Abstract
This paper provides a general method for defining a generalized quantum observable (or POVM) that supplies properly normalized conditional probabilities for the time of occurrence (i.e., of detection). This method treats the time of occurrence as a probabilistic variable whose value is to be determined by experiment and predicted by the Born rule. This avoids the problematic assumption that a question about the time at which an event occurs must be answered through instantaneous measurements of a projector by an observer, common to both Rovelli [22] and Oppenheim et al. [17]. I also address the interpretation of experiments purporting to demonstrate the quantum Zeno effect, used by Oppenheim et al. [17] to justify an inherent uncertainty for measurements of times.
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Notes
- 1.
Some surprisingly powerful results have followed from this assumption. Consider the role of parameter independence in Bell’s Theorem or, more controversially, the so-called Free Will Theorem [7].
- 2.
In particular, requiring that an observable must correspond to a self-adjoint operator (and thus a PVM) is sufficient to guarantee that it returns a valid probability distribution, but this requirement is not necessary. What is necessary for an observable to return a valid probability distribution from the quantum state is that it defines a POVM [4], and every PVM is a POVM. Given Pauli’s Theorem, then, an event time observable will have to be a POVM that is not a PVM (and thus its first moment defines an operator that is symmetric but not self-adjoint).
- 3.
That is, Werner models a screen as a two-dimensional spatial plane, \(\Sigma\), that extends indefinitely in time.
- 4.
Note that this can be seen to subsume the idea of a screen observable: Hoge [10] explicitly demonstrates that this returns the appropriate screen observable as the volume \(\Delta\) becomes an area, \(\vert \Delta \vert \rightarrow 0\).
- 5.
Any bounded positive operator A has a unique positive square root A 1∕2 such that \(\left (A^{1/2}\right )^{2} = A\). A projector P is just a positive operator for which P 2 = P and thus P 1∕2 = P.
- 6.
I first raised this difficulty in [19], which also contains a history of time in early quantum theory.
- 7.
This is just the assumption that we have a well-formed event space.
- 8.
It must be admitted that this restricts the application of this technique to experiments where only a single outcome of some type is expected. However, these experiments are precisely those where probabilistic answers to the question “when does the experiment have an outcome?” makes sense.
- 9.
Note that this operator depends on the Hamiltonian H through the Heisenberg picture family P t = U t † PU t , where U t = e −iHt. It is not surprising that an operator related to the time of an event should depend on the dynamics.
- 10.
For reasonable choices of P and H, Brunetti and Fredenhagen [3] demonstrate that this definition does lead to a positive operator S defined on the domain obtained by taking the orthogonal complement of the subspace of states for which the expectation of S is 0 or infinity.
- 11.
In case this seems only tangentially related to Lüders’ Rule, note that in the temporally extended Hilbert space this expression becomes identical to Lüders’ Rule. See [18, §8.3].
- 12.
Now, Oppenheim et al. [17] point out that there is a technical problem here since the system may be subject to the backflow effect, in which case P(t) can decrease with increasing t and thus fails to be monotonic. There is also the further problem that no unitary evolution that can take a pure state to a mixture. My concern, however, is conceptual rather than technical.
- 13.
In the background here is a simple picture of temporal passage in which a future event becomes present and then past, and tensed statements about that event change their truth value in response. (See [21] for the canonical statement of this view.) For example, if I light the fuse of a firework at time t b and say “the firework will explode,” that statement is true and is made true by the occurrence of the explosion at a later time t e > t b . After the explosion, at times t > t e , the statement “the firework will explode” is false while the statement “the firework did explode” is true, and is made true by the occurrence of the explosion at an earlier time.
- 14.
They also point out that their analysis is completely general and applies to any projector.
- 15.
That is, the subspace onto which \(\mathbb{I} - P\) projects is just the orthogonal complement of the range of P.
- 16.
