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Register a new userNew class of spin projection operators for 3D models
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A new set of projection operators for three-dimensional models are constructed. Using these operators, an uncomplicated and easily handling algorithm for analysing the unitarity of the aforementioned systems is built up. Interestingly enough, this method converts the task of probing the unitarity of a given 3D system, that is in general a time-consuming work, into a straightforward algebraic exercise; besides, it also greatly clarifies the physical interpretation of the propagating modes. In order to test the efficacy and quickness of the algorithm at hand, the unitarity of some important and timely higher-order electromagnetic (gravitational) systems augmented by both Chern-Simons and higher order Chern-Simons terms are investigated.
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We propose a new basis of spin-operators, specific for the case of planar theories, which allows a Lagrangian decomposition into spin-parity components. The procedure enables us to discuss unitarity and spectral properties of gravity models with parity-breaking in a systematic way.
We elaborate on the spin projection operators in three dimensions and use them to derive a new representation for the linearised higher-spin Cotton tensors.
We elaborate on the traceless and transverse spin projectors in four-dimensional de Sitter and anti-de Sitter spaces. The poles of these projectors are shown to correspond to partially massless fields. We also obtain a factorisation of the conformal operators associated with gauge fields of arbitrary Lorentz type $(m/2,n/2 )$, with $m$ and $n$ positive integers.
We utilize a diagrammatic notation for invariant tensors to construct the Young projection operators for the irreducible representations of the unitary group U(n), prove their uniqueness, idempotency, and orthogonality, and rederive the formula for their dimensions. We show that all U(n) invariant scalars (3n-j coefficients) can be constructed and evaluated diagrammatically from these U(n) Young projection operators. We prove that the values of all U(n) 3n-j coefficients are proportional to the dimension of the maximal representation in the coefficient, with the proportionality factor fully determined by its S[k] symmetric group value. We also derive a family of new sum rules for the 3-j and 6-j coefficients, and discuss relations that follow from the negative dimensionality theorem.
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.
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