Abstract
We explore the different meanings of “quantum uncertainty” contained in Heisenberg’s seminal paper from 1927, and also some of the precise definitions that were developed later. We recount the controversy about “Anschaulichkeit”, visualizability of the theory, which Heisenberg claims to resolve. Moreover, we consider Heisenberg’s programme of operational analysis of concepts, in which he sees himself as following Einstein. Heisenberg’s work is marked by the tensions between semiclassical arguments and the emerging modern quantum theory, between intuition and rigour, and between shaky arguments and overarching claims. Nevertheless, the main message can be taken into the new quantum theory, and can be brought into the form of general theorems. They come in two kinds, not distinguished by Heisenberg. These are, on one hand, constraints on preparations, like the usual textbook uncertainty relation, and, on the other, constraints on joint measurability, including trade-offs between accuracy and disturbance.
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Notes
Citations to Heisenberg’s works are prefaced by an asterisk.
In a letter to Pauli [31, p. 105], Heisenberg laments the existence of two separate quantum communities in Göttingen, with very different views of the relationship of Mathematics and Physics. He especially dislikes the group around Hilbert and Weyl embracing matrices as bringing new progress in physics, and he even considers finding a more physicsy term for matrices. The two groups did make rather different contributions. Whereas Heisenberg was satisfied to develop an intuition “for all simple cases” that occurred to him, the mathematicians von Neumann and Weyl searched for and readily found the generalizable structures and interpretations that we still use today. On the other hand, the uncertainty paper is an undeniable success of the heuristic physical, anti-mathematical approach.
However, some of the physicists around Heisenberg did feel the need to bring adequate mathematical tools into the new theory. In 1925/26 Born had already brushed up his operator theory on a visit to Norbert Wiener. He was also the Academy member to submit von Neumann’s paper [56]. Just a short while later Pauli and Jordan were both collaborating with von Neumann.
Heisenberg in his later years expresses his appreciation of the mathematical side by claiming that it was his own work in the first place. Here is how, in 1956 *[8], he summarizes the contribution of his uncertainty paper: the link from an experimental situation to its mathematical representation was to be achieved by the “hypothesis that only such states may appear in nature, or can be realized experimentally, that can be represented by vectors in Hilbert space.” (Italics in the original). Now this and much more could be said about von Neumann’s article [56], in which he actually coins the term “Hilbert space”. Heisenberg’s paper *[2] naturally does not contain the word. Neither does it contain the thing itself. In *[8] he also claims to have corresponded extensively with Pauli about “this kind of solution”, but again, the surviving letters (e.g., [31, p. 115]) have nothing.
In the citations of this section all occurrences of visualizable and intuitive are actually “anschaulich” in the original.
We found two translations of *[2] (see references). Both make a mess of this distinction. Wheeler and Zurek choose “physical content”. The anonymous translation on the NASA website has “actual content”. Another option, found in the translation of a paper by Schrödinger, is “perspicuity”. See also [32].
In contrast to today, linear algebra was not part of the standard curriculum. When he created matrix mechanics in 1925, Heisenberg knew nothing about the mathematics of matrices, and even Born found it worth mentioning from whom he himself had learned this exotic subject.
The published statement, which Heisenberg presumably meant here, came from Schrödinger’s paper [47] in which he shows the equivalence of matrix mechanics and his theory. It is from a footnote in which he recalls why he had initially ignored Heisenberg’s work and why, therefore, his own theory owed nothing to Heisenberg. It is a pity that this is the only part of the paper that Heisenberg mentions. A more mature reaction would have been to accept Schrödinger’s result that the two approaches are two sides of the same coin, with priority granted to matrix mechanics, and then work on the remaining differences. The fight against the “disgusting continuum theorists” as a central issue of *[2] is worked out clearly in [11].
“We create internal virtual images or symbols of the external objects, and we make them in such a way that the logically necessary consequences of the images would invariably be the images of the natural consequences of the depicted objects.” [27]. The role of the Hertzian “images” in Heisenberg’s early works is traced in [29].
“The philosophy is written in that great book, which constantly lies open before our eyes (I speak of the universe) [...]. It is written in the language of mathematics, and the letters are triangles, circles and other geometrical figures” [26].
