Charge conjugation in second quantization

• LayMuon
In summary, under charge conjugation, the current operator reverses sign and is represented by a unitary operator ℂ. The matrix C, which acts on spinor indices, has the property CγμC-1 = - (γμ)T. Therefore, under charge conjugation, ℂ(ψγμψ)ℂ-1 = - ψγμψ.

LayMuon

We know that under charge conjugation the current operator reverses the sign:

$$\hat{C} \hat{\bar{\Psi}} \gamma^{\mu} \hat{\Psi} \hat{C} = - \hat{\bar{\Psi}} \gamma^\mu \hat{\Psi}$$

Here $\hat{C}$ is the unitary charge conjugation operator. I was wondering should we consider gamma matrix here as also an entity undergoing transformation (like when we prove form-covariance of Dirac equation under any unitary transformation): $\hat{C} \gamma^{\mu} \hat{C} = \gamma^{\prime \mu}$? Or gamma matrix is something of a structure ensuring element and should not be changed?

(Forgive me for writing ψ to mean the adjoint.)

In second quantization, charge conjugation is represented by a unitary operator ℂ. Associated with it is a 4 x 4 matrix C that acts on the spinor indices. According to Bjorken and Drell vol 2, the action is

ℂψℂ-1 = C-1ψT
ψ-1 = - ψTC

where the matrix C has the property

μC-1 = - (γμ)T

From this,

ℂ(ψγμψ)ℂ-1 = - (ψTC)γμ(C-1ψT) = + ψTμ)TψT = + (ψγμψ)T = - ψγμψ.

Thank you, Bill. But I still have some points to think about.

1. What is charge conjugation in second quantization?

Charge conjugation in second quantization is a mathematical operation that reverses the sign of all the charges in a quantum system. It is commonly used in particle physics and quantum field theory to study the behavior of particles and their interactions.

2. Why is charge conjugation important in second quantization?

Charge conjugation is important in second quantization because it allows us to describe the symmetries and properties of quantum systems. It is particularly useful in studying particles that have both positive and negative charges, such as electrons and positrons.

3. How is charge conjugation represented in second quantization?

In second quantization, charge conjugation is represented by the C operator, which acts on quantum states and reverses the sign of all the charges. This operator is used in the mathematical formulation of many physical theories, including quantum electrodynamics and the Standard Model.

4. Does charge conjugation always conserve charge?

No, charge conjugation does not always conserve charge. While it reverses the sign of all the charges, it does not change the total amount of charge in a system. Therefore, in some cases, charge conjugation may not conserve charge, but it still remains an important symmetry in quantum systems.

5. What are some practical applications of charge conjugation in second quantization?

Charge conjugation has many practical applications in physics, particularly in studying the properties of particles and their interactions. For example, it is used in calculating scattering amplitudes and cross-sections of particle collisions, as well as in understanding the behavior of antimatter in quantum systems.