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Abstract
We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a “quantum time”, possibly opposed to a “classical” one; there is only one time, and it is a manifestation of entanglement.
Highlights
We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics
The procedure used to identify what time is in quantum mechanics (QM) must have a well-defined classical limit, fully consistent with classical physics and the way time enters the classical equations of motion (e.o.m.)
We tackle the quantum-to-classical crossover via the large-N approach based on Generalized Coherent States (GCS) from refs. 17–21, where it is demonstrated that the theory describing a quantum system for which GCS can be constructed flows into a welldefined classical theory if few specific conditions upon its GCS hold in the N → ∞ limit (N quantifies the number of microscopic quantum components, sometimes referred to as the number of degrees of freedom or dynamical variables, in the literature)
Summary
We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. The procedure used to identify what time is in QM must have a well-defined classical limit, fully consistent with classical physics and the way time enters the classical equations of motion (e.o.m.) We construct such a procedure, demonstrating that it consistently produces the Schrödinger equation for quantum systems and the Hamilton e.o.m. for classical ones, with the parameter playing the role of time being the same in both cases. We take the classical limit for the clock only (see Fig. 2) and derive an equation for the physical states of the evolving system, which is the Schrödinger equation, once the above-mentioned parameter φ is given the role of time. We introduce GCS for the evolving system, take its classical limit (see Fig. 3), and get to our most relevant result: the Hamilton e.o.m. of classical physics are derived, with the same parameter φ as time
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