Massive Particle Quantum Fields

Abstract

With the experience of quantum mechanics on the one hand, where time comes as a real parameter and position as an operator in a Bose quantum algebra Q − (C 2), and special relativity on the other hand, where time and position constitute Minkowski spacetime, one may expect a relativistic quantum structure with both time and position as operators. However both time and position x = (t, , x) are used not as operators, but "only" as real parameters for the translation behavior of relativistic fields. The value space of the fields carries the quantum degrees of freedom, with the example of a free scalar quantum field Φ Φ Φ for massive neutral particles: time translations t ∈ R ↓ x(t) spacetime translations x ∈ R 4 ↓ Φ Φ Φ(x) quantum mechanics R → R 4 Q − (W) → Q − (W Y 3) quantum field theory The quantum algebras come as value spaces for mappings {Y 3 −→ W } of the energy-momentum hyperboloids q 2 = m 2 > 0 for free particles. For each momentum q and spin J, there is a quantum algebra over a representation space W ⊇ C 1+2J for spin and chargelike operations. In retrospect, quantum mechanics is characterizable by quantum orbits of time with the quantum structure implemented by position. The quantum structure of the orbits of relativistic spacetime is implemented by field degrees of freedom, e.g., by spin, electromagnetic charge, isospin, etc. Free particles are characterized by irreducible Hilbert representations of the Poincaré group SL(C 2) × R 4. The infinite-dimensional representations for particles are induced by and embed Hilbert representations of spacetime translations R 4 and of position rotations, i.e., of spin SU(2) for massive particles m 2 > 0 (for convenience m = |m| throughout this chapter) and of circu-larity (helicity, polarization) SO(2) for massless particles (chapter "Massless 95