One of the more counterintuitive aspects of this proposal is that successive measurements of M need not agree—even after an outcome has ostensibly obtained—and thus a sequence of results such as {0, 0, 1, 1, 0} is possible (in the sense that it is assigned non-zero probability). If a measurement of M with result M = 1 takes place at t 0 then the probability of measuring M = 1 at some later time t > t 0 is given by
$$\displaystyle{P(t) =\langle \Psi \vert M_{t_{0}}M_{t}M_{t_{0}}\Psi \rangle.}$$Assume for contradiction that the second measurement of M is guaranteed to return M = 1. Then P(t) = 1 for all t > t 0 and thus \(M_{t_{0}}\vert \Psi \rangle\) is an eigenstate of M t with eigenvalue 1 for any t > t 0, that is
$$\displaystyle{M_{t}M_{t_{0}}\vert \Psi \rangle = M_{t_{0}}\vert \Psi \rangle.}$$Since \(\vert \Psi \rangle\) was arbitrary, this is true for any \(\vert \Psi \rangle \in \mathcal{H}\). In that case, it follows that \(M_{t_{0}} \leq M_{t}\) (i.e., \(M_{t_{0}}\) projects onto a subspace of M t ) and so they commute. But, as Oppenheim, Reznick and Unruh point out (p. 133), in general two projections from the same family \(M_{t},M_{t^{{\prime}}}\) with t ≠ t′ do not commute. (And, as they argue, exceptions to this claim will be rare. For example, if the Hamiltonian is periodic then only if | t − t′ | is equal to (a multiple of) the period will M t , M t ′ commute (since in that case \(M_{t} = M_{t^{{\prime}}}\)).) Therefore, in general there is a non-zero probability that a measurement of M t will return M = 0. (Note that we can run the same argument using the Heisenberg picture family \(M_{t}^{0} = U_{t}^{\dag }M^{0}U_{t} = U_{t}^{\dag }(\mathbb{I} - M)U_{t} = \mathbb{I} - M_{t}\), or any such family.)
- 17.
Oppenheim, Reznick and Unruh critique Rovelli’s proposal on the grounds that:
His scheme only answers the first question: “has the measurement occurred already at a certain time?”, but does not answer the more difficult question “when did the measurement occur?” In other words, it does not provide a proper probability distribution for the time of an event. (108)
I agree, but note that their proposal is subject to precisely this latter critique.
- 18.
They make two additional non-trivial assumptions. First, they assume that the projection P commutes with the one-parameter semi-group of time translations generated by the dynamics. Second, they assume that the semi-group representing the evolution of the system under continuous observation (that results from taking the n → ∞ limit) is continuous for t ≥ 0, not just t > 0.
- 19.
In fact, Itano et al. [11] describe how they did exactly this to prepare the system in the ground state.
- 20.
I note that Ruschhaupt et al. [23] provide a theoretical basis for modeling a detector as involving quantum jumps in a related way, and also make use of the operator normalization technique of Brunetti and Fredenhagen [3] to derive time of arrival POVMs. It seems likely that these POVMs could be interpreted as conditional probabilities (as suggested above), although Ruschhaupt et al. [23] do not interpret them as such.
- 21.
To illustrate: if the interaction term of the Hamiltonian is position dependent then the strength of the coupling may vary with time (through the Schrödinger picture evolution of the joint state) but the Hamiltonian for the joint system will be time-independent.
- 22.
The focus of this paper is the time of this event, which arises from the weak coupling of two systems. An alternative question concerns the correct description of the occurrence of many such events as a stochastic process. The use of a stochastic master equation to describe continuous weak measurements seems promising in this regard [12, §4].
