Understanding the Parity Operator in Dirac Field Theory

In summary, the conversation is about matrices dimensions in the second quantization of the Dirac field. The field operator is written as a linear combination of operators that create and destroy particles with momentum p. The book is trying to find a 4x4 matrix for the Parity operator, and they use a possible phase to make the matrix multiplication make sense. However, there is confusion about the dimensions of the operators and how they are being multiplied.
  • #1
Silviu
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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!
 
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  • #2
Silviu said:
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.
 
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Related to Understanding the Parity Operator in Dirac Field Theory

1. What is the Dirac operator?

The Dirac operator is a mathematical operator in the field of quantum mechanics that was first introduced by physicist Paul Dirac in 1928. It is used to describe the behavior of fermions, which are particles that have half-integer spin, such as electrons and quarks.

2. How does the Dirac operator work?

The Dirac operator is a linear operator that acts on a wave function, representing the state of a particle. It takes into account both the position and momentum of the particle, as well as the spin of the particle. The operator is a matrix of four components and is represented by a combination of partial derivative and matrix multiplication operations.

3. What is the significance of the Dirac operator in physics?

The Dirac operator is a fundamental concept in the field of quantum mechanics and is essential in describing the behavior of fermions. It is used in various areas of physics, such as particle physics, quantum field theory, and condensed matter physics. The operator also plays a crucial role in the development of quantum computing and quantum information theory.

4. How is the Dirac operator related to the Dirac equation?

The Dirac operator is the mathematical representation of the Dirac equation, which is a relativistic wave equation that describes the behavior of fermions. The Dirac equation incorporates special relativity and quantum mechanics, and its solutions provide a complete description of the properties of fermions.

5. What are some applications of the Dirac operator?

The Dirac operator has numerous applications in physics, particularly in the study of particle physics and quantum field theory. It is also used in condensed matter physics to understand the behavior of electrons in solids. Additionally, the operator is used in the development of quantum algorithms and quantum error correction codes for quantum computing.

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