Jerusalem studies in philosophy and history of science, 2020
Einstein's celebrated 1905 argument for light quanta was phrased in terms of the concept of 'stat... more Einstein's celebrated 1905 argument for light quanta was phrased in terms of the concept of 'statistical independence'-wrongly. It was in fact based on the notion of local, non-interacting entities, and as such was neutral on the question of statistical independence. A specific kind of statistical dependence had already been introduced by Gibbs, in connection with the Gibbs paradox, but went unrecognised by Einstein. A modification of Einstein's argument favours the latter. This kind of statistical dependence, as eventually attributed by Einstein to Bose, although expressed, in quantum mechanics, by the (anti)symmetrisation of the wave-function, does not require 'genuine' entanglement. This chapter demystifies quantum indistinguishability; its novel features derive from the finiteness of the state-space in quantum mechanics, not indistinguishability. §1 Overview According to the physicist and historian Abraham Pais, the photon was the first particle to be predicted theoretically. It was also, he noted, met with scepticism on an epic scale: '[i]ts assimilation came after a struggle more intense and prolonged than for any other particle ever postulated'. 2 Yet it was also, in a certain sense, derived from empirical data -specifically, derived by Einstein from the Wien limit of Planck's black-body spectral distribution, taken as a phenomenological given (it was by then accurately confirmed at high frequencies and low temperatures). The derivation was what John Norton has called Einstein's 'miraculous argument', contained in the most revolutionary of all of Einstein's great papers of 1905. 3 If it was inferred from the data, how was it opposed? The answer in part is that it was incomplete: whatever light was really made of, it could not only be Einstein's light quanta, because if it were, it would contravene the Planck distribution in the low frequency hightemperature regime (the Rayleigh-Jeans regime, also by then accurately confirmed). What was the nature of the struggle, that was so intense and prolonged? The answer in part was the difficulty of reconciling a particulate structure to radiation with Maxwell's equations (or any wave equation, needed to explain interference effects), but another part was down to Einstein's insistence on statistical independence. It was Einstein's struggle; and it was only concluded (and then only temporarily) with his quantum theory of the ideal monatomic gas, published in three instalments in the period 1924-5. To see the nature of the difficulty, consider that radiation in thermal equilibrium behaves as a gas of indistinguishable particles, photons, satisfying the Bose-Einstein statistics; and that the latter, as Einstein admitted in his second paper of his gas trilogy, implies that particles are not statistically independent of each other; yet Einstein's 'miraculous argument' of 1905 had shown that thermal radiation, in the Wien regime, is made of statistically independent particles. Contradiction. 1 To appear in Shenker and Hemmo (2020), in a volume dedicated to the memory of Itamar Pitowski. 2 Pais (1982 p.361). 3 Norton (2006).
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Nov 1, 2021
A defence is offered of a version of the branch-counting rule for probability in the Everett inte... more A defence is offered of a version of the branch-counting rule for probability in the Everett interpretation (otherwise known as many-worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule, and deliver a notion of objective probability similar to naïve frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc: it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose-Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.
The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (o... more The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs' notion of 'generic phase'). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. 'Distinguishability', in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.
