Interpreting the Wigner–Eckart Theorem

Abstract

The Wigner–Eckart theorem is central to the application of symmetry principles throughout atomic, molecular, and nuclear physics. Nevertheless, the theorem has a puzzling feature: it is dispensable for solving problems within these domains, since elementary methods suffice. To account for the significance of the theorem, I first contrast it with an elementary approach to calculating matrix elements. Next, I consider three broad strategies for interpreting the theorem: conventionalism, fundamentalism, and conceptualism. I argue that the conventionalist framework is unnecessarily pragmatic, while the fundamentalist framework requires more ontological commitments than necessary. Conceptualism avoids both defects, accounting for the theorem’s significance in terms of how it epistemically restructures the calculation of matrix elements. Specifically, the Wigner–Eckart theorem modularizes and unifies matrix element problems, thereby changing what we need to know to solve them.

Introduction

Shortly after the founding papers on quantum mechanics in 1924 and 1925, Eugene Wigner (1927) and Hermann Weyl (1927) published the first articles detailing the application of group theory to quantum mechanics. By 1931, these ideas had matured into a comprehensive treatment of atomic spectra (Wigner, 1931). Central to this application is what came to be known as the Wigner–Eckart theorem. Developed in a review article by Carl Eckart (1930, p. 358), the seed of the theorem first appeared in a landmark paper by Wigner (1927, p. 645), the first paper to apply group representation theory to the rotational symmetry of atoms (Biedenharn & van Dam, 1965, p. 6). Although originally developed only for the special rotation group $SO\left(3\right)$ and the special unitary group $SU\left(2\right)$, numerous generalizations have extensively extended its scope. Today, it applies to almost every group that has ever found application in physics and chemistry, inspiring a host of applications outside its original domain.¹ In the words of the physicist J. F. Cornwell, “the Wigner–Eckart Theorem provides both the most succinct and the most powerful expression in the whole field of application of group theory in physical problems. Indeed, most physical applications depend directly on it” (1984, p. 108). Similarly, in a wide-ranging review of applied group theory, the chemical physicist B. G. Wybourne argued that many chemists and physicists had misunderstood the significance of applied group theory by failing to appreciate the Wigner–Eckart theorem:

The myth that group theory is limited to qualitative results in its application to physical problems persists with the failure of most expositors of group theory to give adequate attention to the celebrated Wigner–Eckart theorem that completes the group theoretical picture and turns group theory into a practical subject. (Wybourne, 1973, p. 39)

Despite its centrality, the theorem has a puzzling feature: it is in fact dispensable, since elementary methods provide the same information codified by the theorem. This points to a larger puzzle surrounding many symmetry arguments in science. While providing fantastic simplifications, symmetry arguments are often not necessary. When it comes to solving problems, more elementary approaches often suffice. This lack of necessity invites comparisons with other methodological conveniences employed by science. What separates a generic symmetry argument from a faster computer or a fortuitous notation, such as the Einstein summation convention? Or, if there is little difference, why are symmetry arguments routinely touted as deep and intellectually significant?

Given its dispensability, there are prima facie only two possible sources of the theorem’s significance. Perhaps the theorem is significant due to its ontological status, such as carving nature more closely at the joints or playing a privileged nomological or explanatory role. At the other extreme, perhaps the theorem has only pragmatic benefits, determined by the goals and values of scientific agents. This conventionalist or pragmatist response would pinpoint the significance of the theorem to the eye of the beholder. Yet, there is in fact a third possible source of the theorem’s significance, located within its epistemic roles. These epistemic roles give rise to the theorem’s methodological advantages, while making the theorem more than just convenient. Furthermore, the epistemic significance of the theorem does not depend on any particular ontological interpretation. I will thus argue that one can remain neutral—or agnostic—regarding these rival metaphysical interpretations, while nonetheless characterizing how the theorem makes a non-pragmatic and objective difference for understanding. I will refer to such differences in understanding as aspects of intellectual significance (as distinct from the theorem’s methodological or ontic significance).

Section 2 begins by describing a generic class of problems that the Wigner–Eckart theorem solves. I contrast the Wigner–Eckart approach with an elementary approach that eschews the theorem. Next, Section 3 develops a pragmatic interpretation of the theorem, based on its convenience for solving matrix element problems. Section 4 introduces my preferred philosophical framework for interpreting the theorem: conceptualism. I characterize the source of the theorem’s intellectual significance, locating it within what I call epistemic dependence relations. Section 4.3 shows how these epistemic dependence relations give rise to the theorem’s methodological advantages. These pragmatic benefits are thereby separable from the non-pragmatic epistemic benefits of the theorem. Finally, Section 5 considers competing ontological interpretations of the theorem in terms of different conceptions of laws of nature, joint-carving, and explanation. I show that the intellectual significance of the theorem does not depend on privileging any particular ontological interpretation. Indeed, it is not clear that the theorem or its use adjudicates between these competing ontologies. By focusing on its intellectual rather than ontic dimensions, conceptualism provides an empiricist-friendly interpretation of the Wigner–Eckart theorem.