Space-time-symmetric quantum mechanics in 3+1 dimensions
- URL: http://arxiv.org/abs/2308.04376v1
- Date: Tue, 8 Aug 2023 16:27:43 GMT
- Title: Space-time-symmetric quantum mechanics in 3+1 dimensions
- Authors: Eduardo O. Dias
- Abstract summary: In conventional quantum mechanics, time is treated as a parameter, $t$, and the evolution of the quantum state with respect to time is described by $hat H|psi(t)rangle=ihbar fracddt|psi(t)rangle$.
In a recently proposed space-time-symmetric (STS) extension of QM, position becomes the parameter and a new quantum state, $|phi(x)rangle$, is introduced.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In conventional quantum mechanics (QM), time is treated as a parameter, $t$,
and the evolution of the quantum state with respect to time is described by
${\hat {H}}|\psi(t)\rangle=i\hbar \frac{d}{dt}|\psi(t)\rangle$. In a recently
proposed space-time-symmetric (STS) extension of QM, position becomes the
parameter and a new quantum state, $|\phi(x)\rangle$, is introduced. This state
describes the particle's arrival time at position $x$, and the way the arrival
time changes with respect to $x$ is governed by ${\hat
{P}}|\phi(x)\rangle=-i\hbar \frac{d}{dx} |\phi(x)\rangle$. In this work, we
generalize the STS extension to a particle moving in three-dimensional space.
By combining the conventional QM with the three-dimensional STS extension, we
have a ``full'' STS QM given by the dynamic equation ${\hat { P}}^{\mu}|{\phi
}^\mu(x^{\mu})\rangle=- i \hbar~\eta^{\mu\nu}\frac{d}{dx^{\nu}}|{\phi}^\mu
(x^{\mu})\rangle$, where $x^{\mu}$ is the coordinate chosen as the parameter of
the state. Depending on the choice of $x^\mu$, we can recover either the
Schr\"odinger equation (with $x^\mu=x^0=t$) or the three-dimensional STS
extension (with $x^\mu=x^i=$ either $x$, $y$, or $z$). By selecting $x^\mu=x$,
we solve the dynamic equation of the STS QM for a free particle and calculate
the wave function $\langle t,y,z|\phi^1(x)\rangle$. This wave function
represents the probability amplitude of the particle arriving at position
($y$,$z$) at instant $t$, given that the detector occupies the entire
$yz$-plane located at position $x$. Remarkably, we find that the integral of
$|\langle t,y,z|\phi (x)\rangle|^2$ in $y$ and $z$ takes the form of the
three-dimensional version of the axiomatic Kijowski distribution.
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