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:(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# QFT: A Modern Intro, by Michio Kaku

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