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Journal of Mathematical Physics

Must quantum theory assume unrestricted superposition?

Herbert Joseph Bernstein

Published Oct 1, 1974• 16 citations
Publisher

Abstract

A conjecture that the (n −1)2 independent moduli and (2n −1) unphysical phases completely specify all n-dimensional unitary matrices is shown to be true in two and three dimensions, but false in four or more. The implications for quantum theory are discussed.
DOI
10.1063/1.1666523

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It is proposed that an unconstrained quantum system is completely specified by its configuration space CCC and its group of symmetries GGG. It is then shown that the theory of projective unitary representations of GGG on a certain Hilbert space determined by CCC and GGG leads to a definitive resolution of supposed ambiguities relating to superselection rules, choice of wave functions, topological properties of GGG and CCC, anomalies, etc.
The generalized defect D(U) of a unitary NxN matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli |U_ij| as well as the Gram matrix U'*U in the first order. Then the defect d(U) is equal to D(U) - (2N-1), that is the generalized defect diminished by the dimension of the manifold {Dr*U*Dc : Dr,Dc unitary diagonal}. Calculation of d(U) involves calculating the dimension of the space in R^(N^2) spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker product of unitary matrices, thus making it easier to perform calculations numerically. Basing on this, we give a lower bound on D(U) (equivalently d(U)), supposing it is achieved for most unitaries with a fixed Kronecker product structure. Also supermultiplicativity of D(U) with respect to Kronecker subproducts of U is shown.
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