Quantum Mechanics: QM, QCD, QED Explained

[Swapnil]

Hi, I was wondering, what does the realm of quantum mechanics cover? I mean, I always hear a talk about how it would be a holy grail of physics if we could somehow combine GR and QM. But what about QCD, QED, and other theories. Why aren't they mentioned? Is it because QM encompasses them?

 

[vanesch]

Hi, I was wondering, what does the realm of quantum mechanics cover? I mean, I always hear a talk about how it would be a holy grail of physics if we could somehow combine GR and QM. But what about QCD, QED, and other theories. Why aren't they mentioned? Is it because QM encompasses them?

Well there are some confusions in terminology.

"Quantum theory" QT can be seen as a rather abstract formalism of Hilbert spaces, a unitary evolution on it, observable Hermitean operators and so on ; or in equivalent formulations, using path integrals and the like.

One usually first encounters it, applied to a specific kind of problems, which is that of classical non-relativistic mechanics. This is then called "quantum mechanics" QM. But sometimes, one understands by quantum mechanics, quantum theory.

If you (try to) apply quantum theory to another kind of problems, namely those of classical relativistic fields, then you obtain quantum field theory QFT. By chosing a specific set of fields, for instance, the Maxwell equations, Dirac's equation and so on, you can cook up different applications of QFT. The first one was applied to the Maxwell equations and the electron field (Dirac equation). It is called Quantum Electrodynamics QED. Another one, quite more involved, with more fields and tricks, is called the Weinberg-Salam model of electroweak interactions. It incorporates in fact, QED.

Still another one is about the fields of gluons and quarks, and is called quantum chromodynamics QCD.

Both together are called "the standard model". It's a specific application of QFT to a certain set of fields, which, itself, is a specific application of QT to fields.

However, it must be said that in as much as quantum mechanics is a totally mathematically well-found theory, the same cannot be said about QFT. It is mathematically quite messy and nobody has ever found a way to completely make a rigorous formulation of it.

There is a "classical field" however, which has resisted any "attack" by quantum theory, and that is the metric of spacetime (general relativity).
There, things are not only mathematically messy. One doesn't even know how to start, and how to get numbers out.