Elementary particles must have nonnegative mass and finite spin

Elementary particles must satisfy the principles of relativistic quantum field theory. This implies that they are described by nontrivial irreducible unitary representations of the Poincare group, compatible with a vacuum state.

Having a unitary representation of the Poincare group characterizes relativistic invariance, irreducibility corresponds to the elementarity of the particle, and the vacuum is excluded by forbidding the trivial representation.

Causality requires the principle of locality, namely that commutators (or in case of fermions Anticommutators) of the creation and annihilation fields at points with spacelike relative position must commute. Otherwise, the dynamics of distance points would be influenced in a superluminal way.

This rules out many of the irreducible unitary representations, leaving only those with nonnegative mass and finite spin.

By relaxing the assumptions, one can find certain almost acceptable variations of traditional quantum fields:

For the excluded case of zero mas and infinite spin see

Jakob Yngvason,
Zero-mass infinite spin representations of the Poincare group and quantum field theory,
Annals of Physics 64 (1970), 195-203.

G.J. Iverson and G. Mack,
Quantum fields and interactions of massless particles: The continuous spin case,
Comm. Math. Phys 18 (1971), 211-253.

B. Schroer,
Quantization of m^2<0 Field Equations,
Phys. Rev. D 3, 1764 (1971).

A free causal field theory carrying a reducible irreducible representation is described in

R. F. Streater,
Local fields with the wrong connection between spin and statistics,
Comm. Math. Phys. Volume 5, Number 2 (1967), 88-96.