Quantum states of indefinite spins: From baryons to massive gravitino
Conference Paper
Overview
Research
Identity
Additional Document Info
View All
Overview
abstract
One of the long-standing problems in particle physics is the covariant description of higher spin states. The standard formalism is based upon totally symmetric Lorentz invariant tensors of rank-K with Dirac spinor components, ψμ1...μK, which satisfy the Dirac equation for each space time index. In addition, one requires ∂μ1ψ μ1...μK = 0 and γμ1ψ μ1...μK =0. The solution obtained this way (so called Rarita-Schwinger framework) describes has-been spin-(K%2b1/2) particles in therest frame and particles of indefinite (fuzzy) spin elsewhere. Problems occur when ψμ1...μK constrained this way are placed within an electromagnetic field. In this case, the energy of the spin-(K%2b1/2) state becomes imaginary and it propagates acausally (Velo-Zwanziger problem). Here I consider two possible avenues for avoiding the above problems. First I make the case that specifically for baryon excitations there seems to be no urgency so far for a formalism that describes isolated higher-spin states as all the observed nucleon and Δ(1232) excitations (up to Δ(1600)) are exhausted by unconstrained ψμ, ψ μ1...μ3, and ψμ1...μs, structures which originate from rotational and vibrational excitations of an underlying quark-diquark string. Second, I show that the γμ1 ψμ1...μK = 0 constraint is a short-hand of a more general definition of the parity-singlet invariant subspace of the squared Pauli-Lubanski vector, W2. I consider the simplest case of K = 1 and construct the covariant projector onto that very state as -1/3(1/m 2W23/4). I suggest to work in the sixteen dimensional vector space, ψ, of the direct product of the four-vector, Aμ, with the Dirac spinor, ψ, i.e., ψ = Aμ ⊗ ψ, rather than keeping space-time and spinor indices separated and to consider (-1/3(1/m2W2%2b3/4)-1) ψ = 0 as the principal wave equation without invoking any further supplementary conditions. In gauging the equation minimally and, in calculating the determinant, one obtains the energy-momentum dispersion relation. The latter turned out to be well-behaved and free from pathologies, thus avoiding the classical Velo-Zwanziger problem.
