"Physics is like sex.
Sure, it may give some practical results, but that’s not why we do it."
R. P. Feynman




Mathematical Aspects of Quantum and Classical Physical Theories



Department of Mathematics -- University of Trento


Title:
The algebra of classical observables for Maxwell k-forms on a manifold with time-like boundary
Speaker :
Nicolò Drago, University of Trento
When :
Friday 14 June 2019 at 9:30
Where :
Math Seminar Room
Abstract :
We introduce a star-algebra of classical observables A(M) for the solution space Sol(M) of vector potential configurations on a globally hyperbolic manifold M with time-like boundary. The construction of A(M) is well-known for the case of empty boundary while boundary conditions have to be discussed for non-empty boundaries. We show that the algebra A(M) has two important properties: (a) it is separating for Sol(M), that is, observables in A(M) are capable to distinguish all configurations in Sol(M); (b) it is optimal, namely it is the smallest algebra which is separating for Sol(M) - thus there are no redundant observables. We endow A(M) with a pre-symplectic structure and discuss its degeneracy. Joint work with C. Dappiaggi and R. Longhi.
Title:
Causal fermion systems and quantum field theory
Speaker :
Felix Finster, University of Regensburg
When :
Monday 6 May 2019 at 16:30
Where :
Math Seminar Room
Abstract :
The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries. The dynamics is described by the so-called causal action principle. The aim of the talk is to give a simple introduction, with an emphasis on conservation laws (surface layer integrals) and the connection to quantum field theory.
Title:
Background independence in gauge theories
Speaker :
Jochen Zahn, University of Leipzig
When :
Monday 18 May 2019 at 16:30
Where :
Math Seminar Room
Abstract :
In Quantum Field Theory one frequently splits the fields into a classical (background) part, and a perturbation, which is quantised. It is thus a natural question in which sense the theory is independent of this split. We define background independent observables in a geometrical formulation as flat sections of the observable algebra bundle over the manifold of background configurations, with respect to a flat connection which implements background variations. A theory is then called background independent if such a flat (Fedosov) connection exists. We analyze the obstructions to preserve background independence at the quantum level for pure Yang-Mills theory and find that all potential obstructions can be removed by finite renormalization. Based on joint work with M. Taslimi Tehrani.
Title:
Quantum Calculi: from theory to language design
Speaker :
Margherita Zorzi, University of Verona
When :
Thursday 29 November 2018 at 12:30
Where :
Math Seminar Room
Abstract :
In the last 20 years several approaches to quantum programming have been introduced. In this talk we will focus on functional calculi and in particular on the QRAM architectural model. We explore the twofold perspective (theoretical and concrete) of the approach and we will list the main problems one has to face in quantum language design.
Title:
On the geometry of the space of standard subspaces of a Hilbert space
Speaker :
Karl-Hermann Neeb, FAU Erlangen-Nuernberg
When :
Thursday 22 November 2018 at 14:30
Where :
Math Seminar Room
Abstract :
We discuss a representation theoretic approach to some constructions and results in the theory of local observables (Algebraic Quantum Field Theory). Our perspective will be based on representations of Lie groups by unitary and antiunitary operators (antiunitary representations for short). Such representations arise naturally by applying Tomita-Takesaki modular theory to nets of local observables, provided certain "geometric invariance'' conditions are satisfied, that ensure that modular automorphisms can be implemented geometrically. Key building blocks correspond to antiunitary representations of R^x (standard subspaces and modular objects) and to antiunitary representations of the ax+b group (half-sided modular inclusions). On the mathematical side, the geometric structures specified by antiunitary representations of larger groups in terms of these building blocks are presently far from being well-understood.
Title:
Overview on Adiabatic Quantum Computing
Speaker :
Davide Pastorello, University of Trento & TIFPA
When :
Friday 16 February 2018 at 14:00
Where :
Math Seminar Room
Abstract :
AQC originates as an application of quantum adiabatic theorem to solve optimization problems. Indeed it turns out to be polynomially equivalent to quantum circuit model then it is universal for quantum computation. I would like to give an informal introduction to AQC and discuss an example to solve a 3-SAT problem.
