Quantum space and time

Tomislav Živković, Ph.D., Professor
Institute Ruđer Bošković, Croatia

In a classical theory in each reference frame a single particle such as an electron, proton etc. is situated at the well-defined position x in a three-dimensional classical space. Classical transformations are restricted by the condition that in each reference frame such a particle must be localized. In a quantum theory this particle is situated in the state Ψ which is an element of an infinite-dimensional Hilbert space. Depending on the reference frame considered, only some special quantum states such as δ-functions correspond to a classical notion „particle localized in a classical space“. This applies only to a particular reference frame. In another quantum reference frame this state can be substantially delocalized. Quantum transformations are hence not restricted by the classical localizability condition. In quantum theory are possible transformations which connect quantum states with different levels of localizability. Quantum space contains an infinite number of classical spaces. Each of those classical spaces contains an infinite number of reference frames. For each quantum state there is the corresponding classical space where this state is maximally localized. This is a proper classical space of this state. Consider two quantum systems in the mutual interaction. If energies are not extremely high and if those systems interact with velocity independent potentials, in this case those two systems are localized in the same classical space. Therefore those two systems appear to each other as classical systems. To the very good approximation this is the case with gravitational interaction which is mainly velocity independent. Hence all celestial bodies which are close to each other are to a very good approximation localized in the same classical space. However this is only approximately so and it is possible experimentally to detect any deviation from this classical notion. Big Bang is an extreme situation where the state of the Universe appears statistically the same in all possible quantum reference frames. Immediately after Bing Bang effective physical space is hence infinite-dimensional. After some time due to the mutual interaction between various particles this dimension decreases. The most important stable particles are spin 1 /2 particles such as electrons and protons, which satisfy SU(2) symmetry. This is quantum symmetry which is related to the classical SO(3) symmetry (two elements of SU(2) symmetry correspond to one element of SO(3) symmetry). Therefore a quantum infinite-dimensional space at Big Bang finally reduces to a classical three-dimensional space.