Unitary Clothing Transformations Research Articles

A New Family of Interactions between Clothed Particles in QED

The method of unitary clothing transformations (UCTs) has been applied to the quantum electrodynamics (QED) by using the clothed particle representation (CPR). Within CPR, the Hamiltonian for interacting electromagnetic and electron-positron fields takes the form in which the interaction operators responsible for such two-particle processes as e−e− → e−e−, e+e+ → e+e+, e−e+ → e−e+, e−e+ → yy, yy → e−e+, ye− → ye−, and ye+ → ye+ are obtained on the same physical footing. These novel interactions include the off-energy-shell and recoil effects (the latter without any expansion in (v/c)2-series) and their on-energy shell matrix elements reproduce the well-known results derived within the perturbation theory based on the Dyson expansion for the S-matrix (in particular, the Møller formula for the e−e−-scattering, the Bhabha formula for e−e+-scattering, and the Klein–Nishina one for the Compton scattering).

Possible Extension of the Three-body Force by the Unitary Clothing Transformations Method in the Faddeev Equations

Possible Extension of the Three-body Force by the Unitary Clothing Transformations Method in the Faddeev Equations

QED in the clothed-particle representation: a fresh look at positronium properties treatment

We have extended our previous applications of the method of unitary clothing transformations (UCTs) in mesodynamics [1,2] to quantum electrodynamics (QED) [3,4]. An analytical expression for the QED Hamiltonian in the clothed-particle representation (CPR) has been derived. Its distinctive feature is the appearance of a new family of the Hermitian and energy independent interaction operators built up in the e^2e2-order for the clothed electrons and positrons instead the primary canonical interaction between electromagnetic and electron-positron fields. The problem of describing the bound states in QED in case of the positronium system has been considered. The first correction to the energy of the ground state of the para-positronium and its decay rate to two photons has been calculated by using the new interaction operators.

From Quantum Field Theory to the Contemporary Quantum Mechanics

In this talk we remind how the notion of the so-called clothed particles, put forward in relativistic quantum field theory by Greenberg and Schweber, can be used via the method of unitary clothing transformations (shortly, the UCT method) when finding the eigenstates of the total Hamiltonian H in case of interacting fields with the Yukawa - type couplings. In general, the UCT method is aimed at reduction of the exact eigenvalue problem in the primary Fock space to the model-space problems in the corresponding Hilbert spaces of the contemporary quantum mechanics. In this context we consider an approximate treatment of the physical vacuum, the observable one-particle and two-particle bound and scattering states.

Triton Binding Energy of Kharkov Potential

The Kharkov potential is a recent field theoretical model of nucleon–nucleon (NN) interaction that has been built up in the framework of the instant form of relativistic dynamics starting with the total Hamiltonian of interacting meson and nucleon fields and using the method of unitary clothing transformations. The latter connect the representation of “bare” particles and the representation of “clothed” particles, i.e., the particles with physical properties. Unlike many available NN potentials each of which is the kernel of the corresponding nonrelativistic Lippmann–Schwinger (LS) equation this potential being dependent in momentum space on the Feynman-like propagators and covariant cutoff factors at the meson–nucleon vertices is the kernel of relativistic integral equations for the NN bound and scattering states. Therefore we do not need to invent any transform of a given nonrelativistic potential to its relativistic counterpart. As a feasible study, we have started with the so-called 5ch Faddeev calculation for three-nucleon bound state (triton) and obtained a reasonable value of its binding energy ( $$-7.42$$ MeV).

Low-Energy Parameters of the Nucleon–Nucleon Scattering and Deuteron Properties, Electromagnetic Interactions with Bound Systems

One more application of the method of unitary clothing transformations (UCT’s) in the theory of nucleon–nucleon (N − N) interaction has been presented. We have extended our previous analysis (Dubovik and Shebeko in Few-Body Syst 48:109–142, 2010) of the N − N scattering below the pion production threshold to treat the neutron–proton (n − p) scattering at low energies and the deuteron static properties. Our calculations of deuteron magnetic and quadrupole moments have been carried out in the framework of a gauge independent description of electromagnetic (EM) interactions with nuclei (bound systems) using the clothed particle representation of the Hamiltonian, the boost and EM current density operators for the n–p system.

