  An analogy with real Clifford algebras on even-dimensional vector spaces
suggests to assign a couple of space and time dimensions modulo 8 to any
algebra (represented over a complex Hilbert space) containing two self-adjoint
involutions and an anti-unitary operator with specific commutation relations.
  It is shown that this assignment is compatible with the tensor product: the
space and time dimensions of the tensor product are the sums of the space and
time dimensions of its factors. This could provide an interpretation of the
presence of such algebras in PT-symmetric Hamiltonians or the description of
topological matter.
  This construction is used to build an indefinite (i.e. pseudo-Riemannian)
version of the spectral triples of noncommutative geometry, defined over Krein
spaces instead of Hilbert spaces. Within this framework, we can express the
Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative
spectral triple. We exhibit a space of physical states that solves the
fermion-doubling problem. The example of quantum electrodynamics is described.
