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.
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In a classical or quantum local field theory, the generator of Lorentz transformations can be written as an integral
of a local current
Here
requires that
Thus a source of spin-current implies a non-symmetric canonical energy–momentum tensor.
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
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
The curious combination of spin-current components required to make
Belinfante-Rosenfeld and the Hilbert energy-momentum tensor
The Hilbert energy-momentum tensor
or equivalently as
(The minus sign in the second equation arises because
We may also define an energy-momentum tensor
Here
With the vierbein variation there is no immediately obvious reason for
should be zero. As
Once we know that
We can now understand the origin of the Belinfante-Rosefeld modification of the Noether canonical energy momentum tensor. Take the action to be
and the "canonical" Noether energy momentum tensor
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
Now
Thus the Belinfante-Rosenfeld tensor becomes
The on-shell Belinfante-Rosenfeld tensor for the Dirac field is therefore seen to be the symmetrized canonical energy-momentum tensor.
Weinberg's definition
Weinberg defines the Belinfante tensor as
where
and