Interaction between matter and antimatter in Dirac equation

[lagrangman]

TL;DR Summary

Confusing interaction between matter and antimatter in Dirac equation.

I'm new to relativistic quantum mechanics and quantum field theory and was trying to learn about the Dirac equation.

Unfortunately, I got a little stumped by the interaction between matter and antimatter.

It seems like the time derivative of matter is dependent on the spatial derivative of antimatter, but not the spatial derivative of matter. Likewise, the time derivative of antimatter is dependent on the spatial derivative of matter, but not the spatial derivative of antimatter.

To me this means that if there is momentum of matter, then the antimatter field should be changing, which doesn't make sense to me.

I find this counterintuitive and was hoping that someone could explain this to me.

 

[Demystifier]

Summary:: Confusing interaction between matter and antimatter in Dirac equation.

It seems like the time derivative of matter is dependent on the spatial derivative of antimatter, but not the spatial derivative of matter.

What makes you think that?

 

[PeroK]

Summary:: Confusing interaction between matter and antimatter in Dirac equation.

I'm new to relativistic quantum mechanics and quantum field theory and was trying to learn about the Dirac equation.

Unfortunately, I got a little stumped by the interaction between matter and antimatter.

It seems like the time derivative of matter is dependent on the spatial derivative of antimatter, but not the spatial derivative of matter. Likewise, the time derivative of antimatter is dependent on the spatial derivative of matter, but not the spatial derivative of antimatter.

To me this means that if there is momentum of matter, then the antimatter field should be changing, which doesn't make sense to me.

I find this counterintuitive and was hoping that someone could explain this to me.

You could read this and tell us precisely what you don't understand. A liitle mathematics might help:

https://en.wikipedia.org/wiki/Dirac_spinor

 

[lagrangman]

Isn't the first row of the dirac equation
$$i\frac{\partial \psi_1}{\partial t} = m \psi_1 - i \frac{\partial \psi_4}{\partial x} - i \frac{\partial \psi_4}{\partial y} - i \frac{\partial \psi_3}{\partial z}$$

I was under the impression that $\psi_1$ and $\psi_2$ were matter and $\psi_3$ and $\psi_4$ were antimatter.

 

[PeroK]

Isn't the first row of the dirac equation
$$i\frac{\partial \psi_1}{\partial \t} = m \psi_1 - i \frac{\partial \psi_4}{\partial x} - i \frac{\partial \psi_4}{\partial y} - i \frac{\partial \psi_3}{\partial z}$$

I was under the impression that $\psi_1$ and $\psi_2$ were matter and $\psi_3$ and $\psi_4$ were antimatter.

Not quite. The Dirac spinors for both particles and antiparticles have four components. In the non-relativistic case, the solutions simplify to approximately two-component solutions, but not in general.

 

[lagrangman]

Thanks, very helpful.