Mathematical physics

Mathematical physics, including mathematics, is a research area where novel mathematical techniques are invented to tackle problems in physics, and where novel mathematical ideas find an elegant physical realization. Historically, it would have been impossible to distinguish between theoretical physics and pure mathematics. Often spectacular advances were seen with the concurrent development of new ideas and fields in both mathematics and physics. Here one might note Newton's invention of modern calculus to advance the understanding of mechanics and gravitation. In the twentieth century, quantum theory was developed almost simultaneously with a variety of mathematical fields, including linear algebra, the spectral theory of operators and functional analysis. This fruitful partnership continues today with, for example, the discovery of remarkable connections between gauge theories and string theories from physics and geometry and topology in mathematics.

Displaying 913 - 924 of 1033

Conjugates, Correlation and the Jordan Structure of Finite-Dimensional Quantum Theory
May 30, 2013

PIRSA:13050070
Mathematical physics
Resources

Tobias Fritz Universität Innsbruck
May 15, 2013

PIRSA:13050019
Mathematical physics
TQFTs and Topological Phases of Matter

Zhenghan Wang Microsoft Corporation
May 06, 2013

PIRSA:13050022
Mathematical physics
Energy and the Environment - What Physicists Can Do

John Baez University of California, Riverside
April 17, 2013

PIRSA:13040030
Mathematical physics
Mathematica - Lecture 4

Pedro Vieira Perimeter Institute for Theoretical Physics
December 14, 2012

PIRSA:12120022
Mathematical physics
Mathematica - Lecture 3b

Pedro Vieira Perimeter Institute for Theoretical Physics
December 10, 2012

PIRSA:12120021
Mathematical physics
Mathematica - Lecture 3a

Pedro Vieira Perimeter Institute for Theoretical Physics
December 10, 2012

PIRSA:12120020
Mathematical physics
Geometric algebra techniques in flux compactifications

Calin Lazaroiu Horia Hulubei National Institute for R and D in Physics and Nuclear Engineering
November 09, 2012

PIRSA:12110058
Mathematical physics
Is spacetime fundamentally discrete?
- Bianca Dittrich Perimeter Institute for Theoretical Physics
- Seth Major Hamilton College
- Daniele Oriti Complutense University of Madrid
- Roberto Percacci SISSA International School for Advanced Studies
- Tobias Fritz Universität Innsbruck
October 24, 2012

PIRSA:12100100
Mathematical physics
New Directions in Categorical Logic for Classical, Probabilistic and Quantum Physics

Tobias Fritz Universität Innsbruck
October 09, 2012

PIRSA:12100055
Mathematical physics
AdS(3)/CFT(2) correspondence and integrability

Konstantin Zarembo Nordic Institute for Theoretical Physics
October 09, 2012

PIRSA:12100062
Quantum Fields and Strings

Mathematical physics
Hamiltonian Theory of Fractional Chern Bands

Ramamurti Shankar Yale University
September 14, 2012

PIRSA:12090047
Mathematical physics

Conjugates, Correlation and the Jordan Structure of Finite-Dimensional Quantum Theory
May 30, 2013

PIRSA:13050070
Mathematical physics
The last few years have seen a spate of operational or information-theoretic
derivations (or \reconstructions") of nite-dimensional quantum mechanics [1]. These rest of some strong assumptions. In particular, most assume the state of a composite system is entirely determined by the joint probabilities it assigns to the outcomes of measurements on the component systems. This condition, often called local tomography, is satis ed by neither real nor quaternionic quantum theory. If we are interested in the possibilities for theories more general than orthodox quantum mechanics, we might wish to relax this constraint.
In this talk I will discuss a weaker system of assumptions, involving correla-
tions between a probabilistic system and an isomorphic conjugate system, that leads to a representation of such systems in terms of euclidean Jordan algebras [2]. These have a well-known classi cation as direct sums of real, complex or quaternionic quantum systems, possibly the exceptional Jordan algebra, and spin factors. The last are a form of \bit", characterized by a family of two-valued observables, parametrized by antipodal vectors in a sphere of arbitrary dimension. Orthodox quantum mechanics can be singled out by imposing local tomography, plus the existence of a qubit as additional axioms [3]. However, there is a natural way to form non-signaling, but generally non-locally tomo-graphic, composites of systems based on special euclidean Jordan algebras (that is, excluding the exceptional one). This yields a probabilistic theory strictly, but not wildly, more general than orthodox nite-dimensional QM; one that elegantly uni es real, complex and quaternionic quantum theory, has a simple operational basis, and allows for a spectrum of bits more general than permitted in orthodox quantum theory. Parts of this talk reflect ongoing joint work with Howard Barnum, Matthew Graydon and Cozmin Ududec.
References:
[1] G. Chiribella, M. D'Ariano and P. Perinotti, Phys. Rev. A 84, (2011), 012311-
012350; B. Dakic and C. Brukner, in H. Halvorson (ed.), Deep Beauty, Cambridge,
2011; P. Goyal, Phys. Rev. A 78 (2008), 052120-052146; L. Hardy, arXiv:quant-
ph/0101012 (2001); L. Masanes and M. Mueller, New J. Phys. 13 (2011) ; J. Rau,
Annals of Physics 324 (2009) 2622{2637;
[2] A. Wilce, Conjugates, correlation and quantum mechanics, arXiv:1206.2897 (2012)
[3] H. Barnum and A. Wilce, Local tomography and the Jordan structure of quantum
theory, arXiv:1202.4513, (2012)
Resources

