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For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ?

See Weinberg QFT Book Vol.1 page 231.

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- #1

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For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ?

See Weinberg QFT Book Vol.1 page 231.

- #2

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Now the special case ##(A,A)## thus consists of only integer-spin representations ##j \in \{2A,2A-1,\ldots,0 \}##. Now the irreducible integer-spin ##j## representations consist of the symmetric traceless tensors of rank ##j##.

This can be seen by rewriting the spherical harmonics in Cartesian components.

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- #4

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In the local-field realization you can start with a classical field theory and represent a spin-1 field as a four-vector field ##V^{\mu}(x)##. A four-vector builds of course a representation of the full proper orthochronous Poincare group and is easily extended to also describe spatial reflections (P/parity symmetry), time reversal (T symmetry), and in quantized form also charge-conjugation (C symmetry).

Now let's see, what this means in terms of representations of the rotation group as a subgroup. We expect to get an irreducible representation ##J=1##, i.e., one expects ##2J+1=3## spin-components, but of course ##V^{\mu}## has four field-degrees of freedom! That's understandable, because you can easily build a scalar field, even a four-scalar field, which is ##\partial_{\mu} V^{\mu}##. This piece is indeed a scalar field under proper orthochronous Poincare transformations and thus also under the rotations as a subgroup. So in order to describe only spin-1 particles you need to impose a constraint

$$\partial_{\mu} V^{\mu}=0$$

to get rid of the unphysical scalar degrees of freedom.

Another constraint is that the particles should have mass ##m##, i.e., it should fulfill

$$\partial_{\mu} \partial^{\mu} V^{\nu}=-m^2 V^{\nu}.$$

There I've used the fact that the generator for temporal and spacial translations, ##x^{\mu} \rightarrow \bar{x}^{\mu}=x^{\mu}+\delta a^{\mu}## are given by ##\hat{p}_{\mu}=-\mathrm{i} \partial_{\mu}## since

$$\bar{A}^{\nu}(\bar{x})=A^{\nu}(x)=A^{\nu}(\bar{x}-\delta a) = A^{\nu}(x) -\delta a^{\mu} \partial_{\mu} A^{\nu}(x)=A^{\nu} + \mathrm{i} \hat{p}_{\mu} a^{\mu} A^{\nu}(x).$$

There are different ways to achieve this. The most simple one is to consider the Proca equation. Here the idea is to make the constraint automatically being fulfilled by the equations of motion. Using the 4D curl, ##F^{\mu \nu} = \partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu}##, we can write the eom. with some constant ##\lambda##

$$\partial_{\mu} F^{\mu \nu}=\lambda A^{\nu}.$$

The left-hand side expands to

$$\Box A^{\nu} - \partial^{\nu} \partial_{\mu} A^{\mu}=\lambda A^{\nu}.$$

Assuming that the constraint projecting out the scalar degree of freedom, ##\partial_{\mu} A^{\mu}=0## is valid, leads to ##\lambda=-m^2##:

$$\Box A^{\nu} - \partial_{\nu} \partial^{\mu} A^{\mu}=\partial_{\mu} F^{\mu \nu}=-m^2 A^{\nu}.$$

Now contracting this equation with ##\partial_{\nu}## yields

$$-m^2 \partial_{\nu} A^{\nu}=0,$$

and as long as ##m^2>0## we see that the constraint ##\partial_{\nu} A^{\nu}=0## is automatically fulfilled.

So the only thing to go on with quantizing the free spin-1 field is to write down a Lagrangian fulfilling the equations of motion, and this is achieved by

$$\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{m^2}{2} A_{\mu} A^{\mu},$$

where I have also assumed ##A^{\mu} \in \mathbb{R}## and then do the usual steps for canonical quantization.

The only trouble with this is that (a) it's not easy to get the massless limit ##m^2 \rightarrow 0##, if you want to use ##m^2## as a regulator for IR problems with photons and (b) it's not easy to build renormalizable interacting theories with such massive vector bosons, because the propagator in energy-momentum space goes not like ##1/p^2## for large ##p## but has a term ##\propto p^{\mu} p^{\nu}/(m^2 p^2)##.

The way out of both problems is to describe the spin-1 field as massive Abelian vector field by introducing an additional auxilliary scalar field, the socalled Stueckelberg formalism, but that's another story and boils finally down to the same conclusion concerning the spin: You have to project out the spin-0 part of the four-vector field ##A^{\mu}##.

As you know, you can build all irreps of the rotation group and its covering group SU(2) by just using tensor products of the fundamental representation of SU(2), i.e., the spin-1/2 representation. The most simple way is to use just the space ##1/2 \otimes 1/2##, i.e., two spins ##|\sigma_1,\sigma_2 \rangle## with ##\sigma_{j} \in \{-1/2,1/2 \}##. This product vectors however are not transforming according to irreps. of SU(2). Here the reduction into the irreducible pieces is very simple. Just consider the anti-symmetriced and the symmetrices product states,

$$|S=0,M=0 \rangle=\frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle-|-1/2,1/2 \rangle)$$

and

$$|S=1,M=1 \rangle = |1/2,1/2 \rangle,\\

|S=1,M=0 \rangle = \frac{1}{\sqrt{2}} (|1/2,-1/2 \rangle+|-1/2,1/2 \rangle),\\

|S=1,M=-1 \rangle = |-1/2,-1/2 \rangle.$$

The point is to have local fields you cannot start right away in the irred. subspace for ##S=1## but you need to consider the representation ##(1/2,1/2)## of the proper orthochronous Lorentz group, which concerning the rotations contain not only ##S=1## but also ##S=0##, and you have to project out the unwanted ##S=0## piece. This is achieved by imposing corresponding constraints by the equations of motion.

- #5

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- #6

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The rotations as a compact subgroup of the proper orthochronous Lorentz group can be unitarily represented with the generators given by ##\vec{J}=\vec{A}+\vec{B}##. For the rotation subgroup the representation induced by ##(A,B)## is thus containing all representations which turn up in the corresponding problem addition of two independent angular momenta, and it can be split into a direct sum of irreducible parts, containing the irreps with ##J \in \{A+B,A+B-1,\ldots,|A-B| \}##.

The issue is a bit complicated. Perhaps it helps to read the excellent book

R. U. Sexl, H. K. Urbantke, Relativity, Groups, Particles, Springer, Wien (2001).

in parallel to Weinberg's vol. I. It helped me a lot to understand the unitary representations of the proper orthochonous Poincare group, which indeed is the fundamental basis of relativistic QFT.

Another very good source is the original paper by Wigner:

E. P. Wigner, On Unitary Representations of the Inhomgeneous Lorentz Group, Annals of Mathematics 40

(1939) 149.

http://dx.doi.org/10.1016/0920-5632(89)90402-7

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- #8

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What's missing in Weinberg's book?

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