Belinfante–Rosenfeld stress–energy tensor

In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved.

In a classical or quantum local field theory, the generator of Lorentz transformations can be written as an integral

$M_{\mu\nu} = \int \mathrm{d}^3x \, {M^0}_{\mu\nu}$

of a local current

${M^\mu}_{\nu\lambda}= (x_\nu {T^\mu}_\lambda - x_\lambda {T^\mu}_\nu)+ {S^\mu}_{\nu\lambda}.$

Here ${T^\mu}_\lambda$ is the canonical Noether energy–momentum tensor, and ${S^{\mu}}_{\nu\lambda}$ is the contribution of the intrinsic (spin) angular momentum. Local conservation of angular momentum

$\partial_\mu {M^\mu}_{\nu\lambda}=0 \,$

requires that

$\partial_\mu {S^\mu}_{\nu\lambda}=T_{\lambda\nu}-T_{\nu\lambda}.$

Thus a source of spin-current implies a non-symmetric canonical energy–momentum tensor.

The Belinfante–Rosenfeld tensor^[1][2] is a modification of the energy momentum tensor

$T_B^{\mu\nu} = T^{\mu\nu} +\frac 12 \partial_\lambda(S^{\mu\nu\lambda}+S^{\nu\mu\lambda}-S^{\lambda\nu\mu})$

that is constructed from the canonical energy momentum tensor and the spin current ${S^{\mu}}_{\nu\lambda}$ so as to be symmetric yet still conserved.

An integration by parts shows that

$M^{\nu\lambda} = \int (x^\nu T^{0\lambda}_B - x^\lambda T^{0\nu}_B) \, \mathrm{d}^3x,$

and so a physical interpretation of Belinfante tensor is that it includes the "bound momentum" associated with gradients of the intrinsic angular momentum. In other words, the added term is an analogue of the ${\bold J}_\text{bound}= \nabla\times \bold {M}$ "bound current" associated with a magnetization density ${\bold M}$.

The curious combination of spin-current components required to make $T_B^{\mu\nu}$ symmetric and yet still conserved seems totally ad hoc, but it was shown by both Rosenfeld and Belinfante that the modified tensor is precisely the symmetric Hilbert energy–momentum tensor that acts as the source of gravity in general relativity. Just as it is the sum of the bound and free currents that acts as a source of the magnetic field, it is the sum of the bound and free energy–momentum that acts as a source of gravity.

Weinberg defines the Belinfante tensor as^[3]

$T_B^{\mu\nu}=T^{\mu\nu}-\frac{i}{2}\partial_\kappa \left[\frac{\partial\mathcal{L}}{\partial(\partial_\kappa\Psi^\ell)}(\mathcal{J}^{\mu\nu})^\ell_{\,\, m}\Psi^m-\frac{\partial\mathcal{L}}{\partial(\partial_\mu\Psi^\ell)}(\mathcal{J}^{\kappa\nu})^\ell_{\,\, m}\Psi^m-\frac{\partial\mathcal{L}}{\partial(\partial_\nu\Psi^\ell)}(\mathcal{J}^{\kappa\mu})^\ell_{\,\, m}\Psi^m\right]$

where $\mathcal{L}$ is the Lagrangian density, the set {Ψ} are the fields appearing in the Lagrangian, the non-Belinfante energy momentum tensor is defined by

$T^{\mu\nu}=\eta^{\mu\nu}\mathcal{L}-\frac{\partial\mathcal{L}}{\partial(\partial_\mu\Psi^\ell)}\partial^\nu\Psi^\ell$

and $\mathcal{J^{\mu\nu}}$ are a set of matrices satisfying the algebra of the homogeneous Lorentz group^[4]

$[\mathcal{J}^{\mu\nu},\mathcal{J}^{\rho\sigma}]=i\mathcal{J}^{\rho\nu}\eta^{\mu\sigma}-i\mathcal{J}^{\sigma\nu}\eta^{\mu\rho}-i\mathcal{J}^{\mu\sigma}\eta^{\nu\rho}+i\mathcal{J}^{\mu\rho}\eta^{\nu\sigma}$.

References

1. F. J. Belinfante (1940). "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. doi:10.1016/S0031-8914(40)90091-X.
2. L. Rosenfeld (1940). "Sur le tenseur D’Impulsion- Energie". Acad. Roy. Belg. Memoirs de classes de Science 18.
3. Weinberg, Steven (2005). The quantum theory of fields (Repr., pbk. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521670531.
4. Mexico, Kevin Cahill, University of New (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 9781107005211.