What is a Quantum Field?

What exactly a “Quantum Field” is confused me for a long time. Let me attempt to clear up some misconceptions.

When you picture a quantum field, you might picture something like this:


Some field in over all of space that evolve in time. Perhaps at the points where the value of the field is higher, we say that there is a particle there.

This is completely false. What that picture would represent is regular field theory. In regular theory you have just that, a field throughout all of space that evolves in time.

But in quantum field theory what you are studying is much more subtle. It is not a field throughout all of space-time but rather a wave-function over all possible fields.

Basically, in regular quantum mechanics, a particle has different probabilities to be at different points in space. But in quantum field theory, fields have different probabilities to be in certain “configurations.” (The picture above is an example of one such configuration.)

This means that a field has no one actual configuration. As such, you can’t actually ever know what the value of the field is as there is always some quantum uncertainty. Each little part of the field is subject to it’s own analogous version of the Heisenberg uncertainty principle.

This has some complications. For one, the number “degrees of freedom” of a quantum field is infinite. Creating a mathematical framework to easily calculate properties about these quantum fields is the first step of learning quantum field theory.

One major complication is what exactly a “particle” as we experience them is. This is a very subtle issue, but saying that a particle is a “spike” in a possible field configuration is pretty accurate.

Quantum mechanics is what happens when you “quantize” a particle. Quantum Field Theory is what happens when you “quantize” a field. What is interesting is that in Quantum Mechanics, particles behave a lot like waves, but in Quantum Field Theory, waves behave a lot like particles. It is a strange twist.


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