In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen's elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]).
In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen's elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]).
Among the very few books owned by Majorana (fewer than 30 volumes) we find Weyl's Gruppentheorie und Quantenmechanik in its first German edition (1928) [16]. Indeed, as we have already recalled, the clear group-theoretical approach pioneered by Weyl greatly influenced the scientific thought and work of Majorana. Group theory was revealed to be the appropriate mathematical framework for treating symmetries in quantum mechanics, as established in the books by Weyl [16], Wigner [263], and van der Waerden [264], but, despite its recognized relevance, the group-theoretical description of quantum mechanics in terms of symmetries was ignored by almost all theoretical physicists of the time (see Section 7.1), and only in recent times has it been included in physics textbooks. As a matter of fact, although almost every physicist had a copy of Weyl's book in the 1950s, the extensive use of group theory in physics research started, during those years, only in nuclear and particle physics. Remarkably, such indifference did not apply at all to Majorana: the brilliant works on group theory and its applications of that mathematician, as well as those of Wigner, left an unambiguous and fruitful impression on the work of our author. Although this impression is recognizable on any technical page of the present book, in this chapter we will discuss in some detail several studies performed by Majorana on this subject, ranging from exquisitely mathematical results to genuinely physical applications, as was customary for him. Most of these results are not original, as Majorana reproduced what had been obtained by others (mainly by Weyl), but it is nevertheless interesting to see how Majorana obtained those results, and how he used group theory in order to obtain new physical results. As explained in Ref. [265], Majorana can be considered a faithful follower (probably, the only follower) of Weyl's thinking who went well beyond the path taken by Weyl himself, the main difference being in the approach adopted: mathematical for Weyl, physical for Majorana.
Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.
In this paper, I argue that symmetry principles in physics (in particular, in quantum mechanics) have a methodological character, rather than an ontological or an epistemological one. First, I provide a framework to address three related issues regarding the notion of symmetry: (i) how the notion can be characterized; (ii) one way of discussing the nature of symmetry principles, and (iii) a tentative account of some types of symmetry in physics. To illustrate how the framework functions, I then consider the case of the early formulation of quantum mechanics, examining the different roles played by symmetry in this context. Finally, I raise difficulties for ontological and purely epistemological interpretations of symmetry principles, and offer a methodological alternative.
Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.
The kinematics of quantum systems, as is well known, are generally governed by the Heisenberg commutation rules. It was Hermann Weyl who suggested the consideration of the global versions of these rules which define a projective unitary representation of the additive group of the classical phase space. This prompted Weyl to formulate his principle that quantum kinematics is always governed by a projective unitary representation of some abelian group. The ideas of Weyl, and their natural development by his successors, notably Stone, Von Neumann, Mackey, Segal, Shale, Weil, and many others have had and still continue to have a great impact on many areas of mathematics and physics. This paper is an exposition of some of the themes that flow from these ideas.