Dual breaking of symmetries in algebraic models
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000027-4
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Dual breaking of symmetries in algebraic models J. Cseha Institute for Nuclear Research, Pf. 51, 4001 Debrecen, Hungary Received 17 February 2020 / Accepted 28 August 2020 Published online 23 October 2020 Abstract. The general features of the dual symmetry breaking in the algebraic structure models are discussed. Dual breaking indicates here simultaneous dynamical and spontaneous breaking. Several examples are considered, including the multiconfigurational dynamical symmetry (MUSY), which is the common intersection of the shell, collective and cluster models for the multi shell problem.
1 Introduction The basic equation of the time-independent quantum mechanical description is the eigenvalue-equation of the energy. It has two important objects: the (H Hamiltonian) operator and its eigenvectors. When both of them are symmetric (see below for more details), one has an exact symmetry [1]. When the Hamiltonian contains a symmetry breaking interaction, then we speak about a dynamical breaking of the symmetry. In this case, the operator is not symmetric, of course, but the eigenvectors may remain symmetric. This situation requires some special symmetry breaking interaction. When this is the case (non-symmetric operator with symmetric eigenvectors), one has a dynamical (or dynamically broken) symmetry [2–4]. On the other hand, when the operator is symmetric, but the eigenvector (of the ground state) is not symmetric (in the strict sense), then the symmetry is spontaneously broken [5]. We consider here continuous symmetries which are characterized by Lie groups and their associated Lie algebras. (For simplicity, we denote both the group and its algebra by the same letter.) The symmetry of the operators and eigenvectors can be expressed as follows. [1]. An operator H is symmetric, or in other words it is a scalar, if [H, Xq ] = 0, where Xq is any element of the symmetry algebra. The state ψ is symmetric in the strict sense if Xq ψ = 0.1 We call a set of states ψi symmetric in the loose sense if they transform according to a definite irreducible representation (irrep) of the symmetry group: T ψi = Σj Tji ψj , where Tji are elements of the representation matrix of the transformation T . In this paper, a symmetric state is meant in the loose sense, in general. When it is symmetric in the strict sense, we say it explicitly. For illustration: with respect to a
e-mail: [email protected] 1 This
equation is equivalent to the more familiar condition: the state ψ transforms according to the identity representation T ψ = ψ. Here, T is the operator of a symmetry transformation, representing an element of the Lie group, and for the identity representation T = 1.
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The European Physical Journal Special Topics
the rotation a state is strictly symmetric if it has angular momentum L = 0, and it is symmetric in the loose sense if it has well-defined L value. Obviously, these two different kinds o
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