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Quantum time Oppenheim, Jonathan A. 2000

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Q U A N T U M T I M E By Jonathan A . Oppenheim B. Sc., The University of Toronto, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1999 © Jonathan A. Oppenheim, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abst rac t In quantum mechanics, time plays a role unlike any other observable. We find that measuring whether an event happened, and measuring when an event happened are fun-damentally different - the two measurements do not correspond to compatible observables and interfere with each other. We also propose a basic limitation on measurements of the arrival time of a free particle given by 1/Ek where Ek is the particle's kinetic energy. The temporal order of events is also an ambiguous concept in quantum mechanics. It is not always possible to determine whether one event lies in the future or past of another event. One cannot measure whether one particle arrives to a particular location before or after another particle if they arrive within a time of 1/E of each other, where E is the total kinetic energy of the two particles. These new inaccuracy limitations are dynamical in nature, and fundamentally different from the Heisenberg uncertainty relations. They refer to individual measurements of a single quantity. It is hoped that by understanding the role of time in quantum mechanics, we may gain new insight into the role of time in a quantum theory of gravity. 11 Table of Contents Abs t rac t i i Table of Contents i i i Lis t of Figures v i Dedica t ion v i i Acknowledgements v i i i 1 In t roduct ion 1 1.1 Dual Measurements 2 1.2 Differences Between Measurements of Space and Measurements of Time . 7 1.3 Inaccuracies and Uncertainties 10 1.4 What Lies Ahead 11 2 W h e n does an Event Occur 15 2.1 Probabilities at a Time and in Time 16 2.2 Did it Occur vs. When Did it Occur 17 2.3 Time of a Measurement or Arrival 21 2.4 Continual Event Monitoring 25 3 Phys ica l Clocks and Time-of -Arr iva l 30 3.1 A Limitation on Time-of-Arrival Measurements 31 3.2 Free Clocks 33 in 3.3 Measurement of Time-of-Arrival 35 3.3.1 Measurement with a clock 36 3.3.2 Two-level detector with a clock 40 3.3.3 Local amplification of kinetic energy 44 3.3.4 Gradual triggering of the clock 46 3.3.5 General considerations 49 4 T ime-of -Ar r iva l Operators 52 4.1 Indirect Time-of-Arrival Measurements 53 4.2 Conditions on A Time-of-Arrival Operator 54 4.3 Time-of-Arrival Operators vs. Continuous Measurements 56 4.4 The Modified Time-of-Arrival Operator 60 4.5 Normalized Time-of-Arrival States 63 4.6 Contribution to the Norm due to Modification of T 67 4.7 Limited Physical Meaning of Time-of-Arrival Operators 69 5 Traversal T i m e 71 5.1 A Limitation on Traversal Time Measurements . . 72 5.2 Measuring Momentum Through Traversal-Distance 73 5.3 Measuring Traversal Time 74 5.4 General Argument for a Minimum Inaccuracy 77 5.5 From Traversal Time to Barrier Tunneling Time 83 6 Order of Events 85 6.1 Past and Future 86 6.2 Which first? 88 6.3 Coincidence 92 IV 6.4 Coincident States 97 6.5 In Which Direction Does the Light Cone Point 99 7 Conclus ion 101 Bib l iography 106 Appendices 110 A Zero-Current Wavefunctions 110 B Gaussian Wave Packet and Clocks 111 C T ime-o f -Ar r iva l Eigenstates 114 v List of Figures 4.1 Unmodified part of time-of-arrival eigenstate. \ 0 T + ( X , T ) \ 2 VS. X, with A = m (solid line), and A = ^ (dashed line). As A gets smaller, the probability function gets more and more peaked around the origin. . . . 66 4.2 Modified part of time-of-arrival eigenstate. ^ | e r + ( x , r ) | 2 vs. ex, of ij~^x-with A e 2 = ft (solid line) and Ae 2 = ^ (dashed line). As A or e gets smaller, the probability function drops near the origin, and grows longer tails which are exponentially far away 67 6.3 A potential which can be used to measure which of two particles came first (given by V(x,y) = a6(x)0(—y)). The wave function for two incoming particles in one dimension looks like a single wave packet in two dimensions travelling towards the origin 89 6.4 Potential for measuring whether two particles are coincident 93 6.5 Phase shifts for coincidence detector (6i(ka)/60(ka) vs. ka ) 97 vi Dedicat ion In loving memory of my father, Peter Oppenheim (1942-1998) - my first physics teacher, who encouraged my curiosity, patiently answered my questions, and patiently asked his own. He would have loved to flip through this thing, and I had always imagined giving him a copy. vn Acknowledgements I would like to thank my Ph.D. advisor, Bi l l Unruh, not only for teaching me a great deal of physics, but also for teaching me how to attack problems and discover interesting questions. For allowing me to wander off in other directions and subtle guidance. I owe a great deal to Benni Reznik, for continual support, advice and a constant ex-change of ideas. Much of this work is contained in [1] - [7], and was done in collaboration with Yakir Aharanov, Benni Reznik, Sandu Popescu, and Bi l l Unruh. Thanks to my collaborators for their help, for many interesting discussions, insights and paradoxes. Throughout my work, I have had the benefit of discussions with a number of researchers. Carlo Rovelli, Gonzalo Muga, Rick Leavens, and Bob Wald have been particularly helpful. Thanks also to my committee members, Kristin Schleich, Mark Halpern, Gordon Se-menoff, and Ian Affleck for their suggestions, and to fellow graduate students Lori Pani-ak, Adam Monahan, Richard Szabo, Todd Fugleburg, Martin Dube, Sebastian Jaimungal and others for their help. Special thanks to the amazing Victoria Scott for drawing the cartoons for this thesis, as well as for everything else. vm Thanks to all my friends and house mates for putting up with me. To my parents, my brother and the grannies, I thank them for their encouragement their support and their interest, for giving me red boxes to play with, reading to me, and killing the television. And finally, no thanks to A P E C , the Big Cheese, and the Mounties. ix Chapter 1 Int roduct ion 1 Chapter 1. Introduction 2 1.1 D u a l Measurements One of the first lessons of quantum mechanics was that a property of a system does not correspond to an element of reality until it is measured. It makes no sense to talk about the position of a particle or the momentum of the particle, in and of itself. It is only when we measure a physical quantity that we can actually say that a system possesses it. The particle does not have a position until its position is actually measured. Ordinarily in quantum mechanics, one is interested in measuring properties of a sys-tem at a particular time t. One might want to know a particle's position, momentum, or spin, and the measurement of this quantity occurs at a certain time. For experiments at a fixed time, quantum mechanics provides us with a useful formalism to describe reality. Observables are represented by self-adjoint operators, and in the Heisenberg representa-tion they evolve in time. The possible results of any measurement at any instant of time t can be found by applying these operators to the wave function of the system at the time t. This immediately raises the question of what the parameter time t represents in the Heisenberg equations of motion. Since t is a number and not a self-adjoint operator, it does not appear to be an observable in the usual sense. For any measurement of an observable A(t) of a system, one can imagine a dual measurement, where one attempts to measure the time IA at which the system attains a particular value of A. The dual measurement determines the time a certain event occurs, where the event in question is the system attaining a particular value (or values) of an observable. For example, instead of measuring the position of a particle at a certain time, one can consider the dual measurement where the roles of x and t are interchanged. Instead of measuring where the particle is at time t, one measures the time that a particle is found at a particular location XA- In this dual measurement, the position XA is the parameter while Chapter 1. Introduction 3 the time becomes the observable one is trying to measure. Classically, the time of an event can be made into an observable just like any other and this time can be measured in a variety of ways, all of which give the same result. One can simply invert the equations of motion of the system to find the time that an event occurs 1 , and then measure the values of the canonical variables (generalized coordinates and conjugate momenta). Since classically there is no uncertainty relation preventing the measurement of all the coordinates and conjugate momenta simultaneously, there is no limitation for finding the event's time. One could also continually monitor the system to determine the precise time when the event occurred. Since one can make the interaction between the system and the measuring apparatus as small as one likes, this measurement need not disturb the evolution of the system. Finally, one can also couple a clock to the system in such a way that the clock stops when the event occurs. A l l these methods yield the same results, and work to any desired accuracy. Dual measurements, are quite common in modern laboratory experiments. In particle physics one often wants to know the time that certain collisions or decays occurred. However, surprisingly, dual measurements are not easily dealt with using the conventional tools of quantum mechanics. Pauli [8] was the first to demonstrate that there was no operator associated with time. A time operator must be conjugate to the Hamiltonian, and he proved that this is impossible if the Hamiltonian for the system is bounded from above or below. The reason for this is that an operator which is conjugate to the Hamiltonian acts as a shift operator for energy, and one could use it to shift the energy below any lower bound (or above any upper bound). Since then, there have been numerous attempts to circumvent his proof by considering 1 In some systems (especially in the context of general relativity), it is only possible to find the time locally. A global time variable may not exist. Chapter 1. Introduction 4 either modified time operators, or by considering operators which correspond to the "time-of-arrival", i.e. the time that an event first occurs [9][12]. At first glance, the latter operator need not be conjugate to the Hamiltonian, since all that is required is that the time-of-arrival operator not evolve in time 2 . However, in Chapter 4 we show that, in general, time-of-arrival operators also do not exist. Aharonov and Bohm were the first to write clown a time-of-arrival operator [10], and since then most of the work in the field of time-of-arrival has involved interpreting, or modifying this operator[ll][12], or operators associated with it (such as the "current operator" [13]). Allcock [14] [15] was the first to examine a physical model for measuring the time-of-arrival, and this led him to suggest that time-of-arrival may not be measur-able. However, his model did not contain a clock, and he argued that the source of the difficulty in measuring time-of-arrival was in absorbing a particle in an arbitrarily short length of time. In fact, as we will discuss, the source of the difficulty lies elsewhere, and one needs to use models with physical clocks. Peres [16], has used physical clocks to de-scribe measurements of various quantities, although not in the context of time-of-arrival, and physical clocks have also been discussed in the context of barrier tunneling time [17]. Although much of the work in this field (including our own) is done in the context of the Copenhagen interpretation of quantum mechanics, the problem of time-of-arrival has also been studied in the context of the decoherent histories formalism [18]. In this approach, amplitudes are not assigned to events at a certain time, but rather to entire histories in a decohering set of histories. For the case of time-of-arrival, one finds that histories which correspond to different arrival times do not decohere unless the particle is coupled to an environment or a model detector [19]. In many respects, this supports the approach that one must consider physical measurement processes in order to measure 2 by definition, the arrival time is the same at all times - if I will (or did) arrive at 5 p.m., then this statement remains true at 3 p.m., 4 p.m. and 6 p.m. Chapter 1. Introduction 5 the time-of-arrival. The interest in a quantum mechanical time operator stems in part from the troubling notion that elements of reality should be observables. It is hard to understand what the parameter t in the Schrodinger equation means, if it does not correspond to some-thing physically measurable. Recently however, attempts to find a quantum theory of gravity have also inspired numerous authors to examine the problem of time in quantum mechanics. In general relativity (and even in many laboratory experiments), we are often inter-ested in performing experiments which are not fixed in time. For example, if we wish to measure space-time distances, then we will probably want to know how long it takes for a photon to travel between two points. This is a continuous measurement which does not occur at any particular time. We may also want to know whether one event is in the past or future of another event. Both these measurements are, in some sense, a measurement of time itself, and it is these types of measurements which are necessary in order to determine the components of the metric tensor. Another physical property, which appears in the context of quantum cosmology, is the maximum size the universe will attain. This is not a property of the universe at a fixed time, but rather, a property of the universe over all time. In classical physics, one could make measurements on a system at a fixed time in order to predict the evolution of the system for all time. However, as we will.see, in quantum mechanics this is not always the case. It is widely believed that one of the difficulties of constructing a quantum theory of gravity, is that time plays an incompatible role in quantum mechanics and general relativity [20]. In quantum mechanics, time is an external parameter while in general relativity, time is much more a part of the theory. Both time and space bend and twist in the presence of massive objects, and both space and time are represented by coordinates. Chapter 1. Introduction 6 It is space-time which is the element of reality in general relativity. These coordinates are, of course, subject to coordinate transformations, and in par-ticular, the theory is invariant under reparametrization of the time coordinate. One consequence of this, is that if one tries to canonically quantize Einstein's theory of gravi-ty in a closed system, one finds that the wave-function must satisfy the Wheeler-DeWitt equation H*(gab,wab) = 0 (1.1) where the wave function depends on the 3-metric and conjugate momenta and 7i is known as the Hamiltonian constraint and is the generator of time reparametrizations and time-translations when the equations of motion are satisfied. Because the Hamilto-nian constraint must always be satisfied, most standard interpretations require that the only possible observables are those which commute with the constraint [22]. However, observables which commute with the constraints don't evolve in time, making the system rather hard to describe. It is not clear how to best frame physically meaningful questions, if all the observables are static with respect to parameter time. One of the central sources of the the problem of time in quantum gravity (and quan-tum cosmology in particular) is that it attempts to describe the entire universe quantum mechanically. There is no external observer, and therefore, no external parameter time. Many authors have therefore attempted to develop alternative frameworks of quantum mechanics which do not rely on an external time parameter [18] [20] [22] [21]. By re-examining the way in which we think about time, we may be able to construct a consis-tent theory of quantum gravity. In this thesis, we make no pretense of trying to solve the problem of time in quantum gravity. Rather, we take the approach that in order to understand the role of time in quantum gravity, one must first understand the role of time in quantum mechanics. As it turns out, this is far from easy, and there still Chapter 1. Introduction 7 exist many ambiguities in the role of time in quantum mechanics. Our hope is that a better understanding of time in the arena of quantum mechanics will benefit and inform research in the field of quantum gravity. At the end of this thesis, we will discuss some of the connections between the problem of time in quantum gravity and our research. 1.2 Differences Between Measurements of Space and Measurements of T i m e Ever since Einstein's theory of special relativity, we have been encouraged to think of time and space on an equal footing. However, even classically, time and space are quite different as our common experience tells us. Objects move constantly forward in time an a manner very different to the way they move through space. Although we will discuss in more detail the differences between quantum measurements of ordinary observables and measurements of time in Chapter 2, it may be instructive to roughly outline the differences between measurements of a particle's position at a fixed time, and the time a particle is found at a particular location. In standard quantum mechanics, the probability that a particle is found at a given location X at time t is given by Pt(X) = MX,t)\2 . (1.2) If we know ip(x, 0) for all x then the system is completely described and we can easily compute this probability distribution at an instant of time. If we know the Hamiltonian of the system, then using the Schrodinger equation we can also compute ip(x,t) at any time t. This probability distribution corresponds to results of a measurement of position at a particular time. Quantum mechanics gives a well defined answer to the question, "where is the particle at time £?" However, it is also perfectly natural to ask "at what time is the particle at a certain location." Here, quantum mechanics does not seem to provide an unambiguous answer. Chapter 1. Introduction 8 At first sight it seems that the probability distribution PX(T) to find the particle at a certain time at the location x is simply \ip(x,T)\2, However, \ip(x,T)\2, does not represent a probability in time, since it is not normalized with respect to T. One might be tempted therefore, to consider the quantity F ( T ) - J M L fl3) This normalization depends on the particular state being measured, and can only be done if one knowrs the state ip(x,t) at all times t (infinitely far in the past and future). There are also states for which the particle is never found at the position x, in which case the expression above is undefined. Not-withstanding this, one might argue that this quantity gives one a relative probability that the particle is found at the location x at time T (if the measurement is made at that time T) , as opposed to another time T" (if the measurement is made at time T'). However, the expression above certainly does not yield the probability in time to detect the particle. One reason for this failure is that a particle may be detected at a location X at many different times t (e.g. I can be found in my office at many different times in the clay). On the other hand, if at time t a particle is detected at location X, then we can'say with certainty that at the same time t, the particle was not at any other location X' (e.g. at nine a.m. I am in bed, and therefore, I cannot also be in my office). Equation (1.3) does not give a proper probability distribution as the various outcomes are not disjoint. PX(T) is not a probability distribution in time in the sense usually reserved for probability distributions in quantum mechanics. PX(T) is very different from Pt(X) and has different properties (as we will see in the next chapter). This leads us to consider the time of first arrival of a particle, since a particle can only arrive once to a particular location. In order to measure the arrival time one cannot use expression (1.3) since one needs to detect the particle at time t^, and also know that Chapter 1. Introduction 9 the particle was not there at any previous time. In other words, one must continuously monitor the location XA in order to find out when the particle arrives. However, this continuous measurement procedure has it's own difficulty, and also emphasizes the prob-lem with the previous probability distribution. Namely, that the probability to find a particle at t = T is generally not independent of the probability to find the particle at some other time t — T". i.e.. if HXA is the projector onto the position XA, then in the Heisenberg representation 3 Measurements made at different times disturb each other. We will see in Section 2.2 that this is one of the properties of ordinary measurements which measurements in time vio-late. Measurements made at different times do not commute. Therefore the probability distribution obtained from this measurement procedure, although well defined, does not give a probability distribution in time. Von Neumann measurements 4 happen at a certain time. One measures the particle's position at time t. Even a continuous measurement at a particular location is a series of measurements at a certain time. Each instant that the Geiger counter doesn't click, it is measuring the fact that