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Ali Mostafazadeh

Publications and source records attributed to Ali Mostafazadeh.

At least 181 records · Page 10Linked to original sources
arXiv1997

Two-Component Formulation of the Wheeler-DeWitt Equation

The Wheeler-DeWitt equation for the minimally coupled FRW-massive-scalar-field minisuperspace is written as a two-component Schrödinger equation with an explicitly `time'-dependent Hamiltonian. This reduces the solution of the Wheeler-DeWitt equation to the eigenvalue problem for a non-relativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler-DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler-DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt's boundary condition at initial singularity.

gr-qc↗
arXiv1996

On a Class of Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator

Quantum canonical transformations corresponding to the action of the unitary operator $e^{iε(t)\sqrt{f(x)}p\sqrt{f(x)}}$ is studied. It is shown that for $f(x)=x$, the effect of this transformation is to rescale the position and momentum operators by $e^{ε(t)}$ and $e^{-ε(t)}$, respectively. This transformation is shown to lead to the identification of a previously unknown class of exactly solvable time-dependent harmonic oscillators. It turns out that the Caldirola-Kanai oscillator whose mass is given by $m=m_0 e^{γt}$, belongs to this class. It is also shown that for arbitrary $f(x)$, this canonical transformations map the dynamics of a free particle with constant mass to that of free particle with a position-dependent mass. In other words, they lead to a change of the metric of the space.

quant-ph↗
arXiv1996

Quantum Canonical Transformations and Exact Solution of the Schreodinger Equation

Time-dependent unitary transformations are used to study the Schreodinger equation for explicitly time-dependent Hamiltonians of the form $H(t)=\vec R(t).\vec J$, where $\vec R$ is an arbitrary real vector-valued function of time and $\vec J$ is the angular momentum operator. The solution of the Schreodinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case $\vec R=(1,0,ν(t))$. This corresponds to the problem of driven two-level atom for the spin half representation of $\vec J$. It is also shown that by requiring the magnitude of $\vec R$ to depend on its direction in a particular way, one can solve the Schreodinger equation exactly. In particular, it is shown that for every Hamiltonian of the form $H(t)=\vec R(t)\cdot \vec J$ there is another Hamiltonian with the same eigenstates for which the Schreodinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holonomy calculation for SU(2) principal bundles (Yang-Mills gauge theory) is also pointed out.

quant-ph↗
arXiv1996

Inverting Time-Dependent Harmonic Oscillator Potential by a Unitary Transformation and a New Class of Exactly Solvable Oscillators

A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.

quant-ph↗
arXiv1996

Non-Abelian Geometrical Phase for General Three-Dimensional Quantum Systems

Adiabatic $U(2)$ geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of $\xC P^2$ is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians $H=Q^{i_1,\cdots i_n}J_{i_1}\cdots J_{i_n}$ and their linear combinations for spin $j=1$ systems. In particular it is shown that the nuclear quadrupole Hamiltonian $H=Q^{ij}J_iJ_j$ does actually lead to non-Abelian geometric phases for $j=1$. This system, being bosonic, is time-reversal-invariant. Therefore it cannot support Abelian adiabatic geometrical phases.

quant-ph↗
arXiv1996

Scalar Curvature Factor in the Schroedinger Equation and Scattering on a Curved Surface

The scattering of free particles constrained to move on a cylindrically symmetric curved surface is studied. The nontrivial geometry of the space contributes to the scattering cross section through the kinetic as well as a possible scalar curvature term in the quantum Hamiltonian. The coefficient of the latter term is known to be related to the factor ordering problem in curved space quantization. Hence, in principle, the scattering data may be used to provide an experimental resolution of the theoretical factor ordering ambiguity. To demonstrate the sensitivity required of such an experimental setup, the effect of a localized magnetic field in the scattering process is also analyzed.

hep-th↗
arXiv1996

Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators

The general features of the degeneracy structure of ($p=2$) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a forth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for the topological invariants of a class of parasupersymmetric systems is provided.

hep-th↗
arXiv1996

Adiabatic Product Expansion

The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models.

quant-ph↗
arXiv1996

Quantum Adiabatic Approximation, Quantum Action, and Berry's Phase

An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry's adiabatic phase and its semi-classical limit are also discussed.

quant-ph↗
arXiv1996

The Quantum Adiabatic Approximation and the Geometric Phase

A precise definition of an adiabaticity parameter $ν$ of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator $U(τ)=\sum_\ell U^{(\ell)}(τ)$ with $U^{(\ell)}(τ)$ being at least of the order $ν^\ell$. In particular $U^{(0)}(τ)$ corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the $1/τ$-expansion of $U(τ)$. It is also shown that the non-adiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. Some related issues concerning the geometric phase are also discussed.

