The General Boundary Seminar - Semester 2018-2 (01/2018-06/2018)

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

date - time - placespeakertitle (click for abstract)
01-31 - 17:00 - Salón 4 organization session
02-07 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

How shall we describe nature? This question is at the center of the foundations of physics. Traditionally, the answer has been that there is an objective reality laid out in space and time. The primary task of the physicist is thus to describe this external reality in an objective way. The description of any interaction with this external reality must then arise as a mere consequence or special case of the inherent laws of this external reality. This view of nature and what it means to describe it comes natural to us as it accords with our everyday experience. It has been at the center of physics as a science since its inception and dominates most fields of physics to this day. Crowning achievements of this approach to physics include mechanics, electrodynamics and the general theory of relativity.

However, with the advent of quantum mechanics this view on how we ought to model nature has turned out to be unsustainable. Suddenly the observer and the act of observation seemed to play a special role, irrespective of the means or device used to observe. The philosophical fall-out of this revolution started already with the inception of quantum mechanics in the 1920s, but lasts to this way. The reactions have ranged from plain denial ("shut up and calculate", Everett interpretation) through attempts to rescue the old order (Bohmian mechanics, collapse theories, superdeterminism) to cautious acceptance that something qualitatively new is occurring (aspects of the Copenhagen interpretation). This sorry state of affairs has contributed to a major road block on the path to a quantum theory of gravity: The lack (until recently) of a generally covariant formulation of quantum theory. The latter has been, of course, the main original motivation for the research program that is presented in the present seminar series.

In the last decade and a half or so it has been increasingly recognized (principally in the quantum foundations community) that another fruitful approach to modeling nature puts our experience or, in physics language, the notion of measurement at the center. The aim of the physicist is thus to describe and successfully predict correlations of measurement outcomes. Whether or not these arise from an underlying external reality in space and time is then a secondary question. (Spoiler alert: The answer is yes precisely if the theory considered is classical.) It tuns out that with such first principles and few further assumptions a mathematical framework for housing physical theories can be build up, that I have called the positive formalism. This is the subject of the present talk. In subsequent talks of this series we shall see that this framework is sufficiently general to accommodate both classical and quantum theories. On the other hand, it is sufficiently restrictive to encode theories in an efficient way, to the degree of suggesting a useful axiomatization of quantum field theory.

The positive formalism: an operational approach to the foundations of physics [slides]
02-14 - 17:00 - Salón 4
X

After recalling the abstract positive formalism introduced previously I want to add in this talk a new structural element: spacetime. This permits implementing the powerful physical principle of locality, which historically was at the heart of the transition from mechanics to relativistic field theory. This prepares the ground for (this local version of) the positive formalism to serve as a framework for statistical classical and quantum field theory, to be explored in later sessions of this seminar series.

The positive formalism: spacetime and locality [slides]
02-21 - 17:00 - Salón 4
X

After recalling the notion of spacetime in the local positive formalism I want to show how further rigidifying time recovers the notions of state and evolution of the pre-relativistic description of physics in terms of dynamical systems. A special role plays the "state of maximal uncertainty" that permits to codify the common situation of a measurement where the system is afterwards discarded.

The positive formalism: time and evolution [slides]
02-28 - 17:00 - Salón 4
X

I continue the development of the positive formalism in the time-evolution setting. I show how the "state of maximal uncertainty" leads to a natural notion of normalization for states. Crucially, this permits the introduction of temporal causality and its arrow of time. This evolution version of the positive formalism is essentially the convex operational framework for physical theories whose development started in the 1970s and which has seen a resurgence in the last 15 years or so under the name of generalized probabilistic theories. It accommodates classical statistical mechanics and statistical quantum mechanics.

The positive formalism: causality and arrow of time [slides]
03-09 - 17:00 - Salón 3Victor Flores (CCM-UNAM)
X

The general boundary formulation (GBF) is an extension of the standard quantum mechanics and the statistical quantum mechanics, among other theories. One way to be sure that the GBF is viable, is that at least it recovers all the results already known of those theories.

