# QFT basics?

#### Hurkyl

Staff Emeritus
Gold Member
For some reason, I skimmed the wikipedia article on QFT, and I feel like I kind of have an idea what the basic objects are... it would be nice if I can be told if I'm way off base, or something close, or whatever. I'm trying to figure out just what the objects are first, and I'll worry about learning what you do with them later.

Suppose we're work on some differentiable manifold M.

The basic object of study is a vector bundle on M with an algebraic structure. We would like to think of this as an algebra of operators, so we must find something upon which they can operate.

So, the simplest sort of thing that could serve as a basis state is a scalar field on M. (i.e. a complex valued function) (details of cosntructing Hilbert space not supplied -- I think that's something I can ponder independently)

So, it seems the natural sorts of things one would do to a scalar field to produce a scalar field would be:

(2) Multiply by your favorite scalar field
(3) Differentiate with respect to your favorite tangent vector field

So I can lift these operations to operators on the Hilbert space, and form some sort of algebra of operators.

If I'm not mistaken, (2) and (3) would give rise to operators corresponding to position and momentum according to some coordinate chart, so this would be sufficient for QM.

But we could operate on more interesting things. For instance, I could have a SU(1) valued function on M. With the appropriate connection, I can then differentiate these to get a su(1) valued field on M, but su(1) is just R, making it a scalar field. I guess something along these lines is how you're supposed to do electromagnetism?

Or, I could consider vector fields in my favorite vector bundle on M to be basis states. We produce scalar fields by applying a section of the dual vector bundle, but we might, first, want to do all sorts of fun vector operations like:

(1) Do some sort of linear transformation. (Apply a 1,1 tensor, if we're dealing with the tangent bundle!)
(2) Take the covariant derivative with respect to our favorite vector field.
(3) Some more that I don't know!

Or, I can use more exotic Lie Groups, and differentiate to get more exotic Lie Algebras, to which I can apply dual elements to get numbers. Is that what it would mean that color is SU(3)?

Related Quantum Physics News on Phys.org

#### Mike2

Do you know of any books that takes this differentiable manifold approach to QM and QFT? Thanks.

Hurkyl said:
For some reason, I skimmed the wikipedia article on QFT, and I feel like I kind of have an idea what the basic objects are... it would be nice if I can be told if I'm way off base, or something close, or whatever. I'm trying to figure out just what the objects are first, and I'll worry about learning what you do with them later.

Suppose we're work on some differentiable manifold M.

The basic object of study is a vector bundle on M with an algebraic structure. We would like to think of this as an algebra of operators, so we must find something upon which they can operate.

So, the simplest sort of thing that could serve as a basis state is a scalar field on M. (i.e. a complex valued function) (details of cosntructing Hilbert space not supplied -- I think that's something I can ponder independently)

So, it seems the natural sorts of things one would do to a scalar field to produce a scalar field would be:

(2) Multiply by your favorite scalar field
(3) Differentiate with respect to your favorite tangent vector field

So I can lift these operations to operators on the Hilbert space, and form some sort of algebra of operators.

If I'm not mistaken, (2) and (3) would give rise to operators corresponding to position and momentum according to some coordinate chart, so this would be sufficient for QM.

But we could operate on more interesting things. For instance, I could have a SU(1) valued function on M. With the appropriate connection, I can then differentiate these to get a su(1) valued field on M, but su(1) is just R, making it a scalar field. I guess something along these lines is how you're supposed to do electromagnetism?

Or, I could consider vector fields in my favorite vector bundle on M to be basis states. We produce scalar fields by applying a section of the dual vector bundle, but we might, first, want to do all sorts of fun vector operations like:

(1) Do some sort of linear transformation. (Apply a 1,1 tensor, if we're dealing with the tangent bundle!)
(2) Take the covariant derivative with respect to our favorite vector field.
(3) Some more that I don't know!

Or, I can use more exotic Lie Groups, and differentiate to get more exotic Lie Algebras, to which I can apply dual elements to get numbers. Is that what it would mean that color is SU(3)?

#### dextercioby

Homework Helper
Yes, they're called "geometric quantization approaches". Woodhouse is the latest good book on this.

Daniel.

#### Hurkyl

Staff Emeritus
Gold Member
Ooh, keywords are good!

#### dextercioby

Homework Helper
Hurkyl said:
Ooh, keywords are good!
Why? Don't you like them? I have two books on geometric quantization and i said [1] is the latest to my knowledge.

Daniel.

[1]J.Woodhouse, "Geometric Quantization", OUP, 1997.

#### Hurkyl

Staff Emeritus
Gold Member
I wasn't trying to be sarcastic. Keywords help with searches and stuff, so I'm happy to know for what to look now.

#### dextercioby

Homework Helper
Sorry. I'm really sorry. Ok. Those books are more for quantum mechanics, really, i've not seen applications to fields, but maybe searcing for QFT in curved spacetime might help, too.

Daniel.

#### George Jones

Staff Emeritus
Gold Member
A. Sudbery - Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians
https://www.amazon.com/exec/obidos/tg/detail/-/052163265X/qid=1121294640/sr=1-1/ref=sr_1_1/103-1776375-6387846?v=glance&s=books
[URL[/URL]

6387846?%5Fencoding=UTF8&v=glance]N. Landsman - Mathematical Topics between Classical and Quantum Mechanics[/URL]
H. Araki - Mathematical Theory of Quantum Fields
R.Haag - Local Quantum Physics: Fields Particles, Algebras
J. Sniatycki - Geometric Quantization and Quantum Mechanics
N. Woodhouse - Geometric Quantization
[PLAIN][URL[/URL]

6387846?%5Fencoding=UTF8&v=glance]A. Derdzinksi - Geometry of the Standard Model of Elementary Particles[/URL]
[PLAIN][URL[/URL]

6387846?%5Fencoding=UTF8&no=283155&me=ATVPDKIKX0DER&st=books]G. Naber - Topology, Geometry, and Gauge Fields: Foundations
G. Naber _ Topology, Geometry, and Gauge Fields: Interactions[/URL]

The books by Sudbery and Ticciati both have misleading titles, as neither book concentrates on mathematical rigor. They are meant as texts that cover much the same material as standard physics texts, but in a way that mathematics students might find more amenable. The style of both is crisp, clean, and somewhat formal, but not completely rigorous. In particular, Ticciati covers much the same material as the more standard Peskin and Schroeder, but the representation theory of Lie algebras is done *much* better in Ticciati. Ticciati also mentions very briefly the differential geomrtry of gauge field theory. Both Sudbery and Ticicati are favourites of mine, but I haven't worked through nearly as much of Ticciati as I should have.

I not sure what to say about Landsman, but it seems like it might cover some topics of interest.

Araki and Haag are fairly rigorous, but they don't cover many of the standard topics in physics quantum field theory courses.

Snitycki and Woodhouse are books about the specialized topic of geometric quantization (also mentioned by dextercioby) - a way for going from the classical to the quantum, and the choices that must be made when doing this.

Derdzinki uses modern differential geometry to give a fairly succint treatment, at the classical level, of the fields of the standard model. Naber gives a beautiful treatment of differential geometry and gauge field theory, including developing the topolgy and differential geometry from scratch.

Sudbery, Ticiatti, and Naber all contain many exercises/problems. I don't know about the rest.

Tables of Contents are available at the links that I give above. The link for Naber gives Naber I as the titles, but the "Search Inside" feature is for Naber II. This is a shame, because Naber I covers some very interesting material.

Regards,
George

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