DISSERTAÇÃO DE MESTRADO IFT–D.0018/20 The issue of time in Quantum Mechanics Matheus Hrabowec Zambianco Orientador George E. A. Matsas Março de 2021 Zambianco, Matheus Hrabowec Z24i The issue of time in quantum mechanics / Matheus Hrabowec Zambianco. – São Paulo, 2021 126 f. Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: George Emanuel Avraam Matsas 1. Mecânica quântica 2. Relatividade geral (Física). 3. Espaço e tempo. I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). Agradecimentos Aos meus pais, Marcelo Zambianco e Nádia de Oliveira Zambianco. Sou eternamente grato por todo o suporte material e emocional ao longo de todos estes anos, com est́ımulo cont́ınuo ao meu crescimento. À minha famı́lia, por todo o apoio e incentivo à minha vida nos estudos. Em partic- ular, agradeço à minha avó, Dorcelina M. de Oliveira Zambianco, por ter me recebido em sua casa no ińıcio do mestrado, assim como os meus padrinhos, Márcio de Stefano e Kelly Zambianco. Ao meu falecido avô, Antônio Zambianco, que certamente teria muito orgulho desta conquista. Aos meus amigos do IFT, pelas inúmeras conversas produtivas — sobre F́ısica ou não, mas jamais sobre string theory — que certamente tornaram o mestrado mais leve. Aos amigos em geral, tão necessários para a nossa vida enquanto seres humanos. No que se refere ao meu conhecimento de F́ısica, merecem destaque as discussões profundas com Otávio Luiz Canton, Tales Rick Perche e Paulo Sérgio Piva. Obrigado. À minha namorada e companheira, Gabriela de Barros Quirino. Agradeço profundamente todo o carinho e amor, bem como o incentivo cont́ınuo ao meu crescimento profissional e pessoal. Sem dúvidas, você teve um papel importante em tornar este desafio realizável. Ao meu orientador, George E. A. Matsas. Agradeço toda a atenção dedicada a mim. Certamente, levarei um grande aprendizado de nossas discussões ao longo do peŕıodo de mestrado. T ı́πoτα δεν έıναi εúκoλo. Este trabalho foi financiado pela Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, sob o número 2018/24810-5, e pela Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior, CAPES. ii Abstract Time is everywhere. A concept so ubiquitous and natural for us, human beings, that we rarely wonder about its nature. For science, there is a clear need of understanding more and more about the fundamental structure of the world around us, so that an understanding about the concept of time surely is a noble target for physics. In this essay, we explore the concept of time from the perspective of two conflicting paradigms: the relativist and the quantum one. If on the one hand General Relativity allows to characterize time as that measured by honest clocks — pointwise apparatus that neither delay or anticipate irrespective of its past history —, on the other the laws of Quantum Mechanics forbid its existence in nature. Our proposal is to furnish a precise understanding about the reason for such prohibition, as well as a solid perspective about the role of time in quantum theory as a measurable quantity. Key-words: Time, Quantum Mechanics, Relativity Field of knowledge: Quantum Fundamentals iii Resumo O tempo está em todo lugar. Um conceito tão onipresente e natural para nós, seres humanos, que raramente indagamos sobre a sua natureza. Para a ciência, existe a necessidade clara de se entender cada vez mais sobre a estrutura fundamental do mundo a nossa volta, de modo que um entendimento completo sobre o tempo certamente é um objetivo nobre para a F́ısica. Nesta dissertação, investigamos o conceito de tempo a partir da perspectiva de dois paradigmas fundamentalmente conflitantes: o relativ́ıstico e o quântico. Se por um lado a Relatividade Geral permite caracterizar tempo como aquilo medido por relógios honestos — aparatos pontuais que não atrasam e nem adiantam independemente de suas histórias passadas —, por outro, as leis da Mecânica Quântica proibem a existência dos mesmos na natureza. Nossa proposta é fornecer um entendimento preciso da razão de tal proibição, bem como uma perspectiva sólida do papel do tempo na teoria quântica enquanto uma quantidade pasśıvel de medição. Palavras-chaves: Tempo, Mecânica Quântica, Relatividade Áreas do conhecimento: Fundamentos da teoria quântica iv Contents Contents v List of Figures vii Summary of notations and conventions x Introduction 1 1 Time in classical physics 4 1.1 The spacetime of General Relativity . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Using clocks to measure spatial distances . . . . . . . . . . . . . . . . . . . . 13 1.3 Honest clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 What is time? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Essential aspects of Quantum theory 20 2.1 Basic concepts of probability theory . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The probabilistic interpretation of Quantum Mechanics . . . . . . . . . . . . 23 2.3 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 Generalized observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 The time-energy uncertainty relation 43 3.1 The Mandelstamm-Tamm relation . . . . . . . . . . . . . . . . . . . . . . . . 44 v 3.2 Lifetime of a property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 The Aharonov - Bohm point of view . . . . . . . . . . . . . . . . . . . . . . 51 3.4 A derivation using the fact that “energy weighs” . . . . . . . . . . . . . . . . 54 3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Time observables in Quantum Mechanics and the nonexistence of honest clocks 59 4.1 Pauli’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Time as a POVM in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 61 4.2.1 Interpreting the eigenstates of the Aharonov-Bohm time operator . . 69 4.2.2 Example: time of arrival probability distribution for a Gaussian wave packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Ruling out the existence of honest clocks in Quantum Mechanics . . . . . . . 77 4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Quantum Clocks 82 5.1 The Salecker-Wigner-Peres quantum clock . . . . . . . . . . . . . . . . . . . 82 5.2 Larmor clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Harmonic oscillator as a quantum clock . . . . . . . . . . . . . . . . . . . . . 98 6 Conclusion 108 A The pictures of Quantum Mechanics 110 B A result from complex analysis 113 C Mathematical properties of the Aharonov-Bohm time operator 117 D Special functions used 121 Bibliography 122 vi List of Figures 1.1 A representative diagram of the Minkowski spacetime with one spatial dimen- sion, displayed using Cartesian coordinates where the observer O — green line — is at rest. The red dotted-line stands for the trajectory of a light-ray, which determines the causal structure of this spacetime. The blue and purple lines are timelike curves connecting events A and B, the former representing the trajectory of an inertial observer and the latter of an accelerated observer. The events A and C are spacelike-separated: they can happen simultaneously from the point of view of a specific inertial observer, but cannot have any absolute causal relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Minkowski diagram describing the experiment of Martha and Jonas. They want to measure the distance d based on the readings of three clocks. . . . . 15 3.1 Illustrative scheme of the gedanken experiment. The size of the internal clock is exaggerated for the sake of clearness; the clock is located inside the spherical shell that is being ejected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Illustrative diagram of an time-of-arrival experiment. Given a particle pre- pared at a state |ψ〉, all we want to know is the probability that the detector D fires at a time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 vii 4.2 Time of arrival distribution for temporal wave packets centered at t = t0. The parameter δ is inversely proportional to the width of the wave packet. The probability distributions are not normalized (obviously, this does not change the qualitative features that one can grasp by staring at these results). . . . 71 4.3 Time of arrival distribution for a particle with mass m = 1 described by a Gaussian wave packet whose center has position x0 = 1 and momentum p0 = −1. The parameter σ represents the width of the packet in position representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Time of arrival distribution for Gaussian wavepackets approaching the limit of a classical state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Time of arrival distribution for a Gaussian wave packet with x0 = p0 = 0. The particle’s mass was set equal to 1. . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Comparing the readings of the SWP quantum clock with those of a classical clock that measures the evolution parameter t. . . . . . . . . . . . . . . . . . 86 5.2 Quantum clock readings and its confidence interval. . . . . . . . . . . . . . . 87 5.3 Simplified representation of the physical situation corresponding to the Larmor clock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Phase distribution for coherent states of the harmonic oscillator. . . . . . . . 105 5.5 Phase readings and corresponding uncertainties for a coherent state with |α|= 2.107 viii Summary of notations and conventions In this work, we use natural units, setting c = h̄ = G = 1. The spacetime metric has signature (−,+,+,+) • (M, g): spacetime with metric tensor g • TpM: tangent space at the point p. • Einstein summation convention is used, with greek indices running from 0 to 3 and latin ones from 1 to 3. Examples: . Tµνv ν = 4∑ ν=0 Tµνv ν . xix i = 3∑ i=1 xix i . RµνR µν = 3∑ µ=0 3∑ ν=0 RµνR µν • H: complex Hilbert space • L(H): set of all linear operators acting on H. We distinguish operators by using a “hat”: Â, B̂, etc. • L(H)+: the subset of L(H) containing only the positive operators. • †: adjoint of the operator  ix • L2([a, b]): Hilbert space of the square-integrable functions defined over the interval [a, b]. x Introduction The difference between past, present and future is a mere illusion. Time is everywhere. No matter who you are or where you go, it is always with you, a silent and unceasing flow. A concept so ubiquitous and natural for human beings that we rarely wonder about it, except when we face it wickedness on continuing to push us forward to our unavoidable destiny. However, when we are asked about what is time, surely no trivial and short answer readily comes out. As well grasped by Saint Augustine 1: “What then is time? If no one asks me, I know; if I want to explain it to a questioner, I do not know”. Perhaps our perception of time is inseparable from that of change. Systems in nature are in constant change, and those events that happen in a periodic manner — like the sunrise — can be used to mark the passage of time. In this scenario, one could think about time as a human abstraction created to describe succession of events, like the successive positions assumed by the hand of a clock. When it come to physics, time is present in almost every description of natural phenom- ena.2 Before the advent of Einstein’s theory of General Relativity 3, the physical theories were shaped within the absolute time paradigm coined by Isaac Newton (1643 - 1727), ab- solutely befitting with our human intuition. The beginning of the XX century brought the rupture with this paradigm; the concepts of space and time were unified within a more fun- 1Saint Augustine (354 - 430) was a remarkable philosopher for the Christianity. Among his writings, there are reflections about the nature of time. 2If you are a physicist, I bet you to think about what was the largest time you worked without using the concept of time. 3Albert Einstein (1879-1955). 