Quaternionic quantization principle in general relativity and supergravity

Abstract

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space-time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to N = 1 supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

Canonical quantum gravity on noncommutative space–time

“…In this approximation the generalized area operator in loop quantum gravity on noncommutative space-time has finally been determined. The generalized quantization principle as basic constituent of the presented theory, which is implied by the direct combination of the assumption of noncommutative geometry and field quantization, is a different version of a generalized quantization principle as it has been considered with respect to the variables of quantum mechanics as generalized uncertainty principle [56], [57], [58] and in quantum gravity [36], [37], [38], [39], [40]. Usually, noncommutative geometry is mainly interpreted as a possible extension of the structure of space-time, which could possibly cure the appearance of divergencies in quantum field theory, since the presupposed noncommutative geometry implies the existence of a minimal length.…”

“…where the components of the Hamiltonian density are defined explicitly in (33) and (37). This means that for the case of a scalar field the complete constraints containing the interaction between the scalar fields as matter fields and the gravitational field have been obtained.…”

“…To formulate the generalized quantum constraints, the three metric representation of the generalized operators describing the gravitational field is used. Such a generalization of the quantization principle in canonical quantum gravity as it is implied by the noncommutativity of space-time presupposed in this paper, has in other manifestations been considered in [36], [37], [38], [39], [40]. A canonical noncommutativity algebra between the components of the gravitational field and the corresponding generalization of general relativity has been considered in [41].…”

Quaternionic Dirac free particle

“…All these approaches preserve the complex Hilbert space. An authentic quaternionic Hilbert space approach concerning the anti-hermititian ÀQM comprises the formal results given in Adler's book [5], and a few applications that can be genuinely considered relativistic ÀQM [40,41,42,43,44,45,46,47,48,49,50]. Thus, the complete picture of quaternionic applications to the Dirac equation comprises a meaningful number of quaternionic applications in the complex Dirac equation, and a few examples that comprise the relevant references for the real Hilbert space proposal used in this article.…”

Quaternionic quantum harmonic oscillator

“…Furthermore, anti-hermitian solutions of ÀQM are few, involved, and difficult to understand physically [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. We additionally point out that several applications of quaternions in quantum mechanics are not ÀQM because the anti-hermitian framework is not considered [24,[26][27][28][29][30][31] and the quaternions are simply an alternative way to describe specific results of QM.…”