De Rham Operator Research Articles

A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold

We define a regularised version of the de Rham operator over the free loop space. We perform a semi-classical approximation of it, such that the index of the limit operator is equal to the “orbit Euler characteristic” of physicists.

Quantization: Towards a comparison between methods

In this paper it is shown that the procedure of geometric quantiztion applied to Kähler manifolds gives the following result: the Hilbert space ℋ consists, roughly speaking, of holomorphic functions on the phase space M and to each classical observable f (i.e., a real function on M) is associated an operator f on ℋ as follows: first multiply by f+ 1/4 ℏΔdRf (ΔdR being the Laplace–de Rham operator on the Kähler manifold M) and then take the holomorphic part [see G. M. Tuynman, J. Math. Phys. 27, 573 (1987)]. This result is correct on compact Kähler manifolds and correct modulo a boundary term ∫Mdα on noncompact Kähler manifolds. In this way these results can be compared with the quantization procedure of Berezin [Math. USSR Izv. 8, 1109 (1974); 9, 341 (1975); Commun. Math. Phys. 40, 153 (1975)], which is strongly related to quantization by *-products [e.g., see C. Moreno and P. Ortega-Navarro; Amn. Inst. H. Poincaré Sec. A: 38, 215 (1983); Lett. Math. Phys. 7, 181 (1983); C. Moreno, Lett. Math. Phys. 11, 361 (1986); 12, 217 (1986)]. It is shown that on irreducible Hermitian spaces [see S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic, Orlando, FL, 1978] the contravariant symbols (in the sense of Berezin) of the operators f as above are given by the functions f+ 1/4 ℏΔdRf. The difference with the quantization result of Berezin is discussed and a change in the geometric quantization scheme is proposed.

Quantized antisymmetric tensor fields and self-consistent dimensional reduction in higher-dimensional spacetimes

The problem of quantizing an antisymmetric tensor field on a higher-dimensional gravitational background of the form M 4 × S N (where M 4 is an arbitrary 4-dimensional spacetime) is examined. Terms appearing in the effective gravitational action are found for such a theory. This effective gravitational action is then used to find the necessary conditions for the existence of solutions to the higher-dimensional dield equations of the form R 4 × S N . The cases where the background antisymmetric tensor field vanishes or else is of the Freund-Rubin form are both considered. It is mentioned how this analysis is modified when gauge fields are included in a Kaluza-Klein metric on the higher-dimensional space. Two appendices contain details of our conventions concerning differential forms, and results for the eigenvalues (and their multiplicities) of the Hodge de Rham operator acting on exact and coexact forms on a sphere. In a third appendix it is shown how the vacuum energy for photons in the Einstein static universe may be obtained using the methods of this paper.

Electromagnetic phenomena induced by weak gravitational fields. Foundations for a possible gravitational wave detector

In the context of the weak-field approximation to the De Rham wave-vector equation, and applying the standard Lorentz gauge to both electromagnetism and gravitation, a linearized expression for the De Rham operator is derived. The derived equation is applied to solve (1) the interaction between plane, monochromatic, and polarized electromagnetic and gravitational waves and (2) the interaction between a transverse electric ($_{0}\mathrm{T}$ ${\mathrm{E}}_{n}$) mode, propagating in a rectangular wave guide, and a gravitational wave. The second result permits us to consider the possibility of a gravitational-wave antenna based on purely electromagnetic means, because new transverse electric and magnetic modes are generated, whose characteristics depend on the gravitational wave. Evidently this detector is completely different from that based on electromagnetic gravitational resonance designed by Braguinski and Manoukine. Also, in the optic limit of the De Rham equation the known expression for the equivalent index of refraction is obtained and other laws for the amplitude and polarization are derived.