Heisenberg group

Nrnenschwander, D. Probabilities on the Heisenberg Group Limit Theorems and Brownian Motion, Vol. 163," Springer-Verlag, Tnc.. New York, NY, 1996 Perrin. J. Atoms. Ox Bow Press. Woodbridge, MA. 1990. 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

The representation is called unitary if p(g) is a unitary operator for all g. The representation p g) is called irreducible if a non-trivial invariant subspace V does not exist4. We are now ready to introduce the Heisenberg group, which is the main object of this section. 

The Heisenberg group H" is defined by the nature of its elements g and by the law which defines the multiplication between group elements. Each element is a set of 2n + 1 variables... 

The formulation of quantum mechanics requires a representation of the Heisenberg group on the Hilbert space L2 (R") spanned by the functions tp ( ) where the variable indicates a n—dimensional vector = ( 1 , n) whose elelments have physical units of a length [/]. Let us first introduce the set of operators, generators of the Lie group H", I, Xj, and hDj (j = 1,..., n) satisfying the commutation relation... 

We described so far the mathematical apparutus which will be used to obtain a group-theoretical formulation of quantum mechanics (section 3), by means of the Heisenberg group, and to obtain the connection between quantum and classical mechanics (section 4) within the group-theoretical formal-... 

In order to discuss the group-theoretical formulation of quantum mechanics, which is the object of this section, we need to show that the infinitedimensional representation of the Heisenberg group ph(g) can be used as a basis for the vector space of hermitian operators [8]. 

This section is devoted to the derivation of classical mechanics (observables and equations of motion) as a formal limit of quantum mechanics, within the group-theoretical formulation discussed in the previous sections. To this purpose, we need to recall here some properties of the Heisenberg group. 

Finally, when these expressions are inserted into ph(g) to determine the action of the representatives of the Heisenberg group on 1F%, we obtain... 

This result concludes the presentation of the Heisenberg group approach as the powerful tool that allows to derive classical mechanics as a formal limit of quantum mechanics, for h —> 0. The most important ingredients that have been introduced to obtain this result are the Fourier-like representation of observables and equations of motion and the definition of the antiderivative operator. These elements will be used in section 5 to derive a similiar procedure for a mixed quantum-classical mechanics. An ansatz on the quantum-classical equations of motion will be necessary, but the subsequent application of Heisenberg group formalism will be a straightforward generalization of what has been done so far. 

We have thus reconstructed the derivation and interpreted the results of Ref [15], The first two terms, i.e., the commutator and the Poisson brackets, are already present in a theory based on the quantum-classical Liouville representation discussed in section 1. The new term, which appears within the Heisenberg group approach, needs to be explained. In the attempt to provide a physical interpretation to this term we have shown, in Ref. [1], that the new equation of motion is purely classical. This will be illustrated in the following section. 

The generalization of the Heisenberg group formalism to the group D" is not consistent. The comparison between the approaches proposed in section 3 and in section 5 can be summarized in the following scheme ... 

However, the Heisenberg group formalism is a very useful tool to represent quantum and classical dynamical quantities, such as observables and equations of motion, only when a prescription on the generator of the time evolution exists. The comparison with the fully quantum or fully classical dynamics allows us to deduce only the formal properties that the mixed quantum-classical brackets have to satisfy in order to generate a consistent evolution, but does... 

Due to Heisenberg s uncertainty and Pauli s exclusion principles, the properties of a multifermionic system correspond to fermions being grouped into shells and subshells. The shell structure of the one-particle energy spectrum generates so-called shell effects, at different hierarchical levels (nuclei, atoms, molecules, condensed matter) [1-3]. 

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