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Nanosystems Physics Chemistry Mathematics

Dirac operator coupled to bosons

Anton A. Boitsev, Hagen Neidhardt, I. Yu. Popov

Published Apr 20, 2016• 5 citations
Publisher

Abstract

We consider a model of point-like interaction between electrons and bosons in a cavity.The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization.The model fits into the extension theory of symmetric operators.Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators.The approach allows explicit computation of the Weyl function.

DOI
10.17586/2220-8054-2016-7-2-332-339

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We consider a model of point-like interaction between electrons and bosons in a cavity.The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization.The model fits into the extension theory of symmetric operators.Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators.The approach allows explicit computation of the Weyl function.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.
It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.
The plaquette valence bond solid (VBS) phase of the J1−J2J_1-J_2J1​−J2​ antiferromagnetic (AFM) Heisenberg honeycomb model has been studied using plaquette operator theory. Considering six S=12S =\frac{1}{2}S=21​ spins on a single plaquette the plaquette operator representation for the spin operators in terms of plaquette-singlet and plaquette-triplet boson operators has been developed. The formalism has been applied to the J1−J2J_1-J_2J1​−J2​ honeycomb antiferromagnet to derive an effective interacting boson model in terms of those singlets and triplets. The effective model has been analyzed within the harmonic approximation and results have been compared with the exact diagonalization data. The evolution of Dirac and Weyl nodes with respect to J2/J1J_2/J_1J2​/J1​ for this model has been studied.