In Zero Energy Ontology (ZEO), the basic geometric structure is causal diamond (CD), which is a subset of M4× CP2 identified as an intersection of future and past directed light cones of M4 with points replaced with CP2. Poincare symmetries are isometries of M4× CP2 but CD itself breaks Poincare symmetry.
Whether Poincare transformations can act as global symmetries in the "world of classical worlds" (WCW), the space of space-time surfaces - preferred extremals - connecting 3-surfaces at opposite boundaries of CD, is not quite clear since CD itself breaks Poincare symmetry. One can even argue that ZEO is not consistent with Poincare invariance. By holography one can either talk about WCW as pairs of 3-surfaces or about space of preferred extremals connecting the members of the pair.
First some background.
- Poincare transformations act symmetries of space-time surfaces representing extremals of the classical variational principle involved, and one can hope that this is true also for preferred extremals. Preferred extremal property is conjectured to be realized as a minimal surface property implied by appropriately generalized holomorphy property meaning that field equations are separately satisfied for Kähler action and volume action except at 2-D string world sheets and their boundaries. Twistor lift of TGD allows to assign also to string world sheets the analog of Kähler action.
- String world sheets and their light-like boundaries carry elementary particle quantum numbers identified as conserved Noether charges assigned with second quantized induced spinors solving modified Dirac equation determined by the action principle determining the preferred extremals - this gives rise to super-conformal symmetry for fermions.
- The ground states of super-symplectic and super-Kac-Moody representations correspond to spinor harmonics with well-defined Poincare quantum numbers. Excited states are obtained using generators of symplectic algebra and have well-defined four-momenta identifiable also as classical momenta. Quantum classical correspondence (QCC) states that classical charges are equal to the eigenvalues of Poincare generators in the Cartan algebra of Poincare algebra. This would hold quite generally.
- In ZEO one assigns opposite total quantum numbers to the boundaries of CD: this codes for the conservation laws. The action of Poincare transformations can be non-trivial at second (active) boundary of CD only and one has two kinds of realizations of Poincare algebra leaving either boundary of CD invariant. Since Poincare symmetries extend to Kac-Moody symmetries analogous to local gauge symmetries, it should be possible to achieve trivial action at the passive boundary of CD so that the Cartan algebra of symmetries act non-trivially only at the active boundary of CD. Physical intuition suggests that Poincare transformations on the entire CD treating it as a rigid body correspond to trivial center of mass quantum numbers.
- Zero energy states are superpositions of 4-D preferred extremals connecting 3-D surfaces at boundaries of CD, the ends of space-time. One should be able to construct the analogs of plane waves as superpositions of space-time surfaces obtained by translating the active boundary of CD and 3-surfaces at it so that the size of CD increases or decreases. The translate of a preferred extremal is a preferred extremal associated with the new pair of 3-surfaces and has size and thus also shape different from those of the original. Clearly, classical theory becomes an essential part of quantum theory.
- Four-momentum eigenstate is an analog of plane wave which is superposition of the translates of a preferred extremal. In practice it is enough to have wave packets so that in given resolution one has a cutoff for the size of translations in various directions. As noticed, QCC requires that the eigenvalues of Cartan algebra generators such as momentum components are equal to the classical charges.