Mathematical Quantum Field Theory - Lagrangians - Comments

[vanhees71]

The variational derivative becomes the de Rham differential after "transgression" to the space of field histories. This is the content of the section Local observables and Transgression of chapter 7. Observables, the very statement is item 2 of prop. 7.32.

The full proof is given there, but this should be intuitively clear:

The variational derivative measures the change of field values at a point of spacetime, or of change of spacetime derivative of the field value at a point of spacetime, etc. and the field values at spacetime points are precisely the "canonical variables" for the field theory, while the spacetime derivatives of the field values are precisely the "canonical momenta". It it is in this way that the variational derivative on field eventually becomes the de Rham differential on the phase space.

Well, for us normal theoreticians it's always good, if you - from time to time - express everything in our normal notation with indices and usual differential operators (partial derivatives or covariant derivatives or functional derivatives). Often you have quite complicated looking formal equations that become understandable at one glance when translated to old-fashioned 19-century Ricci-calculus notation. I know that's nearly heretic to mathematicians, but it's the common notation in the physics community (and imho for the good reason of clarity and "calculational safety").

 

[Urs Schreiber]

Well, for us normal theoreticians it's always good, if you - from time to time - express everything in our normal notation with indices and usual differential operators (partial derivatives or covariant derivatives or functional derivatives). Often you have quite complicated looking formal equations that become understandable at one glance when translated to old-fashioned 19-century Ricci-calculus notation. I know that's nearly heretic to mathematicians, but it's the common notation in the physics community (and imho for the good reason of clarity and "calculational safety").

I feel there is a lot of standard physics notation in the notes, plenty of examples that unwind the general machine to the standard formulas. I'll be happy to improve further, but please give me more concrete pointers. Which formula do you feel is lacking examples? I'll add them.

By the way, I like to caution against the habit of organizing people into camps, such as "normal theoreticians" here and "mathematicians" there. There is just one subject, QFT, and we need all the tools we can get hold of to understand it. It seems strange to insist that a theoretical physicist is by necessity one who may not be bothered with learning the tools that it takes to make sense of QFT, or that progress in the field should be impossible due to the inertia of "we have always done it differently".

I am a theoretical physicist, and I managed to learn some of what it takes, and here I am trying to explain it. I can try to add more explanation where need be, but for that please let me know which particular bit bothers you. Best to start from the very beginning, since the whole series develops strictly incrementrally: If you start reading the series in chapter 1, page 1 and keep reading, what is the first point where you feel lost?

 

[ftr]

Urs

I don't know what is exactly the purpose of these threads of yours since they seem to be an order of magnitude more complicated than standard textbooks. I think PF readers are vast mix. The majority usually come here for clarification of concepts and simplified setups not mathematical gymnastics. I think you can use your wide expertise to make the physics understandable. Of course, that does not mean it is not useful for some. Please do not take this personal, this is just my opinion and could be wrong.

 

[dextercioby]

It is precisely because physics textbooks are filled with so much mathematical hand-waving, that one needs an authoritary treatment from a mathematician's viewpoint of what is really going on in fundamental physics. From this perspective, I applaud @Greg Bernhardt 's decision to invite Urs here and allow him to share his knowledge with us.

IIRC, it was the famous mathematician's David Hilbert interest in physics that was so magnificiently summed by his (approximately quoted into English) words: "Physics has become too difficult to be left to physicists".

 

[vanhees71]

I must also stress that it's great that Urs posts this series of lecture notes about formal QFT. I know the usual hand-waving approach quite well, and for sure it's a good opportunity to read about it from the more formal mathematical point of view to understand this most important but mathematically least understood methodological framework better. The problem is that the two communities between practitioners and mathematicians are nearly completely disjoint, and I think it's great to have the opportunity in these forums to have somebody you can interact with via the forum.

I only wish, I had the time to start studying the notes seriously :-((.

 

[Urs Schreiber]

more complicated than standard textbooks

If you make this more specific, I can react. Which passage do you find overly complicated?

 

[Boing3000]

The majority usually come here for clarification of concepts and simplified setups not mathematical gymnastics.

It may be then that I am part of some strange minority, but even if standard textbook also eludes me, I still find those articles very interesting, full of lively cross reference, and new terms that are actually very refreshing to read.

 

[vanhees71]

Well, then I'm also a minority, because I think I haven't understood a problem or, even more difficult, the solution of a problem before it is formulated mathematically.