Abstract
This chapter is the first English translation (by E. Crull and G. Bacciagaluppi) of Grete Hermann’s fundamental essay of 1935 on the foundations of quantum mechanics. This essay, written after a protracted visit to Heisenberg’s research group in Lepizig, is possibly the most remarkable and complete early philosophical analysis of quantum mechanics. After criticising known arguments for the completeness of quantum mechanics, including von Neumann’s proof, Hermann argues that the notions of causality and predictability have to be separated, and that only the latter but not the former is lost in quantum mechanics. Indeed, she argues using the Heisenberg microscope as her prime example, that causal chains can be reconstructed after the fact for any quantum mechanical effect, so that quantum mechanics is in fact already causally complete. She goes on to analyse Bohr’s views on complementarity, and to sketch the natural-philosophical picture provided by quantum mechanics along the lines of a neo-Kantian (neo-Friesian) transcendental idealism .
Translated by Elise Crull and Guido Bacciagaluppi. Originally published as Hermann (1935a); this paper was reprinted separately for Verlag Öffentliches Leben (Inh. Erich Irmer, Berlin 1935), with the addition of a Preface by Hermann (included here), and a table of contents (not included) (Hermann 1935b). Excerpts of Sects. 1, 2, 3, 8, 9, 10 and 12 (with minor variants) were also published in Hermann (1935b) under the same title. The original version of the paper will also be included in the forthcoming volume by Herrmann (2017).
Translators’ note: we have translated the following Kantian terms in accordance with Eisler (1930): Anschauung as intuition; Einheit as oneness; Erkennen as perception; Erkenntnis as knowledge; mannigfaltig as manifold; Spaltung as splitting; unmittelbar as direct; Wahrnehmung as sensation. The format of the references has been mostly modernised, and a list of References has been created.
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Notes
- 1.
In the following presentation, I essentially follow the treatment by Heisenberg (1930).
- 2.
If the correctly normalised wave function of an electron is \(\phi (x, y, z)\), then \(\vert \phi (x, y, z) \vert ^2 \varDelta x \cdot \varDelta y \cdot \varDelta z\) is the probability of finding the electron in the interval \(\varDelta x \cdot \varDelta y \cdot \varDelta z\).
- 3.
[Heisenberg 1930, p. 11].
- 4.
This section, as with later passages in small print, contains discussion of physically difficult arguments. These passages are not necessary for the comprehension of the subsequent considerations.
- 5.
Dirac (1930).
- 6.
Neumann: ‘Mathematische Grundlagen der Quantenmechanik’. Berlin, 1930 [sic] [von Neumann (1932)].
- 7.
Von Weizsäcker (1931).
- 8.
- 9.
Compare [Heisenberg 1930, p. 15].
- 10.
See Heisenberg (1930). One has attempted, however, to use such calculations of trajectories to derive predictions in a similarly indirect manner as happens in the interpretation and checking of measurements, and thereby to overcome the limits of the uncertainty relations. One such attempt is found sketched in Popper: ‘Zur Kritik der Ungenauigkeitsrelationen’, Die Naturwissenschaften, volume 22, issue 48 (Berlin 1934). The same issue contains a reply by Weizsäcker that uncovers the physical error in Popper’s thought experiment [Popper and von Weizsäcker 1934]. The real reason for this error, apparent only from the more detailed discussions in Popper’s book Logik der Forschung (Vienna 1935 [sic]) [Popper 1934], lies in a misjudgment of the duality experiments and their consequences. Popper is misled by the probability interpretation of the wave functions to apply these quantum mechanical state descriptions, and the uncertainty relations given with them, only strictly speaking to ensembles of physical systems, and for a single appropriately chosen system to assume instead no restriction through the uncertainty relations. In this he misunderstands that because of the duality experiments the applicability of the classical conceptions is limited according to the uncertainty relations already for every single elementary process, and that accordingly wave functions can in fact be used for describing the state of individual systems. That this use of wave functions is consistent with their probabilistic interpretation is based once again solely on the relative character of the quantum mechanical way of description: on the one hand, the wave function is completely determined by the values of those physical quantities that have a sharp value within the momentary observational context for the system. In this respect it characterises the system quantum mechanically relative to the observational context present. On the other hand, the probability interpretation of the wave functions yields those variables [Bestimmungen] that remain of the classical-intuitive description according to the correspondence principle and that fix which statements can be made for the passage from one observational context into another.
- 11.
- 12.
[Heisenberg 1930, p. 44].
- 13.
The German here reads: ‘daß für gewisse dem Bild notwendig zukommende physikalische Größen, wie etwa dem [sic] Ort oder dem [sic] Impuls einer Partikel, dem Betrachter die Kenntnis ihres genauen Werts mangelt’ (eds.).
- 14.
J. Fr. Fries: “Neue oder anthropologische Kritik der Vernunft”, second edition, page XXIV ff. Heidelberg 1828. New edition Verlag “Öffentliches Leben”. Berlin 1935 (Fries 1828/1968, pp. XXIV ff.).
- 15.
For the following considerations compare [Nelson 1917, pp. 324 ff.].
- 16.
Apelt (1904). The similarity of Apelt’s considerations to quantum mechanical arguments emerges clearly from the images with which the distinctness of possible realms of knowledge are described. As Apelt writes: ‘Human knowledge does not resemble a level surface that one can completely survey with a single glance from any high vantage point; rather it is more like a hilly country, a complete image of which one must assemble only little by little from partial views. There are multiple heights, multiple vantage points one upon the other, each of which presents a different view and where something now shows, now hides itself’. And de Broglie represents the complementarity of position and momentum measurements thus: ‘There are, so to speak, two planes that we cannot see sharply at the same time. We might make a comparison: let there be a figure whose various parts are drawn on two close parallel planes \(\Pi \) and \(\Pi ^{\prime }\). If we observe the figure through a none-too-precise optical instrument, we can focus it on a plane between \(\Pi \) and \(\Pi ^{\prime }\) and obtain an image that still reasonably resembles the figure. We then have the impression that the figure is drawn in one plane. But if we use a very good instrument, then it cannot sharply depict \(\Pi \) and \(\Pi ^{\prime }\) at the same time. The more we focus it on \(\Pi \), the worse we see the parts drawn on \(\Pi ^{\prime }\) and conversely; we are thus forced to recognise that the figure does not lie in one plane. Classical mechanics corresponds to the imprecise instrument; with it we have the impression that we can determine simultaneously position and velocity of the particle exactly. But with the new mechanics, which corresponds to the precise instrument, we come to realise that the spatio-temporal localisation and the energetic description are two different planes of reality that one cannot simultaneously see precisely’ (de Broglie 1929, p. 7 ff.).
- 17.
Heisenberg (1933).
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Hermann, G. (2016). Natural-Philosophical Foundations of Quantum Mechanics. In: Crull, E., Bacciagaluppi, G. (eds) Grete Hermann - Between Physics and Philosophy. Studies in History and Philosophy of Science, vol 42. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0970-3_15
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