The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.
Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.
We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.
Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ (position or otherwise), the distribution ${\ensuremath{\pi}}_{a}(t)$ of time measurement at a fixed state
a can be inferred from the distribution ${\ensuremath{\rho}}_{t}(a)$ of a state measurement at a fixed time
t via the transformation ${\ensuremath{\pi}}_{a}(t)\ensuremath{\propto}|\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}{\ensuremath{\int}}_{\ensuremath{-}\ensuremath{\infty}}^{a}{\ensuremath{\rho}}_{t}(u)du|$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time of arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.
An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A 95, 032133 (2017)]. In this approach, the authors derived a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle) $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ is defined at each point in space. The quantum state $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\ensuremath{\rangle}, \ensuremath{\langle}t|\ensuremath{\phi}(x)\ensuremath{\rangle}$, represents the probability amplitude of the particle's arrival time at
x. In this work we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the initial state $|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, which predicts any measurement on the particle performed by a detector localized at
x0, the SC Schr\"odinger equation provides $|\ensuremath{\phi}(x)\ensuremath{\rangle}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbb{U}}(x,{x}_{0})|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, enabling us to predict measurements when the detector is at $x\ensuremath{\lessgtr}{x}_{0}$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|{P}_{b}(x)\ensuremath{\rangle}$ depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\ensuremath{\psi}(t)\ensuremath{\rangle}={|P\ensuremath{\rangle}|}_{t}$, at time
t, has momentum
P (and indefinite position), the same particle in the state $|\ensuremath{\phi}(x)\ensuremath{\rangle}=|{P}_{b}(x)\ensuremath{\rangle}$ arrives at position
x with momentum
Pb(x) (and indefinite arrival time). By investigating the fact that $|\ensuremath{\psi}(t)\ensuremath{\rangle}$ and $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ describe experimental data of the same observables collected at
t and
x, respectively, we conclude that they provide complementary information about the same particle. Finally, we solve the SC Schr\"odinger equation for an arbitrary space-dependent potential. We apply this solution to a potential barrier and compare it with a generalized Kijowski distribution, showing that they can predict distinct traversal times.