a particle has not entered it. Furthermore, operators which are used to measure the time-of-arrival to the location XA, are not measured at XA, but rather at an instant in time. In quantum mechanics, measurements made at different times can disturb each other, which can make measurements of the time of an event problematic. The probability of detecting a particle at a certain location at time t is not independent 3 T h e exact expression for the commutator of the position projectors at different times is not particular-ly illuminating. However, it is fairly obvious HXA doesn't commute with itself at different times, because the position operator itself doesn't commute with itself at different times. I.e., since x( i ) = xo + pt/rn, we have [x(i),x(0)] = it/m. 4 I n [23] Von Neumann outlined how one goes about measuring observables which correspond to self-adjoint operators. The results one obtains for the measurements are universal, and correspond to actual properties of the system. Chapter 1. Introduction 10 of detecting the particle at some other time t!. 1.3 Inaccuracies and Uncertainties The measurement of an observable corresponding to a self-adjoint operator can be as accurate as one wishes. This is true despite any uncertainty relations which govern various sets of observables. The position, or momentum of a particle (but not both) can be measured to any desired precision. Consider two observables A and B which do not evolve in time, and whose commutator is % (in units where Ti — 1). Imagine that we have an ensemble of identical systems prepared in some initial state. On half the ensemble, we can measure A , and on the other half, we can measure B . , Each individual measurement can be as accurate as we wish. An extraordinary experimentalist can reduce the inaccuracies in the measurement to almost zero, and can get a particular value for each measurement. The experimentalist may have a dial on her device which will point to the value of A after the measurement. She will have to make sure that initially the pointer on her dial points almost exactly to zero, and then after each run of her experiment, she measures the position of the dial very accurately to determine the value of A. If we then plot all of the measurements of A and all of the measurements of B , we will find a distribution of measurements which have a natural width of AA and AB respectively. One then finds that no matter what initial state we choose, AAAB > 1. There is an uncertainty relation between the distributions of A and B, but there are no theoretical limitations on the accuracy of each individual measurement of A or B . The experimentalist does not have to make her measurements totally precise. She could, for example, start off the experiment with her dial in a state where the initial position of the needle is uncertain. An uncertainty in the initial pointer position will result in her measurement being inaccurate. When she measures the final position of her Chapter 1. Introduction 11 pointer, she will not be able to infer the precise value of the measurement of A or B because she will not know exactly what the initial value the pointer was set to. For measurements of conventional observables, there are no limitations on the inac-curacy of measurements. However, we will find that for certain observables relating to the time-of-events, one must make the measurement inaccurate. If one attempts to make the measurement too accurate, one finds that the measurement fails. The inaccuracy limitations we find are not equivalent to the so-called "Heisenberg energy-time uncertainty principle". The limitations- refer to individual measurements of a single quantity. Quantum mechanics places no limitation on how accurately we can make a single measurement of position or momentum (although an accurate measurement of position will disturb the momentum and visa-versa). For measurements of time-of-arrival however, we cannot make a single measurement arbitrarily accurate. If we do so, we may find that the particle never arrives. 1.4 W h a t Lies A h e a d In this thesis, we will find that dual measurements are fundamentally different from measurements of ordinary quantum variables. We will examine a number of different types of dual measurements as well as various methods for making them. In Chapter 2 we will look at measurements where one continually monitors the state to determine whether the event has occurred. This involves a series of measurements at closely spaced time intervals. We find the surprising result that the question of whether an event has occurred and the question of when it occurred are not compatible observables. Describing attributes of a system in time is fundamentally different from describing attributes at a given time. The more difficult question of "when did the event occur?" cannot be measured unambiguously in quantum mechanics. We also critically examine the use Chapter 1. Introduction 12 of the probability, current to measure the time at which a particle arrives to a certain location. The discussion suggests that the difference between time and other observables is not merely formal. The central result of the thesis is contained in Chapter 3 where we discuss the problem of the time-of-arrival of a particle to a particular location. It is argued that the time-of-arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement involving physical clocks, we show that the time-of-arrival for a free particle cannot be measured more accurately then 6t^ ~ l/£t-, where Ek is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device and the resulting probability distribution becomes distorted. This is a new relation which is not equivalent to the so-called "Heisenberg energy-time uncertainty" ° - it places a restriction on each individual measurement of time-of-arrival. The basic reason for the inaccuracy limitation is that while one can construct an arbi-trarily accurate clock, using this clock presents difficulties. The more accurate the clock, the greater the spread in the clock's energy. Accurate clocks are extremely energetic, and this makes it harder for the system to stop the clock. In order to use the clock to measure the time of an event, one needs the system to turn off the clock when the event occurs. For accurate clocks, the system will not always have enough energy to turn it off, and no measurement will occur. Recently, many authors [12] have attempted to construct operators which can be used to measure the time-of-arrival of a particle. In Chapter 4 we present a formal proof that a time-of-arrival operator cannot exist. Still, many believe that one can modify a time-of-arrival operator in such a way as to make the concept useful. We discuss the °For convenience, we will sometimes use the term "Heisenberg's energy-time relation". It should be remembered however, that since time is not represented by a self-adjoint operator, the uncertainty relation is actually between energy, and observables of the system which evolve in time (for example, an atom's life-time becomes uncertain if the atom is close to an eigenstate of energy) Chapter 1. Introduction 13 relationship between these modified operators, and the direct measurements discussed in Chapters 2 and 3, and argue that a measurement of the time-of-arrival operator does not correspond to these continuous measurements. Unlike the classical case, in quantum mechanics the result of a measurement of the time-of-arrival operator may have nothing to do with the time-of-arrival to x = XA-There has been renewed interest in time-of-arrival operators following the suggestion by Grot, Rovelli, and Tate, that one can modify the low momentum behavior of the oper-ator slightly in such away as to make it self-adjoint [9]. We show that such a modification results in the difficulty that the eigenstates are drastically altered. In an eigenstate of the modified time-of-arrival operator, the particle, at the predicted time-of-arrival, is found far away from the point of arrival with probability 1/2. The bound of 1/Ek on the accuracy of time-of-arrival measurements is based on calculations done using numerous measurement models corresponding to specific Hamil-tonians, as well as more general considerations. However, because the limitation is based on dynamical considerations and not kinematic ones, a formal proof of the limitation may not exist. For example, a proof of the Heisenberg uncertainty relation relies only on the properties of specific operators, while our inaccuracy relation is a statement not about operators, but about measurements (and therefore, involves the dynamical considerations of the actual measurement). Perhaps by making certain restrictive assumptions about the Hamiltonian one might be able to construct a formal proof. Such a proof would have to take into account the measurement model which will be discussed in Section 3.3.3 in which we show that if one has prior information about the wavefunction, and if the wave-function is almost an eigenstate of energy (i.e. its time of arrival is completely uncertain), then one can measure the time of arrival to an accuracy better than 1/Ek- One therefore expects that a formal proof will not only have to involve making assumptions about the interaction Hamiltonian, but also the initial state of the wave function. The existence of Chapter 1. Introduction 14 a formal proof for our inaccuracy limitation remains an interesting open question. While we know of no formal proof for the inaccuracy limitation for time-of-arrival, one can make more general statements about measurements of "traversal time". In Chap-ter 5 we consider the problem of a free particle which traverses a distance L and argue that a violation of the above limitation for the traversal-time implies a violation of the Heisenberg uncertainty relation for x and p. This result does not depend on the details of the model being used in the measuring process. Measurements of traversal-time are dual to measurements of traversal distance, and it can be shown that one can measure the distance a particle travels to any desired precision. This chapter also contains a fur-ther discussion on the difference between what we call "inaccuracy" limitations, which constrain the precision with which individual measurements are performed, and "uncer-tainties" which are kinematic quantities which relate to the spread in measurements on ensembles. Chapter 6 contains what may be our most interesting result. In it, we examine whether one can determine the temporal ordering of events. We find that one cannot measure whether one event occurred in the future or past of another event to arbitrary accuracy. The minimum inaccuracy for measuring whether a particle arrives to a given location before or after another particle is given by 1/E where E is the total kinetic energy of the two particles. We discuss the relationship between this type of measurement, and coincident counters, as well as Heisenberg's microscope. We show that in general one cannot prepare a two particle state where the two particles always arrive within a time of 1/E of each other. This has interesting consequences for determining the metric properties of a space-time. In this thesis we will work in units where ft = c = 1 Chapter 2 W h e n does an Event Occur 15 Chapter 2. When does an Event Occur 16 2.1 Probabi l i t ies at a T ime and i n T ime Within quantum mechanics, a complete set of commuting observables can be found which describe the attributes of a system at a given time. However, difficulties arise for at-tributes of a system that extend over time, such as the time of an atomic decay, the time of arrival, etc. As we discuss below, simple extensions of ordinary notions of probabilities at a certain time to probabilities in time give rise to distributions which can no longer be interpreted as probabilities. The reason for this can be understood in simple terms. Consider for example, the event of a particle entering a box. What is the time of the event? Classically, there is no distinction between attributes at one time or in time. One can, for example, measure the position and momentum of the particle at any time with negligible disturbance and use this information to deduce the time of entering the box. Quantum mechanically, however, there are two separate questions. We can either ask at a certain time to, "has the particle already entered the box?", or we can ask "when did the particle enter the box?" To answer the first question we simply measure at time to if the particle is in the box. Although quantum mechanics does provides us with a prediction for the probability P(to) for this event, this probability does not describe a probability in time. The measurement at time to disturbs the evolution of the system in the future and hence the probability distribution at time t > t0 will no longer be given by P(t). In fact by the same consideration, we see that the two questions above, or the corresponding measurements, are not compatible with each other, in the sense that one measurement disturbs the other and visa-versa. One would think that the questions "when did an event occur?" and "has the event occurred?" can be answered simultaneously, however, as we shall see, they are in fact complementary. One cannot necessarily answer both questions simultaneously. In the next section we will formulated these difficulties in a more precise way. Chapter 2. When does an Event Occur 17 We will then examine two specific cases of measuring the time of an event. One is the arrival of a particle to a certain location, and another is a recent proposal of Rovelli [28] to measure the time that a measurement occurred. We argue 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. We also discuss the relationship between Rovelli's measurement scheme, and the use of the probability current for measurements of time-of-arrival. In Section 2.4 we discuss a model and set of operators which can be used to determine a probability distribution for the time of an event. The model is based upon a continuous process akin to a rapid series of measurements. We find that in the limit of high accuracy the system is severely disturbed and the measurement does not work, an effect which is analogous to the Zeno paradox. 2.2 D i d it Occur vs. W h e n D i d it Occur In conventional quantum mechanics, for each observable, we can assign a set of projection operators onto a set of eigenstates </>,- of some operator. At each time t there exists a Hilbert space and inner product which enable one to calculate the probability P,(t) that the system is in one of the states c/>;. In certain cases, one can find a subset a of the set / such that the projection operator n a = ]Tn; ( 2 . 5 ) gives the probability that a certain event a has happened. For example, I I a may project onto the set of states of an atom which has decayed into its ground state and emitted a photon. Or, in one of the examples which we will be discussing in Section 2.3, I I a will be the projector onto the states of a measuring device after a measurement has occurred. Chapter 2. When does an Event Occur 18 For the case of time-of-arrival, I I a will be the projector onto a region of the x-axis (in this case, the index i is continuous). If initially the system is in the state ip then in the Heisenberg representation the probability that the event has happened at any time t is given by One can also compute the "current operator" which gives the rate of change of the probability distribution Pa(t). It is tempting to argue that the probability distribution gives the probability that the event a happens between t and dt, since classically the probability that an event happened some time before time t is just the integral between some initial time t0 and t of the probability that the event happens at that time. However, the probability distribution obtained from J a cannot be thought of as a probability distribution in time. pa(t) is not the probability that the event happened at time t. To see that pa(t) is not a probability distribution in time, let us compare its properties to the properties (1-4) of the conventional quantum 1 probability distribution obtained from the projectors II,-. P rope r ty 1 The probability of finding that the system is in the state & at time t is independent of the probability of finding that the system is in the state <j>j (at the same time t). 1 properties 2-4 are also true of classical probability distributions • ' ' ' . pa(t) = (maim). (2.6) (2.8) Chapter 2. When does an Event Occur 19 i.e.. [nf-(t),ni(t)] = o. ( 2 . 9 ) If we interpret the probabilities Pa(t) as probabilities in time, then our conventional notions of what these probabilities mean, break down. In general, [na(«),na(0]^o. ( 2 . i o ) Measurements made at earlier times influence measurements made at later times. The possible results of an observable at time t will depend on whether there were any previous measurements of that observable. In classical mechanics, one can make the interaction of the measuring device with the system arbitrarily weak, and therefore, not disturb the evolution of the system in time, but this is not true in quantum mechanics. A measurement of position at ti for example, will disturb the momentum of the particle in such a way that future measurements of position at ^2 will give very different results from the case where no measurement was performed at t\. Since Ha{t) does not commute with itself at different times, there is no reason to believe that J a will commute with itself at different times either. It is essentially this difference between conventional probabilities and those obtained from n a which prevents us from determining when an event occurred. In addition, pa{t) and Pa(t) do not have the following other properties of quantum distributions: P rope r ty 2 If i ^ j then the projection operators project onto orthogonal states. i.e.. n,-(*)ni(«) = o ( 2 . i i ) For example, if a particle is found at position x, then it could not have been anywhere else at the same time. On the other hand, a particle may be at the same position at many Chapter 2. When does an Event Occur 20 different times. There is no reason why the event a can not happen at many different times. In general na(*)na(0 o. (2.12) This is also true of classical distributions. For example, I can only be at one place at one time, but I can be at that same place at many different times. P rope r ty 3 The probabilities Pj(t), are normalized at a given time. i.e.. i . However, the operator J a is not necessarily normalized in time /•oo fip (f) / dt—IT1 = , l i m - (2-14) J-oo at t->oo . In special physical circumstances, Pa(t) may initially be zero, and may finally equal one in the distant future, but there is no reason to expect this to be true in general. One might try to renormalize J a , but for each initial state the normalization will in general be different. This property is also true of classical probability distributions. They also must be normalized, and the classical current is not always normalizable. For example, one can have many classical situations in which the event may never occur. The quantum case is more complicated however, since currents which are classically positive definate may be negative in the quantum case [29]. Lastly, P rope r ty 4 The probabilities Pi(t) are positive definite. In general, pa(t) can be negative since P(t) need not be monotonically increasing with time (this is obviously also true for classical probability functions). One can restrict J a Chapter 2. When does an Event Occur 21 to only act on states for which P a is increasing with time, but the restricted domain of definition of Ja may mean that it will no longer be self-adjoint. Furthermore, whether J a is positive or negative will not only depend on the state, but also on the Hamiltonian. For certain Hamiltonians, one may find that there are no states for which pa(t) does not take on negative values. Another interesting aspect of J a and I I a is that in general The operator which measures that the event happened and the operator J a do not com-mute. If one believes that Ja can be used to answer the question "when did the event happen?" then one finds that "when did it happen?" and "has it already happened?" seem to be complimentary (in Bohr's sense) in that they interfere with each other. Naive-ly, it would seem that determining "when did a occur?" would also answer the question "has a occurred?". However the inaccuracy of the determination of "when did a occur?" seems to place limits on our ability to answer "has a occurred?". 