## Wahrscheinlichkeitstheoretischer Aufbau der Quant-enmechanik; Thermodynamik quantenmechanischer Gesamtheiten. Two offprints from: Nachrichten der Gesellschaft der Wissenschaften zu Göttingen. Berlin: Gesellschaft der Wissenschaften zu Göttingen, 1927.

Berlin: Gesellschaft der Wissenschaften zu Göttingen, 1927.

First edition, extremely rare offprint issues, with distinguished provenance, of von Neumann’s mathematical formulation of the probabilistic interpretation of quantum mechanics, and of the quantum-mechanical measurement problem, together with its application to quantum thermodynamics. The two papers were presented to the Göttingen Academy on the same day, 11 November 1927. “In the course of his formulation of quantum mechanics in terms of vectors and operators of Hilbert space von Neumann also gave in complete generality the basic statistical rule of interpretation of the theory. This rule concerns the result of the measurement of a given physical quantity on a system in a given quantum state and expresses its probability distribution by means of a simple and now completely familiar formula involving the vector representing the state and the spectral resolution of the operator which represents the physical quantity. This statistical rule, originally proposed by Born in 1926, was for von Neumann the starting point of a mathematical analysis of quantum mechanics in entirely probabilistic terms. The analysis, carried out in a paper of 1927 [‘Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik’], introduced the concept of statistical matrix for the description of an ensemble of systems which are not necessarily all in the same quantum state. The statistical matrix has become one of the major tools of quantum statistics and it is through this contribution that von Neumann's name became familiar to even the least mathematically minded physicists. In the same paper von Neumann also investigates a problem which is still now the subject of much discussion, viz., the theoretical description of the quantum-mechanical measuring process and of the noncausal elements which it involves. Mathematically speaking von Neumann's study of this delicate question is quite elegant. It provides a clear-cut formal framework for the numerous investigations which were needed to clarify physically the all-important implications of quantum phenomena for the nature of physical measurements, the most essential of which is Niels Bohr's concept of complementarity. The results of the [first] paper were immediately used by the author to lay the foundation for quantum thermodynamics [‘Thermodynamik Quantenmechanischer Gesamtheiten’]. He gave the quantum analogue of the well-known classical formula for the entropy” (van Hove, pp. 97-98). “That von Neumann has been ‘par excellence’ the mathematician of quantum mechanics is as obvious to every physicist now as it was a quarter of a century ago. Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analyzed by one single man in two years time (1927-1929). Conversely, one could almost say in reciprocity, quantum mechanics introduced von Neumann into a field of mathematical investigation, operator theory, in which he achieved some of his most prominent successes” (*ibid*., p. 95). Not on OCLC. Only the Samuel Koslov copies (1996) in auction records.

*Provenance: *front wrappers with the signature of mathematician Aurel Friedrich Wintner (1903-1958) who did important work in probability theory and is considered one of the founders of probabilistic number theory.

“By 1926, [David] Hilbert’s colleagues [at Göttingen] were contending with a proliferation of developments in physics. Heisenberg's new theory of quantum mechanics had been shown by Max Born to be explicable in terms of matrix methods. Schrödinger, at Zurich, had constructed a wave mechanics which, although it led to the same results as Heisenberg, proceeded from an entirely different basis … As was clear to even lay observers, … these conceptual changes in physics resonated deeply in mathematics. In physics, the theory of relativity cast doubt upon many concepts that were central to classical mechanics, such as absolute space and time, and simultaneity, and quantum theory threw mechanistic determinism into question, by demonstrating the impossibility of knowing simultaneously both the position and velocity of a particle, necessary to predicting its future evolution … The first phase of quantum physics had indicated the ‘discontinuous character of all micro-events’ with energy states of complex structures such as atoms and molecules being seen to consist of a set of discrete values, and to ‘jump’ from one state to another. The second phase, centered on Born’s interpretation of the wave equation, suggested that the basic laws of physics were probabilistic laws, allowing for only statistical predictions. Classical determinism, which had been central to Western scientific culture for more than a century, had been shattered.

