Mathematical Physics Seminar

Friday, 10 Apr 2015, 14:00

Francesco Bussola (Masters student—Department of Physics, Trento)

A practical covariant approach to calculate the graviton Feynman propagator on a Schwarzschild spacetime

The aim of this seminar is to outline an approach to calculate the graviton Feynman propagator on a Schwarzschild spacetime. Gravitons are linearized perturbations of the metric field. We expect graviton quanta to be produced during Hawking evaporation of a black hole, similar to the usual example of scalar particles. We will discuss a practical covariant formalism for the mode decomposition of the graviton field and the linearized Einstein equations on Schwarzschild spacetime. Using the de Donder gauge, we will be able to rewrite the Einstein equations as wave equations for the metric perturbation. These wave equations are the starting point for calculating the Feynman propagator.

Aula Seminari (Room 2-85, Povo 0)

Friday, 27 Mar 2015, 14:00

Valter Moretti (Department of Mathematics, Trento)

QFT in curved spacetime, quasifree states and Hadamard states II

Continuation of the topic from the previous week.

Aula Seminari (Room 2-85, Povo 0)

Friday, 20 Mar 2015, 13:30

Valter Moretti (Department of Mathematics, Trento)

QFT in curved spacetime, quasifree states and Hadamard states

Abstract. I will review the *-algebra formulation of QFT in curved spacetime for a general globally hyperbolic spacetime, discussing in particular the notion of quasifree and Hadamard quasifree state and their relevance in QFT.

The talk, very informal, relies on (part of) the recent review paper by Igor and myself [arXiv:1412.5945].

Aula Seminari (Room 2-85, Povo 0)

Friday, 06 Mar 2015, 14:00

Marco Oppio (Department of Mathematics, Trento)

A mathematical formulation of Dirac's formalism

Quantum Mechanics's standard textbooks present the formalism of the theory in a user-friendly way, starting with the so called Dirac's bra-ket notation. This is an intuitive and not rigorous method to treat quantum states and probability amplitudes. Next to the concept of Hilbert basis, physicists usually introduce the one of generalized basis, with respective completeness and orthogonality relations, without caring about the mathematical meaning of this object. To make it meaningful we introduce the concept of generalized Hilbert space, or Gelfand triple, which permits to talk about "generalized vectors". We analyse the concept of generalized eigenvector of a generic self-adjoint operator and show that any finite set of commuting self-adjoint operators admits a common family of these elements, which is complete: every Hilbert vector can be decomposed with respect to it in a very precise way. To conclude, we discuss the standard examples of position and momentum operator.

Aula Seminari (Room 2-85, Povo 0)

Friday, 30 Jan 2015, 11:00

Umbert Lupo (Department of Mathematics, York, UK)

Non-existence of stationary Hadamard states for a black-hole in a box and for the 1+1 massless wave equation to the left of an accelerated mirror

We conjecture that, on the (non-globally hyperbolic!) subspacetime of Kruskal to the left of a constant Schwarzschild-radius surface (representing an enclosing box) and with Dirichlet boundary conditions at that surface, there is no stationary Hadamard state for the Klein-Gordon field. We also prove an analogue of this conjecture for the massless wave equation on the region of 1+1 Minkowski space to the left of an eternally accelerated mirror. In doing so, we must take care of possible infra-red pathologies of the massless field. Existence of a stationary Hadamard state is however conjectured/known to hold when there is also an image box/mirror in the left Schwarzschild/Rindler wedge.

This conjecture/result may further strengthen the conclusion in that the full (Kruskal) maximal extension of the Schwarzschild spacetime is not relevant for the quantum gravity of a black hole in a box, and, instead lends support to 't Hooft's 1985 "brick-wall" model according to which, to the extent that one can have a description in terms of a classical spacetime inside the box, that classical spacetime is something like the exterior Schwarzschild portion of Kruskal only (for radii less than the radius of the box) with a non-classically-describable region around the horizon itself.

