Quantising the Foliation in History Quantum Field Theory

As a preliminary to discussing the quantisation of the foliation in a history form of general relativity, we show how the discussion in \cite{SavQFT1} of a history version of free, scalar quantum field theory can be augmented in such a way as to include the quantisation of the unit-length, time-like vector that determines a Lorentzian foliation of Minkowski spacetime. We employ a Hilbert bundle construction that is motivated by: (i) discussing the role of the external Lorentz group in the existing history quantum field theory \cite{SavQFT1}; and (ii) considering a specific representation of the extended history algebra obtained from the multi-symplectic representation of scalar field theory.