Re: dirac equation in curved spacetime

On Sep 16, 7:11 pm, vivishek <vivishek.sud...@xxxxxxxxx> wrote:
Is there a single, well-accepted form of the dirac equation for a free
particle in curved spacetime?

No

If not, what are the various
possibilities constructed so far?

So many like geometric structures (U, V, non-commutative) for the
spacetime and interactions with fermion and EM fields (e.g. Q-A
interaction) you choose.

For instance, it is usually thought that curved spacetime cannot deal
with spin. Therefore, a first step is generalization of Riemannian
structure (V space) to a Riemann-Cartan one (U).

Now spacetime is not just curved like in GR. The connection Gamma_ab^c
is not just the Christoffel but include the contorsion tensor K_ab^c.
(See equation 2.2 in [gr-qc/9309027])

A popular option is substitution R --> R - Q^2 in the Lagrangian of
GR. Here Q is the torsion (associated to the contorsion K).

Another posibility is to introduce different coupling constants for
curvature and torsion. Then add next term to GR Lagrangian: epsilon
{root -g} Q^2. Where epsilon is the coupling constant.

From variation of generalized Lagrangian (GR + torsion) you can get
the Dirac equation in a Riemann-Cartan spacetime

gamma^b D_b phi = m phi

(see 3.29 in [gr-qc/9309027] for alternative). However, the covariant
derivative D may be computed using the connection Gamma_ab^c
containing tensor corrections to GR connection.

Above equation and the related Dirac Fock Ivanenko equation are under
study in unification and cosmology.

[Adittional references]

New method of integration for the Dirac equation on a curved space-
time. J. Math. Phys. 1992, 33, 2279. Bagrov, V. G; Obukhov, V. V.

http://arxiv.org/pdf/math-ph/0502001

On solutions of the Einstein-Cartan-Dirac theory. Class. Quantum Grav.
1985, 2, 919. Seitz, M.

http://arxiv.org/pdf/gr-qc/9309027

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