Aula Seminari (Room 2-85, Povo 0)

Monday, 15 Dec 2014, 14:30

Igor Khavkine (Trento, CIRM Research in Pairs)

Supergeometry in classical field theory

Ordinary (bosonic) classical field theory consists of "field" bundle on a spacetime manifold, a variational PDE on the field sections, its space of solutions (the "phase space", an infinite dimensional manifold), and the algebra of smooth functions ("observables") on the phase space, with an induced Poisson bracket. Fermionic field theory is defined analogously, except that the fibers of the field bundle are allowed to be supermanifolds instead of ordinary manifolds. In the physics literature, fermionic field theories are usually treated in an essentially algebraic way, at the level of the super-Poisson algebra of observables, with its interpretation as the algebra of functions on a phase space supermanifold lost. I will discuss how a modern, functorial formulation of supergeometry allows us to describe the fermionic phase space as a geometric object and to apply tools from analysis and PDE theory to answer some questions about fermionic theories that were difficult to study or even formulate in the algebraic treatment.

Aula Seminari (Room 2-85, Povo 0)

Friday, 5 Dec 2014, 15:00

Daniel de la Fuente (Department of Applied Mathematics, University of Granada)

Uniformly accelerated rectilinear motion in General Relativity: completeness of inextensible trajectories

The notion of a uniformly accelerated rectilinear motion of an observer in a general spacetime is analysed in detail. Such a observer may be seen as a Lorentzian circle, providing a new characterization of a static standard spacetime. The trajectories of uniformly accelerated rectilinear observers are seen as the projection on the spacetime of the integral curves of a vector field defined on a certain fiber bundle over the spacetime. Using this tool, we find geometric assumptions to ensure that an inextensible uniformly accelerated rectilinear observer does not disappear in a finite proper time.

Aula Seminari (Room 2-85, Povo 0)

Friday, 5 Dec 2014, 14:00

Juan Jesus Salamanca (Department of Geometry and Topology, University of Granada)

Uniqueness of maximal hypersurfaces in GRW spacetimes - Applications to Calabi-Bernstein type problems

The main aim of this talk is to introduce a new technique in order to study maximal hypersurfaces in certain spacetimes. In General Relativity, maximal hypersurfaces are useful to understand some physical aspects of the underlying geometry of the spacetime, as the presence of singularities. On the other hand, a maximal hypersurface appears as a critical point of the area functional. The ambient spacetimes we consider are spatially parabolic generalized Robertson-Walker spacetimes. We will show some features and their relevance in General Relativity. The central technical results concern to assure parabolicity on a complete spacelike hypersurface when some natural extrinsic assumptions are fulfilled. Then, we provide new uniqueness results for complete maximal hypersurfaces. As an application, we solve new Calabi-Bernstein type problems.

Aula Seminari (Room 2-85, Povo 0)

Friday, 28 Nov 2014, 14:00

Igor Khavkine (Department of Mathematics, Trento)

Poisson brackets in field theory via the Peierls formula

In textbook treatments of mechanics and field theory, Poisson brackets are usually only presented in the Hamiltonian framework and between observables at equal times. On the other hand, it has been known since a seminal paper of Peierls (1952) that (even unequal time) Poisson brackets can be obtained directly from the Lagrangian, given the knowledge of advanced and retarded Green functions for the Linearized Euler-Lagrange equations. I will discuss the relation between the covariant symplectic form and Peierls' covariant formula for the Poisson bracket. I will concentrate on the case of theories with unconstrained, well-posed initial value formulations, such as the simple scalar field. Time permitting, I will remark on how the arguments would have to be modified in the presence of constraints and gauge invariance.

