Quantum field theories are the basis of most of modern theoretical physics, but they are not well understood mathematically. Although algebraic quantum field theory provides a framework for describing quantum field theories mathematically, there has been little success at constructing specific models, particularly of gauge theories.
On the other hand, the relationship between classical and quantum mechanics has been abstracted and studied in terms of deformation quantization, and the construction of quantum mechanics has been systematized as geometric quantization, and Berezin-Toeplitz quantization. The objective of the proposed project is to take the first steps toward a similar systematic construction for quantum field theories.
Such a construction would be a powerful tool for understanding quantum field theory. Unfortunately, the phase space (space of solutions) of a classical field theory is infinite-dimensional. I propose that this obstacle may be overcome by treating the phase space through the multisymplectic formulation, which uses the finite-dimensional differential geometry of space-time.
By carrying out the first steps of this program at the host institution, I hope to gain the necessary skill and experience to complete the program and eventually formulate and apply the desired quantization construction.
Call for proposal
See other projects for this call