Einstein’s quantum theory of the monatomic ideal gas: non-statistical arguments for a new statistics

In this article, we analyze the third of three papers, in which Einstein presented his quantum theory of the ideal gas of 1924–1925. Although it failed to attract the attention of Einstein’s contemporaries and although also today very few commentators refer to it, we argue for its significance in the context of Einstein’s quantum researches. It contains an attempt to extend and exhaust the characterization of the monatomic ideal gas without appealing to combinatorics. Its ambiguities illustrate Einstein’s confusion with his initial success in extending Bose’s results and in realizing the consequences of what later came to be called Bose–Einstein statistics. We discuss Einstein’s motivation for writing a non-combinatorial paper, partly in response to criticism by his friend Ehrenfest, and we paraphrase its content. Its arguments are based on Einstein’s belief in the complete analogy between the thermodynamics of light quanta and of material particles and invoke considerations of adiabatic transformations as well as of dimensional analysis. These techniques were well known to Einstein from earlier work on Wien’s displacement law, Planck’s radiation theory and the specific heat of solids. We also investigate the possible role of Ehrenfest in the gestation of the theory.     

Einstein and the Kaluza–Klein particle

In his search for a unified field theory that could undercut quantum mechanics, Einstein considered five-dimensional classical Kaluza–Klein theory. He studied this theory most intensively during the years 1938–1943. One of his primary objectives was finding a non-singular particle solution. In the full theory this search got frustrated, and in the x⁵-independent theory Einstein, together with Pauli, argued it would be impossible to find these structures.     

Modular localization and the holistic structure of causal quantum theory,a historical perspective

Recent insights into the conceptual structure of localization in QFT (modular localization) led to clarifications of old unsolved problems. The oldest one is the Einstein–Jordan conundrum which led Jordan in 1925 to the discovery of quantum field theory. This comparison of fluctuations in subsystems of heat bath systems (Einstein) with those resulting from the restriction of the QFT vacuum state to an open subvolume (Jordan) leads to a perfect analogy; the globally pure vacuum state becomes upon local restriction a strongly impure KMS state. This phenomenon of localization-caused thermal behavior as well as the vacuum-polarization clouds at the causal boundary of the localization region places localization in QFT into a sharp contrast with quantum mechanics and justifies the attribute “holstic”. In fact it positions the E–J Gedankenexperiment into the same conceptual category as the cosmological constant problem and the Unruh Gedankenexperiment. The holistic structure of QFT resulting from “modular localization” also leads to a revision of the conceptual origin of the crucial crossing property which entered particle theory at the time of the bootstrap S-matrix approach but suffered from incorrect use in the S-matrix settings of the dual model and string theory.The new holistic point of view, which strengthens the autonomous aspect of QFT, also comes with new messages for gauge theory by exposing the clash between Hilbert space structure and localization and presenting alternative solutions based on the use of stringlocal fields in Hilbert space. Among other things this leads to a reformulation of the Englert–Higgs symmetry breaking mechanism.     

A probabilistic foundation of elementary particle statistics. Part I

The long history of ergodic and quasi-ergodic hypotheses provides the best example of the attempt to supply non-probabilistic justifications for the use of statistical mechanics in describing mechanical systems. In this paper we reverse the terms of the problem. We aim to show that accepting a probabilistic foundation of elementary particle statistics dispenses with the need to resort to ambiguous non-probabilistic notions like that of (in)distinguishability. In the quantum case, starting from suitable probability conditions, it is possible to deduce elementary particle statistics in a unified way. Following our approach Maxwell-Boltzmann statistics can also be deduced, and this deduction clarifies its status.Thus our primary aim in this paper is to give a mathematically rigorous deduction of the probability of a state with given energy for a perfect gas in statistical equilibrium; that is, a deduction of the equilibrium distribution for a perfect gas. A crucial step in this deduction is the statement of a unified statistical theory based on clearly formulated probability conditions from which the particle statistics follows. We believe that such a deduction represents an important improvement in elementary particle statistics, and a step towards a probabilistic foundation of statistical mechanics.In this Part I we first present some history: we recall some results of Boltzmann and Brillouin that go in the direction we will follow. Then we present a number of probability results we shall use in Part II. Finally, we state a notion of entropy referring to probability distributions, and give a natural solution to Gibbs' paradox.     

Walther Bothe's contributions to the understanding of the wave-particle duality of light

It is little known that during the birth of quantum mechanics Walther Bothe (1891–1957) published from mid-1923 to the end of 1926, partly together with Hans Geiger (1882–1945), as many as 20 papers, all dealing with light quanta (photons). Around half of the publications (11) are of experimental nature; the rest deal with theoretical problems. This paper presents Walther Bothe's experimental and theoretical contributions to the understanding of the particle-wave duality of light in the mid-1920s, for which the interplay between experimental and theoretical ideas plays an essential role.     

