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Why Must We Work in the Phase Space?

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Why Must We Work in the Phase Space? IPPT Reports on Fundamental Technological Research 1/2016 Jan J. Sławianowski, Frank E. Schroeck Jr., Agnieszka Martens WHY MUST WE WORK IN THE PHASE SPACE? Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw 2016 http://rcin.org.pl IPPT Reports on Fundamental Technological Research ISSN 2299-3657 ISBN 978-83-89687-98-2 Editorial Board/Kolegium Redakcyjne: Wojciech Nasalski (Editor-in-Chief/Redaktor Naczelny), Paweł Dłużewski, Zbigniew Kotulski, Wiera Oliferuk, Jerzy Rojek, Zygmunt Szymański, Yuriy Tasinkevych Reviewer/Recenzent: prof. dr Paolo Maria Mariano Received on 21st January 2016 Copyright °c 2016 by IPPT PAN Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk (IPPT PAN) (Institute of Fundamental Technological Research, Polish Academy of Sciences) Pawińskiego 5B, PL 02-106 Warsaw, Poland Printed by/Druk: Drukarnia Braci Grodzickich, Piaseczno, ul. Geodetów 47A http://rcin.org.pl Why must we work in the phase space? Jan J. Sławianowski1, Frank E. Schroeck Jr.2, Agnieszka Martens1 1Institute of Fundamental Technological Research, Polish Academy of Sciences 2Department of Mathematics, University of Denver Abstract We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statis- tical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language. There are some misunderstandings and confusions concerning the concept of induced probability and entropy on the submanifolds of the phase space. First of all, they are restricted only to hypersurfaces in the phase space, i.e., to the manifolds of the defect of dimension equal to one. But what is more important, it was assumed there that the phase-space geometry was metrical-Euclidean and the resulting metric geometry of the microcanonical ensemble was obtained by the reduction of the primary Euclidean geometry to the corresponding sub- manifold. But it is well-known that the phase-space manifold has no natural metric geometry and that all concepts to be used must be of symplectic origin. Otherwise they are just accidental or artificial. So, instead we show that even if the configuration space is endowed with some metric, then in general the true geometry of submanifolds in the corresponding cotangent bundle (phase-space) is of different origin which has nothing to do with the mentioned configura- tion space Riemannian geometry, instead it is of purely symplectic origin. And this is sufficient to constructing microcanonical ensemble and entropy concepts. In any case, the purely symplectic phase-space geometry is sufficient to obtain everything within the completely metric-free language. http://rcin.org.pl Why must we work in the phase space? Jan J. Sławianowski1, Frank E. Schroeck Jr.2, Agnieszka Martens1 1Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk 2Wydział Matematyki, Uniwersytet w Denver Abstrakt Chcemy wykazać, że opis zjawisk mechanicznych oparty na pojęciu przestrze- ni fazowej jest fundamentalny zarówno z klasycznego jak i kwantowego punktu widzenia. Pokazujemy, że liczne problemy mechaniki statystycznej, teorii kwan- tów i mechaniki quasiklasycznej oraz teorii układów całkowalnych mogą być dobrze sformułowane wyłącznie w języku symplektycznej przestrzeni fazowej. Istnieje mnóstwo nieporozumień czy wręcz błędów dotyczących pojęcia praw- dopodobieństwa warunkowego i entropii w przypadku podrozmaitości przestrzeni fazowej. Przede wszystkim są one zazwyczaj definiowane dla przypadku powie- rzchni o defekcie wymiaru jeden. Co jednak dużo ważniejsze, zwykle zakłada się, że przestrzeń fazowa ma jednocześnie metryczną geometrię Euklidesową. Ge- ometria metryczna podrozmaitości, używana w konstrukcji zespołu mikrokanon- icznego, jest otrzymywana jako redukcja, ograniczenie pierwotnej geometrii Eu- klidesowej. Wiadomo jednak, że rozmaitość przestrzeni fazowej nie ma żadnej „wrodzonej” geometrii metrycznej i że wszystkie podstawowe pojęcia, wyjąwszy dynamikę konkretnych modeli, muszą mieć czysto symplektyczną genezę. W prze- ciwnym wypadku są one przypadkowe lub wręcz sztuczne. Zatem, nawet jeśli wyjściowa przestrzeń konfiguracyjna ma zadaną geometrię typu metrycznego, to na ogół właściwa geometria podrozmaitości w wiązce ko-stycznej, przynajm- niej ta istotna dla pojęć statystycznych, nie jest związana z metryką konfigu- racyjną i ma czysto symplektyczną genezę. I to wystarcza dla skonstruowania pojęcia zespołu mikrokanonicznego i entropii. W każdym razie, czysto symplek- tyczna geometria przestrzeni fazowej wystarcza do otrzymania pojęć mechaniki statystycznej w obrębie języka całkowicie niemetrycznego. W przypadku, gdy przestrzeń konfiguracyjna jest Euklidesowa, implikowane przez metrykę pojęcia statystyczne pokrywają się z symplektycznymi. W ogólnym wypadku nie musi tak być. Pokazujemy, że pojęcia te dadzą się wprowadzić w języku czysto sym- plektycznym, niezależnym od metryki konfiguracyjnej. Dotyczy to także uogól- nionych rozkładów mikrokanonicznych. http://rcin.org.pl Contents 1. Introduction 7 2. Phase space aspects of statistical mechanics 17 2.1 Phase space geometry and Poisson manifolds . 