We briefly review the conceptual basis of the recently introduced novel concept of the so-called Local Time in quantum theory. An emphasis is placed on certain cosmological aspects.
We amalgamate three seemingly quite different fields of concepts and phenomena and argue that they actually represent closely related aspects of a more primordial space-time structure called by us wormhole spaces. Connes' framework of non-commutative topological spaces and ``points, speaking to each other'', a translocal web of (cor)relations, being hidden in the depth-structure of our macroscopic space-time and made visible by the application of a new geometric renormalisation process, and the apparent but difficult to understand translocal features of quantum theory. We argue that the conception of our space-time continuum as being basically an aggregate of structureless points is almost surely to poor and has to be extended and that the conceptual structure of quantum theory, in particular its translocal features like e.g. entanglement and complex superposition, are exactly a mesoscopic consequence of this microscopic wormhole structure. We emphasize the close connections with the ``small world phenomenon'' and rigorously show that the micro state of our space-time, viewed as a dynamical system, has to be critical in a scale free way as recently observed in other fields of network science. We then briefly indicate the mechanisms by which this non-local structure manages to appear in a seemingly local disguise on the surface level, thus invoking a certain Machian spirit.
The history of the particle concept is briefly reviewed, with particular emphasis on the `foliation dependence' of many particle creation models, and the possible connection between our notion of particle and our notion of simultaneity. It is argued that the concept of `radar time' (originally introduced by Sir Hermann Bondi in his work on k-calculus) provides a satisfactory concept of `simultaneity' for observers in curved spacetimes. This is used to propose an observer-dependent particle interpretation, applicable to an arbitrary observer, depending solely on that observers motion and not on a choice of coordinates or gauge. This definition is illustrated with application to non-inertial observers and simple cosmologies, demonstrating its generality and its consistency with known cases.
By placing emphasis on conceptual correspondence to the “classical” theory which has been developed within the framework of the time-dependent Hartree-Fock theory, a full quantum theory appropriate for describing large-amplitude collective motion is proposed. A central problem of the quantum theory is how to determine an optimal representation called a dynamical representation; the representation is specific for the collective subspace where the large-amplitue collective motion is replicated as satisfactoryly as possible. As an extension of the classical theory where the concept of an approximate integral surface plays an important role, the dynamical representation is properly characterized by introducing a concept of an approximate invariant subspace of the Hamiltonian.