Response to Dr. Pashby: Time operators and POVM observables in quantum mechanics

doi:10.1016/j.shpsb.2014.08.010

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Volume 52, Part A, November 2015, Pages 39-43

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Abstract

I argue against a general time observable in quantum mechanics except for quantum gravity theory. Then I argue in support of case specific arrival, dwell and relative time observables with a cautionary note concerning the broad approach to POVM observables because of the wild proliferation available.

Introduction

As intended by professor Grosholz, the papers presented at the Workshop on Cosmology and Time have gone through revisions since the original presentation as the participants continued communicating their differing perspectives to one another. For final publication, of course, this process must be brought to a close. Consequently this response may contain some redundancies vis-à-vis Dr. Pashby׳s paper as well as a comment or two that do not address items remaining in Pashby׳s paper. For possible clarification, earlier versions of our papers can be found in the University of Pittsburgh phil-sci archive at: http://philsci-archive.pitt.edu/view/confandvol/

As in my recent papers I will follow the admonitions (which will not be defended here) of Jean Marc Levy-Leblond, 1988, Levy-Leblond, 1999 and Hans Christian Von Baeyer (1997), to drop the term particle and call the bosons and fermions of the world, quantons.

Section snippets

Between Pashby and Hilgevoord

Back in 1998 professor Hilgevoord (Hilgevoord, 1998), extensively referred to by Dr. Pashby (Pashby, 2013), criticized a long paper I co-authored with Jeremy Butterfield (Fleming & Butterfield, 1999), in which we discussed (among other things) Lorentz covariant 4-vector position operators, assigned to space-like hyperplanes, and with operator valued time components. Hilgevoord objected not only to the operator time components, but to the requirement of Lorentz covariance for the position

Time, observables and measurement

There are two brief arguments, other than Pauli’s (Pauli, 1980), that I would mount against a general, canonical, time operator in QM. First, and most importantly: In QM, whether Galilean or Lorentz covariant, space and time or space–time, are not, themselves, dynamical systems. QM is a theory of temporally persistent dynamical systems, indeed of eternal systems, which live in a fixed classical space–time. Unlike Quantum Gravity research or Quantum Cosmology, which seek a QM of space–time and

Time–energy indeterminacy

While we do not have a general time observable in quantum mechanics, we do have a universal time–energy indeterminacy relation (TEIR), $Δ T Δ E \geq ℏ / 2$ , and it is striking how exactly opposite is our interpretation of that relation from Heisenberg’s early interpretation, as described by Pashby, of the original version. While Heisenberg saw $Δ T$ as an indeterminacy in a time of occurrence and $Δ E$ was an interval between precise energy values, we now have $Δ E$ as the standard deviation indeterminacy in the

Case specific time observables

Now I turn to case specific time observables where I agree with Pashby concerning both the possibility and the desirability of identifying and examining such observables in QM for various times of occurrence or durations.

Concepts of quantum observable times come in at least three forms: (1) times of occurrence (arrival times) of specified events, (2) intervals of time (dwell times) spent in specified regions or conditions or (3) (relative times) of occurrence of one event relative to a

The POVM perspective

But just how shall we work with ${\overset{⌢}{T}}_{0}$ in detail, given that it is not self-adjoint? Brunetti et al. tell us that ${\overset{⌢}{T}}_{0}$ is maximally symmetric with deficiency indices of 2 and 0. A possibly more familiar account of the non-self-adjoint character of ${\overset{⌢}{T}}_{0}$ may be acquired by examining it in the momentum representation. The momentum representation of the operator, itself, is given by ${\hat{T}}_{0, m o m} = - (i ℏ m / 2) [p^{- 1} (d / d p) + (d / d p) p^{- 1}],$ and if a square integrable state function, $ψ (p)$ , is written as, $ψ (p) : = | p^{1 / 2} | f (p)$ ,

Comments on the analysis of Brunetti et al.

Brunetti et al. as described by Pashby, explicitly construct the POVM that corresponds to ${\hat{T}}_{0}$ , according to (16a), (16b), (17) (Brunetti & Fredenhagen, 2002) and while natural and physically plausible, their construction would not be uniquely compelling, if they had not known what operator, ${\hat{T}}_{0}$ , they were after. Elsewhere they show (Brunetti & Fredenhagen, 2013) that time translationally covariant POVMs lead to an indeterminacy relation for arrival time observables alone! Not a time–energy

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^☆: Presented at the Workshop on Cosmology and Time, April 16–17, 2013, Pennsylvania State University, University Park, PA, USA.

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Response to Dr. Pashby: Time operators and POVM observables in quantum mechanics☆

Abstract

Introduction

Section snippets

Between Pashby and Hilgevoord

Time, observables and measurement

Time–energy indeterminacy

Case specific time observables

The POVM perspective

Comments on the analysis of Brunetti et al.

Physics Reports

Studies in the History and Philosophy of Modern Physics

Studies in the History and Philosophy of Modern Physics

Time in the quantum theory and the uncertainty relation for time and energy

Physical Review

Time of occurrence observable in quantum mechanics

Physical Review A

Time in quantum physics: From an external parameter to an intrinsic observable

Foundations of Physics

The time–energy uncertainty relation

Operational quantum physics