Probability amplitude waves described by quantum mechanics are globally coherent relativistically space-like space-time structures, having no internal causal links. In this sense these non-classical structures are anticipated by the classical ontology of special relativity, which also delineates (in a completely different way) space-like non-causal intervals in space-time. I discuss and distinguish two broad classes of globally coherent relativistically space-like space-time structure, and suggest a parsimonious extension of the core ontology of physics to include both (rather than only one as at present), as a way of resolving some puzzles in the interpretation of quantum mechanics, and motivating its formalism. This ontological extension yields well known formal features usually postulated ad-hoc, such as separability of the wave function, and explains the puzzling absolute nature of time in quantum mechanics. The curious fact that the ontologies of both special relativity and quantum mechanics delineate (in strikingly different ways) strictly non-causal intervals in space-time suggests the existence of an explanatory “last common ancestor” ontology, at a higher level of abstraction than the ontologies of either. I briefly sketch such an ontology, consisting of world self-mappings, in terms of which there are two distinct concepts of time, intrinsic and extrinsic, which together play an abstract identity role in the world, and which suggests part of the meaning of both special relativity and quantum mechanics, and of the non-causal coherence of non-classical space-time structures, concerns extrinsic abstract structure, in particular extrinsic time.Probability amplitude waves described by quantum mechanics are globally coherent relativistically space-like space-time structures, having no internal causal links. In this sense these non-classical structures are anticipated by the classical ontology of special relativity, which also delineates (in a completely different way) space-like non-causal intervals in space-time. I discuss and distinguish two broad classes of globally coherent relativistically space-like space-time structure, and suggest a parsimonious extension of the core ontology of physics to include both (rather than only one as at present), as a way of resolving some puzzles in the interpretation of quantum mechanics, and motivating its formalism. This ontological extension yields well known formal features usually postulated ad-hoc, such as separability of the wave function, and explains the puzzling absolute nature of time in quantum mechanics. The curious fact that the ontologies of both special relativity and quantum mechanics delineate (in strikingly different ways) strictly non-causal intervals in space-time suggests the existence of an explanatory “last common ancestor” ontology, at a higher level of abstraction than the ontologies of either. I briefly sketch such an ontology, consisting of world self-mappings, in terms of which there are two distinct concepts of time, intrinsic and extrinsic, which together play an abstract identity role in the world, and which suggests part of the meaning of both special relativity and quantum mechanics, and of the non-causal coherence of non-classical space-time structures, concerns extrinsic abstract structure, in particular extrinsic time.

Aug 1, 20200 citations

SPACE-TIME APPROACH TO NON-RELATIVISTIC QUANTUM MECHANICS

Richard P. Feynman

WORLD SCIENTIFIC eBooksAug 1, 200534 citations

Alternative formulation of relativistic quantum mechanics

Koshun Suto

Physics EssaysDec 1, 20112 citations

Space-time-symmetric non-relativistic quantum mechanics: Time and position of arrival and an extension of a Wheeler-DeWitt-type equation

