In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the stress–energy tensor that is constructed from the canonical stress–energy 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 of a local current Here
is the contribution of the intrinsic (spin) angular momentum.
The anti-symmetry implies the anti-symmetry Local conservation of angular momentum requires that Thus a source of spin-current implies a non-symmetric canonical stress–energy tensor.
so as to be symmetric yet still conserved, i.e., An integration by parts shows that 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
"bound current" associated with a magnetization density
The curious combination of spin-current components required to make
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 stress–energy 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.
is defined by the variation of the action functional
with respect to the metric as or equivalently as (The minus sign in the second equation arises because
by varying a Minkowski-orthonormal vierbein
is the Minkowski metric for the orthonormal vierbein frame, and
are the covectors dual to the vierbeins.
With the vierbein variation there is no immediately obvious reason for
should be invariant under an infinitesimal local Lorentz transformation
is an arbitrary position-dependent skew symmetric matrix, we see that local Lorentz and rotation invariance both requires and implies that
is symmetric, it is easy to show that
We can now understand the origin of the Belinfante–Rosenfeld modification of the Noether canonical energy momentum tensor.
via the condition of being metric compatible and torsion free.
is then defined by the variation the vertical bar denoting that the
are held fixed during the variation.
The "canonical" Noether energy momentum tensor
is the part that arises from the variation where we keep the spin connection fixed: Then Now, for a torsion-free and metric-compatible connection, we have that where we are using the notation Using the spin-connection variation, and after an integration by parts, we find Thus we see that corrections to the canonical Noether tensor that appear in the Belinfante–Rosenfeld tensor occur because we need to simultaneously vary the vierbein and the spin connection if we are to preserve local Lorentz invariance.
As an example, consider the classical Lagrangian for the Dirac field Here the spinor covariant derivatives are We therefore get There is no contribution from
if we use the equations of motion, i.e. we are on shell.
Steven Weinberg defined the Belinfante tensor as[3] where
is the Lagrangian density, the set {Ψ} are the fields appearing in the Lagrangian, the non-Belinfante energy momentum tensor is defined by and
are a set of matrices satisfying the algebra of the homogeneous Lorentz group[4]