Infinite-dimensional representations of the rotation group and Dirac monopole problem

Abstract

Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is related to the weight of the Dirac string.

On Observability of Dirac's String

“…32 However, if 2µ ∈ Z, then + µ and − µ are both integers, and the obtained representations become finite-dimensional equivalent representations. In this case the functions ± Y (µ,n) ∓µ vanish on the string and coincide with the monopole harmonics introduced by Wu and Yang.…”

“…29 −µ form the basis of the representation bounded below D + ( , −µ). 32 In general, the obtained solutions are being given by non-square integrable functions on the sphere belong to the extended Hilbert space H η , where the inner product is defined by the bilinear form of the type (ψ, ψ ) η = (ψ, ηψ ) = ψ ηψ dµ(x) (15) such that the norm is positive, (ψ, ψ) η > 0. This provides the standard probabilistic interpretation of the quantum mechanics.…”

“…32,[38][39][40][41] in 34 . However, the situation is quite different for the scattering on the monopole with an arbitrary magnetic charge.…”

“…[27][28][29][30] In the context of the group theory one has to involve infinite-dimensional representations of the rotation group to provide a consistent description of non-quantized Dirac monopole. 29,31,32 Recently, the non-quantized "magnetic" monopoles arising in trapped Λ-type atoms and anisotropic spin systems has been reported in Refs. 5 and 6.…”

Nonassociativity, Dirac Monopole and Aharonov–bohm Effect

“…To this end, one needs to consider the nonassociative generalization of the Hopf bundle, employing nonassociative fibre bundle theory [25][26][27][28]. In the context of the group theory, one has to involve infinite-dimensional representations of the rotation group to provide a consistent description of nonquantized Dirac monopole [28][29][30].…”