2.3 T i m e of a Measurement or A r r i v a l We now examine two specific examples of the determination of when an event occurred. In the first example, one tries to determine when a measurement occurred. In the second example, one wishes to determine the time at which a particle arrives to x = 0 Let us try to measure the time that a measurement occurred (a measurement of a measurement in a sense). Imagine that we want to find out the time of a measurement of the observable A of a quantum system S. The measurement of A . can be accomplished by coupling a macroscopic apparatus O to the system, via an interaction such as '3a(t),Ua(t)] ^ 0 . (2.15) H = g(t)PA (2.16) Chapter 2. When does an Event Occur 22 where P is the conjugate momentum to the pointer Q of the measuring device, and g(t) is a function which is zero everywhere, except during a small interval of time. After the measurement is complete, the measuring apparatus will be correlated with the state of the system. If initially, S is in a superposition of eigenstates \4>,) of the observable A , so that \ips) = J2iCi\<f>i), then we expect the initial state of the combined S — O system to evolve into a correlated state. £ c , - | & ) ® \0) - £> |<fc) <g> \Ot) (2.17) i i where \0) is the original state of the device and the |0,) are orthogonal states of the measuring apparatus which are correlated with the system. If the coupling is small, then the duration of the measurement might need to be long in order to distinguish between the various eigenvalues of A . At any time during the measurement, it is possible to calculate the density matrix of the combined S-0 system. One can imagine that a second apparatus O' measures the state of the first apparatus O to determine whether a measurement has occurred. This has been studied for the case when the measurement is gradual [21] [27]. Rovelli [28] has recently proposed that the apparatus O' might measure the operator M = J2\<t>l)®\Ot)(Ol\®{dl\. (2-18) i This is a projector onto the space of states where a correlation exists between the mea-suring apparatus and the quantum system. In the measurement scheme proposed by R,ovelli, the probability that a measurement has been made at time t is given in the Heisenberg representation by PM(t) = {il>so\M(t)\ipso) (2.19) where \ipso) is the state of the combined S-0 system. The operator which is defined to give the probability that a measurement was made between times t and t + dt is m(t) = —±1. (2.20) Chapter 2. When does an Event Occur 23 In the case of time-of-arrival, one wishes to measure the time a particle arrives to a certain location (say x = 0). Often, the probability current is used to determine the arrival time[13]. One imagines that a particle is localized in the region x < 0 and traveling towards the origin. The projector / ' O O 11+ = / dx\x)(x\ (2.21) Jo is an operator which is equal to one when x > 0 and zero otherwise. The probability of detecting the particle in the positive x-axis is given by P+(t) = (m+(tM- (2.22) In the Schroclinger representation, this expression is just P+(t) = J0°°\ip(x, t)\2dx. It is then claimed that the current J + , given by dJ+ _ dn+(t) dx - dt { Z - Z 6 ) will give the probability that the particle arrives between t and t + dt. It is clear that both the operators M ( i ) and TI+(t) are specific example of the operator TIa discussed in Section 2.2. M ( i ) gives the probability at time t that a measurement has occurred. U+(t) gives the probability that the particle is found at x > 0 at time t. The two operators m(£) and are examples of 3a(t). ui(t) gives the change in the probability that the measurement happened at time t, while 9 3 g i v e s the change in the probability that the particle is found at x > 0. However, one cannot interpret these operators as giving the probability that the measurement occurred (or the particle arrived). None of these operators allow one to measure the precise time at which the event occurred. They do not posses all the Properties 1-4 listed above. Considered as probabilities in time, none of the operators above give distributions which have Property 1. The operators above do not commute with the Hamiltonian, and Chapter 2. When does an Event Occur 24 therefore depend on t. For t — t' <C dH we have for any operator A(t) ~ A(t') + i(t — «')[H,A(«')], and so [A(t), A(t')) ~ i(t - t')[H, A(t')], A(t')] (2.24) For arbitrary Hamiltonians, it is obvious that none of the operators above will commute with themselves at different times. Even for a free particle, one can explicitly calculate that neither 9 3 n o r H+(t) commute with themselves at different times, (the calculation is neither difficult, nor particularly illuminating). For some very specific states, and physical situations, Properties 2-4 may be obeyed, but this is certainly not true in general. For the case of time-of-arrival, even for a free Hamiltonian and wave packets which only contain modes of positive frequency, the current can be negative [29] (a violation of Property 4 - that probabilities must be positive definite). In fact, since the current is simply the time derivative of a projection operator, there is no reason to expect it to always be positive. For free particles which can arrive from the left and right, the current can be zero 2 and hence the probability distribution will be unnormalizable (Property 3). Also, in general, there is no reason why a particle can't be at the same position at many different times (a violation of Property 2). In the case of particles which move in a potential, one may find that there are no states for which Properties 3-4 are obeyed. For example, if there is an infinite potential barrier around the origin, the particle will never arrive, and the current will not be normalizable, and if there is a harmonic oscillator potential, the particle will cross the origin many times from both the left and the right violating Properties 2 and 4. For the case of determining when a measurement occurred, Rovelli restricts the class of measurements he considers to be those for which m(t) obeys Properties 2-4. As a result, m(t) cannot be used for arbitrary measurements. As with the time-of-arrival, there are clearly many Hamiltonians 2See Appendix A where we see that a coherent antisymmetric superposition of left and right moving waves has zero current. Chapter 2. When does an Event Occur 25 for which Property 2-4 will be violated. Nor can m(t) be used for Hamiltonians for which its restricted domain of definition will mean that it is no longer self-adjoint. Although operators such as m and and J + do not commute with themselves at differ-ent times, it is possible to construct an operator which is time-translation invariant, and would give the time of an event in the classical limit. This will be discussed in Chapter 4 where we shall see that such an operator cannot be self-adjoint if the Hamiltonian is bounded from above or below. 2.4 Cont inua l Event M o n i t o r i n g Instead of considering operators, a more physical meaningful method of measuring the occurrence of an event is to consider continuous measurement processes. For example, the operator Jla(t) can be measured continuously or at small time intervals. When one considers such a physical measurement procedure one can see that the time at which an event occurs is not well defined in quantum mechanics. The probability of finding that the system enters one of the states d>i at time ta is given by the probability that it isn't in any of the states & before ta, times the probability that it is in one of the states <pi at To see how such a scheme might work, let us see how one would measure the time of an occurrence of the event corresponding to I I A . A measurement of the operator Ha(t) will tell us whether the event has occurred at time t. We can then measure Ha(t) at times tk = kA for integral k in order to determine when the measurement occurred. A represents the frequency with which we monitor the system, and is therefore the inaccuracy of the measurement in time (it is the coarseness of the measurement in some sense). We will now work in the Schrodinger representation, simply because it is the most Chapter 2. When does an Event Occur 26 natural arena to talk about successive measurements on a system. At time ti, the prob-ability that an event has occurred is given by P(th) = (V>o(0)|UAtnaUA|^o(0)> (2.25) and the probability that it hasn't is P ( i , « i ) = <MO ) |U A t ( l -n a )U A |Vo (0)> (2.26) where j corresponds to detecting that the event has occurred, j corresponds to detecting that an event has not yet occurred, V'o(u) is the initial state of the system and U A is.the evolution operator e~ j H A . If the result is | , we collapse the wave function and evolve it to the next instant. The normalized state before the second measurement is: The probability that an event has occurred at t% is given by the probability that an event didn't occur at ti times the probability that ^ 2 is in one of the states c6,-P ( t t \ - ( ^ Q I U A ^ I - n a ) u A t n a u A ( i - n a ) u A | ^ 0 ) t (^o |U A T ( l - n a )U A |^ 0 ) (2.28) The probability that an event didn't occur is given by P(lt2) = ( ^ o i u ^ i -n a)u At(i - n a ) u A ( i -na)uA|vP) ( 2 . 2 9 ) By repeating this process, we find that at time tk the probability that an event has occurred is given by P{ttk) = (^o\Ak\^0)_ (2.30) where Ak = uAt(i - n a )u A t ( i - n a ) . . .u A tn a u A . . . ( i - n a ) u A ( i - n Q ) u A ( 2 . 3 1 ) Chapter 2. When does an Event Occur 27 and the probability that an event hasn't occurred is P ( U k ) = {ipo\Bk\^o) (2.32) with Bk = uAt(i - n a )u A t(i - n a ) . . .u A t ( i - n a ) u A . . . ( i - n a ) u A ( i - n a ) u A ( 2 . 3 3 ) By allowing the unitary operators to act on the projection operators we can write the Ak or Bk in the Heisenberg representation. For example Ak = {i- n a)(ti)...(i - n a ) (« f c _i )n a (« f c ) ( i - na)(* f c_i)...(i - n a ) ( t i ) ( 2 . 3 4 ) However, while the operators Ua{i) can be found by unitary time-evolution of I I a ( 0 ) , the operators Ak and Bk are not related by a unitary transformation to A0 and BQ. This already signals that they can not give an undisturbed distribution for the time of an event. Nor are the Ak and Bk projection operators. The probabilities derived from Ak and Bk are not universal. In this case, they apply only to the specific measurement scenario under discussion. In particular the probability distribution is sensitive to the frequency at which I I A is measured, a phenomenon which is related to the Zeno paradox [23]. As an example, consider a measurement of the spin of a particle. We wish to know at what time the measurement occurred. The particle is in a state given by and we use a simple measuring device which is also a spin 1/2 particle initially in the state \0) = | T')> which evolves according to the Hamiltonian m = a\ T) + 6| I) (2.35) (2.36) Chapter 2. When does an Event Occur 28 where / g(t)dt = 7r (g(t) is sharply peaked, with width T), and the primed Pauli matrix acts on the measuring device, while the unprimed Pauli matrix acts on the system. After a time T, the spin of the measuring device will be correlated with the system. Since this measurement is rather crude, (the initial state of the device is the same as one of the measurement states), the operator M at t = 0 is not zero. Let us simplify the problem further, by assuming that a = 0 and 6 = 1 . In this case, the only relevant matrix element of 1 - M is | | ) ® | T 'XT' I ® (I I = iVOfaM- We then find the probability that the measuring apparatus has not responded at time tk is P(Utk) = \^0\u\\4>0)\2k ~ | ( V o | l - * A H - A 2 H 2 | ^ ) | 2 f c ~ 1 - A 2 f c ( ( H 2 ) - (H) 2 ) f c (2.37) If we fix a value of r = tk and then make A go to zero, we find P ( j , r ) ~ E - ( W ' * ~ 1, (2.38) which implies that the measuring apparatus becomes frozen and never records a mea-surement. In order not to freeze the apparatus, we need A > 1/dE where dE is the uncertainty in energy of the measuring device O (initially the spacing between energy levels in this case). There is always an inherent inaccuracy when measuring the time that the event (of the measurement) occurred. This inaccuracy is similar to the one which we will find in Chapter 3. Note that as discussed in the Introduction, this inaccuracy is not related to the so-called "Heisenberg energy-time uncertainty relationship" as it applies to every single measurement and not to the width of measurements carried out on an ensemble. Chapter 2. When does an Event Occur 29 One can of course use the set of operators Ak to compute a probability distribution in time, or experimentally determine a probability distribution for the time of an event. However, as we have just seen, this probability distribution is not a function of the system alone, but rather, it is related to the system and the measuring device (or set of operators) For example, the probability distribution will depend on A , and if A is too small, we will find that the event never occurs. The distribution P(f,tfc) does allow as to predict the probabilities of future measurements using a particular measuring device, but the results are not attributes of the system. Chapter 3 Phys ica l Clocks and T ime-of -Ar r iva l 30 Chapter 3. Physical Clocks and Time-of-Arrival 31 3.1 A L i m i t a t i o n on Time-of -Arr iva l Measurements In the previous Chapter, we saw that if we attempt to measure the time of an event using a rapid series of measurements, then our measurement will disturb the very thing which we are trying to measure. For the simple case of a two state system, we were able to show that if we make the measurement accurately enough, the system freezes, and the event never occurs. In this Chapter, we study measurements of the time-of-arrival of a particle to a particular location using physical clocks. The clocks are coupled to the system in such a way that when the particle arrives to the fixed location, the clock will read the time-of-arrival. Unlike the continuous measurement procedure discussed in the last Chapter, we obtain the time of the event using only a single measurement which is made well after the event has occurred. Nonetheless, we will find that if we make the measurement extremely accurate, the measurement will fail. Consider a free particle, upon which a measurement is performed to determine the time-of-arrival to x — XA- The time-of-arrival can be recorded by a clock situated at x = XA which switches off when the particle reaches it. In classical mechanics we could, in principle, achieve this with the smallest non-vanishing interaction between the particle and the clock, and hence measure the time-of-arrival with arbitrary accuracy. In classical mechanics there are other indirect methods to measure the time-of-arrival. One could invert the equation of motion of the particle and obtain the time in terms of the location and momentum, TA{x(t),p(t),XA)- This function can be determined at any time £, either by a simultaneous measurement of x(t) and p(t) and evaluation of T A , or by a direct coupling to TA{x(t),p(t), XA)- One could also measure the time-of-arrival using the method discussed in the previous chapter. By using a weak interaction which doesn't disturb the system, one can continually monitor the point of arrival to see if the particle Chapter 3. Physical Clocks and Time-of-Arrival 32 has arrived. These different methods, namely, the direct measurement, indirect measurement, and continual monitoring are classically equivalent. They give rise to the same classical time-of-arrival. They are not equivalent however, in quantum mechanics In quantum mechanics the corresponding operator T A ( X ( < ) , p(£), %A), if well defined, can in principle be measured to any accuracy. On the other hand, a direct measurement cannot determine the time-of-arrival of free particles to arbitrary accuracy (as was origi-nally argued by Allcock[15]). In Section 3.3.2, we argue that Allcock's arguments are not sufficient to limit the accuracy of time-of-arrival measurements. One needs to consider models with physical clocks. Using these models, we shall argue that the accuracy of time-of-arrival measurements cannot be better than 6tA > 1/Ek, (3.39) where Ek is the initial kinetic energy of the particle. The basic reason is that, unlike a classical clock, in quantum mechanics the uncertainty in the clock's energy grows when its accuracy improves [24]. We find that particles with initial kinetic energy Ek are reflected without switching off a clock if this clock is set to record the time-of-arrival with accuracy better than in eq. (3.39). Furthermore, for the small fraction of the ensemble that does manage to turn off the clock, the resulting probability distribution becomes distorted. A detailed discussion of direct time-of-arrival measurements will be discussed in Chapter 4. We conclude in Section 3.3.5 with a discussion on why we expect eq. (3.39) to hold in general and we discuss the main results. An explicit calculation of the clock's final probability distribution is given in Appendix B. Chapter 3. Physical Clocks and Time-of-Arrival 33 3.2 Free Clocks We will now attempt to make a measurement of the time-of-arrival. In order to do so, we will need a clock. A n ideal clock is linear in time. I.e., the position of the clock's pointer should be proportional to the time t. It is not hard to see that an ideal clock can be represented by the Hamiltonian Kclock = Py • (3.40) To read the time of the clock, we measure the coordinate y conjugate to P y . Using the Heisenberg equations of motion we see that the variable y reads the correct parameter time t found in the Schrodinger equation. y(*)-y(*o) = -i [{y,Hciock}dt = t-t0 (3.41) The Hamiltonian for this clock is unbounded from above and below, nonetheless, using a sufficiently massive particle, we can approximate the ideal situation to arbitrary accuracy 1 . We write p = (p)+<5p, and note that if p >^ <5p, the free particle Hamiltonian is given by P 2 H = 2m = ^ + + 0(*>2/m) . (3.42) 2m m Since the first term is constant and we can ignore the higher order term, we see that the Hamiltonian is approximately linear in momentum. The constant of proportionality is (p)/m which we set to one for convenience. The corrections to equation (3.41) are also of order <5p/p. x One could also consider a Larmor clock with a bounded Hamiltonian Hciock = wJ [16]. The Hilbert space is spanned by 2j+l vectors where j is a natural number, and the clock's resolution can be made arbitrarily fine by increasing j . Chapter 3. Physical Clocks and Time-of-Arrival 34 From (3.41) we see that in order to use this clock to read the time, we need to know the initial position of the clock's dial y(to) and then subtract this from our final reading of y. Quantum mechanics puts no limitation on how accurately this clock can be measured. If we want to accurately infer the time from the final reading of the clock then the clock must initially be prepared in a state with a very small uncertainty in y. At some later point, we can measures the coordinate y(tf) to any degree of accuracy we wish to infer the time from y(tf) — y(to). If initially dy was very small, then we know that the time is given by the final reading of y. However, if initially the state of the clock had a large spread in y, then the time we finally obtain will be inaccurate by an amount dy'. This means that for this clock, the inaccuracy in the time measurement is given by 6T = dy (3.43) If we simply want to use this clock to read the time, then there are no restrictions on how accurate the clock can be. So far, nothing prevents us from making the initial state of the clock's pointer as close to an eigenstate of y(to) as we desire. However, since y(to) and Udock do not commute (and cannot commute if the clock is to operate properly), the smaller the uncertainty in y(to), the greater the uncertainty in Hci0ck- We will see that if we want to use this clock to measure the time of an event, then we will encounter the limitation given by (3.39). We will need to ensure that initially the position of the clock is uncertain in order for our measurements of the time of an event to succeed. The reason for this is that since y is conjugate to Hc/ocfc = P y , accurate clocks (which are narrow in y) have a large spread in Py. This means that in general the energy of an accurate clock can take on fairly large values. For an infinitely accurate clock the energy will almost always be infinite. Accurate clocks therefore, have a large energy uncertainty, and this makes them very hard to use to measure the time of an event. This is because accurate clocks are usually so energetic that they need a large amount Chapter 3. Physical Clocks and Time-of-Arrival 35 of energy to turn them off. To measure the time-of-arrival of a particle, the particle itself will have to turn off the clock when it arrives - the external observer cannot supply any energy since she does not know when to turn the clock off. If the clock is much more energetic than the particle, then it will be impossible for the particle to turn off the clock and no record will exist that the particle arrived. In fact, the particle may be reflected and will never arrive. The situation is very different from usual measurements, where an external observer supplies the energy to make the measurement. If we were to measure the time of the clock, we would have to supply a large amount of energy to make an accurate measurement. However, in order to measure the time-of-arrival, it is the particle which must supply the energy to stop the clock. To see this, let us use the clock to measure the time-of-arrival. 