## Die naturphilosophischen Grundlagen der Quantenmechanik.

Berlin: Verlag Öffentliches Leben, 1935.

First edition, very rare offprint issue, of Hermann’s doctoral thesis. “In 1935 Hermann published an argument demonstrating an apparent flaw in John von Neumann’s 1932 proof which was widely claimed to show that a hidden variable theory of quantum mechanics was impossible. Hermann’s result went unnoticed by the physics community until it was independently discovered and published by John Stewart Bell in 1966, and her earlier discovery was pointed out by Max Jammer in 1974. Some have posited that had her critique not remained nearly unknown for decades, the historical development of quantum mechanics may have been greatly affected; in particular, it has been speculated that a wider awareness of her work would have put in question the unequivocal acceptance of the Copenhagen interpretation of quantum mechanics, by providing a credible basis for the further development of nonlocal hidden variable theories … This work has been referred to as “one of the earliest and best philosophical treatments of the new quantum mechanics.” In this work, she concludes: “The theory of quantum mechanics forces us … to drop the assumption of the absolute character of knowledge about nature, and to deal with the principle of causality independently of this assumption. Quantum mechanics has therefore not contradicted the law of causality at all, but has clarified it and has removed from it other principles which are not necessarily connected to it”” (Wikipedia, accessed 17 October 2017). Carl Friedrich von Weizsäcker referred to her research the “first positive and undeniable contribution to the elucidation of the epistemological implications of quantum mechanics.” Werner Heisenberg devoted an entire chapter of his book *Physics and Beyond*: *Encounters and Conversations *(1971) to a reconstruction of discussions that he had on quantum mechanics and Kantian philosophy with Hermann and von Weizsäcker, and he indicated that both contributed important insights to this topic. In 1936 Hermann was awarded the Richard Avenarius Prize by the Academy of Sciences of Saxony in Leipzig for her work on the significance of quantum theory and field theory of modern physics for the theory of knowledge. COPAC: Cambridge only.

*Provenance*: Manfred Moritz (1909-90), German-Swedish philosopher (signature, ‘Lund 4.4.42’, on front free endpaper). According to the *Routledge Encyclopaedia of Philosophy*, Moritz was “Sweden’s foremost expert on Kant’s ethics after Hägerström”.

“John von Neumann was primarily engaged in the mathematical aspects of quantum mechanics when, during the late 1920s, he was privatdozent at the University of Berlin. But he also took great interest in the discussions on the philosophical implications of the new theory which were held frequently at the end of the famous Physics Colloquia of that University. The main problem that attracted von Neumann’s attention was the precise nature of the statistical character of quantum mechanics, the question why, in spite of the unambiguous definition of the state of a system by the state function, only statistical statements could be made about the value of the physical quantities (observables) involved.

“The fact that ensembles described by the same state function exhibit dispersion suggested to him two a priori conceivable interpretations: either (1) the individual systems, though described by the same [wave function], differ in additional hidden parameters whose values determine the precise outcome of the measurements, or (2) all individual systems of the ensemble are in the same state "but the laws of nature are not causal: dispersion results from nature's disregard of the principle of sufficient reason” (Jammer, pp. 265-6). Possibility (1) became known as the hypothesis of ‘hidden variables’.

In his 1932 work, *Mathematische Grundlagen der Quantenmechanik*, von Neumann gave a mathematical ‘proof’ that hidden variable theories were not consistent with quantum mechanics. This seems to have been accepted until Bell in his 1966 paper ‘On the problem of hidden-variables in quantum mechanics’ (*Reviews of Modern Physics* 38, pp. 447-452) showed, by analyzing the de Broglie-Bohm ‘pilot wave’ theories, that von Neumann’s proof must be incorrect. In 1974, Jammer pointed out (*Philosophy of Quantum Mechanics*, p. 272) that Bell had been anticipated by Hermann in her doctoral thesis, published almost four decades earlier. Hermann’s criticism is part of a wider discussion of the idea of causality in quantum mechanics.

“Having started her academic career with the study of mathematics under Emmy Noether in Göttingen, Grete Hermann became greatly influenced by the philosopher Leonard Nelson, the founder of the Neo-Frisian school, and in the spring of 1934 joined Heisenberg's seminar in Leipzig. Leipzig in the early 1930s was not only, next to Göttingen and Copenhagen, one of the foremost centers for the study of quantum mechanics and its applications (due to the presence of Felix Bloch, Lev Landau, Rudolf Peierls, Friedrich Hund, and Edward Teller), it also became famous for its study of the philosophical foundations and epistemological implications of the quantum theory, particularly after Carl Friedrich von Weizsäcker, at the age of only 18 years, joined the Heisenberg group …

“Although not a specialist in physics by schooling, she was able, assisted by B. L. van der Waerden and C. F. von Weizsäcker, to participate most actively in the work of the seminar. As a result of these discussions Grete Hermann published in March 1935 a long essay on the philosophical foundations of quantum mechanics [offered here], which still deserves attention today. Hermann's point of departure – which led her to the relational conception of quantum mechanical description – was the empirical fact of the unpredictability of precise results in the measurements on microphysical objects. The usual way out of such a situation by searching for a refinement of the state description in terms of additional parameters is denied by the theory.

