Spacetime quantum mechanics for bosonic and fermionic systems

TL;DR

The paper develops a spacetime-symmetric quantum-mechanical framework in which space and time are treated on equal footing via spacetime algebras, replacing an external time parameter with timelike correlations generated by a quantum action $\mathcal{S}$. It provides explicit constructions for both bosonic and fermionic systems, including discrete-time tensor-product and algebraic (spacetime-analytic) formalisms, and proves maps that recover conventional unitary evolution and path-integral results from spacetime correlators. For fermions, it introduces a Grassmann-free fermionic quantum action and a continuum Dirac theory, establishing a direct link to standard propagators in the small-$\tau$ limit and clarifying the Page–Wootters interpretation in a field-theoretic context. The work extends to interacting theories (e.g., Yukawa couplings), shows how spacetime states can be interpreted as weak values, and proposes a quantum-action variational principle that generalizes entropy maximization to timelike correlations. Collectively, these results offer a foundation for a fully spacetime-based QM, enable new computational approaches via tensor networks, and illuminate connections to timelike entanglement, holography, and PaW-type constructions.

Abstract

We provide a Hilbert space approach to quantum mechanics where space and time are treated on an equal footing. Our approach replaces the standard dependence on an external classical time parameter with a spacetime-symmetric algebraic structure, thereby unifying the axioms that traditionally distinguish the treatment of spacelike and timelike separations. Standard quantum evolution can be recovered from timelike correlators, defined by means of a quantum action operator, a quantum version of the action of classical mechanics. The corresponding map also provides a novel perspective on the path integral formulation, which, in the case of fermions, yields an alternative to the use of Grassmann variables. In addition, the formalism can be interpreted in terms of generalized quantum states, codifying both the conventional information of a quantum system at a given time and its evolution. We show that these states are solutions to a quantum principle of stationary action grounded in timelike correlations and pseudo-entropies

Figures (6)

• Figure 1: Asymmetry in the axioms of standard QM in the treatment of spacelike and timelike separations. Panel a) depicts how QM assigns a Hilbert space $h$ of possible states of the system at a given time. As time "flows", $h$ remains the same while the quantum states change within $h$ according to a classical parameterization. Panel b) represents the scenario of two spacelike separated regions $A$, $B$ (or equivalently systems). QM assigns a different Hilbert space to each region while the joint Hilbert space is constructed from their tensor product, i.e. $h=h_A\otimes h_B$.
• Figure 2: Tensor network representation of the map between spacetime traces and conventional traces. The operator $e^{i\epsilon\mathcal{P}}$ allows one to translate traces in $\mathcal{H}=\otimes_t h_t$ to traces in $h$, as easily seen in tensor network notation. The notation is introduced in d) while the planes in a), b) and c) have been added to emphasize that a Hilbert space is assigned to each time slice. The panel a) corresponds to $N=2$ and Eq. \ref{['eq:swap']} while b) is the case $N=3$ of Lemma \ref{['lemma1']}. In c) we show the representation of the trace of just two operators with an arbitrarily larger number of time-slices. This corresponds to Lemma \ref{['lemma1']} with identities places in all but two slices.
• Figure 3: Numerical results for the functional $F[\Gamma]$. In all three panels $H=\lambda Z$ and $N=2$, $\epsilon\equiv 1$ for different test quantum actions. a) $F[\Gamma]$ for a qubit system with $\lambda=1$, $K_1=\alpha (\sigma_1\otimes \mathbbm{1}+\mathbbm{1}\otimes \sigma_1)+\gamma (\sigma_1\otimes \sigma_2+\sigma_2\otimes \sigma_1)$. In both directions $(\alpha,\gamma)$ the functional $F[\Gamma]$ increases when we part from the quantum action (the plane corresponds to $F[e^{-\mathcal{S}_E}]$). b) $F[\Gamma]$ for a qubit system with $\lambda=1$, $K_2=\alpha (\sigma_1\otimes \mathbbm{1}+\mathbbm{1}\otimes \sigma_1)+\gamma (\sigma_1\otimes \sigma_1+\sigma_2\otimes \sigma_2+\sigma_3\otimes \sigma_3)$. In this case, the entangling direction $\gamma$ provides a way to decrease $F[\Gamma]$ showing that $\Gamma=e^{-\mathcal{S}_E}$ is a saddle point of $F$. c) Difference $F[e^{-\mathcal{S}_E}]-F[\Gamma]$ for a spinless fermion. We plot it for $3 \times 10^3$ random values of the parameters defining $\Gamma$, taking possible values $\alpha_{13},\alpha_{14},\alpha_{24}\in (-0.2,0.2)$, $\lambda\in (-2,2)$ and $\alpha_{12}=0$ (chosen such that $K=K^\dag$ defined in Eq. \ref{['eq:Kmajorana']}). In this case, all the test quantum actions yield smaller values of $F$ than the solution.
• Figure 4: Schematic proof of the two point contraction relation of Theorem I. We show schematically the only states that contribute to the trace ${\rm Tr}[Pe^{i\epsilon\mathcal{P}}a_{t_1}a_{t_2}^\dag]$ in the cases of $t_1>t_2$ (panel a)) and $t_2>t_1$ (panel b)). The black dots indicate a full mode while the white dots indicate an empty mode with all modes correpsonding to a different time.
• Figure 5: Graphical proofs of relations involving $X$ for two time slices. The orange lines correspond to traces. On panel a) we show that Eq. \ref{['eq:Xexpl']} satisfies Eq. \ref{['eq:decoherenceX']}. One recognized on the r.h.s. a decoherence functional for two times. On panel b) we prove Eq. \ref{['eq:ishampuri']}. Notice that the ensuing operator on the r.h.s. is precisely the spacetime state: $(\rho \otimes \mathbbm{1}) \,\text{SWAP}\, U\otimes U^\dag=(\rho \,(U^\dag)^2 \otimes \mathbbm{1}) \,\text{SWAP}\,U\otimes U=\rho_0 e^{i\tilde{\mathcal{S}}}$.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 3
• Theorem 4
• Corollary 1
• Theorem 5
• Theorem 6