In fact, the reinterpretation move hardly convinced Heisenberg himself. The “victory footnote” *[2, p. 196] cited above continues by granting Schrödinger an important role in the “mathematical (and in this sense intuitive)” development of the theory [italics by Heisenberg]. This supports our reading that the notion has been extended to include mathematical intuition, here even any mathematical work, but at the same time it seems visualization of a lesser kind. If we apply that to the paper *[2] itself it amounts to affirming the criticism it sets out to rebut: ‘Sorry, we do not have visualizable content, but lots of math’. Therefore the visualizability grapes have to be sour. Indeed, Heisenberg continues by charging wave mechanics with “leading away from the straight path outlined” by Einstein, de Broglie, Bohr, and ‘quantum mechanics’(= himself, with friends) by the poison of “popular visualizability”.
Nevertheless, the microscope yields further interesting aspects upon closer inspection. For example, one can discuss what happens if one places the photographic plate or detector in the focal plane instead of the image plane, so that one detects the direction of the photon, not its point of origin. Moreover, one can make this choice after the Compton scattering has occurred [30].
An example is the online exhibition about Heisenberg by the American Institute of Physics [20]. Following the link “Derivation of the uncertainty relation”, one finds the cited page.
In the “two-men-paper” [13], this is the functional derivative of the formal trace of the Hamiltonian, which requires H to be written as a non-commutative polynomial in a special symmetrized form. In the “three-man-paper” with Heisenberg [12], this is replaced by the directional derivative along scalar shifts. That makes the transition to commutators and to the modern representation considerably simpler. Both forms of partial derivative are forgotten today and only served the explicitly stated purpose of allowing the equations of motion to look like Hamilton’s equations.
This “simplest conceivable assumption” is actually a very poor, even wrong description of the transition. The distinction between diagonal and non-diagonal matrix elements depends on the basis (and none is specified), and just one diagonal element [= Diagonalglied] as a replacement is even crazier. Just a little further down the whole matrix is taken as the replacing object, which makes more sense. It is an interesting project to develop a reading of the quote that makes sense without excessive interpretational bias. In the paper the most helpful passage for this might be *[2, p. 181/182]. Like the quote, it seems to refer to a notion of assigning values to a matrix which is not wholly captured by an expectation value.
In *[7] Heisenberg dates his conversation with Einstein, from which the quote presumably comes, to the spring of 1926. He mentions that, for him, recalling that conversation was essentially the stimulus for the uncertainty paper. In the narrative of *[7], Einstein criticizes the methodology in *[1] although Heisenberg feels that he has taken it from Einstein himself. It is hard to say whether Einstein would have found *[2] any more in agreement with his philosophy.
The observable-centered approach of the older work *[1] of 1925 is sufficient for Heisenberg here to proclaim the conclusiveness of the theory! That any method for establishing a new theory guarantees its truth, must be doubted. Moreover, this confidence is expressed towards a formalism that is still in its infancy. To what extent the matrices of the formalism [12] are “experimentally determined numbers” remains unclear. In [13] and [12], the concept-critical approach plays a subordinate role. Hence, a purpose of this quoted sentence is to mention this aspect and with it Heisenberg’s own contribution once again, and so to defend primacy and priority for the whole of quantum mechanics as well as its ultimate formalism (see also Footnote 2).
If one reads them in isolation, one may exclaim at these sentences: “Has this fellow not read Heisenberg?”. Wasn’t it the whole purpose of §1 to cast doubt upon the word “exact”? But we hope it has become clear that “exact” in this sentence has nothing to do with the opposite of uncertainty or imprecision. Rather “exactly defined” is to be read here as “operationally defined”.
How the uncertainties, which were introduced with the meagre precision of Heisenberg’s tildes, could become the “reason” for precise quantitative probability relations is another unfulfilled burden of proof. It only appears in the abstract. The paper itself has no details about it.
