The electromagnetic current vanishes identically in the Majorana case

In summary, the electromagnetic current vanishes in the Majorana case only if the components of \psi are anticommuting, and the formula for this is \overline{\psi}\gamma^{\mu}\psi = \overline{\psi^{C}}\gamma^{\mu}\psi^{C} = -\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} \overset{?}{=} \overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi = -\overline{\psi}\gamma^{\mu}\psi = 0.
  • #1
Urvabara
99
0
The electromagnetic current vanishes identically in the Majorana case: http://users.jyu.fi/~hetahein/tiede/virta.pdf" .

In the case someone cannot open the pdf, here is the formula:
[tex]\overline{\psi}\gamma^{\mu}\psi = \overline{\psi^{C}}\gamma^{\mu}\psi^{C} =
-\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} \overset{?}{=}
\overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi = -\overline{\psi}\gamma^{\mu}\psi = 0.[/tex]

Does anyone know how is that equality under the ?-sign proved? I just don't know how...

Thanks!
 
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  • #2
Urvabara said:
The electromagnetic current vanishes identically in the Majorana case: http://users.jyu.fi/~hetahein/tiede/virta.pdf" .

In the case someone cannot open the pdf, here is the formula:
[tex]\overline{\psi}\gamma^{\mu}\psi = \overline{\psi^{C}}\gamma^{\mu}\psi^{C} =
-\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} \overset{?}{=}
\overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi = -\overline{\psi}\gamma^{\mu}\psi = 0.[/tex]

Does anyone know how is that equality under the ?-sign proved? I just don't know how...

Thanks!
First off, the electromagnetic current vanishes identically in the Majorana case only if you assume that the components of \psi are anticommuting. If they commute, the transition in question can be performed as follows: the scalar equals its own transposition, C and C-cross change sign under transposition. However, the resulting sign will differ from that in your formula. You obtain the third sign change when you drag anticommuting components of psi through each other.
 
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  • #3


I can provide an explanation for why the electromagnetic current vanishes identically in the Majorana case.

First, let's define what we mean by the Majorana case. In particle physics, a Majorana particle is a type of fermion (a particle with half-integer spin) that is its own antiparticle. This means that a Majorana particle is its own mirror image and has the same properties as its antiparticle. In contrast, most other particles have distinct antiparticles with opposite properties.

Now, let's look at the formula provided in the link. This formula represents the electromagnetic current, which is a mathematical expression that describes the flow of electromagnetic charge in a system. In this case, we are looking at the current for a Majorana particle.

The first part of the formula, \overline{\psi}\gamma^{\mu}\psi, represents the current for a regular fermion. The second part, \overline{\psi^{C}}\gamma^{\mu}\psi^{C}, represents the current for the Majorana particle's antiparticle (which is also a Majorana particle). The third and fourth parts, -\psi^{\text{T}}C^{\dagger}\gamma^{\mu}C\overline{\psi}^{\text{T}} and \overline{\psi}C\gamma^{\mu\text{T}}C^{\dagger}\psi, are equivalent due to the properties of the charge conjugation operator (C) and the transpose operator (T).

Now, the key point here is that since the Majorana particle is its own antiparticle, the current for the particle and the antiparticle must be equal. This means that the first and second parts of the formula must be equal, and the third and fourth parts must also be equal. Therefore, the entire expression is equal to -\overline{\psi}\gamma^{\mu}\psi, which is equivalent to 0.

In other words, in the Majorana case, the electromagnetic current vanishes identically because the particle and antiparticle currents cancel each other out. This is a unique property of Majorana particles and is not seen in other types of particles.

To prove this equality, we can use the properties of the charge conjugation and transpose operators, as well as the fact that a Majorana particle is its own antiparticle. This mathematical proof is beyond the scope of this response
 

Related to The electromagnetic current vanishes identically in the Majorana case

1. What does it mean for the electromagnetic current to vanish identically in the Majorana case?

When we say that the electromagnetic current vanishes identically in the Majorana case, we mean that it is always equal to zero. This occurs because the Majorana fermion is its own antiparticle and has no electric charge, so it does not interact with the electromagnetic field.

2. How is the Majorana fermion different from other fermions in terms of electromagnetic interactions?

The Majorana fermion is different from other fermions because it is its own antiparticle, meaning it has no opposite charge and does not interact with the electromagnetic field. In contrast, other fermions have distinct particles and antiparticles with opposite charges, allowing them to interact with the electromagnetic field.

3. Why is it important that the electromagnetic current vanishes in the Majorana case?

The vanishing of the electromagnetic current in the Majorana case is important because it allows for the possibility of Majorana particles to exist as stable, neutral particles. This has implications for particle physics and cosmology, as well as potential applications in quantum computing.

4. Can the electromagnetic current vanish in other cases besides the Majorana case?

Yes, the electromagnetic current can also vanish in other cases, such as for particles with zero electric charge, like the photon. However, the Majorana case is unique because it involves a fermion that is its own antiparticle.

5. How does the vanishing of the electromagnetic current in the Majorana case relate to the conservation of electric charge?

The vanishing of the electromagnetic current in the Majorana case is related to the conservation of electric charge because, in order for the current to be conserved, it must be zero when there is no electric charge present. This is consistent with the fact that the Majorana fermion has no electric charge and thus does not contribute to the electromagnetic current.

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