Relationship Between QED/QCD and Field Theories

Islam Hassan

Are QED/QCD themselves field theories or sub-field theories of other more general field theories?

What are the formal theories/theoretical frameworks that completely characterise the Standard Model in our present state of knowledge?


IH

The_Duck

QED and QCD are specific quantum field theories. You might read:

http://en.wikipedia.org/wiki/Classical_field_theory
http://en.wikipedia.org/wiki/Quantum_field_theory

"Quantum field theory" is the general mathematical framework that we've found most useful in particle physics. Quantum field theory provides a generic way of discussing particles and their interactions taking into account both quantum mechanics and relativity. The Standard Model is a specific quantum field theory, where we postulate the existence of a certain set of particles and interactions to explain all the observations we've made. QED, which describes the electromagnetic force, and QCD, which describes the strong force, are both "sub-theories" of the standard model, which includes both the electromagnetic and strong forces.

mfb

Well, QED is the low-energy theory of the electroweak interaction, which combines electromagnetic and weak force into one. The SM contains the known elementary particles, QCD, the electroweak interaction and its symmetry breaking via the Higgs mechanism. All is expressed as quantum field theory.

Islam Hassan

Many thanx for the kind feedback. Just to round off my comprehension, can you kindly confirm:

- Quantum field theory in general as well as its QED/QCD incarnations are all necessarily defined as gauge theories;
- There is at present no established (ie, non speculative) quantum gravity field theory and
- Conformal field theory does not concern itself with the SM but is purely a beyond-the-SM theory.


IH

Islam Hassan

Re that comment, just wondering...does that mean that there are other less prevalent/successful mathematical frameworks which, while less useful, still have something to contribute to our understanding of the SM?


IH

The_Duck

No, gauge theories are a specific kind of field theory. Quantum field theories need not be gauge theories. And gauge theories need not be quantum field theories--you can have classical gauge theories (classical electromagnetism is a classical gauge theory).

In the context of quantum field theory, gauge theories are just one way of producing interactions between particles (that is, forces. The electromagnetic, strong, and weak forces are all the results of gauge interactions). But there are other kinds of interactions, like Yukawa interactions (which the Higgs boson experiences): http://en.wikipedia.org/wiki/Yukawa_interaction

Right. We have general relativity, which is a classical field theory, but we don't know the quantum mechanical version.

"Conformal field theory," like "quantum field theory," is a broad class of field theories and is not a specific theory. Some quantum field theories are also conformal field theories. You are right that the standard model is not a conformal field theory.

What I meant to imply was: while all current observations (except gravity) are explained by the specific quantum field theory we call the standard model, and most speculative particle physics involves postulating new quantum field theories that extend the standard model, in principle nature need not be described by a quantum field theory at all. It's possible, perhaps likely, that the "true theory" is not a quantum field theory at all. For instance, I'm not sure whether string theory is strictly speaking a quantum field theory; perhaps someone more knowledgeable can say something about that.

Of course, whatever the "true theory" is, it is clearly well approximated by a quantum field theory, namely the standard model.

K^2

I don't think so. Pretty much everything past Classical Mechanics is a field theory of some sort or another. Even in classical mechanics classical field theory is pretty useful.

If I had to place a bet one way or another, I'd bet on it being a non-quantum field theory.

First of all, both GR and standard model make extremely good predictions. It can't be random. However, GR is a non-linear theory. Pretty much any field theory can be approximated by a linear field theory, in which case you can build a quantum field theory. So given that GR works, and based on general Ocam's razor principle, I would guess that success of SM is due to linear approximation working very well due to relatively low energy scales involved. (Low compared to what? Good question.) In which case, a better field theory would be a non-linear one, and non-linear field theory cannot be a quantum field theory.

But this is more of what we should be aware of, and perhaps looking for in high-energy experiments, rather than any kind of statement on quality of Standard Model. So far, everything we've done confirms SM.

P.S. Oh, and even if we discover that underlying field-theory is non-linear, Quantum Mechanics and Standard Model aren't going anywhere. Linear theories are nice. We can actually compute stuff with them. Compare all the nifty things we've done with QM to stuff we've been able to do with GR. Heck, even in GR, Linearized Gravity is an extremely powerful tool that's used for many predictions.

Islam Hassan

Very interesting your comment "Pretty much everything past Classical Mechanics is a field theory of some sort or another". So field theory really constitutes the bulk of modern elementary/particle physics (SR I take it does not need a field theoretic framework, does it?). How about string and other beyond-the-SM theories, are they also mostly field theories or not?

Finally, regarding your use of the term "non-iinear", is this in the simple intuitive sense that x is a linear coefficient and x^2 is not or it is more complicated than that?


IH

mfb

GR is a (nonlinear) field theory.

I do not remember any non-field theory there.