A Variational Fock-Space Treatment of Quarkonium

The variational method and the Hamiltonian formalism of QCD are used to derive relativistic, momentum space integral equations for a quark-antiquark system with an arbitrary number of gluons present. As a first step, the resulting infinite chain of coupled equations is solved in the nonrelativistic limit by an approximate decoupling method. Comparison with experiment allows us to fix the quark mass and coupling constant, allowing for the calculation of the spectra of massive systems such as charmonium and bottomonium. Studying the results with and without the nonAbelian terms, we find that the presence of the nonAbelian factors yields better agreement with the experimental spectra.Comment: TEX, no figure

Variational Derivation of Relativistic Fermion-Antifermion Wave Equations in QED

We present a variational method for deriving relativistic two-fermion wave equations in a Hamiltonian formulation of QED. A reformulation of QED is performed, in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. The reformulation permits one to use a simple Fock-space variational trial state to derive relativistic fermion-antifermion wave equations from the corresponding quantum field theory. We verify that the energy eigenvalues obtained from the wave equation agree with known results for positronium.Comment: 25 pages, accepted in Journal of Mathematical Physics (2004

Exact two-particle eigenstates in partially reduced QED

We consider a reformulation of QED in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. It is shown that exact few-fermion eigenstates of the resulting Hamiltonian can be obtained in the canonical equal-time formalism for the case where there are no free photons. These eigenstates lead to two- and three-body Dirac-like equations with electromagnetic interactions. Perturbative and some numerical solutions of the two-body equations are presented for positronium and muonium-like systems, for various strengths of the coupling.Comment: 33 pages, LaTex 2.09, 4 figures in EPS forma

Exact spinor-scalar bound states in a QFT with scalar interactions

We study two-particle systems in a model quantum field theory, in which scalar particles and spinor particles interact via a mediating scalar field. The Lagrangian of the model is reformulated by using covariant Green's functions to solve for the mediating field in terms of the particle fields. This results in a Hamiltonian in which the mediating-field propagator appears directly in the interaction term. It is shown that exact two-particle eigenstates of the Hamiltonian can be determined. The resulting relativistic fermion-boson equation is shown to have Dirac and Klein-Gordon one-particle limits. Analytic solutions for the bound state energy spectrum are obtained for the case of massless mediating fields.Comment: 12 pages, RevTeX, 1 figur

Confinement interaction in nonlinear generalizations of the Wick-Cutkosky model

We consider nonlinear-mediating-field generalizations of the Wick-Cutkosky model. Using an iterative approach and eliminating the mediating field by means of the covariant Green function we arrive at a Lagrangian density containing many-point time-nonlocal interaction terms. In low-order approximations of $\phi^3{+}\phi^4$ theory we obtain the usual two-current interaction as well as a three-current interaction of a confining type. The same result is obtained without approximation for a version of the dipole model. The transition to the Hamiltonian formalism and subsequent canonical quantization is performed with time non-locality taken into account approximately. A relativistic three-particle wave equation is derived variationally by using a three-particle Fock space trial state. The non-relativistic limit of this equation is obtained and its properties are analyzed and discussed.Comment: 15 pages, 1 figure, LaTe

Bound-State Variational Wave Equation For Fermion Systems In QED

We present a formulation of the Hamiltonian variational method for QED which enables the derivation of relativistic few-fermion wave equation that can account, at least in principle, for interactions to any order of the coupling constant. We derive a relativistic two-fermion wave equation using this approach. The interaction kernel of the equation is shown to be the generalized invariant M-matrix including all orders of Feynman diagrams. The result is obtained rigorously from the underlying QFT for arbitrary mass ratio of the two fermions. Our approach is based on three key points: a reformulation of QED, the variational method, and adiabatic hypothesis. As an application we calculate the one-loop contribution of radiative corrections to the two-fermion binding energy for singlet states with arbitrary principal quantum number $n$, and $l =J=0$. Our calculations are carried out in the explicitly covariant Feynman gauge.Comment: 26 page

Variational Two Fermion Wave Equations in QED: Muonium Like Systems

We consider a reformulation of QED in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. A simple Fock-state variational trial function is used to derive relativistic two-fermion equations variationally from the expectation value of the Hamiltonian of the field theory. The interaction kernel of the equation is shown to be, in essence, the invariant M-matrix in lowest order. Solutions of the two-body equations are presented for muonium like system for small coupling strengths. The results compare well with the observed muonium spectrum, as well as that for hydrogen and muonic hydrogen. Anomalous magnetic moment effects are discussed

Variational two-particle wave equation in scalar quantum field theory

We study two-particle systems in a model quantum field theory, in which scalar particles of different mass interact via a mediating scalar field. The Lagrangian of the model is reformulated using covariant Green's functions to solve for the mediating field in terms of the particle fields. This results in a Hamiltonian in which the mediating-field propagator appears directly in the interaction term. The variational method, with a simple Fock-state trial state, is used to derive a relativistic momentum-space two-particle wave equation. Non-relativistic and one-particle limits of the equation are determined and discussed brieflyМи вивчаємо двочастинкові системи в модельній квантовій теорії поля, в якій скалярні частинки різної маси взаємодіють через посередкове скалярне поле. Використовуючи коваріантні функції Ґріна для розв’язку посередкового поля в термінах частинкових полів, переформульовано ляґранжіян моделі. В результаті в гамільтоніяні виникає пропагатор посередкового поля прямо в члені, який описує взаємодію. Варіяційна метода з простим пробним фоковим станом використовується для того, щоб вивести релятивістичне двочастинкове хвильове рівняння в імпульс-просторі. Одержуються і обговорюються нерелятивістичні і одночастинкові границі цього рівняння

FEW-PARTICLE EIGENSTATES IN THE YUKAWA MODEL

Розглядається переформулювання моделі Юкави, у якій ферміони взаємодіють через посередництво (масивного або безмасового) скалярного поля. За допомогою коваріянтних функцій Ґріна поле, що переносить взаємодію, виражено у термінах ферміонних полів. Результуючий гамільтоніян теорії містить член, що описує взаємодію, у який явно входить пропагатор поля — носія взаємодії. Показано, що коли можна знехтувати процесами, що включають випромінювання фізичних квантів поля-носія, то з використанням нестандартного означення вакуумного стану можна отримати у рамках канонічного одночасового формалізму точні власні стани результуючого гамільтоніяну для декількох ферміонів. Ці власні стани приводять до двота тричастинкових рівнянь Діракового типу із скалярними взаємодіями. Числовим розв’язуванням рівняння на власні значення отримано двоферміонні зв’язані стани для J = 0 . Проведено порівняння із стандартним розглядом цієї моделі.We consider a reformulation of the Yukawa model, in which fermions interact via a mediating (massive or massless) scalar field. Covariant Green functions are used express the mediating field in terms of the fermion fields. The resulting Hamiltonian of the theory has an interaction term in which the propagator of the mediating field appears directly. We show that if processes involving emission of physical mediating field quanta can be ignored and an unconventional empty vacuum is used, then exact few- fermion eigenstates of the resulting truncated Hamiltonian can be obtained in the canonical equal-time formalism. These eigenstates lead to two- and three-body Dirac-like equations with scalar interactions. Two-fermion bound states are obtained by the numerical solution of the eigenvalue equation for J = 0 states. Comparison is made with conventional treatments of this model