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Suggested Classical Field Theory texts?

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Hey everyone,

I posted this a while back in General Physics without any reply, and it looks like this is actually the germane forum (despite the fact that I'm explicitly NOT looking for QFT) -- but I couldn't find the "move thread" option...

Anyway:

I'm looking for some books that really dig into the topic of classical field theory -- not necessarily just the fields that were known during the heyday of classical physics (electromagnetic / gravitational), but not necessarily all about Yang-Mills and Dirac fields, either.

I'm having some difficulty describing exactly what I'm looking for (which is probably why I'm having trouble finding a book that fits the bill), so maybe the best thing to do would be to list books that I do have, and how closely they fit:

Burgess - Classical Covariant Fields. This is the benchmark. Basically, I'm looking for something that covers the same type of topics that this one does, but perhaps going in-depth on fewer topics.

DeWitt - Dynamical Theory of Groups & Fields. The opening paragraph of this book lays out well exactly what I'm not looking for:


This seems to be a common thread in virtually every book on field theory -- even many of those that are nominally supposed to be about classical field theory in particular.

Binz / Sniatycki / Fischer - Geometry of Classical Fields. Reading the TOC of this on Amazon, I thought WOW, this sounds great. But when I picked it up, not only did I find the unformatted text almost unreadable, but there is almost NO reference to physical applications.
Also, this seems like it's more an advanced differential geometry text than a field theory text, though if the topics were tied back to physical applications, that would probably pass muster.

Soper - Classical Field Theory. I like this one, but it's pretty basic. It was a great primer, but I'm looking for something slightly more advanced (or perhaps at about the same level of 'difficulty' of the non-introductory chapters, but with a broader range of topics).

Barut - Electrodynamics and Classical Theory of Fields and Particles. I'd compare this one to Soper. Really good text, well-written and original, grounded in reality -- but very focused on electrodynamics..which makes sense given the title, but again, I'm looking for maybe this depth on more / different topics.

Doughty - Lagrangian Interaction, Felsager - Geometry, Particles and Fields. Just bought these two. From the TOC, they seem like they touch on classical field theory, but only as a stepping stone to QFT.

Ng - Introduction to Classical and Quantum Fields. Given that classical fields were in the title, I was a little disappointed at how little a role they played.

Lifgarbagez / Landau - The Classical Theory of Fields. I'm probably not gonna make any friends saying this, but I just can't get into the Landau books. They just feel...dated. I was again disappointed by the fact that although it was called "classical theory of fields", which I took to be "fields in general", it was focused on pretty basic electrodynamics / gravity.

....is that enough to go on? Or have I just confused and alienated everyone?

Any suggestions would be great.

Thanks,
Justin
 
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  • #2
dextercioby
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You've pretty much covered the literature. However, I'd like to say a word on the subject: classical field theory = gravitodynamics (the general theory of relativity) + electrodynamics (the theory of the electromagnetic field) + specially/generally relativistic fluid dynamics (using covariant lagrangians and tensorial objects).

So essentially you're searching for a book on all 3 at the same time, right ? As far as I know, Landau & Lifschitz volume 3 of their classical collection should be the right reading/learning instrument (Lifgarbagez / Landau - The Classical Theory of Fields).

I think there's no book which treats classical field theory from the point of view of functional analysis and differential geometry at the same time. As far as I could tell these fields are both distributions over S(R^4) and sections of the first jet bundle of an infinite dimensional manifold. But this consolidated picture is not clear to me, so I won't say more and stop here.
 
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...classical field theory = gravitodynamics (the general theory of relativity) + electrodynamics (the theory of the electromagnetic field) + specially/generally relativistic fluid dynamics (using covariant lagrangians and tensorial objects).
So maybe I'm looking for the wrong topic. I'm looking for something that discusses the dynamics, geometry, analysis, etc of fields in general -- so it would encompass all of those, but it would also encompass Yang-Mills or matter fields (i.e. the Dirac equation).

So essentially you're searching for a book on all 3 at the same time, right ? As far as I know, Landau & Lifschitz volume 3 of their classical collection should be the right reading/learning instrument (Lifgarbagez / Landau - The Classical Theory of Fields).
I think from a purely content point of view, the problem with L&L's Classical Theory of Fields is that it is specific to those classically-known fields.

One topic in particular that seems to have been missing from every treatment I've read (L&L included), is a deep, complete explanation of how the Lagrangian formalism generalizes to fields and how it's used...in virtually all of the books the field Lagrangian density is used, but usually they give a review of particle Lagrangians, then quickly say that it works the same for fields, along with a quick overview of functional differentiation (which is invariably hand-wavey and ambiguous).

Other topics that seem like they'd apply to any field but always get short shrift:

- conservation of energy/momentum in fields

- calculating the intrinsic spin of a field

- field degrees of freedom / constraints...?

- energy/momentum transport/transfer

- interaction between fields

- sources, emission, self-interaction, ...