What does quantum mechanics tell us, if taken realistically, as a fundamental theory, which appli... more What does quantum mechanics tell us, if taken realistically, as a fundamental theory, which applies to everything? After a century of debate, it seems we are no nearer to an answer to this question. But in an important respect we know better what is in contention: it is the Everett interpretation of quantum mechanics, which is also known as the many-worlds interpretation. First advocated by Hugh Everett III in 1957, it is the only realist interpretation of quantum mechanics still standing. On every other approach the answer is that quantum mechanics tells us nothing when applied to everything because it makes no sense taken in this way; the theory must be changed. In the words of John Bell: "Either the wavefunction, as given by the Schrödinger equation, is not everything, or it is not right," 1 seeking to reduce the options to either modifying the Schrödinger equation, or supplementing it with hidden variables. But "everything" for Bell meant "everything in the known universe"; ignored was the alternative that the wavefunction (as given by the Schrödinger equation) describes more than the known universe -that it describes a quantum mechanical multiverse, a superposition of worlds, of which ours is only one. The idea is fantastical, but quantum mechanics is a theory like no other, a revolution still in the making after all these years. The worlds are derived from the unitary formalism; they are not put in by hand. Measurement interactions, we know, lead to macroscopic superpositions -if the unitary dynamics operates untrammelled --where each state in the superposition contains a record of a sequence of events as if obtained by the measurement postulates, as Everett gave a simple model to show. The apparent conflict between quantum theory and locality, as codified in Bell's theorem, is removed: the interpretation extends without modification to relativistic quantum theory. It really is interpretation of the equations: it neither modifies nor supplements them, save that the Schrödinger equation is taken to be universal. It offers a radical and novel understanding of how determinism can be reconciled with indeterminism, solving puzzles in the philosophy of probability that have bedeviled the subject for decades. It provides the basis for quantum cosmology, free of the measurement problem. These claims, of course, are all controversial. In a certain sense, if these arguments all stand up to scrutiny, the case for the Everett interpretation becomes overwhelming. No wonder they are strongly contested: there is simply too much at stake. Much recent literature has been on the probability interpretation, 2 but there is another aspect to the development of the Everett interpretation in the last three decades that is just as important: in terms of
A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's r... more A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995; 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert Space norm. The familiar objection that Everett's approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett's theory, and that this ca...
Many Worlds?: Everett, Quantum Theory, and Reality
Classical and Quantum Gravity, 2011
... Physics. James Ladyman is Professor of Philosophy at the University of Bristol. Tim Maudlin i... more ... Physics. James Ladyman is Professor of Philosophy at the University of Bristol. Tim Maudlin is Professor of Philosophy at Rutgers University. Wayne Myrvold is AssociateProfessor of Philosophy at the University of Western Ontario. ...
What is probability? Physicists, mathematicians, and philosophers have been engaged with this que... more What is probability? Physicists, mathematicians, and philosophers have been engaged with this question since well before the rise of modern physics. But in quantum mechanics, where probabilities are associated only with measurements, the question strikes to the heart of other foundational problems: what distinguishes measurements from other physical processes? Or in more formal terms: when are the unitary dynamical equations suspended in favour of probabilistic ones? This is the problem of measurement in quantum mechanics. The most clearcut solutions to it change the theory: they either add hidden variables (as in the pilot-wave theory), or they give up the unitary formalism altogether (as in state-reduction theories). The two strategies are tied to di¤erent conceptions of probability: probability as in classical statistical mechanics (as formulated by Boltzmann, Gibbs and Einstein), and probability as in Brownian motion (with the dynamics given by a stochastic process, as formulate...
Special relativity is most naturally formulated as a theory of spacetime geometry, but within the... more Special relativity is most naturally formulated as a theory of spacetime geometry, but within the spacetime framework probability appears to be a purely epistemic notion. It is possible that progress can be made with rather different approaches- covariant stochastic equations, in particular- but the results to date are not encouraging. However, it seems a non-epistemic notion of probability can be made out in Minkowski space on Everett’s terms. I shall work throughout with the consistent histories formalism. I shall start with a conservative interpretation, and then go on to Everett’s. 1 Probability in Consistent Histories In the consistent histories approach histories are represented by ordered products of Heisenberg-picture projection operators, of the form C(α(n, −m)) = Pαn(tn)...Pα1(t1)Pα0(t0)Pα−1(t−1)...Pα−m(t−m). (1) Here α(n, −m) is the ordered sequence of variables < αn,...α−m>; its values are histories. (As is usual, variables will on occasion stand for values as well...
A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relati... more A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995, 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert space norm. The familiar objection that Everett’s approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett’s theory, and that this can be transparen...
It is widely accepted that the notion of inertial frame is central to Newtonian mechanics, and th... more It is widely accepted that the notion of inertial frame is central to Newtonian mechanics, and that the correct space-time structure underlying Newton’s methods in Principia is neo-Newtonian or Galilean space-time. I argue to the contrary that inertial frames are not needed in Newton’s theory of motion, and that the right space-time structure for Newton’s Principia requires the notion of parallelism of spatial directions at different times, and nothing more.