Tobias Fritz Universität Innsbruck
May 15, 2013

PIRSA:13050019
Mathematical physics
One of the main challenges that we face both as individual persons and as a species concerns the distribution and use of resources, such as water, time, capital, computing power or negatively valued
resources like nuclear waste. Also within theoretical physics, one frequently deals with resources like free energy or quantum entanglement. I will describe a mathematical theory of resources which makes quantitative predictions about how many resources are required for producing a certain commodity and outline some applications to information theory.
Joint work with Bob Coecke and Rob Spekkens.
TQFTs and Topological Phases of Matter

Zhenghan Wang Microsoft Corporation
May 06, 2013

PIRSA:13050022
Mathematical physics
In two spatial dimensions, there is a good correspondence between TQFTs and topological phases of matter for spin systems. I will discuss this correspondence in one and three spatial dimensions for spin systems. If time permits, I will also discuss the situation for fermion systems.
Energy and the Environment - What Physicists Can Do

John Baez University of California, Riverside
April 17, 2013

PIRSA:13040030
Mathematical physics
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. While politics and economics pose the biggest challenges, physicists are in a good position to help make this transition a bit easier. After a quick review of the problems, we discuss a few ways physicists can help.
Mathematica - Lecture 4

Pedro Vieira Perimeter Institute for Theoretical Physics
December 14, 2012

PIRSA:12120022
Mathematical physics
Mathematica - Lecture 3b

Pedro Vieira Perimeter Institute for Theoretical Physics
December 10, 2012

PIRSA:12120021
Mathematical physics
Mathematica - Lecture 3a

Pedro Vieira Perimeter Institute for Theoretical Physics
December 10, 2012

PIRSA:12120020
Mathematical physics
Geometric algebra techniques in flux compactifications

Calin Lazaroiu Horia Hulubei National Institute for R and D in Physics and Nuclear Engineering
November 09, 2012

PIRSA:12110058
Mathematical physics
Using techniques originating in a certain approach to Clifford bundles known as "geometric algebra", I discuss a geometric reformulation of constrained generalized Killing spinor equations which proves to be particularly effective in the study and classification of supersymmetric flux compactifications of string and M-theory. As an application, I discuss the most general N=2 compactifications of M-theory to three dimensions, which were never studied in full generality before. I also touch upon the connection of such techniques with a certain variant of the quantization of spin systems.
Is spacetime fundamentally discrete?
- Bianca Dittrich Perimeter Institute for Theoretical Physics
- Seth Major Hamilton College
- Daniele Oriti Complutense University of Madrid
- Roberto Percacci SISSA International School for Advanced Studies
- Tobias Fritz Universität Innsbruck
October 24, 2012

PIRSA:12100100
Mathematical physics
Modelling continuum dynamics on discrete space time
We will discuss perfect discretizations which aim at mirroring exactly continuum physics on a given lattice. Such discretizations avoid typical artifacts like Lorentz violation, energy dissipation, particle doubling and in particular breaking of diffeomorphism symmetry. Thus the question arises how to distinguish such lattice dynamics from continuum physics.

____________________________

Turning Weyl’s tile argument into a mathematically rigorous no-go theorem
Weyl's tile argument notes that if space was fundamentally discrete then the set of allowed velocities of a classical particle would not be isotropic. I will generalize Weyl's heuristic argument to a no-go theorem applying to any discrete periodic structure. Since this theorem does not take quantum mechanics into account it should only be regarded as the first step of a program of understanding the phenomenology of discrete spacetimes in a mathematically rigorous way.See arXiv:1109.1963

____________________________

On the Observability of Discrete Spatial Geometry
If quantum geometry is an accurate model of microscopic spatial geometry then two related questions arise, one observational and one theoretical: How and at what scale is the discreteness manifest? And, how is the general relativistic limit achieved? These questions will be discussed in the context of studies of a single atom of geometry. It will be shown that the effective scale of the discreteness could be much larger than the Planck scale. Before this scale can be predicted, the relations between discrete geometry, coherent states, and the semiclassical limit need to be clarified. Work towards this goal, using coherent states in spin foams and the spin geometry theorem of Penrose and Moussouris will be described.