Fritz London and the scale of quantum mechanisms

Fritz London's seminal idea of “quantum mechanisms of macroscopic scale”, first articulated in 1946, was the unanticipated result of two decades of research, during which London pursued quantum-mechanical explanations of various kinds of systems of particles at different scales. He started at the microphysical scale with the hydrogen molecule, generalized his approach to chemical bonds and intermolecular forces, then turned to macrophysical systems like superconductors and superfluid helium. Along this path, he formulated a set of concepts—the quantum mechanism of exchange, the rigidity of the wave function, the role of quantum statistics in multi-particle systems, the possibility of order in momentum space—that eventually coalesced into a new conception of systems of equal particles. In particular, it was London's clarification of Bose-Einstein condensation that enabled him to formulate the notion of superfluids, and led him to the recognition that quantum mechanics was not, as it was commonly assumed, relevant exclusively as a micromechanics.     

Whence the eigenstate–eigenvalue link?

David Wallace has recently argued that the eigenstate–eigenvalue (E–E) link has no place in serious discussions of quantum mechanics on the grounds that, as he claims, the E–E link is an invention of philosophers rather than the community of practicing physicists. This raises an historical question regarding the origin of the link. This paper aims to answer this question by tracing the historical development of the link through six key textbooks of quantum mechanics. In light of the historical evidence from these textbooks, it is argued that Wallace provides insufficient grounds for dismissing the E–E link from discussions of quantum mechanics.     

The emergence and interpretation of probability in Bohmian mechanics

A persistent question about the deBroglie–Bohm interpretation of quantum mechanics concerns the understanding of Born's rule in the theory. Where do the quantum mechanical probabilities come from? How are they to be interpreted? These are the problems of emergence and interpretation. In more than 50 years no consensus regarding the answers has been achieved. Indeed, mirroring the foundational disputes in statistical mechanics, the answers to each question are surprisingly diverse. This paper is an opinionated survey of this literature. While acknowledging the pros and cons of various positions, it defends particular answers to how the probabilities emerge from Bohmian mechanics and how they ought to be interpreted.     

The emergence of selection rules and their encounter with group theory, 1913–1927

In today's quantum mechanics and quantum field theory, the observable signature of a symmetry is often sought in the form of a selection rule: a missing radiation frequency, a particle that does not decay in another one, a scattering process which fails to take place. The connection between selection rules and symmetries is effected thanks to the mathematical discipline of group theory. In the present paper, I will offer an overview of how the productive synergy between selection rules and group theory came to be. The first half of the work will be devoted to the emergence of the idea of spectroscopic selection rules in the context of the old quantum theory, showing how this notion was linked with an interpretive scheme of theoretical nature which, once combined with group theory, would bear many fruits. In the second part of the paper, I will focus on the actual encounter between selection rules and group theory, and on the person largely responsible for it: Eugene Wigner. I will attempt to reconstruct the path which led Wigner, of all people, to be the agent effecting this connection.     

Foundations of quantum gravity: The role of principles grounded in empirical reality

When attempting to assess the strengths and weaknesses of various principles in their potential role of guiding the formulation of a theory of quantum gravity, it is crucial to distinguish between principles which are strongly supported by empirical data – either directly or indirectly – and principles which instead (merely) rely heavily on theoretical arguments for their justification. Principles in the latter category are not necessarily invalid, but their a priori foundational significance should be regarded with due caution. These remarks are illustrated in terms of the current standard models of cosmology and particle physics, as well as their respective underlying theories, i.e., essentially general relativity and quantum (field) theory. For instance, it is clear that both standard models are severely constrained by symmetry principles: an effective homogeneity and isotropy of the known universe on the largest scales in the case of cosmology and an underlying exact gauge symmetry of nuclear and electromagnetic interactions in the case of particle physics. However, in sharp contrast to the cosmological situation, where the relevant symmetry structure is more or less established directly on observational grounds, all known, nontrivial arguments for the “gauge principle” are purely theoretical (and far less conclusive than usually advocated). Similar remarks apply to the larger theoretical structures represented by general relativity and quantum (field) theory, where – actual or potential – empirical principles, such as the (Einstein) equivalence principle or EPR-type nonlocality, should be clearly differentiated from theoretical ones, such as general covariance or renormalizability. It is argued that if history is to be of any guidance, the best chance to obtain the key structural features of a putative quantum gravity theory is by deducing them, in some form, from the appropriate empirical principles (analogous to the manner in which, say, the idea that gravitation is a curved spacetime phenomenon is arguably implied by the equivalence principle). Theoretical principles may still be useful however in formulating a concrete theory (analogous to the manner in which, say, a suitable form of general covariance can still act as a sieve for separating theories of gravity from one another). It is subsequently argued that the appropriate empirical principles for deducing the key structural features of quantum gravity should at least include (i) quantum nonlocality, (ii) irreducible indeterminacy (or, essentially equivalently, given (i), relativistic causality), (iii) the thermodynamic arrow of time, (iv) homogeneity and isotropy of the observable universe on the largest scales. In each case, it is explained – when appropriate – how the principle in question could be implemented mathematically in a theory of quantum gravity, why it is considered to be of fundamental significance and also why contemporary accounts of it are insufficient. For instance, the high degree of uniformity observed in the Cosmic Microwave Background is usually regarded as theoretically problematic because of the existence of particle horizons, whereas the currently popular attempts to resolve this situation in terms of inflationary models are, for a number of reasons, less than satisfactory. However, rather than trying to account for the required empirical features dynamically, an arguably much more fruitful approach consists in attempting to account for these features directly, in the form of a lawlike initial condition within a theory of quantum gravity.     