3.3 Measurement of T ime-of -Arr iva l In this section we consider toy models of a measurement of time-of-arrival. To begin with, assume that the particle interacts with a detector that is located at x = 0 and is coupled to a clock. Initially, as the particle is prepared, the clock is set to show t — 0. Our purpose is to design a particular set-up such that as a particle crosses the point x = 0 the detector stops the clock. Since the masses of the particle detector and the clock are unlimited we can ignore the uncertainty in the position of the measurement device and assume it is properly positioned at x = 0. We shall consider four models. The first model describes a direct interaction of the particle with the clock. In the second model, the particle is detected by a two-level detector, which turns the clock off. To avoid the reflection due to the clock's energy, we look next at the possibility of boosting the energy of the particle in order to turn off the clock. We shall also consider the case of a "smeared" interaction, and conclude with a general discussion. Chapter 3. Physical Clocks and Time-of-Arrival 36 3.3.1 Measurement w i t h a clock The simplest model which describes a direct interaction of a particle and a clock [16], without additional "detector" degrees of freedom, is described by the Hamiltonian # = ^ P x 2 + 0 ( - x ) P y . (3.44) Here, the particle's motion is confined to one spatial dimension, x, and 9(x) is a step function. The clock's Hamiltonian is represented by P y , and the time is recorded on the conjugate variable y. Because of the step function, the clock will stop when the particle is located to the right of the origin. The equations of motion read: x = P x / m , P x = -P y <Kx) (3.45) y = 0(x), P y = 0. (3.46) At t —» oo the clock shows the time of arrival: Yoo = y(*o) + / 6(-x(t))dt (3.47) J to A crucial difference between the classical and the quantum case, can be noted from Equation (3.45). In the classical case the back-reaction can be made negligible small by choosing Py —»• 0. In this case, the particle follows the undisturbed solution, x(t) — x(to) + ^ (t — to). If initial we set y(to) = to a n d x(to) < 0 the clock finally reads: yoo = y(*o)+ re[~x(t0)-^(t-t0)]dt = - ^ ^ - . (3.48) Jt0 m px The classical time-of-arrival is = y<x> = —mx(to)/px. The same result would have been obtained by measuring the classical variable —mx0/px = —mx(t)/px + (t — to), at arbitrary time t. Consequently, the continuous and the indirect measurements alluded to in Section 3.1, are classically equivalent. Chapter 3. Physical Clocks and Time-of-Arrival 37 On the other hand, in quantum mechanics the uncertainty relation dictates a strong back-reaction, i.e. in the limit of A y — At A —> 0, py in (3.45) must have a large uncer-tainty, and the state of the particle must be strongly affected by the act of measuring. Therefore, the two classically equivalent measurements become inequivalent in quantum mechanics. Before we proceed to examine the continuous measurement process in more detail, we note that a more symmetric formulation of the above measurement exists in which knowledge of the direction from which particles are arriving is not needed. We can consider As before, the particle's motion is confined to one spatial dimension, x. Two clocks are represented by P y i and P y 2 , and time is recorded on the conjugate variables y i and y 2 , respectively. The first clock operates only when the particle is located at x < 0 and the second clock at x > 0. For example, if we start with a beam of particle at x < 0, a measurement at t —> oo of y i gives the time-of-arrival. Alternatively we could measure t — y%- As a check we have y i + y 2 = t. It is harder to determine the time-of-arrival if the particle arrives from both directions. If however it is known that initially | . T | < L, we can measure y i and y 2 after t » L/v. The time-of-arrival will then be given by tA = min(y i ,y 2 ) . For simplicity we shall examine in more detail the case of only one clock and a particle initially at x < 0, which travels towards the clock at x = 0. The eigenstates of the Hamiltonian are where k and p, are the momentum of the particle and the clock, respectively, and u(t) = = - l - P x 2 + 0 ( - x ) P V l + 6<(x)Py2. (3.49) Chapter 3. Physical Clocks and Time-of-Arrival 38 |^ + pt. Continuity of <f>kP requires that 2k k + q k — q , A, = ^ (3.51) where q — \Jk2 + 2mp = ^2m(Eic + p). The solution of the Schrodinger equation is /oo /*oo dk dpf{p)g{k)<bkp{x,y,t), • (3.52) -oo JO where N is a normalization constant and f(p). and g(k) are some distributions. For example, with f(p) = e-^2(P-Po)2 g(k) ' = e-Ax2(fc-fc0)2+iteo_ (353) and .To > 0, the particle is initially localized on the left (.r < 0) and the clock (with probability close to 1) runs. The normalization in eq. (3.52) is thus N2 — 4^r*. By choosing po ~ l/Ay, we can now set the the clock's energy in the range 0 < p < 2/ Ay. Let us first show that in the stationary point approximation the clock's final wave function is indeed centered around the classical time-of-arrival. Thus we assume that Ay and are large such that f(p) and g(k) are sufficiently peaked. For x > 0, the integrand in (3.52) has an imaginary phase k2t 9 — qx + kx0 +py — pt. (3.54) 2m ^ | = 0 implies and ^ = 0 gives W r i = - ^ * . + ^ , (3.55) kn m mx ypeak(k) = t - . (3.56) Qo Chapter 3. Physical Clocks and Time-of-Arrival 39 Hence at x = xpeak the clock coordinate y is peaked at the classical time-of-arrival mxQ , y = 1 - . (3.57) To see that the clock yields a reasonable record of the time-of-arrival, let us consider further the probability distribution of the clock p{y,y)*>o = f dx\tl){x > 0 ,y , i ) | 2 . (3.58) In the case of inaccurate measurements with a small back-reaction on the particle A T — 1. The clocks density matrix is then found (see Appendix B) to be given by: p(y,y)>o^ j e (3.59) V 2 7 r K? / ) where the width is j(y) = Ay2 + ( I Z ^ £ ) 2 + ( 2 f c y A a .) 2 - As expected, the distribution is centered around the classical time-of-arrival tc — x0m/k0. The spread in y has a term clue to the initial width A y in clock position y. The second and third term in j(y) is clue to the kinematic spread in the time-of-arrival JJ^ — and is given by d x ^ m where dx(y)2 — Ax2 + (2niAx)2- T n e J dependence in the width in x arises because the wave function is spreading as time increases, so that at later y, the wave packet is wider. As a result, the distribution differs slightly from a Gaussian although this effect is suppressed for particles with larger mass. When the back-reaction causes a small disturbance to the particle, the clock records the time-of-arrival. What happens when we wish to make more accurate measurements? Consider the exact transition probability T = ^ \ A T \ 2 , which also determines the proba-bility to stop the clock. The latter is given by , 2 (3.60) T = lEk + P 2\fEl, r W k + VW+P Chapter 3. Physical Clocks and Time-of-Arrival 40 Since the possible values obtained by p are of the order 1/Ay = 1/At A , the probability to trigger the clock remains of order one only if EkStA > 1. (3.61) Here StA stands for the initial uncertainty in position of the dial y of the clock, and is interpreted as the accuracy of the clock. Ek can be taken as the typical initial kinetic energy of the particle. In measurements with accuracy better then 1/Ek the probability to succeed drops to zero like y/EkSt-A, and the time-of-arrival of most of the particles cannot be detected. Furthermore, the probability distribution of the fraction which has been detected depends on the accuracy StA and can become distorted with increased accuracy. This observation becomes apparent in the following simple example. Consider an initial wave packet that is composed of a superposition of two Gaussians centered around k — k\ and k = k2 » k\. Let the classical time-of-arrival of the two Gaussians be t\ and t2 respectively. When the inequality (3.61) is satisfied, two peaks around t\ and t2 will show up in the final probability distribution. On the other hand, for ^ > StA > f r , the time-of-arrival of the less energetic peak will contribute less to the distribution in y, because it is less likely to trigger the clock. Thus, the peak at ti will be suppressed. Clearly, when the precision is finer than 1 /Ek we shall obtain a distribution which is considerably different from that obtained for the case StA > 1/Ek when the two peaks contribute equally. 3.3.2 Two-level detector w i t h a clock A more realistic set-up for a time-of-arrival measurement is one that also includes a particle detector which switches the clock off as the particle arrives. We shall describe the particle detector as a two-level spin degree of freedom. The particle will flip the state of the trigger from "on" to "off", i.e.. from ] z to [ z . First let us consider a model for the Chapter 3. Physical Clocks and Time-of-Arrival 41 trigger without including the clock: Htrigger = ^-Pl + ^(l + ox)6(x). (3.62) The particle interacts with the repulsive Dirac delta function potential at x = 0, only if the spin is in the | J.?} state, or with a vanishing potential if the state is | lx). In the limit a —> 0 0 the potential becomes totally reflective (Alternatively, one could have considered a barrier of height a2 and width 1/a.) In this limit, consider a state of an incoming particle and the trigger in the "on" state: |^ >)| 1%). This state evolves to 1 | < M I U + I < M I D (3.63) where ipR, and ipr are the reflected and transmitted wave functions of the particle, re-spectively. The latter equation can be rewritten as \\ W(\1>R) + \M) + \\ 1,){\1>R) ~ \rh)) (3-64) Since ] z denotes the "on" state of the trigger, and [ z denotes the "off" state, we have nipped the trigger from the "on" state to the "off" state with probability 1/2 2 . Although this model only works half the time, the chance of success does not depend in any way on the system, and in particular, on the particle's energy. Furthermore, one can construct models where a detector is triggered almost all the time [36], although with some energy dependence in the probability of triggering. So far we have succeeded in recording the event of arrival to a point. We have no infor-mation at all on the time-of-arrival. It is also worth noting that the net energy exchange between the trigger and the particle is zero, i.e.. the particle's energy is unchanged. 2 I t is interesting to see that for some wave functions which represent coherent superpositions of particles arriving from both the left and right, the detector is never triggered. A n example of this is given in Appendix A . Chapter 3. Physical Clocks and Time-of-Arrival 42 This model leads us to reject the arguments of Allcock. He considers a detector which is represented by a pure imaginary absorber Hint = iV9( —x). Allcock's claim is that measuring the time-of-arrival is equivalent to absorbing a particle in a finite region. If you can absorb the particle in an arbitrarily short time, then you have succeeded in transferring the particle from an incident channel into a detector channel and the time-of-arrival can then be recorded. Using his interaction Hamiltonian one finds that the particle is absorbed in a rate proportional to V~l. One can increases the rate of absorption by increasing V , but the particle will be reflected unless V « Ek. He therefore claims that since you cannot absorb the particle in an arbitrarily short time, you cannot record the time-of-arrival with arbitrary accuracy. However, our two level detector is equivalent to a detector which absorbs a particle in an arbitrarily short period of time, and then transfers the information to another channel. The particle is instantaneously converted from one kind of particle (spin up), to another kind of particle (spin down). We therefore see that considerations of absorption alone do not place any restrictions on measuring the time-of-arrival. However, we shall see that when we proceed to couple the trigger to a clock we do find a limitation on the time-of-arrival. A model for this coupling can be given by the Hamiltonian Htrigger+dock = + ^(1 + ^x)^(x) + ^(1 + ( X 2 ) P y . (3.65) Since we can have a » Py it would seem that the triggering mechanism need not be affected by the clock. If the final wave function includes a non-vanishing amplitude of [ z , the clock will be turned off and the time-of-arrival recorded. However, the exact solution shows that this is not the case. Consider for example an initial state of an incoming wave from the left and the spin in the ] z state. Chapter 3. Physical Clocks and Time-of-Arrival 43 The eigenstates of the Hamiltonian in the basis of crz are / V * t r + d>Lie-<kix\ *L(x) = ' e™, (3.66) V (pLle-lk\x J for x < 0 and *R(X) = . t e™, (3.67) for x > 0. Here fcT = \j2m(E - p) = y/2mEk and /cL = \j2mE = ^2m(Ek +p). Matching conditions at x — 0 yields 2fc| fc, , m o fcj — ; T T 7 7 — - 1 + ? m a v fcj ' fc and (3.68) ^ ^ - ( i + fc1) 0L.i = ^ i (3.70) We find that in the limit a —> oo the transmitted amplitude is * « = - * « = ^ . n v k + p - < 3 ' 7 2 ) Precisely as in the previous section, the transition probability decays like \JEkjp. From eqs. (3.70,3.71) we get that 4>L[ —> 0, a n d § L \ —> 1 as the accuracy of the clock increases. Hence the particle is mostly reflected back and the spin remains in the \ z state: i.e., the clock remains in the "on" state. The present model gives rise to the same difficulty as the previous model. Without the clock, we can flip the "trigger" spin by means of a localized interaction, but when we couple the particle to the clock, the probability to flip the spin and turn the clock off decreases gradually to zero as the clock's precision is improved. Chapter 3. Physical Clocks and Time-of-Arrival 44 3.3.3 L o c a l amplification of kinetic energy The difficultly with the previous examples seems to be that the particle's kinetic energy is not sufficiently large, and energy can not be exchanged with the clock. To overcome this difficulty one can imagine introducing a "pre-booster" device just before the particle arrives at the clock. If it could boost the particle's kinetic energy to an arbitrarily high value, without distorting the incoming probability distribution (i.e. amplifying all wave components k equally), and at an arbitrary short distance from the clock, then the time-of-arrival could be measured to arbitrary accuracy. Thus, an equivalent problem is: can we boost the energy of a particle by using only localized (time independent) interactions? Let us consider the following toy model of an energy booster described by the Hamil-tonian Here, a, W, V\ and V2 are positive constants. Let us consider an incoming wave packet propagating from left to right. The role of the term aox6(x) is to flip the spin f z to [ z . The V2 term is the booster, and particles which cross into the region x > 0 will be boosted in kinetic energy by the amount V2. The other parts of the potential serve to damp out undesirable components of the wave function which can interfere with each other if a clock is placed close to the origin. The ] z component of the wave function is damped out exponential by the W term for x > 0. The [ z component is clamped out for x < 0 by the term V\. As we shall see, for a given momentum one can chose the four free parameters above such that the wave is transmitted through the booster with probability 1, while the gain in energy V2 can be made arbitrarily large. The potential barrier W can also be made arbitrarily large. The last requirement means that the unfiipped component decays for x > 0 on arbitrary short scales, which allows us to locate the booster arbitrarily close to the clock, while preventing destructive interference between the nipped and un-flipped H + aax6(x) + y 0 ( x ) ( l + az) + ^ ( - x ) - V 2 0(x) ] ( l (3.73) Chapter 3. Physical Clocks and Time-of-Arrival 45 transmitted waves. The eigenstates of (3.73), in the basis of cr2, are given by ( elkx + (j)L]e-ikx\ * L ( * ) = (3.74) V <t>Lieqx ) for x < 0 and (<t>me-Xx\ = . v } (3-75) for x > 0, where fc2 = 14 — g 2 = — A 2 + W = — V2 + k'2. Matching conditions at x = 0 we find fc'/c + (/A + i(kq - k'X) - a2 k'k- q\-ri{k'\ + kq) + a2' h\ = 4>E;\ ~ 1 = T 7 T — , , , , , o , (3-76) •^1 - = ^ = i]fcr-^(1 + ^ t ) - (3-77) For a given fc, and V2 (or given k, A and fc') we still are free to chose a and V\ (or q). We now demand that a = k'k + q\, q = A-jk (3.78) ft With this choice we obtain for the transmission and reflection probabilities: i2T = 0/ T x = ^ l ^ i l 2 = 1- (3-79) Therefore, the wave has been fully transmitted and the spin has nipped with probability 1. So far we have considered an incoming wave with fixed momentum k. For a general incoming wave packet only a part of the wave will be transmitted and amplified. Fur-thermore one can verify that the amplified transmitted wave has a different form than the original wave function since different momenta have different probabilities of being amplified. Thus, in general, although amplification is possible and indeed will lead to a much higher rate of detection, it will give rise to a distorted probability distribution for the time-of-arrival. Chapter 3. Physical Clocks and Time-of-Arrival 46 There is however one limiting case in which the method does seem to succeed. Consid-er a narrow wave peaked around k with a width dk. To first order in dk, the probability T] that the particle is successfully boosted is given by Tj. ~ 1 - — . (3.80) Therefore in the special case that * << 1, the transition probability is still close to one. If in this case we known in advance the value of k up to dk « k, we can indeed use the booster to improve the bound (3.61) on the accuracy. The reason why this seems to work in this limiting case is as follows. The probability of flipping the particle's spin depends on how long it spends in the magnetic field described by the a term in (3.73). If however, we know beforehand, how long the particle will be in this field, then we can tune the strength of the magnetic field (a) so that the spin gets flipped. The requirement that dk/k << 1 is thus equivalent to having a small uncertainty in the "interaction time" with this field. In some sense, the measurement is possible, because we know the particle's momentum before hand. Of course, if we have prior knowledge of the particle's momentum, then we could just measure x and argue that this allows us to calculate the time of arrival through tA = mx/p. We therefore believe that the reason the measurement procedure described above works in this limiting case is because it assumes prior knowledge of the particle's momentum, and we do not believe that one could improve it so that it works for all states. These "booster" measurements cannot be used for general wave functions, and even in the special case above, one still requires some prior information of the incoming wave function. 3.3.4 G r a d u a l tr iggering of the clock In order to avoid the reflection found in the previous two models, we shall now replace the sharp step-function interaction between the clock and particle by a more gradual Chapter 3. Physical Clocks and Time-of-Arrival 47 transition. When the W K B condition is satisfied d\(x) K = e « 1 (3.81) dx where A ( . T ) - 2 = 2m[Eo — V(x)], the reflection amplitude vanishes as ~ exp( - l /e 2 ) (3.82) Solving the equation for the potential with a given e we obtain "•<*» = * - 2 ^ ? ( 3 ' 8 3 ) Now we observe that any particle with E > EQ also satisfies the W K B condition (3.81) above for the same potential Ve. Furthermore pyVe also satisfies the condition for any py > 1. These considerations suggest that we should replace the Hamiltonian in eq. (3.49) with H = P x 2 / 2 m + V(x)Py (3.84) where V(x) = — £ X < XA s (3.85) — 1 x > XA Here xA2 = 2me2. Thus this model describes a gradual triggering on of the clock which takes place when the particles propagates from x —>• —oo towards x — xA. In this case the arrival time is approximately given by t — y where t = tf — Since without limiting the accuracy of the clock we can demand that py » 1, the reflection amplitude off the potential step is exponentially small for any initial kinetic energy E K . Chapter 3. Physical Clocks and Time-of-Arrival 48 The problem is however that the final value of t — y does not always correspond to the time-of-arrival since it contains errors due to the affect of the potential V(x) on the particle which we shall now proceed to examine. In the following we shall ignore ordering problems and solve for the classical equations of motion for (3.84). We have y(tf)-y(ti) = ft'v(x(t'))dt' (3.86) which can be decomposed to y(tf) - y(tt) = (ti - t 0) + (tf - tt) + r ° V(x(t'))dt' =A+B+C (3.87) where 1 A ^xl+pyxl/E - y/x?+pyXyE\ (3.88) is the time that the particle travels from Xi to XA in the potential pyV(x), B is the total time, and C = -1 + log " + log — y/2mpy L 1 , L , _Ufj_ xA. V PyxA (3.89) The last term C , corresponds to an error due to the imperfection of the clock, i.e. the motion of the clock prior to arrival to XA- By making py large we can minimize the error from this term to-'~ (XA \ogpy/y/2mpy) . Inspecting equation (3.87) we see that by measuring yf — yi and then subtracting B = tf — ti (which is measured by another clock) we can determine the time to — t,;, which is the time-of-arrival for a particle in a potential pyV(x), up to the correction C. However this time reflects the motion in the presence of an external (unknown) potential, while we are. interested in the time-of-arrival for a free particle. Nevertheless, if py/E « 1 we obtain - A = ^ Z3 + o(^) (3.90) V2nJE \EJ v Chapter 3. Physical Clocks and Time-of-Arrival 49 The time-of-arrival can hence be measured provided that EkSt » 1. On the other hand, when the detector's accuracy is St < l/E, the particle still triggers the clock. However the measured quantity, A, no longer correspond to the time-of-arrival. Again, we see that when we ask for too much accuracy, the particle is strongly disturbed and reading of the clock has nothing to do with the time-of-arrival of a free particle. 