“Since Hermann rejected von Neumann's proof of the impossibility of hidden variables, … she raised the question of what justifies this denial. To reject the possibility of such a refinement of the state description merely on the basis of its present unavailability would violate the principle of the incompleteness of experience. The sufficient reason for renouncing as futile any search for the causes of an observed result, she declared, can be only this: *One already knows the causes. *The dilemma which quantum mechanics faces is therefore this: Either the theory provides the causes which determine uniquely the outcome of a measurement – but then why should not the physicist be able to predict the outcome? – or the theory does not provide such causes But then how could the possibility of discovering them in the future be categorically denied? Hermann saw the solution of this dilemma in the relational or, as she expressed it, the ‘relative’ character of the quantum mechanical description, which she regarded as the ‘decisive achievement of this remarkable theory.’

“By renouncing the classical principle of objectivity and replacing it by that of the instrument-dependency in conjunction with the idea that from the factual result of a measurement the physical process, leading to the result, can be causally reconstructed, Hermann explained why the theory prevents predictability without excluding a* post factum *identification of the causes of the particular outcome. How this can be achieved in detail has been described by Hermann for the case of the Weizsäcker-Heisenberg experiment, which was a forerunner of the Einstein-Podolsky-Rosen thought-experiment, which, in its turn, led Bohr to the relational conception of quantum mechanical states. Grete Hermann's resolution of the dilemma was, as she stated, approved by Heisenberg with the words: ‘That's it what we have tried for so long to make clear!’” (Jammer, pp. 207-9).

“The earliest criticism leveled against von Neumann's proof is containedin Grete Hermann's essay [offered here]. Hermann charged von Neumann with having committed a *petitio principii *by introducing into the formal presuppositions of his proof a statement which is logically equivalent to the assertion to be proved. The point in question concerns the additivity of the expectation value for arbitrary ensembles … Grete Hermann now pointed out that since an arbitrary ensemble is a mixture of pure cases it presumably suffices to claim the additivity only for ensembles all elements of which are described by the same (pure state) wave function. For such ensembles, however, von Neumann, according to Hermann, resorted to the mathematical formalism (*φ*, (*R + S*)*φ*) = (*φ*, *Rφ*) + (*φ*,* Sφ*) as a valid relation irrespective of whether [quantum mechanical observables] *R* and *S* commute. Hermann now objected that, as long as the possibility of hidden variables has not yet been disproved, (*φ*, *Rφ*) denotes the expectation value of *R* for such ensembles *E *alone whose elements are described by [the wave function] *φ*. This does not imply that also sub-ensembles *E*_{1}*, *of *E, *defined by perhaps not yet available criteria (hidden variables), have the same expectation value of *R*, nor that the latter satisfies the additivity condition. Thus an important step in von Neumann's proof is lacking. But to retain this assumption, as von Neumann did, is tantamount to assuming that the elements of an ensemble, described by *φ*, cannot be further differentiated by any criteria on which the result of an *R* measurement may depend. Since the denial of the existence of such criteria is precisely the thesis that has to be proved, Hermann concluded that von Neumann's proof is circular” (*ibid*., pp. 272-3).

Four decades later, Bell also pointed to the additivity postulate as the flaw in von Nuemann’s argument. He first constructed a consistent hidden variable theory for particles of spin ½ in which the additivity postulate fails. “Having thus shown that von Neumann's argumentation depended decisively on the validity of the additivity postulate, Bell clarified the situation by pointing out that it was not the ‘objective verified predictions of quantum mechanics,’ as von Neumann asserted, but rather von Neumann's arbitrary additivity assumption, postulated to be valid also for dispersion- free states, that precluded the possibility of hidden variables. Its validity for quantum states, Bell realized, is a peculiarity of quantum mechanics and not at all a priori obvious; for in the case of incompatible observables *R*, and *S* the apparatus to measure *R + S* is in general entirely different from the devices required to measure the individual observables and a priori no statistical relations between the corresponding results can be expected. It is valid for quantum states because, in the words of Frederik J. Belinfante, “it so happens that the other axioms and postulates of quantum theory conspire to make [the expectation value of R] expressible at ∫*φ*Rφdx. *If this expectation value additivity is waived for dispersion-free states … the proof would break down” (*ibid*., p. 305).

A brief summary of this work was published in *Die Naturwissenschaften*, 23. Band, 42. Heft, pp. 718-721 (18 October 1935).

Offprint from Abhandlungen der Fries’schen Schule, Neue Folge, 6. Band, 2. Heft. 8vo (237 x 172 mm), pp. 84. Original printed wrappers. A fine copy.

Item #5044

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