BACK TO THE HISTORY: PARASTATISTICS

BACK TO THE HISTORY: PARASTATISTICS

As we have indicated, as early as 1926 Dirac realized that the symmetric and anti-symmetric functions were merely the two simplest out of the set of possible eigenfunctions for the assembly. Thus he later wrote,

It appears that all particles occurring in nature are either fermions or bosons, and those only anti-symmetrical or symmetrical states for an assembly of similar particles are met with in practice. Other more complicated kinds of symmetry are possible mathematically but do not apply to any known particles. ²²⁶

That other mathematical possibilities exist was also, not surprisingly, noted by Weyl, who insisted that the situation in which these possibilities are not physically realized was 'repugnant' to Nature and wrote that,

She has accordingly avoided this distressing situation by annihilating all these possible worlds except one-or better, she has never allowed them to come into existence! The

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one which she has spared is that one 18218d317s which is represented by anti-symmetric tensors, and this is the content of Pauli's exclusion principle. ²²⁷

The investigation into these alternative symmetry types was pursued along various lines. Perhaps the earliest can be traced back to Gentile, who, in 1941, used Bose's combinatorial methods to obtain an expression for the average number of particles in a group of states which was dependent upon a parameter d giving the maximum number of particles which could occupy any given state. ²²⁸ Fermi-Dirac (F-D) and Bose-Einstein (B-E) statistics were special cases of these 'intermediate' statistics, obtained when d = 1 or ∞, respectively. ²²⁹

Gentile's work represents a straightforward kind of generalization of standard quantum statistics. Two alternative lines of development focused on the implications of Dirac's claim above in the context of quantum mechanics (QM) on the one hand and quantum field theory (QFT) on the other. Before we pursue the historical details, we need to give a preliminary answer to the question, how can there be 'more complicated kinds of symmetry' than the usual symmetrical and anti-symmetrical forms? Within the formalism of QM, the physical state of a system is represented by a vector, or basis function, spanning a subspace of Hilbert space. If we have a system of N-particles, then the relevant subspace will have N! dimensions and will be spanned by the vectors obtained by permuting the particles in the system (we shall return to consider the nature of these permutations shortly). This subspace can then be decomposed into irreducible subspaces each carrying an irreducible representation of the group of permutations. The symmetric and anti-symmetric vectors, corresponding to Bose-Einstein and Fermi-Dirac statistics respectively, span one-dimensional subspaces, related to one-dimensional representations of the permutation group. However, representations of higher dimension are also possible and correspond to higher dimensional subspaces of the relevant Hilbert space spanned by multi-dimensional 'rays'. It is these which represent more general forms of statistics, known as parastatistics.

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Although this possibility was clearly perceived by Dirac and Weyl, the theoretical exploration of these parastatistics can be traced back to a little-known work published by Okayama in Japan in 1952. ²³⁰ Okayama anticipated later discussions in his demonstration that the standard statistics only follow if the relevant subspace is assumed to be one-dimensional and that if this assumption is dropped, and the possibility allowed of a single state corresponding to some larger collection of vectors spanning a multi-dimensional subspace, then other forms of statistics can be admitted. And in particular, he emphasized that these alternative forms are related to various representations of the permutation group and could hence be investigated using the apparatus of group theory. ²³¹ Okayama also attempted to develop a second quantized approach to paraparticle theory, but his generalized commutation relations allowed only the trivial solution that the Hilbert space consists only of null vectors. ²³²

