Ehrenfest Theorem in Precanonical Quantization of Fields and Gravity

TL;DR

This paper extends the Ehrenfest theorem to precanonical quantization of gravity, showing classical Einstein equations emerge as expectation values of quantum operators, linking quantum and classical gravity.

Contribution

It generalizes the Ehrenfest theorem to precanonical quantization of gravity, deriving classical Einstein equations from quantum operator expectations.

Findings

Classical Einstein equations derived as expectation values of quantum operators.

Connection term in covariant Schrödinger equation identified with spin connection.

Demonstrates emergence of classical field equations from precanonical quantum framework.

Abstract

We discuss a generalization of the Ehrenfest theorem to the recently proposed precanonical quantization of vielbein gravity which proceeds from a space-time symmetric generalization of the Hamiltonian formalism to field theory. Classical Einstein-Palatini equations are derived as equations of expectation values of precanonical quantum operators. The preceding consideration of an interacting scalar field theory on curved space-time shows how the classical field equations emerge from the results of precanonical quantization as the equations of expectation values of the corresponding quantum operators. It also allows us to identify the connection term in the covariant generalization of the precanonical Schr\"odinger equation with the spin connection.