Weinberg gives relations of SO(3)

In summary: Ballentine's book is more straightforward and easier to follow than Weinberg's. [...]In summary, Weinberg's book is difficult to follow, as it uses different notation for the same relations. Ballentine's book is more straightforward and easier to follow than Weinberg's.
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(Weinberg QFT, Vol 1, page 68)

He considers Mass-Positive-Definite, in which case the Little Group is SO(3). He then gives the relations
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Is it difficult to derive these relations? I'm asking this mainly because I haven't seen them anywhere other than in Weinberg's book.

Also, I'm finding them difficult to follow, as on one side of the equations the ##J## have two indices whereas on the other side they have only one index.
 

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These are just some standard relations used when dealing with SO(3), but (as usual) Weinberg's treatment/notation makes it seem more difficult that it really is. Generally, Weinberg is able to deepen an existing understanding, but is not good for learning things the first time.

If you haven't already studied angular momentum aspects of ordinary QM, then try Ballentine ch7. In particular, in sect 7.1 Ballentine derives the quantum spectrum applicable to angular momentum (corresponding to the little ##j##'s and ##\sigma##'s). Where Weinberg uses ##\sigma##'s, Ballentine uses the (more common?) symbol ##m##.

After you've studied the material in Ballentine, you should be able to relate it back to Weinberg's formulas more easily.
 
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  • #3
kent davidge said:
Also, I'm finding them difficult to follow, as on one side of the equations the JJJ have two indices whereas on the other side they have only one index.

This is the standard notation in three dimensions. In D spatial dimensions, there are D(D-1)/2 independent rotations, and they are usually arranged in an antisymmetric tensor [itex]J_{ij} = -J_{ji}[/itex] where [itex]i[/itex] and [itex]j[/itex] run from 1 to D. In three dimensions, it is conventional to define an angular momentum pseudovector [itex]J_i \equiv \epsilon_{ijk} J_{jk}[/itex], which extracts the three independent components of this tensor. The latter is the angular momentum vector you've encountered throughout physics, but it's important to keep in mind that angular momentum cannot be expressed as a vector in other dimensionalities.
 
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  • #4
strangerep said:
try Ballentine ch7. In particular, in sect 7.1 Ballentine derives the quantum spectrum applicable to angular momentum
Hey, is what Ballentine does the same or similar as what's in http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter8.pdf ?

I'm asking this because if it's the same derivation, I'll be done with that one, because it's quite difficult for me to get Ballentine's book in my hands.
 
  • #5
kent davidge said:
Hey, is what Ballentine does the same or similar as what's in http://www.cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter8.pdf ?
That's excruciatingly tedious compared to Ballentine's more direct physicist-oriented treatment.

[...] it's quite difficult for me to get Ballentine's book in my hands.
See your private conversations page.
 
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1. What is SO(3)?

SO(3) is a mathematical representation of the special orthogonal group in three dimensions. It is a group of rotations in three-dimensional space that preserve the distance between points.

2. Why is SO(3) important in physics?

SO(3) is important in physics because it is the mathematical framework for describing the rotational symmetry of physical systems. It is used in a variety of areas, including classical mechanics, quantum mechanics, and electromagnetism.

3. What are the "relations" of SO(3)?

The "relations" of SO(3) refer to the mathematical equations that define the group, such as the commutation relations between the elements of the group and the properties of the group's elements under multiplication and inversion.

4. How does Weinberg use SO(3) in his work?

Weinberg uses SO(3) in his work to study the symmetries of physical systems, particularly in the context of quantum field theory. He uses the group's representations to understand the behavior of particles and their interactions.

5. Can you give an example of an application of SO(3) in physics?

One example of an application of SO(3) in physics is in the study of angular momentum. In quantum mechanics, angular momentum is quantized, meaning it can only take on certain discrete values. These values correspond to the irreducible representations of the SO(3) group, and the group's structure is used to understand the properties of angular momentum in physical systems.

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