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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2308.04376 |
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| _version_ | 1866910990475460608 |
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| author | Dias, Eduardo O. |
| author_facet | Dias, Eduardo O. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04376 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Space-time-symmetric non-relativistic quantum mechanics: Time and position of arrival and an extension of a Wheeler-DeWitt-type equation Dias, Eduardo O. Quantum Physics General Relativity and Quantum Cosmology 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. |
| title | Space-time-symmetric non-relativistic quantum mechanics: Time and position of arrival and an extension of a Wheeler-DeWitt-type equation |
| topic | Quantum Physics General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2308.04376 |