Abstract
The quadratic Casimir operator of the special unitary $SU(N)$ group is used to construct projection operators, which can decompose any of its reducible finite-dimensional representation spaces contained in the tensor product of two and three adjoint spaces into irreducible components. Although the method is general enough, it is specialized to the $SU(2N_f) \to SU(2)\otimes SU(N_f)$ spin-flavor symmetry group, which emerges in the baryon sector of QCD in the large-$N_c$ limit, where $N_f$ and $N_c$ are the numbers of light quark flavors and color charges, respectively. The approach leads to the construction of spin and flavor projection operators that can be implemented in the analysis of the $1/N_c$ operator expansion. The use of projection operators allows one to successfully project out the desired components of a given operator and subtract off those that are not needed. Some explicit examples in $SU(2)$ and $SU(3)$ are detailed.
Highlights
The concept of symmetry, and specially gauge symmetry, is crucial in elementary particle physics
Further analyses in nuclear physics struggled to find out how protons and neutrons interact via a strong force to bind together into nuclei
It was discovered that the strong force had an SUð2Þ invariance; it was called isospin symmetry and its irreducible representations were labeled by isospin 1=2; 1; ...A well-known example is the two-dimensional isospin-1=2 representation made up by the proton and neutron
Summary
The concept of symmetry, and specially gauge symmetry, is crucial in elementary particle physics. Various attempts have been made so far to construct grand unified theories of the weak, strong, and electromagnetic interactions It can be pointed out that the method is not limited to the 1=Nc expansion, but it can be used in shell models of atomic and nuclear physics; in this case, the projector method allows one to construct tensor operators which, with the aid of the Wigner-Eckart theorem, can be used to calculate transition amplitudes. V, the method is outlined for the tensor product space of three adjoint spaces In this case, the explicit construction of projection operators becomes a rather involved task, so only a few examples are detailed. The paper is complemented by two appendices, where some supplemental information is provided
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