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First-principle evolution Hamiltonian operator: derivation from ADM quantum constraints and quantum reference-frame conditions

Chun-Yen Lin

Abstract

For any Dirac theory of quantum gravity governed by a set of well-defined quantum constraints, we discover a universal formula for the exact form of the evolution Hamiltonian operator in a variable quantum reference frame of our construction, expressed in terms of the quantum-constraint operators and frame-condition operators as the only inputs. Due to the first-principle nature of the formula, the evolution Hamiltonian operator contains the full interactions encoded in the quantum constraints, and it generates the Schrödinger evolution described by the genuine quantum relational observables associated to the frame and acting in the physical Hilbert space solving the quantum constraints.

First-principle evolution Hamiltonian operator: derivation from ADM quantum constraints and quantum reference-frame conditions

Abstract

For any Dirac theory of quantum gravity governed by a set of well-defined quantum constraints, we discover a universal formula for the exact form of the evolution Hamiltonian operator in a variable quantum reference frame of our construction, expressed in terms of the quantum-constraint operators and frame-condition operators as the only inputs. Due to the first-principle nature of the formula, the evolution Hamiltonian operator contains the full interactions encoded in the quantum constraints, and it generates the Schrödinger evolution described by the genuine quantum relational observables associated to the frame and acting in the physical Hilbert space solving the quantum constraints.
Paper Structure (12 sections, 85 equations)

This paper contains 12 sections, 85 equations.

Table of Contents

  1. Introduction
  2. Quantum reference frame for Dirac theory
  3. Algorithm under general setting
  4. Wigner-Weyl representation
  5. First-integral quantum dynamic in the Weyl-symbol phase space
  6. Weyl symbol of the evolution Hamiltonian under a quantum reference frame
  7. Preliminary series-expansion scheme for $G_{{\!\widehat{H}_t}}$
  8. Series expansion over contractions $\eta^N$ and Fourier modes $k$
  9. Further expansion of $G_{{\!\widehat{H}_t}}$ for semi classical limits
  10. Discussion
  11. Summary and conclusion
  12. Diamond expansion for the $\star$-polynomials