Charge conjugation in Dirac equation

In summary, the mathematical argument shows that the relation $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T}$ is significant in charge conjugation. This can be seen from the complex conjugate of the Dirac equation multiplied by a non-singular matrix $\mathcal{C}$, where if $\mathcal{C} (\gamma^{\mu})^{*} \mathcal{C}^{-1} = - \gamma^{\mu}$, then the field $\psi_{c}\equiv \mathcal{C}\psi$ describes a Dirac particle with opposite charge. This relation can also be derived by setting $\mathcal{C} = C \gamma^{
  • #1
forhad_jnu
2
0
I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ .
Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix and $T$= transposition.


I need to know the significance of these equation in charge conjuration .
 
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  • #2
Start with the Dirac equation
[tex]( i \gamma^{\mu}\partial_{\mu} + e \gamma^{\mu}A_{\mu} - m) \psi = 0[/tex]
Now take the complex conjugate of that and multiply from the left by some non-singular matrix [itex]\mathcal{C}[/itex], you then can write
[tex][(i \partial_{\mu} - e A_{\mu})\ \mathcal{C}\ (\gamma^{\mu})^{*} \mathcal{C}^{-1} + m ] \ \mathcal{C}\psi = 0[/tex]
Thus, if
[tex]\mathcal{C}\ (\gamma^{\mu})^{*} \mathcal{C}^{-1} = - \gamma^{\mu},[/tex]
then the field
[tex]\psi_{c}\equiv \mathcal{C}\psi[/tex]
describes another Dirac particle with opposite charge.
Now, if you write
[tex]\mathcal{C} = C \gamma^{0}[/tex]
then your relation follows in the representation where
[tex]\gamma^{0} = ( \gamma^{0})^{T} = ( \gamma^{0})^{-1}.[/tex]

Sam
 
  • #3
One can see sakurai 'advanced quantum mechanics' for an elegant derivation which describes the relationship between charge conjugated wave function and original one.
 

1. What is charge conjugation in the Dirac equation?

Charge conjugation is a mathematical operation that changes the sign of the electric charge in the Dirac equation, which is a fundamental equation in quantum mechanics that describes the behavior of spin-1/2 particles.

2. Why is charge conjugation important in the Dirac equation?

Charge conjugation is important because it allows us to describe the behavior of antiparticles, which have the opposite charge of their corresponding particles, in the Dirac equation. This is crucial in understanding the symmetries and properties of particles in quantum mechanics.

3. How is charge conjugation related to other symmetries in the Dirac equation?

Charge conjugation is related to other symmetries in the Dirac equation, such as parity and time reversal, through the CPT theorem. This theorem states that the combined operation of charge conjugation, parity inversion, and time reversal must leave the laws of physics unchanged.

4. Can charge conjugation be experimentally observed?

No, charge conjugation cannot be directly observed as it is a mathematical operation. However, its effects can be observed in experiments involving antiparticles, such as the production and annihilation of electron-positron pairs.

5. Are there any exceptions to charge conjugation symmetry in the Dirac equation?

Yes, there are some cases where the charge conjugation symmetry is violated in the Dirac equation. This can occur in certain interactions involving weak nuclear force, which is responsible for radioactive decay. However, overall, the Dirac equation still maintains charge conjugation symmetry in most cases.

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