Exploring the Objects of Quantum Field Theory: A Brief Overview

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In summary: I wasn't trying to be sarcastic. Keywords help with searches and stuff, so I'm happy to know for what to look now.In summary, the conversation discusses the basic objects of study in quantum field theory, which are vector bundles on a differentiable manifold with an algebraic structure. These objects can be thought of as an algebra of operators, with the simplest being a scalar field on the manifold. The conversation also explores different operations that can be applied to these objects, such as differentiation and linear transformations, and the potential applications to electromagnetism and other fields. Various books on geometric quantization and quantum mechanics are mentioned as possible resources for further study.
  • #1
Hurkyl
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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:

(1) Add your favorite scalar field
(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)?
 
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  • #2
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:

(1) Add your favorite scalar field
(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)?
 
  • #3
Yes, they're called "geometric quantization approaches". Woodhouse is the latest good book on this.

Daniel.
 
  • #4
Ooh, keywords are good!
 
  • #5
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.
 
  • #6
I wasn't trying to be sarcastic. Keywords help with searches and stuff, so I'm happy to know for what to look now.
 
  • #7
Sorry.:redface: 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.
 
  • #8
A. Sudbery - Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians

[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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1. What is QFT?

Quantum Field Theory (QFT) is a theoretical framework in physics that combines quantum mechanics with special relativity to explain the behavior of subatomic particles.

2. What are the objects of QFT?

The objects of QFT are fields, which represent the fundamental building blocks of matter and interactions. These fields are described by mathematical equations and can be thought of as tiny vibrations or fluctuations in space-time.

3. How are the objects of QFT explored?

The objects of QFT are explored through experiments and mathematical calculations. Scientists use particle accelerators, such as the Large Hadron Collider, to smash particles together and observe the interactions. They also use mathematical models and equations to make predictions and test the theories of QFT.

4. What are the implications of studying the objects of QFT?

Studying the objects of QFT has led to a deeper understanding of the fundamental nature of matter and energy. It has also led to the development of technologies such as transistors, lasers, and MRI machines. QFT has also been used to make predictions about the behavior of particles and the universe, such as the existence of the Higgs boson.

5. What are some current challenges in exploring the objects of QFT?

One of the current challenges in exploring the objects of QFT is the development of a theory that unifies quantum mechanics with general relativity. Another challenge is understanding the nature of dark matter and dark energy, which are believed to make up a large portion of the universe but cannot be explained by current theories of QFT.

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