This is an informal lectures series on the introduction to the mathematical formulation of the geometric and stochastic quantizations, also on some current development and related topics.
Time：Thu-Fri, November 5-6, 2020
9-1 EST (8-12 Wisconsin, 3-7 Central Europe, 4-8 Finland)
It is organized by :
Eveliina Peltola ([email protected])
Yilin Wang ([email protected])
The zoom invitation will be distributed through the email list a few days prior to the lectures. You can contact Yilin to be added to the mailing list. You may also share it with your colleagues and students, but please do not post it on a publicly viewable website.
Eveliina Peltola ([email protected])
Yilin Wang ([email protected])
The zoom invitation will be distributed through the email list a few days prior to the lectures. You can contact Yilin to be added to the mailing list. You may also share it with your colleagues and students, but please do not post it on a publicly viewable website.
Thursday (EST)
9 : 00 - 10 : 00 Gabriele Rembado (Bonn, Hausdorff center)
Introduction to geometric quantisation [Slides]
The basic picture of classical mechanics involves states, observables, and time-evolution, and a mathematical dictionary for this can be set up by introducing symplectic/Poisson spaces and functions on them. Analogously the basic picture of quantum mechanics admits a mathematical dictionary involving Hilbert spaces and operators on them, and there are then ways to formalise semiclassical limits: these are supposed to produce classical systems from quantum ones, and relate their dynamics.
In these informal lectures we will first set up these dictionaries and introduce the problem of quantisation: the construction of quantum systems with given semiclassical limit. Then we will describe one partial solution to this problem, namely geometric quantisation, and show explicitly how it works in the simplest cases. According to the time remaining we may consider the quantisation of coadjoint orbits of compact Lie groups (one of the original mathematical motivations for geo. quant.), and the far-reaching generalisation of moduli spaces of connections on principal bundles over surfaces.
The basic picture of classical mechanics involves states, observables, and time-evolution, and a mathematical dictionary for this can be set up by introducing symplectic/Poisson spaces and functions on them. Analogously the basic picture of quantum mechanics admits a mathematical dictionary involving Hilbert spaces and operators on them, and there are then ways to formalise semiclassical limits: these are supposed to produce classical systems from quantum ones, and relate their dynamics.
In these informal lectures we will first set up these dictionaries and introduce the problem of quantisation: the construction of quantum systems with given semiclassical limit. Then we will describe one partial solution to this problem, namely geometric quantisation, and show explicitly how it works in the simplest cases. According to the time remaining we may consider the quantisation of coadjoint orbits of compact Lie groups (one of the original mathematical motivations for geo. quant.), and the far-reaching generalisation of moduli spaces of connections on principal bundles over surfaces.
10 : 20 - 11 : 20 Hao Shen (University of Wisconsin-Madison)
Stochastic quantization [Notes]
In the first talk I will give an overview of stochastic quantization, and go through a number of examples. These examples all give rise to stochastic PDEs which look like a gradient of an action plus a white noise; but, the notion of “gradients" and “white” noises depend on choices of metrics on the infinite dimensional spaces. In the second talk I will focus on one particular model, tentatively Yang-Mills, but it could also alter depending on the interest from the first talk. My talks will be a bit more informal than usual and the main purpose is to let everyone ask questions and learn from each other.
In the first talk I will give an overview of stochastic quantization, and go through a number of examples. These examples all give rise to stochastic PDEs which look like a gradient of an action plus a white noise; but, the notion of “gradients" and “white” noises depend on choices of metrics on the infinite dimensional spaces. In the second talk I will focus on one particular model, tentatively Yang-Mills, but it could also alter depending on the interest from the first talk. My talks will be a bit more informal than usual and the main purpose is to let everyone ask questions and learn from each other.
11 : 40 - 12 : 40 Joona Oikarinen (University of Helsinki)
Quantum Fields from Random Fields [Slides]
The aim of this lecture is to explain how quantum fields can be constructed from random fields. First, for motivation, basic concepts of quantum field theory and path integration will be reviewed from the point of view of the Wightman axioms. The random fields then arise by passing to imaginary values of time. This yields a mathematical trick for constructing random fields (probability measures on infinite dimensional spaces) from quantum fields. Since it is often easier to construct random fields than quantum fields (in the sense of Wightman), the question arises: when can one reconstruct a quantum field from a random field? The Osterwalder-Schrader (OS) axioms for random fields and the OS-reconstruction theorem provide an answer to this question. We will briefly review the idea of the reconstruction theorem and consider some simple examples.
The aim of this lecture is to explain how quantum fields can be constructed from random fields. First, for motivation, basic concepts of quantum field theory and path integration will be reviewed from the point of view of the Wightman axioms. The random fields then arise by passing to imaginary values of time. This yields a mathematical trick for constructing random fields (probability measures on infinite dimensional spaces) from quantum fields. Since it is often easier to construct random fields than quantum fields (in the sense of Wightman), the question arises: when can one reconstruct a quantum field from a random field? The Osterwalder-Schrader (OS) axioms for random fields and the OS-reconstruction theorem provide an answer to this question. We will briefly review the idea of the reconstruction theorem and consider some simple examples.
Friday
9 : 00 - 10 : 00 Gabriele Rembado (Bonn, Hausdorff center)
Introduction to geometric quantisation [Slides]
See above.
See above.
10 : 20 - 11 : 20 Hao Shen (University of Wisconsin-Madison)
Stochastic quantization [Slides]
See above.
See above.
11 : 40 - 12 : 40 Masha Gordina (University of Connecticut)
Brownian motion on the group Diff(S^1) [Slides]
Let Diff( S^1) be the group of orientation preserving C^{\infty} diffeomorphisms of S^1. This group has been studied in many fields including mathematical physics. We will start by recalling its significance in the QFT going back to Bowick-Rajeev-Zumino. In 1999 and 2002 P. Malliavin and S. Fang constructed Brownian motion on Homeo(S^1), the group of Holderian homeomorphisms of S^1. They used the H^{3/2} metric related to the group Diff(S^1). The resulting stochastic process does not live in Diff(S^1). We will review these results, and a different method constructing a Brownian motion that lives exactly in the group Diff(S^1). The latter has been the subject of Mang Wu's PhD dissertation, and we will discuss this and related results in infinite-dimensional stochastic analysis.
Let Diff( S^1) be the group of orientation preserving C^{\infty} diffeomorphisms of S^1. This group has been studied in many fields including mathematical physics. We will start by recalling its significance in the QFT going back to Bowick-Rajeev-Zumino. In 1999 and 2002 P. Malliavin and S. Fang constructed Brownian motion on Homeo(S^1), the group of Holderian homeomorphisms of S^1. They used the H^{3/2} metric related to the group Diff(S^1). The resulting stochastic process does not live in Diff(S^1). We will review these results, and a different method constructing a Brownian motion that lives exactly in the group Diff(S^1). The latter has been the subject of Mang Wu's PhD dissertation, and we will discuss this and related results in infinite-dimensional stochastic analysis.
