Coherent states and the quantization of 1+1-dimensional Yang-Mills theory

This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills theory on a spacetime cylinder, from the point of view of coherent states, or equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of "reduced" coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal-Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction

Lower order terms in Szego type limit theorems on Zoll manifolds

This is a detailed version of the paper math.FA/0212273. The main motivation for this work was to find an explicit formula for a "Szego-regularized" determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold. The idea of the Szego-regularization was suggested by V. Guillemin and K. Okikiolu. They have computed the second term in a Szego type expansion on a Zoll manifold of an arbitrary dimension. In the present work we compute the third asymptotic term in any dimension. In the case of dimension 2, our formula gives the above mentioned expression for the Szego-redularized determinant of a zeroth order PsDO. The proof uses a new combinatorial identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This identity is related to the distribution of the maximum of a random walk with i.i.d. steps on the real line. The proof of this combinatorial identity together with historical remarks and a discussion of probabilistic and algebraic connections has been published separately.Comment: 39 pages, full version, submitte

Instability and Chaos in Non-Linear Wave Interaction: a simple model

We analyze stability of a system which contains an harmonic oscillator non-linearly coupled to its second harmonic, in the presence of a driving force. It is found that there always exists a critical amplitude of the driving force above which a loss of stability appears. The dependence of the critical input power on the physical parameters is analyzed. For a driving force with higher amplitude chaotic behavior is observed. Generalization to interactions which include higher modes is discussed. Keywords: Non-Linear Waves, Stability, Chaos.Comment: 16 pages, 4 figure

On the geometry of mixed states and the Fisher information tensor

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant-Kirillov-Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.Comment: 27 pages; Accepted Manuscrip

Invariants of pseudogroup actions: Homological methods and Finiteness theorem

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of k-variants and k-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: We introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie-Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.Comment: v2: some remarks and references addee

Twisting gauged non-linear sigma-models

We consider gauged sigma-models from a Riemann surface into a Kaehler and hamiltonian G-manifold X. The supersymmetric N=2 theory can always be twisted to produce a gauged A-model. This model localizes to the moduli space of solutions of the vortex equations and computes the Hamiltonian Gromov-Witten invariants. When the target is equivariantly Calabi-Yau, i.e. when its first G-equivariant Chern class vanishes, the supersymmetric theory can also be twisted into a gauged B-model. This model localizes to the Kaehler quotient X//G.Comment: 33 pages; v2: small additions, published versio

QSOs and Absorption Line Systems Surrounding the Hubble Deep Field

We have imaged a 45x45 sq. arcmin. area centered on the Hubble Deep Field (HDF) in UBVRI passbands, down to respective limiting magnitudes of approximately 21.5, 22.5, 22.2, 22.2, and 21.2. The principal goals of the survey are to identify QSOs and to map structure traced by luminous galaxies and QSO absorption line systems in a wide volume containing the HDF. We have selected QSO candidates from color space, and identified 4 QSOs and 2 narrow emission-line galaxies (NELGs) which have not previously been discovered, bringing the total number of known QSOs in the area to 19. The bright z=1.305 QSO only 12 arcmin. away from the HDF raises the northern HDF to nearly the same status as the HDF-S, which was selected to be proximate to a bright QSO. About half of the QSO candidates remain for spectroscopic verification. Absorption line spectroscopy has been obtained for 3 bright QSOs in the field, using the Keck 10m, ARC 3.5m, and MDM 2.4m telescopes. Five heavy-element absorption line systems have been identified, 4 of which overlap the well-explored redshift range covered by deep galaxy redshift surveys towards the HDF. The two absorbers at z=0.5565 and z=0.5621 occur at the same redshift as the second most populated redshift peak in the galaxy distribution, but each is more than 7Mpc/h (comoving, Omega_M=1, Omega_L=0) away from the HDF line of sight in the transverse dimension. This supports more indirect evidence that the galaxy redshift peaks are contained within large sheet-like structures which traverse the HDF, and may be precursors to large-scale ``pancake'' structures seen in the present-day galaxy distribution.Comment: 36 pages, including 9 figures and 8 tables. Accepted for publication in the Astronomical Journa

On Bohr-Sommerfeld bases

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions with applications to the non-vanishing of Poincar\'e series of large weight, Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every vector of a BS basis is defined by some half-weighted Legendrian distribution coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying symplectic manifold. The advantage of BS bases (compared to bases of theta functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint 216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information from the skillful analysis of the asymptotics of quantum states. This gives that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply these bases to compare the Hitchin connection with the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory (see, for example, [T. Kohno, Topological invariants for 3-manifolds using representations of mapping class group I, Topology 31 (1992), 203-230; II, Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.Comment: 14 page