I've flipped through Schwinger's Particles, Sources & Fields, which seems like it touches on some of these topics, but it still seems to be very much focused on quantization & scattering.

So...does that ring any more bells?

Thanks again.
 
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dextercioby
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I see your point but we're not really on the same page. So it's fields in general, not only the classical ones. I've not included the scalar field, the spinorial ones and the vector or even graviton multiplets, simply because they don't exist at a classical level. So they are rather quantum fields.

The geometry is incredibly difficult. OTOH, I've never seen a specific mention that the classical fields are really distributions, not (test) functions. But this is already assumed when building the Hamiltonian formalism from the Lagrangian one, but not explicitely acknowledged.

The dynamics is essentially the dynamics and the field equations of HE, Maxwell, or relativistic versions of transport/Cauchy/Navier-Stokes equations, some of them coupled (as is the case of em. fields in the presence of gravity).

Functional differentiation is rather ambigous in most texts in the absence of fields (i.e. particle systems) actually, but can be thoroughly treated using Frechet/Ga^teaux derivatives. These can of course be generalized to fields.

Spin/angular momentum of the field: Nöther theorem for the background symmetry (e.g. Poincare for electrodynamics). For gravity, angular momentum/spin is exhibited in a linearized theory (suitable for gravitational waves).

DOF/constraints: rigorous Hamiltonian formulation starting from the Dirac analysis for finite systems (1st & 2nd class constraints) extended to distributions/fields. The symplectic geometry is extended to fields (don't know details here).

Interaction b/w fields: hmm, usually in the presence of gauge fields, but for flat spacetime. There's a well established theory of interactions through gauge fields: Deser procedure or BRST coupling.

Sources, emission, self-interaction: typically electromagnetism and gravity. Advanced books on GR (Wald, telephone book) and EM (Felsager and JD Jackson) treat these.

I haven't taken a look at Schwinger.

Hope this helps. For books or even articles, keep searching & report your findings.
 
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I see your point but we're not really on the same page. So it's fields in general, not only the classical ones. I've not included the scalar field, the spinorial ones and the vector or even graviton multiplets, simply because they don't exist at a classical level. So they are rather quantum fields.
Maybe I've been misusing terminology; it might be more correct to say I'm looking for the study of fields -- scalar, spinor, vector, tensor, whatever -- outside the context of second quantization. I think the electromagnetic field is still the canonical example, but I'd consider the first-quantized Schrodinger or Dirac fields or Klein-Gordon fields to be "classical" in this sense.

It just seems like every book I've read rushes into second quantization as quickly as possible, when there is so much that is left to learn about the non-(second-)quantized level...

To go another direction, in addition to an "advanced" look at classical field theory, I'd be just as interested in a book putting forward alternative formulations of classical field theories (as I've been using the term -- e.g. first quantized fields), whether either in general or focusing on one field in particular...for instance, if I'm not mistaken, any field can be trivially re-interpreted as a fiber bundle -- but I don't think I've seen any book-length expositions of e.g. electromagnetism as a U(1) bundle, with all of the consequences explicitly derived...I think something like that would be equally interesting / useful, since by necessity it would need to discuss classical fields in a different way. I've seen similar things done for electromagnetism with differential forms (Hehl) or Clifford Algebras (I forget the author), but I've yet to see a similar field theoretic treatment.

...Advanced books on GR (Wald, telephone book)...
I've never seen the term "telephone book" before, but glancing over at my shelf...you mean Misner/Thorne/Wheeler, don't you :)

I haven't taken a look at Schwinger.
They're definitely well over my head, and that's combining with their focus on traditional QED too make me not too enthused by them (particularly compared to how excited I was by the title / author before peeking inside) -- and much of it would probably be review to you.
 
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samalkhaiat
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if I'm not mistaken, any field can be trivially re-interpreted as a fiber bundle---
but I don't think I've seen any book-length expositions of e.g. electromagnetism as a U(1) bundle, with all of the consequences explicitly derived...I think something like that would be equally interesting / useful, since by necessity it would need to discuss classical fields in a different way. I've seen similar things done for electromagnetism with differential forms (Hehl) or Clifford Algebras (I forget the author), but I've yet to see a similar field theoretic treatment.
The mathematical structure of classical field theory is the subject of 300 page "Part One" in the following text

"Quantum Fields and Strings: A Course for Mathematicians" Volume 1
Pierre Deligne, and Daniel S. Freed, ...
American Mathematical Socity, 1999.
 
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dextercioby
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Hi Sam, excellent finding. I totally forgot about Deligne & Freed. I've stepped over it about 7,8 yrs ago while in college (found it useless back then, diff geo was a stranger to me back then) and never read it since.
 
