# Behavior of propagator of quantum field as function of spin

Can we write down a generic expression for propagator for any arbitrary spin? At the least, about the ultraviolet behavior of these propagator. Especially, I would like to know whether there is a specific dependence on spin of the field.

• @HolgerFiedler I meant a generic quantum field of arbitrary spin. Let us also suppose that it is massive. – Truthseeker Aug 14 '16 at 10:34

It is possible to write an expression for the propagator of a massive spin $j$ particle whose ultraviolet limit can be shown to vary as $q^{-2+2j}$.
To see why this holds one has to find a natural generalization of Pauli matrices for $(j,0)$ and $(0,j)$ and of gamma matrices for the one where you deal with parity conserving interaction which involves fields transforming as $(j,0)\oplus (0,j)$ representation the details to which can be found in the reference mentioned.
Momentum space propagator of a particle of spin $j$ (eqn. 5.13 of reference which need not be mentioned in the context) has the structure involving a matrix $\Pi(q)$ written in terms of $2j+1$ dimensional matrix of finite Lorentz transformation $D^j[L (\bf {p})]$ where $L (\bf{ p})$ is lorentz boost (eqn. 4.3 of reference) in the numerator. Explicit calculation of matrix $\Pi(q)$ is given in the appendix of the reference.
In ultraviolet limit, the propagator behaves as $\Pi(q)/q^2$ where the value of monomial $\Pi(q)$ is given in table 1 of the reference. From the table it is easily seen that the monomial’s leading term in ultraviolet region goes as $q^0,q,q^2,q^3$ and so on for respectively $0,1/2,1,3/2$ spin particle and hence contributes a factor of $q^{2j}$ and in total propagator behaves as $q^{-2+2j}$ for a spin $j$ particle.
The case of $(j,0)\oplus (0,j)$ yields similar behavior for massive particle. This analysis actually breaks down for massless particles because of the non semi-simple structure of the little group associated with $m=0$ case and there is not a definite relation between massless particles propagators and the spin as photon and graviton both will behave as $j=0$ in the above relation.
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