How Do Differential Forms Impact the Modeling of Physical Interactions?

[Tyger]

When ever we find a new interaction that needs to be described we tend to model it on the most understood interaction, both theoretically and experimentally, namely Electrodynamics. But that has a hidden danger, and one element of that danger is differential forms. They are certainly very nifty in their working, and they make for neat concise equations. So how can they be dangerous?

First, Electromagnetism, as exemplified by Maxwell's Equations, is scale invariant. This is uaually stated by saying that the Photon has zero rest mass. More correctly, the virtual Photon can have any mass squared from a very high to a very low number, because in any practical apparatus many of the Photon events are virtual. If we build an electrical apparatus that is half or twice the size of another it will probably work just fine.

It is this very scale invariance that makes it possible to use diff forms to describe electromagnetism, but it is the ONLY interaction that is truly scale invariant. Gravity has a modified scale invariance, so they can be used with caution. But all the other interactions are short range, so we should expect that diff forms can not be applied to them, or if they are they will more likely increase the complexity of our equations because we will have to subtract out all the terms that are scale invariant in our description.

So my advice is use diff forms with caution and don't be too intranced by their neatness and concisenes. −

 

[instanton]

Originally posted by Tyger
When ever we find a new interaction that needs to be described we tend to model it on the most understood interaction, both theoretically and experimentally, namely Electrodynamics. But that has a hidden danger, and one element of that danger is differential forms. They are certainly very nifty in their working, and they make for neat concise equations. So how can they be dangerous?

First, Electromagnetism, as exemplified by Maxwell's Equations, is scale invariant. This is uaually stated by saying that the Photon has zero rest mass. More correctly, the virtual Photon can have any mass squared from a very high to a very low number, because in any practical apparatus many of the Photon events are virtual. If we build an electrical apparatus that is half or twice the size of another it will probably work just fine.

It is this very scale invariance that makes it possible to use diff forms to describe electromagnetism, but it is the ONLY interaction that is truly scale invariant. Gravity has a modified scale invariance, so they can be used with caution. But all the other interactions are short range, so we should expect that diff forms can not be applied to them, or if they are they will more likely increase the complexity of our equations because we will have to subtract out all the terms that are scale invariant in our description.

So my advice is use diff forms with caution and don't be too intranced by their neatness and concisenes. −

I don't quite get it. What is scale invariance have to do with differential forms?

Instanton