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Scuola Normale Superiore eBooks

Spontaneous symmetry breaking

Franco Strocchi

Published Jan 1, 2019• 2 citations
Publisher

Abstract

Since the mechanism of spontaneous symmetry breaking has such a relevant impact for the description of physical systems, its explanation should be based on physical considerations with a direct operational meaning. To this purpose, it is useful to recognize that the operational/experimental description of a physical system (not necessarily quantum!) involves the following elements: the set A of its measurable quantities, briefly called observables ; the measurement of the time evolution of the observables; the set ∑ of configurations or states in which the system may be prepared, according to well defined protocols of experimental preparations; the measurement of an observable A , when the system is in the state ω is operationally defined by the experimental expectation 〈A〉 ω

DOI
10.1007/978-88-7642-660-5_1

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Since the mechanism of spontaneous symmetry breaking has such a relevant impact for the description of physical systems, its explanation should be based on physical considerations with a direct operational meaning. To this purpose, it is useful to recognize that the operational/experimental description of a physical system (not necessarily quantum!) involves the following elements: the set A of its measurable quantities, briefly called observables ; the measurement of the time evolution of the observables; the set ∑ of configurations or states in which the system may be prepared, according to well defined protocols of experimental preparations; the measurement of an observable A , when the system is in the state ω is operationally defined by the experimental expectation 〈A〉 ω
How do we describe the state of a quantum mechanical physical system in a mathematically precise way? How do we ascribe physical properties to this system? How does the system evolve in time? How do its properties change when it interacts with another system or a force? How do we measure and determine its properties? These are the questions whose answers are encoded in the laws of quantum mechanics to be set down in this chapter. They have been inscribed in this form after nearly half a century of experimental observations, theoretical modeling, and intellectual gymnastics to tie up the two in a satisfactory and robust structure.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .
Against the conclusion now reached, that the basis of measurement must be a prescribed operational geometry, it can of course be argued that physics has had to abandon the attempt to exhibit its fundamental concepts operationally. For example, wave-functions, whether of single particles or of systems of particles as in the atom or nucleus, are in principle unobservable. To claim that they might somehow be observed, even by as indirect a way as the 'observation' of energy-levels, would be to overthrow the formalism of quantum mechanics in its very origins, and thus to discount all its experimental confirmations.
Over the last several decades, two theoretical tools have been indispensable in the field of statistical physics. Spontaneous symmetry breaking has allowed for the description of systems which exhibit second order phase transitions, while the eigenstate thermalization hypothesis has provided a theoretical framework for understanding how isolated quantum many-body systems come to thermal equilibrium. In this dissertation, we will explore the compatibility of these two paradigms of theoretical physics.We will begin with a brief introduction to the relevant topics discussed in the main body of the dissertation, along with a brief overview of the numerical tools used in the subsequent investigations.Following this, we will numerically explore the compatibility between spontaneous symmetry breaking and eigenstate thermalization through a sequence of papers which have been previously published by myself and a collection of other authors. We will study the compatibility of these two theoretical frameworks through an exact diagonalization and Quantum Monte Carlo study of the Transverse-Field Ising model, a quantum non-integrable system which, we argue, displays both spontaneous symmetry breaking and eigenstate thermalization.Following this exposition, we will briefly comment on several corollaries which follow from these previously published papers, some of which we are currently preparing to incorporate into future publications. These corollaries largely focus on the subject of time evolution in quantum systems which display both spontaneous symmetry breaking and eigenstate thermalization, as well as the possibility that individual eigenstates of such systems may contain information about the critical behaviour of the corresponding finite-temperature phase transition.