Help About charge conjugation of Dirac spinor

In summary, the formula (8.18) in P J Mulders's lecture notes states that the charge conjugation operator (C) applied to a particle spinor (b) and then inverted is equal to the complex conjugate of the antiparticle spinor (d). The presence of a bar on the helicity variable (λ) on the right side may seem confusing, but it is important to note that charge conjugation does not change helicity, spin, or momentum. However, the definition of negative energy or antiparticle spinors may vary depending on the author's conventions, so it is necessary to carefully consider Mulder's approach in this case. It may also be helpful to track through the charge-conjugation
  • #1
snooper007
33
1
The following formula appears in P J Mulders's lecture notes
http://www.nat.vu.nl/~mulders/QFT-0E.pdf [Broken]

[tex]{\cal C}~b(k,\lambda)~{\cal C}^{-1}~=~d(k,{\bar \lambda})[/tex] (8.18)

where [tex]{\cal C}[/tex] is charge conjugation operator.
[tex]\lambda[/tex] is helicity.
I don't know why there is [tex]{\bar {\lambda}}[/tex] on the right side,
as is well known that charge conjugation can not change helicity, spin, and momentum.

Thanks
 
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  • #2
The phases used to define negative energy or anti-particle spinors are often author dependent -- for example, the C matrix is diagonal in Weinberg's book, and off diagonal in Gross's text. Gross uses a different convention for defining antiparticle spinors than do Bjorken and Drell As far as I can figure, you'll have to track through the charge-c process starting with Mulder's conventions for particle and antiparticle spinors. Note that often the CCD relates the complex conjugate antiparticle spinor to the particle spinor. Good luck.
Regards,
Reilly Atkinson
 
  • #3
for providing this interesting formula from P J Mulders's lecture notes. The concept of charge conjugation in quantum field theory is a fundamental one, and it plays a crucial role in understanding the symmetries of particles and their interactions.

First, let's define what charge conjugation is. In quantum field theory, charge conjugation is a symmetry operation that transforms a particle into its antiparticle. It is represented by the operator {\cal C}, which acts on a quantum state to produce its charge-conjugated state. In other words, if we have a state |{\psi}\rangle, then its charge-conjugated state is given by {\cal C}|{\psi}\rangle.

Now, let's look at the formula you provided. This is the charge conjugation transformation for a Dirac spinor, which is a mathematical object that describes spin-1/2 particles like electrons. The spinor is represented by the symbol b(k,\lambda), where k is the momentum and \lambda is the helicity (the projection of the spin along the direction of motion). The formula shows that when we apply the charge conjugation operator to the spinor, we get another spinor d(k,{\bar \lambda}), where {\bar \lambda} is the opposite helicity.

This may seem strange at first, since you correctly pointed out that charge conjugation cannot change helicity, spin, or momentum. However, this formula is not implying that charge conjugation does change these properties. Instead, it is showing that the charge-conjugated spinor is related to the original spinor by a complex conjugation of the helicity variable. In other words, if we have a spinor with helicity \lambda, its charge-conjugated spinor will have helicity {\bar \lambda}.

This may seem like a minor detail, but it is actually very important in understanding the symmetries of particles. The complex conjugation of the helicity variable is necessary to ensure that the charge-conjugated state is still a valid solution to the Dirac equation. This is because the Dirac equation is a complex equation, and the charge conjugation operator needs to account for this complexity.

In summary, the formula you provided is a representation of the charge conjugation transformation for a Dirac spinor. It shows that the charge-conjugated state is related to the original state by a complex conjugation of the helicity variable. This is a necessary component of the
 

1. What is charge conjugation in physics?

Charge conjugation is a fundamental symmetry operation in physics that involves changing the sign of all electric charges in a system. It is represented by the letter "C" and plays a crucial role in theories of particle physics and quantum mechanics.

2. How does charge conjugation apply to Dirac spinors?

In the context of Dirac spinors, charge conjugation involves taking the complex conjugate of the spinor and reversing the order of its components. This operation allows us to describe particles and antiparticles in a unified way and is an important concept in the study of quantum field theory.

3. What is the significance of charge conjugation in particle physics?

Charge conjugation is a fundamental symmetry of the laws of physics and helps us understand the behavior of particles and antiparticles. It also plays a crucial role in the conservation laws of electric charge, which are essential in explaining the behavior of subatomic particles.

4. How is charge conjugation related to other symmetry operations?

Charge conjugation is closely related to other symmetry operations, such as parity (P) and time reversal (T). Together, these operations form the CPT symmetry, which is a fundamental property of quantum field theories. This symmetry states that the laws of physics remain the same when particles are replaced by their antiparticles, space is flipped, and time is reversed.

5. Can the charge conjugation of Dirac spinors be observed in experiments?

Yes, the effects of charge conjugation can be observed in particle physics experiments, such as high-energy collisions at particle accelerators. These experiments have confirmed the existence of antiparticles and have provided evidence for the fundamental symmetries of CPT.

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