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26/02/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática
John Huerta, IST, Lisbon

What can higher categories do for physics? Part II

In this follow up to last year's talk, we briefly review the cobordism hypothesis that formed the subject of our first part, and then outline its use for the existence and construction of field theories, in particular Chern-Simons theory, as discussed in a 2009 paper of Freed, Hopkins, Lurie and Teleman.

22/01/2014, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
, Grupo de Fisica Matemática, Universidade de Lisboa

Introduction to renormalization in QFT (part II)

In the previous talk we gave an overview of the renormalization procedure in Quantum Field Theory. In this lecture we will demonstrate that abstract procedure on a simple explicit example, the so-called ϕ 4 theory of a single real scalar field. We will illustrate the construction of a renormalized state sum using two different regularization schemes, construct the renormalization group equations, and discuss some of their properties.

Ver também

2014-Lisbon-TQFTclub-Renormalization-Lecture.pdf

18/12/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
Marko Vojinovic, Grupo de Fisica Matemática, Universidade de Lisboa

Introduction to renormalization in QFT

We will give an overview of the renormalization procedure in Quantum Field Theory. The emphasis will be on the general idea of constructing a finite QFT from the one plagued by divergencies, in the standard perturbative approach, and discussing the uniqueness of the resulting QFT. The lecture does not assume much background knowledge in QFT, and should be accessible to a wide audience.

Ver também

https://math.ist.utl.pt/seminars/download.php?fid=9

11/12/2013, 17:00 — 18:00 — Sala P4.35, Pavilhão de Matemática
Carlos Guedes, AEI, Golm-Potsdam

The non-commutative Fourier transform for Lie groups

The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. In quantum mechanics on the Euclidean space, the standard Fourier transform gives a unitary map between the position representation -- functions on the configuration space -- and the momentum representation -- functions on the corresponding cotangent space. That is no longer the case for systems whose configuration space is a more general Lie group. In this talk I will introduce a notion of Fourier transform that extends this duality to arbitrary Lie groups.

arXiv:1301.7750

04/12/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
, Universidade Lusófona and GFM, Universidade de Lisboa

Quantum mechanics in phase space: The Schrödinger and the Moyal representations

I will present some recent results on the dimensional extension of pseudo-differential operators. Using this formalism it is possible to generalize the standard Weyl quantization and obtain, in a systematic way, several phase space (operator) representations of quantum mechanics. I will present the Schrodinger and the Moyal phase space representations and discuss some of their properties, namely in what concerns the relation with deformation quantization.

27/11/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

What can higher categories do for physics?

We describe Baez and Dolan's cobordism hypothesis - a deep connection between topological quantum field theory, higher categories, and manifolds. Physically, this encodes the idea that quantum field theories, even "topological" ones, should be local: no matter how we cut up the spacetime on which they are defined in order to perform the path integral, the net result must be the same. Recently, this hypothesis was formulated and proved by Jacob Lurie using the tools of homotopy theory. We describe the version of the hypothesis he proved. Finally, we touch on Freed, Hopkins, Lurie and Teleman's recent work on Chern-Simons theory, and on Urs Schreiber's ideas for using Lurie's toolkit in full-fledged quantum field theory.

30/09/2013, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Nuno Freitas, Univ. Bayreuth

The Fermat equation over totally real number fields

Jarvis and Meekin have shown that the classical Fermat equation \(x^p + y^p = z^p\) has no non-trivial solutions over \(\mathbb{Q}(\sqrt{2})\). This is the only result available over number fields. Two major obstacles to attack the equation over other number fields are the modularity of the Frey curves and the existence of newforms in the spaces obtained after level lowering.

In this talk, we will describe how we deal with these obstacles, using recent modularity lifting theorems and level lowering. In particular, we will solve the equation for infinitely many real quadratic fields.

29/08/2013, 15:00 — 16:00 — Sala P4.35, Pavilhão de Matemática
Travis Willse, The Australian National University

Groups of type G 2 and exceptional geometric structures in dimensions 5, 6, and 7

Several exceptional geometric structures in dimensions 5, 6, and 7 are related in a striking panorama grounded in the algebra of the octonions and split octonions. Considering strictly nearly Kähler structures in dimension 6 leads to prolonging the Killing-Yano (KY) equation in this dimension, and the solutions of the prolonged system define a holonomy reduction to a group of exceptional type G 2 of a natural rank-7 vector bundle, which can in turn be realized as the tangent bundle of a pseudo-Riemannian manifold, which hence relates this construction to exceptional metric holonomy. In the richer case of indefinite signature, a suitable solution ω of the KY equation can degenerate along a (hence 5-dimensional) hypersurface Σ, in which case it partitions the underlying manifold into a union of three submanifolds and induces an exceptional geometric structure on each. On the two open manifolds (which have common boundary Σ), ω defines asymptotically hyperbolic nearly Kähler and nearly para-Kähler structures. On Σ itself, ω determines a generic 2-plane field, the type of structure whose equivalence problem Cartan investigated in his famous Five Variables paper. The conformal structure this plane field induces via Nurowski's construction is a simultaneous conformal infinity for the nearly (para-)Kähler structures.

