Representation Heisenberg group

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 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... 

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 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 group-theoretical formalism we have introduced so far is particularly suited to formulate quantum mechanics in the Heisenberg representation, where the time dependence is shifted from the wavefunctions to the operators. As we shall show in the next section, the formalism allows to show in a straightforward way that the Poisson brackets are obtained as a formal limit of the commutator when h —> 0. 

The derivation of a consistent mixed quantum-classical dynamics discussed in this paper was first proposed in Ref. [15] and commented and clarified in Ref. [1], This derivation is based on a group-theoretical formulation of quantum and classical mechanics, which introduces a very elegant and formally rigorous mathematical apparatus and allows to directly obtain classical mechanics as the limit for h —> 0 of quantum mechanics, in the Heisenberg representation of quantum dynamics. 

Weyl thus answered the second question in his investigations of the mathematical analysis of the space problem. His researches with the representation theory of Lie groups started because of his diverse background, from the philosophy of mathematics to the natural sciences. Further, he clarified Heisenberg s non-commuting physical quantities in quantum mechanics, which were initially stated in a mathematical form rather than a physical form. 

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