Aula Seminari (Room 2-85, Povo 0)

Tuesday, 25 Nov 2014, 16:30

Florian Hanisch (Potsdam, CIRM Research in Pairs)

Introduction to the basic ideas of supergeometry

The concept of a supermanifold was developped by Berezin and others in order to describe a classical analogue of physical systems containing fermionic degrees of freedom. The function algebras of these spaces also contain elements, which anticommute with each other. We will first give an introduction to the ringed space approach to the subject and discuss some geometric constructions in this framework. In a second step, we will introduce the functorial approach to supergeometry which is more useful to describe infinite-dimensional objects, e.g. spaces of (fermionic) fields. If time allows, we will give an easy example of a classical field theory containing Fermions, formulated in this formalism.

Aula Seminari (Room 2-85, Povo 0)

Friday, 21 Nov 2014, 14:00

Alberto Melati (Department of Physics, Trento)

Curvature fluctuations in asymptotically de Sitter spacetimes via the semiclassical Einstein's equations

It has been proposed recently to consider in the framework of cosmology an extension of the semiclassical Einstein's equations in which the Einstein tensor is considered as a random function. This paradigm yields a hierarchy of equations between the n-point functions of the quantum, normal ordered, stress energy-tensor and those associated to the stochastic Einstein tensor. Assuming that the matter content is a conformally coupled massive scalar field on de Sitter spacetime, this framework has been applied to compute the power spectrum of the quantum fluctuations and to show that it is almost scale-invariant. We test the robustness and the range of applicability of this proposal by applying it to a less idealized, but physically motivated, scenario, namely we consider Friedmann-Robertson-Walker spacetimes which behave only asymptotically in the past as a de Sitter spacetime. We show in particular that, under this new assumption and independently from any renormalization freedom, the power spectrum associated to scalar perturbations of the metric behaves consistently with an almost scale-invariant power spectrum.

Aula 7 (Povo 0)

Friday, 31 Oct 2014, 14:00

Igor Khavkine (Department of Mathematics, Trento)

Variational bicomplex in the calculus of variations

Recapping from last week, I will summarize the relation between calculus of variations and the algebra of differential forms on jet space, including horizontal and vertical differentials, and evolutionary vector fields. I will then show how the horizontal and vertical differentials give rise to the so-called variational bicomplex. The structure of the bicomplex can be used to give precise meaning to the formal construction of the symplectic structure directly from a Lagrangian. Time permitting, I will also discuss conservation laws and Noether's theorem.

Aula Seminari (Room 2-85, Povo 0)

Friday, 24 Oct 2014, 14:00

Igor Khavkine (Department of Mathematics, Trento)

Jets and the variational bicomplex

Variational calculus, as used in Lagrangian mechanics, is often presented in terms of formal manipulations with functional derivatives. These manipulations can in fact be given a rigorous interpretation using the finite dimensional geometry of jets and forms on jet spaces. I will briefly introduce jets and jet bundles as geometric objects that capture the information about higher derivatives, as tangent and cotangent bundles do for first derivatives. I will then discuss special features of the algebra of differential forms on jet bundles, which gives rise to the so-called variational bicomplex. The variational bicomplex gives a rigorous interpretation to formal manipulations with functional derivatives and captures a lot of information about the local geometry of variational calculus.

Aula Seminari (Room 2-85, Povo 0)

Friday, 17 Oct 2014, 14:00

Igor Khavkine (Department of Mathematics, Trento)

Covariant phase space method in mechanics and field theory

Canonical (aka symplectic and Poisson) structure in mechanics and field theory is usually introduced by using the Legendre transform to pass from the Lagrangian to the Hamiltonian formalism. Unfortunately, the Legendre transform can be ambiguous for field theories or for singular Lagrangians (gauge theories) and, by selecting a preferred time variable, the Hamiltonian formalism breaks manifest relativistic invariance. On the other hand, it is well-known that the symplectic and Poisson structure can be obtained directly from the Lagrangian, avoiding the above inconveniences. This is sometimes known as the "covariant phase space method". I will make an informal introduction to the subject. Time permitting, I will discuss how this method is mathematically formalized using the language of jets and the variational bicomplex.

Aula Seminari (Room 2-85, Povo 0)