1 damental underlying structure: the spacetime. This new abstract structure was posed as the “background” where all physical theories should be defined. In general lines, the message of General Relativity is that “different observers extract different space and time contents from the same spacetime.”4 The effect of gravity is to change the spacetime geometrical features, defining in an univocal manner how different events (spacetime points without any temporal or spatial extension) relate to each other and, thence, the values of the physical quantities that different observers will measure when performing certain experiments. Within the classical physics paradigm — which considers General Relativity — it is possible to supply an objective definition for the concept of time: Time is what honest clocks measure. At first sight, such a definition may seem circular. However, since physics is a experimental science, it does not make sense to define an observable without telling how to measure it; in this sense, there is no problem to outscore the definition of time for that of an honest clock, provided we give a clear prescription of such apparatus using the variables of the theory. Honest clocks are pointwise apparatus that irrespective of its past histories attribute the same real number to each pair of events arbitrarily close they visit on spacetime. As we shall argue, there is no problem with this definition within the scenario of General Relativity. The issues come out when one realizes the honest clocks, while apparatus that supposedly exist in nature, must ultimately be subject to the laws of Quantum Mechanics. Thus, we have the following questions: How Quantum Mechanics impacts on the physical reliability of honest clocks? If, somehow, Quantum Mechanics forbids the existence of honest clocks, how one can use the theory to predict the time interval between two events? Furthermore, it is important to realize that insofar as these questions explore the concept of time in the context of both quantum and relativity theories, they lead us to pursue an even more fundamental scientific problem: what would be time in a (real) quantum theory of gravity? 4I must admit that this statement, which is a kind of catchphrase of my advisor, is perhaps the most meaningful way of talking about Relativity in just one line. 2 This essay is organized as follows: Chapter 1 is devoted to the definition of time according the the Classical Physics paradigm. After a brief review about some fundamental aspects of General Relativity, we define the concept of an honest clock and, then, formulate the main issue of the essay: How Quantum Mechanics impacts on the definition of honest clocks? For the reader who is not familiar with General Relativity and is only interested in Quantum Mechanics, we stress that this first chapter can be safely skipped without any loss to the understanding of the main topic. Chapter 2 contains a review on the subject of Quantum Mechanics, focusing on the ex- tended formulation of observables as described by Positive Operator Value Measures (POVMs). In chapter 3, we begin our analysis on the issue of time in Quantum Mechanics through a discussion about formulations and interpretations of the time-energy uncertainty relation. Chapter 4 is concerned with a celebrated no-go theorem due to Pauli that points out the impossibility of constructing a self-adjoint operator canonically conjugated to the Hamilto- nian of any physical system. However, given the interpretation of observables as described most generally by POVMs, we show how to use such formalism to construct a time observable in the quantum realm for a particular — but rather interesting — case. Finally, we discuss a fundamental result due to Wald and Unruh that rules out the realizability of honest clocks in Quantum Mechanics. In the last chapter, we address the issue of using particular quantum systems as quantum clocks. The conclusion then follows. 3 1 Time in classical physics This first chapter is concerned with a precise definition for the concept of time within the classical physics paradigm. By “classical physics”, we mean all theories that do not take Quantum Mechanics into account. Among these theories, General Relativity surely is the one that — until now — best describes the fundamental notions of space and time. Thus, we begin by giving an overview about General Relativity, focusing on the conceptual features that will be crucial for defining time itself. Since the present work focuses on issues arising from Quantum Mechanics, the discussion about General Relativity will be brief. For those willing to read more about the subject of General Relativity and its underlying mathematical structure, we recommend references [1] and [2]. 1.1 The spacetime of General Relativity The spacetime is the background where all the physical theories are to be defined. Before the advent of General Relativity, all classical physics can be seen as taking place in the Galileo spacetime, where time is an absolute entity. Different observers could disagree about space measurements — what is really intuitively based on our own daily experience — but never about time measurements. 4 This paradigm is changed when Albert Einstein (1879 - 1955) introduces the special theory of Relativity in the year of 1905, thereby showing that the Minkowski spacetime is the “correct” background to describe all physical phenomena in the absence of gravity. Later on 1915, General Relativity would teach us how gravity can be understood as the own geometric structure of spacetime. In this new scenario, different observers extract different space and time contents from the same spacetime. Both space and time measurements outcomes are dependent on the observer who is asking the experimental questions. As a consequence, notions like simultaneity, past and future — which seem to be absolute based on our everyday experiences — also need to be reformulated in an observer-dependent way. Formally, the spacetime of General Relativity is a four-dimensional pseudo-Riemannian manifold M equipped with a non-degenerated bi-linear form g, the so-called metric tensor. The set M, which has the usual structure of a differentiable manifold,1 is the collection of the spacetime events, whereas the mathematical object g is what allows one to ascribe a real number— the distance — between each pair of events, being understood as the ultimate description of gravity itself. The adjective “pseudo-Riemannian” comes from the fact that the metric g is not positive-definitive — in which case we would have a Riemaniann manifold —, thus meaning that distances between different events ofM can be null or even negative. Since distances between events on a spacetime cannot be interpreted as the usual “pos- itive distances” of our everyday life Euclidian geometry, it is necessary to give a proper physical interpretation to the new possibilities unraveled when one uses all the tools of pseudo-Riemanninan geometry to describe the spacetime. Indeed, vectors in a spacetime can be divided into three classes, according to the value of its length: given a vector v ∈ TpM, we say that • v is timelike if g(v, v) < 0 • v is spacelike if g(v, v) > 0 1Intuitively, a differentiable manifold of dimension n is a space that locally behaves like Rn, being the proper generalization to the concept of surfaces. If you want to review the mathematical definition of a manifold, see chapter 2 of reference [2]. 5 • v is null if g(v, v) = 0 This same nomenclature applies to vector fields defined in an open set ofM. In particular, if v is the tangent vector field to some curve γ : I ⊂ R → M, γ is to be named after the classification of v. Physically, such a classification is important to determine the causal structure of the spacetime: two events that can be joined by a timelike curve are causally connected. That is: if A, B ∈ M can be joined by a timelike curve and some observer says that A occurred before B, then all observes will agree on this causal relation. In this way, timelike curves are the worldlines of massive particles. On the other hand, events joined by a spacelike curve have no causal relation, so that no object — massive or not — can follow a spacelike trajectory on the spacetime. As for the null curves, they define the trajectories of the massless particles that travel at the speed of light. In principle, it should be possible to discuss all relativity issues without using coordinates. After all, physical phenomena do not care about which coordinates one chooses to describe them. Nonetheless, one cannot deny that without coordinates the physicist task would be much harder2 or even impossible. For this reason, we follow the common trend and introduce coordinates in order to discuss a few more issues in a clearer way. Given a generic spacetime (M, g), one can always introduce a system of coordinates (xµ) covering some open set U ⊂M. Thus, the description of the metric tensor becomes g = gµνdx µ ⊗ dxν . (1.1) The components gµν are to be understood as real functions of the coordinates describing each point of U , whereas the objects dxµ, when considered at some point p ∈ U , are the one- form basis for the cotangent space T ∗pM. The corresponding coordinate basis for the tangent space TpM is denoted by ∂ ∂xµ (the evaluation point is omitted by the sake of simplicity), being fully characterized by the relation 2It is already hard enough! 6 dxµ ( ∂ ∂xν ) = δµν . (1.2) When there is no room for ambiguities regarding the coordinate system being used, the coordinante basis vectors are denoted by ∂µ. In a more general fashion, a tensor field T defined over U can be written as T = Tα1...αr β1...βs ∂ ∂xα1 ⊗ . . .⊗ ∂ ∂xαr ⊗ dxβ1 ⊗ . . .⊗ dxβs . (1.3) The actual description of the tensor T is then read from its components. In particular, the action of the metric over two vector fields v = vµ∂µ and w = wµ∂µ can be written as g(v, w) = gµνv µwν . (1.4) Within this formulation, the distance between arbitrarily close events is given by ds2 = gµνdx µdxν . (1.5) This is the so-called line element. Usually, this object is referred as the metric itself. Of course, such abuse of nomenclature is no harmful: given the relation (1.5), we can extract all physical features from spacetime. The actual dynamics of free particles — here understood as particles subject solely to the action of gravity — in the spacetime is described by geodesics, trajectories that minimize the line element (1.5). Let γ(λ) = (xµ(λ)) be a curve with affine parameter λ and tangent field uµ = dxµ dλ . If γ represents a valid trajectory for a real particle on spacetime, it needs to be either timelike or null. In the first case we have uµuµ = −1, whereas for null curves we have uµuµ = 0. On both cases, uµ is called the four-velocity of the particle. 