Path Integrals from Spacetime Quantum Actions

TL;DR

The paper develops a spacetime-symmetric formulation of quantum mechanics by introducing the Quantum Action (QA) operator and embedding histories in an extended Hilbert space, recasting Feynman Path Integrals as quantum traces. It shows how discrete and continuum time constructions lead to PI representations that emerge from different extended bases, such as trajectory, coherent, and nonlocal-in-time bases, including Matsubara-like expansions. A key result is the tau-invariance of generating functionals, which links the QA trace to classical actions and, in the large-τ limit, to spacetime vacuum mean values in relativistic QFT. The framework preserves explicit spacetime symmetries and opens avenues for quantum-computation of PI evaluations, as well as new regularization and interpretation of correlators in a covariant, spacetime-wide setting.

Abstract

The possibility of extending the canonical formulation of quantum mechanics (QM) to a space-time symmetric form has recently attracted wide interest. In this context, a recent proposal has shown that a spacetime symmetric many-body extension of the Page and Wootters mechanism naturally leads to the so-called Quantum Action (QA) operator, a quantum version of the action of classical mechanics. In this work, we focus on connecting the QA with the well-established Feynman's Path Integral (PI). In particular, we present a novel formalism which allows one to identify the "sum over histories" with a quantum trace, where the role of the classical action is replaced by the corresponding QA. The trace is defined in the extended Hilbert space resulting from assigning a conventional Hilbert space to each time slice and then taking their tensor product. The formalism opens the way to the application of quantum computation protocols to the evaluation of PIs and general correlation functions, and reveals that different representations of the PI arise from distinct choices of basis in the evaluation of the same trace expression. The Hilbert space embedding of the PIs also discloses a new approach to their continuum time limit. Finally, we discuss how the ensuing canonical-like version of QM inherits many properties from the PI formulation, thus allowing an explicitly covariant treatment of spacetime symmetries.

Figures (3)

• Figure 1: Pictorial representation of the operators involved in the extended representation of $\langle q',\text{\footnotesize$T$}|\hat{T}[q_\text{\tiny$H$}(t_1)q_\text{\tiny$H$}(t_2)]|q\rangle$ for $t_2>t_1$. A proper operator has been "inserted" in each Hilbert $\mathfrak{H}_t$. A contribution from a discrete trajectory $\tilde{q}(t)$ is also depicted in order to emphasize the resemblance between the usual interpretation and the Hilbert space construction.
• Figure 2: Diagrammatic representation of the map connecting traces in the extended version of QM to the conventional ones. The map which connects traces of operators within $\mathcal{H}=\otimes_t \mathfrak{h}_t$ with traces of operators in $\mathfrak{h}$ is here represented in tensor network notation (the conventions are depicted in $d)$). In a) we show essentially a SWAP test. In b) a generalization from two to three "slices". Instead in $c)$ we show the example of a two-point correlation function, as treated by the formalism. Interestingly, the many sums on the l.h.s. of the diagram, represented by the lines on each vertical row (corresponding each to a different time slice), are precisely the sum over histories. Notice also that the diagrammatic expansion shows that $e^{i\mathcal{P}_t\epsilon}$ can be represented as the composition of many SWAPs.
• Figure 3: Protocol for evaluation of PIs via Hadamard test. $a)$ A generic Hadamard test. The trace ${\rm Tr}\,[{\cal U}\rho]$, where $\rho$ is a an arbitrary (pure or mixed) state of $n$ qubits and ${\cal U}$ an arbitrary unitary operator on $n$ qubits (hence involving $2^n\times 2^n$ matrix representations) can be evaluated by measuring the averages $\langle\sigma_x\rangle$ and $\langle \sigma_y\rangle$ of the auxiliary top qubit, initially in an eigenstate of $\sigma_z$. The quantum circuit involves just a Hadamard ($H$) and a controlled ${\cal U}$ gates. $b)$ The application of the protocol to the r.h.s. of Eq. (\ref{['eq:meanval']}). When applied to PIs $({\cal U}\rightarrow e^{i{\cal S}}$, Eq. (\ref{['eq:S']})) the "sum over histories" is implicit on the full mixed states at the entry. One may also employ a subset of states, covering a subset of trajectories. Thermal correlation functions and/or "imaginary time evolution" (non-implementable through unitary gates) can be computed by replacing $\rho$ and the maximally mixed states $\mathbbm{1}/d$ on the left by suitable thermal states (see sec. \ref{['sec:corrfunc']}).

Theorems & Definitions (2)

• proof
• proof