Title:
Geometric and analytic problems in the study of contact interactions
Speaker :
Vladimir Lotoreichik, NPI CAS Prague - Alessandro Michelangeli, SISSA Trieste
When :
Tuesday 21 November 2018 at 14:00
Where :
Math Seminar Room
Abstract :
We plan to give a very informal survey of a few topical problems in the rigorous treatment of quantum interactions with zero range. This is part of the research activity of our present visit at the CIRM Trento. In the first half we present an outlook on the spectral optimization problems arising when a quantum particle moves free in the space or in a domain in the presence of a contact interaction. In the second half we flash the problem of the construction of physically relevant and mathematically rigorous Hamiltonians of point interaction for systems of particles (such as the "unitary gases") and the study of their spectral features (such as the Thomas and the Efimov effect).
Title:
On the classification of static vacuum metrics in presence of a cosmological constant. Part II - Riemannian Penrose Inequality for static metrics with positive cosmological constant
Speaker :
Lorenzo Mazzieri, University of Trento
When :
Wednsday 19 April 2017 at 14:00
Where :
room A107
Abstract :
Static vacuum metrics probably represent the most basic objects in General Relativity. In fact they are solutions to the Einstein Field Equations with vanishing Stress-Energy tensor (vacuum), featuring a very special metric structure (warped product). Such a structure induces a natural foliation of the spacetime into space-like slices which are all isometric to each other, so that the corresponding physical universe is static. For static metrics with positive cosmological constant, we recall the notion of virtual mass introduced in the first part of the seminar, where a Positive Mass Statement was also proved. Building on this, we provide in this context also the precise analog of the Riemannian Penrose Inequality. Time permitting, we show how to employ these instruments in order to obtain a Uniqueness Theorem for the Schwarzschild-de Sitter solution among the static black holes obeying to some natural conditions.
Title:
On the classification of static vacuum metrics in presence of a cosmological constant. Part I – Solutions with zero mass
Speaker :
Stefano Borghini , University of Trento
When :
Wednsday 29 March 2017 at 14:00
Where :
Math Seminar Room
Abstract :
Static vacuum metrics probably represent the most basic objects in General Relativity. In fact they are solutions to the Einstein Field Equations with vanishing Stress-Energy tensor (vacuum), featuring a very special metric structure (warped product). Such a structure induces a natural foliation of the spacetime into space-like slices which are all isometric to each other, so that the corresponding physical universe is static. For this class of solutions we discuss the definition of an appropriate notion of mass. This is particularly relevant when the cosmological constant Λ is positive and the model solutions are compact so that – unlike in the asymptotically flat (Λ = 0) and asymptotically hyperbolic (Λ < 0) situation – a general notion of mass is not available in the literature. Building on this, we characterise the De Sitter solution as the only static vacuum metric with zero mass. Finally, exploiting some particular features of our formalism, we show how the same analysis can be fruitfully employed to treat the case of negative cosmological constant, leading to a uniqueness theorem for the Anti-De Sitter spacetime.
Title:
The viewpoint of differential geometry on the notion of quantum controllability
Speaker :
Davide Pastorello , University of Trento and TIFPA
When :
Thursday 2 March 2017 at 15:00
Where :
Math Seminar Room
Abstract :
In this talk I propose a new geometric approach to study the controllability of quantum systems in terms of symplectic and Riemannian structures on projective Hilbert spaces, exploiting some tools of classical control theory. In particular the notion of accessibility algebra for classical non-linear systems in affine form can be adapted to study quantum controllability within geometric Hamiltonian formulation of quantum mechanics. Moreover operator controllability of a quantum system will be completely characterized in terms of Killing vector fields on the complex projective space w.r.t. Fubini-Study metric.
Title:
Phase space Feynman path integrals as analysis on path space
Speaker :
Naoto Kumano-go , Kogakuin University
When :
Wednesday 15 Feb 2017 at 14:00
Where :
Math Seminar Room
Abstract :
We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.
Title:
Ideas to control magnetic micro-robots
Speaker :
Marta Zoppello , University of Trento
When :
Friday 16 Dec 2016 at 14:00
Where :
Math Seminar Room
Abstract :
Controlling artificial devices that mimic the motion of real microorganisms is attracting increasing interest, both from the mathematical point of view and applications. A model for a magnetically driven micro-swimmer is presented, supported by a feasibility study for its realization. Using the well known Resistive Force Theory (RFT) approach the micro-swimmer can be described by a driftless affine control system where the control is an external magnetic field. Moreover we discuss how to realize different kind of paths.
