding that the total angular-momentum current be conserved forces the antisymmetric part of the canonical stress tensor to equal minus the divergence of the spin current. An asymmetric $T^{\mu\nu}$ — the disease diagnosed in the stress-energy article — is thus not a bug but a ledger entry: wherever spin flows, momentum bookkeeping tilts to compensate. That identity is exactly what the Belinfante-Rosenfeld article will exploit to symmetrize the tensor, and its charge-level content — only the total $J = L + S$ is conserved, with orbital and spin parts free to trade — is a working fact of modern optics and spintronics laboratories.

Key takeaways

Lorentz invariance (6 parameters: 3 rotations + 3 boosts) yields via Noether a conserved rank-three current $M^{\mu\alpha\beta} = x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha} + S^{\mu\alpha\beta}$ , antisymmetric in $\alpha\beta$ .
For scalars the current is purely orbital ( $x \wedge T$ , the field version of $\mathbf{r}\times\mathbf{p}$ ); fields with Lorentz indices add the spin current $S^{\mu\alpha\beta} = -\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}(\Sigma^{\alpha\beta})_a{}^b\phi_b$ .
Spin is representation content: it comes from the matrices $\Sigma^{\alpha\beta}$ that rotate field components into each other, and exists even for spatially structureless plane waves.
The charges split as $J^{\alpha\beta} = L^{\alpha\beta} + S^{\alpha\beta}$ , but only the total is conserved in general — orbital and spin angular momentum exchange.
Conservation $\partial_\mu M^{\mu\alpha\beta} = 0$ forces the identity $T^{\alpha\beta} - T^{\beta\alpha} = -\partial_\mu S^{\mu\alpha\beta}$ : stress-tensor asymmetry is compensated spin flow.
The rotation charges $J^{ij}$ are angular momentum; the boost charges $J^{0i}$ enforce uniform motion of the center of energy.
The EM spin current, computed explicitly, is not gauge-invariant by itself — a warning shot for the orbital/spin split's physical status.
Key idea: Lorentz symmetry conserves one total angular momentum current with two faces — orbital ( $x \wedge T$ ) and spin (component rotation) — locked together by the stress tensor's asymmetry.

The core insight

Lorentz invariance gives the conserved current $M^{\mu\alpha\beta} = x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha} + S^{\mu\alpha\beta}$ ; its conservation ties the spin current to the antisymmetry of the canonical stress tensor.

Where this lives today

The orbital-spin anatomy of this article is a working laboratory subject. Structured-light optics manufactures beams at every point of the $L$ - $S$ ledger: circularly polarized Gaussian beams (pure spin), Laguerre-Gaussian vortex modes carrying $\ell\hbar$ of orbital angular momentum per photon with $\ell$ into the hundreds (pure orbital, the worked example's mode), and engineered superpositions used for optical tweezers that spin and orbit trapped microspheres simultaneously — the two rotation modes distinguishable by eye under a microscope, orbital circulating the particle around the beam axis while spin rotates it about its own. OAM multiplexing packs independent data channels onto coaxial vortex beams; quantum-communication experiments entangle the OAM degree of freedom to reach high-dimensional Hilbert spaces. Spintronics runs the same ledger in solids: spin currents — literal transport of $S$ without charge transport — are generated, steered and detected (spin Hall effects), and the mechanical face of the exchange survives as micro-scale Einstein-de Haas experiments where demagnetizing a cantilever twists it. On the theory frontier, the split's ambiguities went from nuisance to research subject: the "proton spin crisis" — measurements showing quark spins supply only about a third of the proton's $\hbar/2$ — turned the gauge-invariant decomposition of angular momentum into quark/gluon spin and orbital pieces into a two-decade program (Ji's and Jaffe-Manohar's decompositions differ by exactly the improvement-class reshufflings of this article), with lattice QCD and the upcoming Electron-Ion Collider tasked to measure the pieces. Within the chapter: identity (5) is the input the Belinfante-Rosenfeld article converts into a symmetric stress tensor; the six charges $J^{\alpha\beta}$ join $P^\mu$ to complete the Poincaré algebra whose closure the Poisson-brackets article exhibited and the Lorentz-covariance article audits.