17 2.2 Symplectic versus Riemannian geometry . 29 3. Symplectic foundations of quantum mechanics and mechanical-optical analogy 45 3.1 Towards quantum mechanics and mechanical-optical analogy . 45 3.2 The message of the Weyl-Wigner-Moyal-Ville formalism . 59 3.3 Lagrange and Legendre submanifolds . 66 3.4 Geometrization of the Huygens prescription . 70 3.5 The message of contact geometry . 75 3.6 Relation between contact geometry and Hamilton-Jacobi equations . 81 4. Hamiltonian systems on groups 85 4.1 Hamiltonian systems on Lie groups and their coadjoint orbit represen- tations . 85 4.2 The rigid body and the affine body from the symplectic and Poisson point of view . 103 4.3 Lattice aspects of the phase-space description of affine dynamics . 121 5. Classical mechanics formulated in quantum terms 129 5.1 Groups and phase spaces . 129 5.2 Quantum mechanics and classical mechanics on Hilbert spaces . 135 5.2.1 The Hilbert spaces . 136 5.2.2 A physical interpretation of ´ ........................ 145 5.2.3 Informational completeness of the A´(f) ................. 145 ² 2 5.2.4 The connection of [g2GL ¹(G=HR¤(g¡1)!) with the classical representation spaces of classical mechanics . 147 5.2.5 An application to solid state physics . 149 5.2.6 The value of working in quantum mechanics on phase space . 153 Bibliography 155 Author Index 161 http://rcin.org.pl http://rcin.org.pl 1 Introduction By abuse of language one can claim that our modern phase-space concepts have some roots in elementary human thinking, in elementary experience and in ancient philosophy. In a sense, the phase space ideas go back to the ancient Greeks, namely they seem to have their origin in the famous Zeno’s Paradox of Aporia. In spite of a manifold of mathematical and philosophical arguments aimed at justifying the reality of motion, all of them seem to be somehow superficial and unconvincing (hypocritical, one could say by abuse of language). It seems that the intuition of Greek sophists, the first critical positivists in European thought, was on the right track in spite of their deliberately provocative mode of expression. One cannot resist the feeling that “common sense” was more reliable here. There is something wrong in considering motion “in vitro”, killing it and try- ing to recover it again as a sequence of inanimate rest states. As an example, take a sequence of pictures which simulate some motion, as in a movie picture. You have a stage coach which seems to be going in some direction. But then you look at its wheels which appear to be rolling in the opposite direction! The elementary human intuition conceives motion “in vivo” as something primary and non-reducible to the sequence of instantaneous positions. According to the animalistic and naive but also non-corrupted pre-scientific human perception, it is not so that motion is a change of position; unlike this, change is a result of motion, the latter being perceived as something primary. And no modern mathematical distinctions between zero-measure and finite-measure subsets are able to overwhelm our doubts and bad feelings. One feels intuitively that the mentioned formally-logical shields offer a rather weak protection against at least some arguments of ancient sophists. Motion does not resurrect from instanta- neous positions. When trying to solve the “problem” of motion, one is faced immediately with another one. Namely, if it is not the instantaneous position but rather the instantaneous state of motion that is to be a primary concept, then our description must be shifted from the physical space M in which a particle moves http://rcin.org.pl 8 1. Introduction to some other manifold the points of which are labelled, roughly speaking, by instantaneous positions in M and velocities. More generally, if one considers a multiparticle system, e.g., one consisting of N material points, and subject it perhaps to some constraints, then M is replaced by some submanifold Q ½ M N , the configuration space of the system. And when analyzing motion we must lift the description from Q to some byproduct TQ the points of which contain the information about instantaneous configurations q 2 Q and systems of instantaneous velocities when q is passed. This manifold TQ of mechanical states in the sense of Newton is what mathematicians call the tangent bundle over
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