The action for gravity and the standard model includes, as well as the
positive energy fermion and boson fields, negative energy fields. The Hamiltonian
for the action leads through a positive and negative energy symmetry of the
vacuum to a cancellation of the zero-point vacuum energy and a vanishing
cosmological constant in the presence of a gravitational field solving the
cosmological constant problem. To guarantee the quasi-stability of the vacuum, we
postulate a positive energy sector and a negative energy sector in the universe
which are identical copies of the standard model. They interact only weakly
through gravity. As in the case of antimatter, the negative energy matter is
not found naturally on Earth or in the universe. A positive energy spectrum
and a consistent unitary field theory for a pseudo-Hermitian Hamiltonian is
obtained by demanding that the pseudo-Hamiltonian is ${\cal P}{\cal T}$
symmetric. The quadratic divergences in the two-point vacuum fluctuations and the
self-energy of a scalar field are removed. The finite scalar field self-energy
can avoid the Higgs hierarchy problem in the standard model.
Abstract: Energy-parity has been introduced by Kaplan and Sundrum as a
protective symmetry that suppresses matter contributions to the cosmological
constant [KS05]. It is shown here that this symmetry, schematically Energy --> -
Energy, arises in the Hilbert space representation of the classical phase
space dynamics of matter. Consistently with energy-parity and gauge symmetry, we
generalize the Liouville operator and allow a varying gauge coupling, as in
"varying alpha" or dilaton models. In this model, classical matter fields can
dynamically turn into quantum fields (Schroedinger picture), accompanied by a
gauge symmetry change -- presently, U(1) --> U(1) x U(1). The transition
between classical ensemble theory and quantum field theory is governed by the
varying coupling, in terms of a one-parameter deformation of either limit.
These corrections introduce diffusion and dissipation, leading to decoherence.
In this paper we study a new symmetry argument that results in a vacuum
state with strictly vanishing vacuum energy. This argument exploits the
well-known feature that de Sitter and Anti- de Sitter space are related by analytic
continuation. When we drop boundary and hermiticity conditions on quantum
fields, we get as many negative as positive energy states, which are related by
transformations to complex space. The paper does not directly solve the
cosmological constant problem, but explores a new direction that appears
worthwhile.
We outline the evaluation of the cosmological constant in the framework of
the standard field-theoretical treatment of vacuum energy and discuss the
relation between the vacuum energy problem and the gauge-group spontaneous
symmetry breaking. We suggest possible extensions of the 't Hooft-Nobbenhuis
symmetry, in particular, its complexification till duality symmetry and discuss the
compatible implementation on gravity. We propose to use the discrete
time-reflection transform to formulate a framework in which one can eliminate the
huge contributions of vacuum energy into the effective cosmological constant
and suggest that the breaking of time--reflection symmetry could be responsible
for a small observable value of this constant.