There's constant name dropping and random historical information. Stories about the discovery of various mesons, or all the notes about nobel prizes that were won, are just pointless. And descriptions of the confusion before the Standard Model do not aid in understanding the modern theory at all.

There are some major diversions explaining old theories that aren't really necessary. For example he goes through a big section on the "Eightfold Way" which is a historical artifact that doesn't need to be taught in modern explanations of the Standard Model, as it introduces an apparent SU(3) symmetry of u,d,s which is not the real SU(3) symmetry of quarks and is thus unnecessarily confusing. One of the big mistakes is that he spends a lot of time talking about "isospin" as nuclear isospin (proton-neutron identity rotation invariance), but then changes and finally admits that is not the isospin of the standard model and introduces us to Weak isospin.

One of the big mistakes in the book is that he is constantly introducing not-quite-right simplified explanations of things which are really not any simpler, and wind up taking more text to explain the same thing.

He also randomly uses non-standard notation, such as calling the group of rotations in 3 dimensions R3 instead of SO(3) , and he weirdly refuses to explain things, such as using the term SU(2) but noting "the S stands for something technical that we don't need to bother with here". What? Just say it means length-preserving.

I think the explanation of group theory in the book is disappointing. I think lay people can easily understand a lot about groups, and more time should have been spent on this. Even concepts like building up macroscopic rotations by applying infinitesimal ones over and over could be explained.

Worst of all I think a great opportunity is missed. Feynman's QED is a brilliant shining star of explaining the quantum field theory of U(1) in a non-mathematical way, which actually builds up a physical intuition for the reader in a very non-intuitive topic. The author could have focused on the geometry of gauge fields and fiber bundles, and what an SU(2) gauge field is like intuitively. We have an intuitive for what electromagnetic forces are like because we can see them at macroscopic scales, but what would an SU(2) gauge force field be like at macroscopic scales?

If you want an intro to particular physics without mathematics, I can still only recommend "QED". If you want an intro to gauge fields, I recommends Baez's "Gauge Fields, Knots and Gravity".

## 4 comments:

"Too often DDT simply takes the mathematical approach, but then leaves out the actual mathematical details that would make it clear."

That's something that a lot of popular science books do and I find it utterly infuriating. Not dropping complex operators on the casual reader is a perfectly sensible editing decision, but excising all traces of maths, even if it's at the level of high school algebra, that's just talking down to the reader. The whole point of good science is that it produces quantitative, measurable, reproducible predictions instead of wishy-washy qualitative hand-waving. A lot of key concepts like derivatives are fairly easy to explain geometrically and will get you quite far.

Related pet peeve: Presentation of mathematical proofs, which is usually completely bottom-up (when it's done at all). For complicated proofs, you'll sometimes get a few lemmas which then magically imply the main theorem. And you almost always get the "polished", "simplified", as-short-as-possible but fully general versions of everything that make it very hard to see

whysomething is done (behold: the magic quadratic form apropos of nothing whose nonnegativity implies our inequality! Gee, thanks!).Math isn't as hard as most people think it is, it's just that a lot of the literature is didactically awful, and a frightening percentage of math teachers doesn't have any deep understanding of what they're talking about, so they can't explain it properly.

'using the term SU(2) but noting "the S stands for something technical that we don't need to bother with here". What? Just say it means length-preserving.'All transforms in U(2) are length-preserving (|det(A)|=1). For O(3) vs. SO(3), the geometric interpretation is obvious (reflections+rotations vs. just rotations), but in U(2) you can't just have sign flips but an arbitrary complex phase. I don't have any geometric understanding of what det(A)!=+-1 really means in that context. Is there even an obvious interpretation in Euclidean Geometry? If yes, is a transform with determinant i halfway between a rotation and a reflection, whatever that means?

"Is there even an obvious interpretation in Euclidean Geometry? If yes, is a transform with determinant i halfway between a rotation and a reflection, whatever that means?"

A reflection is a scale of -1 along a chosen axis. So reflection and rotation are orthogonal to each other. Half a reflection is a scale of 0 along the axis, i.e. a squash to a plane.

This answer is pretty basic and you know your stuff, so I suspect I didn't understand the question :-)

"All transforms in U(2) are length-

preserving (|det(A)|=1)."

Eh.. good correction.

"but in U(2) you can't just have sign flips but an arbitrary complex phase. I don't have any geometric understanding "

eh.. you would have to talk about the geometry of C2 which is not very intuitive to me. You're multiplying by an overall phase on C2, which is a matrix with phase on the diagonals :

[ e^i phi 0 ]

[ 0 e^ i phi ]

You can use

U(2) = U(1) x SU(2) = SO(1) x SO(3)

(ignoring factors of Z2 and using = instead of twiddle equals)

C2 is the same as R4 , so we can think of the SO(1) x SO(3) as working on R4 (note that it's smaller than SO(4))

BTW C2 is equivalent to a quaternion, the four components are two independent complex numbers :

q = (a+ib) + j*(c+id)

and you can also see that q is close to SO(1) x SO(3) , because :

q = q_w + |q_xyz| * [ijk] dot q_xyz (normal)

that is, the SO(1) rotates the scalar part (q_w) into vector magnitude (|q_xyz|) , while the SO(3) rotates around the normal q_xyz

But this hasn't actually helped my geometric picture yet.

In R4 this phase matrix is applying the same rotation two independent R2xR2 subspaces.

Someone needs to write a good book about this stuff, I certainly need it.

The U(1) x SU(2) factorization seems to be the most useful to me in terms of figuring out what's going on (since I have some intuition about what goes on in the SU(2) =~ Z_2 x SO(3) part from Quaternions).

I got a bit thrown off by having Quaternions in two different places (as model of elements of C^2 and unit quaternions as representatives of elements in SU(2)). But of course the latter operate as (q*rq) on other Quaternions r which leaves their real part invariant - that's where the extra DOF from U(1) goes into (as you explained).

It just boils down to "unlike SU(2), U(2) has no nice geometric interpretation in 3-dimensional euclidean space". It's really just an operation on a pair of complex numbers (which tells me nothing in terms of underlying geometry - except for C=C^1, the effect of the complex superstructure in geometric terms is completely opaque to me) or alternatively constrained rotation in R^4 (which seems easier conceptually but doesn't give any obvious geometric interpretation of the constraints).

But then, the same is true in complex inner product spaces in general - I have some reasonable geometric intuition of what's happening in real inner product spaces, even infinite-dimensional ones, but I have no geometric intuition at all about inner products with imaginary parts.

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