3.3.5 Genera l considerations We have examined several models for a measurement of time-of-arrival and found a limitation, St A > 1/Ek, (3.91) on the accuracy that tA can be measured. Is this limitation a general feature of quantum mechanics? First we should notice that eq. (3.91) does not seem to follow from the uncertainty principle. Unlike the uncertainty principle, whose origin is kinematic, (3.91) follows from the nature of the dynamic evolution of the system during a measurement. Furthermore here we are considering a restriction on the accuracy (not uncertainty) of a single mea-surement. While it is difficult to provide a general proof, in the following we shall indicate why (3.91) is expected to hold under more general circumstances. Let us examine the basic features that gave rise to (3.91). In the toy models considered in Sections 3.3.1 and 3.3.2, the clock and the particle had to exchange energy py ~ l/StA-As a result, the effective interaction by which the clock switches off, looks from the point of view of the particle like a step function potential. This led to "non-detection" when (3.91) was violated. Can we avoid this energy exchange between the particle and the clock? Let us try to deliver this energy to some other system without modifying the energy of the particle. Chapter 3. Physical Clocks and Time-of-Arrival 50 For example consider the following Hamiltonian for a clock with a reservoir: P 2 # = _2E_ + 8(-x)Hc + Hrea + Vres6(x) (3.92) The idea is that when the clock stops, it dumps its energy into the reservoir, which may include many other degrees of freedom, instead of delivering it to the particle. In this model, the particle is coupled directly to the clock and reservoir, however we could as well use the idea of Section 3.3.2 above. In this case: H = ^ - + ^ ( l + <7x)6{x) + ^(1 + <Tz)Hc + Hres + hi - az)Vres. (3.93) 2m 2 2 2 The particle detector has the role of providing a coupling between the clock and reservoir. Now we notice that in order to transfer the clock's energy to the reservoir without affecting the free particle, we must also prepare the clock and reservoir in an initial state that satisfies the condition Hc - Vres = 0 (3.94) However this condition does not commute with the clock time variable y. We can measure initially y — R, where R is a collective degree of freedom of the reservoir such that [R, Vres] = i, but in this case we shall not gain information on the time-of-arrival y since R is unknown. We therefore see that in the case of a sharp transition, i.e. for a localized interaction with the particle, one cannot avoid a shift in the particle's energy. The "non-triggering" (or reflection) effect cannot be avoided. We have also seen that the idea of boosting the particle "just before" it reaches the detector, fails in the general case. What happens in this case is that while the detection rate increase, one generally destroys the initial information stored in the incoming wave packet. Thus although higher accuracy measurements are now possible, they do not reflect directly the time-of-arrival of the initial wave packet. Chapter 3. Physical Clocks and Time-of-Arrival 51 Finally we note that in reality, measurements usually involve some type of cascade effect, which lead to signal amplification and finally allows a macroscopic clock to be triggered. A typical example of this type would be the photo-multiplier where an initially small energy is amplified gradually and finally detected. Precisely this type of process occurs also in the model of section 3.3.4. In this case the particle gains energy gradually by "rolling down" a smooth step function. It hence always triggers the clock. The basic problem with such a detector is that when (3.91) is violated, the "back reaction" of the detector on the particle, during the gradual detection, becomes large. The relation between the final record to the quantity we wanted to measure is lost. We have examined various models for the measurement of time-of-arrival, £4, and found a basic limitation on the accuracy that tA can be determined reliably: St A > 1/Ek-This limitation is quit different in origin from that due to the uncertainty principle; here it applies to a single quantity. Furthermore, unlike the kinematic nature of the uncertainty principle, in our case the limitation is essentially dynamical in its origin; it arises when the time-of-arrival is measured by means of a continuous interaction between the measuring device and the particle. Chapter 4 T ime-of -Arr iva l Operators 52 Chapter 4. Time-of-Arrival Operators 53 4.1 Indirect T ime-of -Ar r iva l Measurements In the previous chapter, we saw that one cannot measure the time-of-arrival of a free particle to arbitrary accuracy by coupling the particle to a clock. Still, one can imagine an indirect determination of arrival time by a measurement of some regularized time-of-arrival operator T(x(t),p(t),xA) [9]. In quantum mechanics, ordinary observables like position and momentum are represented by operators at a fixed time t. However, we will show that there is no operator associated with the time it takes for a particle to arrive to a fixed location. In Section 4.2 we will prove formally that in general a Hermitian time-of-arrival operator with a continuous spectrum can only exist for systems with an unbounded Hamiltonian. This is because the existence of a time-of-arrival operator requires the existence of a time operator which is conjugate to the Hamiltonian. As is argued in Section 4.3, since T can be measured with arbitrary accuracy it does not correspond to the result obtained by the direct measurement discussed in Chapter 3 . In Section 4.4 we show why the time-of-arrival operator for a free particle is not self-adjoint, and explore the possible modifications that can be made in order to make it self-adjoint. The idea is that by modifying the operator in a very small neighborhood around k = 0, one can formally construct a modified time-of-arrival operator which behaves in much the same way as the unmodified time-of-arrival operator. We then explore some of the properties of the modified time-of-arrival states. In Section 4.5 we examine normalizable states which are coherent superpositions of time-of-arrival eigenstates, and discuss the possibility of localizing these states at the location of arrival at the time-of-arrival. Our results for the "unmodified" part of the time-of-arrival state seem to agree with those of Muga, Leavens and Palao who have studied these states independently [30]. In Section 4.6 we show that in an eigenstate of the modified time-of-arrival operator, the particle, at the predicted time-of-arrival, is found far away from Chapter 4. Time-of-Arrival Operators 54 the point of arrival with probability 1/2. We also calculate the average energy of the states, in order to relate them to our proposal in Chapter 3 that one cannot measure the time-of-arrival to an accuracy better than l/Ek- We end with concluding remarks in Section 4.7. 4.2 Condi t ions on A Time-of -Arr iva l Operator As discussed in the previous section, although a direct measurement of the time-of-arrival may not be possible, one can still try to observe it indirectly by measuring some operator T (p ,x ,x A ) . In the next two sections we shall examine this operator and its relation to the continuous measurements described in the previous chapters. First in this section we show that an exact time-of-arrival operator cannot exist for systems with bounded Hamiltonian. To begin with, let us start with the assumption that the time-of-arrival is described, as other observables in quantum mechanics, by a Hermitian operator T . Here the subscript ) t denotes the time dependence of the eigenkets, and T may depend explicitly on time. Hence for example, the probability distribution for the time-of-arrival for the state is continuous and unbounded: — oo < tA < oo. Should T correspond to time-of-arrival it must satisfy the following obvious condition. T must be a constant of motion and in the Heisenberg representation T{t)\tA)t = tA\tA) (4.95) (4.96) will be given by prob(tA) = |c/(£^)|2. We shall now also assume that the spectrum of T dT ~dt dT ~dt + -[T,H] = 0. (4.97) Chapter 4. Time-of-Arrival Operators 55 That is, the time-of-arrival cannot change in time. If, for example, I measure that the bus is supposed to arrive at 7 p.m., then if I make another measurement at some other time, I should still find that the bus should (or did) arrive at 7 p.m For a time-independent Hamiltonian, time translation invariance implies that the eigenkets \tA)t depends only on t — tA, i.e. the eigenkets cannot depend on the absolute time t. This means for example that at the time of arrival: \tA)t=tA = \^A)t=t'A- Time-translation invariance implies |*A>* = e- l ' G |0) o . (4.98) where G = G(t — tA) is a hermitian operator. Therefore, l^)^ satisfies the differential equations • d . , dG. . dG. , .d. . dG. . / ( rtrtX ^ h = W J ^ = - ^ \ ^ ^ t \ t A h = ^ \ t A ) , (4.99) Now act on the eigenstate equation (4.95) with the differential operators idtA and idt. This yields dG dG - T — \ t A ) t = - t A — \ t A ) t + i \ t A ) u (4.100) and <9T dG dG •i-fr\tA)t + T — | ^ ) , = tA — \tA)t. (4.101) By adding the two equations above, the dependence on dG/dt drops off, and after using the constancy of T (eq. 4.97) we get ([T,H] + i)\tA) = 0. (4.102) Since the eigenkets \tA) span, by assumption, the full Hilbert space [T,H] = -i. • (4.103) Hence T is a generator of energy translations. From equation (4.97) we have T = t — T , where T is the "time operator" of the system whose Hamiltonian is H. It is well know Chapter 4. Time-of-Afrival Operators 56 that equation (4.103) is inconsistent unless the Hamiltonian is unbounded from above and below [8]. 4.3 T ime-o f -Ar r iva l Operators vs. Continuous Measurements Although formally there cannot exist a time-of-arrival operator T, it may be possible to approximate T to arbitrary accuracy [9]. This modified operator will be discussed more fully, in the next section, but for now, assume that we can define the regularized Hermitian operator T ' = 0(p)TO(p) (4.104) where O(p) is a function which is equal to 1 at all values of p except around a small neighborhood of k = 0. For \p\ < e, O(p) goes rapidly to zero (at least as fast as \fk). T" is thus an operator which behaves just like T except in a very small neighborhood around k = 0. It's eigenvalues are complete and orthogonal, and it circumvents the proof given above, because it satisfies [T',H] = -?:0 (4.105) i.e. it is not conjugate to H at p = 0. Although T is not always the shift.operator of the energy, the measurement can be carried out in such a way that this will not be of consequence. To see this, consider the interaction Hamiltonian Hmeas = 5(<)qT", (4.106) which modifies the initial wave function ip —• exp(—iqT')ip. We need to demand that T ' acts as a shifts operator of the energy of ip during the measurement. Therefore we need that q > —Emin, where Emin is the minimal energy in the energy distribution of ?/>. In this way, the measurement does not shift the energy clown to E = 0 where T ' is no longer Chapter 4. Time-of-Arrival Operators 57 conjugate to H. The value of T ' is recorded on the conjugate of q - call it Pq. Now the uncertainty is given by dT'A = d(Pq) = l/dq, thus naively from dq — l/dT'A < Emin, we get EmindT' > 1. However here, the average (q) was taken to be zero. There is no reason not to take (q) to be much larger than Emini so that (q) — dq » —Emin. If we do so, the measurement increases the energy of tp and T ' is always conjugate to H. The lim-itation on the accuracy is in this case dT'A > l/{q) which can be made as small as we like. However, even small deviations from the commutation relation (4.103) are problemat-ic. Not only is the modification arbitrary, it will also result in inaccurate measurements. For example, since § = l - 0 , (4.107) T'(t) = T ' ( 0 ) - « ( l - O ) . (4.108) For the component of the wave function ip(k) which has support in the neighborhood of k — 0, the time-of-arrival will no longer be a constant of motion. The average value of T'(t) for the state tp(k) is given by . (T(t)) = (T(0)) - t f dk [1 - 0(k)\ mk)\2 . (4.109) The second term on the right hand side will be non-zero if ip(k) has support for < e. Even if ip(k) is negligibly small around k = 0, the second term will grow with time. Thus, one only needs to wait a sufficiently long period of time before measuring T ' to find that the average time-of-arrival will change in time. As mentioned in the previous section, this does not correspond to what one would want to call a "time-of-arrival". The greater |i/>(fc)|2 is around k — 0, the greater the deviation from the condition that the time-of-arrival be a constant of the motion. Another difficulty with the time-of-arrival operator, is that if one makes a measurement at a time t' before the particle arrives, then Chapter 4. Time-of-Arrival Operators 58 one needs to know the full Hamiltonian from time t! until tA. Even if one knows the full Hamiltonian, and can find an approximate time-of-arrival operator, one has to have faith that the Hamiltonian will not be perturbed after the measurement has been made. On the other hand, the continuous measurements we have described can be used with any Hamiltonian. A further difficulty is that a measurement of the time-of-arrival operator is not equiv-alent to continuously monitoring the point-of-arrival I.e., measuring the time-arrival-operator is not equivalent to the measurement procedures discussed in Chapter 2. If P 0 is the projector onto re = 0, one finds that A measurement of the time-of-arrival operator does not commute with the projection operator onto the point of arrival. It is also clear from the discussion in Section 3.3, that in the limit of high precision, continuous measurements respond very differently to the time operator. When measure-ments are made with physical clocks, then in the limit dtA —> 0 all the particles bounce back from the detector. Such a behavior does not occur for the time of arrival operator which can be measured to arbitrary accuracy. Nevertheless, one may still hope that since the eigenstates of T have an infinitely spread in energy, they do trigger a clock even if dtA —>• 0. For the type of models we have been considering, we can show however that this will not happen. Let us assume that the interaction of one eigenstate of T with the clock (of, say, Section 3.3.1) evolves as Here, \y — t0) denotes an initial state of the clock with dtA —» 0, |X (*A)) denotes the final (4.110) (4.111) Chapter 4. Time-of-Arrival Operators 59 state of the particle if the clock has stopped, and | X ' ( * A ) ) the final state of the particle if the clock has not stopped. Since the eigenstates of T form a complete set, we can express any state of the particle as \tp) = J dtAC(tA)\tA)- We then obtain : JdtAC(tA)\tA)\y = t0). -* IdtAC(tA)\x(tA))\y = tA) + (|dt AC{t A)\x'{t A))}\y = t). (4.112) The final probability to measure the time-of-arrival is hence /dt a\C(t a)x(t a)\ 2• On the other hand we found that for a general wave function ip, in the limit of dta —>• 0, the probability for detection vanishes. Since the states of the clock, \y = ta), are orthogonal in this limit, this implies that x{ta) = 0 in eq. (4.111) for all tA. Therefore, the eigenstates of T cannot trigger the clock. It should be mentioned however, that one way of circumventing this difficulty may be to consider a coherent set of T eigenstates states instead of the eigenstates themselves. These normalizable states will no longer be orthogonal to each, so they may trigger the clock if they have sufficient energy l . In this regard it is of interest to prematurely quote a result which we will show in Section 4.5 - the average energy of a Gaussian distribution of time-of- arrival eigenstates is proportional to 1/A where A is the spread of the Gaussian. This puts us at the edge of the limitation given in Equation (3.91). 1An arbitrary wave packet can be written as a superposition of normalized eigenstates, and yet we know that arbitrary wave packets do not trigger the clocks of Section 3.3.1. This creates a somewhat interesting situation if normalized eigenstates trigger a clock, but wave packets made of superpositions of them do not. Chapter 4. Time-of-Arrival Operators 60 4 . 4 The Mod i f i ed T ime-of -Arr iva l Operator Kinematically, one expects that the time-of-arrival operator for a free particle arriving at the location XA — 0 might be given by „, m 1 1 . T = -T7Px(0)7i- ( 4' 1 1 3 ) The operator —m(^x + x^ ) is equivelant to the one above as can be seen by use of the commutation relations for x and p. In the k representation this operator can be written as m /, * . 1 d 1 . A d d 1, T(k) = ~vm—j= — ^= =-im(T— + —T) 4.114 y/kdky/k kdk dkk' v where \fk = i\J\k\ for k < 0. If one solves the eigenvalue equation, one finds a set of anti-symmetric states 2 for this operator given by rfrk) = (0(k)-6(-k))-^y/\k\eii& . (4.115) The symmetric states are 9tjk) = (6(k) + 6(-k)) - ^ y f i k i e ^ (4.116) V27rra However, the operator is not self-adjoint and these states are not orthogonal. ^ = ^ -n.Cde^m"-"° = ^ - ^ - ^ r ^ ) - < 4 1 1 7 ) It is important to recall that a symmetric operator which is not self-adjoint always has complex eigenvalues and eigenfunctions [46]. If in (4.115) we choose tA complex, having positive imaginary part, then the eigenstate is a square integrable function (i.e.. it is a true eigenstate of the operator) which has complex eigenvalues. 2 T o find the actual generalized eigenfunctions of this operator, one needs to first specify a domain of definition. We will not deal with this issue here, since the modified time-of-arrival operator is self-adjoint, and therefore one does not encounter the same problems as with the time-of-arrival operator which is not self-adjoint. Chapter 4. Time-of-Arrival Operators 61 Trying to make T self-adjoint by defining boundary conditions at k = 0 leads to the requirement on square integrable wave functions u(k),v(k) such that (u, Tv) — (T*u, v) = im v(k)u(k) v(k)u(k) km v , ' + hm v , ' 0 \ (4.118) i.e.. the boundary conditions must be chosen so that ^ f c ^ f c ) is continuous through k = 0. This continuity condition cannot force u(k) to have the same boundary conditions as v(k) for any choice of boundary condition on v(k). For example, if we choose v(k)/Vk to be continuous through the origin, then u{k)/y/k must be anti-continuous through the origin. I.e. the domain of definition of T and T* differ and T cannot be self-adjoint. This is not at all surprising, given the proof in Section 4.2. One might however, try to modify T in order to make it self-adjoint in the manner shown in Section 4.3 [9]. Consider the operator Te{k) = -im^W)^4W) (4 .H9) where fe(k) is some smooth function which differs from 1/A- only near k = 0. Since u(k) and v(k) could diverge at the origin at a rate approaching 1/y/k and still remain square-integrable, if fe(k) goes to zero at least as fast as A, then T e will be self-adjoint and defined over all square integrable functions. However, as we show in Sections 4.5 and 4.6, these eigenstates do not behave as one would expect a time of arrival eigenstate to behave. It can be verified that T e has a degenerate set of eigenstates for k > 0 and \tA, —} for k < 0, given by gfA(k) = (k\tA,±) = 6(±k)^L=-7l=e^ ^7^dk' (4.120) v 2 7 r m \]Uk) which are orthonormal as expected. Grot, Rovelli, and Tate [9] choose to work with the Chapter 4. Time-of-Arrival Operators 62 states given by \k\ < e (4.121) |fc|>€ If e —• 0, one might expect T e to be a good approximation to the time of arrival operator when acting on states that do not have support around k = 0 [9]. As we show in Appendix C, when these states are examined in the x-representation, and if one only considers the contribution to the Fourier transform of gtA{k) from > e (i.e.. the "unmodified" part of the eigenstate), then one finds that at the time-of-arrival, the states are not delta functions 8(x) but are proportional to x~3^2; they have support over all x. However, although the state has long tails out to infinity, the quantity J dx'\x'~3^2\2 ~ x~2 goes to zero as x —» oo. Furthermore, the modulus squared of the eigenstates diverges when integrated around the point of arrival .T = 0. As a result, one might expect that the normalized state will be localized at the point-of-arrival at the time-of-arrival. In Section 4.5 we show that this is indeed so. However, the full eigenstate, is made up both of this "unmodified" piece, and a modified piece. The modified part of the eigenstate is not well localized at the time-of-arrival. The contribution to the Fourier-transform of the state gfA{k) from 0 < k < e is given by <~9+(x)tA = -jL=f ^ e - ^ e ^ - . (4.122) •\f2-Km Jo \Jk Because T e is no longer the generator of energy translations for |A;| < e, gtA(k) is not time-translation invariant. For the tA — 0 state, (4.4) can be integrated to give <9+(x)u = -7==*(>/R (4-123) y Zxim where $ is the probability integral. For large re, f ^ + ( x ) ^ goes as and the quantity /dx'\€g^A(x')\2 ~ ln:r diverges as x —• oo. For small x, fgtA(x) is proportional to e~KX. Chapter 4. Time-of-Arrival Operators 63 Its modulus squared vanishes when integrated around a small neighborhood of x = 0. eg+(x)tA then, is not localized around the point of arrival, at the time-of-arrival. This will also be verified in the next section where we examine normalizable states. Although eg+(x)tA is not localized around the point of arrival at the time of arrival, one might hope that this part of the state does not contribute significantly in time-of-arrival measurements when e —• 0. However, we will now see that for coherent superpositions of these eigenstates, half the norm is made up of the modified piece of the eigenstate. 