# Fermion commutation relations

## Homework Statement

Consider left-handed fermions in two spacetime dimensions ##(t,x)##: ##\psi_L=\frac{1}{2}(1-\gamma_5)\psi_D## with ##J_0^\epsilon(t,x)=\psi_L^+(x+\epsilon)\psi_L(x-\epsilon)##.

(a). Use canonical equal-time anti-commutation relations for fermions to compute
##[J_0^\epsilon(t,x),J_0^\epsilon(t,y)]##

(b). Take ##\langle 0 \mid\psi_L^+(t,x)\psi_L(t,y)\mid 0 \rangle=\frac{1}{x-y}## and evaluate
##\langle 0 \mid[J_0^\epsilon(t,x),J_0^\epsilon(t,y)]\mid 0 \rangle## and its limit for ##\epsilon \rightarrow 0##.

## The Attempt at a Solution

For part (a) I got ##[J_0^\epsilon(t,x),J_0^\epsilon(t,y)]=\delta^3(x-y-2\epsilon)\psi_L^+(x+\epsilon)\psi_L(y-\epsilon)-\delta^3(y-x-2\epsilon)\psi_L^+(y+\epsilon)\psi_L(x-\epsilon)##.

Using this expression means for (b) I get in the limit ##\epsilon \rightarrow 0##

##\langle 0\mid [J_0^\epsilon(t,x),J_0^\epsilon(t,y)]\mid 0 \rangle=\frac{2\delta^3(x-y)}{x-y}##

which seems a bit too easy. What's going wrong?