The physical picture

Two ways a field can carry angular momentum, and the eye can tell them apart. The first is orbital: energy-momentum circulating around an axis. A vortex-shaped wave whose phase winds around the origin carries momentum density $T^{0i}$ with a swirl in it, and the moment of that swirl — position wedge momentum, integrated — is angular momentum exactly as a planet's is. Nothing about the field's type matters; a pure scalar does this perfectly well, and its angular-momentum current is nothing but $x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha}$ . The second way needs indices. Take a vector field — at each point an arrow — and let every arrow rotate in place, uniformly, everywhere: a circularly polarized plane wave does precisely this. There is no swirl in the energy flow; the wave marches straight ahead, spatially featureless. Yet rotating the system costs a phase, the configuration responds to rotations, and Noether duly assigns it angular momentum — stored not in circulation of energy through space but in the turning of the field's own components. That is spin: angular momentum of representation content. Its existence in classical field theory is the article's first surprise, and the 1936 Beth experiment made it mechanical fact — circularly polarized light, passed through a birefringent plate hanging on a torsion fiber, twists the plate.

Figure

Two ways to carry angular momentum — energy circulating through space, and field components rotating in place

The third piece of the picture is the coupling between the two faces. Orbital and spin angular momentum are not separately sacred: pass a circularly polarized beam through a suitably shaped lens and spin converts to orbital swirl; let an electron's spin relax in a magnet and the crystal lattice picks up rotation (the Einstein-de Haas effect — a magnet, suddenly demagnetized, physically turns). Field theory's accounting for this exchange is exact and local: the total current $M^{\mu\alpha\beta}$ is conserved; its orbital divergence equals the stress tensor's antisymmetric part; its spin divergence equals minus that — the two leaks cancel identically. An asymmetric canonical $T^{\mu\nu}$ , which looked like a defect in the stress-energy article, is revealed as the pivot of the exchange: momentum bookkeeping tilts precisely where spin is flowing.

The Lorentz current

An infinitesimal Lorentz transformation acts on coordinates with an antisymmetric parameter matrix:

x'^\mu = x^\mu + \omega^\mu{}_\nu\,x^\nu, \qquad \omega_{\mu\nu} = -\omega_{\nu\mu} \qquad (1)

where: the six independent components of $\omega$ are the three rotation angles ( $\omega_{ij}$ ) and three boost rapidities ( $\omega_{0i}$ ) — the SR chapter's transformations, linearized. Fields respond in two ways at once, and keeping the two responses distinct is the whole game:

\delta\phi_a(x) = -\tfrac{1}{2}\,\omega_{\alpha\beta}\,(\Sigma^{\alpha\beta})_a{}^b\,\phi_b(x)\; -\; \omega^\mu{}_\nu\,x^\nu\,\partial_\mu\phi_a(x) \qquad (2)

where: the second term is orbital — the field is dragged along the rotated coordinates, exactly as translations dragged it, just with a position-dependent shift $\delta x^\mu = \omega^\mu{}_\nu x^\nu$ ; the first term is intrinsic — the matrices $\Sigma^{\alpha\beta}$ encode how the field's own components mix under Lorentz transformations. For a scalar, $\Sigma = 0$ (one component, nothing to mix). For a vector field $A^\rho$ , the $\Sigma$ 's are the rotation/boost matrices acting on the index $\rho$ ; for spinors, they are built from Dirac matrices. The $\Sigma$ 's are the field's Lorentz representation, chosen when you declare what kind of field you have — and everything called "spin" descends from them.

Derivation AThe angular-momentum current and its orbital-spin decomposition

We run Noether's machine on the Lorentz transformation (1)-(2) and obtain the conserved rank-three current, watching it split into an orbital piece built from the stress tensor and a spin piece built from the $\Sigma$ matrices. Strategy: this is a spacetime symmetry, so we use the general current "momenta × total field response − dragged density," with the same $K^\mu$ mechanics as translations; the new labor is bookkeeping the two response terms and peeling off the six antisymmetric parameters. Watch how the orbital piece assembles itself out of the translation current — angular momentum is the moment of momentum, in field theory as in mechanics.