A simplified genesis of quantum mechanics

The bewildering complexity of the history of quantum theory tends to discourage its use as a means to understand or teach the foundations of quantum mechanics. The present paper is an attempt at simplifying this history so as to make it more helpful to physicists and philosophers. In particular, Heisenberg's notoriously difficult derivation of the fundamental equations of quantum mechanics, or later derivations of its statistical interpretation are replaced with shorter and more direct arguments to the same purpose. As the implied amputations and distortions do not imply major anachronisms, they should facilitate the grasping of the main historical steps without excluding a reasonable assessment of their historical or logical necessity.     

The making of an intrinsic property: “Symmetry heuristics” in early particle physics

Mathematical invariances, usually referred to as “symmetries”, are today often regarded as providing a privileged heuristic guideline for understanding natural phenomena, especially those of micro-physics. The rise of symmetries in particle physics has often been portrayed by physicists and philosophers as the “application” of mathematical invariances to the ordering of particle phenomena, but no historical studies exist on whether and how mathematical invariances actually played a heuristic role in shaping microphysics. Moreover, speaking of an “application” of invariances conflates the formation of concepts of new intrinsic degrees of freedom of elementary particles with the formulation of models containing invariances with respect to those degrees of freedom. I shall present here a case study from early particle physics (ca. 1930–1954) focussed on the formation of one of the earliest concepts of a new degree of freedom, baryon number, and on the emergence of the invariance today associated to it. The results of the analysis show how concept formation and “application” of mathematical invariances were distinct components of a complex historical constellation in which, beside symmetries, two further elements were essential: the idea of physically conserved quantities and that of selection rules. I shall refer to the collection of different heuristic strategies involving selection rules, invariances and conserved quantities as the “SIC-triangle” and show how different authors made use of them to interpret the wealth of new experimental data. It was only a posteriori that the successes of this hybrid “symmetry heuristics” came to be attributed exclusively to mathematical invariances and group theory, forgetting the role of selection rules and of the notion of physically conserved quantity in the emergence of new degrees of freedom and new invariances. The results of the present investigation clearly indicate that opinions on the role of symmetries in fundamental physics need to be critically reviewed in the spirit of integrated history and philosophy of science.     

Symmetry and asymmetry in electrodynamics from Rowland to Einstein

Halfway through the paper in which he laid down the foundations for the theory of special relativity, Einstein concludes that “the asymmetry mentioned in the Introduction … disappears.” Making asymmetry disappear has proved to be one of Einstein's many significant moves in his annus mirabilis of 1905. This elimination of asymmetry has led many commentators to claim that Einstein was motivated by either an aesthetic or an epistemic argument which gives priority to symmetry over asymmetry. Following closely the development of electrodynamics in the period from 1880 to 1905 and the usage of the related terms reciprocity and symmetry, we suggest a different way of understanding Einstein's motivation and the path he took. In contrast to the received view, we argue that Einstein responded to a debate in the literature on electrodynamics and that he was concerned neither with an aesthetic nor with an epistemic argument; rather, his reasoning was physical in the best sense, and most original. We will show that by providing a new perspective on the relation between electricity and magnetism, Einstein succeeded in bringing the discussion of symmetry in electrodynamics to an end.     

