SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

Abstract

A relation between the precanonical quantization of pure Yang-Mills fields and the standard functional Schrödinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter κ introduced in precanonical quantization goes to infinity. In this limiting case, the Schrödinger wave functional can be expressed as the trace of the Volterra product integral of Clifford-algebra-valued precanonical wave functions restricted to a field configuration, and the canonical functional derivative Schrödinger equation together with the quantum Gauß constraint can be derived from the Dirac-like precanonical Schrödinger equation.

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

“…The issue we would like to address here is a relation between the description of quantum fields derived from precanonical quantization and the standard QFT based on canonical quantization. More specifically, we would like to generalize the relation found in flat space-time in the case of quantum scalar field theory [34,37,38] and quantum YM theory [29,31] to curved space-time. Our presentation here follows [41][42][43].…”

“…and γ µ → γ µ being a local Clifford algebra automorphism. From (31) it follows that Φ Σ can be taken in the form…”

“…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].…”

Polysymplectic formulation for BF gravity with Immirzi parameter

“…However, we must emphasize that the manifestly covariant polysymplectic formalism allowed us to recover Einstein equations in terms of the tetrads directly from the BF action and without recursively introducing the Holst action. Besides, as the polysymplectic phase space is finite dimensional, contrary to the standard canonical Hamiltonian formalism where the field variables are defined on some space-like hypersurface thus implying an infinite dimensional phase space, our formalism may pave the way to analyze novel quantum aspects of General Relativity thought as a BF theory from the perspective of an explicitly covariant canonical formulation [26,36,41].…”