“Working initially with Hilbert’s assistant, [Lothar] Nordheim, von Neumann entered mathematical physics in this, the second phase, beginning with the axiomatisation of Heisenberg's work, and elaborating a mathematical basis for quantum mechanics in Hilbert space” (Leonard, pp. 52-53), contained in his paper ‘Mathematische Grundlagen der Quantenmechanik’ (pp. 1-57 of the same journal volume as the offered papers).

“In another paper the same year [‘Thermodynamik Quantenmechanischer Gesamtheiten’], he pushed this probabilistic interpretation in physics further, introducing an idea that remained important in the literature — that of a statistical matrix describing an ensemble of systems of different quantum states. He also broached the question of how to construct a mathematical formalism that would provide an adequate theoretical description of the observation process in quantum mechanics, in which the relationship between the observer and the subatomic phenomenon being observed took on special importance. When a measuring instrument was coupled to a very small particle, the very act of measuring affected the object, the characteristics of which were being measured, thereby introducing a certain ambiguity or uncertainty into the process. One could not be sure that the atom coupled with the instrument behaved similarly to the atom on its own. Here, the ambiguity took the particular form of the uncertainty principle, formulated by Heisenberg, which said that, as a matter of principle, not just of empirical fact, one could not know both the position and the momentum of the atom. The surer one was about one, the less sure one could be about the other. At the time, the leading interpretation of quantum theory was that represented by he complementarity principle of Niels Bohr and the Copenhagen School. Bohr attributed this feature of complementarity to other pairs of variables in quantum physics: the energy of an atom and the time at which it has that energy; the wavelike and particle-like properties of the electron or of light. He even expected that the principle of complementarity was some kind of universal, extending to other realms of nature, such as the biological and psychological. As Steve Heims points out, the theory also assumed that the measurements read on the measuring instrument were ‘objective public facts’, independent of the particular person doing the reading. Lastly, it assumed that the observer had no objective view of the isolated atom itself but saw only the combination of atom plus instrument.

“In the paper in question, von Neumann gave a formulation in terms of an algebra of operators in Hilbert space, and a philosophical interpretation that was different from Bohr's. Von Neumann's formalism divided the world into the system under observation, the measuring instrument, and the observer. The last remained outside the theory, and von Neumann pointed out that how one chose to describe the observer — where one draws the line between consciousness, eyes, nervous system, brain, and equipment external to the body — was arbitrary. In his system, the observer could reduce a superposition — that is to say, change the state of the system merely by observing it — but the question of inter-subjective agreement — that is, whether two observing subjects would reach similar observational conclusions, was left unresolved. We had only the single knowing subject and the world that he alone observes and affects. Unlike previous work, von Neumann's formalism was fully axiomatic and logically rigorous, given the particular assumption that measurement was made infinitely fast, rather than in finite time. This strong assumption von Neumann acknowledges to be incorrect, but he argues that it is essential to preserving the formalism and should not matter in the end. The philosophical details, therefore, are not spelled out. The emphasis is on the mathematical achievement, with the implicit idea that perfecting the formalism is more conducive to progress than is attending to the epistemological details. As Heims correctly puts it, the ‘algebra interested von Neumann, but not the metaphysics’.”

Leonard, *Von Neumann, Morgenstern, and the Creation of Game Theory*, 2010. Van Hove, ‘Von Neumann’s contributions to quantum theory,’ *Bulletin of the American Mathematical Society *64 (1958), 95-99.

Two vols., 8vo (241 x 168 mm), pp. [ii: separate title], [245], 246-272, [2: blank]; [273], 274-291, [1: blank]. Original light green printed wrappers (some chipping and a few small tears to the extremities), Very rare.

Item #5654

**
Price:
$12,500.00
**