It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to
the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and
‘indistinguishability’ of particles has come to be required.
However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle
the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s
first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its
dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or
Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject.
Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace.
In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go
through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics
became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the
twentieth century the mutual influence between both disciplines was profuse.
We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often
known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to
the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and
‘indistinguishability’ of particles has come to be required.
However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle
the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s
first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its
dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or
Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject.
Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace.
In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go
through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics
became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the
twentieth century the mutual influence between both disciplines was profuse.
We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often
known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.
The suggestion that particles of the same kind may be indistinguishable in a fundamental sense, even so that challenges to traditional notions of individuality and identity may arise, has first come up in the context of classical statistical mechanics. In particular, the Gibbs paradox has sometimes been interpreted as a sign of the untenability of the classical concept of a particle and as a premonition that quantum theory is needed. This idea of a ‘quantum connection’ stubbornly persists in the literature, even though it has also been criticized frequently. Here we shall argue that although this criticism is justified, the proposed alternative solutions have often been wrong and have not put the paradox in its right perspective. In fact, the Gibbs paradox is unrelated to fundamental issues of particle identity; only distinguishability in a pragmatic sense plays a role (in this we develop ideas of van Kampen [10]), and in principle the paradox always is there as long as the concept of a particle applies at all. In line with this we show that the paradox survives even in quantum mechanics, in spite of the quantum mechanical (anti-)symmetrization postulates.
The Gibbs paradox has frequently been interpreted as a sign that particles of the same kind are fundamentally indistinguishable; and that quantum mechanics, with its identical fermions and bosons, is indispensable for making sense of this. In this article we shall argue, on the contrary, that analysis of the paradox supports the idea that classical particles are always distinguishable. Perhaps surprisingly, this analysis extends to quantum mechanics: even according to quantum mechanics there can be distinguishable particles of the same kind. Our most important general conclusion will accordingly be that the universally accepted notion that quantum particles of the same kind are necessarily indistinguishable rests on a confusion about how particles are represented in quantum theory.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
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.
Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the microlevel, and from the relation S=k ln W leads to a nonextensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable and that quantum mechanics is indispensable for making sense of this inconsistency. In contrast, we argue that the classical statistics of distinguishable particles and the resulting nonextensive entropy function are perfectly acceptable from both a theoretical and an experimental perspective. The inconsistency with thermodynamics can be removed by taking into account that the entropy concept in statistical mechanics is not completely identical to the thermodynamical one. We observe that even identical quantum particles are in some cases distinguishable, and conclude that quantum mechanics is irrelevant to the Gibbs paradox.
Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistinguishability that is crucial for the treatment of the Gibbs’ problem and the consistency of its solution with conventional thermodynamics. Quantum mechanics provides a quantitative criterion, not possible in the classical picture, for the degree of indistinguishability in terms of visibility of quantum interference, or overlap of the states as pointed out by von Neumann, thereby endowing the entropy expression with mathematical continuity and physical reasonableness.