Yang-Mills covariant derivative

[Phrak]

In developing the Yang-Mills Lagrangian, Wikipedia defines the covariant derivative as

$$\ D_ \mu = \partial _\mu + A _\mu (x)$$.

Is A_mu to be taken as a 1-form, so that

$$\ D _\mu \Phi = \partial _\mu \Phi + A _\mu (x)$$

or an operator on \Phi, such that

$$\ D _\mu \Phi = \partial _\mu \Phi + A _\mu (x) \Phi$$

 

[malawi_glenn]

operator

 

[hamster143]

Classically, it's a 1-form field that gets multiplied by Phi. When you quantize, you promote it to operator.

 

[Phrak]

thanks guys

 

[Phrak]

As an exercise, I derived the (classical) connection given by the Wikipedia article. But now I'm wondering what it is that I've derived?

They claim "the gauge field A(x) is defined to have the transformation law:

$$A_\mu (x) \mapsto A_\mu (x) - \frac{1}{g}(\partial_\mu G(x))G^{-1}(x)$$

This is all very nice, but the connection $$A_\mu$$ is zero valued before the transformation isn't it?

If this is the case it would seem to be saying

$$0 \mapsto 0 - \frac{1}{g}(\partial_\mu G(x))G^{-1}(x)$$

 

[hamster143]

No, it's not zero valued, it's completely arbitrary.

Yang-Mills field consists of a set of N matter fields (boson or fermion) and a gauge field. For simplicity, consider scalar fields. Classically, you can describe Yang-Mills field with a set of N numbers and a connection (basically a N*N*4 array of complex numbers) in every point of space. This description has a "gauge symmetry", meaning that many different combinations of numbers describe identical physics. Your transformation law says that, you can start with $$A(x)$$ and $$\Phi(x)$$, apply gauge transformation using arbitrary $$G(x)$$, and you'll have an identical state, in the sense that all physical observables are the same as before the transformation. One consequence is that the Lagrangian must be invariant under the transformation. The other is that all physical observables must be invariant. For example, the field itself is not invariant, therefore it's not measurable. But you can construct invariant quantities, such as the trace of an integral of the connection around any closed path.

 

[Phrak]

No, it's not zero valued, it's completely arbitrary.

Yes, that was my point. A temporary mental block had me stuck.

[/quote]Yang-Mills field consists of a set of N matter fields (boson or fermion) and a gauge field. For simplicity, consider scalar fields. Classically, you can describe Yang-Mills field with a set of N numbers and a connection (basically a N*N*4 array of complex numbers) in every point of space. This description has a "gauge symmetry", meaning that many different combinations of numbers describe identical physics. Your transformation law says that, you can start with $$A(x)$$ and $$\Phi(x)$$, apply gauge transformation using arbitrary $$G(x)$$, and you'll have an identical state, in the sense that all physical observables are the same as before the transformation. One consequence is that the Lagrangian must be invariant under the transformation. The other is that all physical observables must be invariant. For example, the field itself is not invariant, therefore it's not measurable. But you can construct invariant quantities, such as the trace of an integral of the connection around any closed path.[/QUOTE]

Thanks for the clarifications. Can you recommend a text? Wikipedia is proving insufficient of course.

 

[Phrak]

No, it's not zero valued, it's completely arbitrary.

Yes, that was my point. A temporary mental block had me stuck.

And thanks for the clarifications. Can you recommend a text? Wikipedia is proving insufficient of course.

 

[hamster143]

What kind of text do you need?

If you want general QFT, there's no good text, they are all equally bad. It still amazes me that we don't have a single textbook that would bother to keep the distinction between numbers and operators (is it so hard to keep hats?), or to define the Hilbert space of quantum fields, or to maintain some kind of rigor when dealing with loops of virtual particles. Many texts even use dimensional regularization (aka "magic") as the primary device to show how to cancel infinities that arise in those loops. (Which, in my opinion, is an absolute crime against humanity punishable by a lifetime ban on publishing) No one even tries to make the connection between QFT and higher-level theories (classical field theory or nonrelativistic QM). Grassmann numbers (the epitome of unphysical) are the centerpiece of every book, but you have to go into specialized literature to find out about polymer models. Ugh.

If you just want the basics of Yang-Mills theory, chapter 15 of Peskin & Schroeder does a decent job while staying readable.

 

[Phrak]

What kind of text do you need?

If you want general QFT, there's no good text, they are all equally bad. It still amazes me that we don't have a single textbook that would bother to keep the distinction between numbers and operators (is it so hard to keep hats?), or to define the Hilbert space of quantum fields, or to maintain some kind of rigor when dealing with loops of virtual particles. Many texts even use dimensional regularization (aka "magic") as the primary device to show how to cancel infinities that arise in those loops. (Which, in my opinion, is an absolute crime against humanity punishable by a lifetime ban on publishing) No one even tries to make the connection between QFT and higher-level theories (classical field theory or nonrelativistic QM). Grassmann numbers (the epitome of unphysical) are the centerpiece of every book, but you have to go into specialized literature to find out about polymer models. Ugh.

If you just want the basics of Yang-Mills theory, chapter 15 of Peskin & Schroeder does a decent job while staying readable.

Thanks, I'll keep Peskin in mind. I have Weinberg, but I've been loathe to open it, for similar reasons you give above. Guess I'll have to. I really need a good development of classical, local gauge fields, (for the slow learner, of course :) that I can apply to a Kaluza-Klein topology, and see what breaks--or doesn't. The quantum development would be extra icing.

 

[Phrak]

I found Weinberg too thick and he doesn't cover gauge theory much. Kaku's Quantum Field Theory is better, first covering Lie algebra, then devoting a chapter to gauge theory.

 

[Haelfix]

Weinbergs text is probably the most intellectually satisfying, in that its pretty logically consistent. However the notation is atrocious (why carry so many Clebsch coefficients?) and justs serves to mask some of the physics.

Id recommend reading through Zee quickly without doing the exercises and just getting a feel of things, and then find Colemans lectures or just take a class.

 

[Phrak]

Weinbergs text is probably the most intellectually satisfying, in that its pretty logically consistent. However the notation is atrocious (why carry so many Clebsch coefficients?) and justs serves to mask some of the physics.

Id recommend reading through Zee quickly without doing the exercises and just getting a feel of things, and then find Colemans lectures or just take a class.

Thanks Haelfix. After pecking around, I was suprized to find this folderhttps://www.physicsforums.com/forumdisplay.php?f=151"

Some are downloadable texts, or lecture notes, so I can actually see from the table of contents if they contain what I'm after.

The Coleman Lectures look very cool, but my limited bandwidth places them out range.