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February 28, 2007
QFT of Charged n-Particle: Gauge Theory Kinematics
Posted by Urs Schreiber
Some basic remarks on how gauge theory in -dimensions fits into the general framework of the charged -particle, followed by a semi-close look at how
Christian Fleischhack
Representations of the Weyl Algebra in Quantum Geometry
math-ph/0407006
realizes, in a continuous (instead of smooth) version of gauge theory, the algebra of observables.
QFT of Charged n-Particle: Algebra of Observables
Posted by Urs Schreiber
What is the “algebra of observables”, really?
February 27, 2007
Quantization and Cohomology (Week 16)
Posted by John Baez
This week in our course on Quantization and Cohomology we considered some fancier path integrals. Then, fortified by these examples, we returned to the more abstract issues this course is really about:
- Week 16 (Feb. 27) - More examples of path-integral quantization. The particle in a potential on the real line. The Lie-Trotter Theorem. The particle in a potential on a complete Riemannian manifold. Back to general questions: how do we get a Hilbert space from a category equipped with an action functor? The problem of Cauchy surfaces.
February 25, 2007
This Week’s Finds in Mathematical Physics (Week 246)
Posted by John Baez
In week246 of This Week’s Finds, read about Peter Woit’s Not Even Wrong and Lee Smolin’s The Trouble With Physics:
Amplimorphisms and Quantum Symmetry, II
Posted by Urs Schreiber
In the last entry in this series, Amplimorphisms and Quantum Symmetry, I, I talked about algebras of physical observables and their Doplicher-Haag-Roberts representations.
Here I make a remark on how this is related to the statement
February 23, 2007
The Health Book
Posted by David Corfield
How to Write Mathematics Badly
Posted by John Baez
Everyone who cares about mathematics should watch this hilarious and educational video:
- Jean-Pierre Serre, How to Write Mathematics Badly.
If you don’t know who Serre is, read a bit about him before watching the video. You’ll enjoy it more.
Classical vs Quantum Computation (Week 15)
Posted by John Baez
In this week’s class on Classical vs. Quantum Computation, we continued to work through an example of how typed -calculi give cartesian closed categories:
- Week 15 (Feb. 22) - The λ-theory of commutative rings and the cartesian closed category it generates: the "free cartesian closed category on a commutative ring object". What is a cartesian closed functor from this to Set? Guess: just a commutative ring! Blog entry.
Last week’s notes are here.
February 21, 2007
Noncommutative Geometry Blog
Posted by David Corfield
Cake Talk
Posted by David Corfield
An Introduction to Algebraic Topology
Posted by John Baez
This quarter, besides my seminars on Quantization and Cohomology and Classical vs. Quantum Computation, I’m also teaching the graduate qualifier course on algebraic topology. While a bit elementary for some Café regulars, it might be fun for other folks:
- John Baez, Mike Stay and Christopher Walker, Algebraic Topology.
Quantization and Cohomology (Week 15)
Posted by John Baez
This week in our course on Quantization and Cohomology, we finished off the path-integral quantization of the free particle:
- Week 15 (Feb. 20) - The free particle on a line (part 2). Showing the path-integral approach agrees with the Hamiltonian approach. Fourier transforms and Gaussian integrals.
February 17, 2007
Congratulations!
Posted by David Corfield
It’s John’s wedding day today!
I’m sure all the Café regulars will join me in wishing you and Lisa a happy continuation of your life together.
February 16, 2007
Classical vs Quantum Computation (Week 14)
Posted by John Baez
This time in our course on Classical vs. Quantum Computation, we sketched how a typed λ-calculus serves as a presentation of a cartesian closed category, and how every cartesian closed category arises this way. Since the students seemed to be struggling with the levels of abstraction involved, we slowed down to tackle an example:
-
Week 14 (Feb. 15) - The cartesian closed category generated by a typed λ-calculus, and how this construction gives a functor . The ‘internal language’ of a cartesian closed category, and how this gives a functor . and are adjoint,
and in fact give an equivalence between typed λ-calculi and
and cartesian closed categories. Example 1: the λ-theory of
commutative rings.
Supplementary reading:
- Joachim Lambek and Phil Scott, Introduction to Higher-Order Categorical Logic, Cambridge U. Press, 1988. Part 1, Section 11: the cartesian closed category generated by a typed λ-calculus.
QFT of Charged n-Particle: T-Duality
Posted by Urs Schreiber
Last time I described how the idea of pull-push propagation in quantum mechanics should look like when we refine the formalism to quantization on a category, or even to quantization on an -category, i.e. when we systematically replace spaces by categories and regard, for instance, a string not just as an interval but as a poset propagating not just on a target space but on the corresponding category of 2-paths
In particular, I drew a pasting diagram that descibed the pull-push of a section,
of an -bundle with connection through a suitable correspondence.
supposed to describe the quantum evolution of the state corresponding to that section over the worldvolume .
I claim that this is the natural operation of a worldvolume on a state. And I claim that it is once again crucial that we have understood a section as a transformation (1) between transport functors. Notice that, by passing to the components of (2), the 2-morphisms filling this diagram– which are forced upon us by the transformation nature of sections – turn the bare correspondence into a correspondence with an -bundle on the correspondence space Moreover, by the rules for composition of transformations of functors, the pull-push through this correspondence automatically and naturally incorporates the action of that bundle on the section pulled up the correspondence space.
Such a transformation is known to categorify ordinary linear operations, as recalled in Fourier-Mukai, T-Duality and other linear 2-Maps.
In particular, (topological) T-duality for 2-particles is an example for such a transformation, as described in Mathai on T-Duality, II: T-dual K-classes by Fourier-Mukai.