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On Quantization and Point Canonical Transformations in Phase-Space Functional Integrals

F. Lanoouche

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

where 7 stands for the discretization and H~ depends on i t (1). Knowledge of 3' means that one is given one of the possible functions hv(p, q', q) satisfying (M is the dimension of q-space) (3) = f ~ exp [~p.(q'--q)] h,(p, q', q) . Then (2) is defined as I = lira I:v, ~T-+ 0% where (e : (t--to)/(3T+ 1), Aq~ = q~--qJ-x) (4) z ~ = l r l d q , I [ I ~ e x p [ ~ p , . ~ ) h v ( p , , q , , q j _ 0 J ~i j=x lV~m

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where 7 stands for the discretization and H~ depends on i t (1). Knowledge of 3' means that one is given one of the possible functions hv(p, q', q) satisfying (M is the dimension of q-space) (3) = f ~ exp [p.(q'--q)] h,(p, q', q) . Then (2) is defined as I = lira I:v, T-+ 0% where (e : (t--to)/(3T+ 1), Aq = q--qJ-x) (4) z ~ = l r l d q , I [ I ~ e x p [ ~ p , . ~ ) h v ( p , , q , , q j _ 0 J i j=x lVm
In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.
This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position x^ and a momentum operator p^ (called the deformed momentum), which becomes a function F of the same operators: x^,p^=F(x^,p^), that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra x^,p^0=iℏI. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian x^ coordinate and its non-deformed conjugate momentum p^0, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum p^ and a new position coordinate y^, which is its canonical conjugate variable, so y^ and p^ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation.