Abstract
The indistinguishability of bosons and fermions has been an essential part of our ideas of quantum mechanics since the 1920s. But what is the mathematical basis for this indistinguishability? An answer was provided in the group representation theory that developed alongside quantum theory and quickly became a major part of its mathematical structure. In the 1930s such a complex and seemingly abstract theory came to be rejected by physicists as the standard functional analysis picture presented by John von Neumann (in his book Mathematical Foundations of Quantum Mechanics) took hold. The purpose of the present account is to show how indistinguishability is explained within representation theory.
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Notes
In Leibniz’s own words this is the principle that ‘It is not true that two substances may be exactly alike and differ only numerically, solo numero [only in number].’ Discourse on Metaphysics, IX, from 1678.
This was then argued for bosons in Muller and Seevinck (2009). Such ideas were initially worked out by Saunders in a series of papers (see his 2003b) and presented for philosophers in Saunders (2006) and, more recently, (2016). Saunders believes bosons, in contrast to fermions, are not objects at all. Muller has come to adopt a structuralist view of both bosons and fermions (Muller 2011, 2015). They are united in their belief in non-reductive relations.
A common feature of many of these simplifications came from dealing with only two identical particles, with the result that only two permutations in the symmetric group \(\mathfrak {S}_{2}\) needed to be considered, one of these being the identity, the other being an exchange. But \(\mathfrak {S}_{2}\) has properties not shared by symmetric groups of higher order, because it is a commutative group with a trivial representation theory in which all irreducible representations are one-dimensional. This is picked up in Sects. 2 and 3.
The copying of this text in Muller and Saunders is not exact. Though the discrepancies are not serious, they have been corrected here.
The book that Muller and Saunders seize upon was in fact written in 1926 and scarcely revised for its 1949 english translation, as Weyl noted in essay 9, p. 220, of Weyl (2009). It doesn’t properly represent his view in 1930, let alone 1949.
It is symptomatic of the quality of the English translation that the word ‘Eigenwert’, i.e. eigenvalue, is used four times in the German original and not at all in the English. However the more accurate modern term would be ‘weight’, which had little currency in 1930, being coined by Cartan and made popular later by Bourbaki (see section 4). In the disputed paragraph Robertson repeats the word ‘permuted’ which is definitely a mistake, and does not occur in the German.
Adding the half-integer spin cases by tensoring an even-dimensional module with that coming from \(\,GL_{n}\) will give one the complete pure states required for electrons: two electrons being allowed for any given value of the energy. Then the Pauli exclusion principle says that a measurement cannot result in two particles in the same complete eigenstate—because, again, the eigenvalues do not have any multiplicity in the space.
Thus he speaks, also in Weyl (1949), of the ‘Leibniz–Pauli exclusion principle’: eigenstates must be distinct: distinguishability of eigenvalues implies their non-identity.
The antisymmetric space is also often called, and will often be hereafter, the exterior power, \(\bigwedge ^{r} (V^{n})\), and the symmetric space is often called the symmetric power, \(\text{ Sym }^{r} (V^{n})\).
Nor does the field need to be \(\,\mathbb {C}\,\)—the theorem will continue to hold in an algebraically closed field of any characteristic.
This was generalised to countable dimension vector spaces in Dixmier (1981).
Burnside’s work was first presented in Burnside (1905) and Frobenius and Schur’s extension in their (Frobenius and Schur 1906). See Burnside (1951) appendix I for a summary; also foundational is Schur (1905). However Burnside’s theorem was known to Weyl, von Neumann, Wigner and other German mathematicians largely through Frobenius and Schur’s paper, so I call it, as they sometimes did, the Burnside–Schur–Frobenius theorem. The ring-theoretic version of the theorem may have been due originally to Emmy Noether or Emil Artin. See Artin’s later (1950) for discussion. It was known to von Neumann in his (1929), and is the basis for his work in “rings of operators”; and, of course, to Weyl in his (1929) (1931) and (1939)—but note Weyl avoids using the word ‘ring’ and throughout prefers the word ‘algebra’. See footnote 9 to ch. III of Weyl (1931). The theorem was then generalised by Jacobson and Chevalley into what was called the Density theorem, a fundamental result in ring theory. For modern proofs of the theorem see Lam (1998) or the more recent Radjabalipour et al. (2004).
The importance of Burnside’s theorem is hard to overstate. Von Neumann gave a copy of the Frobenius and Schur paper to Eugene Wigner, of which the latter said that it “had an enormous effect on me.” (From an interview that Wigner gave on his years at the IAS, at https://web.archive.org/web/20121005061854/https://www.princeton.edu/~mudd/finding_aids/mathoral/pmc44.htm.) Van der Waerden, in his book on group theory, reproduced the proof. Dirac uses it in his Dirac (1958). Weyl reproduced the proof in his (Weyl 1930) and (1939) and relied on it throughout the work. This, combined with Schur’s lemma and the double commutant theorem—also given by Schur—were the main results underpinning the group theoretic view.
This material is best given in a modern form in Goodman and Wallach (2009) ch. 9.
For r = 3, there are 3 distinct irreducible representations (there are four if one counts multiplicities.) This corrects French and Rickles (2003) who give the number to be 3! = 6 (p. 218). The mistake is fundamental, for there is a confusion there between permutations (which number 6 for r = 3) and the irreducible representations—and most importantly what the irreducible representations are acting on. This invalidates many of their claims concerning representations.