Step 1. The general spacetime-symmetry current (Noether article's template, with $\Delta\phi$ the full response and the transport term for $\delta x^\mu \neq 0$ ):

J^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\,\delta\phi_a + \mathcal{L}\,\delta x^\mu

with $\delta x^\mu = \omega^\mu{}_\nu x^\nu$ ; the $\mathcal{L}\,\delta x^\mu$ term is the dragged density, legitimate because $\partial_\mu(\mathcal{L}\,\omega^\mu{}_\nu x^\nu) = \omega^\mu{}_\nu x^\nu\partial_\mu\mathcal{L}$ (the trace $\omega^\mu{}_\mu = 0$ kills the extra piece — Lorentz transformations are volume-preserving, which is why they enter the $K^\mu$ game as cleanly as translations).

Step 2. Insert the two-part response (2) and group. The orbital part of $\delta\phi$ combines with the dragged density into the stress tensor contracted with the coordinate moment:

J^\mu = -\tfrac{1}{2}\omega_{\alpha\beta}\,\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}(\Sigma^{\alpha\beta})_a{}^b\phi_b \;-\; \omega^\alpha{}_\beta\, x^\beta\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\partial_\alpha\phi_a - \delta^\mu_\alpha\mathcal{L}\right]

The bracket is exactly $T^\mu{}_\alpha$ — the canonical tensor, reappearing as the carrier of the orbital piece.

Step 3. Peel the parameters. Using the antisymmetry of $\omega_{\alpha\beta}$ to antisymmetrize the coefficient, write $J^\mu = \tfrac12\omega_{\alpha\beta}M^{\mu\alpha\beta}$ with

M^{\mu\alpha\beta} = x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha} + S^{\mu\alpha\beta}, \qquad S^{\mu\alpha\beta} = -\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)}\,(\Sigma^{\alpha\beta})_a{}^b\,\phi_b \qquad (3)

Six independent $\omega$ 's, six conserved currents: $\partial_\mu M^{\mu\alpha\beta} = 0$ on-shell.

Step 4. Read the anatomy. The orbital piece $x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha}$ is the field-theoretic $\mathbf{r}\times\mathbf{p}$ : moments of the momentum ledger. The spin piece $S$ is momenta contracted with the representation matrices — structurally the same template as the internal-symmetry current of the internal-symmetries article, with $\Sigma^{\alpha\beta}$ in place of the internal generators $T^A$ . Spin is, in this precise sense, "internal charge under rotations": angular momentum carried by the field's index structure, no moment arm involved. For scalars $S = 0$ and the current is purely orbital; the charges are

J^{\alpha\beta} = \int\mathrm{d}^3x\; M^{0\alpha\beta} = L^{\alpha\beta} + S^{\alpha\beta} \qquad (4)

Go deeper · Derivation B · 1 minConservation forces the spin-asymmetry identity

Derivation BConservation forces the spin-asymmetry identity

We differentiate the total current (3) and demand conservation, obtaining the article's pivotal identity: the antisymmetric part of the canonical stress tensor equals minus the divergence of the spin current. This one line explains why scalar theories have symmetric $T$ (no spin, no asymmetry), why spinful theories cannot (their spin flows), and hands the Belinfante-Rosenfeld article exactly the object it needs to symmetrize with. Strategy: product rule on the orbital piece, using translation conservation to kill the $x$ -weighted terms; what survives is pure algebra.

Step 1. Divergence of one orbital term, by the product rule ( $\partial_\mu x^\alpha = \delta^\alpha_\mu$ ):

\partial_\mu\big(x^\alpha T^{\mu\beta}\big) = \delta^\alpha_\mu T^{\mu\beta} + x^\alpha\,\partial_\mu T^{\mu\beta} = T^{\alpha\beta}

Why the second term dies: $\partial_\mu T^{\mu\beta} = 0$ is the translation conservation law of the stress-energy article — angular-momentum bookkeeping consumes momentum bookkeeping as an input, exactly as $\mathrm{d}(\mathbf{r}\times\mathbf{p})/\mathrm{d}t = \mathbf{r}\times\mathbf{F}$ consumed $\dot{\mathbf{p}} = \mathbf{F}$ in mechanics.

