TQFT Club Seminars
9th - 13th October, 2006
TQFT home
Mark Gotay (Univ. of
Hawai at Manoa)
"Stress-Energy-Momentum Tensors"
J. Marsden and I present a new method of constructing a
stress-energy-momentum tensor for a classical field theory based on
covariance considerations and Noether theory. Our
stress-energy-momentum tensor T^mu\nu is defined using the
(multi)momentum map associated to the spacetime diffeomorphism group.
The tensor T^mu\nu is uniquely determined as well as
gauge-covariant, and depends only upon the divergence equivalence class
of the Lagrangian. It satisfies a generalized version of the classical
Belinfante--Rosenfeld formula,
and hence naturally incorporates both the canonical
stress-energy-momentum tensor and the "correction terms" that are
necessary to make the latter well behaved. Furthermore, in the presence
of a metric on spacetime, our T^mu\nu coincides with the Hilbert
tensor and hence is automatically symmetric.
References:
[1] Gotay, M. J. and J. E. Marsden [1992], Stress-energy-momentum
tensors and the Belinfante--Rosenfeld formula, Contemp. Math. 132,
367--391.
[2] Forger, M. and H. R\"omer [2004], Currents and the energy-momentum
tensor in classical field theory: A fresh look at an old problem, Ann.
Phys.
309, 306--389.
Monday 9th October 2006, 11.00-12.00, Room 3.10
"Obstructions to Quantization"
Quantization is not a straightforward proposition, as demonstrated by
Groenewold's and Van~Hove's discovery, sixty years ago, of an
"obstruction" to quantization. Their "no-go theorems" assert that it is
in principle impossible to consistently quantize every classical
polynomial observable on the phase space R^{2n} in a physically
meaningful way. Similar obstructions have been recently
found for S^2 and T^*S^1, buttressing the common belief that
no-go theorems should hold in some generality. Surprisingly, this is
not so - it has just been proven that there are no obstructions to
quantizing either T^2 or T^*R_+.
In this talk we conjecture - and in some cases prove -
generalized Groenewold-VanHove theorems, and determine the maximal Lie
subalgebras of observables which can be consistently quantized. This
requires a study of the structure of Poisson algebras of symplectic
manifolds and their representations. To these ends we review known
results as well as recent theoretical work.
Our discussion is independent of any particular method of quantization;
we concentrate on the structural
aspects of quantization theory which are common to all Hilbert
space-based quantization techniques. (This is joint work with J.
Grabowski, H. Grundling and A. Hurst.)
References:
[1] Gotay, M. J. [2000], Obstructions to Quantization, in:
Mechanics: From Theory to Computation. ( Essays in Honor of Juan-Carlos
Simo), J. Nonlinear Science, Editors, 271--316 (Springer, New York).
[2] Gotay, M. J. [2002], On Quantizing Non-nilpotent Coadjoint Orbits
of Semisimple Lie Groups. Lett. Math. Phys. 62, 47--50.
Thursday 12th October 2006, 16.00-17.00, Room 3.10
"Experimental Star-Product
Quantization"
Let (L,\nabla) be a prequantum line bundle over a symplectic manifold
X, and S its symplectization. Kostant showed that the classical Poisson
bracket on S is simply prequantization on X. C. Duval and I have taken
this a step farther to obtain a quantization of X using a generalized
star-product on S.
References:
[1] Kostant, B. [2003], Minimal coadjoint orbits and symplectic
induction, arXiv: SG/0312252.
Friday 13th October 2006, 14.00-15.00, Room 3.31
FINANCIAL SUPPORT:
Centro de Análise
Matemática,
Geometria e Sistemas Dinâmicos
FEDER
DATES:9-13 October, 2006
VENUE: 3.10, 3.31, 3rd Floor Mathematics Building
URL: http://www.math.ist.utl.pt/~rpicken/tqft
rpicken@math.ist.utl.pt