This project is a collaboration with Rod Gover and Roberto Panai.

20/06/2013, 11:00 — 12:00 — Sala P4.35, Pavilhão de Matemática
John Huerta, IST, Lisbon

QFT V

In the final lecture of our gentle introduction to quantum field theory, we discuss the renormalization of phi cubed theory at one loop.
Please note that the seminar lasts an hour and a half, 11h00 - 12h30.

14/06/2013, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

QFT IV

We will introduce Feynman diagrams by studying finite-dimensional Gaussian integrals and their perturbations, leading up to phi-cubed theory.

24/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

QFT III

Last time, we talked about quantization of the free scalar field by replacing the modes of the field by quantum oscillators. Now, we put this field into the form used by physicists, and talk about the Wightman axioms, which allow a rigorous treatment of free fields.

17/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

QFT II

We continue our gentle introduction to quantum field theory for mathematicians. We discuss the Klein-Gordon equation, and how it decomposes into oscillators. We quantize this system by quantizing the oscillators, obtaining the free scalar field, the simplest quantum field there is.

10/04/2013, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

QFT I

This series of lectures will be a gentle introduction to quantum field theory for mathematicians. In our first lecture, we give a lightning introduction to quantum mechanics and discuss the simplest quantum system: the harmonic oscillator. We then sketch how this system is used to quantize the free scalar field.
Note: room 4.35 is on the 4th floor at the end of the main corridor

03/04/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

Anomalies IV

We will introduce the notion of stable isomorphism for gerbes, and talk about how stable isomorphism classes are in one-to-one correspondence with Deligne cohomology classes. We define WZW branes and discuss how the basic gerbe on a group trivializes when restricted to the brane.

20/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Aleksandar Mikovic, Univ. Lusófona

Categorification of Spin Foam Models

We briefly review spin foam state sums for triangulated manifolds and motivate the introduction of state sums based on 2-groups. We describe 2-BF gauge theories and the construction of the corresponding path integrals (state sums) in the case of Poincaré 2-group.

References

  • J. F. Martins and A. Mikovic, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15 (2011) 1059, arxiv:1006.0903
  • A. Mikovic and M. Vojinovic, Poincaré 2-group and quantum gravity, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694
Room 3.10 is confirmed

13/03/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

Anomalies III

We continue examining Gawedzki and Reis's paper:

WZW branes and gerbes, http://arxiv.org/abs/hep-th/0205233

We define a gerbe, and show gerbes can be "transgressed" to give line bundles over loop space. Trivial gerbes give trivial bundles on loop space, whose sections are thus mere functions. Any compact, simply connected Lie group comes with a god-given gerbe whose curvature is the canonical invariant 3-form. Restricting this gerbe to certain submanifolds, we get trivial gerbes who thus transgress to trivial line bundles, "cancelling" the anomaly of a nontrivial line bundle.

Room 3.10 is now confirmed

27/02/2013, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

Anomalies II

We continue our informal discussion of anomalies by talking about global anomalies on branes, and their relationship with gerbes.
Room 3.10 is confirmed

06/02/2013, 14:00 — 15:00 — Sala P5.18, Pavilhão de Matemática
John Huerta, Instituto Superior Técnico

Introduction to anomalies

In physics, an "anomaly" is the failure of a classical symmetry at the quantum level. Anomalies play a key role in assessing the consistency of a quantum field theory, and link up with cohomology in mathematics, a general tool by which mathematicians understand whether a desired construction is possible. In this informal series of talks, we aim to understand what physicists mean by an "anomaly" and their mathematical interpretation.

19/12/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
Sara Tavares, Univ. of Nottingham

Observables in 2D BF theory

BF theory in two dimensions has been the subject of intensive study in the last twenty five years. I will readdress it by highlighting the TQFT interpretation of the spinfoam approach to its quantisation. I will also introduce the mathematical model that allows us to treat surfaces with inbuilt topological defects and how we expect them to relate to operators in the quantum field theory.

28/11/2012, 11:30 — 12:30 — Sala P4.35, Pavilhão de Matemática
, Univ. of Illinois at Chicago

Non-Commutative Worlds and Classical Constraints

This talk shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one (i) arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces t (temporal quantity) by it. In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator [P,Q]=i is explained. We discuss iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of non-commutative worlds. This generalization depends upon the definitions of derivatives via commutators and upon the way the non-commutative calculus mimics standard calculus. We examine constraints that link standard and non-commutative calculus and show how asking for these constraints to be satisfied leads to some possibly new physics.

Ver também

https://www.math.ist.utl.pt/seminars/qci/index.php.en?action=show&id=3243
Note also another seminar session by the same speaker on Friday 30th November

Páginas de sessões mais antigas: Anterior 5 6 7 8 9 10 11 12 Mais antiga


Organizadores correntes: Roger Picken, Marko Stošić.

Projecto FCT PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry.

CAMGSD FCT