Step 2. The full orbital divergence is therefore the stress tensor's antisymmetric part:

\partial_\mu\big(x^\alpha T^{\mu\beta} - x^\beta T^{\mu\alpha}\big) = T^{\alpha\beta} - T^{\beta\alpha}

Step 3. Add the spin piece and impose total conservation:

0 = \partial_\mu M^{\mu\alpha\beta} = \big(T^{\alpha\beta} - T^{\beta\alpha}\big) + \partial_\mu S^{\mu\alpha\beta} \quad\Longrightarrow\quad T^{\alpha\beta} - T^{\beta\alpha} = -\,\partial_\mu S^{\mu\alpha\beta} \qquad (5)

Step 4. Read it in both directions. Left to right: a canonical stress tensor is asymmetric exactly where spin current diverges — the asymmetry is not an error but the local record of spin-orbital exchange. Right to left: if a theory's spin current happens to be divergence-free (or absent, as for scalars), its canonical tensor is symmetric and $L$ and $S$ are separately conserved. Charge-level corollary: $\dot L^{\alpha\beta} = \int(T^{\alpha\beta} - T^{\beta\alpha})\,\mathrm{d}^3x = -\dot S^{\alpha\beta}$ — orbital and spin angular momentum trade at a rate set by the integrated asymmetry, with the total (4) fixed.

The electromagnetic spin current — and a warning label

The showcase spinful field is electromagnetism. From $\mathcal{L} = -\tfrac14 F_{\alpha\beta}F^{\alpha\beta}$ , the canonical momenta are $\frac{\partial\mathcal{L}}{\partial(\partial_\mu A^\rho)} = -F^\mu{}_\rho$ , and the vector representation's generators $(\Sigma^{\alpha\beta})^\rho{}_\sigma = \delta^\alpha_\sigma\eta^{\beta\rho} - \delta^\beta_\sigma\eta^{\alpha\rho}$ give, from the definition in (3),

S^{\mu\alpha\beta}_{\text{EM}} = -F^{\mu\alpha}A^\beta + F^{\mu\beta}A^\alpha \qquad (6)

where: the structure is transparent — field strength times bare potential, antisymmetrized. Its spatial charge density $S^{0ij}$ evaluates, in vector language, to $(\mathbf{E}\times\mathbf{A})_k$ (with $ij$ dual to $k$ ): the classical spin density of light, the very quantity Beth's torsion fiber measured. But note the warning label stamped on (6): it contains the bare potential $A^\beta$ , so the spin current — like the canonical stress tensor it partners — is not gauge-invariant. Under $A \to A + \partial\lambda$ the spin and orbital pieces reshuffle, while the total $M^{\mu\alpha\beta}$ changes only by improvement-class terms with invariant charges. The physical lesson, sharpened in the FAQ: for light, the total angular momentum is unambiguous; the $L$ / $S$ split is convention-dependent in general and acquires operational meaning in specific settings (paraxial beams, where "spin per photon $\pm\hbar$ , orbital per photon $\ell\hbar$ " is measured daily).

Figure

The exchange ledger — orbital and spin angular momentum trade through the stress tensor's asymmetry, total fixed

Limiting cases

Case	Result	Meaning
scalar field	$\Sigma = 0$ : $M = x\wedge T$ only	no spin; canonical $T$ symmetric automatically
point-particle limit	$J^{ij} \to (\mathbf{r}\times\mathbf{p})_k$	the mechanics chapters' angular momentum recovered
plane wave, circular polarization	$L = 0$ , $S \neq 0$	spin with zero spatial structure — pure representation content
vortex (phase-winding) scalar wave	$S = 0$ , $L = \ell E/\omega$	orbital with zero spin — pure swirl (worked example)
divergence-free spin current	$T^{[\alpha\beta]} = 0$ ; $L$ , $S$ separately conserved	the exchange channel closed
rotation charges $J^{ij}$	conserved angular momentum	the 3 rotations' Noether harvest
boost charges $J^{0i}$	center of energy moves uniformly	the 3 boosts' harvest — see FAQ
after Belinfante improvement	$S$ absorbed into $\Theta^{\mu\nu}$ ; $M = x\wedge\Theta$	all angular momentum reads as orbital of the improved tensor