______________________________

Asymptotic safety and minimal length
Since asymptotic safety - if true - would make a quantum field theory of gavity consistent "up to arbitrarily high energy", it would seem that this notion is incompatible with the existence of a minimal length. I will argue that this is not necessarily the case, due to ambiguity in the notion of minimal length.
New Directions in Categorical Logic for Classical, Probabilistic and Quantum Physics

Tobias Fritz Universität Innsbruck
October 09, 2012

PIRSA:12100055
Mathematical physics
I will give an idea of what category theory is and how it can be successfully applied in mathematics and the mathematical sciences by means of example. The example is a notion from mathematical logic formalizing the intuitive concept of "property". The new category-theoretical definition of this notion can physically be interpreted as a measurement. Unraveling this definition in particular categories can be regarded as defining the concept of "property" in different context, e.g. in classical, probabilistic and quantum physics, and in each case this recovers familiar things.
Disclaimer: Up to some differences in the technical details, all of this is work of Bart Jacobs. (See arXiv:1205.3940)
AdS(3)/CFT(2) correspondence and integrability

Konstantin Zarembo Nordic Institute for Theoretical Physics
October 09, 2012

PIRSA:12100062
Quantum Fields and Strings

Mathematical physics
Integrability has been successfully used to compute the non-perturbative spectrum, Wilson loops and scattering amplitudes in the AdS/CFT correspondence. Most of these results apply to N=4, D=4 SYM / strings on AdS(5)xS(5). Strings on AdS(3)xM, where M is either S(3)xT(4) or S(3)xS(3)xS(1), are also integrable and potentially solvable by the same methods (Bethe ansatz, Y-system, TBA etc). An interesting aspect of string theory on AdS(3) is a large number of parameters that preserve integrability. One parameters, in particular, interpolates between the integrable RR AdS(3) background and conformal NSNS background described by a non-compact WZW model. Another interesting aspect of string theory on AdS(3) is emergence of target-space Viraroso symmetry. I will review how integrability arises in the AdS(3)/CFT(2) correspondence, and will describe what is (not) know about the Bethe ansatz solution of the relevant sigma-models.
Hamiltonian Theory of Fractional Chern Bands

Ramamurti Shankar Yale University
September 14, 2012

PIRSA:12090047
Mathematical physics
It has been known for some time that a system with a filled band will have an integer quantum Hall conductance equal to its Chern number, a toplogical index associated with the band. While this is true for a system in a magnetic field with filled Landau Levels, even a system in zero external field can exhibit the QHE if its band has a Chern number. I review this issue and discuss a more recent question of whether a partially filled Chern band can exhibit the Fractional QHE. I describe the work done with Ganpathy Murthy in which we show how composite fermions, which were so useful in explaining the usual FQHE, can be introduced here and with equal success by adapting our Hamiltonian Theory of CFs developed for the FQHE in the continuum.

Title	Speaker(s)	Date	Talk Type	Info link
Conjugates, Correlation and the Jordan Structure of Finite-Dimensional Quantum Theory		2013-05-30	Conference	View details
Resources	Tobias Fritz	2013-05-15	Scientific Series	View details
TQFTs and Topological Phases of Matter	Zhenghan Wang	2013-05-06	Conference	View details
Energy and the Environment - What Physicists Can Do	John Baez	2013-04-17	Scientific Series	View details
Mathematica - Lecture 4	Pedro Vieira	2012-12-14	Course	View details
Mathematica - Lecture 3b	Pedro Vieira	2012-12-10	Course	View details
Mathematica - Lecture 3a	Pedro Vieira	2012-12-10	Course	View details
Geometric algebra techniques in flux compactifications	Calin Lazaroiu	2012-11-09	Scientific Series	View details
Is spacetime fundamentally discrete?	Bianca Dittrich, Seth Major, Daniele Oriti, Roberto Percacci, Tobias Fritz	2012-10-24	Conference	View details
New Directions in Categorical Logic for Classical, Probabilistic and Quantum Physics	Tobias Fritz	2012-10-09	Scientific Series	View details
AdS(3)/CFT(2) correspondence and integrability	Konstantin Zarembo	2012-10-09	Scientific Series	View details
Hamiltonian Theory of Fractional Chern Bands	Ramamurti Shankar	2012-09-14	Scientific Series	View details

Mathematical physics

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