In this first talk I will introduce some basic concepts of the quantum mechanics (quantum operation, observable, measurement process, density operator, etc.) and their relation with the GBF. In particular, I will show how we calculate some possible results of a measurement applied to any experiment described in both: in quantum mechanics and in the GBF.

I will finish this talk showing that the observables in quantum mechanics form a partially ordered vector space (a space of self-adjoint operators on a Hilbert space), also that a measurement consists in a completely positive map between two spaces of mixed states.

General Boundary Formulation: recovering (standard) quantum mechanics
03-14 - 17:00 - Salón 4
X

In this talk the concepts we saw in the last talk will be applied to the case of the quantum theory of information. Specifically, I am going to introduce the concept of entropy of a quantum system and I will show the no-cloning theorem and a teleportation protocol (in a system of spins). As fundamental part of this talk, I am going to discuss the problem of hidden variables, I will show a theorem of why it is impossible to describe a quantum system in terms of (local) classical hidden variables (the Bell's inequality). This is a way to introduce and to base the concept of entanglement, the fundamental block in quantum information. As a last part, I will give a clue of how quantum information is well incorporated in the general boundary formulation.

GBF: recovering quantum mechanics (quantum information)
03-21 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

The convex operational framework originated in the 1970s as a means to unify notions such as state, measurement, expectation and intervention between classical and quantum physics. This is also the framework that is recovered from the positive formalism when restricting to a time-evolution picture of physics. I want to show how both classical mechanics and quantum mechanics can be expressed in this unified language.

The convex operational framework: classical and quantum theory [slides]
04-04 - 17:00 - Salón 4Juan Orendain (CCM-UNAM)
X

String diagrams are fundamental objects in what is known as the pictorial calculus for monoidal categories. In this theory objects in monoidal categories are represented by decorated 'strings' and morphisms between tensor products of objects are respresented by 'boxes' or 'nodes' between the corresponding sets of decorated strings. We show how structure on monoidal categories is implemented by geometrical constraints on this calculus and we show how in certain instances geometric representation offers simple interpretations of otherwise obscure identities and relations. We make special emphasis on the pictorial representations of objects in the theory of symmetric monoidal rigid tensor categories where every object is self dual and we apply this calculus to interpret diagrams appearing in the positive formalism as morphisms in suitable categories.

String diagrams: Pictorial calculus on monoidal categories and the positive formalism
04-11 - 17:00 - Salón 4
X

We present the pictorial calculus associated to braided and symmetric monoidal categories. We interpret constraints present in the theory of braided monoidal categories in terms of identities related to the theory of knots and we interpret the corresponding constraints on symmetric monoidal categories in terms of combinatorial conditions on our theory. We use the ideas presented to introduce the pictorial calculus for the category of partially ordered real vector spaces with inner product. Finally we use this theory to interpret theories in the positive formalism as pivotal symmetric monoidal subcategories of the category of partially ordered real vector spaces with inner product. We will discuss as well, time permitting, the corresponding interpretation of theories implementing time dependence and causality.

String diagrams: Pictorial calculus on monoidal categories and the positive formalism II
04-18 - 17:00 - Salón 4José Antonio Zapata (CCM-UNAM)
X

We present a formulation of Classical Field Theory appropriate for a spacetime that is subdivided into a collection of confined spatiotemporal regions each corresponding to the spatiotemporal location of a laboratory during the time a measurement takes place. When a region UM can be foliated by Cauchy surfaces for M , we recover the Lagrangian version of the Covariant Phase Space formulation of Classical Field Theory. For more general regions, the formulation involves a refinement of the notion of gauge. We consider observables associated to measurements localized in spacetime, and we also consider observables calculated by integration of gauge invariant conserved currents.