7 One way to impose γ to be a geodesic is to demand that its intrinsic acceleration must be zero. By intrinsic acceleration, we mean the covariant derivative of the four-velocity u = uµ∂µ along the curve γ, namely ∇uu = 0 =⇒ d2xµ dτ 2 + Γµαβ dxα dτ dxβ dτ = 0. (1.6) Here, the quantities Γµαβ are the Levi-Civita connection symbols and ∇ is the connection itself (see reference [2] if you want to review the subject of connections and covariant deriva- tives). In the usual formulation of General Relativity, the connection — an abstract rule that allows one to take derivatives of tensor fields on manifolds — chosen is the Levi-Civita’s, the only that is symmetric and compatible with the metric tensor g. This is a derivative rule that is rather natural if one assumes that the manifold can be immersed in a large Euclidean space. In terms of the metric components, we have (with ∇∂µ ≡ ∇µ) ∇α∂β = Γµαβ∂µ =⇒ Γµαβ = gµσ 2 (∂αgσβ + ∂βgασ − ∂σgαβ) . (1.7) Now, let us turn our attention to more fundamental issues regarding how an observer can extract space and time contents from spacetime. Consider an observer characterized by the four-velocity uµ. Any vector field X = Xµ∂µ admits a decomposition of the form Xµ = Υµ νX ν︸ ︷︷ ︸ timelike vector + Πµ νX µ︸ ︷︷ ︸ spacelike vector , (1.8) where Υ and Π are the projectors onto the timelike and spacelike subspaces with respect to the observer uµ. That is: we are decomposing the vector X into a portion that lies on what the observer calls space and another portion over what he calls time. Since such a construction is observer dependent, we can take it as the mathematical realization of the statement that different observers extract different space and time contents from the same 8 spacetime. The expression for the time projector Υ is Υµ ν = −uµuν , (1.9) so that the spacelike part of X according to the observer with four-velocity u can be calculated through the relation Πµ νX ν = Xµ + (uνX ν)uµ. (1.10) A concept that is central to Relativity is that of proper time. Given two events A and B joined by a timelike geodesic on M whose coordinate description is γ(λ) = (xµ(λ)), with A = γ(λ1) and B = γ(λ2), the proper time between the events can be calculated as ∆τAB = ∫ λ2 λ1 dλ √ −gµν(λ)uµuν , (1.11) where uµ = dxµ dλ . Notice that if λ is taken to be an affine parameter τ along γ, then the previous expression reduces to ∆τAB = τ2 − τ1. (1.12) Proper time has a straightforward interpretation: if a free-falling observer carrying a clock follows the trajectory γ, the quantity ∆τAB would be the time, as told by his clock, between the events A and B. Nonetheless, any observer following a distinct trajectory on the spacetime would say that the time elapsed between events A and B is different. Indeed, an observer with four-velocity vµ says that the time elapsed between the events A and B is 9 ∆tAB = ∫ λ2 λ1 dλ √ −gµν(Υµ αvα)(Υν βv β) = ∫ λ2 λ1 dλ (vαuα) √ −gµνuµuν . (1.13) For spacelike-separated events, one can always find some observer O that says the events are simultaneous. Mathematically, this amounts to choosing a coordinate description nat- urally associated with the rest frame of this observer, where both events lie on the same spacelike surface determined by fixing the value of x0. In this case, those events can be used to define the proper length of some object. Indeed: given two events A and B linked by a spacelike geodesic γ(λ) = (x0, xi(λ)), with A = γ(λ1) and B = γ(λ2), the proper distance between them reads ∆LAB = ∫ λ2 λ1 dλ √ gij(λ) dxi dλ dxj dλ . (1.14) From the point of view of another observer, the events A and B may not be simultaneous. Indeed, let w = ( 0, dxi dλ ) and consider an observer P following a timelike worldline with four- velocity v = vµ∂µ such that vµwµ 6= 0. In this case, the observer K believes that the events A and B are separated in time by an amount ∆t (K) AB = ∫ λ2 λ1 dλ (vαw α) √ −gµνvµvν . (1.15) The simplest spacetime model is the Mikowski spacetime, whose line element in Cartesian coordinates reads ds2 = −dt2 + dx2 + dy2 + dz2. (1.16) Both null and timelike geodesics in this spacetime are straight-lines. Curved lines rep- 10 resent non-geodesic (accelerated) observers. Below, we display a diagram representing some basic features of the Minkowski spacetime. A B C t x O Figure 1.1: A representative diagram of the Minkowski spacetime with one spatial dimension, displayed using Cartesian coordinates where the observer O — green line — is at rest. The red dotted-line stands for the trajectory of a light-ray, which determines the causal structure of this spacetime. The blue and purple lines are timelike curves connecting events A and B, the former representing the trajectory of an inertial observer and the latter of an accelerated observer. The events A and C are spacelike-separated: they can happen simultaneously from the point of view of a specific inertial observer, but cannot have any absolute causal relation. Such a spacetime is flat, meaning that it posses no intrinsic curvature. That is: if one computes all the components of the Riemman curvature tensor, one finds that they all vanish: Rµναβ = 0. Physically, this means that this is a spacetime without gravity. Including gravity, the space- time in the vicinity of a spherical massive object is well described by the Schwarzschild metric: 11 ds2 = − ( 1− 2M r ) dt2 + ( 1− 2M r )−1 dr2 + r2(sin2 θdθ2 + dφ2) (1.17) Studying the timelike and null geodesics of this spacetime, it is possible to predict two important phenomena that are important tests for the theory: the perihelion precession of Mercury and the deflection of light by the sun. Contrary to (1.16), the Schwarzschild metric describes a curved spacetime. Mathemati- cally, this is represented by a non-vanish Riemann tensor, as outlined above. Both Minkowski and Schwarzschild metrics are solutions of the Einstein’s equations: Rµν − R 2 gµν = 8πTµν . (1.18) Also known as the field equations of General Relativity, the equations above relate the matter-energy content of the spacetime — described by the energy-momentum tensor Tµν — with its intrinsic geometric strucuture. Recall that the Ricci tensor is the only independent contraction of the Riemman tensor; in components notation, we have Rµν = Rα µαν . (1.19) A fundamental point that is worth mentioning here is the Equivalence Principle, the General Relativity’s conceptual cornerstone. Essentially, this principle states that the physics in a freely-falling frame is the same of special relativity. Formally, this principle has its mathematical counterpart in the existence of the so-called Riemann normal coordinates (see chapter 5 of reference [2] for more details). Indeed, around each point p of a generic spacetime (M, g), one can introduce a system of coordinates (nµ) where, up to second order in the curvature, the metric components takes the form gµν(n σ) = ηµν(p) + 1 3 Rµαβν(p)n αnβ +O(R2), (1.20) 12 where ηµν = diag(−,+,+,+) is the Minkowski metric. That is: it is always possible describe the spacetime metric at a particular point as the Minkowski metric. Of course, locally a curved spacetime still remains distinguishable from a spacetime without gravity through experiments sensible to Rαβµν . Now, it is time to state our main issue. First of all, we claim that if somehow one has access to the value of the proper time (1.11) between every pair of events belonging to a generic spacetime (M, g), then one has a complete description of the geometric structure of the spacetime. That is: measuring the proper time between two events can, in principle, distinguish a particular spacetime from all other. This statement will be sustained on the next section, where we show how clocks can be used to also measure spacelike distances as well. Next, it is important to recall that in physics, observables have no meaning at all if it is not possible to measure them. Then, concerning theoretical fundamentals, we need to introduce an apparatus that allows one to measure the proper time between two events belonging to a given spacetime. In doing so, notice that such an apparatus will allows one to test the spacetime itself, insofar as the recording of proper time measures will reveal the geometric structure of spacetime itself. Before introducing the aforementioned apparatus, we need to convince the reader that there is a “hidden” asymmetry between space and time concepts on a classical spacetime, making time measurements more fundamental — in a sense to be specified afterwards — than spatial ones. 1.2 Using clocks to measure spatial distances There is no doubt that General Relativity treats space and time in a unified picture. Nonetheless, this does not mean that they have the same nature. Indeed, the fact all geo- metrical features are determined by a pseudo-Riemannian metric with signature (−,+,+,+) 13 already indicates that all the four spacetime dimensions cannot be interpreted as sharing the same features. The own distinction between timelike and spacelike distances also made clear the difference regarding space and time. Now, we are going to argue that everything we need to extract space and time mea- surements from the spacetime is a clock. At this point, the shall call a clock any pointwise apparatus that allows one to measure the proper time along some timelike trajectory. Notice that the hypothesis of being pointwise is necessary in order to assure that the clock follows this same trajectory. Later on, we shall make this notion more precise introduction the concept of an honest clock. In the Minkowski spacetime, consider the following experiment: two observers, Jonas and Martha, are at rest in the same inertial frame and located a distance d apart. Consider a system of Cartesian coordinates (x, t) such that x = 0 is Jonas’s position and x = d is Martha’s. At t = 0 (event A), Jonas sends a clock to Martha with constant velocity v1. Martha grabs the clock at the event B and, immediately, sends back a clock moving with constant velocity v2, which reaches Jonas at event C. During all this process, Jonas has a clock with him that measures a proper time τ between the events A and C. As for the traveling clocks, the first one attributes a proper time of τ1 between the events A and B, whereas the clock sent by Martha reads τ2 between B and C. This situation is described on the spacetime diagram of figure 1.2 below. 14 x t A B C τ1 τ2 τ d Figure 1.2: Minkowski diagram describing the experiment of Martha and Jonas. They want to measure the distance d based on the readings of three clocks. Our claim is that the distance d can be expressed exclusively in terms of the proper times τ , τ1 and τ2. Let us prove this. The first thing to do is to recall how proper time is related to the coordinate time t. From the Minkowski metric in one spatial dimension, we have dτ 2 = −ds2 = −dt2 + dx2 =⇒ dτ = √ 1− v2dt, (1.21) where v = dx dt is the velocity as measured in the rest frame of Jonas and Martha. Thus, if tB denotes the coordinate time of the event B, we can write τ1 = tB √ 1− v2 1 (1.22) and τ2 = (τ − tB) √ 1− v2 2. (1.23) Notice that v1 = D tB and v2 = D τ − tB . Then, using relation (1.23), we get 15 tB = τ − √ τ 2 2 +D2. Substituting this on equation (1.22) and performing further algebraic manipulations, we obtain our result: d = 1 2τ √ (τ 2 1 + τ 2 2 − τ 2)2 − 2(τ1τ2)2. (1.24) Therefore, we see that it is possible to design a measurement scheme where spatial dis- tances are measured only with clocks. For a generic spacetime, this same conclusion holds provided one works locally. Thus, using solely clocks, only can, in principle, determine the geometric structure of the spacetime. At this point, one can ask: why not try to use rules to measure time intervals? After some thinking, you should convince yourself that there is no natural way of doing that without resorting to some other object (like a light-ray). So far, we do not know any example in the specialized literature showing that this is even possible. Thus, we shall work under the hypothesis that clocks are the most fundamental measuring tools that one can introduce on spacetime. Recall that we have introduced a clock as being just an apparatus that can be used to measure the proper time between two timelike-separated events. Moreover, we have shown that clocks can also be used to measure spacelike distances. Now, it remains to discuss more deeply what we mean by a clock. Additional restrictions on this apparatus will led us to introduce the concept of honest clock. 1.3 Honest clocks Honest clocks are pointwise apparatus that attribute the same real number to each pair of arbitrarily close events they visit irrespective of its past history. Equivalently, they can be characterized as periodic dynamical systems whose evolution does not depend upon the tra- 16 jectory described by the clock on the spacetime. Furthermore, in order to avoid ambiguities, we always take this trajectory to be a timelike geodesic. More precisely, let γ : I ⊂ R →M be a timelike geodesic followed by a clock. Without loss of generality, we can always assume that γ can be parameterized by the proper time τ . Moreover, consider that the trajectory of the clock in its phase space is described by the map Γ(λ) = (qi(λ), pi(λ)), where qi and pi are canonically conjugated dynamical variables, with i = 1, . . . , n. Then, for such a clock to be called honest there must exist some function f(qi(λ), pi(λ), λ) linear with respect to the real parameter λ whose output is the time as told by the clock. That is: we are assuming that it is possible to construct an one-to-one association between the parameter λ that keeps track of the clock’s internal evolution and the real number ∆τAB ≡ τB−τA that the clock ascribes for the events A = γ(τA) and B = γ(τB). Hence the internal dynamics of the clock does not depend upon the timelike trajectory, thus fulfilling the condition that the clock readings ∆τAB cannot depend on the clock’s past history. Additionally, we also assume that the classical Hamiltonian governing the evolution of the clock in its phase space does not depend explicitly on the parameter λ. Physically, such a condition ensures that the clock will not suffer deteriorations, so that its recordings will always advance at the same pace. Now, we need to describe a protocol to decide whether a given clock is honest or not. In fact, from the experimental point of view it is not possible to decide if an isolated clock is honest. The best we can do is to decide whether a clock is honest compared to another honest clock. At first, this might seem a kind of tautology; however, you shall see in a moment that what we are really constructing here is an equivalency class of honest clocks. Before describing the protocol in detail, it is worth making a small digression about such a mathematical concept. Let X be generic a non-empty set. An equivalence relation on X is a binary relation ∼ satisfying the following conditions: 1. a ∼ a, ∀ a ∈ X 17 2. If a ∼ b then b ∼ a, ∀ a, b ∈ X 3. If a ∼ b and b ∼ c then a ∼ c, ∀ a, b, c ∈ X The equivalence class of an element a ∈ X, denoted by [a], is defined as the set of all other elements of X that are ∼ related with a: [a] = {x ∈ X : x ∼ a}. Now comes the protocol: suppose you are given a collection of clocks (the set X). As a fundamental hypothesis, we assume that X contain at least two honest clocks. Indeed, there is no reason to believe that classical General Relativity forbids the existence of honest clocks. Also, the existence of only one honest clock in the whole universe does not make sense: if there exists one, you can just reply it to construct an infinite amount of honest clocks! Next, we separate those clocks whose readings advance at the same pace, obtaining a new set of clocks A ⊂ X. It is important to realize that one cannot guarantee that all clocks belonging to the set A are honest: some clocks could be devised to exhibit an equally dishonest bevahior. So, the proper procedure is to submit all those clocks to different conditions like, e.g., moving them, turning them off for arbitrary periods, and so on. The idea is to force all the clocks to follow quite distinct worldlines on the spacetime they are placed in. After this, the clocks that still advance at the same pace surely are the honest ones, belonging to the same equivalence class. 1.4 What is time? Throughout this introductory chapter, we have briefly reviewed some fundamental as- pects of the theory of General Relativity, focusing on how one can extract spatial and time measurements from a given spacetime. Now, we turn back to the promise made at the be- ginning of the chapter and give a precise definition of time within the paradigm of classical 18 physics: time is what honest clocks measure. At first, one may be disappointed (or not?) with this definition. Nonetheless, from the point of view of physics as an experimental science, we defend once more that observables are meaningless if one cannot state clearly how to measure them. Henceforth, the definition of a measurable quantity cannot be dissociated from the measuring apparatus itself. After all, even though we have outsourced the definition of time to that of honest clocks, we were able to give a precise description of such apparatus using classical physics. Another point that is worth thinking about is that questions regarding the “true” nature of some concept often lie outside the scope of physics. At the end of the day, all that we have are models that predict the values of quantities to be observed experimentally. For instance, one cannot be sure that gravity “really” is a “manifestation” of the spacetime curvature; all we can be sure is that by using the model of gravity proposed by the theory of General Relativity we can make predictions that are observed in nature. Therefore, we can safely state that we do have a precise characterization for the concept of time within the paradigm of classical physics. However, we cannot forget that although we assume that an honest clock is an object that exists in nature, its ultimate description must be given by Quantum Mechanics. Thus, it is unavoidable to ask: how quantum theory impacts on our definition of a honest clock? In the next chapters, we shall approach this question by analyzing different views about the role of time in Quantum Mechanics and how one can predict time measurements within the quantum formalism. 19 2 Essential aspects of Quantum theory The purpose of this chapter is to review some aspects of Quantum Mechanics, focusing on what will be used to discuss the issue of time. After presenting topics that are more traditional, which can be found in more detail on references like [3, 4, 5, 6, 7], we discuss the concept of positive operator-valued measures (POVMs), a notion that is often used on quantum information theory [8]. This will help us to construct a generalized view of quantum observables [10, 13, 14], which will be important when we address some attempts of defining an operator for time in Quantum Mechanics. 2.1 Basic concepts of probability theory Since Quantum Mechanics is a theory where all predictions are made within the framework of probability, we begin this review chapter by presenting the basic language of probability theory. In the present exposition we define some concepts aiming to outline the core of the probabilistic interpretation of Quantum Mechanics. For more details regarding probability theory, see e.g reference [9]. A more complete discussion about the probability concepts used in quantum theory may be found in reference [10]. In the same way that Linear Algebra happens in vector spaces, the background for prob- ability theory is the so-called probability space. To construct such a space, one needs three 20 ingredients: • Sample space • σ - algebra • Probability measure A sample space is simply any set Ω 6= ∅ that contains all the events of our probabilistic theory. To give it a “soul”, we define a σ- algebra A on it, which is a set of subsets of Ω satisfying the following conditions: 1. Ω ∈ A and ∅ ∈ A; 2. If A1, A2 ∈ A then A1 ∪ A2, A1 ∩ A2, and A1 − A2 ∈ A; 3. If An ∈ A ∀n ∈ N, then ∞⋃ n=1 An ∈ A. The structure (Ω,A) is called a measurable space. The mathematical theory of these spaces is the Measure Theory, whose treatment is completely out of the scope of this essay. If a subset A ⊂ Ω is in A, we shall call it an event. Given a measurable space (Ω,A) we can turn it into a probability space by defining a probability measure on it, which is a map P : A → R that satisfies: 1. 0 ≤ P (A) ≤ 1, ∀A ∈ A; 2. P (Ω) = 1; 3. For A1, A2, . . . , An, . . . ∈ A, with Ai ∩ Aj = ∅ whenever i 6= j, one has P ( ∞⋃ n=1 An ) = ∞∑ n=1 P (An). Notice that these conditions (also known as the Kolmogorov axioms) just formalize our intuitive understanding about probability: (1) the probability of any event is a number 21 between 0 and 1; (2) the probability of all the sample space (“anything happens”) is 1; (3) the probability of the union of disjoint events is the sum of the probability of each event (the “or rule”). A basic tool in the framework of probability is the concept of random variable, which can be defined as a map X : Ω → R that associates real numbers to “elementary events” (points in the sample space). The key feature of random variables is that they allow us to work directly on the real line R, thus being an important quantitative tool. At this point, it is important to mention that the set R itself can be made a measurable space with the aid of the σ-algebra of Borel sets B, defined as the smallest σ-algebra of R that contains all subsets of the form (−∞, x]. Thus, for X to be a well-defined random variable, the following condition must holds: B ∈ B =⇒ X−1(B) ∈ A. Assuming this condition we can define a probability distribution for the random variable X as PX(B) = P (X−1(B)). (2.1) Let us work a little bit more with definitions to construct the probability density function of a random variable, a central quantitative tool for Quantum Mechanics. Consider the set Ax = {ω ∈ Ω : X(ω) ≤ x}. Notice that Ax is the pre-image of the Borel set (−∞, x]. The cumulative distribution function of X is then defined by FX(x) = P (Ax) ≡ P (X ≤ x) (2.2) 22 Finally, we say that X has a probability density function pX if we can write FX(x) = ∫ x −∞ dλ pX(λ). (2.3) Given this definition, one can show that the following properties hold: • FX(x1) ≤ FX(x2) whenever x1 < x2. • lim ε→0+ FX(x+ ε) = FX(x) (continuity from the right) • lim x→∞ FX(x) = 1 • lim x→−∞ FX(x) = 0 Conversely, any function satisfying these four properties is a cumulative distribution func- tion. Notice that definition (2.3) implies on pX(x) = dFX(x) dx . (2.4) Moreover, the relation with the probability distribution of X is straightforward: PX(B) = ∫ B dx pX(x). (2.5) 2.2 The probabilistic interpretation of Quantum Me- chanics In the framework of Quantum Mechanics, the degrees of freedom of each physical system are represented by a complex Hilbert space H. Pure states of the system are described by normalized vectors |ψ〉 ∈ H. For most physical situations of interest, one assumes that H 23 is separable, meaning that it is possible to find an countable orthonormal basis {|φn〉} which allows us to write any vector in the form |ψ〉 = ∑ n 〈φn|ψ〉 |φn〉 . (2.6) Usually, one assumes that measurable quantities (i.e., the observables of the theory) are represented by self-adjoint operators acting on H. For now, we restrict our discussion within this view. Later on, we shall present some mathematical tools that allow for a more generic approach towards the issue of measurement in Quantum Mechanics. For a self-adjoint operator  : H → H with discrete spectrum {λn}, the spectral theorem allows us to write  = ∑ n λnΠ̂n, (2.7) where Π̂n is the projector onto the eigenspace belonging to the eigenvalue λn, i.e., the projec- tor onto the space Ker(Â− λnÎ)1. The family of projectors {Π̂n} is orthogonal, in the sense that Π̂iΠ̂j = δijΠ̂i. (2.8) This means that eigenvectors belonging to different eigenvalues are orthogonal. From the mathematical point of view, a proper generalization of (2.7) to the case where  has a continuous spectrum needs some tools from functional analysis. Since such a subject is out of our scope, we will just give a formal motivation for the generalization, aiming to understand how to operationally associate a random variable to a self-adjoint operator in Quantum Mechanics. Still considering the operator (2.7), let us define 1The kernel of a linear operator  is defined as the set Ker(Â) = {|ψ〉 ∈ H :  |ψ〉 = 0}. 24 Ê(λ) = ∑ λn≤λ Π̂n. (2.9) The collection of operators {Ê(λ)}λ∈R is called the spectral family of the operator Â. Notice that • lim ε→0+ Ê(λ+ ε) = Ê(λ) • lim λ→−∞ Ê(λ) = 0 (null-operator) • lim λ→∞ Ê(λ) = Î (identity operator) • [Ê(λ), Â] = 0, ∀λ ∈ R • 〈ψ|Ê(α)|ψ〉 ≤ 〈ψ|Ê(β)|ψ〉 whenever α ≤ β, for all |ψ〉 ∈ H. Moreover, ∆Ê(λn) ≡ Ê(λn)− Ê(λn−1) = Π̂n. (2.10) So, the self-adjoint operator  can be written as  = ∑ n λn∆Ê(λn). (2.11) It turns out that this is the expression that should be used for a generalization to the continuous case. Indeed, the general form of the spectral theorem reads  = ∫ ∞ −∞ λ dÊ(λ). (2.12) Here, the spectral family satisfies the same conditions outlined previously. For our pur- poses, it will not be necessary to give a rigorous definition for the “spectral measure” dÊ(λ). (for more details regarding technical issues of functional analysis, see reference [11]). Let us 25 treat it operationally, keeping in mind that it is the generalization of (2.10) for the continuous case. As an example consider H = L2(R), the Hilbert space of the square-integrable functions on the real line. Let X̂ be the operator that takes a function f ∈ H and returns the function g : x 7→ xf(x). The spectral family of this operator can be defined as Ê(λ)(f)(x) = Θ(x− λ)f(x), (2.13) where Θ(x) is the Heaviside step function. In this case, the spectral measure dÊ(λ) acts on each function f to generate a measure on the real line, so that the integral (2.12) can be computed by the usual rules of calculus. Formally, dÊ(λ)(f)(x) = δ(λ− x)f(x)dλ. (2.14) Thus, we see that X̂(f)(x) = ∫ dλ λδ(λ− x)f(x) = xf(x). (2.15) Returning to the generic case, it remains to address the question regarding the eigenvalues of the operator  represented by (2.12). It turns out that one can give a rigorous definition for the spectrum of a self-adjoint operator solely by the properties of its spectral family. First of all, notice that in the discrete case the eigenvalues of  correspond to “jumps” in the operators of the spectral family. For the sake of illustration, suppose that the spectrum of  is Spec(A) = {λ0, λ1, . . . λn}. In this case we have Ê(λ) = Π̂0 λ0 ≤ λ < λ1 Ê(λ) = Π̂0 + Π̂1 λ1 ≤ λ < λ2 . 26 . . Ê(λ) = Î λ ≥ λn. The key concept here is that the presence of eigenvalues is related to changes in the spectral family. When the spectral family is constant in some neighborhood of a point λ ∈ R, we say that this point is a stationary one. Precisely: λ ∈ R is a stationary point of the spectral family {Ê(λ)} if there is some ε > 0 such that Ê(α) = Ê(β) ∀α, β ∈ (λ− ε, λ+ ε). (2.16) From the definition (2.12), one sees that for the interval J = (λ− ε, λ+ ε) the expression ∫ J λ dÊ(λ) represents a null operator. Thus, it does not make sense to include the point λ in the spectrum of Â. In this way, the spectrum may be defined as Spec(Â) = {λ ∈ R : λ is a non-stationary point of the spectral family{ ˆE(λ)}}. In a moment, we shall interpret the spectrum of a self-adjoint operator as the collection of all the possible outcomes that can be observed in a measurement of the physical quantity described by the operator. But first, we must figure out some way to connect all these mathematical definitions with the probability tools we have considered previously. At first, consider a physical system completely characterized by a pure state |ψ〉 ∈ H (a normalized vector in a complex Hilbert space). It turns out that each self-adjoint operator  gives raise to a real-valued random variable A whose cumulative distribution function is 27 FA(λ) = 〈ψ|Ê(λ)|ψ〉. (2.17) The proof that FA is indeed a cumulative distribution function follows directly from the properties of the spectral family Ê(λ). Now, assume that λ /∈ Spec(Â). Then, λ is a stationary point of the spectral family {Ê(λ)}, i.e, there is some ε > 0 such that Ê(x) does not change for x ∈ (λ − ε, λ + ε). So, the cumulative distribution FA is constant in this same interval, which means that the probability associated with I = (λ − ε, λ + ε) is zero. Thus, not only λ, but all the real values pertaining to the interval I are not valid outcomes for a measurement of Â. On the other hand, if λ ∈ Spec(Â), the function FA will exhibit changes in every neighborhood of λ, so that there is always a non-zero probability of observing values sufficiently close to this point (of course, the probability of a specific point is always zero). Therefore, within such probabilistic paradigm we see that the spectrum of a self-adjoint operator can indeed be associated to the possible outcomes for the measurement of some physical quantity described by Â. Furthermore, given the cumulative distribution function (2.17) the probability that the outcomes of a measurement of  will lie in some interval J ⊂ R is PA(J) = ∫ J dFA(λ) = ∫ J 〈ψ|dÊ(λ)|ψ〉. (2.18) From the experimental point of view, the quantity that deserves more attention is the mean value of  over the state |ψ〉, to be denoted by 〈Â〉ψ. Indeed, the real number 〈Â〉ψ is what Quantum Mechanics predicts the experimentalist will obtain if he performs infinite measurements of the physical property described by  in a system prepared in the state |ψ〉. Of course, since no laboratory in the real world can deal with infinite repetitions of the 28 same experiments, one needs to establish some error criteria to claim that the measurements match or not with the previsions of theory. Concerning Quantum Mechanics, today there is no doubt that it is a theory that describe very well all natural phenomena apart the ones that involves gravity. Now, we are in position of stating the principal result of this section: the formula that encodes the core of the probabilistic interpretation of Quantum Mechanics. Recall that the mean value of a random variable X is given by 〈X〉 = ∫ ∞ −∞ dx xpX(x) = ∫ ∞ −∞ xdFX(x). (2.19) So, using the random variable A associated with the self-adjoint operator Â, we can write 〈Â〉ψ ≡ 〈A〉 = ∫ ∞ −∞ λdFA(λ) = ∫ ∞ −∞ λd〈ψ|Ê(λ)|ψ〉 = 〈ψ|Â|ψ〉. (2.20) As already mentioned, this equation is the core of the probabilistic interpretation of Quantum Mechanics, and will be the basis for further generalizations for the measurement concept as represented by an operator. This will be the subject of section 2.5. Another statistical quantity that deserves mentioning is the variance (or dispersion) of a self-adjoint operator over a state |ψ〉. Using the random variable A naturally associated with the operator  again, such a quantity is defined by ∆ψ ≡ √ 〈A2 − 〈A〉2〉 = √ 〈ψ|Â2|ψ〉 − 〈ψ|Â|ψ〉2. (2.21) Before finishing the present issue, we shall discuss how to extend the relation (2.20) to the case where it is not possible to describe the system under study by a vector in the Hilbert space. As is well known from the standard formalism of Quantum Mechanics (see [3], for instance), in this case one need to resort to the concept of density matrix. Indeed: the most general characterization of a physical system in Quantum Mechanics is given by an operator ρ̂ : H → H that satisfies to the following conditions: 29 • ρ̂ = ρ̂† • 〈ψ|ρ̂|ψ〉 ≥ 0, for all |ψ〉 ∈ H • Tr(ρ̂) = 1 One way to motivate the introduction of such a mathematical object is by considering a statistical mixture of quantum states. For instance, consider a case where there is ignorance about which state the system was prepared in. Suppose that everything we know is that there is a probability pi that the state of the system is |ψi〉, for i = 1, . . . , N . Then, in a measurement of Â, the mean value would be 〈Â〉 = N∑ n=1 pi〈ψi|Â|ψi〉. (2.22) Introducing the operator ρ̂ = N∑ n=1 pi|ψi〉〈ψi|, (2.23) the quantity 〈Â〉 can be cast into 〈Â〉 = Tr [ ρ̂ ] . (2.24) Notice that from the condition ρ̂ = ρ̂†, every density matrix in a finite-dimensional Hilbert space can always be diagonalized, so that expression (2.23) is quite general. For an infinite dimensional Hilbert space there are more mathematical subtleties to be taken into account, but from a pragmatic point of view things work in the same way. Moreover, the equation (2.24) must be taken as the proper extension of (2.20) to a state described by a density matrix ρ̂. Finally, if all the coefficients pi vanish but one — say, pk = 1 — we will have ρ̂ = |ψk〉〈ψk|⇐⇒ |ψk〉 , so that the usual description of a pure state is recovered. 30 2.3 Quantum dynamics The dynamics of quantum systems is generated by the Schrödinger equation: i d dt |ψ(t)〉 = Ĥ(t) |ψ(t)〉 . (2.25) In this equation, the self-adjoint operator Ĥ : H → H is the system’s Hamiltonian, whose eigenstates are the states with defined energy value. This equation is completely deterministic: if we know the state of the system at some particular time t = t0, equation (2.25) determines it for later all later times t > t0. It is important to mention that here time is just a parameter that keeps track of the evolution of the system under study. Of course, it could be interpreted as the time as told by an external (classical) clock that guides the experimentalist to formulate statements like “at time t = t0 the system was prepared in state |ψ0〉” or “the measurement was carried at instant t”. The evolution of states can be given in terms of the temporal evolution operator Û(t, t0), which takes the state of the system at t = t0 and evolves it to some other instant t according to: Û(t, t0) |ψ(t0)〉 = |ψ(t)〉 (2.26) In the case where the Hamiltonian does not depend upon the evolution parameter t, such an operator has a simple closed expression, namely Û(t, t0) = e−i(t−t0)Ĥ (2.27) More generally, it can be described by a time ordered exponential: Û(t, t0) = T exp ( −i ∫ t t0 ds Ĥ(s) ) (2.28) The time ordering symbol T is an operation that can be defined as follows: given a product 31 of time-dependent operators Â(t1)Â(t2)...Â(tn), the operation T reorganizes the order of the factors so that T [Â(t1)Â(t2)...Â(tn)] = Â(s1)Â(s2)...Â(sn), s1 > s2 > . . . > sn. (2.29) For the case where the state of our system is a mixture, the evolution of the density matrix reads ρ̂(t) = Û(t, t0)ρ̂(t0)Û †(t, t0). (2.30) All these results were presented within the Schrödinger picture, a description where the operators are fixed and the states evolve with time. In the Heisenberg picture, we have a description where the roles are reversed: states are fixed and operators evolve in time. We also have the interaction picture, whose limiting cases are just the two pictures already mentioned. For the sake of completeness — since we will eventually use the Heisenberg picture — we review these pictures on appendix A. Certainly, the physical predictions of Quantum Mechanics do not depend upon the picture we choose to perform our mathematical description. 2.4 Symmetries In the context of Quantum Mechanics, symmetries are understood as operations realized over a system that do not change the physical predictions of the theory. In this sense, a sym- metry operation must be implemented on the Hilbert space of the system by operators that preserve probabilities. Mathematically speaking: let |ψ〉 and |φ〉 be two allowable quantum states of some physical system. Assume that the outlined operation can be represented by the action of an operator Ŝ : H → H. Define the transformed states |Ψ〉 = Ŝ |ψ〉 (2.31) 32 and |Φ〉 = Ŝ |φ〉 . (2.32) Thus, for the operator Ŝ to be regarded as the proper implementation of a symmetry on the Hilbert space, the following condition must holds: |〈ψ|φ〉|2= |〈Ψ|Φ〉|2. (2.33) The Wigner’s theorem states that in order to an operator to satisfy relation (2.33), it must be either unitary or anti-unitary. Recall that an operator  is said to be anti-unitary if for any vectors |ψ〉 , |φ〉 ∈ H and scalars α, β ∈ C the following conditions holds: 〈ψ|†Â|φ〉 = 〈ψ|φ〉∗ and Â(α |ψ〉+ β |φ〉) = α∗ |ψ〉+ β∗ |φ〉 . The main example of a symmetry represented by an anti-unitary operator is the time reversal. Now, we restrict the discussion to unitary operators: to each element g belonging to some symmetry group G we associate an unitary operator Û(g) : H → H. That is: formally speaking, the implementation of a symmetry in Quantum Mechanics means to construct a representation of the group G that encapsulates the symmetry operations on the physical space. Given any self-adjoint operator  ∈ L(H), its transformation under the action of the symmetry described by g ∈ G is given by the relation 33 Â′ = Û(g)ÂÛ †(g). (2.34) The representation of G on the Hilbert space is called projective if for any g1, g2 ∈ G one can write Û(g1)Û(g2) = eiφ(g1,g2)Û(g1g2), (2.35) for some real function φ. When it is possible to choose this phase to be zero, we say that we have a complete representation. Example 1: Galileo invariance. The invariance under Galileo transformations is a symmetry of non-relativistic Quantum Mechanics. Its representation on the Hilbert space of a spinless particle moving along one dimension, e.g., is the most notable case of a projective representation. Indeed, if p̂ and x̂ denote the usual momentum and position operators, the action of a Galileo boost with velocity v over the system’s state at some instant t is described by the operator Ĝ(v) = e imv2t 2 eimvx̂e−ivtp̂. (2.36) When we combine two boosts with velocities v1 and v2 we obtain: Ĝ(v1 + v2) = e2imv1v2tĜ(v1)Ĝ(v2). (2.37) Example 2: Continuous symmetries. In physics, continuous symmetries are described by the so-called Lie groups, which essen- tially are differentiable manifolds with a group structure [12]. Let G be a connected Lie Group. Consider each element g ∈ G to be described by n real variables xa ≡ (x1, . . . , xn): g = g(xa). Certainly, the identity element IG corresponds to xa = 0. 