Abstract
This chapter considers the new relationships between physics and mathematics that emerge with Heisenberg’s discovery of matrix mechanics and its development in the work of Born, Jordan, and Heisenberg himself, and in Dirac’s version of the formalism. Taking as its point of departure Einstein’s view of “the Heisenberg method” as “a purely algebraic method of description of nature,” Section 4.1 examines the shift from geometry to algebra in quantum mechanics as a reversal of the philosophy that governed classical mechanics by grounding it mathematically in the geometrical description of the behavior of physical objects in space and time. Heisenberg’s matrix mechanics abandons any attempts to develop this type of description and instead offers essentially algebraic machinery for predicting the outcomes of experiments observed in measuring instruments. By the same token, a new nonrepresentational type of relationship between mathematics and physics is established, compelling Bohr to speak, in the wake of Heisenberg’s discovery, of “a new era of mutual stimulation of mechanics and mathematics.” Section 4.2 addresses these relationships and their implications.
… a geometrical interpretation of such quantum-theoretical phase relations in analogy with those of classical theory seems at present scarcely possible…
—Werner Heisenberg, “On Quantum-TheoreticalRe-Interpretation of Kinematic and MechanicalRelations” (Heisenberg 1925, SQM, p. 265)
… a new era of mutual stimulation of mechanics and mathematics has commenced.
—Niels Bohr, “Atomic Theory and Mechanics”(Bohr 1925b, PWNB 1, p. 51)
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Notes
- 1.
Einstein made an intriguing contribution to this part of the old quantum theory in one of his lesser known papers, published in 1917. It contains penetrating insights into the problems of the Bohr–Sommerfeld quantization scheme, in part anticipating recent developments in the so-called “quantum chaos” (Einstein 1917). The paper was cited by de Broglie and mentioned by Schrödinger as the closest work of the old quantum theory to his wave mechanics. For a discussion of Einstein’s paper, see Stone (2005).
- 2.
Both letters are cited by Mehra and Rechenberg (MR 2, p. 266), who offer a useful discussion of Pauli’s views of Heisenberg’s new mechanics (MR 2, pp. 262–273).
- 3.
It would be difficult to fault Pauli in this regard, let alone to question his achievements as a major figure of twentieth-century physics. It is also worth noting that, while never shy to criticize Bohr, he offered strong support to Bohr’s epistemological ideas and helped to propagate them. Nevertheless, some “misses” on Pauli’s part, such as those just mentioned, are intriguing. Earlier Pauli missed the potential discovery of spin, virtually contained in his manipulation of quantum numbers that led him to his exclusion principle. He also initially missed the significance of the EPR argument and related arguments of Einstein, which he just about dismissed, even though later on he astutely criticized Born’s rejoinder to Einstein as missing Einstein’s real point (Born 2005, pp. 216–220). In the early 1950 s, Pauli did not pursue or publish his findings concerning what became known as the Yang–Mills theory, a non-abelian gauge theory that forms the foundation of the present-day quantum field theory. One might argue that this miss is, again, due to a refusal to see that the theory was a way to the solution of some of the key problems of quantum field theory, in spite of the difficulties the approach contained, which worried Pauli. It is true that the effectiveness of the Yang–Mills theory became apparent only later; and it is also true that in this case Pauli’s creativity, including in his use of mathematics, was of the highest order. His other major contributions to quantum theory, such the exclusion principle or his neutrino proposal, for which he was awarded a Nobel Prize, his arguably greatest contributions were not based on the kind of mathematically innovative thinking that defined the work of Heisenberg and Dirac. As is well known, especially from his criticism of Born, Pauli in general distrusted mathematical thinking in physics.
- 4.
I, again, leave aside the question of nature itself. For example, to what degree do the space–times of relativity correspond to the ultimate structure of nature? How realist is such a description, even as an idealization, as opposed to serving as a mathematical tool for correct predictions (in this case, exact rather than probabilistic)? These questions can also be posed in the case of special relativity, or even classical mechanics, where, however, the descriptive idealizations used are more in accord with our phenomenal experience. For some of these complexities, see (Butterfield and Isham 2001) and other articles in (Callender and Huggett 2001).
- 5.