A much more successful and more well-known attempt to construct a form of parafield theory was made by Green in 1953, following Wigner's demonstration that the standard commutation relations were not uniquely determined by the equations of motion. ²³³ Green's aim in this work was to see if certain fundamental problems with quantum field theory could be resolved by relaxing the theory's formal structure. ²³⁴ By regarding any quantization scheme to be deemed satisfactory if it simply ensured the equations of motion, Green was able to introduce whole families of alternative commutation schemes obeyed by paraparticles. Each such scheme can be labelled by an integer p and following Green's work, if one quantizes with the p'th scheme, one talks of a parafermion or paraboson field of order p. When p = 1, one recovers the standard bosonic and fermionic schemes, of course. From the perspective of the themes we are interested in here, it is interesting that Green argued that although his generalized theory implied that a new state would result from the interchange of two particles, the particles would always divide up into groups such that permutations within a group would not lead to a new state and furthermore, that interactions can be devised which prevent the formation of new states by permutations between the groups. This allowed him to conclude that the 'Principle of Indistinguishability of Identical Particles' would be retained in parafield theory. ²³⁵

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Despite all this theoretical work, there did not appear to be any evidence for the existence of paraparticles in nature. This issue was addressed by Greenberg and Messiah in their famous construction of a consistent first-quantized para-particle theory. ²³⁶ By this time-the early 1960s-Heisenberg's and Dirac's crucial insight that the wave function for an assembly of particles should be either symmetric or anti-symmetric had become enshrined in the so-called 'Symmetrization Postulate' (SP). Greenberg and Messiah pointed out that this is ". an extremely strong condition, very much stronger than what is implied by the indistinguishability of identical particles". ²³⁷ In its place they proposed a weaker 'Indistinguishability Postulate' (IP). This states that if the particles of the assembly are permuted, then there is no way of observably distinguishing the wave function which results from the original unpermuted wave function. ²³⁸ We shall consider IP in more detail, both formal and philosophical, in the next chapter but for the moment we simply wish to note Greenberg and Messiah's observation that SP is sufficient but not necessary for IP. The former can be regarded as a restriction on the states for all observables-namely that they must be suitably symmetrized-whereas the latter can be understood as a restriction on the observables for all states-namely that they must commute with the operator representing a particle permutation. Greenberg and Messiah went on to insist that the usual ways of introducing SP into the quantum mechanical formalism were suspiciously ad hoc in nature and that many experimental results which had been taken to be tests of SP were actually just tests of the weaker IP. ²³⁹

Their overall approach to this issue was to take the usual assumption in quantum mechanics that the wave functions are represented by one-dimensional state vectors in Hilbert space and generalize it, to allow states to be represented by vectors spanning a multi-dimensional subspace of the Hilbert space. They called the set of vectors spanning such a multi-dimensional subspace a 'generalized ray' and showed that the indeterminacy associated with such a move (since there is now more than one vector associated with a given state) actually causes no difficulty in the interpretation of the theory because measurable results do not depend on precisely which state vector spanning the multi-dimensional subspace is chosen to represent the state. ²⁴⁰ Furthermore-and this will be important for our discussion in Chapter 4-Greenberg and Messiah demonstrated that the Indistinguishability Postulate (together with a formal result known as Schur's Lemma) imposes what is known as a 'superselection rule' on Hilbert space which effectively decomposes it into a number of subspaces such that once a particle is in a particular subspace-corresponding to bosonic, fermionic, or some form of paraparticle statistics-the particle must remain in that subspace.

Greenberg and Messiah also made it clear that those arguments which purport to demonstrate that state vectors must be either symmetrical or anti-symmetrical involve an implicit assumption to the effect that states which cannot be distinguished by any observation must be represented by the same vector to within a phase factor. ²⁴¹ The introduction of 'generalized rays' shows how such an assumption can be abandoned while retaining the Indistinguishability Postulate. ²⁴² Hartle and Taylor subsequently showed how Greenberg and Messiah's generalized rays could be eliminated and the usual connection between states and rays restored by moving to a subspace of lower dimension and using the one-dimensional ray belonging to such a subspace to label the mixed symmetry states. ²⁴³

In the second half of their paper, Greenberg and Messiah gave a very detailed discussion of the direct experimental tests of the Symmetrization Postulate and concluded that in many cases what such experiments actually tested was the Indistinguishability Postulate (IP) instead. ²⁴⁴ After surveying the experimental evidence for SP for various kinds of particles they concluded that although their statistical characters were accepted and well established, there were still some for which it was not entirely clear what form of statistics they should obey.

