Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From quantum time to manifestly covariant QFT: on the need for a quantum-action-based quantization

N. L. Diaz

Abstract

In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action--based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.

From quantum time to manifestly covariant QFT: on the need for a quantum-action-based quantization

Abstract

In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action--based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.
Paper Structure (18 sections, 2 theorems, 93 equations, 2 figures, 1 table)

This paper contains 18 sections, 2 theorems, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical and quantum dynamical time
  3. Single Particle: From time in phase-space to quantum time via Dirac quantization
  4. Second quantization: From the universe operator to the quantum action operator
  5. Limitations of a constrained equation in the extended Fock space
  6. Spacetime classical and quantum mechanics
  7. Classical mechanics with spacetime Poisson Brackets
  8. Quantization: the problem with Dirac's approach
  9. Quantization: The quantum action approach
  10. Interacting field theories within the quantum action approach
  11. Quantum actions and the concept of quantum state in spacetime
  12. Conclusions
  13. Discrete time formalism: constraints with initial/final conditions
  14. Classical case
  15. Quantum case
  16. ...and 3 more sections

Key Result

Theorem 1

Dirac quantization scheme applied to the spacetime classical formalism leads to standard QFT.

Figures (2)

  • Figure 1: Scheme on the relation between the QT scheme of a particle and QFT. On top we depict a single particle formulated according to the PaW mechanism with time as a degree of freedom. This defines an extended notion of quantum particle. On the other hand, according to QFTs, a particle is a quantum excitation of an underlying field. However, the particles corresponding to standard QFT do not include the time degree of freedom. Instead, according to a formulation of QFT where fields are independent in spacetime a field has as excitations extended PaW-like particles.
  • Figure 2: Different causality induced behavior of partial traces of a spacetime state $\mathcal{R}$ of a relativistic theory. For the observer of the Minkowski diagram canonical quantization is imposed at fixed time and leads to a quantum state $\rho_1$. Another observer quantizing on a different "tilted" surface also defines its corresponding notion of state $\rho_2$. While these two states are conventionally associated with different (isomorphic) Hilbert spaces, from the spacetime perspective both $\rho_1$ and $\rho_2$ can be recovered from partial traces of a single object $\mathcal{R}$. One can also consider an operator $\rho_3$ defined on a timelike surface also determined by $\mathcal{R}$. This operation does not yield a standard quantum state as causality requires $\rho_3^\dag \neq \rho_3$. Even more generally, if one considers a region which is not entirely spacelike (for some pair of points the interval is timelike) the corresponding object $\rho_4$ is also not a standard quantum state.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof