Understanding the Parity Operator in Dirac Field Theory

[Silviu]

Hello! I am a bit confused about matrices dimensions in the second quantization of the Dirac field. The book I am using is "An Introduction to Quantum Field Theory" by Peskin and Schroder and I will focus in this question mainly on the Parity operator which is section 3.6. The field operator (one of them) is written as: $\psi(x) = \int{\frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}\sum_s(a_p^su^s(p)e^{-ipx}+b_p^{s\dagger}\bar{u}^s(p)e^{ipx})}$. $b_p^{s\dagger}$ acting on vacuum creates an antifermion with momentum p, while $a_p^s$ destroys a fermion state. $u^s(p)$ is a column vector. Now, they are trying to find a $4\times4$ matrix of the parity operator $P$. They say $Pa_p^sP=\eta_\alpha a_{-p}^s$ where $\eta$ is a possible phase. Based on this, I conclude that $a_p^s$ is a $4x4$ matrix, so that the matrix multiplication makes sense (same for b's). Then they calculate $P\psi(x)P$ and get it equal to $P\psi(x)P = \int{\frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}\sum_s(\eta_\alpha a_{-p}^su^s(p)e^{-ipx}+\eta_b^* b_{-p}^{s\dagger}\bar{u}^s(p)e^{ipx})}$. This implies that $\psi(x)$ is also a $4\times4$ matrix but here I get confused. Inside the integral, we have terms of the form $a_{p}^su^s(p)$ based on what I said above this should lead to a $(4\times4)\times(4\times1)=4\times1$ column vector, so the whole integral would be a column vector, but it is equal to $\psi(x)$ which is a $4\times4$ matrix, which doesn't make sense to me. Also, when they multiply $P$ on both sides, you would have inside the integral terms of the form $Pa_{p}^su^s(p)P$ which wouldn't make sense because of the column vector $u^s(p)$, but somehow they just move P to the left of it and solve the problem. So I don't really understand how they do this. I am obviously missing something. Can someone explain this to me please? Thank you!

 

[stevendaryl]

Hello! I am a bit confused about matrices dimensions in the second quantization of the Dirac field. The book I am using is "An Introduction to Quantum Field Theory" by Peskin and Schroder and I will focus in this question mainly on the Parity operator which is section 3.6. The field operator (one of them) is written as: $\psi(x) = \int{\frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}\sum_s(a_p^su^s(p)e^{-ipx}+b_p^{s\dagger}\bar{u}^s(p)e^{ipx})}$. $b_p^{s\dagger}$ acting on vacuum creates an antifermion with momentum p, while $a_p^s$ destroys a fermion state. $u^s(p)$ is a column vector. Now, they are trying to find a $4\times4$ matrix of the parity operator $P$. They say $Pa_p^sP=\eta_\alpha a_{-p}^s$ where $\eta$ is a possible phase. Based on this, I conclude that $a_p^s$ is a $4x4$ matrix, so that the matrix multiplication makes sense (same for b's). Then they calculate $P\psi(x)P$ and get it equal to $P\psi(x)P = \int{\frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}\sum_s(\eta_\alpha a_{-p}^su^s(p)e^{-ipx}+\eta_b^* b_{-p}^{s\dagger}\bar{u}^s(p)e^{ipx})}$. This implies that $\psi(x)$ is also a $4\times4$ matrix but here I get confused. Inside the integral, we have terms of the form $a_{p}^su^s(p)$ based on what I said above this should lead to a $(4\times4)\times(4\times1)=4\times1$ column vector, so the whole integral would be a column vector, but it is equal to $\psi(x)$ which is a $4\times4$ matrix, which doesn't make sense to me. Also, when they multiply $P$ on both sides, you would have inside the integral terms of the form $Pa_{p}^su^s(p)P$ which wouldn't make sense because of the column vector $u^s(p)$, but somehow they just move P to the left of it and solve the problem. So I don't really understand how they do this. I am obviously missing something. Can someone explain this to me please? Thank you!

No. The as and bs are not matrices. They are just abstract operators that raise and lower the number of particles.