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# A forgotten female pioneer of quantum physics - The mathematician and philosopher Grete Hermann

Those who are more concerned with the early development of quantum theory will quickly realize that the pioneers of modern physics appear to be a pure men’s club: Max Planck, Albert Einstein, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Wolfgang Pauli, Max Born, Paul Dirac, Enrico Fermi, later Richard Feynman or Freeman Dyson, to name only […]

Those who are more concerned with the early development of quantum theory will quickly realize that the pioneers of modern physics appear to be a pure men’s club: Max Planck, Albert Einstein, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Wolfgang Pauli, Max Born, Paul Dirac, Enrico Fermi, later Richard Feynman or Freeman Dyson, to name only the most well-known of them. All modern physics is occupied by men. All modern physics? No, a single woman managed to break through the men’s crew: Grete Hermann. “Grete who?”, most readers are likely to ask. In fact, Grete Hermann is completely unknown even to most physicists today. But it pays to do a little research on the work of this impressive woman, as it opens up fascinating perspectives on the thought of the pioneers of quantum theory. Furthermore, with her philosophical and political commitment, Grete Hermann was a great example of the impact of a deep thinker on society as a whole.

The development of quantum theory was surely one of the most significant intellectual challenges of the 20th century. It put millennia-old philosophical paradigms and assumptions into question. In 1932, the heat of the debate about its interpretation seemed to have come down somewhat. The great mathematician John von Neumann had presented a mathematical proof that quantum physics cannot contain so-called “hidden variables”. What sounds like an esoteric sideshow question for mathematicians deals in fact with one of the fundamental problems in the interpretation of quantum theory until today: How does the quantum world relate to reality? Is there any objective physical reality in the microcosm in which events take place deterministically, as we know them to do in our world, and in which the local character of physics (there is no instantaneous remote action, i.e. only adjacent events in space-time can have an impact on each other) is maintained? These were exactly the points that had been questioned by quantum theory. And this to the chagrin of Albert Einstein in particular, whose discussions with his colleague Niels Bohr on the interpretation of the new quantum physics are among the most important philosophical discussions of the 20th century, and in which Einstein had to admit defeat at first. But in 1935 he resumed the intellectual fight and published jointly with two of his colleagues the so-called “EPR paradox”, with which he challenged once again with a very tricky thought experiment the prevailing interpretation of quantum theory, according to which there should be no objective reality in the microcosm (the ” Copenhagen interpretation “, represented mainly by Bohr and Heisenberg). This in turn brought Erwin Schrödinger, the other skeptic of the Copenhagen interpretation, in the same year to formulate the most famous thought experiment of modern physics: the “Schrödinger cat”. Both essays led deeply into the philosophical heart of the quantum world. John von Neumann’s proof was intended to clarify this discussion mathematically in the sense of Bohr and Heisenberg. And indeed, the Copenhagen interpretation, despite the strong resistance of Einstein and Schrödinger, became the most common philosophical interpretation of quantum theory, in many circles to this day.

But von Neumann’s proof came with a problem: it was wrong. More precisely, it turned out that the mathematician had made some very specific assumption and his proof was thus not universally valid. But in the 1930s to mathematically contradict the great John von Neumann came equal to heresy. Thus, the physicists accepted the proof as valid only too docilely. It was not until 1966 that John Bell pointed out that while Neumann’s argument was mathematically correct, it was based on unrealistic physical assumptions. The most important physicists of the twentieth century, from Bohr to Heisenberg, Schrödinger to Einstein, had been blindly following an insufficient mathematical argument for more than thirty years!

Only one person had looked more closely, and this as early as 1935, just around the second round of the Bohr-Einstein debate on the interpretation of quantum reality: Grete Hermann. Her point was mostly about philosophy: In her 1935 publication *The Fundamentals of the Natural Philosophy of Quantum Mechanics* she first and foremost explored the question of whether the causality principle can be maintained even for atomic processes. That is exactly what quantum physics called into question. Hermann followed the argumentation of Immanuel Kant and his critical philosophy and its further development in the tradition of the early 19th century philosopher Jakob Friedrich Fries. Her reply to the statement of the quantum theorists that in the micro world processes can only be described with probabilities and therefore a strict causality no longer applies in this realm, was that quantum theory cannot rule out that there could be yet unknown variables controlling the processes, whereby the principle of causality would apply again. This was precisely where Von Neumann’s proof of impossibility, which he had formulated in 1931 in his groundbreaking book *Mathematical Foundations of Quantum Mechanics*, came into play: His proof was intended to show with mathematical rigor and exactness that quantum theory by its very mathematical structure excludes such “hidden variables”. This in turn provided the quantum physicists with the argumentative bulwark that protected their Copenhagen interpretation from the claim that determinism can be restored by yet unknown variables. And it was this very bulwark that was torn down by Grete Hermann showing that the premises on which Neumann’s proof rested are too specific to apply to the physical realm in general.

