Abstract
This paper is mainly focused on the description of an approach for establishing a spinorial representation of linear canonical transformations. It can be considered as a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. The said method is based on the development of an adequate parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations. Obtaining this pseudo-orthogonal representation makes it possible to establish the spinorial representation exploiting the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theories are both studied. The design of the pseudo-orthogonal transformation associated to a linear canonical transformation is achieved by introducing adequate operators which are linear combinations of reduced momentum and coordinate operators. It is shown that a linear canonical transformation is equivalent to a special pseudo-orthogonal transformation defined in the set formed by these adequate operators. The spinorial representation is then deduced by defining a composite operator which is linear combinations of the tensorial products of the generators of the Clifford algebra with the adequate operators defining the special pseudo-orthogonal representation. It is established that unlike the case of a spinorial representation associated with an ordinary commutative vector space, the main invariant corresponding to the transformation is not the square of the composite operator but a higher degree polynomial function of it.
Highlights
In our previous papers [1,2,3,4], we have performed a series of study on a phase space representation of quantum theory and Linear Canonical Transformations (LCTs)
The approach described in this work shows that it is possible to establish a spinorial representation of Linear Canonical Transformations
The spinorial representation can be established using the relations between special pseudo-orthogonal group and spin group
Summary
In our previous papers [1,2,3,4], we have performed a series of study on a phase space representation of quantum theory and Linear Canonical Transformations (LCTs). In the work [2], we have introduced operators defined from the momentum and coordinate operators of a particle. Some of these operators will be used throughout the present paper. These operators are the reduced operators and , the reduced dispersions operators , and and their multidimensional generalization , , , and. Operators defined from the momentum and coordinate operators are written in bold. Raoelina Andriambololona et al.: Study on a Spinorial Representation of Linear Canonical Transformation
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More From: International Journal of Applied Mathematics and Theoretical Physics
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