Covariance of Dirac's modified canonical commutation relations for fermions interacting with gravity and bosons
For fermions in curved space-time, where ihpsi/sup dagger/ is no longer the canonical conjugate of psi, canonical quantization according to Dirac requires use of ''modified'' graded commutation relations, in which the canonical conjugates to the tetrad field components are no longer commutative with each other or with the fermion field. As shown earlier by the DeWitts, these modified commutation relations may be understood as a canonical quantization of new field variables that can conventionally be interpreted in terms of creation and annihilation operators. Because of the horrible transformation properties of these new variables, covariance of this quantization of fermion fields and of the gravitational field with which the fermions interact is best proved in terms of the old variables. Like canonical quantization, also this modified quantization requires alteration of the theory destroying general invariance of the Lagrangian. As discussed in a preceding paper, covariance of this modified quantization under a group of finite transformations will follow, under the conditions that (1) it can be proved for a generator T/sub 1/, linear in infinitesimal descriptors of the transformations, that by these modified commutation relations it will generate the substantial variations of the canonical field variables, while (2) the second-order generator T/sub 12/, constructed from T/sub 1/ as previously discussed, will be conserved and invariant. It is here proved that these two conditions are satisfied, provided that the transformations leave the Lagrangian function L invariant. This restriction limits the proof of covariance of quantization to affine transformations only. It is discussed why this yet leaves the possibility of general covariance of the ''physical part'' of this quantization procedure.
- Research Organization:
- Department of Physics, Purdue University, Lafayette, Indiana 47907
- OSTI ID:
- 5221356
- Journal Information:
- J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 26:11; ISSN JMAPA
- Country of Publication:
- United States
- Language:
- English
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