Common trapThe orbital-spin split is not two conservation laws — only the total is sacred

The single most seductive error this subject invites: treating $L^{\alpha\beta}$ and $S^{\alpha\beta}$ as separately conserved quantities because they are separately named. They are not. Identity (5) is explicit about the exchange: whenever the spin current diverges, the canonical stress tensor tilts and orbital angular momentum is created or destroyed at exactly the compensating rate — only $J = L + S$ is a Noether charge. Concrete manifestations: a quarter-wave plate converts spin to nothing mechanical except by absorbing the difference as torque (Beth); a spiral phase plate converts spin to orbital swirl within the beam; the Einstein-de Haas effect trades electron spin for crystal rotation. Compounding the trap, the split itself is convention-dependent: improvement transformations — including the gauge transformations of (6) — reshuffle $L$ and $S$ while preserving the total, so "the spin part of the field's angular momentum" is well-defined only after fixing a convention or an operational context (paraxial optics being the famous benign case). The error's exam form: computing $S$ for the EM field in one gauge, $L$ in another, and reporting a "violation" of angular-momentum conservation. The books balance only when kept in one convention, and only for the total.

Worked examples

1 · Orbital angular momentum of a vortex wave (confidence builder)

A scalar field carries a rotating vortex mode: $\phi = A\,f(r)\cos(\ell\theta - \omega t)$ in cylindrical coordinates, with winding number $\ell = 2$ and $\omega = 1.25$ . Show that the cycle-averaged angular momentum per unit energy is $\ell/\omega$ , and evaluate the angular momentum for total wave energy $E = 0.0500$ .

Step 1. The $z$ -angular-momentum density is the moment of the azimuthal momentum density: $m_z = -T^0{}_\theta = -\dot\phi\,\partial_\theta\phi$ (scalar field: purely orbital, $S = 0$ ). Compute the two factors: $\dot\phi = A\omega f\sin(\ell\theta - \omega t)$ , $\partial_\theta\phi = -A\ell f\sin(\ell\theta - \omega t)$ .

Step 2. So $m_z = A^2\omega\ell f^2\sin^2(\ell\theta - \omega t)$ , with cycle average $\langle m_z\rangle = \tfrac12 A^2\omega\ell f^2(r)$ — positive everywhere: the wave circulates in the $+z$ sense set by the sign of $\ell$ .

Step 3. The energy density's oscillating structure gives (dominant terms for a slowly varying envelope) $\langle T^{00}\rangle = \tfrac12 A^2\omega^2 f^2(r)$ — the traveling-wave result of the stress-energy article, applied azimuthally.

Step 4. Take the ratio, which the envelope $f^2$ drops out of upon integration:

\frac{J_z}{E} = \frac{\int\langle m_z\rangle\,\mathrm{d}^3x}{\int\langle T^{00}\rangle\,\mathrm{d}^3x} = \frac{\ell}{\omega} = \frac{2}{1.25} = \mathbf{1.60}

For $E = 0.0500$ : $J_z = 1.60 \times 0.0500 = \mathbf{0.0800}$ . The quantum reading falls out for free: each quantum carries energy $\hbar\omega$ , hence angular momentum $\hbar\omega \times \ell/\omega = \ell\hbar$ — vortex modes carry $\ell\hbar$ of orbital angular momentum per photon, the operating principle of the OAM light beams below.

2 · Spin of a circularly polarized wave, and the photon's spin (exam level)

A circularly polarized electromagnetic plane wave propagates along $z$ : in the radiation gauge, $\mathbf{A} = \frac{E_0}{\omega}(\cos(kz - \omega t),\,\sin(kz - \omega t),\,0)$ with $E_0 = 0.10$ , $\omega = k = 2.00$ (vacuum). (i) Compute the spin density $s_z = (\mathbf{E}\times\mathbf{A})_z$ and the energy density $u$ . (ii) Form the ratio and interpret per quantum.

Step 1. Fields: $\mathbf{E} = -\partial_t\mathbf{A} = E_0(-\sin(kz - \omega t),\,\cos(kz - \omega t),\,0)$ — magnitude $E_0$ at every instant (circular polarization never dims). The magnetic field likewise has magnitude $|\mathbf{B}| = E_0$ (vacuum wave).

