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Thor Shen
- 17
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How to use the standard techniques of projection operators to obtain the equation (1) by the first formula? Thanks
ChrisVer said:How to get that [itex] 3 \pi^a \pi^b - \delta^{ab} \pi^2 [/itex] is seen by the fact that this combination is the only traceless : (a=b and sum) [itex] 3 \pi^a \pi^a - \delta^{aa} \pi^2 = 3 \pi^2 - 3 \pi^2 =0 [/itex] and symmetric: (interchanging a,b you get the same result)...
In general these type of combinations can be easier obtained (I think) from Young Tableaux...
For the other question, you just have:
[itex] (C+D) \delta^{ac} \delta^{bd} - \frac{1}{3} (C+D) \delta^{ab} \delta^{cd} [/itex]
[itex] \frac{1}{3}(C+D) [3\delta^{ac} \delta^{bd} - \delta^{ab} \delta^{cd} ] =\frac{1}{3}(C+D) [2\delta^{ac} \delta^{bd} + \delta^{ac} \delta^{bd} - \delta^{ab} \delta^{cd} ] [/itex]
The last two terms in the brackets cancel out...
vanhees71 said:How to derive these "projections" is not such a simple thing, because you need representation theory of the rotation group. A great book about this is Lipkin's "Lie groups for pedestrians".
ChrisVer said:I didn't understand the first questions...
the cancelation of deltas you give are a general feature and doesn't apply for a given representation...
A fast way is to use mathematica to show you that the combination is zero for all a,b,c,d which was the way I used to give a fast answer... As for how to see that, well you could play with symmetries? or try to put some given values on a,b,c,d... ?
Sure, you can as well treat the entire business also with the usual isospin states. Then you represent your pions with ##\pi^{\pm}## and ##\pi^0## fields rather than in the real SO(3) representation of isospin. Of course, this is entirely equivalent. The relation between the fields isThor Shen said:Thank you for your recommendation. By the way , [itex]T^{I=2}(s,t)=<I=2,I_3=0|T^{abcd}|I=2,I_3=0> [/itex]? I and I3 are isospin and component of isospin for initial final state,respectively
Isospin is a quantum number that describes the properties of elementary particles in a similar way to spin, but with reference to the strong nuclear force instead of the electromagnetic force. It is used to classify particles that have similar properties, such as protons and neutrons, into isospin multiplets.
Isospin decomposition involves separating a quantum system into different isospin states, with each state having a different isospin value. This allows for the study of isospin symmetry breaking and the effects of the strong nuclear force on particles.
The projection operator technique is a mathematical method used to obtain specific states or properties of a quantum system. In the context of isospin decomposition, it is used to obtain the isospin states of a system by projecting out all other states.
Eq.(1) refers to the mathematical equation used to represent the isospin projection operator technique. It is significant as it allows for the calculation of isospin states and the study of isospin symmetry breaking in a quantum system.
Decomposed isospin is an important concept in nuclear and particle physics, as it helps to understand the properties and interactions of particles in the nucleus. It is also relevant in the study of isospin symmetry breaking and the search for new physics beyond the Standard Model.