Idealization Second Quantization of Composite Particles

A practical method is developed to deal with the second quantization of the many-body system containing the composite particles.In our treatment,the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles.The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian.To be emphasized,the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model III

A classical nonrelativistic U(1)×U(1) gauge field model that describes the topologically massive electromagnetic interaction of composite particles in (2+1) dimensions is proposed. The model, generalization of a previously postulated one, contains a Chern-Simons U(1) field and the topologically massive electromagnetic U(1) field, and it uses both a composite boson system or a composite fermion one. The second case is considered explicitly. By using the Dirac Hamiltonian method for constrained systems, the canonical quantization is carried out. By means of the Faddeev-Senjanovic formalism, the path integral quantization is developed. Consequently, the Feynman rules are established and the diagrammatic structure is treated. The application of the Becchi-Rouet-Stora-Tyutin algorithm is discussed. The present and previous models are compared.     

Two-component theory of fermions

A variation of the theory of fermions is proposed in which the fermions are described by two-component spinors obeying a relativistic equation of the second order. In order to make the probability density of the spin-1/2 particles positive definite, the rule is established that complex conjugation of functions and Hermitian conjugation of operators are accompanied by the operation of spatial reflection. A one-particle theory in Hamiltonian form, a Lagrangian formalism for a free two-component field, and the second quantization of the theory are derived. For calculations in quantum electrodynamics a Hamiltonian is proposed similar to the interaction Hamiltonian of spinless particles with the electromagnetic field but containing a spin-dependent part.Translated from Izvestiya VUZ. Fizika, No. 5, pp. 53–58, May, 1971.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model II

By generalizing a model previously proposed, a classical nonrelativistic U(1)×U(1) gauge field model for the electromagnetic interaction of composite particles in (2+1) dimensions is constructed. The model contains a Chern–Simons U(1) field and the electromagnetic U(1) field, and it describes both a composite boson system or a composite fermion one. The second case is considered explicitly. The model includes a topological mass term for the electromagnetic field and interaction terms between the gauge fields. By following the Dirac Hamiltonian formalism for constrained systems, the canonical quantization for the model is realized. By developing the path integral quantization method through the Faddeev–Senjanovic algorithm, the Feynman rules of the model are established and its diagrammatic structure is discussed. The Becchi–Rouet–Stora–Tyutin formalism is applied to the model. The obtained results are compared with the ones corresponding to the previous model.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Klein Gordon Equations on a Time Lattice

We consider the second quantization procedure for a KLEIN GORDON equation with time dependent Hamiltonian and with replaced second order time derivative by the appropriate difference operator. With each time step there is a Bogoljubov transformation which describes particle creation and annihilation and the accompanied change of the Fock vacuum.     

Superexchange interaction in rare earth dielectrics

The second quantization method is applied to the analysis of superexchange interaction in rare earth dielectrics. Introduction of double tensor operators associated with the second quantization operators makes it possible to write the superexchange interaction Hamiltonian in the effective form of the total magnetic moment interaction of ions . As an example, the obtained Hamiltonian is used to explain the magnetic properties of rare earth pnicudes of the yttrium group.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 78–82, December, 1973.The authors thank B. V. Karpenko and V. Ya. Mitrofanov for useful discussions.     

Second Quantization of the Elliptic Calogero-Sutherland Model

We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian.Work supported by the Swedish Natural Science Research Council (NFR).     

Can Dirac quantization of constrained systems be fulfilled within the intrinsic geometry?

For particles constrained on a curved surface, how to perform quantization within Dirac’s canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. In this paper, we take a minimum surface, helicoid, on which the motion is constrained, to explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that not only an inconsistency within Dirac theory occurs, but also an incompatibility with Schrödinger theory happens. In contrast, in three-dimensional Euclidean space, the Dirac quantization turns out to be satisfactory all around, and the resultant geometric momentum and potential are then in agreement with those given by the Schrödinger theory.     

Quantization of a Hamiltonian system with an infinite number of degrees of freedom

We propose a method of quantization of a discrete Hamiltonian system with an infinite number of degrees of freedom. Our approach is analogous to the usual finite-dimensional quantum mechanics. We construct an infinite-dimensional Schrödinger equation. We show that it is possible to pass from the finite-dimensional quantum mechanics to our construction in the limit when the number of particles tends to infinity. Rigorous mathematical methods are used.     

On Covariant Quantization of the Massive Self-dual 3-forms in 7 Dimensions

Massive self-dual 3-forms in 7 dimensions are analyzed from the point of view of the Hamiltonian path integral quantization. The quantization procedure relies on the quantization of a first-class system equivalent with the original theory. The first-class system is constructed in the framework of gauge unfixing approach and Batalin-Fradkin method. The Hamiltonian path integral of the first-class system takes a manifestly Lorentz-covariant form.     

Branes,Quantum Nambu Brackets and the Hydrogen Atom

The Nambu-bracket quantization of the hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the quantum Nambu-brackets comprise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the hydrogen atom, the results coincide with those furnished by Hamiltonian quantization - but the method is not limited to Hamiltonian systems.     

A General Theory of Self-consistent Field Beyond Hartree-Fock

A general scheme is devised in the formalism of second quantization to make the Hamiltonian of an electronic system diagonal by a unitary transformation together with a suitable choice of the basis of one-electron wave functions. The scheme can be regarded as a generalization of the HartreeFodr approximation of self-consisten t field, and can be carried to any order of approximation in principle. Formulae for the lowest order approximation beyond Hartree Fodr are given.     

Hamiltonian evolution and quantization for extremal black holes

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes.     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model

A classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of composite particles is proposed and the quantum formalism is constructed. This gauge model containing a Chern–Simons U(1) field and the electromagnetic U(1) field can be coupled to both a bosonic or a fermionic matter field. We explicitly consider the second case, a composite fermion system in the presence of an electromagnetic field, and we carry out the canonical quantization by the Dirac method. The path integral approach is developed and the Feynman rules are established. A simplified model is considered. As an alternative path integral method, the BRST formalism for this gauge model is also treated.     

Towards the deformation quantization of linearized gravity

We present a first attempt to apply the approach of deformation quantization to linearized Einstein's equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and the discrete energy spectrum of linearized gravity.     

Polarization of spin-1/2 particles in an axisymmetric magnetic field

The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed. Zh. éksp. Teor. Fiz. 114, 1153–1161 (October 1998)     

Vertex calculation for a composite particle

The classical treatment and the quantization of composite relativistic systems is given a manifestly covariant formulation in presence of constraints. A particular formulation of Feynman's quantum mechanics is used to treat the scattering of composite relativistic systems. A covariant harmonic oscillator model is employed to calculate vertices of interactions: the results are similar to the corresponding ones in the usual field theories, but the presence of some convergence factors gives hope that a theory with composite particles may be finite.