Mathematical Physics (MP)http://hdl.handle.net/20.500.11824/162022-08-15T15:50:04Z2022-08-15T15:50:04ZCOHOMOLOGY OF CONTACT LOCIBudur, N.Fernández de Bobadilla, J.Le, Q.Nguyen, D.http://hdl.handle.net/20.500.11824/14942022-07-08T22:20:18Z2022-01-01T00:00:00ZCOHOMOLOGY OF CONTACT LOCI
Budur, N.; Fernández de Bobadilla, J.; Le, Q.; Nguyen, D.
We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.
2022-01-01T00:00:00ZClassical dynamics from self-consistency equations in quantum mechanicsBru, J.-B.de Siqueira Pedra, W.http://hdl.handle.net/20.500.11824/14872022-07-06T22:20:07Z2022-05-09T00:00:00ZClassical dynamics from self-consistency equations in quantum mechanics
Bru, J.-B.; de Siqueira Pedra, W.
During the last three decades, P. Bóna has developed a non-linear generalization of quantum mechanics, based on symplectic structures for normal states and offering a general setting which is convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bona's one, with much brother domain of applicability. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. We build a Poisson bracket for the polynomial functions on the hermitian weak∗ continuous functionals on any C∗-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C∗-algebra, by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C∗-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C0-groups of contractions. As a matter of fact, in general commutative C∗-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak ∗ Gâteaux derivative, state-dependent C∗-dynamical systems and the weak∗-Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convex weak∗-compact sets generically have weak∗-dense extreme boundary in infinite dimension. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here. Note that the present paper is an extended version of the published one.
2022-05-09T00:00:00ZTime Dynamics in Quantum Field Theory SystemsLIll, S.http://hdl.handle.net/20.500.11824/14822022-06-24T22:20:01Z2022-12-23T00:00:00ZTime Dynamics in Quantum Field Theory Systems
LIll, S.
In this doctoral thesis, we develop and investigate new mathematical tools that are intended to allow for a rigorous description of non–perturbative quantum field theory (QFT) dynamics. Here, the term QFT is to be understood as describing a quantum system with particle creation and annihilation that can, but does not need to, comply with special relativity. The tools aim at cases where a formal Hamiltonian exists but is ill–defined.
2022-12-23T00:00:00ZImplementing Bogoliubov Transformations Beyond the Shale-Stinespring ConditionLIll, S.http://hdl.handle.net/20.500.11824/14812022-06-23T22:20:00Z2022-04-28T00:00:00ZImplementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition
LIll, S.
We provide two extensions of a dense subspace of Fock space, such that Bogoliubov transformations become implementable on them, even though they violate the Shale-Stinespring condition, so they are not implementable on Fock space. Both the bosonic and fermionic case are covered. Conditions for implementability in the extended sense are stated and proved. From these, we derive conditions for a quadratic Hamiltonian to be diagonalizable by a Bogoliubov transformation that is implementable in the extended sense. Three examples illustrate situations, in which an implementation in the extended sense is possible although the Shale-Stinespring condition fails to hold.
2022-04-28T00:00:00Z