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:
The quantum operator
T^3, corresponding to the projection of the toroidal moment on the
z axis, admits several self-adjoint extensions, when defined on the whole
R3 space.
T^3 commutes with
L^3 (the projection of the angular momentum operator on the
z axis) and they have a \textit{natural set of coordinates}
(k,u,ϕ) where
ϕ is the azimuthal angle. The second set of \textit{natural coordinates} is
(k1,k2,u), where
k1=kcosϕ,
k2=ksinϕ. In both sets,
T^3=−iℏ∂/∂u, so any operator that is a function of
k and the partial derivatives with respect to the \textit{natural variables}
(k,u,ϕ) commute with
T^3 and
L^3. Similarly, operators that are functions of
k1,
k2, and the partial derivatives with respect to
k1,
k2, and
u commute with
T^3. Therefore, we introduce here the operators
p^k≡−iℏ∂/∂k,
p^(k1)≡−iℏ∂/∂k1, and
p^(k2)≡−iℏ∂/∂k2 and express them in the
(x,y,z) coordinates. One may also invert the relations and write the typical operators, like the momentum
p^≡−iℏ∇ or the kinetic energy
H^0≡−ℏ2Δ/(2m) in terms of the "toroidal" operators
T^3,
p^(k),
p^(k1),
p^(k2), and, eventually,
L^3. The formalism may be applied to specific physical systems, like nuclei, condensed matter systems, or metamaterials. We exemplify it by calculating the momentum operator and the free particle Hamiltonian in terms of \textit{natural coordinates} in a thin torus, where the general relations get considerably simplified.
Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.