  By replacing the field of complex numbers with the commutative ring of
bicomplex numbers, we attempt to construct interacting scalar quantum field
theories that feature both positive- and negative-energy states. This work
places the tentative ideas proposed in [R. Dickinson, J. Forshaw and P.
Millington, J. Phys. Conf. Ser. 631 (2015) 012059] on more solid and general
mathematical foundations and incorporates the "energy-parity" symmetry
introduced in [A. D. Linde, Phys. Lett. B 200 (1988) 272; D. E. Kaplan and R.
Sundrum, JHEP 0607 (2006) 042]. The interplay of the positive- and
negative-energy states allows for cancellations of the vacuum energy. Both the
positive- and negative-energy states have positive norms, and their direct
mixing is prevented by virtue of the zero divisors of the bicomplex numbers,
thereby eliminating the possibility of negative-energy cascades. We suggest
that the same interplay of positive- and negative-energy states may allow
Haag's theorem to be circumvented, removing the associated criticism of the
Fock representation. We consider scalar theories with cubic and quartic
interactions and describe how this construction may yield transition
probabilities consistent with the standard scattering theory. Whilst these
results are intriguing, we draw attention to potentially serious limitations in
relation to perturbative unitarity.
