Preprint Series of Department of Mathematics, Hokkaido University
A class of deformations of the Schrdinger representation of the Heisenberg commutation relation and exact solution to a Heisenberg equation and a Schrdinger equation
We consider a class of representations {(Hv, Fv )}v of the Heisenberg commutation relation (HvFv - FvHv = -i) which are deformations of the Schrodinger representation, where the index parameter V is in a class of realvalued C00-functions on R. The Schrodinger representation is given by the case V = l. We classify the representations in terms of properties of V and show that they are divided into three subclasses. All elements of one of these subclasses are unitarily equivalent to the Schrodinger representation. But the others contain no elements unitarily equivalent to the Schrodinger representation. In particular a subclass consists of representations in each of which Hv has no self-adjoint extension. Moreover, we obtain exact solutions to the Heisenberg equation and the Schrodinger equation associated with Hv. We show that, even in the case where Hv has no self-adjoint extension, exact solutions local in time can be constructed.
We consider a class of representations {(Hv, Fv )}v of the Heisenberg commutation relation (HvFv - FvHv = -i) which are deformations of the Schrodinger representation, where the index parameter V is in a class of realvalued C00-functions on R. The Schrodinger representation is given by the case V = l. We classify the representations in terms of properties of V and show that they are divided into three subclasses. All elements of one of these subclasses are unitarily equivalent to the Schrodinger representation. But the others contain no elements unitarily equivalent to the Schrodinger representation. In particular a subclass consists of representations in each of which Hv has no self-adjoint extension. Moreover, we obtain exact solutions to the Heisenberg equation and the Schrodinger equation associated with Hv. We show that, even in the case where Hv has no self-adjoint extension, exact solutions local in time can be constructed.
Dedicated to the memory of Gerda Dorfmeister. Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X]=Id.
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.
We examine the possibility of a direct Fock representation of the recently obtained non-trivial central extensions CEHeis of the Heisenberg algebra, generated by elements a, a†, h and E satisfying the commutation relations [a, a†]CEHeis = h, [h, a†]CEHeis = z E and [a, h]CEHeis = ̄z E, where a and a† are dual, h is self-adjoint, E is the non-zero selfadjoint central element and z 2 C \ {0}. We define the exponential vectors associated with the CEHeis Fock space, we compute their Leibniz function (inner product), we describe the action of a, a† and h on the exponential vectors and we compute the moment generating and characteristic functions of the classical random variable corresponding to the self-adjoint operator X = a + a† + h.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory