# Physical interpretation of Lorentz invariant fermion field product?

• blue2script
In summary, the Lorenz invariant quantity \bar\psi\psi does not have a probabilistic interpretation as it represents a fermion field which is an operator. However, the 1-particle wave function \psi^{\dagger}\psi can be interpreted as the probability density of a fermion field. The field operator \psi should be distinguished from the wave function, which is a c-number function representing a quantum state.
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

No, because it is not positive.

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

blue2script said:
Hmm.. right. But then, what is the physical interpretation of the product above? And what is the probability density of a fermion field?

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

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

blue2script said:
Hmm.. right. But then, what is the physical interpretation of the product above? And what is the probability density of a fermion field?
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
$$\psi^{\dagger}\psi$$

Demystifier said:
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
$$\psi^{\dagger}\psi$$

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

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

## 1. What is the physical significance of the Lorentz invariant fermion field product?

The Lorentz invariant fermion field product is a mathematical concept used in the study of quantum field theory. It represents the interaction between two fermion particles and is essential for understanding the behavior of particles at high energies and speeds.

## 2. How does the Lorentz invariance property affect the fermion field product?

The Lorentz invariance property of the fermion field product means that it remains unchanged under Lorentz transformations, which are changes of reference frame that preserve the laws of physics. This property allows for the consistent treatment of particles in different reference frames.

## 3. What is the physical interpretation of the Lorentz invariant fermion field product in terms of particle interactions?

The Lorentz invariant fermion field product represents the probability amplitude for two fermion particles to interact with each other. It is a crucial component in the calculation of scattering amplitudes, which are used to predict the outcomes of particle collisions.

## 4. How does the Lorentz invariant fermion field product relate to the concept of spin in quantum mechanics?

The Lorentz invariant fermion field product is closely related to the concept of spin in quantum mechanics. It is a fundamental property of fermion particles, and the Lorentz invariance of the product is crucial for properly describing the spin interactions of particles.

## 5. Can the Lorentz invariant fermion field product be experimentally observed?

No, the Lorentz invariant fermion field product is a theoretical concept used to describe the behavior of particles at the quantum level. It cannot be directly observed in experiments, but its predictions can be tested through various high-energy physics experiments.

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