Step 2. Spin density from the EM spin current (6) in this gauge:

s_z = (\mathbf{E}\times\mathbf{A})_z = E_x A_y - E_y A_x = \frac{E_0^2}{\omega}\big(-\sin\cdot\sin - \cos\cdot\cos\big) = -\frac{E_0^2}{\omega}

Magnitude $|s_z| = (0.10)^2/2.00 = \mathbf{0.00500}$ , constant — no cycle-averaging needed; the sign flips with the handedness (this wave is left-circular in the chosen convention; the mirror wave gives $+E_0^2/\omega$ ).

Step 3. Energy density: $u = \tfrac12(E^2 + B^2) = \tfrac12(E_0^2 + E_0^2) = E_0^2 = \mathbf{0.0100}$ , also constant.

Step 4. The ratio: $|s_z|/u = 1/\omega = \mathbf{0.500}$ . Per quantum of energy $\hbar\omega$ , the wave carries spin $\hbar\omega/\omega = \hbar$ — the photon's spin is $\pm\hbar$ , sign set by handedness, read off a classical field computation. Beth's 1936 measurement is Step 4 with a torsion balance: a half-wave plate flips the handedness, absorbing $2\hbar$ per photon as mechanical torque, and the measured twist matched this classical ledger. Note what is absent: no spatial swirl anywhere — $L_z = 0$ for this beam — the angular momentum is pure representation content, the $\Sigma$ term working alone.

Origins · 1 minEugene Wigner and how this was discovered

Eugene Wigner

1902–1995 · Princeton

Wigner made symmetry the primary language of physics — this chapter's harvest of currents and charges is organized along lines he drew. Budapest-born (one of the "Martians," with von Neumann, Szilard and Teller), trained as a chemical engineer, he imported group theory into quantum mechanics in the late 1920s when most physicists dismissed it as "Gruppenpest" — the group-theory plague. His 1939 classification theorem is the deepest statement behind this article: the physically distinct relativistic fields correspond to irreducible unitary representations of the Poincaré group, labeled by exactly two invariants — mass and spin. Spin, in other words, is not a quantum afterthought bolted onto point particles; it is a representation label, fixed the moment you specify how a field transforms under the Lorentz group — precisely the role the $\Sigma^{\alpha\beta}$ matrices play in this article's current, and the reason the spin current exists classically. The 1963 Nobel Prize cited his symmetry principles; his other legacies run from the Wigner-Eckart theorem (how angular momentum constrains matrix elements) through random-matrix theory to the first reactor engineering. His essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" named the mystery this chapter keeps exhibiting: that objects invented for bookkeeping — representation matrices, Noether currents — turn out to be the furniture of the world.

History

1909

— Poynting argues circularly polarized light should carry angular momentum and exert torque — a classical-field prediction.

1918

— Noether's theorem folds rotations and boosts into the general symmetry-current framework.

1936

— Beth measures the torque of circularly polarized light on a birefringent plate: the photon's

\hbar

of spin, detected mechanically.

1939

— Wigner classifies relativistic particles by mass and spin as Poincaré representation labels; Belinfante ties the spin current to stress-tensor symmetrization.

1992

— Allen and coworkers create Laguerre-Gaussian laser beams carrying

\ell\hbar

of orbital angular momentum per photon — the vortex worked example, realized.

Going further

References. Landau & Lifshitz, The Classical Theory of Fields, §32–33 · Weinberg, The Quantum Theory of Fields, Vol. 1, §7.3–7.4 · Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., Ch. 13 · Soper, Classical Field Theory, Ch. 9 (angular momentum of fields) · Allen, Barnett & Padgett, Optical Angular Momentum (2003) — the modern optics of $L$ and $S$ Key identities to keep in mind: $\partial_\mu(x^\alpha T^{\mu\beta}-x^\beta T^{\mu\alpha})=T^{\alpha\beta}-T^{\beta\alpha}$ follows from $\partial_\mu T^{\mu\nu}=0$ ; the EM spin current follows from $\pi^\mu{}_\nu=-F^\mu{}_\nu$ and the vector generators; the vortex ratio is $J_z/E=\ell/\omega$ and the circular-polarization ratio is $s_z/u=1/\omega$ . Worked-example anchor numbers: $J_z/E = 2/1.25 = 1.60$ , $J_z = 0.0800$ ; $|s_z| = E_0^2/\omega = 0.00500$ , $u = E_0^2 = 0.0100$ , ratio $0.500 = 1/\omega$ , giving $\hbar$ per photon.

