Construction of Local Quantum Fields Describing Many Masses and Spins

Abstract

We discuss the possibility of constructing, out of particle creation and destruction operators, local quantum fields that transform as representations of the homogeneous Lorentz group. Our immediate goal is to write down a consistent local quantum field theory which can simultaneously describe many particles with different masses and spins. In the case that the field is a finite-dimensional irreducible Lorentz tensor, we are able to carry through our program with no restrictions on the masses considered as functions of the spin, provided the usual connection between spin and statistics is satisfied. However, when the field transforms as a unitary irreducible representation of the homogeneous Lorentz group (an infinite-dimensional representation), the requirement of locality, along with the physical assumption that the masses are bounded below, $m\left(j\right)\mathrm{}\ge m_0>0\mathrm{}$, leads to the restriction that the masses are independent of the spin. This property is shown to hold when the transformation law of the field is taken to be an irreducible finite-dimensional representation ⊗ a unitary irreducible representation. The physical consequences of this result and possible methods for evading it are discussed. Finally, an Appendix is included, where the related problem of orthogonality properties of timelike solutions to infinite-component wave equations is examined. In particular, we show that when the solutions of such wave equations transform as unitary irreducible representations of the homogeneous Lorentz group, only the Majorana representations support a scalar product, which is orthogonal for different spins.