On the Topological Approach to the Metaphysical Problem of Indistinguishable Quantum Particles

Tomasz Bigaj (University of Warsaw)

On the topological approach to the metaphysical problem of indistinguishable quantum particles

Abstract  

The standard quantum theory of many particles imposes an important restriction on the available states of particles of the same type. This restriction takes on the form of the symmetrization postulate, according to which the state of a system of “indistinguishable” particles has to be either symmetric (for bosons) or antisymmetric (fermions) with respect to the permutation of individual particles. The symmetrization postulate is applied to reduce the number of accessible states that can be identified in a full tensor product of N individual (labeled) Hilbert spaces. However, there are some alternative ways of representing mathematically states of indistinguishable particles. In the topological approach, the configuration space is obtained by identifying all the elements of the full tensor product of individual spaces that differ only with respect to the permutation of the elements (this procedure is known as “quotienting out”). The resulting configuration space appears to have new interesting topological properties due to the existence of singularities at points where two or more particles possess the same state. In particular, it can be shown (Leinaas & Myrheim 1977) that the difference in global topology between configuration spaces for distinguishable and indistinguishable particles naturally leads to the symmetry constraints on the states of particles of the same type. In this article the topological approach will be compared to yet another method of representing the states of indistinguishable particles recently suggested by Ladyman et al. (2013). In this latest proposal the states of N particles of the same type are constructed with the help of the symmetric or antisymmetric “wedge” product rather than the full tensor product. The wedge product of two vectors is defined as the equivalence class that contains all and only vectors of the ordinary tensor product for which the operation of symmetrization (antisymmetrization) gives the same result as when applied to the direct product of the initial vectors. Both the topological approach and the wedge formalism will be analyzed with respect to their ability to shed new light on the metaphysical problem of indistinguishable particles, which is the question whether quantum particles of the same type can be discerned by any meaningful physical properties or relations, and whether they can thus achieve the status of individual objects equipped with their own unique identities. Another question addressed in the paper will be the problem of the redundancy of some parts of the mathematical formalism used in the description of physical reality (the problem of “surplus structure” in Michael Redhead’s terminology). It turns out that both the topological approach and the wedge formalism present us with their own unique ways of eliminating such surplus structures in the case of the quantum theory of many particles.
ReferencesLadyman, J., Linnebo, Ø., and Bigaj, T. (2013). Entanglement and non-factorizability. Studies in History and Philosophy of Modern Physics, 44:215–221.Leinaas, J. and Myrheim, J. (1977). On the theory of identical particles. Il Nuovo Cimento B Series 11, 37(1):1–