Heisenberg uses a variant of this line also in his Nobel lecture *[4]. The error of taking the uncertainty relation as an additional element, which is independent enough of quantum mechanics so that one could even think of taking or leaving it, has also been committed by some people without such personal motives. For example, following his semi-classical discussion of the double-slit experiment, Richard Feynman writes: “The uncertainty principle ‘protects’ quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, the quantum mechanics would collapse. So he proposed that it must be impossible. Then people sat down and tried to figure out ways of doing it, and nobody could [...]. Quantum mechanics maintains its perilous but still correct existence.” [25, Section 1–8] That completely warps the role of a theorem in the theory and is also historical nonsense, just like Feynman’s fictitious reasons for why Einstein did not accept quantum mechanics [25, 18-8].
Don Howard has an entirely different interpretation here [33]. He would possibly let this be true for Heisenberg but is of the opinion that Bohr had, from the beginning, a deep understanding of entanglement, particularly the entanglement between a system and measuring apparatus caused by the measurement interaction.
It is also unclear for me (R.F.W.) how the American Kennard came to write in a German style so reminiscent of Heisenberg’s. Maybe Heisenberg’s contribution to this paper was bigger than Kennard’s acknowledgement reveals. In a letter to Pauli [31, p. 164], dated May 31 in Copenhagen, Heisenberg writes: “I am still very unhappy about the work of an American, which he began with me, and which is in that touchy subject, which I would like to stay away from right now”. This could be a reference to Kennard, but the normally very thorough editors of the letters give no hint about his identity.
A relatively early acknowledgement of this problem came from Karl Popper [41, 42]. In these papers we see him fighting the ambiguity in *[2]. He did not achieve a complete clarification, and later even had to retract a proposed experiment. But the distinction between the “statistical scatter relations” (Popper’s own translation of “Streuungsrelationen”, i.e., variance relations) of preparation uncertainty from questions of measurement precision is quite clear: the measurement of a distribution presupposes the sharp measurability of the outcomes in each single case. He comes very close to the concepts of preparation and measurement uncertainty with his distinction of non-prognostic measurements, i.e., those in which one does not care about the particle afterwards, and prognostic measurements, which serve to prepare a new initial state. In his immediate response, von Weizsäcker gives an argument why the measurement uncertainty relations (which Popper is in some sense asking for) must be the same as the preparation uncertainty relations: he sees one kind directed towards the future and the other towards the past, and invokes the time inversion symmetry of quantum mechanics. If it were really that simple, we could have saved ourselves a lot of mathematical work.
This will usually be a generalized observable, for which the probability of each outcome is not given by a projection, but by a general positive operator (POVM, positive operator valued measure). In a sequential measurement, this is anyhow what comes out, so we cannot avoid this kind of observable. However, it is a point that von Neumann missed. And here we have to say that great men cast long shadows. Even though his role in the early days of quantum mechanics is systematically ignored in the textbook literature on quantum theory, and most physicists have not even heard of him, von Neumann was in the long run hugely influential. His 1932 book was widely recognized as the mathematical basis, and his choice of projections as the yes/no observables, the related identification of observables with hermitian operators, and finally the projection postulate are almost universally accepted. However, these assumptions are unnecessarily restrictive. They are often false in experiments, and almost always when one analyzes an indirect measurement. Various authors came to this realization, beginning with Holevo in estimation theory and Ludwig [38] for reasons of axiomatic parsimony. Now the quantum information community operates entirely in the wider framework. It would be interesting to study von Neumann’s reasons, but that is beyond the scope of this article.
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Acknowledgements
R.F.W. is grateful for the feedback from several audiences, beginning with the Heisenberg Society. Several errors in an earlier arXiv version were spotted by Blake Stacey.
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Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SFB DQ-mat/A06).
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Dedicated to the memory of Paul Busch.
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This article is based on a talk in German, given by one of us (R.F.W.) at the 2016 annual meeting of the Heisenberg Society in Munich [57]. It contained some expressions of personal opinion or experience, which we left in, marked by (R.F.W.).
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Werner, R.F., Farrelly, T. Uncertainty from Heisenberg to Today. Found Phys 49, 460–491 (2019). https://doi.org/10.1007/s10701-019-00265-z
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DOI: https://doi.org/10.1007/s10701-019-00265-z