Space and Time in Quantum Theory
Tomislav Živković
Institute Ruđer Bošković, Zagreb
Abstract Symmetry principles contained in the Euclidean group of transformations (translations and rotations) produce a classical notion of a three-dimensional space. This space is almost automatically, without any further analyse or justification, applied to quantum theory. However, detailed analyse of the formalism of quantum theory shows that those very same principles allow for a huge number of additional transformations which are not possible in a classical approach. Each such additional transformation determines a transition to some new “quantum” reference frame. Classical three-dimensional space hence generalises to a quantum space which contains all such reference frames. Essential property of those quantum transformations is that they dissolve the notion of a “point”. In a classical theory the point is a primitive indestructible element of space, and the point in one reference frame is the point in all other reference frames. This is not true in quantum theory. An infinite number of reference frames which in classical approach allow for the same description of physical reality is in quantum theory generalized to an infinite number of “classical” spaces which are all isomorphic to the Euclidean three-dimensional space and in each of those spaces the laws of quantum theory are the same. The points contained in one of those spaces are in other spaces in general delocalised and occupy some finite or even some infinite volume. The notion of a particle trajectory hence loses its absolute meaning. All attempts to reformulate quantum theory in terms of some hidden variables are hence obsolete.
Symmetry principles contained in the Poincare group applied to quantum theory produce similar results. Instead of a single space Minkowski one finds a multitude of such spaces and all those spaces are mutually interlaced. The notion of a four-dimensional event accordingly generalises, and this applies to its space as well as to its time component. In particular, classical notion of a moment of time (which is associated to a definite position in space) in quantum theory loses its absolute meaning and the moment of time in one reference frame can appear as some period of time in another reference frame.