4.5 Normalized Time-of-Arrival States Since the time-of-arrival states are not normalizable, we will examine the properties of states |TA ) which are narrow superpositions of the modified time-of-arrival eigenstates. These states are normalizable, although they are no longer orthogonal to each other 3 . We can now consider coherent superpositions of these eigenstates I T ! ) = N J dtA\tA,±)e-ilA^}1. (4.124) where N is a normalization constant and is given by N = (-^)x/4. The spread dtA in arrival times is of order A . We now examine what the state r(x,t)+ = (X\T£) looks like at the point of arrival as a function of time. In what follows, we will work with the state centered around r = 0 for simplicity. This will not affect any of our conclusions. r + ( . r , i ) is given by r+(x,t) = N l(x\e=^1\tA,+)e-!^dtA = N e-^e—'e^gt^dtAdk +N eSe—felkxg+A(k)dtAdk JO Je EE €T+(x,t) +0T+(x,t) (4.125) 3These coherent states form a positive operator valued measure ( P O V M ) . The measurement of time-of-arrival using P O V M s has been discussed in [26]. Chapter 4. Time-of-Arrival Operators 64 As argued in the previous section, the second term should act like a time-of-arrival state. The first term is due to the modification of T and has nothing to do with the time of arrival. We will first show that the second term can indeed be localized at the point-of-arrival x — 0 at the time of arrival t — tA. We will do this by expanding it around x — 0 in a Taylor series. After taking the limit e —* 0, it's n'th derivative at x = 0 is given by fjn 1\J r roc (24 , 2 y ^ 0 r + ( . r , t ) | , = o = - ^ = \ \ e-^e(k)Vk(ik)ne^-^dtAdk dxn y/2ft"m J 3 I 3 n /•CO e i s ™ 2 e~^~ki+ndk = - ^ - r < 4 + 2 ) ( A ) : r + T e _ 5 Z T B - f - » < ^ - » < 4- 1 2 6> where Dp(z) are the parabolic-cylinder functions. For any finite t, we can choose A small enough so that the argument of Dp(z) is large, and can be written as D p ( z ) ^ e - ^ z ^ l - ^ ^ + ...) . (4.127) We can now write o r + ( 0 , i ) as a Taylor expansion around x = 0 0 r + ( .T,t) ~ >/A(5)i £ «„(#*)" (4-128) r n = 0 V 1 where a„ is a numerical constant given by 3 i n _ n — 1 1 „ , 3 77-. . a„ = i-* + 2 2—vr-*r(- + - ) (4.129) We can now see that for any finite t the amplitude for finding the particle around x = 0 goes to zero as A goes to zero. The probability of being found at the point of arrival at a time other than the time-of-arrival can be made arbitrarily small. On the other hand, at the time-of-arrival t = 0, we will now show that the state 0T+(x,t) can be as localized as one wishes around x = 0. From (4.126), we expand 0r+(x, 0) as a Taylor series OTHX,0) = (^Y,K(J^XT (4.130) Chapter 4. Time-of-Arrival Operators 65 w here 6„ = i " 2 " - i * - i r ( ! + (4.131) We see then that o T + ( . T , 0 ) is a function of \J^x (with a constant of (^•)1 /'4 out front). As a result, the probability of finding the particle in a neighborhood 6 of x is given by r s , , , ffn „ s l 0 . I~A r6^ I s | o T + ( V Z * ' ° ) | 2 d 3 ; = V m /« A '°r+(U' °)|2fk' (4-132) Since | 0 T + ( U , , 0)| 2 is proportional to and is square integrable, we see that for any <*), one need only make A small enough, in order to localize the entire particle in the region of integration. 0T+(x,t) is localized in a neighborhood 6 around the point-of-arrival at the time-of-arrival as A —> 0. The state is localized in a region 6 of order <J~^. This is what one would expect from physical grounds, since we have (k) dx ~ dtA-m (4.133) ((/e) is calculated in the following section and is proportional to yJm/A): The probability distribution of 0r+(x,t) at t = r is shown in Figure 4.1. This behavior of the unmodified piece of the time-of-arrival state, 0 r + ( :c , £), as a function of time appears to agree with the results of Muga, Leavens and Palao, who have studied these coherent states independently [30]. The modified part of the time-of-arrival state, eT+(x,0), is not found near the origin at * = tA = 0. We find e r + ( . r ,0) = N^L= r f e ' ^ ^ e ^ ^ e ^ d k d t A V27rm - / - o o Jo y/k = N . \ e " A 2 7 ± + --iex)(-iex)s--^dtA. 4.134 V 2 7 r m J-oo m 2 Chapter 4. Time-of-Arrival Operators 66 0 . 5 A 0 . 4 A 0.3-\ 0.2-\ y Figure 4.1: Unmodified part of time-of-arrival eigenstate. | 0 T + ( . T , T ) | 2 VS. X, with A = m (solid line), and A = ^ (dashed line). As A gets smaller, the probability function gets more and more peaked around the origin. If iex is not large, we can use the fact that for A and e very small, teH^/m <C 1/2 so that we have 4 . / / x 1 / e 3 A §{J—iex) (r+(x,0) ~ 2 T T W - K - y = 1 . (4.135) y Zm y/—iex Note the similarity between this state (the form above is not valid for large x), and that of the modified part of the eigenstate (4.123). We are interested in the case where ^ goes to zero, in which case ET+(x,0) vanishes near the origin. For large ex, it goes as / o A z m V xm • From (4.134) we can also see that if ex > e < 2 A then the last factor in the integrand oscillates rapidly and the integral falls rapidly for larger x. Thus, as we make ^ smaller, the value of the modulus squared decrease around x = 0, but the tails, which extend out to e ^ A / e , get longer. Jx | e r + (a : ,0) | 2 goes as ^Mn.r up to ex ~ e ^ A . As ^ —»• 0, the particle is always found in the far-away tail. The.state er+(x,0) is not found near the point of arrival at the time-of-arrival. It's probability distribution at Chapter 4. Time-of-Arrival Operators 67 0 . 0 0 2 5 . 0 . 0 0 1 5 0 . 0 0 0 5 - 1 5 — 1 0 5 1 0 1 5 y Figure 4.2: Modified part of time-of-arrival eigenstate. ]\ET+(x, T)\2 V S . ex, o f ^ f r r . with A e 2 = ^ (solid line) and A e 2 = (dashed line). As A or e gets smaller, the probability function drops near the origin, and grows longer tails which are exponentially far away. t — tA = 0 is shown in Figure 4.2. 4.6 Contribution to the Norm due to Modification of T We now show that the modified part of | r £ ) contains at least half the norm, no matter how small e is made. The norm of the state \T£) can be written as j \(k\T^}\2dk = N2 j^\e-3gtA{k)dtA\2dk + N2 j \e~^ g+A{k)dtA\2dk = Nf + N2 (4.136) where N2 is the norm of the modified part of the time-of-arrival state, and N2 is the norm of the unmodified part. The second term can be integrated to give 2 N ° 2irm J J e-^eik^-^-^dtAdt'Adk N2A2ir r00 m Jo dk ke s m - ' Chapter 4. Time-of-Arrival Operators 68 = \ (4.137) where without loss of generality, we are looking at the state centered around r = 0 at t = 0. • The unmodified piece can contain only half the norm. The rest is found in the modified piece. N2 2 - t 2 t'? N2 = f- rdkjdUdf'e^e^ 2rcm Jo J k N2A2 f€ - e 4 A 2 i n 2 k/c e2 r Jo dke~ 2m Jo k = \ (4-138) The reason for this, is that essentially, the modification 1/k —> fe(k) involves expanding the region 0 < k < e into the entire negative k-axis. I.e. we see from (4.117) that in order to make the eigenstates orthogonal, one needs the integration variable to go from — oo to oo and this involves making the modification rk dk' The orthogonality condition then becomes /o o 1 ± o o d Z ± 2^e'itA~t,A)^ = 6 { t A " t > A ) • ( 4 - 1 4 0 ) No matter how small we make e, half the norm comes from the contribution z± < Q which is the modified part of the eigenstate. As a result, if one makes a measurement of time-of-arrival, then one finds that half the time, the particle is not found at the point of arrival at the predicted time-of-arrival. Modified time of arrival states do not always arrive on time. From (4.138), one can also see that if f€(k) goes to zero faster than k, then Ne will diverge as A or e go to zero. If fe{k) = k1+s, then we find 1 i°-m2 e 2 ! - t ( - f e 2 A ^ ) m (4.141) Chapter 4. Time-of-Arrival Operators 69 As e or A go to zero, N€ diverges, and if we renormalize the state, the entire norm will be made up of the modified part of the eigenstate. It is also of interest to calculate the average value of the kinetic energy for these states, in order to see whether these states will trigger the physical clocks discussed in Chapter 3. In calculating the average energy, the modified piece will not matter since k2 goes to zero at k = 0 faster than -4= diverges. We find We see therefore, that the kinematic spread in arrival times of these states is proportional to l/Ek- Since the probability of triggering the model clocks discussed in Chapter 3 decays as ^EkStA, where St^ is the accuracy of the clock, we find that the states \T£) will not always trigger a clock whose accuracy is 6tA = A . 4.7 Limited Physical Meaning of Time-of-Arrival Operators We have seen that formally, a time-of-arrival operator cannot exist. If one modifies the time-of-arrival operator so as to make it self-adjoint, then its eigenstates no longer behave as one expects time-of-arrival states to behave. Half the time, a particle which is in a time-of-arrival state will not arrive at the predicted time-of-arrival. The modification also results in the fact that the states are no longer time-translation invariant. For wave functions which don't have support at k = 0, measurements can be carried out in such a way that the modification will not effect the results of the measurement. Nonetheless, after the measurement, the particle will not arrive on time with a probability of 1/2. One cannot use T e to prepare a system in a state which arrives at a certain time. (r+|H f c |r+) = j elk—(Tt\k)(k\T£) 4 (4.142) Chapter 4. Time-of-Arrival Operators 70 We would also like to stress that continuous measurements differ both conceptually and quantitatively from a measurement of the time-of-arrival operator. Operationally one performs here two completely different measurements. While the time-of-arrival operator is a formally constructed operator which can be measured by an impulsive von-Neumann interaction, it seems that continuous measurements are much more closer to actual experimental set-ups. Furthermore, we have seen that the result of these two measurements do not need to agree, in particular in the high accuracy limit, continuous measurements give rise to entirely different behavior. This suggests that as in the case of the problem of finding a "time operator" [20] for closed quantum systems, the time-of-arrival operator has a somewhat limited physical meaning. Chapter 5 Traversal Time 71 Chapter 5. Traversal Time 72 5.1 A Limitation on Traversal Time Measurements In Chapter 3, we considered various clock models for measuring the time it takes for a free particle to arrive to a given location xA- Because the energy of the clock increases with its precision, we argued that the accuracy of a time-of-arrival detector cannot be greater than 1/Ek, where Ek is the kinetic energy of the particle. Measurements of traversal time [16] are analogous to that of time-of-arrival. One tries to measure how long it takes a particle to travel between two fixed locations x\ and x%. Although no proof has yet been found for the restriction on time-of-arrival accuracy, in this Chapter we provide model independent arguments that a necessary minimum inaccuracy on traversal time measurements is given by 6TF > 1/E.P. (5.143) We do this by arguing that a traversal time measurement is also a simultaneous mea-surement of position and momentum, and that (5.143) is required in order to preserve the Heisenberg uncertainty relationship. Once again (5.143) is not analogous to the Heisenberg Energy-time uncertainty relationship. It reflects the inherent inaccuracy of every individual measurement, while the Heisenberg uncertainty relationships refer to well-defined and perfectly accurate measurements made on ensembles. This chapter proceeds as follows. In section 5.2 we motivate the notion that traversal time is a measurement of momentum by looking at measuring the traversal-distance. In section5.3 we discuss a physical model for measuring the traversal time, and show the relation between (5.143) and the uncertainty principle. The main result of this Chapter is given in Section 5.4, where we provide a model independent argument for (5.143), as well as a qualitative proof. Chapter 5. Traversal Time 73 5.2 Measuring Momentum Through Traversal-Distance The measurement of traversal-distance may be considered the space-time dual of the measurement of traversal time: instead of fixing x\ and x2 and measuring tp — t2 — £i, one fixes t\ and t2 and measures xp = x2 — X \ . It is instructive to examine first this simpler case of traversal-distance and point out the similarities and the differences. Unlike the case of traversal time, a measurement of traversal-distance can be described by the standard von Neumann interaction. For a free particle the Hamiltonian is ~ 2 H = ^ + Qx 2m 6(t - h) - 6(t - t2) (5.144) where Q is the coordinate conjugate to the pointer variable P. The change in P yields the traversal-distance: P(t > t2) - P 0 = x(t 2) - x(«i) = x f . (5.145) However the measurement of the traversal-distance provides additional information: it also determines the momentum p of the particle during the time interval t\ < t < t2. From the equations of motion we get: p (0 p 0, t < t\ or t > t2 Po - Q, h <t <t2 and x(<) x0 + %t, t< h { xo + ^h + ^ i t - h ) , h<t<t2 and therefore, P(t > t2) - P 0 _ m V . = po - Q = p(ti < t < t2). (5.146) (5.147) (5.148) . * 2 - * l . . Thus, one can determine simultaneously and to arbitrary accuracy the traversal-distance and the momentum in intermediate times. This, of course, does not contradict the Chapter 5. Traversal Time 74 uncertainty relations, because p commutes with X f , and x remains completely uncertain. Similarly, in the case of the traversal time we shall see that the measurement determines also the momentum during the traversal, however unlike the present case, since the particle has to be in the interval x2 — x\ during the traversal, it is also a measurement of the location. This indicates that, in the latter case, in order not to violate the kinematic uncertainty principle for x and p, the accuracy with which the traversal time, Tp, or the momentum may be measured must be limited. 5.3 Measuring Traversal Time For traversal time the classical equations of motion suggest that a traversal time operator might be given by (5.149) where L = x2 — x\. This operator is self-adjoint, but like the time-of-arrival operator, we shall see that different outcomes are found in a direct measurement of TV and a measurement of the operator T F - Furthermore, one can measure the quantity T F at any time, so there is no reason to believe that the particle actually traveled between the two points in the time tp. Since T F is only a function of p, the measurement will result in the particle's position being spread over all space, so there is no finite amount of time one could wait before being certain that the particle went between the two fixed points. For example, after the measurement of Tp, the potential between x\ and x2 might change. General traversal time operators would require that one knows the Hamiltonian not only in the past, but also in the future. If one measures the traversal time operator above, then one has to have faith that the Hamiltonian will not change after the time of the measurement t0 to t —»• oo. Chapter 5. Traversal Time 75 It is also commonly accepted that the dwell time operator [32], given by / • o o r D = / dtUXA{t) (5.150) where n^ (0 )= r\x)(x\ (5.151) can be used to compute the traversal time 1. Such a quantity however, cannot be mea-sured, since, as we saw in Chapter 2, the operator HXA(t) does not commute with itself at different times. [IlXA(t),UXA(t')]^0. (5.152) Therefore, one must measure the traversal time in a more physical way. One must demand that if we measure the traversal time to be tp, then the particle must, actually traverse the distance between x-i and x2 in the time given by the traversal time measurement. For example, one could have a clock which runs when the particle is between Xi and .T2 given by the Hamiltonian [16] [33] 2 ' H = £ - + V(x)Q (5.153) 2m where the traversal time is given by the variable P conjugate to Q and the potential V is equal to 1 when xi < x < x2 and 0 everywhere else . In the Heisenberg picture, the equations of motion are x = p/ra, p = -Q(<5(x - xi) - <5(x - x2)) (5.154) P = F(x) , Q = 0. (5.155) The particle's momentum is disturbed during the measurement p'.= v / p 2 - 2 m Q (5.156) 1 i n our case, where there is no potential barrier, the dwell time and traversal time are equivalent Chapter 5. Traversal Time 76 where p' is the particle's momentum during the measurement, and p is.the undisturbed momentum. However if the interaction is weak Q <C Ep, then after a sufficient time, the clock will read the undisturbed traversal time P(t oo) - P(0) ~ r V fx(0) - —) dt Jo V m J m(x2 - xx) (5.157) P If we require an accurate measurement of the traversal time, then a small dP will result in large values of the coupling Q. If Q is too large, the clock can reflect the particle at Xi and one will obtain a traversal time equal to 0. This therefore imposes a restriction on the accuracy with which one can measure the traversal time. As in Chapter 3 we find that 6TF>1/EP (5.158) is required in order to be able to measure the traversal time, and 6TF > 1/EP (5.159) in order to measure the undisturbed value of the traversal time. Let us show that the above conditions are consistent with the uncertainty relations for the position and momentum. If (5.159) is satisfied, we have Q <£L E, and by eq. (5.156) the momentum during the measurement is TTi p ' ~ p Q. (5.160) P Thus during the measurement, the momentum will be uncertain by an amount 77? dp'~—dQ. (5.161) Po In order to know whether the particle entered our detector, we need to be able to dis-tinguish between the case where the pointer is at its initial position P = 0, and the case Chapter 5. Traversal Time 77 where the particle has gone through the detector P = 'tp — — . We therefore need the Po condition dP<—. (5.162) Po Since at best we have dP = 1/dQ, we find dp'dx = dp'L > 1 . (5.163) The uncertainty relation (5.158) only applies to this particular model clock - it might be possible to accurately measure the traversal time in some clever way. In the following section we will argue that this cannot be done, by demonstrating that this uncertainty applies to all measurements of traversal time. Finally, we should note that a traversal time detector could be made by measuring the time-of-arrival to X\ and the time-of-arrival to x2. This would require two time-of-arrival clocks each with its own inaccuracy, whereas the model above only has one clock. 5 . 4 General Argument for a Minimum Inaccuracy We now consider general measurements of traversal time. We will however, impose some fairly unrestrictive requirements on the measurement. We will assume that the measurement does not prevent the particle from actually traversing the distance between X\ and x2. I.e. we want to be able to say that the particle did indeed traverse the distance L - otherwise, it is unclear what it is that we are measuring. We also demand that the result of the measurement corresponds in some sense with the classical notion of traversal time. I.e., we are measuring something like mL/p. Our measuring device will consist of a pointer, which is set to some initial position P0 with an uncertainty in the initial value of the pointer of dP. At the end of the measurement, we assume that the value of the traversal time is inferred accurately by reading the final Chapter 5. Traversal Time 78 value of the pointer. The measured traversal time is then proportional to P / — P,. The relative accuracy of the traversal time will then be given by STf/Tf — dP/(Pf — Pi) Another condition we will impose is that the inaccuracy of the measurement, 6 T F , is less than the quantity we are trying to measure TF (i.e. we are looking at accurate measurements). Finally, we assume that the experimentalist has no knowledge of the state of the particle, and thus must set the initial state of the measuring device (and its inaccuracy 6 P ) with no prior knowledge of the ensemble. Before proceeding with the argument, we should be clear to distinguish between different types of uncertainties. For an operator A , there exists a kinematic uncertainty which we will denote by dA given by dA2 = ( A 2 ) - ( A ) 2 . (5.164) This is the uncertainty in the distribution of the observable A. There is also the inherent inaccuracy in the measuring device. This is the relevant quantity in equations (5.143) and (5.158). It refers to the uncertainty in the initial state of the measuring device's pointer position P , and we will denote it by dA. For our measuring devices we have 6 A = dP0 (5.165) This inaccuracy applies to each individual measurement. Lastly, there is the uncertainty A A which applies to the spread in measurements made on the ensemble. Given a set AM of experiments i — 1, 2, 3... which yield results A, :, we have AA2 =< A2M > - < AM >2 . (5.166) This uncertainty includes a component due-to the kinematic uncertainty of the attribute of the system, and also the inaccuracy of the device. For our measuring device, the kinematic spread in the pointer position at the end of each experiment gives A A A A = dPj (5.167) Chapter 5. Traversal Time 79 The Heisenberg uncertainty relationship dAdB > 1 applies to measurements on en-sembles. Given an ensemble, we measure A on half the ensemble and B on the other half. The uncertainty relation also applies to simultaneous measurements 2 . If we measure A and B simultaneously on each system in the ensemble, then the distributions of A and B must still satisfy the uncertainty relationship. Returning now to the traversal time, we see that it can be interpreted as a simulta-neous measurement of position and momentum. We know the particle's momentum p during the time that it was between x — x\ and x = x2 from the classical equations of motion mL • tF = . 