Simultaneous emergence of curved space–time and quantum mechanics

It is shown in this paper that the geometrically structureless space–time manifold is converted instantaneously to a curved, a Riemannian, or may be a Finslerian space–time with an associated Riemannian space–time, on the appearance of quantum Weyl spinors dependent only on time in a background flat manifold and having the symplectic property in the abstract space of spinors. The scenario depicts simultaneous emergence of gravity in accord with general relativity and quantum mechanics. The emergent gravity leads to the generalized uncertainty principle, which in turn ushers in discrete space–time. The emerged space–time is specified here as to be Finslerian and the field equation in that space–time has been obtained from the classical one due to the arising quantized space and time. From this field equation we find the quantum field equation for highly massive (of the Planck order) spinors in the associated Riemannian space of the Finsler space, which is in fact, the background homogeneous and isotropic Friedmann–Robertson–Walker space–time of the universe. These highly massive spinors provide the mass distribution complying with the Einstein equivalence principle. All these occurred in the indivisible minimum time considered as zero time or spontaneity.

DYNAMIC REARRANGEMENT OF VACUUM AND THE PHASE TRANSITIONS IN THE GEOMETRIC STRUCTURE OF SPACE-TIME

It is shown that in the case of spontaneous breaking of the original gauge symmetry, a dynamic rearrangement of vacuum may lead to the formation of some anisotropic condensates. The appearance of such condensates causes the respective phase transitions in the geometric structure of space-time and creates a flat anisotropic, i.e. Finslerian event space. Actually there arises either a flat relativistically invariant Finslerian space with partially broken 3D isotropy, i.e. axially symmetric space, or a flat relativistically invariant Finslerian space with entirely broken 3D isotropy. The fact that any entirely anisotropic relativistically invariant Finslerian event space belongs to a 3-parameter family of such spaces gives rise to a fine structure of the respective geometric phase transitions. In the present paper the fine structure of the geometric phase transitions is studied by classifying all the metric states of the entirely anisotropic event space and the respective mass shell equations.

PSEUDO-FINSLERIAN SPACE–TIMES AND MULTIREFRINGENCE

Ongoing searches for a quantum theory of gravity have repeatedly led to the suggestion that space–time might ultimately be anisotropic (Finsler-like) and/or exhibit multirefringence (multiple signal cones). Multiple (and even anisotropic) signal cones can be easily dealt with in a unified manner, by writing down a single Fresnel equation to simultaneously encode all signal cones in an even-handed manner. Once one gets off the signal cone and attempts to construct a full multirefringent space–time metric the situation becomes more problematic. In the multirefringent case we shall report a significant no-go result: in multirefringent models there is no simple or compelling way to construct any unifying notion of pseudo-Finsler space–time metric, different from a monorefringenent model, where the signal cone structure plus a conformal factor completely specifies the full pseudo-Riemannian metric. To throw some light on this situation we use an analog model where both anisotropy and multirefringence occur simultaneously: biaxial birefringent crystal. But the significance of our results extends beyond the optical framework in which (purely for pedagogical reasons) we are working, and has implications for any attempt at introducing multirefringence and intrinsic anisotropies to any model of quantum gravity that has a low energy manifold-like limit.