# Chiral currents of a Dirac plane wave

• Slereah
In summary, the conversation is about checking the formula for the chiral current of the Dirac equation for a plane wave solution. The right-handed current is calculated using hyperbolic identities and a spin vector, and the spatial components are found to be missing a term according to the book. The question is where this term is coming from.
Slereah
I am currently trying to check the formula for the chiral current of the Dirac equation for a plane wave solution (found here ), that is,

$j_{R}^\mu = \psi_R^\dagger \sigma^\mu \psi_R$

$j_{L}^\mu = \psi_L^\dagger \sigma^\mu \psi_L$

With

$\psi_R = I( \cosh(\frac{\theta}{2}) + \sigma^i n_i \sinh(\frac{\theta}{2})) \xi e^{ix^\mu p_\mu}$

$\psi_R = I( \cosh(\frac{\theta}{2}) - \sigma^i n_i \sinh(\frac{\theta}{2})) \xi e^{ix^\mu p_\mu}$

χ being the rest frame spinor, θ the rapidity and n the direction of the momentum.

So for the right-handed current, I get

$\psi_R^\dagger \sigma^\mu \psi_R = ([\cosh(\frac{\theta}{2}) + \vec{\sigma} \cdot \vec{n} \sinh(\frac{\theta}{2}) ] \xi)^\dagger \sigma^\mu [\cosh(\frac{\theta}{2}) + \vec{\sigma} \cdot \vec{n} \sinh(\frac{\theta}{2}) ] \xi$

$\psi_R^\dagger \sigma^\mu \psi_R = \cosh^2(\frac{\theta}{2}) \xi^\dagger \sigma^\mu \xi + \cosh(\frac{\theta}{2}) \sinh(\frac{\theta}{2}) \xi^\dagger( \sigma^\mu (\vec{\sigma} \cdot \vec{n}) + (\vec{\sigma} \cdot \vec{n}) \sigma^\mu ) \xi + \sinh^2(\frac{\theta}{2}) \xi^\dagger (\vec{\sigma} \cdot \vec{n}) \sigma^\mu (\vec{\sigma} \cdot \vec{n}) \xi$

Using the following hyperbolic identities,

$\sinh^2(\frac{\theta}{2}) = \frac{1}{2} [\cosh(\theta) - 1] = \frac{1}{2} [\gamma - 1]$

$\cosh^2(\frac{\theta}{2}) = \frac{1}{2} [\cosh(\theta) + 1] = \frac{1}{2} [\gamma + 1]$

$\cosh(\frac{\theta}{2}) \sinh(\frac{\theta}{2}) = \frac{1}{2} \sinh(\theta) = \frac{1}{2} \beta \gamma$

And defining the spin vector

$\xi^\dagger \sigma^\mu \xi = s^\mu = (1,\vec{s})$

I get the following :

$\psi_R^\dagger \sigma^\mu \psi_R = \frac{1}{2} [ \gamma \xi^\dagger (\sigma^\mu + \sigma^i \sigma^\mu \sigma_i ) \xi + \xi^\dagger (\sigma^\mu -\sigma^i \sigma^\mu \sigma_i ) \xi + \beta_i \gamma \xi^\dagger \{\sigma^\mu, \sigma^i\} \xi ]$

$\beta_i$ being $\beta n_i$, and $\sigma^i \sigma^\mu \sigma_i$ coming from $(\vec{\sigma} \cdot \vec{n}) \sigma^\mu (\vec{\sigma} \cdot \vec{n})$ (I can't find the exact proof to show it to be so, but I think this is correct - though it may be the problem). So far this is pretty much the results indicated. But then, if I try finding out the results using this identity :

$\sigma^i \sigma^\mu \sigma_i = \bar{\sigma}^\mu$

I finally get this formula :

$\psi_R^\dagger \sigma^\mu \psi_R = \frac{1}{2} [ \gamma \xi^\dagger (\sigma^\mu + \bar{\sigma}^\mu) \xi + \xi^\dagger (\sigma^\mu -\bar{\sigma}^\mu ) \xi + \beta_i \gamma \xi^\dagger \{\sigma^\mu, \sigma^i\} \xi ]$

Which gives the following components for the current :

$\psi_R^\dagger \sigma^0 \psi_R = \gamma \xi^\dagger \xi + \beta_i \gamma \xi^\dagger \sigma^i \xi = \gamma + \gamma \beta_i s^i$

Which is the correct result, but for the spatial components I find

$\psi_R^\dagger \sigma^j \psi_R = \xi^\dagger \sigma^j \xi + \beta^j \gamma \xi^\dagger \xi = s^j + \beta^j \gamma$

Which according to the book lacks a term in $+ (\gamma-1) \beta^i s_i \beta^j$, unless I am misinterpreting the notation used (the results are page 16).

So, my question is, where is that term coming from ? I can't seem to find it. Thank you for your help !

## 1. What are chiral currents of a Dirac plane wave?

Chiral currents of a Dirac plane wave refer to the flow of electric charge or energy in a particular direction in a system described by the Dirac equation. They are a consequence of the chiral nature of the Dirac equation, which describes particles with spin. Chiral currents are important in various fields of physics, including particle physics and condensed matter physics.

## 2. How are chiral currents related to the Dirac equation?

The Dirac equation, which describes the behavior of particles with spin, has a chiral symmetry. This means that the equation is invariant under a transformation that exchanges left-handed and right-handed particles. This chiral symmetry leads to the existence of chiral currents in systems described by the Dirac equation.

## 3. What is the significance of chiral currents in particle physics?

In particle physics, chiral currents are important for understanding the behavior of elementary particles and their interactions. For example, chiral currents play a crucial role in the weak interaction, one of the four fundamental forces of nature. The discovery of chiral currents in the 1960s was a major breakthrough in our understanding of the weak interaction.

## 4. How do chiral currents impact condensed matter systems?

In condensed matter physics, chiral currents are relevant to the study of materials with topological properties. These materials can exhibit exotic behavior, such as the quantum Hall effect, which is a result of chiral currents flowing along the edges of the material. Chiral currents also play a role in the behavior of certain types of superconductors.

## 5. Can chiral currents be observed experimentally?

Yes, chiral currents can be observed experimentally through various techniques, such as scattering experiments and transport measurements. In particle physics, chiral currents are indirectly observed through the weak interaction. In condensed matter systems, they can be directly observed through the measurement of electric or thermal currents in the material.

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