Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results:
Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results:
Abstract. It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space H of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U, respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[SN]. Let U (r1,r2)
It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[ S N ] such as decompositions of ideals of the group ring C[ S N ], idempotents of C[ S N ], discrete Fourier transforms of S N , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[ S N ]. Let U (r1,r2) μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight ( r 1 , r 2 ). If one determines the smallest symmetry class for every v U (r1,r2) μ then one can observe jumps of the symmetry behaviour. For "generic" v U (r1,r2) μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U (r1,r2) μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ( J -self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨ n , n ⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J -self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
Structured as a dialogue between a mathematician and a physicist, Symmetry and Quantum Mechanics unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry. Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.
We examine the possibility of a direct Fock representation of the recently obtained non-trivial central extensions CEHeis of the Heisenberg algebra, generated by elements a, a†, h and E satisfying the commutation relations [a, a†]CEHeis = h, [h, a†]CEHeis = z E and [a, h]CEHeis = ̄z E, where a and a† are dual, h is self-adjoint, E is the non-zero selfadjoint central element and z 2 C \ {0}. We define the exponential vectors associated with the CEHeis Fock space, we compute their Leibniz function (inner product), we describe the action of a, a† and h on the exponential vectors and we compute the moment generating and characteristic functions of the classical random variable corresponding to the self-adjoint operator X = a + a† + h.