On the “spin connection foam” picture of quantum gravity from precanonical quantization

Abstract

Precanonical quantization is based on a generalization of the Hamiltonian formalism to field theory, the so-called De Donder-Weyl (DW) theory, which does not require a spacetime splitting and treats the space-time variables on an equal footing. Quantum dynamics is described by a precanonical wave function on the finite dimensional space of field coordinates and space-time coordinates, which satisfies a partial derivative precanonical Schrödinger equation. The standard QFT in the functional Schrödinger representation can be derived from the precanonical quantization in a limiting case. An analysis of the constraints within the DW Hamiltonian formulation of the Einstein-Palatini vielbein formulation of GR and quantization of the generalized Dirac brackets defined on differential forms lead to the covariant precanonical Schrödinger equation for quantum gravity. The resulting dynamics of quantum gravity is described by the wave function or transition amplitudes on the total space of the bundle of spin connections over space-time. Thus, precanonical quantization leads to the "spin connection foam" picture of quantum geometry represented by a generally non-Gaussian random field of spin connection coefficients, whose probability distribution is given by the precanonical wave function. The normalizability of precanonical wave functions is argued to lead to the quantumgravitational avoidance of curvature singularities. Possible connections with LQG are briefly discussed.Keywords: Quantum gravity; precanonical quantization; De Donder-Weyl theory; vielbein gravity; Gerstenhaber algebra; Dirac brackets; Clifford algebra; Lévy processes. Precanonical quantization of fieldsContemporary quantum field theory originates from the canonical quantization which is based on the canonical Hamiltonian formalism. The latter dictates a picture of fields as infinite dimensional Hamiltonian systems. It also restricts the consideration to the globally hyperbolic space-times, as it implies a different role of the time dimension, along which the evolution proceeds, and the space dimensions, which label the continuum of degrees of freedom of fields. Many problems we encounter in quantum gravity theories can be traced back to this very origin of QFT.However, the canonical Hamiltonian formalism is not the only possibility to extend the Hamiltonian formalism from mechanics to field theory. The alternative "Hamiltonizations" (i.e. writing the field equations in the first order form using some generalization of the Legendre transform) known in the calculus of variations 1 are, in fact, inherently more geometrical than the canonical formalism, and they treat the space and time variables (i.e. the independent variables of the multiple integral variational problem) on the equal footing, i.e. essentially as multidimensional generalizations of the one-dimensional time parameter in mechanics.

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

“…The latter point of view is supported by the observation that κ actually disappears from the observable quantities of the theory in the case of free fields. On the other hand, the fundamental nature of the scale κ is supported by our recent estimation of the mass gap in the precanonical formulation of quantum pure SU(2) gauge theory [37] and a naive estimation of the cosmological constant from the precanonically quantized pure Einstein gravity [36]. Surprisingly, both estimations point to a subnuclear scale of κ, thus a further research is required to understand this fact.…”

“…One of the features of this formulation of quantized fields is that it allows us to reproduce the classical field equations as the equations of expectation values of operators defined by and evolving according to the equations of precanonical quantization [10,11]. Moreover, by treating the space-time variables on an equal footing and leading to a formulation on a finite-dimensional space of field and space-time variables the precanonical quantization approach provides a natural and promising framework for quantization of gravity [35,36] and even for the consideration of the mass gap problem in quantum gauge theory [37]. On the classical level, the DW Hamiltonian formulation and the underlying polysymplectic structure have been used recently in order to construct highly effective numerical integrators for PDEs and general relativity [38,39].…”

Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

“…We hope that these results can stimulate new approaches to the treatment of effects of quantum fields in curved space-time [48], such as the Hawking radiation, and they also can lead to better understanding of the nature of states of quantum fields in arbitrary space-times when a separation to positive-and negative-frequency modes is not possible. They also may help to clarify the connections between the existing approaches to quantum gravity originating from the canonical quantization of general relativity [39,40] and precanonical quantization of gravity (see [20][21][22][23][24][26][27][28]).…”

“…Applications of precanonical quantization so far include quantum gravity in metric [20][21][22][23] and vielbein variables [24][25][26][27][28], and quantum Yang-Mills theory [29][30][31]. However, the connection with the standard techniques and concepts of QFT still remains not sufficiently explored.…”

“…On the quantum level, the connection between precanonical quantization and the functional Schrödinger representation of QFT [36] was found for scalar fields [34,37,38] and YM fields [29][30][31] in flat space-time. As a step towards understanding the connections between precanonical quantization of gravity [20][21][22][23][24][26][27][28] and the existing approaches based on canonical quantization [39,40] we have also explored a possible extension of those results to curved space-time [41][42][43]. The present discussion is a concise presentation of those papers.…”