hep-th↗
arXiv1995

Parabose-Parafermi Supersymmetry

The ($p=2$) parabose-parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the ($p=2$) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose- -parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose-parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose- parafermi supersymmetry is offered, thus introducing ($p=2$) Supersymmetric Paraquantum Mechanics. The spectrum degeneracy structure of general ($p=2$) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose-parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.

hep-th↗
arXiv1995

Para-Generalization of Peierls Bracket Quantization

A convenient formalism is developed to treat classical dynamical systems involving $(p=2)$ parafermionic and parabosonic dynamical variables. This is achieved via the introduction of a parabracket which summarizes the paracommutation relations of the corresponding Green components in a unified manner. Furthermore, it is shown that Peierls quantization scheme may be applied to such systems provided that one uses the above mentioned parabracket to express the quantum paracommutation relations. Application of the Peierls scheme also provides the form of the parafermionic and parabosonic kinetic terms in the Lagrangian.

hep-th↗
arXiv1995

Lattice Topological Field Theory on Non-Orientable Surfaces

The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a one-to-one correspondence between real associative $*$-algebras and the topological state sum invariants defined on such surfaces. The partition and $n$-point functions on all two-dimensional surfaces (connected sums of the Klein bottle or projective plane and $g$-tori) are defined and computed for arbitrary $*$-algebras in general, and for the the group ring $A=\R[G]$ of discrete groups $G$, in particular.

hep-th↗
arXiv1995

Topological Aspects of Parasupersymmetry

Parasupersymmetric quantum mechanics is exploited to introduce a topological invariant associated with a pair of parameter dependent Fredholm (respectively elliptic differential) operators satisfying two compatibility conditions. An explicit algebraic expression for this topological invariant is provided. The latter identifies the parasupersymmetric topological invariant with the sum of the analytic (Atiyah-Singer) indices of the corresponding operators.

hep-th↗
arXiv1994

Spectrum Degeneracy of General (p=2)--Parasupersymmetric Quantum Mechanics and Parasupersymmetric Topological Invariants

A thorough analysis of the general features of $(p=2)$ parasupersymmetric quantum mechanics is presented. It is shown that for both Rubakov--Spiridonov and Beckers--Debergh formulations of $(p=2)$-parasupersymmetric quantum mechanics, the degeneracy structure of the energy spectrum can be derived using the defining parasuperalgebras. Thus the results of the present article is independent of the details of the Hamiltonian. In fact, they are valid for arbitrary systems based on arbitrary dimensional coordinate manifolds. In particular, the Rubakov--Spiridonov (R-S) and Beckers--Debergh (B-D) systems possess identical degeneracy structures. For a subclass of R-S (alternatively B-D) systems, a new topological invariant is introduced. This is a counterpart of the Witten index of the supersymmetric quantum mechanics.

hep-th↗
arXiv1994

Supersymmetry, Path Integration, and the Atiyah-Singer Index Theorem

A new supersymmetric proof of the Atiyah-Singer index theorem is presented. The Peierls bracket quantization scheme is used to quantize the supersymmetric classical system corresponding to the index problem for the twisted Dirac operator. The problem of factor ordering is addressed and the unique quantum system that is relevant to the index theorem is analyzed in detail. The Hamiltonian operator is shown to include a scalar curvature factor, $\hbar^2R/8$. The path integral formulation of quantum mechanics is then used to obtain a formula for the index. For the first time, the path integral "measure" and the Feynman propagator of the system are exactly computed. The derivation of the index formula relies solely on the definition of a Gaussian superdeterminant. The two-loop analysis of the path integral is also carried out. The results of the loop and heat kernel expansions of the path interal are in complete agreement. This confirms the existence of the scalar curvature factor in the Schrödinger equation and validates the supersymmetric proof of the index theorem. Many other related issues are addressed. Finally, reviews of the index theorem and the supersymmetric quantum mechanics are presented.

hep-th↗
arXiv1993

Geometric Phase, Bundle Classification, and Group Representation

The line bundles which arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the representation theory. The remarkable relation of the geometric phase to the classification of complex line bundles provides the necessary tools for establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry's connection. The results about the fibre bundles and group theory are used to introduce a procedure to reduce the problem of the non-adiabatic (geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-abelian monopoles is pointed out.

hep-th↗