34 Suppose the composition law of G is g(xa)g(x̄b) = g(f(xa, x̄b)), where f : Rn×Rn → Rn is a differentiable function. We shall denote its component functions by fa : Rn × Rn → R. For the sake of consistency, notice that fa(xj, 0) = fa(0, xj) = xa. As a quite natural ansatz, the representation of each g(xa) on the Hilbert space will be described by the operator Û(g(xa)) = Î + ixaM̂a + 1 2 xaxbM̂ab + . . . . (2.38) The operators M̂a, M̂ab, etc., are self-adjoint operators that do not depend on the real coordinates xa. Now, assuming a complete representation we have Û(g(xa))Û(g(x̄b)) = Û(g(f(xa, x̄b))). (2.39) Consider then the Taylor expansion of the component functions fa: (2.40) fa(xi, x̄j) = fa(xi, 0) + fa(0, x̄j) + faijx ix̄j + . . . = xa + x̄a + faijx ix̄j + . . . Applying this last relation on (2.39) and then using the expansion outlined in equation (2.38), we obtain the Lie algebra for the symmetry generators on the Hilbert space, namely [M̂b, M̂c] = iCa bcM̂a. (2.41) Here, Ca bc = facb − fabc are the structure constants of G [12]. Thus, we see that the commutation relations of the symmetry generators on the Hilbert space are determined by the structure constants of the Lie group that describe the symmetry operations on the physical space. Let us see two examples. Exemple 2.1: Linear momentum. 35 The linear momentum concept arises naturally on the context of spatial translations. Thus, its associated symmetry group can be taken as G = (R,+), the additive group of real numbers. In this case, all the structure constants vanish. As is well known, the generators of the translation symmetry in the Hilbert space of Quan- tum Mechanics are the momentum operators p̂a, a = 1, 2, 3, which satisfy to the commutation relations [p̂a, p̂b] = 0. (2.42) Finite translations by a vector ~a ∈ R3 are implemented by the unitary operator Û(~a) = e−i~a· ~̂p. (2.43) Exemple 2.2: Angular momentum. Since the concept of angular momentum is closely related to that of rotations, the symme- try group that gives rise to the angular momentum algebra is G = SO(3), the multiplicative group of 3× 3 orthogonal real matrices. In this case, the commutation relation of the gener- ators — the angular momentum operators Ĵa — is given by [Ĵa, Ĵb] = iεabcĴ c. (2.44) In order to keep the notation’s elegance (in the context of Einstein’s summation conven- tion), we have defined Ĵa ≡ Ĵa. The rotation by an angle φ ∈ [0, 2π] around the axis defined by the unit vector ~n ∈ R3 — on the counter-clockwise sense and adopting the passive view — is implemented by the unitary operator Û(~n, φ) = e−iφ~n· ~̂ J . (2.45) 36 2.5 Generalized observables In this section, we use the concepts outlined previously to discuss how one can describe quantum measurements in a more general way, showing how to predict probabilities for outcomes in experiments where the measured quantity cannot be represented by a self-adjoint operator. For the sake of completeness, a generic system will always be assumed to be described by a density matrix ρ̂. Adapting the approach of references like [8, 10, 14], we generalize the concept of ob- serbables so that they now are represented by positive operator-valued measures (POVMs). A general definition of a POVM goes as follows: let (Ω,A) be a measurable space, as defined in section 2.1. A POVM over this space is a map E : A → L(H)+ such that 1. 0 ≤ 〈ψ|Ê(X)|ψ〉 ≤ 1 ∀ X ∈ A, ∀ |ψ〉 ∈ H 2. Ê(∪iXi) = ∑ i Ê(Xi), for any countable collection {Xi} of disjoint sets belonging to the σ-algebra A 3. Ê(Ω) = Î If the state of the system is represented by a density matrix ρ̂, then the POVM E is supposed to describe a measurement process whose outcomes can be associated with the sets X ∈ A. Furthermore, the probability distribution is P (X) = Tr [ ρ̂Ê(X) ] (2.46) The key point here is that properties 1-3 above ensure that this function is a truly prob- ability distribution. Indeed, we can always find an orthonormal basis {|φn〉} of H such that the density matrix assumes a diagonal form ρ̂ = ∑ n wn|φn〉〈φn|, (2.47) 37 where wn ≥ 0 for each n and ∑ n wn = 1. Using this decomposition, relation (2.46) can be cast into P (X) = ∑ n ∑ j wj 〈φn|φj〉 〈φj|Ê(X)|φn〉 = ∑ n wn〈φn|Ê(X)|φn〉 (2.48) From property 1, we have 0 ≤ 〈φn|Ê(X)|φn〉 ≤ 1 for each n. Thus, 0 ≤ P (X) ≤ ∑ n wn = 1. Property 2 ensures the usual addition law for the probability of union of disjoint sets, whereas the condition Ê(Ω) = Î implies on P (Ω) = 1. In practice, we usually are concerned with the case where Ω = R and A = B(R). What we really do here is to associate each open set J ⊂ R to a positive operator Ê(J) whose properties ensure the probabilistic interpretation of (2.46). Then, the quantity P (J) is interpreted as the probability that the measurement outcome lies in the set J . For the case where possible measurement outcomes can be labeled by a set of discrete indexes J , we proceed in the same way by associating an positive operator Êi to each i ∈ J . In this case, ∑ i Êi = Î, (2.49) and the probability of obtaining the outcome labeled by i is to be given by P (i) = Tr [ ρ̂Êi ] . (2.50) In the remainder of this section, we shall assume the discrete case for the sake of simplicity. A projective measurement, i.e., a measurement described by a self-adjoint operator Â, surely is a particular case of a POVM. Indeed, considering again the spectral decomposition  = ∑ i λiΠ̂i, 38 it suffices to take Êi = Π̂i. Essentially, a POVM can be viewed as a particular case of the general measurement for- malism where one is not concerned with the state of the system after the measurement but solely with its statistical description [8], since the probability distribution of the measure- ment’s outcomes is still well defined. At the same time, it provides us with a theoretical framework that is more general than the one of projective measurements, thus allowing us to describe experimental situations that are not covered by the latter. The point here is that a projective measurement presupposes repeatability: if the measurement of the observable  yields the result i, then when  is measured again in the same system the result i will be observed with probability 1. It turns out that not all experimental measurements schemes meet this condition; e.g, when one uses a silvered screen to measure the position of a photon, the particle is destroyed. Example2: suppose we were given a qubit prepared in one of the states |ψ1〉 = 1√ 2 (|0〉+ |1〉) or |ψ2〉 = |0〉 (2.51) since these states are not orthogonal, there is no measurement that can safely distinguish between them (for a rigorous proof of this claim, see [8]). However, one can perform a POVM measurement to avoid misidentifications. Indeed, consider the operators Ê1 = √ 2 1 + √ 2 |1〉〈1|, Ê2 = √ 2 2(1 + √ 2) (|0〉〈0|−|0〉〈1|−|1〉〈0|+|1〉〈1|), 2This example was taken from reference [8]. 39 and Ê3 = Î− Ê1 − Ê2. One can easily verify that M ≡ {Ê1, Ê2, Ê3} is a POVM measurement. Now, suppose that the state of the system is |ψ〉 = |ψ2〉. Because Ê1 |ψ2〉 = 0, if the outcome of the measurement M is i = 1, then we can be sure that the qubit that was given to us is not in the state |ψ2〉. In the same token, since Ê2 |ψ1〉 = 0 if the resultant outcome is i = 2 we have certainty that |ψ〉 6= |ψ1〉. Therefore, the measurement scheme described by M allow us to distinguish between the states |ψ1〉 and |ψ2〉 whenever the outcome is i = 1 or i = 2. Of course, if the outcome is i = 3 nothing can be said, but at least we have ruled out the possibility of misidentifications. Notwithstanding, one could object against the understanding of POVMs as associated to distinct outcomes of the physical observable to be measured in the real experiment due to the lack of orthogonality. Indeed, for the example just exposed, one can verify that in general ÊiÊj 6= δijÊi, as opposed to what happens to the projectors that can be used to construct the self-adjoint operator in relation (2.12). However, it turns out that there is a way to circumvent such a “problem” by enlarging the Hilbert space of the system to include the measuring apparatus. Indeed, suppose now that the Hilbert space H of the system can be written as 40 H = HS ⊗HA, whereHS is the Hilbert space of the system of interest — i.e., the one where the measurements are to be carried on — and HA describes the degrees of freedom of the measuring apparatus. Furthermore, assume that dimHA = N and let {|a1〉 , . . . , |aN〉} be a basis of the same space such that each |ai〉 is associated with a possible outcome of the measurement we want to perform in HS. Define the projectors Π̂i = |ai〉〈ai|, i = 1, . . . , N. (2.52) Initially, suppose the global state of the extended system is describe by separable density matrix ρ̂ = ρ̂S ⊗ ρ̂A. (2.53) Notice that such a description stands for a particular instant before the experiment starts, where it is reasonable to suppose that there are no correlations between the system S and the apparatus A. The measurement process will correspond to a interaction between the quantum subsys- tems S and A. Indeed, in such a scenario, one generally works with a Hamiltonian of the form Ĥ(t) = ĤS ⊗ ÎA + ÎS ⊗ ĤA + ĤI(t). (2.54) Due to the presence of the interaction Hamiltonian ĤI(t), the temporal evolution as determined by the total Hamiltonian will create correlations between S and A. The study of how such correlations impact on local measurements on the system of interest — in this case, S — is very important, for instance, in decoherence models. 41 Hence, if Û(t) is the time evolution operator arising from the complete Hamiltonian Ĥ(t), the probability for the outcome i in the measurement scheme under consideration is given by the formula P (i) = Tr [ ρ̂(t)(ÎS ⊗ Π̂i) ] = Tr [ Û(t)ρ̂Û †(t)(ÎS ⊗ Π̂i) ] . (2.55) Using expression (2.53), this last result can be cast into P (i) = TrS[ρ̂SÊi], (2.56) where TrS denotes the partial trace over HS and the operator Êi is defined by Êi = TrA[ρ̂AÛ †(t)Π̂iÛ(t)]. (2.57) It turns out that the collection {Êi} represents a POVM. Indeed, it is quite clear that each operator Êi is positive. Moreover, N∑ i=1 Êi = TrA [ ρ̂AÛ †(t) ( N∑ i=1 Π̂i ) Û(t) ] = TrA[ρ̂AÛ †(t)ÎAÛ(t)] = TrA[ρ̂A] = 1. (2.58) Therefore, a set of orthogonal projectors acting in the Hilbert space of the measuring appa- ratus induce a POVM in HS. As a concluding remark, we mention here that POVMs constitute the most general defi- nition of an observable that is compatible with the probabilistic interpretation of Quantum Mechanics. Since a mathematically rigorous statement of this result is out of our scope, we just refer the reader to reference [13]. The chapter 2 of the book [14] also deals with this same issue. 42 3 The time-energy uncertainty relation The time-energy uncertainty relation ∆E∆t ≥ h̄ 2 , (3.1) has been a matter of controversies since its proposal at the early days of quantum theory. The main reason for this is that there is no direct way of interpreting (3.1) in the same fashion as one interprets the uncertainty relation between position and momentum. Indeed, a general result from linear algebra states that for any operators Â, B̂ ∈ H one has the inequality ∆ψÂ∆ψB̂ ≥ 1 2 |〈[Â, B̂]〉ψ|, (3.2) Here, the quantity ∆Âψ is the dispersion of the operator  over the state |ψ〉, as defined previously on equation (2.21). Choosing  = x̂ and B̂ = p̂ we have (for any state |ψ〉): ∆ψx̂∆ψp̂ ≡ ∆x∆p ≥ h̄ 2 . (3.3) Surely, there is no doubt about the meaning of the quantity ∆E on relation (3.1): it is just the dispersion of the Hamiltonian Ĥ when it is measured over the state |ψ〉 under study, 43 namely ∆E = √ 〈ψ|Ĥ2|ψ〉 − 〈ψ|Ĥ|ψ〉2. (3.4) Nonetheless, in principle it is not clear how to define the quantity ∆t. In this chapter, we shall discuss some formulations and interpretations of the time-energy uncertainty relation. 