How Do Fermion Commutation Relations Affect Current Operators in 2D Spacetime?

[Maybe_Memorie]

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 $\epsilon \rightarrow 0$.

Homework Equations

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?

 

[Maybe_Memorie]

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