However, during the next 30 years very few scientists questioned the proof in its generality and applicability. One who nonetheless adhered to hidden variables (without having unmasked Neumann’s proof as incomplete) was Albert Einstein. Nevertheless, and despite Hermann’s insights, it took until 1966 before the physics community finally acknowledged that Neumann’s proof did *not *imply the impossibility of any hidden variables in the quantum world. And anyone who reads John Bell’s 1966 essay quickly realizes that his argument is not significantly different from Hermann’s. Thus Grete Hermann writes: “Von Neumann demands that for the expected value function E (R), which is defined for an ensemble of physical systems and gives a number for every physical quantity, the following holds: E (R + S) = E (R) + E(T). In words: The expected value of a sum of physical quantities is equal to the sum of the expectation values of both quantities. With this assumption, Neumann’s proof holds or it fails. “John Bell, for his part, writes: “His [i.e. von Neumann’s] assumption is: Any linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also”. In mathematical terms these statements are almost identical. In his paper, John Bell succeeded in providing a measurable criterion under which circumstances hidden variables can occur in a quantum theory, the well-known “Bell’s inequality”. Thus, only in the last decades has the struggle over the interpretation of quantum theory been decided, and this *experimentally *(first in 1982 by an experiment of Alain Aspect). In the end von Neumann’s statement that hidden variables do not exist is correct: the Bell inequality does not hold in the experimental results.

The fact that the physicists did not notice Hermann’s insights is regrettable in that the discussion about hidden variables was only seriously taken up again in the 1960s and 1970s. And it is the resulting accurate understanding of the all underlying quantum property of *entanglement*, which today leads to very exciting possibilities of completely new quantum technologies (see L. Jaeger, The Second Quantum Revolution, Springer (2018))! If Grete Hermann’s insights had only been taken up more broadly, who knows how much faster this development would have been.

But why is it that Hermann’s work was ignored, while more than thirty years later Bell’s work spurred a revival in the study of the fundamental properties of quantum mechanical systems? Here we find an example that the progress of knowledge in the sciences is anything but always linear and optimal, but comes with a lot of strings attached. Hermann herself published her work with a rather insignificant publisher (*Verlag Öffentliches Leben*) and only a much shortened version in a scientific journal (*Naturwissenschaften*, Volume 23, Issue 42, pp. 718-721 (1935)). The latter did not contain the refutation of von Neumann’s proof. And in later publications she did not elaborate on her criticism of the proof any further. For in her argument she was mainly concerned with the philosophical discussion, which in her view can receive “valuable stimulation and fertilization” by the physical knowledge, but in its fundamental questions such as that of causality may not refer to empirical sciences and their theories. In addition, Hermann’s work was never translated from German into other languages, which was a major obstacle for later scientists. For the almost all German-speaking pioneers of quantum theory, however, this was not the case. Therefore, it remains unclear: Why did Bohr, Heisenberg and other fail to take notice of Grete Hermann’s work? Heisenberg himself devoted even an entire chapter (the 10th) of his famous autobiography *Physics and Beyond* to his philosophical discussion with Grete Hermann, without giving any reference to von Neumann’s proof and its assumptions, however. Well, for the representatives of the orthodox Copenhagen interpretation of quantum mechanics von Neumann’s proof fit wonderfully into their thinking. Even the greatest physicists occasionally lack skepticism about their own views.

Grete Hermann also drew from her philosophical studies and intellectual proximity to Kantian philosophy very practical, respectively political consequences: she actively opposed National Socialism in Germany. In her philosophical seminars she offered reasoning for why to resist the Nazi regime. In addition, she became editor of a magazine against Hitler. Thus in 1936 she had to flee Germany, first to Denmark, then via Paris to London, where she continued to work in resistance to the Nazi regime. After the war she returned to Germany and began a political career in the SPD. She worked together with Willi Eichler on the 1959 Godesberg program of her party. The “de-ideologization” of the SPD towards the foundation of social democratic politics through values and ethical considerations carried out therein goes back not least to the mediation of the philosophical thoughts of Grete Hermann. Unfortunately, this multifaceted woman was no longer given an opportunity to resume her mathematical and physical studies. Thus, regrettably her impressive work remains largely unknown today.

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