4D conformally invariant QFT and the geometry of locally conformally flat space-time
Evan WIlliams Theatre
Peter Hall Building
In this talk classical (1st quantised) and 2nd quantised QFT (quantum field theory) will be discussed in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e. are manifolds) and hence are Möbius structures. We describe natural principal bundles associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields.
Classical quantum field theory (the Dirac equation and Maxwell's equations) is obtained by considering representations of the structure group \(K\subset SU(2,2)\) of a principal bundle associated with a given Möbius structure where \(K\), while being a subset of \(SU(2,2)\), is also isomorphic to \(SL(2,C)\times U(1)\). The analysis requires the use of an intertwining operator between the action of \(K\) on Minkowski space and the adjoint action action of \(K\) on \(su(2,2)\) and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.
Well defined 2nd quantised (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering a natural unitary representation of K on a state space of Schwartz spinors, a Fock space of multiparticle states and a space of fermionic multiparticle states which forms a Grassman algebra.
Scattering processes are associated with intertwining operators between various algebras, which are encoded in an associated bundle of kernel algebras. Kernels can be generated using K invariant matrix valued measures given a suitable definition of invariance. Feynman propagators, fermion loops and the electron self energy can be given well defined interpretations as measures invariant in this sense. An example is given in which the first order Feynman amplitude of electron-electron scattering \(ee\to ee\) is derived from a simple order (2,2) kernel.
Dr John Mashford, The University of Melbourne