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The mathematical structure of classical field theory is the subject of 300 page "Part One" in the following text

"Quantum Fields and Strings: A Course for Mathematicians" Volume 1
Pierre Deligne, and Daniel S. Freed, ...
American Mathematical Socity, 1999.
Hi Sam, excellent finding. I totally forgot about Deligne & Freed. I've stepped over it about 7,8 yrs ago while in college (found it useless back then, diff geo was a stranger to me back then) and never read it since.
Wow -- from the name of the book, and even the *first* table of contents, I'd have totally skipped it...I have enough trouble with the well-founded and well-understood aspects of our physical world that delving into the speculative worlds of supersymmetry and string theory just seems like it would be sort of like someone who pukes out the card window on the way home after a wine cooler deciding to start the next night with a pint of Gin.

But your recommendation convinced me to flip a few pages further, and looking at the actual table of contents, this looks like it could be pretty interesting...maybe not exactly what I was looking for, but it looks like it will at least get me some new questions to ask.

And, on top of that, it's not only available on Amazon, but it's actually pretty affordable. I'll definitely be checking it out.

Thanks again.
 
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Siegel's Fields might be relevant. I've never found it very readable, though.

http://arxiv.org/abs/hep-th/9912205
Excellent -- the colors are a bit...distracting...but the table of contents looks like it's got some pieces of what I'm looking for.

As an aside, I hope more authors / publishers start to see the value of this publishing model.
 
  • #13
Sean Carroll's 'Spacetime and Geometry'
Chapter 1
section 1.10 Classical Field Theory
page 37
pretty much gets right to the point of explaining fields very clearly
 
  • #16
dextercioby
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No, thanks! Rubakov has that pesky convention not to use „upstairs indices”. So, even if he uses the natural (+ - - -) notation for the trace of the Minkowski metric tensor, he writes ##F_{\mu\nu}F_{\mu\nu}## for ##F_{\mu\nu}F^{\mu\nu}##. And as if this wasn't bad enough (believe me, it is), he mistakes Lorentz for Lorenz for the name of the gauge condition. :(
 
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he mistakes Lorentz for Lorenz for the name of the gauge condition. :(
Well, that's not attractive.
 
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dextercioby
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Well, that's not attractive.
This is ignorance by the author, I find it unacceptable. There is even a special note on this subject by JD Jacskson in the 3rd edition of his textbook.
 
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This is ignorance by the author, I find it unacceptable. There is even a special note on this subject by JD Jacskson in the 3rd edition of his textbook.
It was a pun on Lorenz attractor. Usually a bad idea, but I couldn't help myself.
 
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Demystifier
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No, thanks! Rubakov has that pesky convention not to use „upstairs indices”. So, even if he uses the natural (+ - - -) notation for the trace of the Minkowski metric tensor, he writes ##F_{\mu\nu}F_{\mu\nu}## for ##F_{\mu\nu}F^{\mu\nu}##. And as if this wasn't bad enough (believe me, it is), he mistakes Lorentz for Lorenz for the name of the gauge condition. :(
If you cannot stand when someone writes ##F_{\mu\nu}F^{\mu\nu}## as ##F_{\mu\nu}F_{\mu\nu}##, then how do you feel about writing it as ##F_{\mu\nu}^2##? I am a tolerant person, but the latter notation really drives me mad.
 
  • #21
vanhees71
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Well, some abuse of notation is tolerable, but not these two examples. All the advantages of the Ricci calculus get spoiled by not using a clear distinction between components of vectors and co-vectors which transform contra- and covariantly respectively. Such a book, I'd never buy. The author shouldn't get a penny for such unjustified sloppiness!

The only place where lower-indices-only is somewhat justified is in Euclidean space using Cartesian bases only.
 
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To go another direction, in addition to an "advanced" look at classical field theory, I'd be just as interested in a book putting forward alternative formulations of classical field theories (as I've been using the term -- e.g. first quantized fields), whether either in general or focusing on one field in particular...for instance, if I'm not mistaken, any field can be trivially re-interpreted as a fiber bundle -- but I don't think I've seen any book-length expositions of e.g. electromagnetism as a U(1) bundle, with all of the consequences explicitly derived...I think something like that would be equally interesting / useful, since by necessity it would need to discuss classical fields in a different way. I've seen similar things done for electromagnetism with differential forms (Hehl) or Clifford Algebras (I forget the author), but I've yet to see a similar field theoretic treatment.
So this is probably not that useful to you anymore, but I wanted to comment that there's also this book that does a similar thing using Clifford algebra (the author writes "algebraic geometry") to cover the material that one would see in their first one or two courses on formal mechanics. It's called "New foundations for classical mechanics," by David Hestenes. Anybody know about it? To be honest, I've glanced through it and don't know what to make of the approach. https://www.springer.com/de/book/9789027725264

From what I can tell, I think José/Saletan does something kind of similar but at a higher level. This is probably symplectic geometry. https://www.cambridge.org/at/academ...mporary-approach?format=PB&isbn=9780521636360

I've seen great suggestions so far in this thread, but I am also intrigued about other books on certain topics (classical mechanics, electrodynamics) that introduce fields at a high level, as discussed by OP. I can only think of these ones andm of course, Jackson for the EM-
 

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