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 an alternative perspective on the path integral formulation that, in the case of fermions, does not require 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.
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 nonseparability 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.
Space-time is one of the most essential, yet most mysterious concepts in physics. In quantum mechanics it is common to understand time as a marker of instances of evolution and define states around all the space but at one time; while in general relativity space-time is taken as a combinator, curved around mass. Here we present a unified approach on both space and time in quantum theory, and build quantum states across spacetime instead of only on spatial slices. We no longer distinguish measurements on the same system at different times with measurements on different systems at one time and construct spacetime states upon these measurement statistics. As a first step towards non-relativistic quantum field theory, we consider how to approach this in the continuous-variable multi-mode regime. We propose six possible definitions for spacetime states in continuous variables, based on four different measurement processes: quadratures, displaced parity operators, position measurements and weak measurements. They are motivated by the pseudo-density matrix formulation among indefinite causal structures and the path integral formalism. We show that these definitions lead to desirable properties, and raise the differences and similarities between spatial and temporal correlations. An experimental proposal for tomography is presented, construing the operational meaning of the spacetime states.
What i Really RepresentsFrom Spacetime Action to Quantum Phase: The Imaginary Unit as aMarker of Action-Phase GeometryPeter Davies, v1.0230th April, 2026AbstractQuantum mechanics differs from classical probability theory by assigning complex amplitudes rather than real probabilities to physical alternatives. The appearance of the imaginary unit i in the Schr¨odinger equation, unitary time evolution, path integrals, gauge transformations, and geometric formulations of quantum mechanics suggests that i is not merely a calculational artifact.This paper argues that i is best understood as the algebraic marker of action-phase geometry: the phase-rotation structure through which action, measured in units of ℏ, becomes a probability-preserving quantum amplitude. In this interpretation, spacetime geometry determines the action accumulated along possible histories, while quantum mechanics maps this action into phase through eiS/ℏ. The constants ℏ and h acquire complementary interpretations: ℏ is action per radian of phase, while h = 2πℏ is action per full phase cycle. Observable probabilities arise only after interference among phase-weighted histories, with the final quadratic product converting the surviving amplitude structure into measurable probability. The paper develops this interpretation through phase gradients, action periodicity, proper-time phase, gauge connections, gravitational phase modulation, light–matter phase matching, and multi-photon interference. The proposed contribution is not a replacement of quantum mechanics, quantum field theory, or general relativity, but a conceptual synthesis: the repeated appearance of i, ℏ, h, action, proper time, and interference may be understood as different aspects of a single action-phase structure linking spacetime and quantum mechanics. The proposal is interpretive rather than a Bell-local hidden-variable completion of quantum mechanics. It does not assign pre-existing local outcome values to measurement events, and it is intended to remaincompatible with the nonclassical correlation structure required by Bell-type results. - Preprint version v1.02. This manuscript presents a conceptual framework and has not yet undergone peer review. Later versions may incorporate corrections, expanded references, and feedback on framing.
Quantum mechanics is traditionally formulated using the reversible coordinate time
t, which enforces a globally reversible unitary group structure and time‑symmetric propagators. This conflicts with physically irreversible processes such as decoherence, measurement, entropy production, causal propagation, and curvature‑driven focusing in general relativity. We resolve this mismatch by replacing
t with the monotonic temporal field
T, previously developed in the context of global causal structure, temporal focusing, and worldline ordering in curved spacetime. Once
T is taken as the fundamental evolution parameter, quantum dynamics cannot be reversible: the inverse operator
U−1 is not physically realizable, and unitary evolution becomes a forward‑directed semigroup. We formulate **forward unitarity**, in which physical evolution is generated only by forward‑directed operators
Uphys(T2,T1) with
T2>T1. Pure states evolve according to a forward Schrödinger equation with an intrinsic, monotonic decoherence functional
D(T), while mixed states evolve under a forward Lindblad structure. Probability is conserved, but coherent recoverability is forbidden by geometric irreversibility, causal structure, and temporal focusing. We construct a Forward‑QM Lagrangian, a forward Heisenberg picture, and a forward path integral restricted to histories with non‑decreasing
T, showing how operator algebras, propagators, correlation functions, and creation/annihilation operators inherit the arrow of time. Forward‑QM provides a unified, forward‑directed formulation of quantum mechanics compatible with general relativity, causal geometry, and the irreversible structure of spacetime. It replaces reversible unitary groups with physically meaningful semigroup evolution, aligns quantum dynamics with measurement, decoherence, and entropy production, and establishes the foundation for Forward Quantum Field Theory (FQFT). The extension to FQFT—covering forward propagators, interacting fields, renormalization in the presence of monotonic decoherence, and causal compatibility in curved spacetime—will be developed in future work.
Abstract The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works also in interacting quantum field theories and sheds new light on the foundations of quantum physics.