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Quantum probability, geometry and gravity


The main theme of the proposed research is the study of quantum probability and geometry and their relation. The longer term aim is to develop a quantisation scheme that will be employed in the context of quantum gravity. There are three main objectives to the proposed research. Firstly, in continuation of work in the previous Marie Curie grant, we will elaborate on the theory of quantum processes. This is a formalism of quantum theory, which was developed in analogy to stochastic processes but includes the information of the quantum interference phases. We will study quantum differential equations (the analogues of stochastic differential equations), systems in discrete time (which generalise quantum random walks) and the relation to quantum information geo metry. Secondly, we will elaborate on the relation of phase space geometry to quantum probability, which also was a major theme of the previous Marie Curie grant. We will study the second quantisation procedure using as our basic tool the coherent states o f the Poincare and apos; group and focus on the appearance of geometric phases, the geometric description of gauge interactions and the role ofspacetime symmetries. With the successful conclusion of these two objectives, we shall have fully established the geometric description for all known fundamental quantum systems, and we shall be able to develop a quantisation scheme based on quantum processes, which highlights the role of geometry, and consequently of the spacetime symmetries. Such a scheme will be a ble to deal rigorously with theories with non-trivial temporal structure. This will lead us to our third objective, which is the quantisation of models that are relevant for quantum gravity and are characterised by non-trivial temporal structure. We shall try to identify the basic principles of quantum growth processes, which are thought to implement the dynamics in the causal set description of gravity.

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