Dispersion relations and dynamic characteristics of bound states in the model of a Dirac field interacting with a material plane

Symanzik’s approach for construction of quantum field model in inhomogeneous space-time is used as a basis for modeling the interaction of a macroscopic material body with quantum fields. In quantum electrodynamics it enables one to establish the most general form of the action functional describing the interaction of 2-dimensional material objects with photon and fermion fields. Results obtained within this approach for description of the interaction of the spinor field with a material plane are presented.


Introduction
The proposed by Symanzik approach for modeling of interaction of a macroscopic material body with quantum fields is considered. Its application in quantum electrodynamics enables one to establish the most general form of the action functional describing the interaction of 2-dimensional material surface with photon and fermion fields. The models making it possible to calculate the Casimir energy and Casimir-Polder potential for non-ideal conducting material are presented. Applications of the models to descriptions of of interaction of the spinor field with a material plane are considered. From the basic principles of QED ( gauge invariance, locality, renormalizability ) it follows that for thin film without charges and currents, which shape is defined by equation The action S Φ (A) is a surface Chern-Simon action

Formulation of model
The full action of the model, which satisfies the requirement of locality, gauge invariance and renormalizability, has the form Due to the requirements of renormalizability the fields interaction is described by standard contribution eψÂψ to the QED action.
Yury M.Pismak Dispersion relations and dynamic characteristics of bound states Casimir energy for two parallel planes The energy density E 2P of defect is defined as where T = dx 0 is time of existence of defect, and S = dx 1 dx 2 , is area of one. It is expressed in an explicit form in terms of polylogarithm function Li 4 (x ) in the following way: Here E j is an infinite constant, which can be interpreted as self-energy density on the j-th planes, and E Cas is an energy density of theirs interaction. Function Li 4 (x ) is defined as For identical defect planes (a 1 = a 2 = a) the force F 2P (r, a) between them is given by

Statement of problem for Dirac field
We will consider the material plane x 3 = 0 as a defect. In this case, in the Dirac part of the action the interaction of the spinor field with the plane is described with Since Ω(x 3 ) and δ(x 3 ) have the dimension of mass, the matrix Q is dimensionless. For homogeneous isotropic material plane in more general case, the matrix Q could be presented in the form: with I -identity 4x4 matrix, γ 3 , γ 5 = iγ 0 γ 1 γ 2 γ 3 are Dirac matrices.
Yury M.Pismak Dispersion relations and dynamic characteristics of bound states Movement of spinor particle in the field of defect Ω(x 3 ) is described by a modified Dirac equation It is one of the Euler-Lagrange equations, which is obtained by variational differentiating of the action over ψ(x ). Taking the derivative over ψ(x ) we obtain the second equation Let us introduce the convenient notations. For 2 × 2 -matrix M with elements M ij , i, j = 1, 2, we define the 4 × 4 -matrices M (+) , M (−) in the following way Let M (±) is the set of all the matrices M (±) , then for arbitrary matrices M

Yury M.Pismak
Dispersion relations and dynamic characteristics of bound states If we denote τ 0 the unit 2 × 2 -matrix, τ j , j = 1, 2, 3, the Pauli matrices, and τ the corresponding 4 × 4 -matrices, then the matrix Q can be presented as We denote ψ(x ) the solution of the modified Dirac equation, and The spinors ψ(x ) ± for x 3 = 0 satisfy the free Dirac equation and boundary condition lim One can choose the regularization procedure for δ(x 3 ) in such a way that the matrix S is expressed in terms of Q as Yury M.Pismak Dispersion relations and dynamic characteristics of bound states
Yury M.Pismak Dispersion relations and dynamic characteristics of bound states

Free Dirac equations
The free Dirac equation in coordinate space reads For real p 3 the considered spinor ψ(x ) describes the scattering state and the imaginary p 3 -the bound state.

The general solution ψ(p) of the Dirac equation can be presented as an arbitrary linear combination of linear independent spinors
for p 0 < 0.

Yury M.Pismak
Dispersion relations and dynamic characteristics of bound states Substituting p 3 → ± iκ with κ = |κ| = m 2 + p 2 1 + p 2 2 − p 2 0 we obtain the spinors describing the bound states They can be presented as follows The spinors ψ ± , ψ ± fulfill the relations which can be presented as systems of linear equations for coefficients a 1 , a 2 , d 1 , d 2 , a 1 , a 2 , d 1 , d 2 contained in ψ ± , ψ ± .
We consider the quantities of the form with 4 × 4 basic Dirac matrices Γ using convenient notations with totally antisymmetric ε jkl , ε 123 = 1.
For scalar and pseudoscalar invariant densities components of electric and axial 4-currents for anomalous electric and magnetic dipole moments we obtain the following results

Yury M.Pismak Dispersion relations and dynamic characteristics of bound states
Properties of bound states Yury M.Pismak Dispersion relations and dynamic characteristics of bound states We see also that The solvability condition (dispersion relation) can be presented as In virtue of p 2 0 + κ 2 − p 2 1 − p 2 2 − m 2 = 0 ,it follows from dispersion relation that p 0 , κ, m satisfy the equation describing the relation between the dimensionless magnitudes p o /m, κ/m characterizing the bound state. Its solution is presented on the (p 0 − κ)-plane by hyperbola or by two straight lines.
Since for Dirac particle the physical value of p 0 , κ are positive, the bound state can be realized if there are points of the (p 0 − κ)-plot presenting the dispersion relation in the region p 0 > 0, κ > 0. This part of plot can be connected or disconnected, and the possible values of p 0 , κ for bound state can be both restricted and non-restricted from above.
Yury M.Pismak Dispersion relations and dynamic characteristics of bound states

Dispersion relation
By ς 0± = ς 2± = 0 the dispersion law has the form It describe the propagation of massless particle in the defect plane with the Fermi-velocity v F . The motion of such particles explains numerous effects in graphene.
In the framework of the Symanzik approach, we build the model of QED field interaction with 2D material. The action of the model consist of the usual QED action and extra defect contribution. The action contains parameters, that characterize the material property.
The characteristics of photons and Dirac particles scattering on the defect plane can be calculated in the model, also the properties of states localized near the defect plane can be investigated.
The model and obtained on its basis results could be used for the theoretical description of the interaction of electrons, positrons and neutrons with two-dimensional materials (graphite, thin films, sputters, sharp boundaries of a solid body). Simple modifications of the model allows to take into account the effects of external electromagnetic fields.