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 self- interactions 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 two- particle 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.