3.1 The Mandelstamm-Tamm relation We begin our discussion regarding the time-energy uncertainty relation by revisiting a formal derivation due to Mandelstamm and Tamm [20] (see also [19]). At first, consider a classical clock equipped with a dynamical variable C = C(t) from which the time measurements are determined. E.g., C may represent the angular position of a pointer in an analog clock. Let ∆C be the uncertainty in a measurement of C. Then, it is reasonable to suppose that the corresponding uncertainty in a time measurement, ∆t, is related with the first one by ∆C = ∣∣∣∣dCdt ∣∣∣∣∆t. (3.5) The fundamental hypothesis of the derivation considered here is based on the extrapola- tion of the relation (3.5) for the quantum case, where the variable C must be replaced by the observable Ĉ. In doing so, we use the Correspondence Principle to replace the variable C(t) in (3.5) by the mean value of Ĉ over the state of the system. The clock must now be understood as a quantum system whose time evolution (followed by the parameter t) is determined by some Hamiltonian Ĥc. By simplicity, we suppose that such Hamiltonian does not depend on t. So, in the quantum case one can write ∆Ĉ = ∣∣∣∣∣d〈Ĉ〉dt ∣∣∣∣∣∆t(Ĉ). (3.6) 44 In the previous expression, the quantity ∆Ĉ must be read as the dispersion of the operator Ĉ over the state of the system, as defined on (2.21). The change of notation ∆t → ∆t(Ĉ) was made to stress out that equation (3.6) must be understood as a definition of the quantity ∆t(Ĉ) that, a priori, depends on the dynamical behavior of the observable represented by the operator Ĉ. A clear way to interpret such a quantity is to say that it represents a measure of the characteristic time scale during which a significant change of the expectation value of Ĉ — that is, a change by an amount of the order of the dispersion ∆Ĉ — takes place on a specified system’s state. Thus, in this scenario the equation (3.6) defines what we understand by “time uncertainty” in Quantum Mechanics. Further assuming that the operator Ĉ does not depend explicitly on the parameter t, we can use Ehrenfest’s theorem [3, 4] to write ih̄ d〈Ĉ〉 dt = 〈[Ĉ, Ĥ]〉. (3.7) In this way, we have ∆Ĉ = 1 h̄ |〈[Ĉ, Ĥ]〉|∆t(Ĉ). (3.8) Then, applying the result (3.2) with  = Ĥ and B̂ = Ĉ, we get ∆E∆t(Ĉ) ≥ h̄ 2 . (3.9) It is crucial to realize that the explicit dependence upon the observable Ĉ in this relation does not alter its universal validity: no matter which operator Ĉ we use to construct the quantity ∆t(Ĉ), the uncertainty relation will hold. Example: Free particle described by a Gaussian wave packet. Let us see what definition (3.6) tell us in the simplest particular case possible: a free particle. Here, we work in the Heisenberg picture (see appendix A) with Ĉ(t) = x̂(t). To simplify the notation, we write ∆t instead of ∆t(x̂). 45 The Hamiltonian of the system is Ĥ = p̂2 2m , so that p̂ is a constant of motion. Thus, we write p̂(t) = p̂(0) ≡ p̂0. Using the Heisenberg equation (A.9), we can have x̂(t) = p̂0t m + x̂(0) ≡ p̂0t m + x̂0. (3.10) Moreover, ∆x̂(t) = √ t2 m2 (∆p̂0)2 + (∆x̂0)2 + t m (〈x̂0p̂0 + p̂0x̂0〉 − 〈p̂0〉〈x̂0〉). (3.11) Now, assume that the state of our particle is described by a Gaussian wave packet centered at x0 = 0 and with width σ in position space. Let p0 be the mean value of the momentum operator over this sate. In this way, the normalized wave function for t = 0 can be written as 〈x|ψ0〉 ≡ ψ0(x) = ( 1 2πσ2 )1/4 e−( x 2σ ) 2 eip0x. (3.12) As for the wave function in the momentum space (that is, the Fourier transform) we have 〈p|ψ0〉 ≡ ψ̃0(p) = ( 2σ2 π )1/4 e−σ 2(p−p0)2 . (3.13) Switching back to the Schrödinger picture, we understand the mean and dispersion values of the operators x̂0 and p̂0 in equation (3.11) as the corresponding values for the Schrödinger picture operators x̂ and p̂ computed over the state |ψ0〉. Thus, we have ∆x̂0 = 〈ψ0|x̂|ψ0〉 = σ, 46 ∆p̂0 = 〈ψ0|p̂|ψ0〉 = 1 2σ , 〈x̂0〉 = 〈ψ0|x̂|ψ0〉 = 0, and 〈p̂0〉 = 〈ψ0|p̂|ψ0〉 = p0 For the sake of completeness, we outline the computation of the last quantity 〈x̂0p̂0 + p̂0x̂0〉. First of all, notice that 〈x̂0p̂0 + p̂0x̂0〉 = 〈ψ0|x̂p̂+ p̂x̂|ψ0〉 = 2〈ψ0|p̂x̂|ψ0〉+ i 〈ψ0|ψ0〉 = 2〈ψ0|p̂x̂|ψ0〉+ i. Inserting the identity operator (in position representation) into the first term of this last expression, it follows that 〈ψ0|p̂x̂|ψ0〉 = ∫ dx 〈ψ0|p̂|x〉〈x|x̂|ψ0〉 = ∫ dx 〈ψ0|p̂|x〉xψ0(x) ≡ ∫ dx J(x)xψ0(x). It turns out that the function J(x) is basically the derivative of ψ∗0(x). Indeed, J(x) = ∫ dp 〈ψ0|p̂|p〉〈p|x〉 = 1√ 2π ∫ dp pe−ipxψ̃0(p) = i ∂ ∂x [ 1√ 2π ∫ dp e−ipxψ̃0(p) ] = i ∂ψ∗0(x) ∂x = ( p0 − i x 2σ2 ) ψ∗0(x). 47 So, 〈ψ0|p̂x̂|ψ0〉 = ∫ dx x ( p0 − i x 2σ2 ) |ψ0(x)|2 = 1 σ √ 2π ∫ ∞ −∞ dx xe− x2 2σ2 ( p0 − i x 2σ2 ) =⇒ 〈ψ0|p̂x̂|ψ0〉 = − i 2 . This means that 〈x̂0p̂0 + p̂0x̂0〉 = 0. Thus, for a free particle initially prepared in a state described by the Gaussian wave packet (3.12), the dispersion of the position operator x̂ at any time t reads ∆x̂(t) = σ √ 1 + ( t 2mσ2 )2 . (3.14) The picture here is the following: as the center of the wave packet moves like it were a classical particle with constant velocity, the packet spreads out and the uncertainty about its position increases with the parameter t. Finally, using this result on the relation (3.6) with d〈x̂〉 dt = p0 m , the definition of the time uncertainty ∆t reads ∆t = mσ p0 √ 1 + ( h̄t 2mσ2 )2 . (3.15) In order to give a proper interpretation to this quantity, we have restored the fundamental constant h̄. The first point to notice here is that the factor mσ p0 48 is the ratio between the uncertainty about the initial position of the particle, ∆x0 = σ, and the group velocity v0 = p0 m , which is to be interpreted as the (constant) velocity of the free classical particle associated to our quantum particle. So, in the classical limit h̄ → 0, ∆t could be read as the time of flight through a region with spatial extent ∆x0, just as expected. In the quantum case, the presence of the term √ 1 + ( h̄t 2mσ2 )2 allows us to interpret ∆t as the time interval where the probability of finding the particle within the spatial range defined by (3.14) is appreciable. That is: there is an appreciable change of detecting the particle at position p0t m + ∆x̂(t) on the time interval t+ ∆t. 3.2 Lifetime of a property Following [15], we present a derivation of a slightly different type of the time-energy uncertainty relation concerning the lifetime of a property (see also [17, 20]). By property, we mean a particular instance of some physical observable that can be represented by a projector operator Π̂. E.g., the physical observable represented by the self-adjoint operator  = N∑ n=1 λnΠ̂n, has N properties described by each one of the projectors onto its eigenspaces. Let Π̂ be a projector operator representing some property P . We say that a state |ψ〉 has property P if Π̂ |ψ〉 = |ψ〉 . (3.16) Once more, we work within the Heisenberg picture. Consider the differential equation that gives the evolution of the projector Π̂(t): 49 dΠ̂(t) dt = i[Ĥ, Π̂(t)]. Now, the state |ψ〉 of the system is fixed. Define the function p(t) = 〈ψ|Π̂(t)|ψ〉. (3.17) Since Π̂ is a self-adjoint operator, p(t) is a real function of the time parameter t. Furthermore, we assume that the property P holds at the initial time instant t = 0, so that p(0) = 1. Given that, we can define the lifetime τP of property P to be the first value of t > 0 such that p(τP ) = 1 2 . (3.18) Taking the time derivative of p(t) and using relation (3.2), it follows that dp(t) dt = 1 h̄ |〈[Π̂(t), Ĥ]〉|≥ 2 h̄ ∆Ĥ∆Π̂. (3.19) In the expression above, the quantities ∆Ĥ ≡ ∆E and ∆Π̂ are to be calculated in the Heisenberg picture state |ψ〉. Since ∆Π̂ = √ p(t)(1− p(t)), we can write dp dt ≥ 2 h̄ ∆E √ p(t)(1− p(t)). (3.20) Recalling that p(0) = 1, the integration of (3.20) yields: p(t) ≥ cos2 ( ∆Et h̄ ) , for 0 ≤ t ≤ πh̄ 2∆E . Therefore, we must conclude that τP∆E ≥ h̄π 4 > h̄ 2 (3.21) 50 3.3 The Aharonov - Bohm point of view In the paper [21], Aharonov and Bohm criticize the following statement:“In a measure- ment of energy carried out in a time interval ∆t, there must be a minimum uncertainty ∆E in the transfer of energy to the observed system”. In doing so, they refute this particular interpretation of the time-energy uncertainty relation. Next, we shall discuss a model for measurement of energy that illustrates the core of such criticism. The model is the following: a bipartite system H = HA⊗HB composed by two interact- ing particles, one representing the object of interest (the “observed system”) and the other – the probe — playing the role of a “measurement apparatus”. Let x̂, p̂ and m be the po- sition, momentum and mass of the first particle (A), whereas X̂, P̂ and M stands for the corresponding quantities of the probe (B). The total Hamiltonian of the system is Ĥ(t) = p̂2 2m ⊗ ÎB + ÎA ⊗ P̂ 2 2M + g(t)p̂⊗ X̂, (3.22) where g(t) =  g0, if 0 ≤ t ≤ ∆t 0, otherwise (3.23) On the following, we omit the tensor product symbol for local operators (the first two terms of the total Hamiltonian). Notice that the presence of the interaction term, g(t)p̂⊗ X̂, enables us to interpret this model as representing a measurement scheme of the particle’s momentum (p̂) by means of a coupling with the probe’s position (X̂). Such a coupling has a constant strength g0 and is “turned on” during a time ∆t. Since measuring the momentum of a free particle is equivalent to determining the value of its energy, the model indeed represents 51 a measurement scheme for the particle’s energy that lasts for a time interval ∆t. Now, in order to see how this time can be made as short as one pleases without disturbing the measurement’s outcome, we analyze the dynamics of the relevant operators using the Heisenberg picture. Applying the Heisenberg equation (A.9) to each one of the four operators that compose the total Hamiltonian Ĥ(t), we get dx̂(t) dt = p̂(t) m + g(t)X̂(t), (3.24) dX̂(t) dt = P̂ (t) M , (3.25) dP̂ (t) dt = −g(t)p̂(t), (3.26) and dp̂(t) dt = 0. (3.27) From the last equation, we see the the particle’s momentum p̂ is a constant of motion. So, we write p̂(t) = p̂(0) ≡ p̂0. Restricting the time parameter t in the range 0 ≤ t ≤ ∆t, so that g(t) = g0 in equation (3.26), the time evolution of the probe’s momentum P̂ (t) is described by the relation P̂ (t) = P̂ (0)− g0tp̂0. (3.28) Using these results in equations (3.24) and (3.25), we have (still for the limited time range 0 ≤ t ≤ ∆t) 52 X̂(t) = X̂(0) + P̂ (0) M − g0tp̂0 M , (3.29) x̂(t) = p̂0 m + g0X̂(0) + g0P̂ (0) M − g2 0tp̂0 M . (3.30) Now, let us focus on the measurement of the particle’s energy. According to what we know so far, during the time interval ∆t the energy of the particle is constant, being described by the Hamiltonian Ĥ0 = p̂2 0 2m . (3.31) So, all we need is to find a way to measure the observable p̂0. Notice that it can be achieved by a measurement of the change on the probe’s momentum in the course of the time interval ∆t. Indeed, if one considers an ensemble of identically prepared systems (“particle” + “probe”), equation (3.28) tells us that the outcomes of measurements in both particle and probe’s momenta will have mean values related by 〈P̂ (∆t)− P̂ (0)〉 = −g0∆t〈p̂0〉. (3.32) Without loss of generality, from now on we assume that P̂ (0) = 0. In the same token, the dispersion of these quantities in the same measurement process satisfies ∆P̂ (∆t) = −g0∆t∆p̂0. (3.33) This result is the mathematical core of Aharonov - Bohm’s argument: for a given resolu- tion ∆P̂ (∆t) on the measurement of the probe’s momentum, the value of the product ∆t∆p̂0 53 can be made as small as one pleases just by adjusting the coupling strength g0. Therefore, in the present scenario one can measure the particle’s energy Ĥ0 with any desired accuracy in an arbitrarily small time interval. It is important to recognize that the Aharonov-Bohm’s energy measurement scheme only refutes a particular interpretation of the time-energy uncertainty relation, according to which there is an intrinsic “uncertainty relation between the duration of the measurement and the energy transfer to the observed system” [21]. However, as the authors of [21] point out, “it is commonly realized, of course, that the “inner” times of the observed system do obey an uncertainty relation”. In this statement, they endorse that there is no doubt on the physical validity of the time-energy uncertainty relation as formulated in the same fashion we did on section 3.1. Therefore, the argument presented by Aharonov and Bohm shows that there is no scope for a generic time-energy uncertainty relation concerning time intervals that are defined by objects external to the system where the energy measurement is carried on. Surely, this is not the case of the formulation due to Maldestamm and Tamm, where the time uncertainty is defined by means of some inner observable of the system (see equation (3.6)). 