Title:
On the construction of the Green operators and of the ground state for a massive scalar field theory in AdS
Speaker :
Claudio Dappiaggi , University of Pavia
When :
Friday 2 December 2016 at 11:00
Where :
Math Seminar Room
Abstract :
We consider a real, massive scalar field on the Poincaré domain of the (d+1)-dimensional AdS spacetime. Since the background is not globally hyperbolic, first we determine all admissible boundary conditions that can be applied on the conformal boundary and, subsequently, we address the problem of constructing the advanced and retarded Green operators as well as the two-point function for the ground state satisfying those boundary conditions. We unveil that, in some cases, bound states exist, while, when they are absent, the two-point function can be explicitly written in terms of special functions. In addition, we investigate the singularities of the resulting state, showing that they are consistent with the requirement of being of Hadamard form in every globally hyperbolic subregion of the Poincaré patch.
Title:
Renormalization of vector fields in locally covariant AQFT
Speaker :
Alberto Melati , University of Trento
When :
Friday 25 Nov 2016 at 16:00
Where :
Math Seminar Room
Abstract :
In a fundamental work, Hollands&Wald characterised finite renormalization of scalar fields. Despite the great importance of this result, they used a very unnatural assumption: the analytic dependence on metric of Wick powers. Recently Khavkine&Moretti elaborated a new proof without this unnatural requirement. In this talk I present the natural evolution of this work, discussing the finite renormalization of vector fields.
Title:
Mathematical definition of Feynman path integrals
Speaker :
Sonia Mazzucchi , University of Trento and TIFPA
When :
Friday 11 Nov 2016 at 14:00
Where :
Math Seminar Room
Abstract :
In 1942 R. Feynman proposed an heuristic functional integral representation for the solution of the Schrödinger equation, computing the wave function of a non-relativistic quantum particle as a "sum over all possible histories of the system''. Despite the lack of mathematical rigor, "Feynman path integrals'' are widely used in many areas of theoretical physics, not only as a computational tool, but also as a particularly effective quantization method. From a mathematical point of view, their precise definition is rather difficult and requires the implementation of an integration theory on infinite dimensional spaces alternative to the Lebesgue "traditional" one, in order to handle the lack of absolute convergence of Feynman integrals. In this talk I shall give an overview of of the main mathematical problems in the definition of Feynman path integrals as well as the possible solution. Some recent results and applications will also be discussed.
Title:
Geometry of tensor decomposition starting from spin squeezed states
Speaker :
Alessandra Bernardi , University of Trento
When :
Friday 28 Oct 2016 at 14:00
Where :
Math Seminar Room
Abstract :
My intent for this talk is to start from a state of physical interest like the spin squeezed state, to compute its symmetric rank and to show the geometry that is behind this result. It is something very classical in algebraic geometry and it goes back to XIX century due to the work of JJ Sylvester. I would like to present the geometry of this theory and to show possible generalizations.
Title:
Why don't we formulate quantum theories in real Hilbert spaces? (Part II)
Speaker :
Marco Oppio , University of Trento and TIFPA
When :
Thursday 20 Oct 2016 at 15:00
Where :
Math Seminar Room
Abstract :
As shown in the previous seminar, a "naive" attempt to define a relativistic particle on a real Hilbert space by borrowing the well-known approach of Wigner leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does.
Trying a more solid approach we end up with three possibilities: an equivalent description in terms of Wigner unitary representations on a real, complex or quaternionic Hilbert space. At this point the "naive" result turns out to be a definitely important technical lemma, for it forbids the extreme options. Conclusion, the real theory is actually complex.
Title:
Why don't we formulate quantum theories in real Hilbert spaces?
Speaker :
Valter Moretti, University of Trento
When :
Thursday 13 Oct 2016 at 14:30
Where :
Math Seminar Room
Abstract :
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solér's theorem (1995) closing a long standing problem that can be traced back to von Neumann's mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in real Hilbert spaces. A physical attempt to justify this absence was proposed by Stuckuelberg in the 60s without a mathematically rigorous approach and relying upon not so fundamental principles in our view.
I show that elementary relativistic (but also Galileian) systems (in Wigner's approach) cannot be described in real Hilbert spaces as a consequence of some peculiarity of continuous unitary projective representations of SL(2,C) (the orthochronous Poincaré group) related with the theory of polar decomposition of operators in real, complex or quaternionic Hilbert spaces.
This is the first of a couple of talks regarding a recent result obtained in collaboration with Marco Oppio. The second talk, by Marco, will prove how the obtained result is robust, since it holds also assuming very weak physical hypotheses on the von Neumann algebra of observables thanks to a characterisation of (infinite) dimensional composition algebras, and it implies strong constraints also for formulations of quantum theories in quaternionic Hilbert spaces.