Belinfante–Rosenfeld stress–energy tensor

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In mathematical physics, the BelinfanteRosenfeld 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

of a local current

Here is the canonical Noether energy–momentum tensor, and is the contribution of the intrinsic (spin) angular momentum. Local conservation of angular momentum

requires that

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

that is constructed from the canonical energy momentum tensor and the spin current so as to be symmetric yet still conserved.

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 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.

Belinfante-Rosenfeld and the Hilbert energy-momentum tensor[edit]

The Hilbert energy-momentum tensor 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 .)

We may also define an energy-momentum tensor by varying a Minkowski-orthonormal vierbein to get

Here 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 to be symmetric. However, the action functional should be invariant under an infinitesimal local Lorentz transformation , , and so

should be zero. As is an arbitrary position-dependent skew symmetric matrix, we see that local Lorentz and rotation invariance both requires and implies that .

Once we know that is symmetric, it is easy to show that , and so the vierbein-variation energy-momentum tensor is equivalent to the metric-variation Hilbert tensor.

We can now understand the origin of the Belinfante-Rosefeld modification of the Noether canonical energy momentum tensor. Take the action to be where is the spin connection that is determined by via the condition of being metric compatible and torsion free. The spin current is then definied by the variation

and 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.

Now if are distinct and zero otherwise. As a consequence is totally antisymmetric. Now, using this result, and again the equations of motion, we find that

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[edit]

Weinberg defines 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]

.

References[edit]

  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.