This opened up the possibility that the behaviour of some particles might be described by parastatistics. Hartle and Taylor may have expressed the views of many physicists when they wrote, ". although there is no theoretical reason to exclude para-particles, their properties are sufficiently disagreeable for one to hope sincerely that there will continue to be no evidence in their favor". ²⁴⁵ Nevertheless, in a famous and historically important paper

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Greenberg suggested that the newly introduced quarks might be regarded as paraparticles of a certain type. ²⁴⁶ However, it was subsequently shown that this was equivalent to the introduction of a three-fold degree of freedom for quarks that came to be called colour and which led to the development of quantum chromodynamics. ²⁴⁷ Following the demonstration that the colour model could be made gauge invariant, whereas the paraquark model could not, interest in paraparticle theory declined, at least among mainstream physicists. ²⁴⁸

There still remains the question of the relationship between Greenberg and Messiah's theory of paraparticles and Green's parafield theory. The answer is somewhat delicate since, as Greenberg had emphasized, a permutation of the particle labels cannot be represented within Quantum Field Theory where, on the standard formulation, such labels are not introduced to begin with. This meant that IP is ill-defined within Quantum Field Theory. However, this problem can be overcome through the use of 'state' or 'place' permutations which act on the indices labelling the different states and which are well-defined in parafield theory. This distinction goes back to Dirac, ²⁴⁹ who noted that, unlike the particle permutation operators, the place permutation operators depend upon the basis with respect to which they are defined. Although the difference between these two kinds of operation does not emerge for two particles distributed over two states (one might think of permuting two balls between boxes or permuting the boxes around the balls, for example) and is generally not significant for ordinary statistics, it is apparent once we consider three particles and the possibility of parastatistics; in particular, it turns out to be crucial for establishing the relationship between paraparticle theory and parafield theory.

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Thus it was demonstrated that IP is equivalent to the requirement that all observables that distinguish among states differing only in the order of the variables are functions of the place permutation operators. ²⁵⁰ On this basis the equivalence between parafield and paraparticles of 'finite order' could be established. ²⁵¹

With the Indistinguishability Postulate at the heart of paraparticle theory, the shift back to a field theoretic perspective allowed for the possibility of further theoretical generalizations. Kamefuchi and Ohnuki, in particular, embarked upon a general group theoretical investigation of 'indistinguishability', or loss of identity in this context. ²⁵² They subsequently went on to consider the necessary and sufficient conditions for particles to be considered indistinguishable (in the quantum sense). Arguing that the premises used to establish IP were too restrictive, they adopted the following field-theoretic condition: those particles are regarded as indistinguishable which result from the second quantization of a given field, since they share the same intrinsic properties. ²⁵³ The quantum mechanics of such particles then follows from the corresponding field theory and Kamefuchi and Ohnuki obtained a general theory of indistinguishable particles by simply translating the field theoretic results concerning many particle systems into the language of quantum mechanics. This formed the basis of a further generalization covering forms of statistics different from both ordinary and parastatistics. ²⁵⁴ The exploration of such forms of non-standard statistics has continued to the present day and has proceeded in both experimental and theoretical directions. ²⁵⁵

At this point we need to stop. Our central aim in this chapter was to sketch the emergence of the Received View of particle non-individuality in its historical context. As we hope to have illustrated, a concern with particle indistinguishability runs through the history of quantum statistics like a thread,

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connecting Ehrenfest's puzzlement over Planck's non-classical counting with Kamefuchi and Ohnuki's analysis of paraparticle theory. We have included the later material on parastatistics because this offers a broader formal framework in which to represent this notion of indistinguishability. As we shall see in the next chapter, the Indistinguishability Postulate offers a useful basis on which to explore the philosophical implications of quantum statistics in more detail. In particular, it opens up the possibility of an alternative metaphysics to the Received View, one in which quantum particles can still be regarded as individuals.

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