Builds on: Noether's Theorem for Continuous Symmetries · The Canonical Stress-Energy Tensor · Conserved Currents and Charges · Lorentz Covariance of Classical Field Theory Leads to: Symmetrization and the Belinfante-Rosenfeld Tensor · Internal Symmetries and Conserved Charge · Hamiltonian Formulation for Fields

How can a classical field have spin? I thought spin was quintessentially quantum.

What quantum mechanics contributes to spin is the quantization — the $\hbar/2$ units, the discrete spectrum — not the existence. Classically, spin is representation content: a field with components responds to rotations partly by mixing those components (the $\Sigma$ term of the transformation law), and Noether's theorem duly books angular momentum against that response. The circularly polarized wave of the worked example carries angular momentum density $E_0^2/\omega$ with literally zero spatial structure — nothing orbits anything — and that angular momentum exerts measurable torque (Beth, 1936). What the classical theory cannot do is explain why measuring a photon's spin along any axis yields exactly $\pm\hbar$ rather than a continuum: that discreteness arrives with quantization, when the classical ratio $s/u = 1/\omega$ per unit energy meets energy quantization $E = \hbar\omega$ to give exactly $\hbar$ per photon. The historical irony is worth savoring: the electron's spin was discovered in quantum spectroscopy and then recognized as Lorentz representation theory (Wigner), while light's spin was predicted classically (Poynting, 1909) and measured classically — the two routes converging on the same object, the $\Sigma^{\alpha\beta}$ of this article.

What do the boost charges J⁰ⁱ actually conserve? "Boost momentum" sounds meaningless.

They conserve something with a crisp Newtonian reading: the uniform motion of the center of energy. Write the charge out: $J^{0i} = \int(x^0 T^{0i} - x^i T^{00})\,\mathrm{d}^3x = t P^i - \int x^i\,T^{00}\,\mathrm{d}^3x$ . Conservation, $\mathrm{d}J^{0i}/\mathrm{d}t = 0$ , rearranges to $\frac{\mathrm{d}}{\mathrm{d}t}\langle x^i\rangle_E = P^i/P^0$ where $\langle x^i\rangle_E$ is the energy-weighted centroid: the center of energy moves in a straight line at constant velocity $\mathbf{P}/E$ — the field-theoretic upgrade of Newton's first law for the center of mass, with energy replacing mass as the weight (as relativity requires). This is a genuinely useful theorem: it forbids a closed system from displacing its own centroid (no reactionless drives), underlies the "photon in a box" thought experiments that derive $E = mc^2$ from bookkeeping, and its quantum operator version is why boosts, unlike rotations, have explicitly time-dependent generators. The boost charges also complete the Poincaré algebra: their brackets with $P^\mu$ and among themselves (closing on rotations — the Wigner rotation of the SR chapter's velocity-addition article) are the last entries of the algebra the covariance article audits.

Is the split of angular momentum into spin and orbital parts measurable or not? Optics labs seem to measure both daily.

Both horns are true, and the resolution is context. The general theorem is negative: improvement transformations (for gauge fields, ordinary gauge transformations — see the bare $A$ in the EM spin current) reshuffle $L$ and $S$ while fixing the total, so there is no universal, convention-free split. The operational escape is that specific physical settings fix the convention. In paraxial optics — beams nearly collimated along one axis — the natural transverse-gauge split is unambiguous at leading order, and the two parts couple to different degrees of freedom of a probe: spin torques a birefringent particle about its own axis; orbital angular momentum orbits it around the beam axis. Labs measure both because the paraxial context has already made the choice. The instructive hard case is the proton: quarks and gluons are relativistic, interacting and gauge-coupled, no paraxial mercy applies, and the community maintains two standard decompositions (Ji's, which packages quark orbital angular momentum gauge-invariantly but leaves gluon $J$ unsplit; Jaffe-Manohar's, natural in light-cone gauge) that genuinely differ — by exactly the reshuffling terms this article identifies. Neither is wrong; they answer differently posed questions. The portable lesson: always ask "in which decomposition?" before comparing spin fractions across papers.

