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Fermion commutation relations

  1. Mar 28, 2015 #1
    1. The problem statement, all variables and given/known data
    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##.

    2. Relevant equations


    3. 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?
     
  2. jcsd
  3. Mar 29, 2015 #2
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