General boundary formulation of Classical Field Theory [slides]
04-25 - 17:00 - Salón 4
05-02 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

Text books on quantum field theory generally start by telling us stories of Hilbert spaces, of time-evolution, of instantaneous observables acting as operators, of commutators. Then, through a series of more-or-less convincing steps, from these the calculational tools are build up which form the actual backbone of modern quantum field theory and its stunning empirical success. However, from the point of view of the original structures these tools exhibit properties that appear rather unnatural: I focus on the path integral, crossing symmetry and the time-ordered product. In this talk I want to turn this logic around. That is, I want to take serious the tools and their empirical success, but consider the supposedly fundamental structures as unnatural. Rather from the tools I want to reverse engineer what the natural fundamental structures are. This leads me to the general boundary formulation (and subsequently to the positive formalism).

Reverse engineering quantum field theory [slides]
05-09 - 17:00 - Salón 4Juan Orendain (CCM-UNAM)
X

In this talk we introduce the categorical foundations of topological quantum field theory and the theory of quantum invariants of closed oriented manifolds. We offer interpretations of Atiyah and Segal's original set of axioms in terms of monoidal functors between certain symmetric monoidal categories and we use this interpretation to associate, to every topological quantum field theory, a set of invariants for closed oriented manifolds. We perform relevant computations and we use our presentation to explain how topological quantum field theories naturally organize into a groupoid. Finally, we present classification theorems, within our framework, for 1- and 2-dimensional topological quantum field theories.

A brief introduction to topological quantum field theory and the theory of quantum invariants
05-16 - 17:00 - Salón 4
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In this talk we present classification theorems for topological quantum field theories of dimensions 1 and 2 and we present explicit examples of topogical quantum field theories in dimension 3. We define equivalences between categories of structures of algebraic nature, dual pairs and commutative Frobenius algebras in dimensions 1 and 2 respectively, and the corresponding groupoids of topological quantum field theories. We provide examples of topological quantum field theories of dimension 3 by presenting a simplified version of the Turaev-Viro-Barret-Westubry state-sum construction associated to any fusion category. This construction provides an interpretation of the Levin-Wen model fitting the framework developed thus far.

A brief introduction to the theory of topological quantum field theories and quantum invariants II. Classification and examples.
05-23 - 17:00 - Salón 4Robert Oeckl (CCM-UNAM)
X

Working towards the aim of axiomatizing realistic quantum field theories in a TQFT-type framework I focus on the simplest class of examples: linear field theories and their perturbation theory. The problem of quantization in this setting is much richer than ordinary canonical quantization. It turns out to be useful to introduce an axiomatization of classical field theory also, on manifolds with boundary. I show how geometric quantization together with the Feynman path integral then leads to a quantization functor from (augmented) classical field theories to quantum field theories.

Local functorial quantization of field theory (I) [slides]
05-30 - 17:00 - Salón 4Local functorial quantization of field theory (II) [slides]
06-06 - 17:00 - Salón 4Daniele Colosi (ENES-UNAM Morelia)
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A general boundary quantum field (GBQFT) is a qft that satisfies the axioms of the General Boundary Formulation of quantum theory. The two existing schemes to quantize a classical field theory that produce a GBQFT are the Schrödinger-Feynman and the holomorphic quantization. In the talk we present the first quantization scheme, derive the general expression for the main ingredients of the quantum theory (vacuum state, field propagator, amplitudes) and provide examples for qft in different spaces (Minkowski, Rindler, de Sitter).

Schrödinger-Feynman quantization for general boundary quantum field theory [slides]
06-13 - 17:00 - Salón 4Daniele Colosi / Robert Oeckl
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The Unruh effect is among the most studied effects in QFT in curved space. The effect has been interpreted as establishing a relation between the vacuum state for a quantum field in Minkowski space and a mixed state for the corresponding field in Rindler space. In the talk we present the quantization of a scalar field in both Minkowski and Rindler space based on the general boundary formulation of quantum theory and provide an analysis of the Unruh effect by comparing expectation values of local observables in the two quantum theories.

A derivation of the Unruh effect based on general boundary quantum field theory [slides]

Last updated 13 June 2018.