Download Bethe-Salpeter Equation with Spin

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.