Angular Momentum, Spin Currents, and Symmetric Stress-Energy

Local spin alters the Lorentz current by adding an intrinsic spin current to its orbital part. Conservation then relates the antisymmetric part of canonical energy-momentum to spin transport; Belinfante improvement repackages both contributions into a symmetric tensor without erasing the total angular momentum.

Assume a Poincaré-invariant first-order field theory in Cartesian Minkowski coordinates. Let $T_{\rm can}^{\lambda\nu}$ be its translation current and let $S^{\lambda\mu\nu}=-S^{\lambda\nu\mu}$ be the current arising from the Lorentz action on the field’s component indices.

Required background. Classical Fields and Their Actions supplies field actions and Noether currents, while Events, Intervals, and Minkowski Geometry fixes Lorentz transformations and index conventions. Helpful background. Calculus and Linear Algebra for Relativity supplies tensor algebra; Presymplectic Potentials and Currents develops boundary-current ambiguities.

Lorentz current and local exchange

The Noether current for rotations and boosts can be written

$$J^{\lambda\mu\nu} =x^\mu T_{\rm can}^{\lambda\nu} -x^\nu T_{\rm can}^{\lambda\mu} +S^{\lambda\mu\nu}.$$

The first two terms are orbital; $S^{\lambda\mu\nu}$ is intrinsic.

The first index labels the current direction: $J^{0\mu\nu}$ is a density and $J^{i\mu\nu}$ its spatial flux. For example, rotation in the $x$–$y$ plane has density

$$J^{012}=x\,T_{\rm can}^{02}-y\,T_{\rm can}^{01}+S^{012}.$$

The first two terms are the familiar $x p_y-y p_x$ per unit volume. The last term accounts for how the field’s vector or spinor components themselves transform; it does not require modeling the field as little rotating solid bodies.

Using $\partial_\lambda T_{\rm can}^{\lambda\nu}=0$ on shell,

$$\partial_\lambda J^{\lambda\mu\nu} =T_{\rm can}^{\mu\nu}-T_{\rm can}^{\nu\mu} +\partial_\lambda S^{\lambda\mu\nu}.$$

Lorentz invariance therefore implies

$$\boxed{ T_{\rm can}^{\mu\nu}-T_{\rm can}^{\nu\mu} =-\partial_\lambda S^{\lambda\mu\nu}.}$$

An asymmetric canonical tensor is not by itself a failure of angular-momentum conservation: it measures local exchange between orbital and intrinsic pieces. For a scalar, $S^{\lambda\mu\nu}=0$; Tong, §1.3.3, printed pp. 16–18, Eqs. (1.51)–(1.57) derives precisely that orbital special case. The component-index term and its bosonic Belinfante completion are developed in Pons, §4.1, printed pp. 11–14.

Belinfante construction

Define

$$B^{\lambda\mu\nu} =\frac12\left( S^{\lambda\mu\nu} +S^{\mu\nu\lambda} +S^{\nu\mu\lambda} \right).$$

The antisymmetry of $S$ in its last two indices implies $B^{\lambda\mu\nu}=-B^{\mu\lambda\nu}$. Hence

$$T_{\rm B}^{\mu\nu} =T_{\rm can}^{\mu\nu} +\partial_\lambda B^{\lambda\mu\nu}$$

has the same divergence identically. Moreover,

$$T_{\rm B}^{\mu\nu}-T_{\rm B}^{\nu\mu} =T_{\rm can}^{\mu\nu}-T_{\rm can}^{\nu\mu} +\partial_\lambda S^{\lambda\mu\nu}=0$$

on shell. This is the Belinfante tensor. Its integrated four-momentum and angular momentum agree with the canonical-plus-spin charges only when the improvement’s boundary integrals vanish. Pons, §4.1, printed pp. 11–14, Eqs. (25)–(30) also shows that the on-shell qualification is essential.