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2020
The concept of indistinguishable particles in quantum theory is fundamental to questions of ontol... more The concept of indistinguishable particles in quantum theory is fundamental to questions of ontology. All ordinary matter is made of electrons, protons, neutrons, and photons and they are all indistinguishable particles. Yet the concept itself has proved elusive, in part because of the interpretational difficulties that afflict quantum theory quite generally, and in part because the concept was so central to the discovery of the quantum itself, by Planck in 1900; it came encumbered with revolution. I offer a deflationary reading of the concept 'indistinguishable' that is identical to Gibbs' concept 'generic phase', save that it is defined for state spaces with only finitely-many states of bounded volume and energy (finitely-many orthogonal states, in quantum mechanics). That, and that alone, makes for the difference between the quantum and Gibbs concepts of indistinguishability. This claim is heretical on several counts, but here we consider only the content of the claim itself, and its bearing on the early history of quantum theory rather than in relation to contemporary debates about particle indistinguishability and permutation symmetry. It powerfully illuminates that history.
The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (o... more The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.
The concept of objective probability — chance — has three distinctive features: chances are measu... more The concept of objective probability — chance — has three distinctive features: chances are measured by statistics, but only with high chance, increasing with the number of trials involved; when the chances of the outcomes of a chance process are known, even in the context of a single trial, then one's subjective probabilities — credences — in those outcomes should be set equal to those chances; and finally, chance processes involve uncertainty as to the outcomes of those processes. The second of these features is established for ratios in the squared norms of branches by the Born-rule theorem, proved in Chapter 8. This chapter shows that the first and third hold for those same quantities. All three characteristics of chance thus apply to branching in the Everett interpretation. In consequence, branching processes should be identified with chance processes, and ratios in the squared norms with chances.
Jerusalem studies in philosophy and history of science, 2020
Einstein's celebrated 1905 argument for light quanta was phrased in terms of the concept of 'stat... more Einstein's celebrated 1905 argument for light quanta was phrased in terms of the concept of 'statistical independence'-wrongly. It was in fact based on the notion of local, non-interacting entities, and as such was neutral on the question of statistical independence. A specific kind of statistical dependence had already been introduced by Gibbs, in connection with the Gibbs paradox, but went unrecognised by Einstein. A modification of Einstein's argument favours the latter. This kind of statistical dependence, as eventually attributed by Einstein to Bose, although expressed, in quantum mechanics, by the (anti)symmetrisation of the wave-function, does not require 'genuine' entanglement. This chapter demystifies quantum indistinguishability; its novel features derive from the finiteness of the state-space in quantum mechanics, not indistinguishability. §1 Overview According to the physicist and historian Abraham Pais, the photon was the first particle to be predicted theoretically. It was also, he noted, met with scepticism on an epic scale: '[i]ts assimilation came after a struggle more intense and prolonged than for any other particle ever postulated'. 2 Yet it was also, in a certain sense, derived from empirical data -specifically, derived by Einstein from the Wien limit of Planck's black-body spectral distribution, taken as a phenomenological given (it was by then accurately confirmed at high frequencies and low temperatures). The derivation was what John Norton has called Einstein's 'miraculous argument', contained in the most revolutionary of all of Einstein's great papers of 1905. 3 If it was inferred from the data, how was it opposed? The answer in part is that it was incomplete: whatever light was really made of, it could not only be Einstein's light quanta, because if it were, it would contravene the Planck distribution in the low frequency hightemperature regime (the Rayleigh-Jeans regime, also by then accurately confirmed). What was the nature of the struggle, that was so intense and prolonged? The answer in part was the difficulty of reconciling a particulate structure to radiation with Maxwell's equations (or any wave equation, needed to explain interference effects), but another part was down to Einstein's insistence on statistical independence. It was Einstein's struggle; and it was only concluded (and then only temporarily) with his quantum theory of the ideal monatomic gas, published in three instalments in the period 1924-5. To see the nature of the difficulty, consider that radiation in thermal equilibrium behaves as a gas of indistinguishable particles, photons, satisfying the Bose-Einstein statistics; and that the latter, as Einstein admitted in his second paper of his gas trilogy, implies that particles are not statistically independent of each other; yet Einstein's 'miraculous argument' of 1905 had shown that thermal radiation, in the Wien regime, is made of statistically independent particles. Contradiction. 1 To appear in Shenker and Hemmo (2020), in a volume dedicated to the memory of Itamar Pitowski. 2 Pais (1982 p.361). 3 Norton (2006).