The Method of Unitary Clothing Transformations in Relativistic Quantum Field Theory: Recent Applications for the Description of Nucleon–Nucleon Scattering and Deuteron Properties

The clothing procedure, put forward in quantum field theory by Greenberg and Schweber, is applied for the description of nucleon–nucleon (N–N) scattering below the pion production threshold and deuteron properties. We consider pseudoscalar (π and η), vector (ρ and ω) and scalar (δ and σ) meson fields interacting with N and \({\bar N}\) ones via the Yukawa-type couplings to introduce trial interactions between “bare” particles. The subsequent unitary clothing transformations (UCTs) are found to express the total Hamiltonian through new interaction operators that refer to particles with physical (observable) properties, the so-called clothed particles. The corresponding analytic expressions in momentum space are compared with the separate meson contributions to the one-boson-exchange potentials in the meson theory of nuclear forces. We will also show a worked example where the UCTs method is used in the framework of a gauge-independent field-theoretical treatment of electromagnetic interactions of deuterons (bound systems).

The Method of Unitary Clothing Transformations in the Theory of Nucleon–Nucleon Scattering

The clothing procedure, put forward in quantum field theory (QFT) by Greenberg and Schweber, is applied for the description of nucleon–nucleon (N–N) scattering. We consider pseudoscalar (π and η), vector (ρ and ω) and scalar (δ and σ) meson fields interacting with 1/2 spin (N and $${\bar{N}}$$ ) fermion ones via the Yukawa-type couplings to introduce trial interactions between “bare” particles. The subsequent unitary clothing transformations are found to express the total Hamiltonian through new interaction operators that refer to particles with physical (observable) properties, the so-called clothed particles. In this work, we are focused upon the Hermitian and energy-independent operators for the clothed nucleons, being built up in the second order in the coupling constants. The corresponding analytic expressions in momentum space are compared with the separate meson contributions to the one-boson-exchange potentials in the meson theory of nuclear forces. In order to evaluate the T matrix of the N–N scattering we have used an equivalence theorem that enables us to operate in the clothed particle representation (CPR) instead of the bare particle representation with its large amount of virtual processes. We have derived the Lippmann–Schwinger type equation for the CPR elements of the T-matrix for a given collision energy in the two-nucleon sector of the Hilbert space $${\mathcal{H}}$$ of hadronic states.

The method of unitary clothing transformations in the theory of nucleon–nucleon scattering

The clothing procedure, put forward in quantum field theory by Greenberg and Schweber, is applied for the description of nucleon-nucleon (N-N) scattering. We consider pseudoscalar, vector and scalar meson fields interacting with 1/2 spin fermion ones via the Yukawa-type couplings to introduce trial interactions between "bare" particles. The subsequent unitary clothing transformations are found to express the total Hamiltonian through new interaction operators that refer to particles with physical (observable) properties, the so-called clothed particles. In this work, we are focused upon the Hermitian and energy-independent operators for the clothed nucleons, being built up in the second order in the coupling constants. The corresponding analytic expressions in momentum space are compared with the separate meson contributions to the one-boson-exchange potentials in the meson theory of nuclear forces. In order to evaluate the T-matrix of the N-N scattering we have used an equivalence theorem that enables us to operate in the clothed particle representation (CPR) instead of the bare particle representation with its huge amount of virtual processes. We have derived the Lippmann-Schwinger-type equation for the CPR elements of the T-matrix for a given collision energy in the two-nucleon sector of the Hilbert space of hadronic states and elaborated a code for its numerical solution in momentum space.

Relativistic interactions for meson-nucleon systems in the clothed particle representation

The method of unitary clothing transformations (see [1–3] and references therein) enables one to represent the initial Hamiltonian H for interacting fields in terms of the creation (destruction) operators for the so-called clothed (physical) particles and their relativistic interactions. Using this clothed particle representation (CPR) of H and starting from the Yukawa-type couplings between fermions (nucleons and antinucleons) and bosons (π-, η-, ρ-, ω-mesons, etc.), we have derived a new family of the Hermitian and energy-independent interactions for physical processes πN → πN, NN → NN, NN ↔ πNN and NNN → NNN. Each of the obtained analytic expressions in momentum space is the sum of a Feynman-like part and some off-energy-shell correction term. The latter cannot appear when calculating the corresponding S-matrix elements by Feynman rules.

Clothing of particles in a meson-nucleon system: Operators of relativistic interactions. Diagram technique

A diagram technique is represented, which makes it possible to construct operators of relativistic interactions in the method of unitary clothing transformations. With the use of this technique, operators of physical processes in the second and third orders with respect to the coupling constant have been obtained.

