arXiv subjects

Brian C. Hall

Publications and source records attributed to Brian C. Hall.

At least 37 records · Page 2Linked to original sources

arXiv2013

The Large-$N$ Limit of the Segal--Bargmann Transform on $\mathbb{U}_N$

We study the (two-parameter) Segal--Bargmann transform $\mathbf{B}_{s,t}^N$ on the unitary group $\mathbb{U}_N$, for large $N$. Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit $\mathscr{G}_{s,t}$ as $N\to\infty$, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of {\em trace polynomials}, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal--Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform $\mathbf{B}_{s,t}^N$ to its limit $\mathscr{G}_{s,t}$. We characterize the operator $\mathscr{G}_{s,t}$ through its inverse action on the standard polynomial basis. Finally, we show that, in the case $s=t$, the limit transform $\mathscr{G}_{t,t}$ is the ``free Hall transform'' $\mathscr{G}^t$ introduced by Biane.

math.FA↗

arXiv2013

Dimension-independent estimates for heat operators and harmonic functions

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace--Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.

math.DG↗

arXiv2012

Coherent states for a 2-sphere with a magnetic field

We consider a particle moving on a 2-sphere in the presence of a constant magnetic field. Building on earlier work in the nonmagnetic case, we construct coherent states for this system. The coherent states are labeled by points in the associated phase space, the (co)tangent bundle of S^2. They are constructed as eigenvectors for certain annihilation operators and expressed in terms of a certain heat kernel. These coherent states are not of Perelomov type, but rather are constructed according to the "complexifier" approach of T. Thiemann. We describe the Segal--Bargmann representation associated to the coherent states, which is equivalent to a resolution of the identity.

math-ph↗

arXiv2012

Complex structures adapted to magnetic flows

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g,$ and let $β$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^*M$ that describes a charged particle moving in the magnetic field $β$. Following an idea of T. Thiemann, we construct a complex structure on a tube inside $T^*M$ by pushing forward the vertical polarization by the Hamiltonian flow "evaluated at time $i$." This complex structure fits together with $ω-π^*β$ to give a Kaehler structure on a tube inside $T^*M$. We describe this magnetic complex structure in terms of its $(1,0)$-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney-Bruhat and Grauert, which is a magnetic analogue to the analytic continuation of the geometric exponential map. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by $-β$, and we give two formulas for local Kaehler potentials, which depend on a local choice of vector potential 1-form for $β$. When $β=0$, our magnetic complex structure is the adapted complex structure of Lempert-Szőke and Guillemin-Stenzel. We compute the magnetic complex structure explicitly for constant magnetic fields on $\mathbb{R}^{2}$ and $S^{2}.$ In the $\mathbb{R}^{2}$ case, the magnetic adapted complex structure for a constant magnetic field is related to work of Krötz-Thangavelu-Xu on heat kernel analysis on the Heisenberg group.

math.SG↗

arXiv2010

The Segal-Bargmann transform for compact quotients of symmetric spaces of the complex type

Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let Γ be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal--Bargmann transform, defined in terms of the heat equation, on the compact quotient Γ\G/K. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula in this setting, our proofs make use of double coset integrals and a holomorphic change of variable.

math-ph↗

arXiv2010

Adapted complex structures and the geodesic flow

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the "imaginary-time geodesic flow" to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by "composition with the imaginary-time geodesic flow." We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.

math.SG↗

arXiv2009

Toeplitz operators on generalized Bergman spaces

We consider the weighted Bergman spaces HL^2(B^d,μ_λ), where dμ_λ(z)=c_λ(1-|z|^2)^lambda dτ, τbeing the hyperbolic volume measure. These spaces are nonzero if and only if λ>d. For 0<λ\leq d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert--Schmidt operators on the generalized Bergman spaces.

math.CV↗

arXiv2008

Berezin-Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and Kc its complexification. The generalized Segal-Bargmann space for Kc, is a space of square-integrable holomorphic functions on Kc, with respect to a K-invariant heat kernel measure. This space is connected to the "Schrodinger" Hilbert space L^2(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L^2(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on Kc. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.

math-ph↗

arXiv2007

The heat operator in infinite dimensions

Let (H,B) be an abstract Wiener space and let μ_{s} be the Gaussian measure on B with variance s. Let Δbe the Laplacian (*not* the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H. I will show that the heat operator \exp(tΔ/2) is a contraction operator from L^2(B,μ_{s} to L^2(B,μ_{s-t}), for all t<s. More generally, the heat operator is a contraction from L^p(B,μ_{s}) to L^q(B,μ_{s-t}) for t<s, provided that p and q satisfy (p-1)/(q-1) \leq s/(s-t). I give two proofs of this result, both very elementary.

math-ph↗

arXiv2007

Isometry theorem for the Segal-Bargmann transform on noncompact symmetric spaces of the complex type

We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.

math-ph↗

arXiv2007

Unitarity in "quantization commutes with reduction"

Let M be a compact Kahler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M//G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515--538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M//G and the G-invariant subspace of the quantum Hilbert space over M. We prove that in general the natural map of Guillemin and Sternberg is not unitary, \textit{even to leading order in Planck's constant}. We then modify the quantization procedure by the "metaplectic correction" and show that in this setting there is still a natural invertible map between the Hilbert space over M//G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin--Sternberg map is asymptotically unitary to leading order in Planck's constant.

math.SG↗

arXiv2005

The range of the heat operator

This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic continuation to the appropriate complexified manifold with growth in the imaginary directions at most like that of the time-t heat heat kernel.

math.DG↗

arXiv2005

The Segal-Bargmann transform for noncompact symmetric spaces of the complex type

We consider the generalized Segal-Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal-Bargmann transform is a unitary map onto a certain L^2 space of meromorphic functions. For general functions, we give an inversion formula for the Segal-Bargmann transform, involving integration against an "unwrapped" version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities.

quant-ph↗

arXiv2004

Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform

We consider the generalized Segal-Bargmann transform C_t for a compact group K, introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. Let K_C denote the complexification of K. We give a necessary-and-sufficient pointwise growth condition for a holomorphic function on K_C to be the image under C_t of a C-infinity function on K. We also characterize the image under C_t of Sobolev spaces on K. The proofs make use of a holomorphic version of the Sobolev embedding theorem.

math-ph↗

arXiv2002

The large radius limit for coherent states on spheres

This paper concerns the coherent states on spheres studied by the authors in [J. Math. Phys. 43 (2002), 1211-1236]. We show that in the odd-dimensional case the coherent states on the sphere approach the classical Gaussian coherent states on Euclidean space as the radius of the sphere tends to infinity.

quant-ph↗

arXiv2002

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.

quant-ph↗

arXiv2001

Coherent states on spheres

We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere S^d. The coherent states are labeled by points in the associated phase space T*(S^d). These coherent states are NOT of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal-Bargmann representation for the system, the associated unitary Segal-Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of B. Hall and M. Stenzel, we give here a substantially different description based on ideas of T. Thiemann and of K. Kowalski and J. Rembielinski. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to be able to carry out the calculations in a self-contained and explicit way.

quant-ph↗

arXiv2001

Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type

Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kahler polarization obtained by identifying T*(K) with the complexified group K_C. The first main result is that the Hilbert space obtained by using the Kahler polarization is naturally identifiable with the generalized Segal-Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal-Bargmann transform introduced by the author. This means that in this case the pairing map is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kahler polarization. Together with results of the author with B. Driver, these results may be seen as an instance of "quantization commuting with reduction."

quant-ph↗