# Operator equations vs. field equations

1. Jun 8, 2013

### mpv_plate

If my understanding is correct, the equations of QFT (Dirac, Klein-Gordon) govern the behavior of operator fields (assigning operator to each point in space). Does it mean there are no equations governing the behavior of fields (assigning a number / vector/ spinor to each point in space)? Is QFT somehow using the concept of non-operator fields, or everything is only and solely operator field?

2. Jun 8, 2013

### dextercioby

Fields (classical or quantum) are just mathematical models of >reality<, they are tools and nothing more. The concept of <field equation> pertains rather to classical field theory (such as electromagnetism or GR), since quantum fields are objects whose evolution or dynamics is not sought, they are tools to get to the observables (scattering probabilities), so that the Dirac equation for the quantized Dirac field is only the start, not the finality of the theory.

And the word <behavior> is pretty vague. I'd use in sociology and psychology, not physics.

3. Jun 9, 2013

### stevendaryl

Staff Emeritus
Let me explain in terms of the analogy with single-particle physics.

In classical particle physics, a particle's position and momentum are dynamic variables, with definite values at every moment.

In quantum mechanics, position and momentum become operators, and there is a probability distribution (described by the wave function) for measuring various values of those operators.

Now, the case of field theory is analogous.

In classical field theory, a field and its canonical momentum have definite values at every moment, and at every location.

In quantum field theory, a field and its canonical momentum become operator, and there is a probability distribution (described by a wave function) for measuring various values of those operators.

In quantum field theory, the "wave function" is never written down explicitly, except in the abstract state notation, if then.

4. Jun 9, 2013

### mpv_plate

Thank you both for the answers.

This question actually came to me when I was thinking about the path integral formulation, where I'm supposed to integrate over all possible field configurations. I'm assuming that includes also such field configurations that are not solutions to any reasonable equation of motion (like Dirac, K-G, Proca, etc.). So even such clasically impossible configurations seem to play a role in QFT.

So I was wondering if these "impossible" configurations also have some non-zero probability amplitude in the wave functional formulation. Whether the quantum field, seen as a superposition of infinitely many configurations, also encompasses clasically unthinkable configurations (though with very small probability).

Based on what you said, this is not an important question in QFT, because we need to predict the observables - and the fields are not observables. But still, the specific field configurations play a role in the path integral approach, so the question seems not to be completely beside the point.

Thanks for the correction. I'm not an English native speaker and I'm still learning.