Eduardo O. Dias

We generalize a space-time-symmetric (STS) extension of non-relativistic quantum mechanics (QM) to describe a particle moving in three spatial dimensions. In addition to the conventional time-conditional (Schrödinger) wave function $ψ(x, y, z | t)$ , we introduce space-conditional wave functions such as $ϕ(t, y, z | x)$ , where $x$ plays the role of the evolution parameter. The function $ϕ(t, y, z | x)$ represents the probability amplitude for the particle to arrive on the plane $x = \text{constant}$ at time $t$ and transverse position $(y, z)$ . Within this framework, the coordinate $x^μ\in \{t, x, y, z\}$ can be conveniently chosen as the evolution parameter, depending on the experimental context under consideration. This leads to a unified formalism governed by a generalized Schrödinger-type equation, $\hat{P}^μ |ϕ^μ(x^μ)\rangle = -i\hbar \, η^{μν} \frac{d}{dx^ν} |ϕ^μ(x^μ)\rangle$ . It reproduces standard QM when $x^μ= t$ , with $|ϕ^0(x^0)\rangle = |ψ(t)\rangle$ , and recovers the STS extension when $x^μ= x^i \in \{x, y, z\}$ . For a free particle, we show that $ϕ(t, y, z | x) = \langle t, y, z | ϕ(x) \rangle$ naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states $|ϕ^μ(x^μ)\rangle$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^μ\rangle$ , leading to constraint equations of the form $\hat{\mathbb{P}}^μ|Φ^μ\rangle = 0$ . This formulation generalizes the spirit of the Wheeler-DeWitt-type equation: instead of privileging time as the sole evolution parameter, it treats all coordinates on equal footing.We generalize a space-time-symmetric (STS) extension of non-relativistic quantum mechanics (QM) to describe a particle moving in three spatial dimensions. In addition to the conventional time-conditional (Schrödinger) wave function $ψ(x, y, z | t)$, we introduce space-conditional wave functions such as $ϕ(t, y, z | x)$, where $x$ plays the role of the evolution parameter. The function $ϕ(t, y, z | x)$ represents the probability amplitude for the particle to arrive on the plane $x = \text{constant}$ at time $t$ and transverse position $(y, z)$. Within this framework, the coordinate $x^μ\in \{t, x, y, z\}$ can be conveniently chosen as the evolution parameter, depending on the experimental context under consideration. This leads to a unified formalism governed by a generalized Schrödinger-type equation, $\hat{P}^μ |ϕ^μ(x^μ)\rangle = -i\hbar \, η^{μν} \frac{d}{dx^ν} |ϕ^μ(x^μ)\rangle$. It reproduces standard QM when $x^μ= t$, with $|ϕ^0(x^0)\rangle = |ψ(t)\rangle$, and recovers the STS extension when $x^μ= x^i \in \{x, y, z\}$. For a free particle, we show that $ϕ(t, y, z | x) = \langle t, y, z | ϕ(x) \rangle$ naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states $|ϕ^μ(x^μ)\rangle$ can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate $|x^μ\rangle$, leading to constraint equations of the form $\hat{\mathbb{P}}^μ|Φ^μ\rangle = 0$. This formulation generalizes the spirit of the Wheeler-DeWitt-type equation: instead of privileging time as the sole evolution parameter, it treats all coordinates on equal footing.

arXiv:2308.04376Aug 8, 20230 citations

Remarks on Relativistic Quantum Mechanics

K. V. Roberts

Physical ReviewJan 1, 19512 citations

ψ(x, y, z | t)

, we introduce space-conditional wave functions such as

ϕ(t, y, z | x)

, where

x

plays the role of the evolution parameter. The function

ϕ(t, y, z | x)

represents the probability amplitude for the particle to arrive on the plane

x = \text{constant}

at time

t

and transverse position

(y, z)

. Within this framework, the coordinate

x^μ\in \{t, x, y, z\}

can be conveniently chosen as the evolution parameter, depending on the experimental context under consideration. This leads to a unified formalism governed by a generalized Schrödinger-type equation,

\hat{P}^μ |ϕ^μ(x^μ)\rangle = -i\hbar \, η^{μν} \frac{d}{dx^ν} |ϕ^μ(x^μ)\rangle

. It reproduces standard QM when

x^μ= t

, with

|ϕ^0(x^0)\rangle = |ψ(t)\rangle

, and recovers the STS extension when

x^μ= x^i \in \{x, y, z\}

. For a free particle, we show that

ϕ(t, y, z | x) = \langle t, y, z | ϕ(x) \rangle

naturally reproduces the same dependence on the momentum wave function as the axiomatic Kijowski distribution. Possible experimental tests of these predictions are discussed. Finally, we demonstrate that the different states

|ϕ^μ(x^μ)\rangle

can emerge by conditioning (i.e., projecting) a timeless and spaceless physical state onto the eigenstate

|x^μ\rangle

, leading to constraint equations of the form

\hat{\mathbb{P}}^μ|Φ^μ\rangle = 0

. This formulation generalizes the spirit of the Wheeler-DeWitt-type equation: instead of privileging time as the sole evolution parameter, it treats all coordinates on equal footing.