QFT: A Modern Intro, by Michio Kaku

[Jimmy Snyder]

I have obtained a copy of this book and I am having a very difficult time understanding it. My problems start at the bottom of page 38 and continue through the middle of page 41. For instance:

The last thing on page 38 is a definition of a scalar field in terms of an equation. The first thing on page 39 is a proof of the equation. As one does not normally prove definitions, I assume that what is meant is the following:

If a field has a Taylor expansion, then it is a scalar field.

and the text at the top of page 39 is a proof of it. Am I correct about this? Anyway, this is not my main sticking point.

My next problem comes at the bottom of page 39 where it says: D(g_i) can be split up into smaller pieces, with each piece transforming under a smaller representation of the same group. I thought that the smaller pieces D_1(g_i) WERE representations, not transformed under them. Am I wrong? Does the author mean that the underlying space that D(g_i) operates on can be split up with each piece transforming under the smaller representations?

This is not made clearer by the statement on the second and third lines of page 40: the basic fields of physics transform as irreducible representations... Does the author mean that fields are representations? I would have expected something like: the basic fields of physics transform UNDER irreducible representations.

Equation (2.29) makes no sense to me at all. When you compare it to equation (2.25), there are two differences, one just annoying, the other confusing. In (2.29) the O operators operate on the unprimed tensor, but in (2.25) on the primed vector. This is fixed by multiplying both sides by O(-\theta). However, the U operators have disappeared so that under the definition of a tensor, a vector is not a tensor. Is this just a typo?

Next is the statement near the bottom of page 40 that \epsilon ^{ij} is a genuine tensor, while at the top of page 41 it is referred to as a pseudotensor. I assume the author means that it is a genuine tensor under SO(2) and a pseudotensor under O(2). Is this correct?

Finally, we have equation (2.34) where I assume U is being overloaded with a meaning different from the one in eqn (2.22).

In general, I think that the author is using language that is ambiguous between a representation and the space upon which a representation operates, but I haven't been able to pick apart the meanings in the text. For example, when the author says on page 40:

within the collection of elements that compose the tensor, we can find subsets that by themselves form representations of the group.

he means:

within the collection of elements that compose the tensor, we can find subsets that form a space which is the underlying space for a representation of the group.

I would appreciate any help I can get on this stuff.

 

[Physics Monkey]

Hi jimmy,

I can't say I envy you reading Kaku's book, I didn't find it very appealing myself. I don't have a copy of the book, but I will try to answer your general questions.

1) Scalar fields
Usually scalar fields are defined in terms of their transformation properties under the Lorentz group: [tex] \phi'(x') = \phi(x) [/tex]. I don't know what Kaku is referring to with his comment about Taylor expansions. As an aside, what you really want to study is the covering group of the Lorentz group.

2) Reps
The set of matrices [tex] \{ D(g) \} [/tex] form a representation of the group. The general representation can be decomposed into irreducible pieces which are themselves representations. A representation acts on some vector space, and we say the vector space carries the representation. A representation carried by some vector space is reducible if there are proper subspaces that transform into themselves.

3) Fields
Fields are not representations, the carry representations. In other words, the components of the fields get scrambled by the matrices of the representation. Basic fields typically transform under irreducible representations of the group. A noteworthy exception is the Dirac field which transforms under a reducible representation.

A quick glance suggests that the rest may be too specific for me to answer without the book. I hope this helps some.

 

[Perturbation]

A scalar is a Lorentz invariant function, so it satisfies [tex]\psi (x)=\psi (\Lambda^{-1}x)[/tex] ([tex]\Lambda^{-1}\in SL(1, 3)[/tex]). One can expand it as a taylor series, yes, but this isn't a definition of a scalar.

I've read reviews that say Kaku is pretty bobbins.

 

[vanesch]

I can't say I envy you reading Kaku's book, I didn't find it very appealing myself. I don't have a copy of the book, but I will try to answer your general questions.

I second that. I have it on my bookshelf, and never understood it. I find that Kaku has, in general, the habit of expanding on what is easy, and obscuring what is difficult, with some pseudo-simple argument. Usually, Kaku's writings make more sense once you know the material - then you also see the shortcuts he takes.

 

[Jimmy Snyder]

Usually scalar fields are defined in terms of their transformation properties under the Lorentz group

Thanks, Physics Monkey, for taking the time to look at this.
For the benefit of those who do not have a copy of the book at hand, I will need to add some detail. I will take care of that tonight. For the time being, I will note that on pages 38 to 41, Professor Kaku is working with what he calls the simplest non-trivial group, O(2) (although he goes back and forth between O(2) and SO(2) without always explicitly saying so). His definitions of scalar, vector, tensor, and general fields are, in these few pages, all in terms of O(2). He will work with the Lorentz and Poincare groups in the following pages and of course, these definitions will be changed to be in terms of the Lorentz group then.