From aether impulse to QED: Sommerfeld and the Bremsstrahlen theory

The radiation that is due to the braking of charged particles has been in the focus of theoretical physics since the discovery of X-rays by the end of the 19th century. The impact of cathode rays in the anti-cathode of an X-ray tube that resulted in the production of X-rays led to the view that X-rays are aether impulses spreading from the site of the impact. In 1909, Arnold Sommerfeld calculated from Maxwell׳s equations the angular distribution of electromagnetic radiation due to the braking of electrons. He thereby coined the notion of “Bremsstrahlen.” In 1923, Hendrik A. Kramers provided a quantum theoretical explanation of this process by means of Bohr׳s correspondence principle. With the advent of quantum mechanics the theory of bremsstrahlung became a target of opportunity for theorists like Yoshikatsu Sugiura, Robert Oppenheimer, and–again–Sommerfeld, who presented in 1931 a comprehensive treatise on this subject. Throughout the 1930s, Sommerfeld׳s disciples in Munich and elsewhere extended and improved the bremsstrahlen theory. Hans Bethe and Walter Heitler, in particular, in 1934 presented a theory that was later regarded as “the most important achievement of QED in the 1930s” (Freeman Dyson). From a historical perspective the bremsstrahlen problem may be regarded as a probe for the evolution of theories in response to revolutionary changes in the underlying principles.     

Physics from Fisher information

B. R. Frieden uses a single procedure, called extreme physical information, with the aim of deriving ‘most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations and quantum gravity’. His method, which is based on Fisher information, is given a detailed exposition in this book, and we attempt to assess the extent to which he succeeds in his task.     

Engineering Entanglement,Conceptualizing Quantum Information

Proposed by Einstein, Podolsky, and Rosen (EPR) in 1935, the entangled state has played a central part in exploring the foundation of quantum mechanics. At the end of the twentieth century, however, some physicists and mathematicians set aside the epistemological debates associated with EPR and turned it from a philosophical puzzle into practical resources for information processing. This paper examines the origin of what is known as quantum information. Scientists had considered making quantum computers and employing entanglement in communications for a long time. But the real breakthrough only occurred in the 1980s when they shifted focus from general-purpose systems such as Turing machines to algorithms and protocols that solved particular problems, including quantum factorization, quantum search, superdense code, and teleportation. Key to their development was two groups of mathematical manipulations and deformations of entanglement—quantum parallelism and ‘feedback EPR’—that served as conceptual templates. The early success of quantum parallelism and feedback EPR was owed to the idealized formalism of entanglement researchers had prepared for philosophical discussions. Yet, such idealization is difficult to hold when the physical implementation of quantum information processors is at stake. A major challenge for today's quantum information scientists and engineers is thus to move from Einstein et al.'s well-defined scenarios into realistic models.     

Zero-Point Energy: The Case of the Leiden Low-Temperature Laboratory of Heike Kamerlingh Onnes

In this paper we examine the reaction of the Leiden low-temperature laboratory of Heike Kamerlingh Onnes to new ideas in quantum theory. Especially the contributions of Albert Einstein (1906) and Peter Debye (1912) to the theory of specific heat, and the concept of zero-point energy formulated by Max Planck in 1911, gave a boost to solid state research to test these theories. In the case of specific heat measurements, Kamerlingh Onnes's laboratory faced stiff competition from Walter Nernst's Institute of Physical Chemistry in Berlin. In fact, Berlin got the better of it because Leiden lacked focus. After the liquefaction of helium in 1908, Kamerlingh Onnes transformed his laboratory into an international facility for low temperature research, and for this reason it was impossible to make headway with the specific heat measurements. In the case of zero-point energy, Leiden developed a magnetic research programme to test the concept. Initially the balance of evidence seemed to be tipping in favour of zero-point energy. After 1914, however, Leiden would desert the theory in fovour, of a concept from calssical physics. A curious move that illustrates Kamerlingh Onnes's discomfort with the new quantum theory.     

Einstein and Singularities

Except for a few brief periods, Einstein was uninterested in analysing the nature of the spacetime singularities that appeared in solutions to his gravitational field equations for general relativity. The existence of such monstrosities reinforced his conviction that general relativity was an incomplete theory which would be superseded by a singularity-free unified field theory. Nevertheless, on a number of occasions between 1916 and the end of his life, Einstein was forced to confront singularities. His reactions show a strange asymmetry: he tended to be more disturbed by (what today we would call) merely apparent singularities and less disturbed by (what we would call) real singularities. Einstein had strong a priori ideas about what results a correct physical theory should deliver. In the process of searching through theoretical possibilities, he tended to push aside technical problems and jump over essential difficulties. Sometimes this method of working produced brilliant new ideas—such as the Einstein–Rosen bridge—and sometimes it lead him to miss important implications of his theory of gravity—such as gravitational collapse.