Srednicki — Quantum Field Theory · Problem 18.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · renormalizability-dirac-fields

Problem

In any number

d

of spacetime dimensions, a Dirac field

\Psi_\alpha(x)

carries a spin index

\alpha

, and has a kinetic term of the form

i\overline{\Psi}\gamma^\mu \partial_\mu \Psi

, where we have suppressed the spin indices; the gamma matrices

\gamma^\mu

are dimensionless, and

\overline{\Psi} = \Psi^\dagger \gamma^0

. a) What is the mass dimension

[\Psi]

of the field

\Psi

? b) Consider interactions of the form

g_n(\overline{\Psi}\Psi)^n

, where

n \geq 2

is an integer. What is the mass dimension

[g_n]

g_n

? c) Consider interactions of the form

g_{m,n}\varphi^m(\overline{\Psi}\Psi)^n

, where

\varphi

is a scalar field, and

m \geq 1

and

n \geq 1

are integers. What is the mass dimension

[g_{m,n}]

g_{m,n}

? d) In

d = 4

spacetime dimensions, which of these interactions are allowed in a renormalizable theory?