Dirac Field quantization and anti-commutator relation

  • #1
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Main Question or Discussion Point

Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to
$$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$

My question is how the above mentioned terms can be written as
$$Σ_sΣ_r[ \{b_p^s, b_q^r †\} u^s(p)u^r†(q)e^{i.(p.x-q.y)}]$$.And why not
$$Σ_sΣ_r[ \{b_p^s, b_q^r †\} u^r†(q)u^s(p)e^{i.(p.x-q.y)}]$$
As per my knowledge $$u^s(p)u^r†(q)!= u^r†(q) u^s(p)$$
please see equation 5.6 in david tong notes. $s$ and $r$ are spinor index and summed over 1-2.
http://www.damtp.cam.ac.uk/user/tong/qft/five.pdf
 

Answers and Replies

  • #2
vanhees71
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The u's are spinors (usually written as 4d columns). So for sure the last equation doesn't make sense, because
##u^s(p) u^{r \dagger}(q)## is ##4 \times 4##-matrix while ##u^{r \dagger}(q) u^s(p)## is a complex number.
 

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