Courses

The two-fold role of observables in Classical and Quantum Kinematics
by Federico Zalamea (EFREI Paris)

 

In this course, we will explore some of the mathematical structures underlying the kinematical description of classical and quantum systems.  As we get familiar with the geometry of the space of states and the algebraic structures of observables, we will try to understand the conceptual meaning of these structures: why is it the case that the space of states of classical or quantum systems is a symplectic manifold? Why do complex numbers play a more prominent role in the formulation of quantum mechanics?

 

During this promenade, I will emphasise the striking similarities between the classical and quantum worlds. In fact, from the kinematical point of view, it will be clear that they share much more than is usually stressed. Hopefully, this common language will allow us to pinpoint more precisely the meaning of the structural differences between the two sides of the walkway.

 

Some bibliography for the valiant reader:

    • N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Chapter 1.
    • N.P. Landsman, Foundations of Quantum Theory - From Classical Concepts to Operator Algebras.

 

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Timeless statistical mechanics and applications to quantum gravity
by Isha Kotecha (Max Planck Institute for Gravitational Physics, AEI Postdam)

 

Constrained dynamical systems are notoriously well-known for their absence of notions of absolute time and energy. To formulate a statistical mechanical framework in these settings is an open problem. In this course, we will discuss the issue of characterising statistical equilibrium in background independent systems, first for spacetime relativistic systems and then for a candidate quantum gravity system of spin networks devoid of any spacetime structures. In this context we will explore topics like: the problem of time; the thermal time hypothesis; presymplectic mechanics; the maximum entropy principle and the Kubo-Martin-Schwinger condition; and applications to statistical mechanics of quanta of geometry.

 

References:

 

https://iopscience.iop.org/article/10.1088/0264-9381/11/12/007

 

https://doi.org/10.1088/0264-9381/10/8/015

 

http://arxiv.org/abs/arXiv:1801.09964

 

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Cosmological field theories
by Scott Melville (University of Cambridge)

 

Historically, classical/quantum field theory has enjoyed much success in describing condensed matter and particle physics. As a result, most introductions to the subject focus on these regimes. However, the language of field theory is far more powerful. Recent advances in cosmology have allowed us to build accurate field theory descriptions of virtually every stage of our Universe's evolution: describing inflation in the early Universe, the subsequent growth of large scale structures, and the late-time domination of dark energy.
Not only does this allow forecasting for a number of current and upcoming experiments and a wealth of new data, but it has also taught us important lessons about how to construct and apply field theories under diverse conditions.

In this short lecture series, we will introduce the main concepts which underpin the Effective Field Theories of Inflation, Large Scale Structure and Dark Energy, and bridge the gap between a first course in QFT and its modern application to cosmology.

 

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Fun and Games with the Choi-Jamiolkowski Isomorphism
by Emily Adlam (Basic Research Community for Physics)

 

 

The Choi-Jamiolkowski isomorphism is a remarkable mathematical correspondence between quantum channels and entangled bipartite states. This mathematical trick has a range of important uses in quantum information theory and has also played a part in a number of recent `operational' reformulations of quantum mechanics. However, the interpretation of the isomorphism remains a contentious issue.

In this course we will examine some existing proposals for interpreting the Choi-Jamiolkowski isomorphism, including the operational procedure known as `gate teleportation’ and the possibility that quantum mechanical operators may be understood in terms of propagation of subjective degrees of belief according to Bayesian laws for belief propagation. We will then consider a general formulation of the Choi-Jamiokowski  isomorphism which can be applied within any operational theory, and explore the implications for any operational theory in which this isomorphism holds. Finally we will move to open discussion on the nature of the isomorphism.

 

References:

 

https://arxiv.org/abs/quant-ph/0611233

 

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Am I a statement about the physical world?
by Titouan Carette (LORIA, Université de Lorraine)

 

dixitque Deus fiat lux et facta est lux

Genesis 1:3

 

My word shapes my ideas so that they can be grasped, modified, copied or discarded by my peers. Logic can formalise this process. This course is an invitation to shatter the thin border between abstract logical statements and concrete physical systems. I see it as a three-step journey. First, we'll discover the joy of formal reasoning through sequent calculus from classical to linear logic. Then, we'll explore the incarnation of those abstract systems into the quantum realm. Finally, we will attempt to reconstruct quantum mechanics in this setting. Additional optional time will be dedicated to the fine art of quantum pictorialism.

 

Objectives: at the end of the course attendee should be able to:

    • derive simple classical and linear formulas from the sequent calculus
    • give physical interpretations of the classical and linear logical rules
    • state the main ideas behind the categorical quantum mechanics program.
    • (optional) manipulate the syntax and semantic of ZX-calculii.

 

Things one can look at as an appetizer (by decreasing order of physicist friendliness):

 

Unfortunately, Cohl cannot assist the summer school.

 

Towards a more efficient model of particle physics
by Cohl Furey (University of Cambridge)

 

Grand unified theories envision the standard model of particle physics as a piece of a larger system. However, in these lectures we will ask the opposite question:  Could the standard model result from a set of algebras much smaller than itself? By the late 1930s, Arthur Conway knew that the complex quaternions (just a 4 complex-dimensional algebra) could single-handedly encode the notion of rotations and boosts, in addition to the degrees of freedom of electric and magnetic fields, energy and momentum, fermionic spin and chirality.  Here we will demonstrate hints that the octonions might be capable of similar feats in efficiency.

 

 

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