Covariant basis induced by parity for the(j,0)⊕(0,j)representation

Abstract

In this work, we build a covariant basis for operators acting on the ðj; 0Þ È ð0; jÞ Lorentz group representations. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank 2j. The covariant properties of parity involve the Jordan algebra of anticommutators of the Lorentz group generators which unlike the Lie algebra is not universal. We make the construction explicit for j ¼ 1=2, 1 and 3=2, reproducing well-known results for the j ¼ 1=2 case. We provide an algorithm for the corresponding calculations for arbitrary j. This covariant basis provides an inventory of all the possible interaction terms for gauge and nongauge theories of fields for these representations. In particular, it supplies a single second-rank antisymmetric structure, which in the Poincaré projector formalism implies a single Pauli term arising from gauge interactions and a single (free) parameter g, the gyromagnetic factor. This simple structure predicts that for an elementary particle in this formalism all multipole moments, Q l E and Q l M , are dictated by the complete algebraic structure of the Lorentz generators and the value of g. We explicitly calculate the multipole moments, for arbitrary j up to l ¼ 8. Comparing with results in the literature we find that only the electric charge and magnetic moment of a spin j particle are independent of the Lorentz representation under which it transforms, all higher multipoles being representation dependent. Finally we show that the propagation of the corresponding spin j waves in an electromagnetic background is causal.

Iterating free-field AdS/CFT: higher spin partition function relations

“…This zero mode factor is essential to reproduce the correct value for the a-anomaly giving extra − 1 2 shift: using (4.7) to find a 1⊥ (3) we get a(T) = 2 a 1⊥ (3) − 1 2 = − 19 60 . 28 Fields transforming in the (p, 0) ⊕ (0, p) representation are Weyl-like tensors (see, e.g., [57]). They may be represented as rank 2p tensors T µ 1 ν 1 µ 2 ν 2 ···µ p ν p antisymmetric in each pair µ i ν i , totally symmetric with respect to the exchange of the pairs (µ i ν i ) and (µ j ν j ), traceless, and obeying a generalized "Bianchi identity" T ···[µνρ] = 0.…”

Spin portal to dark matter

“…The covariant basis for the ð1; 0Þ ⊕ ð0; 1Þ representation space is given by the set of 6 × 6 matrices f1; χ; S μν ; χS μν ; M μν ; C μναβ g where 1 is the identity matrix. The first principles construction of these matrices can be found in [38] and their explicit form depends on the basis chosen for the states in the ð1; 0Þ ⊕ ð0; 1Þ representation. All the calculations in this work are representation independent and rely only on their algebraic properties.…”

“…Using Eqs. (25), (38), we numerically solve the Boltzman equation (18) for different values the couplings g t , g s and g p , matching the solution YðxÞ with the equilibrium solution Y eq ðxÞ in Eq. (20) at high temperatures, i.e., in the relativistic regime x ≪ 1.…”

“…Spin-one matter fields are the generalization of Dirac construction to j ¼ 1, i.e., fields transforming in the ð1; 0Þ ⊕ ð0; 1Þ representation. The basic object is a six-component "spinor" ψðxÞ and the corresponding quantum field theory was studied in [36], taking advantage of the general construction of a covariant basis for ðj; 0Þ ⊕ ð0; jÞ representation space introduced in [38]. For j ¼ 1, the covariant basis is given by the set of 6 × 6 matrices f1; χ; S μν ; χS μν ; M μν ; C μναβ g where χ is the chirality operator, S μν stands for a symmetric traceless (S μ μ ¼ 0) matrix tensor transforming in the (1,1) representation of the HLG, M μν are the HLG generators and C μναβ is a matrix tensor transforming in the ð2; 0Þ ⊕ ð0; 2Þ representation of the HLG.…”

“…It was shown there that a consistent quantum field theory of spin-one matter fields requires a constrained dynamics formalism but the constraints are second class and can be solved along Dirac conventional method [37]. In order to solve the constraints, however, we need to know the algebraic structure of a covariant basis for the operators acting in the ð1; 0Þ ⊕ ð0; 1Þ representation space, which was previously worked out in [38]. This basis naturally contains a chirality operator, χ, and spin-one matter fields can be decomposed into chiral components transforming in the (1,0) (right) and (0,1) (left) representations.…”