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volume II: Synopsis

part III: Modern Physics

page 21: Symmetry

One of the most startling features of quantum mechanics for a deterministic classical physicist is that it does not always make definite predictions. Although it enables us to compute the eigenvalues of operators to a high degree of precision (consistent with observation), it merely assigns probabilities (in accordance with the 'Born Rule') to the different possible outcomes of an initial situation. Born rule - Wikipedia

This led some physicists (notably Einstein) to believe that the quantum theory is not complete. Alternatively, it may be that the quantum theory truly represents reality and that the world is not deterministic as the Newtonian world it replaced was believed to be. Most physicists accept this second idea, following the interpretation of quantum mechanics suggested by Born: 'The motion of particles follows probabilistic laws but the probability itself propagates according to the law of causality.' Here the theory of probability is fundamental. Pais, Inward Bound page 258

The source of randomness is symmetry or lack of differentiation. Once symmetry is broken, things become differentiated, confusion is removed and communication errors can be eliminated. Deterministic (causal, classical) process becomes possible. Such a process can be modelled by a Turing machine (abstract, formal computer, ie transformer). Symmetry - Wikipedia

In our model, symmetry is the name for nothingness, no difference. We do not need to ask where nothingness comes from, since it is nothing. The interesting point is when symmetry is broken and nothing becomes something. A spinning die is symmetrical. Once it has stopped it is (relative to the table) asymmetrical, one number is up. The equiprobability of the faces has 'collapsed' into a fixed number which can be observed by all (with whatever consequences, financial or other).

Where there is nothingness there is randomness, since there is no more reason for the 6 to show than the 1, or any other element of the set {1 2 3 4 5 6}. We may call symmetry breaking creation, bringing something definite to be where previously there was nothing. One of the most patronised creators in the Australian alcohol scene is the poker machine. Online Pokies

The mathematical theory of probability was axiomatized by Kolmogorov in 1933. In this theory, we understand randomness as arising from the symmetry or featurelessness of the continuous line running from 0 to 1. The probability of an event may then be measured by the length of a corresponding segment of this continuum. Kolmogorov, Probability axioms - Wikipedia

Probability is the mathematics of tolerance. A probability is a measure or estimate of the degree of confidence one may have in the occurrence of a particular event, measured on a scale running from 0 (the event is will never happen) to 1 (the event will always happen). A set of events may be called symmetrical when they are equiprobable, like the faces of a coin or die. Symmetry is broken when the coin or die is loaded so one number becomes more probable than another. The degree to which the symmetry is broken may be called its resolution. Perfect resolution, like perfect orthogonality, means perfect distinction and perfect symmetry breaking, one event having the probaility 1, all others 0.

The theory of probability is of particular interest to gamblers, and its first use was to calculate the probabilities of various outcomes in gambling games with cards, dice etc as a guide to rational investment. Since then probability theory has become an important branch of mathematics particularly concerned with signal detection: is something happening, or are we seeing something purely random?

As with other mathematical theories, the theory of probability begins with a set of points and axioms (rules) governing the relationships of those points. In the case of probability theory, we begin with a set of elementary events. Such a set may contain, for instance, two events: a coin falling head up and a coin falling head down. It is a feature of axiomatic (or abstract) mathematical theories that each may have unlimited number of concrete interpretations. So the theoretical structure which governs the tossing of coins may be applied to any situation whose elementary events comprise p and not-p.

The formal study of probability seeks to establish relationships between the probabilities of elementary events and the probabiities of random events which comprise certain sets of elementary events. Thus we can build up a theory which relates the outcome of tossing a coin one hundred times to the outcome of tossing it once.

Probability theory is particularly important in physics. The classical application of probability theory gives us statistical mechanics. Classical statistical mechanics is a development of thermodynamics. Thermodynamics is concerned with the macroscopic behaviour of large numbers of individuals such as atoms or molecules. Classical statistical mechanics faces a similar task, of linking large numbers of microscopic elementary events to macroscopic phenomena such as the freezing and melting of water, or the formation of raindrops. Statistical mechanics - Wikipedia

In the deterministic Newtonian world, classical statistical mechanics is seen as a method of compressing the vast amount of information available at the microscopic level into a smaller volume of information meaningful at the macroscopic level. In quantum mechanics, however, we find that definite information is often not available, and that the outcomes of quantum mechanical calculations are probabilities rather than certainties.

