Time Operator Time Operator Innovation and Complexity I. Antoniou B. Misra Z. Suchanecki A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto Contents Preface ix Acknowledgments xi Introduction xiii 1 Predictability and innovation 1 1.1 Dynamical systems 2 1.2 Dynamical systems associated with maps 4 1.3 The ergodic hierarchy 6 1.4 Evolution operators 9 1.5 Ergodic properties of dynamical systems – operator approach 12 1.6 Innovation and time operator 14 2 Time operator of Kolmogorov systems 21 3 Time operator of the baker map 29 4 Time operator of relativistic systems 33 5 Time operator of exact systems 35 v vi CONTENTS 5.1 Exact systems 35 5.2 Time operator of unilateral shift 38 5.3 Time operator of bilateral shift 44 6 Time operator of the Renyi map and the Haar wavelets 47 6.1 Non-uniform time operator of the Renyi map 48 6.2 The domain of the time operator 52 6.3 The Haar wavelets on the interval 54 6.4 Relations between the time operators of the Renyi and baker maps 56 6.5 The uniform time operator for the Renyi map 58 7 Time operator of the cusp map 61 8 Time operator of stationary stochastic processes 65 8.1 Time operator of the stochastic processes stationary in wide sense 68 8.2 Time operators of strictly stationary processes - Fock space 73 9 Time operator of diffusion processes 77 9.1 Time Operators for Semigroups and Intertwining 78 9.2 Intertwining of the Diffusion Equation with the Unilateral Shift 79 9.3 The Time Operator of the Diffusion Semigroup 82 9.4 The Spectral Resolution of the Time Operator 85 9.5 Age Eigenfunctions and Shift Representation of the Solution of the Diffusion Equation 86 9.6 Time Operator of the Telegraphist Equation 87 9.7 Nonlocal Entropy Operator for the Diffusion Equation 88 10 Time operator of self-similar processes 91 11 Time operator of Markov processes 97 11.1 Markov processes and Markov semigroups 99 11.2 Canonical process 101 11.3 Time operators associated with Markov processes 102 12 Time operator and approximation 107 CONTENTS vii 12.1 Time operator in function spaces 108 12.2 Time operator and Shannon theorem 113 12.3 Time operator associated with the Haar basis in L 2 [0,1] 114 12.4 Time operator associated with the Faber-Schauder basis in C [0,1] 120 13 Time operator and quantum theory 125 13.1 Self-adjoint operators, unitary groups and spectral resolution 125 13.2 Different definitions of time operator and their interrelations 126 13.3 Spectrum of L and T 130 13.4 Incompatibility between the semiboundeness of the generator H of the evolution group and the existence of a time operator canonical conjugate to H 130 13.5 Liouville-von Neumann formulation of quantum mechanics 131 13.6 Derivation of time energy uncertainty relation 134 13.7 Construction of spectral projections of the time operator T 136 14 Intertwining dynamical systems with stochastic processes 143 14.1 Misra-Prigogine-Courbage theory of irreversibility 143 14.2 Nonlocality of the Misra-Prigogine-Courbage semigroup 153 15 Spectral and shift representations 159 15.1 Generalized spectral decompositions of evolution operators 159 15.2 Relation Between Spectral and Shift Representations 170 Appendix A Probability 175 A.1 Preliminaries - probability 175 A.2 Stochastic processes 181 A.3 Martingales 184 A.4 Stochastic measures and integrals 187 A.5 Prediction, filtering and smoothing 189 A.6 Karhunan-Loeve expansion 191 viii CONTENTS Appendix B Operators on Hilbert and Banach spaces. 195 Appendix C Spectral analysis of dynamical systems 215 References 223 Preface This is an example preface. 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PRIGOGINE Brussels, Belgium ix [...]... properties Such time operator corresponds to the interpolation of continuous functions by polygonal lines We give the explicit form of the eigenprojectors of this time operator and characterize the functions from its domain in terms of their modulus of continuity 1 Predictability and innovation Consider a physical system that can be observed through time varying quantities xt , where t stands for time that... f − t→∞ = 0 , for each f ∈ L2 f dµ L2 14 1.6 PREDICTABILITY AND INNOVATION INNOVATION AND TIME OPERATOR We begin with the concept of information and the strictly related concept of entropy as the measure of disorder Both these concepts have a long history The concept of entropy was introduced in thermodynamics already in 1854 by Clausius and then Boltzmann found its logarithmic form The first step towards... on P (Ai0 ) and can be defined as some function of P (Ai0 ), say f (P (Ai0 )) Using the above postulate about the information carried by two σ-algebras corresponding to independent random variables we can find the explicit form of the INNOVATION AND TIME OPERATOR 15 function f Note that the independence expressed in terms of partitions π1 and π2 means P (A ∩ B) = P (A)P (B) for each A ∈ π1 and B ∈ π2... by 2 and stretches X along the x-axis by 2 Such compressed and stretched rectangle is then vertically divided on two equal parts and the right-hand part is placed on the left hand part The inverse of the baker transformation  x 1  for 0 ≤ x < 1, 0 ≤ y < 2   2 , 2y ,  S −1 (x, y) =  x 1  1  + , 2y − 1 , for 0 < x ≤ 1, 2 < y < 1  2 2 1 is defined everywhere on X except the lines y = 2 and y... connections of time operator with wavelets, especially those restricted to the interval [0, 1], and the corresponding multiresolutions analysis We establish a link between the Shannon sampling theorem and the eigenprojectors of the time operator associated with the Shannon wavelet We construct the time INTRODUCTION xv operator associated with the Faber-Schauder system on the space C[0,1] and study its... on initial conditions such time reversal is practically possible only for short time intervals t − s Our aim is to introduce criteria that will allow to distinguish innovative systems Then we shall show that systems with innovations have their internal time that can be expressed by the existence of time operator First, however, let us introduce rigorously some basic concepts and tools 1.1 DYNAMICAL SYSTEMS... each xt is a random variable We are interested in the global evolution of the system, not particular realizations xt , from the point of view of innovation We call the evolution innovative if the dynamics of the system is such that there is a gain of information about the system when time increases Our purpose is to associate the concept of internal time with such systems The internal time will reflect... knowledge of the 1 2 PREDICTABILITY AND INNOVATION position xt0 of some point at the time instant t0 determines its future positions xt for t > t0 In principle, there is no place for innovation for such deterministic dynamical system Let us however, consider two specific examples Consider first the dynamics of pendulum (harmonic oscillator) The knowledge of its initial position x0 and the direction of the movement... significantly our prediction The above examples show that while there is no innovation in the harmonic oscillator there must be some intrinsic innovation in highly unstable systems like billiards Innovation is also connected with the observed direction of time Indeed, suppose that knowing the position xt of an evolving point at the time instant t we want to recover the position xs , for some s < t This... see the problems of prediction and filtering of stochastic processes from a new perspective and connect them directly with physical problems In this article we shall present an interesting connection of time operator with approximation theory The first connection, although indirect, of the time operator with the approximation theory has been obtained through wavelets [AnGu,AStime] An arbitrary wavelet multiresolution . Time Operator Time Operator Innovation and Complexity I. Antoniou B. Misra Z. Suchanecki A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane. to any given accuracy, while in the case of billiards this is impossible in practice because initial accuracy can get amplified due to sensi- tivity on initial conditions. It is also obvious that. there must be some intrinsic innovation in highly unstable systems like billiards. Innovation is also connected with the observed direction of time. Indeed, suppose that knowing the position x t of