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 briefly


Introduction
In earlier papers [1,2], a relativistic wave equation for a scalar particle-antiparticle system, interacting via a mediating scalar field, was derived variationally for the scalar Yukawa model (which is also called the Wick-Cutkosky model [3][4][5][6]).We consider the complementary problem of two scalar particles with different masses in the present paper.The model quantum field theory being studied is defined by the Lagrangian density (h = c = 1) where φ 1 (x) and φ 2 (x) are the scalar fields corresponding to the particles of masses m 1 and m 2 respectively, while g 1 , g 2 , λ 1 , λ 2 are positive coupling constants.The mec J.Darewych diating "chion" field can be massive (µ = 0) or massless (µ = 0).The fields φ k and χ satisfy the Euler-Lagrange equations where ρ and the conjugates of (3).Equation ( 2) has the formal solution where dx = d N x dt in N + 1 dimensions, and χ 0 (x) satisfies the homogeneous (or free field) equation (equation ( 2) with ρ = 0), while D(x − x ′ ) is a covariant Green function (or chion propagator, in the terminology of QFT ), such that Substitution of the formal solution (4) into equation (3) yields the equations Equations ( 6) are derivable from the action principle δ dx L = 0, corresponding to the modified Lagrangian density (We suppress the Lagrangian density of the free chion field.) The QFTs based on ( 1) and ( 7) are equivalent in the sense that they lead to the same invariant matrix elements in various order of covariant perturbation theory.The difference is that, in the formulation based on (7), the interaction term, which contains the propagator leads to Feynman diagrams involving virtual chions, while the term that contains χ 0 corresponds to diagrams that cannot be generated using the term with D(x − x ′ ), such as those with external (physical) chion lines.
The Hamiltonian density corresponding to the Lagrangian (7) is given by where and To specify our notation, we quote the Fourier decomposition of the fields in N +1 dimension: where ω(p, m) = √ p 2 + m 2 , q • x = q ν x ν and q ν = (q 0 = ω(q, m k ), q), that is with k 2 = µ 2 .The momentum-space operators obey the usual commutation relations.The nonvanishing operators are These operators have the usual interpretation, namely that A † k are creation operators of the (free) scalar particles of mass m k (k = 1, 2), B † k are the corresponding antiparticle creation operators, while d † is the creation operator of the mediating-field quantum (which may be massive, µ > 0, or massless, µ = 0).
The Hamiltonian operator, Ĥ = d N x Ĥ(x), of the QFTheory is expressed in terms of the creation and the annihilation operators the usual way.Since we are not interested in vacuum-energy questions in this work, we commute these operators so that they stand in normal order in the Hamiltonian.

Two-particle trial state and variational equations
We seek approximate two-particle states variationally by evaluating the expectation value of the Hamiltonian operator of the QFT given in equation (8).The simplest possible two-particle trial state is where |0 is the vacuum state annihilated by all the annihilation operators, A k , B k , d, of the theory, and F (p 1 , p 2 ) is an adjustable function to be determined variationally.Note that the commutation properties of the operators, together with the definition of |ψ 2 , imply that We shall consider the simplified case with λ k = 0 in this paper.The relevant matrix elements needed to implement the variational principle are and where We have normal-ordered the entire Hamiltonian, since this circumvents the need for mass renormalization which would otherwise arise.Not that there is a difficulty with handling mass renormalization in the present formalism (as shown in various earlier papers; see, for example, [6] and citations therein).It is simply that mass renormalization has no effect on the two-body states that we obtain in this paper.Furthermore, the approximate trial state (18), which we use in this work, is incapable of sampling loop effects.
We now specialize to the rest frame of the two-particle system.The momentum operator of this quantum field theory is given by : P : = d N q q d † (q)d(q) + The requirement that : P : |ψ 2 = 0 implies that in the rest-frame of the two-particle system.Then, the matrix elements (19) and (21) reduce to and where p 2 (k) = m 2 k .We evaluate the matrix elements at t = 0, and choose f (p) in accordance with the variational principle whereupon we find that f (p) must be a solution of the momentum-space wave equation where the Lagrange multiplier E represents the total rest-frame energy of the twoparticle system, that is the total mass of a bound two-particle system.Note that the kernel (momentum-space potential) in this equation contains terms corresponding to one-chion exchange (this is perhaps more obvious from the manifestly covariant terms in equation ( 25)).