# Physical interpretation of Lorentz invariant fermion field product?

1. Nov 24, 2009

### blue2script

Hey all!
Just a very short question: May I interpret the Lorenz invariant quantity

$$\bar\psi\psi$$

as being the probability density of a fermion field? Thanks!
Blue2script

2. Nov 24, 2009

### Demystifier

No, because it is not positive.

3. Nov 24, 2009

### blue2script

Hmm.. right. But then, what is the physical interpretation of the product above? And what is the probability density of a fermion field?

4. Nov 24, 2009

### hardphysics

as far as i know there is no propability interpretation of fermion fields only charge density.

5. Nov 24, 2009

### hardphysics

the interpretation of $$\bar\psi\psi$$ is that it transforms under lorentz boosts like a scalar.

6. Nov 25, 2009

### Demystifier

Fermion field is an operator, so it does not have a probabilistic interpretation. However, one should distinguish the field operator from the wave function which is a c-number function representing a quantum state. For a 1-particle wave function $$\psi$$, the probability density is
$$\psi^{\dagger}\psi$$

7. Nov 25, 2009

### hardphysics

you mean $$\psi^{*}\psi$$ because $$\psi$$ is a scalar ? but eitherway its okay to write it with a dagger.

8. Nov 26, 2009

### Demystifier

I mean dagger because psi a spinor, i.e., a 4-component wave function.