5.168) V In other words, eigenstates of momentum must have traversal times given by equation (5.168). This measurement of momentum is analogous to the measurement described in Section 5.2. Instead of measuring the change in position at two specified times t\ and t2, we are now measuring the difference in arrival times after specifying two different posi-tions x\ and x2. During the measurement, we also know that the particle is somewhere between x = xi, and x = x2. i.e.. we know that x = X] +X2 ± L/2. The uncertainty relationship also applies to these measured quantities AxAp > 1. This essentially means that a detector of size L will disturb the momentum by at least 2/L, so that repeated measurements on an ensemble will give Ap > 2/L. The position of the detector X commutes with the momentum of the particle p [10] however, we demand that the particle actually travel the distance L. The particle must actually be inside the detector during the measurement. As a result, X must be coupled to the position x of 2 For a discussion of how the uncertainty relation applies to simultaneous measurements, see for ex-ample, Arthurs and Kelly[39] They propose a model for simultaneous measurements using a Hamiltonian H = 8{t)(F\A + P2B) which measures the variables A and B using two measurement pointers Qi and Q 2 which are conjugate to P i and P 2 . They show how the Heisenberg uncertainty relation applies to the uncertainty in the outcomes of the measurement of A and B . Chapter 5. Traversal Time 80 the particle and so a measurement of X is also a measurement of x. This is what we mean by a local interaction. We now show why we expect (5.143) to be true. During the measurement of traversal time, the momentum will be disturbed by an amount dp>2/L. (5.169) If this disturbance is small, then from (5.168) we expect this will cause an inaccuracy given by rrr, mL , STF = —dp P > 1/EP (5.170) For measurements where the disturbance to the system is minimal (i.e.. when dp and ST are small) we see that (5.143) gives the minimum inaccuracy on traversal time measure-ments. We now proceed with the slightly more rigorous argument. We imagine a traversal time detector which has an inaccuracy given by ST p. Measurements can then be carried out on arbitrary ensembles with arbitrary Hamiltonians. We will show that by choosing this ensemble appropriately, the uncertainty relationship ArrAp > 1 ,can be violated, unless the traversal time obeys the relationship given by (5.143). We assume that initially, the pointer on our traversal time detector is given by P0 = e (5.171) where e is a small random number which arises because of the initial inaccuracy of the clock, i.e.. the distribution of e is such that (e) = 0 and the clock's initial inaccuracy in pointer position is dP2 = (e2). It is important to note that this inaccuracy is fixed as an initial condition before any measurements are made. It is a property of the device, Chapter 5. Traversal Time 81 and does not depend on the nature of the ensemble upon which we will be making measurements. For a free Hamiltonian, a measurement of the traversal time will result in a final pointer position given by „ . „ mL Pf = P0 + (5.172) p ' where p is the momentum of the particle in the absence of any measuring device. For eigenstates of p (or states peaked highly in p), we demand that the traversal time be given by the classically expected value 3 . Recall that the kinematic spread in the particle's momentum is given by dp2 = (p2) — (p)2. A measurement of the traversal time for a particular experiment i can take on the values Ti = Pf mL „ „„. = + e (5.173) P A given measurement Ti will allow us to infer the momentum of the particle pi during the measurement mL mLp „ . Pi(Ti) = — = • / ' . 5.174 Ti mL + pe The average value of any power a of the measured momentum is <*> = / ( ^ ) ° ^ e ) * * ( 5 1 7 5 ) where f(p) gives the distribution of the particle's momentum and g(e) is the distribution of the fluctuations. We now choose the mass m of the ensemble so that we always have ep < mL. (5.176) In other words, we consider measurements on ensembles where the measurement is much more accurate than the quantity being measured, i.e.. 6Tp « Tp. Indeed for the 3 I t is possible to include small deviations from the classical value, by including an additional term in (5.172). These fluctuations need to average to zero in order to satisfy the correspondence principle. For small fluctuations, the following discussion is not altered. Chapter 5. Traversal Time 82 example given in the previous section, for every given e and p, we can increase Ep by choosing a sufficiently large m, and reach this limit. This limit ensures that (pM) never diverges, and simplifies our calculation by. allowing us to write „2 \ a (paM)^ j [P-^Z) f(P)^)dpde (5.177) For a — 1 we find For a = 2 we find mL = (P)- (5.178) 04) ~ [ ^ - 2 ^ + (^)^f(p)g(e)dpde (5.179) <P>> + W - (5-180) {mL)1 This gives us A p 2 = ( ^ ) - ( P M ) 2 (mL)-+ dp2 (5.181) Since we find (dE) 2 = ^ - ( E ) 2 (5.182) A p 2 = {^)2{dE2 + (£)2) + dp2. ' (5.183) Finally, we arrive at the relation (A.rAp) 2 = STp((E}2 + dE2) + ~^dp2. (5.184) The uncertainty relation A . r A p > l ' (5.185) Chapter 5. Traversal Time 83 then implies Now we note that we can arrange our experiment with Ldp arbitrarily small, by choosing dp of the ensemble arbitrarily small, i.e.. the uncertainty in the traversal time is small. As a result, in order to ensure that Heisenberg's uncertainty relation is never violated, we must have 8TF > 1 (5.187) ^(E)2 + dE2 Since dp is small, we can write STF > -^y- (5-188) It is interesting to note that since the momentum operator commutes with the free Hamiltonian, the restriction on traversal time measurements only comes from the dy-namical considerations given above. 5.5 From Traversal Time to Barrier Tunneling Time We have seen that the measurement of the traversal time given two positions cannot be made arbitrarily accurate. We have argued this by looking at a simple model for measuring traversal time, and we have also given a model independent, qualitative proof of this which applies when the measurement only disturbs the system slightly. Finally we have given a more rigorous argument which applies when the uncertainty in traversal time is small. This strongly suggests that the limitation on measurements of arrival times is a general rule and not just an artifact of the types of models considered so far. Operators for both the traversal time and the arrival time don't seem to correspond to physical (continuous) processes. The case of traversal time is different from time-of-arrival in that there does exist a self-adjoint traversal time operator, and the semi-bounded spectrum of the Hamiltonian does not seem to play an important role in the Chapter 5. Traversal Time 84 restriction on measurement accuracy. The accuracy restriction on traversal time may be particularly important for experiments on barrier tunneling time. If one uses a physical clock to measure the time it takes for a particle to travel from one location to another, with a barrier situated somewhere between the two locations [17] [33], then the accuracy of this clock may affect the tunneling particle. The limitation presented in this Chapter seem to indicate that measurements of barrier tunneling times would also need to be inherently inaccurate, because if one tries to measure the tunneling time too accurately, the particle may be unable to tunnel. Our result concerning traversal time indicates that the barrier tunneling time also cannot be precisely defined. Chapter 6 Order of Events W H O K ' U E D S H R O V E R S C M ? Chapter 6. Order of Events 86 6.1 Past and Future The notion that events proceed in a well defined sequence is unquestionable in classical mechanics. Events occur one after the other, and our knowledge concerning the events at one time allows us to predict what will occur at another time. One can unambiguously determine whether events lie in the past or future of other events. Given two events, A and B, one can compute which event occurred first. It may be, that event A causes event B, in which case, event A must have preceded event B. In this chapter, we are interested in whether the well defined classical concepts of tem-poral ordering have a quantum analogue. In other words, given two quantum mechanical systems, can we measure which system attains a particular state first. Can we decide whether an event occurs in the past or future of another event. The problem of measur-ing the time ordering of two events is in some sense more primitive and fundamental a concept than that of measuring the time of an event. We saw previously that one cannot measure the time-of-arrival to an accuracy better than 1/Ek where Ek is the kinetic energy of the particle. This leads one to suspect that the ordering of events may not be an unambiguous concept in quantum mechanics. However, for a single quantum event A, although one cannot determine the time an event occurred to arbitrary accuracy, it can be argued that one can often measure whether A occurred before or after a fixed time tB to any desired precession. Consider a quantum system initially prepared in a state ipA a n d a n event A which corresponds to some projection operator acting on this state. For example, we could initially prepare an atom in an excited state, and could represent a projection onto all states where the atom is in its ground state i.e. the atom has decayed. ipA could also represent a particle localized in the region x < 0 and Y1A could be a projection onto the positive x-axis. In this case, the event A corresponds to the particle arriving to x — 0. Chapter 6. Order of Events 87 If the state evolves irreversible to a state for which nA\I/(t) = 1, then we can easily measure whether the event A has occurred at any time t. We could therefore measure whether a free particle arrives to a given location before or after a classical time t#. Of course, for many systems, the system will not irreversible evolve to the required state. For example, a particle influenced by a potential may cross over the origin many times. However, for an event such as atomic decay, the probability of the atom being re-excited is relatively small, and one can argue that the event is more or less irreversible. For the case of a free particle which is traveling towards the origin from x < 0 one can argue that if at a later time I measure the projection operator onto the positive axis and find it there, then the particle must have arrived to the origin at some earlier time. This is in some sense a definition, because we know of no way to measure the particle being at the origin without altering its evolution or being extremely lucky and happening to measure the particle's location when it is at the origin. While measuring whether an event happened before or after a fixed time may be possible, we will find that for two quantum systems, one cannot in general measure whether the time tA of event A, occurred before or after the time to of event B. In Section 6.2, confining ourselves to a particular example of order of events, we will consider the question of order of arrival in quantum mechanics. Given two particles, can we determine which particle arrived first to the location xa. Using a model detector, we find that there is always an inherent inaccuracy in this type of measurement, given by 1/E where E is the typical total energy of the two particles. This seems to suggest that the notion of past and future is not a well defined observable in quantum mechanics. Note that the measurements we are considering here are continuous measurements, as opposed to the impulsive measurement of an operator. One could, for example, determine Chapter 6. Order of Events 88 the order of arrival by measuring the operator O = sgn(Tx - Ty) (6.189) where Tx and Ty are the time-of-arrival operators associated with each particle. In Section 6.3 we discuss measurements of coincidence. I.e., can we determine whether both particles arrived at the same time. Such measurements allow us to change the accuracy of the device before each experiment. We find that the measurement fails when the accuracy is made better than 1/E. In Section 6.4 we discuss the relationship between ordering of events and the resolving power of Heisenberg's microscope, and argue that in general, one cannot prepare a two particle state which is always coincident to within a time of 1/E. 6.2 Which first? We now examine a case where the time t# is not given by a classical clock, but rather a quantum system. Consider two free particles (which we will label as x and y) initially localized to the right of the origin, and traveling to the left. We then ask whether one can measure which particle arrives to the origin first. The Hamiltonian for the system and measuring apparatus is given by H = ^ + ^ + H l - (6.190) zm zm where H, is some interaction Hamiltonian. For example, a promising interaction Hamil-tonian is Hi = a6(x)6(-y) . (6.191) with a going to infinity. If the y-particle arrives before the x-particle, then the x-particle will be reflected back. If the y-particle arrives after the x-particle, then neither particle Chapter 6. Order of Events 89 y II I X V(x,y) III IV Figure 6.3: A potential which can be used to measure which of two particles came first (given by V(x,y) = aS(x)0(— y)). The wave function for two incoming particles in one dimension looks like a single wave packet in two dimensions travelling towards the origin. sees the potential, and both particles will continue traveling past the origin. One can therefore wait a sufficiently long period of time, and measure where the two particles are. If both the x and y particles are found past the origin, then we know that the x-particle arrived first. If the y-particle is found past the origin while the x-particle has been reflected back into the positive x-axis then we know that the y-particle arrived first. Classically, this method would appear to unambiguously measure which of the two particle arrived first. However, in quantum mechanics, this method fails. From (6.190) we can see that the problem of measuring which particle arrives first is equivalent to deciding where a single particle traveling in a plane arrives. Two particles localized to the right of the origin is equivalent to a single particle localized in the first quadrant (see Figure 6.3). The question of which particle arrives first, becomes equivalent to the question of whether the particle crosses the positive x-axis or the positive y-axis. The Hamiltonian (6.191) is therefore equivalent to the problem of scattering off a Chapter 6. Order of Events 90 thin edge. Classically, particles which do not scatter off the edge will travel to the third quadrant (x arrived first), while particles which scatter off the edge will be found in the fourth quadrant (y arrived first). However, quantum mechanically, we find that sometimes the particle is found in the two classically forbidden regions, I and II. If the particle is found in either of these two regions, then we cannot determine which particle arrived first. The solution for a plane wave which makes an angle 9Q with the x-axis is well known[47]. If the boundary condition is such that ip(r,0) = 0 on the negative y-axis, then the solution is ^ ( r > 0) = _ L \e-lkrco<d-0^^V2k~rcos(-—-)] - e , : f c r c o s ( 0 + ^ ) $ [ - \ / 2 l 7 s i n ( -27T • ;"1 (6.192) where $(z) is the error function. Asymptotically, this solution looks like e-ikrcos(0-eo) _j_ f(Q^e^ e~ikr cos(0-0o) _ et'fcrcos(0+0o) _j_ f(Q)  e ' k r -90 < 9 < 7T + 9 -90>9> - T T / 2 7T - 9Q < 9 < 3TT /2 (6.193) where /(*) = 87T/C + (6.194) [ s i n ( ^ ) ' c o s ( ^ ) . where the above approximation is not valid when c o s ( ^ a ) or s i n ( ^ t t ) is close to zero. Since we demanded that the particle was initially localized in the first quadrant, the initial wave cannot be an exact plane wave, but we can imagine that it is a plane wave to a good approximation. We see from the solution above that the particle can be found in the classically forbidden regions of quadrant I and II. For these cases, we cannot determine which particle arrived first. This is due to interference which occurs when the particle is close Chapter 6. Order of Events 91 to the origin (the sharp edge of the potential). The amplitude for being scattered off the region around the edge in the direction 9 is given by |/(r,0)\2. It might be argued that since these particles scattered, they must have scattered off the potential, and therefore they represent experiments in which the y-particle arrived first. However, this would clearly over count the cases where the y-particle arrived first. We could have just as easily have placed our potential on the negative x-axis, in which case, we would over count the cases where the x-particle arrived first. In the "interference region" we cannot have confidence that our measurement worked at all. We should therefore define a "failure cross section" given by r2ir / •Z7T Of = J \.fW 1 - t c o s ( E j < " 9 S > From (6.195) we can see that cross section for scattering off the edge is the size of the particle's wavelength multiplied by some angular dependence. Therefore, if the particle arrives within a distance of the origin given by 6x > 2/Jfc (6.196) the measurement fails. We have dropped the angular dependence from (6.195) - the angular dependence is not of physical importance for measuring which particle came first, as it depends on the details of the potential (boundary conditions) being used. The particular potential we have chosen is not symmetrical in x and y. From this we can conclude that if the particle arrives to within one wavelength of the origin, then there is a high probability that the measurement will fail. If we want to relate this two-dimensional scattering problem back to two particles traveling in one dimensional, we need to use the relation Chapter 6. Order of Events 92 In other words, our measurement procedure relies on making an inference between time measurements and spatial coordinates. The last two equations then give us One will not be able to determine which particle arrived first, if they arrive within a time 1/E of each other, where E is the total kinetic energy of both particles. Note that Equation (6.198) is valid for a plane wave with definite momentum k. For wave functions for which dk « k, one can replace E by the expectation value (E). However, for wave functions which have a large spread in momentum, or which have a number of distinct peaks in k, then to ensure that the measurement almost always works, one must measure the order of arrival with an accuracy given by where E is the minimum typical total energy . Although it seemed plausible that one could measure which particle arrived first, we found that if the particles are coincident to within 1/E, then the measurement fails. 6.3 Coincidence In the previous model for measuring which particle arrived first, we found that if the two particles arrived to within 1/E of each other, the measurement did not succeed. The width 1/E was an inherent inaccuracy which could not be overcome. However, in our simple model, we were not able to adjust the accuracy of the measurement. It is therefore instructive to consider a measurement of "coincidence" alone for which one can quite naturally adjust the accuracy of the experiment. Given two particles 1 For example, one need not be concerned with exponentially small tails in momentum space, since the contribution of this part of the wave function to the probability distribution will be small. If however, ip(E) has two large peaks at Esmaii and Eug spread far apart, then if 6t does not satisfy St > 1/E s m au one will get a distorted probability distribution. For a discussion of this, see Chapter 3. 6t > 1 (6.198) E (6.199) Chapter 6. Order of Events 93 y II i X V(x,y) III rv Figure 6.4: Potential for measuring whether two particles are coincident. traveling towards the origin, wTe ask whether they arrive within a time 6tc of each other. If the particles do not arrive coincidently, then we do not concern ourselves with which arrived first. The parameter 5tc can be adjusted, depending on how accurate we want our coincident "sieve" to be. We will once again find that one cannot decrease Stc below A simple model for a coincidence measuring device can be constructed in a manner similar to (6.191). Mapping the problem of two particles to a single particle in two dimensions, we could consider an infinite potential strip of length 2a and infinitesimal thickness, placed at an angle of 7r/4 to the x and y axis in the first quadrant (see Figure 6.4). Particles which miss the strip, and travel into the third quadrant are not coincident, while particles which bounce back off the strip into the first quadrant are measured to be coincident. I.e. if the x-particle is located within a distance a of the origin when the y-particle arrives (or visa versa), then we call the state coincident. Classically, one expects there to be a sharp shadow behind the strip. Quantum l/E. Chapter 6. Order of Events. 