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Conservation Laws and the Quantization of Gravity

Tianfeng Feng, Chiara Marletto, Vlatko Vedral

TL;DR

This work introduces a general quantum-classical hybrid framework to analyze interactions between quantum matter and gravitational fields, focusing on how global conservation laws constrain local observables. It proves a no-go theorem: under a specified decomposition, quantum-classical dynamics cannot change local observables while preserving a global conserved quantity, implying that a purely classical gravity cannot induce momentum or energy changes in a quantum system. Applying this to gravity, the authors argue that classical gravity cannot account for observed momentum/energy shifts, whereas a quantum gravitational field can, supporting gravity's quantumness in a non-relativistic regime. The paper connects conservation laws, expectation-value formalism, and gravity’s nature, offering a framework to interpret free-fall and related experiments as evidence for gravity’s quantum character.

Abstract

Adopting general frameworks for quantum-classical dynamics, we analyze the interaction between quantum matter and a classical gravitational field. We point out that, assuming conservation of momentum or energy, and assuming that the dynamics obeys Hamiltonian formalism or a particular decomposition property set out in the paper, the classical gravitational field cannot change the momentum or energy of the quantum system, whereas the quantum gravitational field can do so. Drawing upon the fundamental relationship between conservation laws and the quantum properties of objects, our analysis offers new perspectives for the study of quantum gravity and provides a novel interpretation of existing experimental observations, such as free fall.

Conservation Laws and the Quantization of Gravity

TL;DR

This work introduces a general quantum-classical hybrid framework to analyze interactions between quantum matter and gravitational fields, focusing on how global conservation laws constrain local observables. It proves a no-go theorem: under a specified decomposition, quantum-classical dynamics cannot change local observables while preserving a global conserved quantity, implying that a purely classical gravity cannot induce momentum or energy changes in a quantum system. Applying this to gravity, the authors argue that classical gravity cannot account for observed momentum/energy shifts, whereas a quantum gravitational field can, supporting gravity's quantumness in a non-relativistic regime. The paper connects conservation laws, expectation-value formalism, and gravity’s nature, offering a framework to interpret free-fall and related experiments as evidence for gravity’s quantum character.

Abstract

Adopting general frameworks for quantum-classical dynamics, we analyze the interaction between quantum matter and a classical gravitational field. We point out that, assuming conservation of momentum or energy, and assuming that the dynamics obeys Hamiltonian formalism or a particular decomposition property set out in the paper, the classical gravitational field cannot change the momentum or energy of the quantum system, whereas the quantum gravitational field can do so. Drawing upon the fundamental relationship between conservation laws and the quantum properties of objects, our analysis offers new perspectives for the study of quantum gravity and provides a novel interpretation of existing experimental observations, such as free fall.
Paper Structure (11 sections, 3 theorems, 40 equations, 3 figures)

This paper contains 11 sections, 3 theorems, 40 equations, 3 figures.

Table of Contents

  1. Definition of a Classical system
  2. General Quantum-classical Dynamics
  3. Observables in Hybrid systems
  4. A No-go theorem
  5. Quantumness of gravitational field
  6. Other conservation laws
  7. Conclusions
  8. Appendix
  9. Hamiltonian formalism of a closed quantum-classical hybrid system
  10. Observable in general quantum-classical hybrid systems
  11. Multiple-time evolution of hybrid systems

Key Result

Theorem 1

Any quantum-classical dynamics that satisfies the decomposition in figure fig1 can not change the local observable $\langle O_{\mathcal{C/Q}}\rangle$ of either a classical or quantum system under the conservation law of a global quantity of the hybrid system $\mathcal{C}\oplus\mathcal{Q}$.

Figures (3)

  • Figure 1: Operational definition of general quantum-classical dynamics. Quantum-classical dynamics of multi-time evolution: Each time interval is infinitesimal, that is, $t_i - t_{i+1} = dt$. The dynamics of a hybrid system consist of sequentially infinitesimal slices of local evolution and global evolution. This decomposition satisfies the causality.
  • Figure 2: Momentum change of two quantum matters in the interaction of gravitational field $E$. (a) At the moment of $T = t$, the initial positions of the two stationary masses are $x_S$ and $x_M$, and the momentum are $\langle P_{\mathcal{Q/C}}\rangle =p_S=p_M=0$. (b) At the moment of $T = t + \Delta t$, the positions of masses $S$ and $M$ have changed, and their respective momentum changes in the same amount and in opposite directions while the whole satisfies the conservation of momentum.
  • Figure 3: Quantum-classical dynamics of multi-time evolution. Each time interval is infinitesimal, that is, $t_i-t_{i+1}=\delta t$. The dynamics of a hybrid system consist of these infinitesimal slices of evolution. For simplicity, here the local operation on a quantum system can be incorporated into the $\Lambda$ since local operations do not change the conserved quantity $O_{\mathcal{Q/C}}$

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Corollary 2