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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

Noninvariance groups for quantum-mechanical systems

R. Aldrovandi, Paulo Leal Ferreira

Published Feb 1, 1969• 11 citations
Publisher
DOI
10.1007/bf02753730

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Physical Review A•Feb 16, 2007•3 citations
We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.
Elementary quantum mechanical principles are applied to bipartite or multipartite systems to test the nonseparability of subsystems (or nonadditivity in the system Hamiltonian) through interchanging the ordering of subsystem (local) operators that cause transitions between energy eigenstates, whether applied to pure or mixed states. A tentative measure for quantifying nonseparability is proposed, based on complete sets of subsystem transition operators. This serves as an upper bound for experimental determinations of the intersubsystem coupling. Two- and three-qubit systems are numerically tested to establish criteria for the noninvariance of transition probabilities in them with respect to the sequence of excitations, and it is conjectured that sequence noninvariance can be used as a probe for subsystems' nonseparability.