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Nov 1, 2021
A defence is offered of a version of the branch-counting rule for probability in the Everett inte... more A defence is offered of a version of the branch-counting rule for probability in the Everett interpretation (otherwise known as many-worlds interpretation) of quantum mechanics that both depends on the state and is continuous in the norm topology on Hilbert space. The well-known branch-counting rule, for realistic models of measurements, in which branches are defined by decoherence theory, fails this test. The new rule hinges on the use of decoherence theory in defining branching structure, and specifically decoherent histories theory. On this basis ratios of branch numbers are defined, free of any convention. They agree with the Born rule, and deliver a notion of objective probability similar to naïve frequentism, save that the frequencies of outcomes are not confined to a single world at different times, but spread over worlds at a single time. Nor is it ad hoc: it is recognizably akin to the combinatorial approach to thermodynamic probability, as introduced by Boltzmann in 1879. It is identical to the procedure followed by Planck, Bose, Einstein and Dirac in defining the equilibrium distribution of the Bose-Einstein gas. It also connects in a simple way with the decision-theory approach to quantum probability.
The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (o... more The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs' notion of 'generic phase'). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. 'Distinguishability', in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.
What does quantum mechanics tell us, if taken realistically, as a fundamental theory, which appli... more What does quantum mechanics tell us, if taken realistically, as a fundamental theory, which applies to everything? After a century of debate, it seems we are no nearer to an answer to this question. But in an important respect we know better what is in contention: it is the Everett interpretation of quantum mechanics, which is also known as the many-worlds interpretation. First advocated by Hugh Everett III in 1957, it is the only realist interpretation of quantum mechanics still standing. On every other approach the answer is that quantum mechanics tells us nothing when applied to everything because it makes no sense taken in this way; the theory must be changed. In the words of John Bell: "Either the wavefunction, as given by the Schrödinger equation, is not everything, or it is not right," 1 seeking to reduce the options to either modifying the Schrödinger equation, or supplementing it with hidden variables. But "everything" for Bell meant "everything in the known universe"; ignored was the alternative that the wavefunction (as given by the Schrödinger equation) describes more than the known universe -that it describes a quantum mechanical multiverse, a superposition of worlds, of which ours is only one. The idea is fantastical, but quantum mechanics is a theory like no other, a revolution still in the making after all these years. The worlds are derived from the unitary formalism; they are not put in by hand. Measurement interactions, we know, lead to macroscopic superpositions -if the unitary dynamics operates untrammelled --where each state in the superposition contains a record of a sequence of events as if obtained by the measurement postulates, as Everett gave a simple model to show. The apparent conflict between quantum theory and locality, as codified in Bell's theorem, is removed: the interpretation extends without modification to relativistic quantum theory. It really is interpretation of the equations: it neither modifies nor supplements them, save that the Schrödinger equation is taken to be universal. It offers a radical and novel understanding of how determinism can be reconciled with indeterminism, solving puzzles in the philosophy of probability that have bedeviled the subject for decades. It provides the basis for quantum cosmology, free of the measurement problem. These claims, of course, are all controversial. In a certain sense, if these arguments all stand up to scrutiny, the case for the Everett interpretation becomes overwhelming. No wonder they are strongly contested: there is simply too much at stake. Much recent literature has been on the probability interpretation, 2 but there is another aspect to the development of the Everett interpretation in the last three decades that is just as important: in terms of
A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's r... more A variety of ideas arising in decoherence theory, and in the ongoing debate over Everett's relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995; 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert Space norm. The familiar objection that Everett's approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett's theory, and that this ca...
Many Worlds?: Everett, Quantum Theory, and Reality
Classical and Quantum Gravity, 2011
... Physics. James Ladyman is Professor of Philosophy at the University of Bristol. Tim Maudlin i... more ... Physics. James Ladyman is Professor of Philosophy at the University of Bristol. Tim Maudlin is Professor of Philosophy at Rutgers University. Wayne Myrvold is AssociateProfessor of Philosophy at the University of Western Ontario. ...