A signed sum, or signum, means that the sign of the permutation (\(+1\) for a permutation which consists of an even number of inversions of elements, \(-1\) for a permutation which is an odd number of inversions of elements) is the coefficient of the permutation.
Our convention will be to take multiplication on the right for \(\,\mathbb {C}[\mathfrak {S}_{r}]\,\).
Schur’s first formulation of what is now called Schur–Weyl duality appeared in his 1901 doctoral dissertation under Frobenius. Under the impetus of Weyl’s (and other’s) work in quantum mechanics, Schur revisited this topic in 1927, giving a more concise version relying on the double commutant theorem. Weyl was aware of this 1927 paper in writing Weyl (Weyl 1930), and cites it in ch. V note 1. But he does not use it, preferring the 1901 formulation. Weyl (1939) used Schur’s (1927) paper more extensively and there gives the double commutant theorem. He compares the two formulations pp. 96 ff.
Saunders in (2003b) 301, says that ‘one is not thereby led to local symmetry groups (symmetries which, viewed as Lie groups, are infinite dimensional).’ It is not clear what he might mean, as almost all Lie groups which are local symmetry groups are finite dimensional.
If V is irreducible then \(A = \,\mathbb {C}\, \), and so the dim \(V \otimes _{\mathbb {C}[G]} S\) is either 0 or 1. If V is not irreducible then, since it is simple, reduce it and and go back to the first case.
Fulton and Harris (1991) lemma 6.23 and following exercises. The Zariski topology is coarser than the euclidean metric topology.
Here \(\mathbb {C}^{*}\) is \(\mathbb {C} - \{0\}\). The components \(\,\varepsilon _{}\,\) of the eigenvalue sum are determined by N. Weyl interprets them to be energy-values.
For the general partition mentioned before \([3, 2^{2}, 1^{3}]\) it would be (1, 1, 1).
The English translation of this passage and the entire section is misprinted and at variance with the german original, so I have corrected it here. (See Weyl 1931 368 for comparison.) In the two particle case the symmetric space has the eigenvalue \(2 E_{1}\) (\(f_{1}' = f\)) which does not occur in the (lower) antisymmetric space. But \(\textit{E}_{1} + \textit{E}_{2}\) occurs in both (because \(f_{1} = f_{2} = 1\)).
This analogy originated in Schrödinger (1950) and was given currency in Mary Hesse’s (1963, pp. 49–50). It was given a further twist by Dieks and Versteegh (2008), who adapted it to differentiate between fermions and bosons. Dieks (2010) states that this analogy is closer to the QM situation than Black’s two spheres. I agree.
I thank an anonymous referee for pointing out this presupposition in the Muller and Saunders paper.
In the same way there seems to have been a convergence of both sides to a “structuralist interpretation” of QM. For some animadversions on structuralism see Heathcote (2014).
This only seems plausible, I believe, because of the exclusive focus on the case where there is only one permutation other than the identity: the two particle case. Further: It is pertinent to ask whether, in all this talk of permuting particles, anyone has thought to wonder how these particles are being moved around by such a dynamical transformation. But surely to ask the question should make it clear that it could not be taken literally.
The minimal dimension for the existence of such indecomposable tensors is four.
It is worth noting in this respect that Einstein et al. (1935) had stated more circumspectly than was acknowledged by their critics that the system in such a complete state is not further analysable: ‘We cannot, however, calculate the state in which either one of the two systems is left after the interaction.’ (p. 779).
Von Neumann also had interests that encompassed philosophy, but it is fair to say that these were more on the side of logic and set theory.
In ch. IV, §9, he says: ‘If two such individuals [that are fully equivalent] unite to form a single physical system \(\textit{I}^{2}\)...’ the vectors will be tensors of order two. Weyl (Weyl 1931 p. 239). This is the basis for the electrons being indistinguishable later in the same section.
Infinite permutation groups have been studied but not everything will carry over, particularly such things as the trace. For finite groups every representation is equivalent to a unitary representation. But this is not so for an infinite group. This raises the question as to whether our current picture will carry over to an infinite set of particles in a field theory. Whether this pessimism is justified it may be too early to tell. For example see the work of Vershik and Okounkov (2005) and Tsilevich et al. (2019). (For early background see Cameron 1990).
It should be noted in this context that ‘Hilbert space’ is rarely ever mentioned in Weyl (1931). It is an idiosyncrasy of the work.
To the best of my knowledge it has never been explained why this project was left incomplete, but in a letter to Pascual Jordan, dated Dec. 1949 (Rédei 2005) von Neumann indicated that his work Von Neumann (1960) most closely represented his finished thoughts on quantum logic. Von Neumann believed that the ring \(\mathrm {II}_{1}\) was the correct ring in which to do QM and it was also his foremost example of a continuous geometry.
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My thanks to the two anonymous referees; their suggestions have led to numerous improvements.
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Adrian Heathcote has retired from Department of Philosophy, University of Sydney, Sydney, Australia.
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Heathcote, A. Multiplicity and indiscernibility. Synthese 198, 8779–8808 (2021). https://doi.org/10.1007/s11229-020-02600-8
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DOI: https://doi.org/10.1007/s11229-020-02600-8