text: Q:
  
  Dirac spinor and field quantization
  
  Can we simplify  $$  Σ_s Σ_r [b_p^s u^s(p)\mathrm e^{ipx} (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} + (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} b_p^s u^s(p)\mathrm e^{ipx}]\tag{1}$$ as $$Σ_sΣ_r[ \{b_p^s, (b_q^r)^†\} u^s(p)(u^r)^†(q)\mathrm e^{i(p\cdot x-q\cdot y)}]~?$$
  If we can how do you justify rewriting $$(u^r)^†(q)  u^s(p)$$ as $$u^s(p)(u^r)^†(q)$$ in the second term in the sum of Equation 1?
  
  A:
  
  It depends on what you are doing with the hidden indices on the $u^s(p)$, which are really $u^s_\alpha(p)$   -- i.e vectors labeled by $s$ with components labeled by $\alpha$. If the LHS of your last equation is 
  $$
  u^s_\alpha(p) (u^r_\beta)^*
  $$ 
  then you can interchange them. If you mean the dagger to refer to matrix transpose as well, then you you have the usual issue that for vector $x$, $y$ the product $x^\dagger y$ is a number, but $y x^\dagger$ is matrix.  
  So the solution is to write out the hidden indices explicitly. 
  
  