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Recently I've started to study Quantum Field Theory and I think that since then I've been missing the whole point with the theory, and as a result I've been getting really confused.

Let's start with Classical Field Theory and its most well known example: Electrodynamics.

What is the basic problem in Electrodynamics? Well it is to understand how charges affect each other. This problem has two steps: the first is to understand how charges are affected by others by mean of fields and the second is to understand how these fields are generated and how they evolve in time.

Historically we can say that one started discovering the influences on charges as forces and then introduced the field viewpoint. The whole historical process on how one got to the Maxwell equations for the fields is covered in EM textbooks like Griffiths.

That is all fine, but now let's get to QFT. At first sight my thought was:

In analogy to Classical Field Theory, the basic problem in QFT is probably obviously be to find the fields $\phi(x)$, which are now operator-valued, how they evolve, and how they interact with other systems.

This would be the natural generalization of what we see in Classical Electrodynamics textbooks.

This lead me to a bunch of confusion when I started reading books and finding out that authors never, ever, end up defining or finding the fields $\phi(x)$. In truth, based on this discussion actually they suppose the field exists and obeys some commutation relations and start working with this.

This is in sharp contrast with CFT, where the whole point is to find out what is the functional dependency $\phi(x)$ of the field on events $x\in M$ on spacetime.

This led me to question the following: what is really the main objective of Quantum Field Theory, if it is not finding the field and its time evolution $\phi(x)$ as one does, e.g., in Electrodynamics? What is the whole point with the theory by the way? What is really the problem being tackled and what kind of thing is being sought?

If possible, I think some historical explanation could clarify this matter, but every help is appreciated.

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  • $\begingroup$ What do you mean by saying that QFT never finds the fields while CFT finds them? You yourself say that CFT finds out the functional dependency of fields, which is not the same as finding a field. $\endgroup$
    – Helen
    Apr 1, 2017 at 6:03
  • $\begingroup$ Possible duplicate of What does it actually mean "to define a field" in QFT? $\endgroup$
    – Helen
    Apr 1, 2017 at 6:07
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    $\begingroup$ That's like asking what the main objective of classical mechanics (including statistical mechanics and the like!) is, since "quantum field theory" is really a rather broad field. The obvious answer is that it depends on the problem you're trying to solve. Sometimes you want scattering amplitudes, sometimes you want time-evolution of certain states, sometimes you want order parameters, sometimes other things entirely. There's no main objective because it's a tool that can be applied to many different settings. I don't see how one could give any kind of definite answer to this. $\endgroup$
    – ACuriousMind ♦
    Apr 1, 2017 at 8:04

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I'll answer this as a spectator of QFT rather than a practitioner, so bear in mind I'm not answering with any great authority.

Once we move from the free quantum field to an interacting quantum field we no longer know what the states of the quantum field are because we can't solve the equations. The best we can do is use a perturbative approach, but this limits what we can do - for example it's hard to study solitons with perturbation theory.

However what we can do is use the perturbative approach to calculate scattering amplitudes. This is now pretty straightforward, if tedious, for weakly interacting fields though still difficult for strongly interacting fields like QCD.

So for most end users the point of QFT is to develop the tools for calculating scattering amplitudes. I say end users because I'm excluding the Arkani-Hamed end of the mathematical physics spectrum.

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    $\begingroup$ "I say end users because I'm excluding the Arkani-Hamed end of the mathematical physics spectrum" :))) $\endgroup$
    – Helen
    Apr 1, 2017 at 7:30
  • $\begingroup$ To add: and the point of having expressions for scattering amplitudes is that it is the most accurate description of how elementary particles interact with each other which is currently at our disposal. $\endgroup$
    – Prof. Legolasov
    Apr 1, 2017 at 8:05
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    $\begingroup$ As a hep person myself, this is a hopelessly narrow description of QFT. Condensed matter theorists certainly are interested in things other than scattering amplitudes, and in fact couldn't, I suspect, care less about them. Thermal and non-equilibrium field theory do care about time evolution and observables other than scattering, and I'm sure there are applications I don't even know of. Your answer is specific to the high-energy application of QFT as it relates to colliders and the interaction of approximately free particles, and not about "QFT" as the more general toolset it is. $\endgroup$
    – ACuriousMind ♦
    Apr 1, 2017 at 8:06
  • $\begingroup$ The answer's explanation of what is the point of QFT is very shortsighted. It's like saying the point of general relativity is so that astronomers can calculate better orbits and distances to galaxies. QFT is the best we have to deal with all three forces (not gravity). It led to gauge theories and symmetries which led us to discover a bunch of new particles. It led us to the Higgs boson, strange quarks and electroweak unification. None of that was due to Arkani-Hamed, or whatever that 'end of mathematical physics' means. $\endgroup$
    – Bob Bee
    Apr 1, 2017 at 19:17
  • $\begingroup$ @BobBee I think the last part was referring to highly speculative areas of physics that do use mathematics, but in a highly speculative way. $\endgroup$
    – Helen
    Apr 6, 2017 at 4:19

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