I hope this helps some.

It helps a lot. I will have more questions tonight.

 

[Jimmy Snyder]

This is reposted below as message #10

 

[selfAdjoint]

Firstly, our local verion of LaTeX does not recognize the itex tags. Use tex instead.

Secondly Kaku is saying that as a complex function, a scalar is invariant under rotations in the complex plane, which he denotes as multiplication by [tex]e^{i\theta}[/tex] and also as a member of the group SO(2). If you want to express som field given to you by nature as something over the complex numbers, then the way you set the complexes up is independent of where you put the real and imaginary axes. Clearly if you change the orientation of the real and complex axes "rigidly", so they rotate through an arbitrary angle, any complex number with a non-zero magnitude is going to change. But a scalar won't; this gives us an analytic way to descibe a scalar.

 

[Hurkyl]

Firstly, our local verion of LaTeX does not recognize the itex tags. Use tex instead.

What do you mean? Chroot made the [ itex ] tags roughly the same time he made the [ tex ] tags.

 

[Jimmy Snyder]

Firstly, our local verion of LaTeX does not recognize the itex tags. Use tex instead.

I edited my post so that the first instance of itex was changed to tex. This seems to have had no effect. In the past I had no trouble with either tag and on occasion, used them both in the same post.

Secondly Kaku is saying that as a complex function, a scalar is invariant under rotations in the complex plane, which he denotes as multiplication by [tex]e^{i\theta}[/tex] and also as a member of the group SO(2).

Actually, I had no trouble with any of the individual definitions of scalar, vector, or tensor. I have a minor issue with what I perceive as a discrepency between the definitions of vector and tensor and would like to clear it up, but is is not my main issue.

PS: Your tex shows up in my post, but not in yours.

I am reading the appendix in Zee (Nutshell) that covers similar ground and I find it much more readable. One it has cleared up for me is this (unless I got it wrong): The space of vectors (or the space of tensors) is the space upon which the representation acts. With a slight fuzziness of language, the invariant subspaces that are the underlying space of irreducible representations are themselves called irreducible representations of the vector (or tensor). If this is wrong, or could be said more clearly, please jump in.

 

[Jimmy Snyder]

I reposted this as a quote so that the tex would show up.

Here is Professor Kaku on fields:

Let [tex]L = i(x\partial y - y\partial x)[/tex]
Let [tex]U(\theta) = L^{i\theta L}[/tex]

Then define a scalar field [tex]\phi[/tex] as one that transforms under SO(2) as:
[tex]U(\theta)\phi(x)U^{-1}(\theta) = \phi(x')[/tex]

define a vector field [tex]\phi^i[/tex] as one that transforms as:
[tex]U(\theta)\phi^i(x)U^{-1}(\theta) = O^{ij}(-\theta)\phi^j(x')[/tex]

and define a general field [tex]\phi^A[/tex] as one that transforms as:
[tex]U(\theta)\phi^A(x)U^{-1}(\theta) = \mathcal{D}^{AB}(-\theta)\phi^j(x')[/tex]

This progression makes sense when you realize that the implicit 1, and the explicit O and [tex]\mathcal{D}[/tex] are all representations of O. However, he defines a tensor [tex]A^iB^j[/tex] as:

[tex]A^{i'}B^{j'} = \left[O^{i'i}(\theta)O^{j'j}(\theta)\right](A^iB^j)[/tex]

Aside from the minor annoyance that instead of [tex]O(-\theta)[/tex] on the primed side of the eqn, we now have [tex]O(\theta)[/tex] on the unprimed side, the real problem is that U has disappeared. As a result, according to these definitions, a vector is not a tensor. Is this a erratum, or is this the proper way to define things?

 

[Jimmy Snyder]

I quote a customer review from amazon.com concerning this book:

It is too sketchy with group theory, although it at first sight looks like an introduction. Only someone very well versed with group theory and representation can understand this chapter. Unfortunately, this is the mark of a rather sloppy mathematics writer, as further reading confirms.

It seems that I just burned $80. I'm reading Zee now and trying to get a handle on groups and representations. Anyone have a better suggestion?

 

[selfAdjoint]

PS: Your tex shows up in my post, but not in yours.

It seems they fixed whatever was wrong with the LaTeX interpreter.

 

[Jimmy Snyder]

It seems they fixed whatever was wrong with the LaTeX interpreter.

In message #7, I still see this message:

Latex image generation failed, the LaTeX source for this image is invalid.

while the same latex appears correctly when quoted in message #9.

This is a test: [tex]e^{i\theta}[/tex]
and so is this: [itex]e^{i\theta}[/itex]

[tex]e^{i\theta}[/tex]

[itex]e^{i\theta}[/itex]