Leibniz’s algebraic way of thinking concerning calculus or the nature of thought itself is worth noting here, as part of the history of algebraic thinking that eventually led to the “algebraic method of description of nature.” Leibniz’s work on the systems of linear equations was one of the origins of the idea of the matrix, and his project of universal characteristic anticipates the project of formalization of mathematics, undertaken from the end of the nineteenth century on. This project was given an expected twist in and, in its highest aspiration (of the possibility of formalizing mathematical thinking), was brought to its end by Gödel’s incompleteness theorems. Leibniz’s view was Platonist insofar as the formalization by a given form of calculus (differential, propositional, or other) was seen as a way of at least approximating the ultimate reality. However, Leibniz’s reversal of the hierarchy of geometry and algebra, which governed the preceding thinking in physics and mathematics, or, from Plato on, in philosophy, was a powerful move, including vis-à-vis Descartes’s and Newton’s geometrical philosophy. Descartes’s analytic geometry, for example, still dealt with the ultimately geometrical reality, in part correlatively with the idea of mechanical motion. Descartes had a concept of a “mechanical” curve. Algebra, say, in the Hamiltonian formalism in classical mechanics, allows one to bypass engaging with geometrical aspects of physical reality, even under the assumption of its geometrical nature. Algebra liberates theoretical physics from dealing with these aspects at least in working with equations and doing calculations.
- 6.
See, however, Mehra and Rechenberg’s discussion of the subject (MR 6, pp. 20–36).
- 7.
Einstein’s broader thoughts on the subject are another matter, however. Einstein pondered the question of geometry in the context of special and general relativity, including in relation to the constitution of our measuring instruments (rods and clocks), throughout his life (e.g., Einstein 1921, 1949a), and his thinking on the subject and its implications would require a treatment beyond my scope here. Cf., (Brown and Pooley 2001) and also Heisenberg’s comments (Heisenberg 1989, pp. 82–83).
- 8.
On these connections see Pierre Cartier’s article (Cartier 2001). Both Connes and Cartier hold strongly Platonist views. Cartier’s Platonism also interestingly manifests itself in Cartier’s discussion of physics from Kepler to Bohr in his “Kepler et la musique du monde” in the special issue, Nombres, of La Recherche on numbers (Cartier 1995).
- 9.
The actual architecture of nature may, again, be still something else, thus making the relationships considered here tri-partite, which, however, further amplifies my point here.
- 10.
It is peculiar and in a way remarkable that in commenting on Bohr’s passage in their introduction to Volume 3 (“The Formulation of Matrix Mechanics and Its Modifications, 1925–1926”) of their treatise and even in building this introduction around it, Mehra and Rechenberg miss the epistemological meaning and implications of Bohr’s statement. They see the passage as indicating primarily or even exclusively that “the unsolved problem[s?] of atomic theory would still require the use of appropriate mathematical tools” (MR 3, p. 4). This point is of course trivially correct, but it is hardly germane to Bohr here. They say nothing about Bohr’s statement that “in atomic problems we have apparently met with such a limitation of our usual means of visualization,” again, manifestly crucial to this statement and to all of Bohr’s thinking. This instance is not unsymptomatic of Mehra and Rechenberg’s reading of Bohr in their treatise, which often misses the deeper and more radical aspects of Bohr’s thought.
- 11.
The letter is cited and discussed, from a rather different perspective than the one offered here, by Mehra and Rechenberg (MR 3, pp. 118–121).
- 12.
For a discussion of Galileo in this context, see (Plotnitsky and Reed 2001).
- 13.
Cf., Julian B. Barbour’s concept of “Platonia,” an underlying reality without change and motion (Barbour 1999), the idea apparently originating with Parmenides, who inspired Plato. Barbour’s conception appears to originate in the idea that it does not appear possible by means of quantum theory to describe the motion of the ultimate constituents of nature. From a nonclassical viewpoint, however, while this is true, it does not follow that everything “stands still” at that level, since, as just explained, the latter concept would not apply any more than that of “motion” (or “object” and “quantum”) to quantum objects.
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Plotnitsky, A. (2010). From Geometry to Algebra in Physics, with Heisenberg. In: Epistemology and Probability. Fundamental Theories of Physics, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85334-5_4
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