The General Boundary Seminar - Semester 2019-1 (08/2018-12/2018)

The seminar is organized by Robert Oeckl and held at the CCM-UNAM in Morelia.

date - time - placespeakertitle (click for abstract)
08-22 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

In this talk I want to review the quantization of the free scalar field in the general boundary formulation, for some simple geometries. In this scenario many of the novel phenomena arising when going from spacelike to timelike hypersurfaces and from transition amplitudes to generalized amplitudes become transparent and easy to explain. Phenomena discussed include: The in/out-quantum number, evanescent waves, S-matrix at spacelike infinity, probabilities for processes in regions with connected boundary.

Klein-Gordon theory and the generalized S-matrix [slides]
08-29 - 17:00 - Salón 4Albert Much (CCM-UNAM)
X

One relative easy and quite rigorous path to quantize a field theory on an arbitrary curved spacetime is given by geometric quantization. In principle one uses the wave equation to construct a phase space (+symplectic structure) and a corresponding complex structure. Once we have this structure a simple formula gives us an inner product that allows the construction of a Hilbert space and thus (by taking the infinite symmetric tensor product) a Fock space. In this talk we explicitly and rigorously construct complex structures for explicit examples. Moreover, we prove that the complex structures are topological linear isomorphisms from Sobolev Spaces that we explicitly construct as well.

Rigorous Applicable Quantum Field Theory and the General Boundary Formulation
09-05 - 17:00 - Salón 4Homero Díaz (UMSNH)
X

As a toy example of a quantization of gauge field theories within the GBF formalism we consider abelian Yang-Mills fields on compact Riemannian manifolds with boundary. We start from an axiomatic semiclassical formalism and then use the elementary tools of geometric holomorphic quantization properly adapted to infinite dimensional symplectic linear spaces.

Quantum abelian Yang-Mills fields on Riemannian manifolds [slides]
09-12 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

In two spacetime dimensions Yang-Mills theory is solvable, even in the non-abelian case. A description of the quantum theory in terms of a topological quantum field theory was given first around 1990 by Witten and others. This fits perfectly into the general boundary formulation and I will give a description from this perspective. What is more, I describe the generalized case that includes corners. That is, I allow hypersurfaces to have boundaries and to glue regions along pieces of boundaries. I emphasize how non-abelian gauge symmetries lead to a modification of the tensor product composition rule for hypersurfaces.

2-dimensional quantum Yang-Mills theory with corners [slides]
09-26 - 17:00 - Salón 4
X

It turns out that the classical data of a fermionic field theory can be encoded in a general boundary setting quite analogous to the bosonic case. However, the fermionic grading leads to some surprising features. Thus, Lagrangian subspaces are replaced by hypermaximal nul subspaces and inner products become necessarily indefinite. This leads not only to a novel and beautiful mathematical picture, but also has striking physical implications. In particular, the indefinite inner product can be linked to an emerging notion of time, even if there is no metric present on the manifolds representing spacetime.

The field theory of free fermions - classical theory [slides]
10-03 - 17:00 - Salón 4
X

In this talk I present the quantization of free fermionic field theory in the general boundary formulation. In contrast to the bosonic case, quantization in terms of holomorphic wave functions is not available, but Fock quantization is. As in the classical theory, Hilbert spaces are generalized to Krein spaces. I show how the algebraic notion of time seen in the classical theory is inherited by the quantum theory. I also show how the gluing anomaly can be renormalized and discuss the superselection rules relevant for the probability interpretation.

The field theory of free fermions - quantum theory [slides]
10-10 - 17:00 - Salón 4Daniele Colosi (ENES-UNAM Morelia)
X

We apply the general boundary formulation of quantum theory to a real massless scalar field in two-dimensional Minkowski spacetime subject to Dirichlet boundary condition at timelike hyperplanes. The quantum theory is constructed in two different kind of regions: one with spacelike boundary (given by equal time hyperplanes, corresponding to the standard treatment) and three regions with timelike boundaries. These last quantization schemes exploit the new possibilities provided by the general boundary formulation. In particular, the observable maps defined in two regions sharing a common timelike boundary are composed in order to obtain a functional integral from which the energy density is calculated. The Casimir effect is recovered in all the regions considered by computing the vacuum expectation value of the energy density of the field and subtracting the corresponding quantity computed for the free field (i.e. not subject to boundary conditions).

Casimir effect from composition of amplitudes
10-31 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

Unlike in classical physics, measurement is a primitive notion in quantum theory. Explaining how measurement outcomes can be predicted is thus a fundamental task of any quantum theory, including any quantum theory of gravity. Existing approaches have struggled with this basic requirement. However, recent progress on the foundations of quantum theory has led to a much better understanding of how predictions in quantum gravity can be made in principle. Rather than discussing the underlying framework, as I have done in many previous talks, in this talk I want to focus on a concrete example: A black hole bounce. This also illustrates where certain traditional approaches fail.

Operational quantum gravity: The black hole to white hole transition [slides]
11-07 - 17:00 - Salón 4
X

In the quantization of field theory in curved spacetime appears an ambiguity that can be understood as the freedom to choose the vacuum. This can be encoded in terms of a complex structure and relates notions of time-evolution and energy. I present recent insights obtained in joint work with Daniele Colosi that place this traditional understanding of the vacuum in a much wider context and yield powerful generalizations of the concepts involved. Thus, for a spacetime region the vacuum can be understood as a choice of complexified exterior solution space, realized as a Lagrangian subspace of the space of germs of solutions on the boundary. In a global setting, it encodes asymptotic (but not necessarily temporal!) boundary conditions. This also involves a new understanding of the notion of Wick rotation.

The vacuum as a Lagrangian subspace
11-14 - 17:00 - Salón 4
11-28 - 17:00 - Salón 4Juan Orendain (CCM-UNAM)
X

We present an introduction to area-dependent topological quantum field theory. We show how area-dependency circumvents the issue of finite-dimensionality of spaces of states in Atiyah and Segal's formulation of topological quantum field theory. We further show how the value of area-dependent topological quantum field theories on identities can be interpreted as a completion of certain topological spaces. We finally present, time permitting, examples of area-dependent topological quantum field theories via the Runkel-Szegedy version of Abrams' theorem on Frobenius algebras and 2-dimensional topological quantum field theories. The chief example of the theory is 2-dimensional quantum Yang-Mills theory.

Duality, limits, and cylinders in area-dependent topological quantum field theory
12-05 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

In this talk I would like to give a general overview of quantum gravity in the general boundary formulation: motivation, existing approaches, possible approaches.

Quantum gravity in the general boundary formulation [slides]

Last updated 7 December 2018.