Interacting Fermion System Commutation

[Xyius]

Problem Question
My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple.

The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class for this problem and the part I am stuck on is this.

$$<0,0|a_{2 \uparrow} a_{1 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}a_{1 \uparrow}a_{2 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=-1$$

My question is, why is this equal to -1?

Attempt at Solution
Here is my logic on how to evaluate this.

$$a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>$$
then
$$a_{1 \uparrow}a_{2 \uparrow}|\uparrow \uparrow>=|0,0>$$
then
$$a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>$$
finally
$$a_{2 \uparrow} a_{1 \uparrow}|\uparrow \uparrow>=|0,0>$$
Thus all together I get 1, not -1. Why is this equal to -1?

 

[TSny]

Here is my logic on how to evaluate this.

$$a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>$$
then
$$a_{1 \uparrow}a_{2 \uparrow}|\uparrow \uparrow>=|0,0>$$

I believe the mistake is in the second equation above. There are sign conventions for Fermion operators operating on Fock states.

See equations 4, 5, and 6 here: http://www.phys.ufl.edu/~kevin/teaching/6646/03spring/2nd-quant.pdf

 

[Xyius]

I believe the mistake is in the second equation above. There are sign conventions for Fermion operators operating on Fock states.

See equations 4, 5, and 6 here: http://www.phys.ufl.edu/~kevin/teaching/6646/03spring/2nd-quant.pdf

Thank you! I got it!