Noncanonical Commutation Relations Research Articles

Supersymmetry, quark confinement and the harmonic oscillator

We study some quantum systems described by noncanonical commutation relations formally expressed as , where is the associated (harmonic-oscillator-like) Hamiltonian of the system and is a Hermitian (constant) operator, i.e. . In passing, we also consider a simple ( canonical) model, in the framework of a relativistic Klein–Gordon-like wave equation.

Effective QED in dielectric media

A general method, which allows to obtain an effective Hamiltonian for the QED in a dielectric medium, is developed. No unphysical limitation common to usual approaches, like the absence of dispersion, is assumed. In this paper we are interested mainly in the physical interpretation for the effective theory. We show that the transcription from the full theory into the effective one is non-unitary and that it introduces non-canonical commutation relations among field operators. This fact explains why usually a quantization procedure based on an effective classical theory fails.

Bose realization of a non-canonical Heisenberg algebra

The authors find out the Bose realization of a generalized Heisenberg algebra, in which the bracket of the annihilation and creation operators is proportional to a polynomial function of the number operator. The eigenvalues of the corresponding oscillator are derived in a special case. They stress also the connection between non-canonical commutation relations and q-algebras.

Comments onq-algebras

A critical study of some elementary aspects ofq-algebras is presented. The results are: (i) theq-algebras are related to para-Bose (para-Fermi) algebras only when both reduce to the usual Bose (Fermi) case, (ii) after performing a linear transformation of the operatorsA andA† that satisfy theq-algebra relation AA†, a generalized version of Penney's theorem (in the sense that the new operators satisfy noncanonical commutation and anticommutation relations) is obtained, (iii) the spectrum of one of the Hamiltonians of the system is obtained from the correspondence principle, and (iv) a whole family ofq-algebra Hamiltonians is exhibited. This family has the property that the noncanonical commutation relation is stable.

Absence of noncanonical modification in the balance equation theory of resistivity

Relative electron momenta p i ′ and coordinates r i ′, introduced in the balance equation theory of resistivity, are constrained and obey noncanonical commutation relations [r′ iα, p′ jβ]= iδ αβ(δ ij− N l . Thus it is improper to use them as translation generation. Nevertheless the contribution to the frictional force from the noncanonical 1/N term is exactly cancelled by that arising from the noncommutability of the center-of-mass coordinates R( t) at different times, such that the relative electrons can be treated as an unconstrained, canonical system and the center-of-mass as a classical single particle in the calculation of the resistivity of the system.

Consistent derivation of impurity resistivity from the force-balance equation.

The adiabatic impurity resistivity calculation based upon the force-balance equation is studied with use of the method of closed-time-path Green's functions. In this, the effects due to both the noncommutability of the center-of-mass fluctuations at different times and the exact noncanonical commutation relations between the coordinates and momenta of relative electrons are considered. We show that the leading higher-order resistivity terms generated by the noncommutability of center-of-mass fluctuations fully cancel among themselves and make no contribution to the impurity resistivity. As a result the center-of-mass fluctuations can be regarded as classical variables. When canonical commutation relations are approximated for relative electrons, the resulting adiabatic resistivity vanishes. We also demonstrate that the standard adiabatic resistivity formula can be obtained only if the exact noncanonical commutation relations are employed for relative electrons. In the presence of a momentum-conserving inelastic scattering time due to a heat bath and/or direct electron-electron interactions, we confirm that the impurity resistivity (isothermal) is devoid of the divergences of Van Hove's ``${\ensuremath{\lambda}}^{2}$t''-series expansion. This result is in agreement with a phenomenological study based on the Boltzmann equation.

FORMAL PATH INTEGRAL FOR THEORIES WITH NONCANONICAL COMMUTATION RELATIONS

For theories with noncanonical commutation relations, the generating functional (S-matrix) is constructed at a formal level, as a Hamiltonian path integral.

Quantized non-Abelian monopole.

The Hamiltonian for the quantized non-Abelian monopole is found using the Dirac constraint method. This Hamiltonian is correct to O(h-dash-bar) since we do not consider operator ordering. We choose the covariant background-gauge condition for the fluctuations. The collective-coordinate gauge constraints are chosen to give canonical commutation relations. We find that the constraints chosen by Abouelsaood lead to noncanonical commutation relations.

Extension of the caldirola procedure in space

Based on results known until now of Caldirola’s chronon theory, we present an extension of this theory to the space co-ordinates with the use ofq-representation. First the new momentum operators are defined. These depend on a constant with dimension of length and their commutators with the position operator satisfy noncanonical commutation relations. The eigenfunctions and eigenvalues of the operators are found. We also study with this new extension the problems of free particles and the discrete time-dependent and time-independent Schrodinger equations are found. We next determine the coherent states of the new annihilation operators and finally calculate the Bloch density matrices for both cases&—Caldirola-Montaldi procedure and its new extension. These are noncanonical.

Remarks on the noncanonical commutation relations of the four-vectors of the energy-momentum operators

Taking as an example the noncanonical commutation relations of the four-vectors of the energy-momentum operators, we study the noncanonical character of hadronic mechanics.