# Why are the “coupling constants” constant?

The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which are determined by some numbers. In other words, the set of all ad-invariant inner products form a space of more than one dimension and to fix a specific inner product we need to choose some numbers which are the coupling constants. This is the story which happens over each point of the space-time. Mathematically one can produce a theory in which these numbers (and so inner product) changes from a point to another (similar to Riemann metric on a general Riemannian manifold). In other words we can have coupling fields rather that constants.

What is the physical reasoning which disallow us having such a theory (with coupling fields)? And what would be the physical implications if the nature follows such a model?

• I'm not sure about this, but I think the result would be that physics would vary from point to point in spacetime, in contradiction of observations - eg an electron is the same wherever it's observed (mass, charge, magnetic moment etc) . – Dr Chuck Jul 24 '14 at 20:09
• @Qmechanic Thanks Qmechanic for the tag edit. I am a mathematican and new here. It will take me time to get used to all the tags. – Asghar Ghorbanpour Jul 24 '14 at 22:20
• @DrChuck I don't see how changing of a coupling constant can change electron charge? (e.g charge is not defined using coupling constant rather the other way, is n't it?) – Asghar Ghorbanpour Jul 24 '14 at 22:34
• Ashgar - charge is a coupling constant. – Dr Chuck Jul 25 '14 at 8:16

You may always promote "couplings constants" (charge, mass, etc...) to fields. Now, as a physicist, you need to make some contact with reality. So you have to tell why and which field you are using (for instance the Higgs field (up to a constant), which has a $SU(2)$ charge, is used to replace a constant mass coupling in the interaction $m (\bar e_R e_L + \bar e_L e_R)$, this is because the left electron has a $SU(2)$ charge while the right electron has not), and how an experiment can test your hypothesis.

Finally, even coupling constants, in a Quantum field theOry, are not really constant, and depenD on the energy scale $e = e(\Lambda)$

• Thank you. As a beginner, where can I find more details about your last point that the coupling constants depend on the energy scale? and how does the theory work with such a dependence? – Asghar Ghorbanpour Jul 26 '14 at 21:20
• This is a characteristic of coupling constants in Quantum Field Theory (QFT) (for instance Quantum Electrodynamics (QED)). Any physics book on the subject (for instance, A.Zee, Quantum Field Theory in a Nutshell) will help you. Keywords here are Renormalization, Renormalization group. – Trimok Jul 27 '14 at 11:57

The Einstein equivalence principle states :

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

Emphasis added. Note that this principle has done well in explaining quite a few things about gravity. So there is no a priori reason why you couldn't have changing constants, it's an empirically motivated principle that seems to lead to a good description of how the stress-energy tensor sources curvature in the metric tensor, so people are quite tempted to keep it when developing quantum gravity theories, rather than violate it with "coupling fields" as you call them.

This doesn't prevent others from actively seeking for observations that would defy this. It just means that theoretically and observationaly it remains, for the time being, a well motivated principle. It doesn't "have" to be true due to purely logical considerations, it just seems to be true given theoretical and observational advances in the field.

• Can't one see this in the same way as change in the volume form of the curved space-time (which is not in contrary of the equivalence principle). Indeed, the change caused by introducing "coupling field" is very similar to change in volume form. Although the latter one is not a field for which the Lagrangian is written. – Asghar Ghorbanpour Jul 24 '14 at 22:46