The Jordan-Wigner transformation (JWT) is a method used in quantum mechanics to map spin operators, which are typically associated with spin-1/2 particles, to fermionic operators, which describe fermions such as electrons. This transformation is particularly useful in the study of one-dimensional quantum systems, such as spin chains and quantum wires.
We use JWT and write spin operators like creation and annihilations operators $$d_j = \exp(i\phi_j) S_j^- \\ d_j^{\dagger}= \exp(-i\phi_j) S_j^+$$ we have this: $$\phi_j = \pi \sum_{l}^{j-1} d_l^{\dagger} d_l.$$
Now, I want to find the value of this anti-commutation: $$ \left\{{d_i,d_j^{\dagger}}\right\} = \delta_{i,j}$$ I have wriiten $Q_j$ like this: $$Q_j = \exp(i \pi \sum_{l}^{j-1} d_l^{\dagger} d_l).$$ Using the exponential functions, I transformed the summation into multiplication: $$Q_j = \prod_{l=1} ^{j-1}\exp(i \pi d_l^{\dagger} d_l).$$ Now, because we haven't got summation, we can use Taylor Series: $$Q_j = \sum_{n=0}^{\inf} \frac{(i\pi)^n}{n!} d_l^{\dagger} d_l = 1 +(\exp(i\pi) -1) d_l^{\dagger} d_l = 1 -2d_l^{\dagger} d_l.$$ Now, I can Write the operators I want to transform it in fermionic operators like: $$d_j = \prod_{l=1} ^{j-1}(1 - 2d_l^{\dagger}d_l) S_j^-.$$ I'm uncertain whether I can write these for $d_l^\dagger$: $$d_j^{\dagger} = \prod_{l=1} ^{j-1}(-1)(1 - 2d_l^{\dagger}d_l) S_j^+.$$ Now, I don't know exactly how to find the anti-commutation relation for $$d_l , d^{\dagger}_l.$$ I don't know what I can do with $\prod$ in finding anti-commutation. I have tried and I didn't find $\delta_{ij}.$
