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Quantum Mechanical Observers and Time Reparametrization Symmetry

Eiji Konishi

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

We propose that the degree of freedom of measurement by quantum mechanical observers originates in the Goldstone mode of the spontaneously broken time reparametrization symmetry. Based on the classification of quantum states by their nonunitary temporal behavior as seen in the measurement processes, we describe the concepts of the quantum mechanical observers via the time reparametrization symmetry.

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We propose that the degree of freedom of measurement by quantum mechanical observers originates in the Goldstone mode of the spontaneously broken time reparametrization symmetry. Based on the classification of quantum states by their nonunitary temporal behavior as seen in the measurement processes, we describe the concepts of the quantum mechanical observers via the time reparametrization symmetry.
The formal time symmetry of the quantum measurement process is extensively discussed. Then, the origin of the alleged association between a fixed temporal direction and quantum measurements is investigated. It is shown that some features of such an association might arise from epistemological rather than purely physical assumptions. In particular, it is brought out that a sequence of statements bearing on quantum measurements may display intrinsic asymmetric properties, irrespective of the location of corresponding measurements in time t of the Schrödinger equation. The situation of an observer performing two measurements in two opposite directions of t is eventually investigated. Essential differences are found between two descriptions of this situation: the internal one (taking only into account what is recorded in the observer's memory) and the external one (whereby the observer is considered as a quantum system ruled by the Schrödinger equation). Finally, a method allowing several observers to establish a correspondence between their memory sizes is analyzed. The most important facts that usually lead to the associating of a preferential temporal direction with quantum measurements may be inferred from this correspondence.
The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ``clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ``test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers. \textcopyright{} 1996 The American Physical Society.
We study the long-time aging dynamics of spin-glass models with two-spin interactions by performing a renormalization group (RG) transformation on the time variable in the nonequilibrium dynamical generating functional. We obtain the RG equations and find that the flow converges to an exact fixed point. We show that this fixed point is invariant under reparametrizations of the time variable. This continuous symmetry is broken, as evidenced by the fact that the observed correlations and responses are not invariant under it. We argue that this gives rise to the presence of Goldstone modes, and that those Goldstone modes shape the behavior of fluctuations in the nonequilibrium dynamics.