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Bethe-Salpeter equation with Spin-1 constituents In a quantum field theory we are able to define the two-particle Green’s function as the time ordered product of each of the fields given by • Where G is the two-particle Green’s function and all the fields are in the Heisenberg representation. These are the fully dressed fields with their self-interactions. The fields ‘a’ and/or ‘b’ can be spin-1/2, spin-0 or spin-1. • The original derivation (done by both Bethe & Salpeter and Schwinger as well) was done in 1951 for spin-1/2 particles which turns out to be the simplest case. • I shall show how this case relates to relativistic quantum mechanics. I then show the next generalization of it were one of the particles is a boson. In principle, in this case, a boson-boson bound states can be discussed, but we shall as an example, the boson fermion case. • From here, we consider the next generalization where new material introduced. That is, bound states in which one of the constituents is a spin-1 particle. Thus, the formalism supports consideration of a bound state of two spin-1 particles; a bound state of spin-1 particle and a boson as well as the case, where one particle is of spin-1 and the other, spin-1/2. • Where S-matrix takes us from the Heisenberg picture to the interaction picture. • Using this, we can write the two-particle Green’s function as: (the particle selfinteractions can be included in the interaction kernel, so for these considerations, we shall ignore the difference between free propagators and dressed ones). • Where the form of the interaction kernel can be determent by the S-matrix expansion. • Expanding the S-matrix occurring in the twoparticle Green’s function, we can write • This clearly a necessary condition. • The inhomogeneous term vanishes when we consider a bound-state as we are considering a localized system. The first term (the inhomogeneous term) takes the particles from –infinity to +infinity. A more mathematical argument can be made in terms of the analytic structure of the interaction kernel. • Which satisfies the above condition • Where |n> are intermediate bound states • Where the numbers correspond to the the coordinates • Where we have ignored the first term and repeated numbers are integrated over repeated numbers. • Where f^k(1,2) is the two-particle bound state. • Which is the Bethe-Salpeter equation for the interaction of two-fermions Where we can write • As a concrete example, we illustrate the formalism for the Coulomb interaction although, the formalism is completely general. • Which is the static Coulomb interaction • This separation is key to relating the equation to the relativistic quantum mechanics. The inverse of the last factor is the link while the other two-terms provide the ‘relativistic quantum recoil’ from the quantum field theory. • From a one-photon exchange The first term propagates positive energy particles forward in time and the second term, negative energy particles, backwards in time. • Upper component represents positive energy states while the lower one, negative energy. • Where one of the constituents is a boson and the other a fermion.Both components could be chosen to be bosons. • Which is the Bethe-Salpeter equation for fermion and a boson. • The term on the right contains the recoil contribution arising from the Bethe-Salpeter equation. Spin-1 Particles and the Bethe-Salpeter equation • The occurrence of charge spin-1 particles is evident by examining the electro-weak lagrangian • The first one, we have already encountered, the charged boson, while the second, is for a spin-1 particle • Which represents the W-wavefunction • • We could as well, considered the case, where the bound state consists of two-particles of spin-1 or the other case, where one of the particles is a boson and there other is a spin-1 particle.