94 mechanically, we once again find an interference region around the strip which scatters particles into the classically forbidden regions of quadrant two and four. The shadow is not sharp, and we are not always certain whether the particles were coincident. A solution to plane waves scattering off a narrow strip is well known and can be found in many quantum mechanical texts (see for example [47] where the scattered wave is written as a sum of products of Hermite polynomials and Mathieu functions). However, for our purposes, we will find it convenient to consider a simpler model for measuring coincidence, namely, an infinite circular potential of radius a, centered at the origin. Hi = a.V(r/a) (6.200) where V(x) is the unit disk, and we take the limit a —> oo. It is well known that if a < 1/k, then there will not be a well-defined shadow behind the disk. To see this, consider a plane wave coming in from negative x-infinity. It can be expanded in terms of the Bessel function Jm(kr) and then written asymptotically ( r > l ) as a sum of incoming and outgoing circular waves. e t o = ^emimJm{kr)cosme m=0 i r o o o o eikr e™ c o s m 0 + i e ' i k r E £ m cos m(6 - TT) • . (6.201) 1 V 2mkr L m = 0 m = 0 where em is the Neumann factor which is equal to 1 for m = 0 and equal to 2 otherwise. Since it can be shown that £ e m cos m<9 = V . l Q 2 ' 6.202 The two infinite sums approach 2n8(8) and 2nS(9 — z) respectively, and so the incoming wave comes in from the left, and the outgoing wave goes out to the right. The presence of the potential modifies the wave function and in addition to the plane wave, produces Chapter 6. Order of Events 95 a scattered wave ip = eikx + ^=f{r9) (6.203) where pikr o o - p / M ) = -iY, s i n£ m # m ( f c r ) cos ro0 , (6.204) V R ra=0 Hm{kr) are Hermite polynomials and JVm(/ca) (Nm(ka) are Bessel functions of the second kind). For large values of r, the wave function can.be written in a manner similar to (6.201), except that the outgoing wave is modified by the phase shifts 6m. i o o —ikr „ikr rl> ^ -?==i E emcosm(0 - T T ) — = - + -7=f{r,6) . (6.206) V2mk m = 0 V 7 ' V r where 1 o o f(r, 9) ~ - = = £ e m e- 2 i 5 m ( f c a ) cos mfl ' (6.207) y/2'Kik m = 0 In the limit that b > m the phase shifts can be written as 6m~ka-^(m + ^) . (6.208) In the limit of extrememely large a (but a < r), the outgoing waves then behave as , ( r , , ) s ! l i m + (6.209) V V A # - O O y^nfc sin | ( 0 - T T ) V where once again we see that the angular distribution goes as the delta function 6(9 — TT). The disk scatters the plane wave directly back, and a sharp shadow is produced. We see therefore, that in the limit of ka >^ 1, our measurement of coincidence works. The differential cross section can in general be written as o o = I £ e m e ~ 2 i S ^ cos m9\2 (6.210) m=0 Chapter 6. Order of .Events 96 For ka » 1 (but still finite), (6.210) can be computed using our expression for the phase shifts from (6.208), and is given by a 0 1 9 '• a(0) ~ - s i n - + — - cot 2 - sin 2 ka9 (6.211) K ' 2 2 • 2irk 2 v -The first term represents the part of the plane wave which is scattered back, while the second term is a forward scattered wave which actually interferes with the plane-wave. The reason it appears in our expression for the scattering cross section is because we have written our wave function as the sum of a plane-wave and a scattered wave, and • so part of the scattered wave must interfere with the plane-wave to produce the shadow behind the disk. For ka <C m, the phase shifts look like 6m(ka)~—[-j) m ^ O (6.212) and tan<5 0 (fca)~-f^- (6.213) 2m ka As a result, for ka -C 1, S0 is much greater than all the other Sm and the outgoing solution is almost a pure isotropic s-wave. For ka <C 1 the only contribution to (6.210) comes from So and the differential cross section becomes a ( 0 ) c — \ - (6.214) zkin ka and is isotropic. In other words, no shadow is formed at all, and particles are scattered into classically forbidden regions. We see therefore, that as long as the s-wave is dominant, our measurement fails. The s-wave will cease being dominant when So is of the same order as S\. As can be seen from Equation 6.208, 8i/80 approaches a limiting value of 1 when a sharp shadow is produced. It is only when SI/SQ — 1 that the cross-section no longer Chapter 6. Order of Events 97 0 . 8 -0 . 6 -y 0 . 4 -0 . 2 -0 5 1 0 1 5 2 0 k a Figure 6.5: Phase shifts for coincidence detector (8i(ka) / 8o(ka) vs. ka ) depends on k. This is what we require then, for the probability of our measurement to succeed independently of the energy of the incoming particles. From a plot of 6I/6Q we see that this only occurs when ka 3> 1 (Figure 6.5). Our condition for an accurate measurement is therefore that a 3> 1/k. Since 8tc ~ am/k we find < f t c > l / £ (6.215) 6.4 Coincident States We have seen that we can only measure coincidence to an accuracy of 8tc = 1/E. We shall now show that one cannot prepare a two particle system in a state ipc which always arrives coincidentally within a time less than 8tc. In other words, one cannot prepare a system in a state which arrives coincidentally to greater accuracy than that set by the limitation on coincidence measurements. Chapter 6. Order of Events 98 i Preparing a state xbc corresponds to preparing a single particle in two dimensions which always arrives inside a region 6r — p6tc/m of the origin. In other.words, suppose ; we were to set up a detector of size fir at the origin. If a state ipc exists, then it would always trigger the detector at some later time. Our definition of coincidence requires that the state ipc not be a state where one . particle arrives at a time t > 6tc before the other particle. In other words, if instead, we were to perform a measurement on ipc to determine whether particle x arrived at least . Stc before particle y, then we must get a negative result for this measurement. : This latter measurement would correspond to the two-dimensional experiment of . placing a series of detectors on the positive y-axis, and measuring whether any of them . are triggered by tpc. If ipc is truly a coincident state, then none of the detectors which 1 are placed at a distance greater than y = 6r can be triggered. One could even consider a single detector, placed for example, at (0, <5r), and one would require that tpc not trigger ! this detector. Now consider the following experiment. We have a particle detector which is either , placed at the origin, or at (0,6r) (we are not told which). Then after a sufficient length ' of time, we observe whether it has been triggered. If we can prepare a coincident state ' ipc, then it will always trigger the detector when the detector is at the origin, but never trigger the detector when the detector is at (0,<5r). This will allow us to determine whether the detector was placed at the origin, or at (0,<5r). For example, if we use the detectors described in Section 6.3 (namely, just a scattering potential), then some of the time, the particle will be scattered, and some of the time it won't be, and if it is scattered, we can conclude that the potential was centered around the origin rather than around : ( 0 » . However, as we know from Heisenberg's microscope, a particle cannot be used to resolve anything greater than it's wavelength. In other words ipc cannot be used to Chapter 6. Order of Events 99 determine whether the detector is at the origin, or at (0, 8r) if 6r < 2ir/k. As a result, ipc can only be coincident to a region around the origin of radius less than 6r or, coincident within a time Stc ~ 1/E. It is also interesting to consider the situation where we have an event B which must be preceded by an event A . For example, B could be caused by A , or the dynamics could , be such that B can only occur when the system is in the state A . One can then attempt to i force B to occur as close to the occurrence of event A as possible. This problem has been ; studied in relation to the maximum speed of quantum computers [40] and it was found i that one cannot force the system to evolve at a rate greater than the average energy. 6.5 In Which Direction Does the Light Cone Point I i ' | We have argued that we cannot measure the order of arrival for two free particles, if they j arrive within a time of 1/E of each other, where E is their typical total kinetic energy. If we try to measure whether they arrive within a time btc of each other, then our measurement fails unless we have at least 8tc > 1/E. Furthermore, we cannot construct j a two particle state where both particles arrive to a certain point within a time of 1/E ! of each other. The results in this Chapter support the limitations we have found for j measurements of arrival time. i In general, it appears that for two quantum mechanical events, one cannot determine i j which one precedes the other to arbitrary accuracy. Determining what order events ' occur in is not a trivial problem. This is interesting in light of attempts to construct | a quantum theory of gravity. In classical general relativity, the metric allows you to j determine whether two space-time events are space-like separated or time-like separated, ' and determine, relative to a coordinate system, which event occurred first. However, J when quantum mechanics is taken into account, it does not appear that there is a way Chapter 6. Order of Events 100 to do this to arbitrary accuracy, without affecting the system. In some sense, one may not be able to determine which way the light cone points. One could use the arrival of arbitrarily energetic particles in order to denote space-time events, and although one can increase the energy of the particles in order to increase the accuracy with which one is able to measure the order of events, at some point the energy of the particles will effect the curvature of the neighboring space time. Chapter 7 Conclusion 1 0 1 Chapter 7. Conclusion 102 We have argued that time plays a unique role in quantum mechanics. It is unlike other observables and one cannot naively assume it to be measurable. We have examined a number of different types of measurements of the time of an event, including measure-ments which involve continual monitoring of the system, coupling to physical clocks, measuring of current operators, and time-of-arrival operators. These various types of measurements given different results, and there does not appear to be any canonical method for measuring the time of an event. In the context of the time-of-arrival tA, we have found a basic limitation on the accuracy (as opposed to uncertainty) that tA can be determined reliably: 6tA > 1/Ek-This limitation is quit different in origin from that due to the uncertainty principle; here it applies to the inference of the value of time for a single event. Furthermore, unlike the kinematic nature of the uncertainty principle, in our case the limitation is essentially dynamical in its origin; it arises when the time-of-arrival is measured by means of a continuous interaction between the measuring device and the particle. While we know of no formal proof that this relation holds for time-of-arrival, our arguments are fairly general in nature. For the case of traversal time, we have argued that the limitation does not depend on any particular measurement procedure. We have also argued that monitoring whether the particle is at the location of arrival xA at various times, and also measuring the current operator, do not allow one to con-struct a probability distribution which one could interpret as representing the probability that the particle will arrive at a certain time. We would also like to stress that continuous measurements differ both conceptually and quantitatively from a measurement of the time-of-arrival operator. While the time-of-arrival operator is a formally constructed operator which can be measured by an impulsive von-Neumann interaction, continuous measurements are much closer to actual experiments. Furthermore, we have seen that the result of these two measurements do Chapter 7. Conclusion 103 not need to agree. In particular, at high accuracy, continuous measurements give rise to entirely different behavior - the particle never arrives. The time-of-arrival on the other hand, can be measured to any accuracy. However, the time-of arrival operator is not self-adjoint. Attempts to modify the time-of-arrival operator in such a way as to make it self-adjoint result in the problem that the particle does not arrive on time with probability 1/2. Operators which classically might give the time of an event cannot be given a physical interpretation. While several authors [9] [31] have maintained that the problems with defining an operator for the time of an event are technical, and can be circumvented by slightly modifying these operators. We have argued that probabilities in time are fundamentally different from traditional probabilities in quantum mechanics, and that there is a limitation on these measurements. As is the case with "time operators" [20] in closed quantum systems, the time-of-arrival operator has a somewhat limited physical meaning. We have also seen that one cannot determine the temporal ordering of events to arbitrary accuracy. The limitation on these measurements is once again given by 1/E where E is the typical total energy of the system. Nor can one prepare a two particle system in a state in which the two particles always arrive within a time 1/E of each other. However as with most research, this thesis raises more questions than it answers. Does a formalism exist where time is an element of reality? If not, does there exist a proof of the minimum inaccuracy bounds we have proposed? More intriguing, are some of the connections between this research, and the problem of time in quantum gravity and quantum cosmology. In the Introduction we briefly discussed the canonical approach to quantizing gravity. One immediately encounters the problem that relative to the external parameter time in the Schrodinger equation, the state of the universe does not evolve. This is because the Chapter 7. Conclusion 104 system must satisfy constraints which are equivalent to reparametrization of the time variable. The situation is somewhat analogous to being inside a box, and having some external observer weigh the box with high accuracy [41]. In order to keep the box at this fixed weight, the external experimenter cannot measure observables which evolve in time. Quantum mechanics also dictates that the observer will see people inside the box in a superposition of many different ages. This is because observables which would allow one to infer the time are (in a sense) conjugate to energy (they can't be exactly conjugate to the energy as we learned in Chapter 4). This gives us a rather interesting way to perform the Schrodinger cat experiment [42] (see the Figure at the beginning of this Chapter). Take an animal (Schrodinger's poodle, for example), stick her in a box, and weigh the box accurately. If the box is sufficiently isolated from the environment (a very difficult task), the poor poodle will be in a superposition of herself at different stages of her life. If we weigh the box very accurately, and later look at the age of the poodle, we will sometimes find that the poodle is so old that she died (or so young that she was just matter waiting to be born) - she is in many superpositions of being alive and dead. If we measure the weight of the box with infinite accuracy, then essentially any time we look inside the box, we will find nothing but poodle dust. In general relativity, which describes the entire universe, all of us, observers and the observed, are in some sense living inside a box of fixed energy. In this regard, the recent results of Aharonov and Reznik [43] are interesting. They have shown that if one attempts to measure the energy of a closed system from within that system, then the time required to make this measurement must be at least 1/6E where 6E is the precision with which one desires to make the measurement of energy. This result is very closely related to the measurements discussed in this thesis. Of course, an observer outside the box can measure the box's energy in as short a time and as accurately as desired [44]. Chapter 7. Conclusion 105 These examples, which arise out of trying to understand quantum gravity, have led us to examine the role that time plays in ordinary quantum mechanics. We have argued that measurements of the time of an event are fundamentally different from ordinary observ-ables in quantum mechanics. Certainly quantum mechanics is entirely self-consistent, and yet, questions remain about the role of time in the theory. 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Morse and Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, New York, 1953 Appendix A Zero-Current Wavefunctions One interesting aspect of the detector discussed in Section 3.3.2, is that while it can be used for wave-packets arriving from the left or the right, it will not always be triggered if the wavefunction is a coherent superposition of right and left moving modes. Consider for example, the superposition if}{x) = Ae'kx + Ae -ikx (A.216) One can easily verify that the current 1 j(x,t) = -i-2m ^dtp(x,t) dip*(x,t) ^ ( x ^ ) — 5 1 ^z,—#M) (A.217) dx dx is zero in this case. |^(0, t)\2 is non-zero, although the state is not normalizable. As in eq. (3.63) this state evolves into 0*#>l Tx> V2 feikx + e-ikxy y + ( e , * x + e - , t o ) | [ x ikx i ikx} (A.218) Which, when rewritten in the az basis, is just A{elkx + e-lkx)\ lz). (A.219) i.e. the detector is never triggered. This wavefuntion is similar to the antisymmetric wavefunctions discussed by Yamada and Takagi in the context of decoherent histories [37] and Leavens [38] in the context of Bohmian mechanics, where also one finds that the particles never arrive. How to best treat these cases is an interesting open question. 110 Appendix B Gaussian Wave Packet and Clocks Using the simple model of Section 3.3.1 (3.44), we now calculate the probability distri-bution of a clock which measures the time-of-arrival of a Gaussian wave packet. We will perform the calculation in the limits when the clock is extremely accurate and extremely inaccurate. The wave function of the clock and particle is given by (3.52) and the dis-tributions are both Gaussians given by (3.53). In the inaccurate limit, when p0 « k, AT ~ 1. We trace over the position of the particle on the condition that the clock was triggered, ie. x > 0. p{y,y)*>o = J dx\iP(x>0,y,t)\2 (B.220) ~ N2 I" dkdk' Hdpdp' dxg(k)g%k')f{p)f%p')el(q-q')x+,(p-p')y-'(q^ J-oo Jo After a sufficiently long time, ie. / >> ta the wave function has no support on the negative x-axis, and if pa > I/Ay, then it will not have support in negative p. We can thus integrate p and x over the entire axis. Integrating over x gives a delta-function in q. We can then integrate over p' to give p(y,y)x>0 ~ ±±- / dkdk'dpyjk* + 2mpg(k)g*(k>)f(p)r(p+ — y ^ - * 8 ) * m J v 2m where we have used the fact that 8(f(z)) = P(~J?\ when f(za) = 0. The square root term varies little in comparison with the exponential terms and can be replaced by its average value y / ^ + 2mp 0 ~ k0. Integrating over p gives p(y,y)x>0 * l^^Jdfcdfc'e^(fc+^a^^^(iby(jfc0e«^^. (B.221) 111 Appendix B. Gaussian Wave Packet and Clocks 112 Since Ay k » 1, for a wave packet peaked around kQ we can approximate the argument - A y 2 k 2 m 2 of the first exponential by A v2f c p (fc — A;')2. This allows us to integrate over fc and fc' p{y, y)>0 ~ -F=^=e (B.222) \J'^i{y) where the width is 7 (y ) = A y 2 + ( ^ ) 2 + ( ^ ) 2 . As expected, the distribution is centered around the classical time-of-arrival tc = x0m/k0. The spread in t/ has a term due to the initial width Ay in clock position y. The second and third term in j(y) is due to the kinematic spread in the time-of-arrival 1/dE = ^ and is given by d x ^ m where dx(y)2 = A . T 2 + i-jj^)2- The y dependence in the width in x arises because the wave packet is spreading as time increases, so that at later y, the wave packet is wider. As a result, the distribution differs slightly from a Gaussian although this effect is suppressed for particles with larger mass. When the clock is extremelv accurate ie. pQ » kQ we have AT ~ kj—. 1 u u j. y m.p 2N~ f°° . I°° . fcfc , * / , / \ ft \ / • * / / \ i(n-n')T.-t-i(n-n'Ui-p(y,y)x>0 / dkdk' dpdp'dx-j^giky/ik'Jf^rip'y* m . / - o o Jo y j v v Ipp 4TT7V2 r „ , , fcfc' ' m k2 + 2mP n(^nVy^ f ( r A f*fn M k--k'- r(k'2-k^ 2m } f dkdk'dp—. * +,:Za(k)g*(k')f(p)r(p + Since pa >> k0, we can approximate this integral as A p(y, y)x>0 ~ - | fdkk g(k)e-^ | 2 (B.223) m J where A = A-ir^^N2 J ^\f(p)\2- We can approximate p by p0 to take it outside the integrand, giving A* (B.224) mp0 7T2 The final integration over fc yields Appendix B. Gaussian Wave Packet and Clocks 113 where the width 7(2/) = Ax2 + (9kvAx)2 is independent of Ay because the kinematic spread in the time-of-arrival 1/dE is much larger than the spread in the position of the clock. In this limit we see two additional factors. The amplitude decays like yE0/p0 so that improved accuracy decreases our chances of detecting the particle. Also, there is a minor correction of More energetic particles with faster arrival times are more likely to trigger the clock. Appendix C Time-of-Arrival Eigenstates We will now show that the eigenstates of [9] and those of the unmodified time of arrival operator do not correspond to a delta function at the time-of-arrival tA but are instead 3 proportional to x~~?. Using the Schrodinger representation, we see that at time tA, these eigenstates (eg egfA(x,t = tA) of eqn. (4.4)) in the x-representation are given by dke-^e-^e9lA(k) - o o Jo V 27rm vfc \ rn J+e k) (C.226) As e goes to zero the first integral goes to zero. So we get < < ^ = ^ = & 3 / 2 f <c-2 2 7 ) where we have added a small imaginary part to x to make the integral converge, and then set it to zero at the end. One finds the same behaviour for the eigenstates of the unregularized time-of-arrival operator |T). The reason is that as e goes to zero, the modification of the eigenstates only occurs at k — 0 which is a set of measure zero. 114 

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