Spacetime quantum and classical mechanics with dynamical foliation

TL;DR

The paper develops a spacetime-symmetric extension of classical and quantum mechanics by promoting the time variable in the Legendre transform to a dynamical foliation and by symmetrizing the phase-space structure across spacetime. It then performs a full quantization in which the foliation becomes a quantum degree of freedom and the action becomes a quantum operator, yielding off-shell extended particles and a covariant set of transformation rules. A general map between this spacetime formulation and conventional QM is established through spacetime correlators, with evolution emerging from generalized states and conditioning on foliation, in the spirit of Page–Wootters, and standard propagators recovered in appropriate limits. The formalism remains Lorentz-covariant and offers a natural framework to discuss generalized states, spacetime entanglement, and potential quantum foliation effects, while pointing to future directions in curved foliations, quantum gravity, and quantum reference frames.

Abstract

The conventional phase space of classical physics treats space and time differently, and this difference carries over to field theories and quantum mechanics (QM). In this paper, the phase space is enhanced through two main extensions. First, we promote the time choice of the Legendre transform to a dynamical variable. Second, we extend the Poisson brackets of matter fields to a spacetime symmetric form. The ensuing "spacetime phase space" is employed to obtain an explicitly covariant version of Hamilton equations for relativistic field theories. A canonical-like quantization of the formalism is then presented in which the fields satisfy spacetime commutation relations and the foliation is quantum. In this approach, the classical action is also promoted to an operator and retains explicit covariance through its non-separability in the matter-foliation partition. The problem of establishing a correspondence between the new noncausal framework (where fields at different times are independent) and conventional QM is solved through a generalization of spacelike correlators to spacetime. In this generalization, the Hamiltonian is replaced by the action, and conventional particles by off-shell particles. When the foliation is quantized, the previous map is recovered by conditioning on foliation eigenstates, in analogy with the Page and Wootters mechanism. We also provide an interpretation of the correspondence in which the causal structure of a given theory emerges from the quantum correlations between the system and an environment. This idea holds for general quantum systems and allows one to generalize the density matrix to an operator containing the information of correlators both in space and time.

Figures (4)

• Figure 1: Standard phase space & quantization vs the spacetime approach. a) In Hamiltonian classical mechanics a symplectic structure is defined for a fixed choice of time. The quantization is thus performed in a given $d$ dimensional hypersurface by promoting $\phi(\textbf{x})$ and $\pi(\textbf{x})$ to quantum operators. One possible basis of the ensuing Hilbert space is given by field configurations in the hypersurface, detoned by $|\phi(\textbf{x})\rangle$. b) In the spacetime approach, both Poisson brackets & commutators are spacetime symmetric and the foliation is "dynamical". A basis of the Hilbert space is given by the tensor product between spacetime configurations of the field $|\phi(x)\rangle$ and the foliation eigenstates $|n\rangle\equiv |n^0,n^1,\dots n^d\rangle$. General operators, such as the spacetime quantum actions and ensuing ladder operators (associated with extended off-shell particles) are nonseparable in the matter-foliation partition. Their explicit covariant features become feasible in the complete Hilbert space only.
• Figure 2: Scheme of the correspondence between the QM formulations (fixed $n^\mu)$. In the spacetime formulation, we can codify all the information about a given system and its evolution in generalized states, encompassing an environment correlated with the system. By "measuring" on the system only (see the remarks on weak values) conventional propagators and Feynman rules are recovered. The example of the Feynman propagator is depicted, which corresponds to the hermitian observable $\phi(x)\phi(y)$. Contrary to canonical QM (CQM) where $[\phi_H(x),\phi_H(y)]\neq 0$ inside the light-cone, in the spacetime formulation every field $\phi(x)$ is independent from the others and $[\phi(x),\phi(y)]=0$ for any spacetime points (a much stronger statement than microcausality). The information about evolution and causality is contained in the generalized system-environment state $R_\tau$ and one can think that it emerges from the ("generalized/pseudo") correlations between the two. Since the environment is ignored, one can also work with the partial "state" of the system $e^{i\mathcal{S}_\tau}$ directly, as described in section \ref{['sec:stcorr']}. It was shown in diazp.21 that particular evaluations of the ensuing traces lead to the PI formulation (see also Appendix \ref{['sec:Apmap']}).
• Figure 3: Quantum circuit for computing a quantity of the form$\langle \varphi|\mathcal{O}|\psi\rangle$. The scheme is a Hadamard test where measurements are performed in the ancillary qubit (on top) to estimate the real and imaginary parts of $\langle \psi| \mathcal{V}\mathcal{O}|\psi\rangle$, and where we choose $\mathcal{V}$ so that $|\varphi\rangle=\mathcal{V}^\dag|\psi\rangle$. By using states $|\psi\rangle$, $|\varphi\rangle$ that define a generalized state, one can compute spacetime correlation functions (see also Appendix \ref{['sec:Apmap']}).
• Figure 4: Tensor network representation of the correspondence. The operator $e^{i\epsilon\mathcal{P}_0}$ allows one to translate traces in $\mathcal{H}=\otimes_i h_i$ to traces in $h_i$, as easily seen in tensor network notation. The notation is introduced in c) while the planes in a) and b) have been added to emphasize that a Hilbert space is assigned to each time slice. For QFTs each plane represents the Hilbert space of fields quantized in a given hypersurface. In a) we represent Eq. \ref{['eq:trab']}. In b) we show the representation of the same trace with a larger number of time-slices.