3.4 A derivation using the fact that “energy weighs” In the last section we have seen that there is no scope for a time-energy uncertainty when one speaks of measuring the energy of a system during a time interval defined by a clock external to the system. Nonetheless, this situation changes when one considers energy measurements that are performed from within the system itself. In the present section, we discuss a gadanken experiment proposed on [22] that uses the fact that energy weighs — a fundamental conclusion of Relativity — to derive a uncertainty relation between the measured energy and the duration of the measurement as defined by a clock internal to the system. 54 In general lines, the ideia is to weigh the total mass of a closed system from within the system itself. In doing so, it is possible to argue that the time τ an internal observer takes to measure the system’s energy with an accuracy ∆E is bounded by the relation ∆Eτ ≥ h̄ 2 . (3.34) The proposed gedanken experiment consists of measuring the mass M of a spherical shell with radius R by means of tracking the time it takes for an ejected spherical shell (test shell), with mass m � M , to reach a certain height z � R, to be specified below. The point here is that this time must be measured from a clock that is inside the test shell. According to General Relativity, the time as told by such clock will be affected by gravity. So, the peculiarities of the mass ejection event will alter the way the clock ticks inside the shell. Indeed: working within the post-Newtonian approximation, a clock located at a height z in a gravitational field with potential φ(z) will have its time measurements dilated according to the relation τ(z) = t ( 1 + φ(z) c2 ) . (3.35) Next, assume that the internal clock is located somewhere inside the test shell. Setting the ground level of the potential at the surface of the original spherical shell, we have z = r−R, where r is the radius of the test shell. The situation is represented in the figure below. 55 R r M m Internal clock Figure 3.1: Illustrative scheme of the gedanken experiment. The size of the internal clock is exaggerated for the sake of clearness; the clock is located inside the spherical shell that is being ejected. Recalling the assumption z � R, we can write the change on the gravitational potential felt by the clock when the shell moves from z = 0 to z = r −R as Φ ≡ φ(z)− φ(0) = Gmz R2 . (3.36) Now, we turn to Quantum Mechanics and state that the uncertainty in the height deter- mination is ∆z. Of course, this amounts to a uncertainty in the radial momentum ∆pz such that ∆z∆pz ≥ h̄ 2 . (3.37) It turns out that we can relate the quantity ∆z to the uncertainty ∆τ in the duration τ 56 of the experiment. Since τ is the tine recorded by the internal clock between the events “test shell with radius R” and “test shell with radius r > R”, equation (3.35) allows us to write τ = t ( 1 + Φ c2 ) = t ( 1 + Gmz c2R2 ) . (3.38) Thus, ∆τ = Gm∆z c2R2 t. (3.39) Furthermore, assuming that Φ c2 � 1, one can make the following approximation: ∆τ τ ≈ Gm∆z c2R2 =⇒ ∆z ≈ c2R2∆τ τGm . (3.40) As for ∆pz, we can give an upper bound to it by means of the following reasoning: if the mass of the original shell is to be measured within an accuracy ∆M , the change in the test shell’s momentum due to the impulse of the gravitational force F = mg = mGM R2 is well approximated by δpz ≈ Gm∆Mτ R2 . (3.41) So, we take the above quantity as an upper bound to the uncertainty about the determi- nation of pz: ∆pz ≤ δpz. Therefore, in the context of the outlined assumptions, a direct application of the relation (3.37) yields c2∆M∆τ ≥ h̄ 2 . (3.42) Finally, using that the energy of the spherical shell relates to its mass according to the celebrated relation E = Mc2, as well as the requirement τ ≥ ∆τ , we get the claimed result: 57 ∆Eτ ≥ h̄ 2 . 3.5 Concluding remarks At first, the absence of a general self-adjoint operator for time makes it harder to formulate an universal uncertainty relation regarding simultaneous measurement of energy and time. Nonetheless, apart from the content of the last section — which tends to be more “heuris- tic” due to the incorporation of gravitational effects in quantum systems — the discussion presented in this chapter supports the view that the Mandelstamm-Tamm formulation is — to the best of our knowledge — the most precise formulation of the time-energy uncertainty relation. According to such a formulation, any attempt to keep track of the dynamics of some quantum system’s observable to define elapsed time will be subject to the limitation (3.9). This fact per se already leads one to suspect that objects like honest clocks are forbidden by quantum theory. In the next chapter we go deeper into this discussion, arguing that any quantum system is indeed forbidden to play the role of an honest clock due to more fundamental issues. 58 4 Time observables in Quantum Mechanics and the nonexistence of honest clocks In standard Quantum Mechanics textbooks it is usually said that time cannot be repre- sented by a self-adjoint operator (like energy, position and momentum do); instead, it must be regarded simply as a parameter that labels the dynamics of states (or operators). We begin this chapter by presenting a no-go theorem — Pauli’s theorem — that supports such a statement. Nonetheless, since the requirement that only self-adjoint operators can represent observ- ables in the quantum realm is not true at all — as supported by our exposition of POVMs on section 2.5 —, Pauli’s theorem by itself is not sufficient to rule out the possibility of a consistent definition for quantum measurements of time. Indeed, as we shall present in this chapter, there is an approach that leads to the construction of a time observable in a quan- tum system, namely, a POVM that allows one to extract predictions about time of arrival measurements performed over a free particle. After discussing this construction, we finish the chapter by presenting a fundamental 59 result due to Unruh and Wald that completely rules out the realization of honest clocks in the quantum realm. 4.1 Pauli’s theorem Wolfgang Pauli quotes in [24] a celebrated argument that rules out the possibility of constructing a self adjoint operator canonically conjugated to the Hamiltonian of a system that admits ground state. In this section, we state and proof Pauli’s theorem. Theorem: Consider a physical system whose Hamiltonian Ĥ ∈ L(H) is bounded from below. Then, it is not possible to construct a self adjoint operator T̂ ∈ L(H) such that [Ĥ, T̂ ] = îI. (4.1) Proof: First of all, the hypothesis over Ĥ means that there exists an energy eigenstate |E0〉 with energy E0 such that E ≥ E0 for any other value of energy E allowed in this system. It is important to recognize that such assumption is quite natural, since in nature we do not observer systems without an energy ground state. Now, suppose by contradiction that there is a self-adjoint operator T̂ that satisfies equa- tion (4.1). For α ∈ R, consider the following unitary operator: Ûα = eiαT̂ . (4.2) Notice that unitariness of Ûα follows from the assumption of T̂ being self-adjoint, so that T̂ = T̂ †. For the sake of completeness, we recall the following identity: eλÂB̂e−λ = B̂ + λ[Â, B̂] + λ2 2 [Â, [Â, B̂]] +O(λ3). (4.3) In the previous equation,  and B̂ are any linear operators, and λ ∈ C. Adapting such 60 equation to our context, we have ÛαĤÛ † α = Ĥ + αÎ. (4.4) Since the parameter α can assume any real value, the spectrum of the operator Ĥ + αÎ is the whole real line. Moreover, the unitariness of Ûα ensures that the spectrum of Ĥ and ÛαĤÛ † α are the same. Indeed, one can show that their characteristic polynomials are equal by reasoning as follows: det ( ÛαĤÛ † α − λÎ ) = det ( Ûα(Ĥ − λÎ)Û †α ) = det ( Ĥ − λÎ ) . Therefore, we must conclude that the spectrum of the Hamiltonian Ĥ is the whole set R, what gives the contradiction that completes the proof of Pauli’s theorem. � 4.2 Time as a POVM in Quantum Mechanics Following the approach of [26] (see also [27, 28, 31, 32, 33]), we discuss the possibility to define a time observable for a particular quantum system — the free particle — within the formalism of POVMs (refer to section 2.5). Let us begin by defining the concept of time POVM for a generic quantum system. Consider G to be a group representing some symmetry in the physical space (spatial translations, rotations, etc.). For each g ∈ G, we denote by Û(g) ∈ L(H) the unitary representation of the group. We say that a POVM P : (Ω,A) → L+(H) is covariant with respect to this unitary representation if 61 Û(g)P̂(X)Û †(g) = P̂(Xg), ∀ X ∈ A, (4.5) where Xg denotes the set of A obtained through the action of the group element g on X. For instance, if X = [a, b) and G is the translation group, with Û(g) = e−igp̂, then Xg = [a+ g, b+ g). Thereby, we define a time POVM simply as a POVM that is covariant with respect to the group of time translations. Explicitly: T is a time POVM if, and only if eiτĤT̂(X)e−iτĤ = T̂(X + τ), (4.6) where X + τ = {x+ τ : x ∈ X}. Now, consider that H is the Hilbert space of a particle in one dimension. Moreover, suppose the particle is free and has mass m, so that the Hamiltonian is Ĥ = p̂2 2m . (4.7) If we were dealing with a classical particle with velocity v whose location at the instant t = t0 = 0 is x = x0 6= 0, then it would arrive at the origin x = 0 when t = −x0 v . (4.8) This is the time of arrival at the origin. Notice that it can be negative, meaning that the particle was at the origin before the instant t = 0. A quite natural extension of (4.8) to the quantum case is T̂ = −m 2 (x̂p̂−1 + p̂−1x̂). (4.9) Some authors make reference to this operator as the “Arahonov-Bohm time operator”, since it was first proposed — with opposed signal — as a time operator in the paper [21]. 62 Notice that T̂ as defined in this way is canonically conjugated to the free Hamiltonian: [Ĥ, T̂ ] = îI. (4.10) Of course, this relation must be understood in the domain D(T̂ ) of T̂ . For more details concerning the nature of D(T̂ ), as well as related mathematical issues, see appendix C. Nonetheless, due to Pauli’s theorem one can be sure that such an operator cannot be self-adjoint. The idea, then, is to construct a POVM in R from which we can compute probabilities regarding time measurements. A natural way of beginning such a task is to look for the eigenstates of the operator T̂ . Formally, we shall employ the notation T̂ |t〉 = t |t〉 , (4.11) with t ∈ R. In momentum representation, we have 〈p|T̂ |ψ〉 = im 2 ( ψ(p) p2 − 2 p dψ(p) dp ) , (4.12) where, as usual, ψ(p) = 〈p|ψ〉. Introducing the notation φt(p) = 〈p|t〉, relation (4.11) yields the following differential equation: dφt(p) dp = ( 1 2p + ipt m ) φt(p). (4.13) Due to the divergence at p = 0, this equation has two families of eigenfunctions: one for the branch p > 0 and other for p < 0. Indeed, for α = ±1, we can write these eigenfunctions in the form 63 φt(p;α) = 1√ 2πm Θ(αp) √ |p|e ip2t 2m , (4.14) where Θ(x) is the Heaviside step function, which takes the value 1 when x > 0 and vanishes otherwise. The multiplicativ