Belinfante–Rosenfeld stress–energy tensor

In mathematical physics, the BelinfanteRosenfeld 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 canonical stress–energy tensor satisfying , and 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.

The Belinfante–Rosenfeld tensor[1][2] 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, 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.

Belinfante–Rosenfeld and the Hilbert energy–momentum tensor

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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   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–Rosenfeld 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 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.

Now

 

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

 

Thus the Belinfante–Rosenfeld tensor becomes

 

The Belinfante–Rosenfeld tensor for the Dirac field is therefore seen to be the symmetrized canonical energy–momentum tensor.

Weinberg's definition

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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]

 .

References

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  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 (5): 449. Bibcode:1940Phy.....7..449B. CiteSeerX 10.1.1.205.8093. doi:10.1016/S0031-8914(40)90091-X.
  2. ^ L. Rosenfeld (1940). "Sur le tenseur d'impulsion-énergie" (PDF). Mémoires Acad. Roy. De Belgique. 18 (6): 1–30.
  3. ^ Weinberg, Steven (2005). The quantum theory of fields (Repr., pbk. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521670531.
  4. ^ Cahill, Kevin, University of New Mexico (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 9781107005211.{{cite book}}: CS1 maint: multiple names: authors list (link)