Maxwell vector example

For the free Maxwell field, the canonical tensor found on the preceding page is

$$T_{\rm can}^{\mu\nu} =F^{\mu\alpha}\partial^\nu A_\alpha -\frac14\eta^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}.$$

Because $A_\mu$ is a Lorentz vector, its component transformation contributes the nonzero spin current

$$\boxed{ S^{\lambda\mu\nu} =F^{\lambda\mu}A^\nu-F^{\lambda\nu}A^\mu.}$$

It is antisymmetric in $\mu,\nu$. Using the vacuum field equation $\partial_\lambda F^{\lambda\mu}=0$ and the symmetry of $F^\mu{}_{\alpha}F^{\nu\alpha}$ gives

$$\partial_\lambda S^{\lambda\mu\nu} =-\left(T_{\rm can}^{\mu\nu}-T_{\rm can}^{\nu\mu}\right),$$

so the orbital and intrinsic divergences cancel exactly. Substitution into the Belinfante formula simplifies its superpotential to

$$B^{\lambda\mu\nu}=F^{\lambda\mu}A^\nu.$$

Consequently,

$$T_{\rm B}^{\mu\nu} =T_{\rm can}^{\mu\nu} +\partial_\lambda\!\left(F^{\lambda\mu}A^\nu\right) =F^\mu{}_{\rho}F^{\nu\rho} -\frac14\eta^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}$$

on the free equations of motion. The separate canonical and spin expressions depend on the gauge potential, whereas their Belinfante repackaging depends only on $F_{\mu\nu}$. With sources, the field sector alone has an exchange term and the full matter-plus-field system is required.

What happens to spin?

Using $T_{\rm B}$, the Lorentz current can be represented, up to a superpotential, as

$$J_{\rm B}^{\lambda\mu\nu} =x^\mu T_{\rm B}^{\lambda\nu} -x^\nu T_{\rm B}^{\lambda\mu}.$$

Intrinsic spin has been absorbed into the local distribution of momentum; it has not become physically absent. For the standard first-derivative scalar treated in this chapter, there is no component-index Lorentz generator, $S=0$, and the explicit canonical tensor is already symmetric. More generally, $S=0$ together with Lorentz invariance forces canonical symmetry on shell. Vector and spinor fields can have nonzero spin currents. The Maxwell calculation above produces the familiar gauge-invariant symmetric tensor

$$T_{\rm EM}^{\mu\nu} =F^{\mu\rho}F^\nu{}_\rho -\frac14\eta^{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}.$$

The original all-field construction is in Belinfante, Physica 7 (1940), pp. 449–474. Hilbert stress-energy may coincide with $T_{\rm B}$ under standard matter and covariantization assumptions, but remains defined by metric variation, not by this formula.

Angular momentum also depends on the chosen origin. Under $x^\mu\mapsto x^\mu+b^\mu$, the integrated charge changes by $J^{\mu\nu}\mapsto J^{\mu\nu}+b^\mu P^\nu-b^\nu P^\mu$, while intrinsic spin does not acquire an origin term. This supplies both a dimensional check and a reminder that only the complete Lorentz charge has the stated transformation law.

Exercise: verify local exchange

At some region suppose

$$T_{\rm can}^{12}-T_{\rm can}^{21}=2a(x), \qquad \partial_\lambda S^{\lambda12}=-2a(x).$$

Check conservation of $J^{\lambda12}$ and symmetry of the improved $12$ components.

Solution

The divergence of the Lorentz current is

$$\partial_\lambda J^{\lambda12} =\left(T_{\rm can}^{12}-T_{\rm can}^{21}\right) +\partial_\lambda S^{\lambda12} =2a-2a=0.$$

For the Belinfante tensor, the difference of the two improvement terms is precisely $\partial_\lambda S^{\lambda12}$. Therefore

$$T_{\rm B}^{12}-T_{\rm B}^{21} =2a-2a=0.$$

If $a=0$, spin current may still be present but must be divergence-free in this component. If $S=0$, Lorentz invariance forces the canonical tensor itself to be symmetric on shell.

Boundaries and scope

Boundary spin flow can make subsystem orbital and spin charges change even while the combined charge is conserved. Gauge theories may need boundary degrees of freedom, and spinors require the appropriate Lorentz representation and often a tetrad formulation. The construction here is classical and flat-space; it does not define gravitational angular momentum.

Stress-Energy Tensors for Particles, Fields, and Fluids next compares symmetric tensors while making their material assumptions explicit.

References

• F. J. Belinfante, “On the Current and the Density of the Electric Charge, the Energy, the Linear Momentum and the Angular Momentum of Arbitrary Fields,” Physica 7, 449–474 (1940), doi:10.1016/S0031-8914(40)90091-X.
• Josep M. Pons, “Noether Symmetries, Energy-Momentum Tensors and Conformal Invariance in Classical Field Theory,” Journal of Mathematical Physics 52, 012904 (2011), doi:10.1063/1.3532941, Open PDF, §4.1, printed pp. 11–14, Eqs. (25)–(30).
• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes (2006/2007), official PDF, §1.3.3, printed pp. 16–18, Eqs. (1.51)–(1.57).