Clothing of particles in the meson-nucleon system: Mass and charge renormalization

Generators of the Poincare group have been constructed in the instant form of relativistic dynamics in the second and third orders in the coupling constant, using the method of unitary clothing transformations. It is shown that the algebra of group generators is satisfied in the corresponding orders. The relativistic invariance of the calculated corrections to the particle masses and coupling constant is substantiated.

Clothed particle representation in quantum field theory: mass renormalization

The method of unitary clothing transformations is used to handling the so-called clothed particle representation (CPR) (see [A.V. Shebeko and M.I. Shirokov, Phys. Part. Nucl. 32 (2001) 31; nucl-th/0102037 , V.Yu. Korda and A.V. Shebeko, Phys. Rev. D 70 (2004) 085011, V.Yu. Korda, L. Canton and A.V. Shebeko, doi:10.1016/j.aop.2006.07.010 , Ann. Phys. (2006) in press; nucl-th/060325 ] and refs. therein), where the total field Hamiltonian H and the three boost operators in the instant form of relativistic dynamics take on the same sparse structure in the Hilbert space of hadronic states. In this approach the mass counterterms are cancelled by commutators of the generators of clothing transformations and the field interaction operator. This allows the pion and nucleon mass shifts to be expressed through the corresponding three-dimensional integrals whose integrands are proved to be dependent on certain covariant combinations of the relevant three-momenta. The property provides the momentum independence of mass renormalization.

Relativistic interactions for the meson-two-nucleon system in the clothed-particle unitary representation

The method of unitary clothing transformations put forward in relativistic quantum field theory (QFT) by Greenberg and Schweber and developed by Shirokov is applied to construct a new family of interactions in the meson-two-nucleon system. Along with a brief exposition of its basic elements we show a specific transition from the initial “bare” one-meson and one-nucleon operators and states to their physical “clothed” counterparts. We emphasize that the clothing transformations in question do not alter the original total Hamiltonian, but provides a conceptually more transparent representation of the same Hamiltonian in terms of a new set of operators for particles with physical properties and their relativistic interactions. The Hermitian and energy-independent interaction operators for the processes π N → π N, NN → NN and NN ↔ π NN are derived starting from the Yukawa-type couplings between fermions (nucleons and antinucleons) and bosons (π−, η−, ρ−, ω− mesons, etc.). These types of interaction have a distinctive off-energy-shell structure which is naturally generated by the unitary transformation that removes from the Hamiltonian the (three-leg) π NN vertex coupling.

Clothed particle representation in quantum field theory: Mass renormalization

We consider the neutral pion and nucleon fields interacting via the pseudoscalar (PS) Yukawa-type coupling. The method of unitary clothing transformations is used to handle the so-called clothed particle representation, where the total field Hamiltonian and the three boost operators in the instant form of relativistic dynamics take on the same sparse structure in the Hilbert space of hadronic states. In this approach the mass counterterms are cancelled (at least, partly) by commutators of the generators of clothing transformations and the field interaction operator. This allows the pion and nucleon mass shifts to be expressed through the corresponding three-dimensional integrals whose integrands depend on certain covariant combinations of the relevant three-momenta. The property provides the momentum independence of mass renormalization. The present results prove to be equivalent to the results obtained by Feynman techniques.

An extended Lee model with indefinite metric

The question on the existence of a physical Hilbert subspace in a field theory with indefinite metric is investigated in an extended Lee model. The interaction Hamiltonian of the original Lee model is generalized to contain an infinite set of terms representing the basic interaction processes, V+nθ↔N+(n+1)θ,n=1, 2, 3, ..., besides the usual trilinear term. The bare field operators are transformed into «clothed» field operators according to the unitary clothing transformations of Greenberg and Schweber, so that the conditions for separating out the «ghost» states are easi y sought. It is found that a physical Hilbert subspace exists in this model provided that the cut-off functions introduced have certain special forms.

Clothed particle operators in simple models of quantum field theory

The idea of using clothed operators in the quantum theory of fields, i.e. operators which create and annihilate one particle eigenstates of the total Hamiltonian, is examined. In cut-off theories, these clothed operators are related to the unrenormalized Schrodinger picture operators by a similarity transformation. In terms of these clothed operators the virtual cloud of particles surrounding a physical particle appearing in present discussions of field theory no longer appears, corresponding to the elimination, in the terminology of Van Hove, of persistent effects. Mass and wave function renormalization are automatically accomplished by unitary clothing transformations, and the usual renormalization program is extended to the renormalization of the states appearing in the theory. The form of the Hamiltonian expressed in terms of clothed operators is discussed and these ideas are applied to the static scalar model, the Lee model, and the Ruijgrok-Van Hove model.