References
L.E. Ballentine, Comment on “quantum Zeno effect”. Phys. Rev. A 43(9), 5165 (1991)
A. Beige, G.C Hegerfeldt, Projection postulate and atomic quantum Zeno effect. Phys. Rev. A 53(1), 53 (1996)
R. Brunetti, K. Fredenhagen, Time of occurrence observable in quantum mechanics. Phys. Rev. A 66(4), 044101 (2002)
P. Busch, Quantum states and generalized observables: a simple proof of Gleason’s theorem. Phys. Rev. Lett. 91(12), 120403 (2003)
P. Busch, M. Grabowski, P.J. Lahti, Time observables in quantum theory. Phys. Lett. A 191(5), 357–361 (1994)
P. Busch, M. Grabowski, P.J. Lahti, Operational Quantum Physics (Springer, Berlin, 1995)
J. Conway, S. Kochen, The free will theorem. Found. Phys. 36(10), 1441–1473 (2006)
P.A.M. Dirac, The Principles of Quantum Mechanics. (Clarendon Press, Oxford, 1930)
I. Egusquiza, J. Muga, A. Baute, “Standard” quantum–mechanical approach to times of arrival, in Time in Quantum Mechanics (Springer, New York, 2002), pp. 305–332
M.O. Hoge, Relationale Zeit in der Quantenphysik. Master’s Thesis, University of Hamburg (2008)
W.M. Itano, D.J. Heinzen, J.J. Bollinger, D.J. Wineland, Quantum zeno effect. Phys. Rev. A 41(5), 2295 (1990)
K. Jacobs, D.A. Steck, A straightforward introduction to continuous quantum measurement. Contemp. Phys. 47(5), 279–303 (2006). http://dx.doi.org/10.1080/00107510601101934
J. Kijowski, On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time. Rep. Math. Phys. 6(3), 361–386 (1974)
G.W. Mackey, The Mathematical Foundations of Quantum Theory (WA Benjamin, New York, 1963)
B. Misra, E.C. George Sudarshan. The zeno’s paradox in quantum theory. J. Math. Phys. 18(4), 756–763 (1977)
W. Nagourney, J. Sandberg, H. Dehmelt, Shelved optical electron amplifier: observation of quantum jumps. Phys. Rev. Lett. 56(26), 2797 (1986)
J. Oppenheim, B. Reznik, W.G. Unruh, When does a measurement or event occur? Found. Phys. Lett. 13(2), 107–118 (2000)
T. Pashby, Time and the foundations of quantum mechanics. Ph.D. Thesis, University of Pittsburgh (2014). http://philsci-archive.pitt.edu/10723/
T. Pashby. Time and quantum theory: a history and a prospectus. Stud. Hist. Phil. Sci. Part B: Stud. Hist. Phil. Mod. Phys. 52, 24–38 (2015)
Y.S. Patil, S. Chakram, M. Vengalattore, Measurement-induced localization of an ultracold lattice gas. Phys. Rev. Lett. 115(14), 140402 (2015)
A.N. Prior, Changes in Events and Changes in Things. (Department of Philosophy, University of Kansas, Lawrence, 1962)
C. Rovelli, “Incerto tempore, incertisque loci”: can we compute the exact time at which a quantum measurement happens? Found. Phys. 28(7) 1031–1043 (1998)
A. Ruschhaupt, J. Gonzalo Muga, G.C. Hegerfeldt, Detector models for the quantum time of arrival, in Time in Quantum Mechanics-Vol. 2 (Springer, Berlin, 2009), pp. 65–96
M. Srinivas, R. Vijayalakshmi, The ‘time of occurrence’ in quantum mechanics. Pramana 16(3), 173–199 (1981)
A. Sudbery, Diese verdammte quantenspringerei. Stud. Hist. Phil. Sci. Part B: Stud. Hist. Phil. Mod. Phys. 33(3), 387–411 (2002)
R. Werner, Screen observables in relativistic and nonrelativistic quantum mechanics. J. Math. Phys. 27, 793 (1986)
A.S. Wightman, On the localizability of quantum mechanical systems. Rev. Mod. Phys. 34, 845–872 (1962)
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Pashby, T. (2017). At What Time Does a Quantum Experiment Have a Result?. In: Renner, R., Stupar, S. (eds) Time in Physics. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-68655-4_9
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