What is probability? Physicists, mathematicians, and philosophers have been engaged with this que... more What is probability? Physicists, mathematicians, and philosophers have been engaged with this question since well before the rise of modern physics. But in quantum mechanics, where probabilities are associated only with measurements, the question strikes to the heart of other foundational problems: what distinguishes measurements from other physical processes? Or in more formal terms: when are the unitary dynamical equations suspended in favour of probabilistic ones? This is the problem of measurement in quantum mechanics. The most clearcut solutions to it change the theory: they either add hidden variables (as in the pilot-wave theory), or they give up the unitary formalism altogether (as in state-reduction theories). The two strategies are tied to di¤erent conceptions of probability: probability as in classical statistical mechanics (as formulated by Boltzmann, Gibbs and Einstein), and probability as in Brownian motion (with the dynamics given by a stochastic process, as formulate...
Special relativity is most naturally formulated as a theory of spacetime geometry, but within the... more Special relativity is most naturally formulated as a theory of spacetime geometry, but within the spacetime framework probability appears to be a purely epistemic notion. It is possible that progress can be made with rather different approaches- covariant stochastic equations, in particular- but the results to date are not encouraging. However, it seems a non-epistemic notion of probability can be made out in Minkowski space on Everett’s terms. I shall work throughout with the consistent histories formalism. I shall start with a conservative interpretation, and then go on to Everett’s. 1 Probability in Consistent Histories In the consistent histories approach histories are represented by ordered products of Heisenberg-picture projection operators, of the form C(α(n, −m)) = Pαn(tn)...Pα1(t1)Pα0(t0)Pα−1(t−1)...Pα−m(t−m). (1) Here α(n, −m) is the ordered sequence of variables < αn,...α−m>; its values are histories. (As is usual, variables will on occasion stand for values as well...
A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relati... more A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relative-state theory, can be linked to issues in relativity theory and the philosophy of time, specifically the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995, 1996a); in what follows we shall consider the same circle of ideas, but specifically in relation to the interpretation of probability, and its identification with relations in the Hilbert space norm. The familiar objection that Everett’s approach yields probabilities different from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as defined by Everett’s theory, and that this can be transparen...
It is widely accepted that the notion of inertial frame is central to Newtonian mechanics, and th... more It is widely accepted that the notion of inertial frame is central to Newtonian mechanics, and that the correct space-time structure underlying Newton’s methods in Principia is neo-Newtonian or Galilean space-time. I argue to the contrary that inertial frames are not needed in Newton’s theory of motion, and that the right space-time structure for Newton’s Principia requires the notion of parallelism of spatial directions at different times, and nothing more.
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2020
The concept of indistinguishable particles in quantum theory is fundamental to questions of ontol... more The concept of indistinguishable particles in quantum theory is fundamental to questions of ontology. All ordinary matter is made of electrons, protons, neutrons, and photons and they are all indistinguishable particles. Yet the concept itself has proved elusive, in part because of the interpretational difficulties that afflict quantum theory quite generally, and in part because the concept was so central to the discovery of the quantum itself, by Planck in 1900; it came encumbered with revolution. I offer a deflationary reading of the concept 'indistinguishable' that is identical to Gibbs' concept 'generic phase', save that it is defined for state spaces with only finitely-many states of bounded volume and energy (finitely-many orthogonal states, in quantum mechanics). That, and that alone, makes for the difference between the quantum and Gibbs concepts of indistinguishability. This claim is heretical on several counts, but here we consider only the content of the claim itself, and its bearing on the early history of quantum theory rather than in relation to contemporary debates about particle indistinguishability and permutation symmetry. It powerfully illuminates that history.
The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (o... more The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.
The concept of objective probability — chance — has three distinctive features: chances are measu... more The concept of objective probability — chance — has three distinctive features: chances are measured by statistics, but only with high chance, increasing with the number of trials involved; when the chances of the outcomes of a chance process are known, even in the context of a single trial, then one's subjective probabilities — credences — in those outcomes should be set equal to those chances; and finally, chance processes involve uncertainty as to the outcomes of those processes. The second of these features is established for ratios in the squared norms of branches by the Born-rule theorem, proved in Chapter 8. This chapter shows that the first and third hold for those same quantities. All three characteristics of chance thus apply to branching in the Everett interpretation. In consequence, branching processes should be identified with chance processes, and ratios in the squared norms with chances.
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Papers by Simon Saunders