In classical statistical mechanics, every particle is has a definite position and identity. They are all definite and separate objects. In quantum statistical mechanics, on the other hand, particles are not so well defined, and we speak of identical or indistinguishable particles. These identical particles fall into two separate classes called fermions and bosons, each with special statistical properties. This identity is a a form of symmetry, and further suggests that quantum mechanics works at the edge of reality, between symmetry and definition, that is between nothingness and being. Quantum statistical mechanics - Wikipedia, Fermi-Dirac statistics - Wikipedia, Bose-Einstein statistics - Wikipedia

[revised 23 May 2013]

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Further reading

Books

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Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands 
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Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' 
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Khinchin, A Y, The Mathematical Foundations of Quantum Statistics, Dover 1998 'In the area of quantum statistics, I show that a rigorous mathematical basis of the computational formulas of statistical physics ... may be obtained from an elementary application of the well-developed limit theorems of the theory of probability' 
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Kolmogorov, A N , and Nathan Morrison (Translator) (With an added bibliography by A T Bharucha-Reid), Foundations of the Theory of Probability, Chelsea 1956 Preface: 'The purpose of this monograph is to give an axiomatic foundation for the theory of probability. . . . This task would have been a rather hopeless one before the introduction of Lebesgue's theories of measure and integration. However, after Lebesgue's publication of his investigations, the analogies between measure of a set and mathematical expectation of a random variable became apparent. These analogies allowed of further extensions; thus, for example, various properties of independent random variables were seen to be incomplete analogy with the corresponding properties of orthogonal functions ... ' 
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Pais, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Preface: 'I will attempt to describe what has been discovered and understood about the constituents of matter, the laws to which they are subject and the forces that act on them [in the period 1895-1983]. . . . I will attempt to convey that these have been times of progress and stagnation, of order and chaos, of belief and incredulity, of the conventional and the bizarre; also of revolutionaries and conservatives, of science by individuals and by consortia, of little gadgets and big machines, and of modest funds and big moneys.' AP 
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Tanenbaum, Andrew S, Computer Networks, Prentice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. ... This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.'  
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Links
Born rule - Wikipedia Born rule - Wikipedia, the free encyclopedia 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see Bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <psi,|Pi|psi> where Pi is the projection onto the eigenspace of A corresponding to λi'. back
Bose-Einstein statistics - Wikipedia Bose-Einstein statistics - Wikipedia, the free encyclopedia 'In statistical mechanics, Bose–Einstein statistics (or more colloquially B–E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.' back
Fermi-Dirac statistics - Wikipedia Fermi-Dirac statistics - Wikipedia, the fre encyclopedia 'In statistical mechanics, Fermi-Dirac statistics is a particular case of particle statistics developed by Enrico Fermi and Paul Dirac that determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. In other words, it is the distribution of the probabilities that each possible energy levels is occupied by a fermion.; back
Online Pokies Australian Poker Machine Information 'The word "pokies" is Australian slang for slot machines. They can be a range of different gaming machine types but today the most popular and widely found variety are five video reel, multi-line slot games all of which have a bonus game of some sort. The most frequently seen bonus game is the chance to win free spins but other types are available.' back
Probability axioms - Wikipedia Probability axioms - Wikipedia, the free encyclopedia 'In probability theory, the probability P of some event E, denoted P(E), is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below. These assumptions can be summarised as: Let (Ω, F, P) be a measure space with P(Ω)=1. Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.' back
Quantum statistical mechanics - Wikipedia Quantum statistical mechanics - Wikipedia, the free encyclopedia 'Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.' back
Statistical mechanics - Wikipedia Statistical mechanics - Wikipedia, the free encyclopedia 'Statistical mechanics (or statistical thermodynamics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. . . . The essential problem in statistical thermodynamics is to determine the distribution of a given amount of energy E over N identical systems. The goal of statistical thermodynamics is to understand and to interpret the measurable macroscopic properties of materials in terms of the properties of their constituent particles and the interactions between them. This is done by connecting thermodynamic functions to quantum-mechanic equations. Two central quantities in statistical thermodynamics are the Boltzmann factor and the partition function.' back
Symmetry - Wikipedia Symmetry - Wikipedia, the free encyclopedia 'Symmetry in common usage generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise' back

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