Why does angular-momentum conservation require the stress tensor to be symmetric — mechanics never needed that?

Mechanics did need it; it just hid the requirement inside a modeling assumption. In point mechanics, $\mathrm{d}\mathbf{L}/\mathrm{d}t = \sum\mathbf{r}_i\times\mathbf{F}_i$ vanishes for internal forces only if they are central — Newton's third law in its strong form — which is the particle-mechanics ancestor of tensor symmetry. In continuum mechanics, the Cauchy stress tensor's symmetry is derived from the requirement that a vanishing volume element not experience infinite angular acceleration from surface torques — again angular momentum policing the momentum ledger. Field theory clarifies the logic: identity (5) shows symmetry of $T$ is equivalent to the absence of net spin flow, so theories with spin degrees of freedom legitimately carry asymmetric canonical tensors, the asymmetry booking the spin-orbital exchange. Even continuum mechanics knows the exception: "micropolar" or Cosserat media — materials whose grains carry intrinsic rotation, like liquid crystals — have asymmetric stress tensors and internal spin currents, a nineteenth-century curiosity now standard in soft-matter physics. So the honest statement is not "T must be symmetric" but "T plus the spin ledger must balance"; symmetric $T$ is the special case of spinless constituents, and the Belinfante construction shows any theory can be presented that way by absorbing the spin flow into the momentum bookkeeping.

Does the spin current mean angular momentum can flow without anything moving?

Yes — and that is neither paradox nor idle formalism. The spin current $S^{i\alpha\beta}$ books transport of intrinsic angular momentum in direction $i$ without any accompanying mass, charge or even energy current being required in principle. Solid-state physics realizes this cleanly: in a spin current, equal numbers of spin-up and spin-down electrons drift in opposite directions — zero charge current, zero net particle flow, finite angular-momentum flow — and such pure spin currents are generated (spin Hall effect), transmitted through insulators via magnon exchange, and detected (inverse spin Hall) as a matter of routine. The electromagnetic version is subtler because of the gauge caveat but has measurable proxies: the transverse spin of evanescent waves ("photonic wheels") produces lateral optical forces with no beam deflection. There is also a conceptual dividend for the chapter: the possibility of pure spin flow is exactly why the canonical stress tensor can be asymmetric while total angular momentum stays conserved — identity (5) is the local IOU between the two ledgers. What spin flow cannot do, by the boost-charge theorem above, is move the center of energy: intrinsic angular momentum is transportable, but the centroid's straight-line motion is protected by its own conservation law.

How does this article's machinery connect to the spin of the electron and the Dirac equation?

Structurally, completely — only the representation changes. The electron field is a spinor: its $\Sigma^{\alpha\beta} = \tfrac{i}{4}[\gamma^\alpha, \gamma^\beta]$ are built from Dirac matrices, and dropping them into this article's template (3) yields the Dirac field's spin current, whose charge density integrates to the familiar $\hbar/2$ per particle upon quantization. Every classical structure survives: the canonical stress tensor of the Dirac field is asymmetric with (5) booking the compensation; the Belinfante procedure symmetrizes it; the total $J = L + S$ generates rotations in the Poisson-bracket (then commutator) algebra. Two famous quantum wrinkles attach to the story. First, half-integer spin: the spinor representation is double-valued under rotations (a $2\pi$ rotation flips the spinor's sign), which is invisible in classical observables but becomes physical in quantum interference — neutron interferometry has directly observed the $4\pi$ periodicity. Second, the spin-statistics connection: fields with half-integer $\Sigma$ content must be quantized with anticommutators — a theorem whose inputs are exactly this chapter's Lorentz covariance and locality structures, and which has no classical-mechanical shadow at all. The classical spin current is thus the last station before genuinely quantum territory: the geometry of spin is classical; its statistics are not.