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My research is in mathematical physics and involves Lie groups, functional analysis, probability, and geometry. Specifically, I study generalizations of the Segal-Bargmann transform. The idea is as follows. In classical mechanics, one typically has a "configuration space," which is some manifold M. One then has the associated "phase space," which is the cotangent bundle of M. The cotangent bundle arises because Newton's equations are second order in time: a second-order equation on M becomes a first-order equation on the cotangent bundle of M. In the corresponding quantum system, one tries to construct a Hilbert space associated to the classical system. The simplest such space is the "position Hilbert space," which is a the space of square-integrable functions over M with respect to some measure. Alternatively, one can look for some nice complex structure on the cotangent bundle of M and then build a space of square-integrable holomorphic function on the cotangent bundle, again with respect to some (hopefully natural) measure. Such a space is called a (generalized) Segal-Bargmann space. A natural unitary map between the position Hilbert space and the Segal-Bargmann space is called a Segal-Bargmann transform. Associated to such a Segal-Bargmann transform is a collection of special quantum states called "coherent states."

Segal and Bargmann themselves worked on the case in which the configuration space M is R^n and the phase space (cotangent bundle of M) is identified with C^n. In my Ph.D. thesis, I introduced a generalization of this in which the configuration space M is a compact Lie group K and the phase space is the "complexification" of K. (For example, on may take K = SU(n), in which case the complexification is the complex group SL(n,C). The complexification of K can also be identified with the cotangent bundle of K.) This work was extended by Stenzel to the case in which M is an arbitrary compact symmetric space; for example, M could be a sphere of arbitrary dimension. See the survey article, Publication 11 on my publications page, for an overview of these and related results.

I should point out that there are many other sorts of generalizations, in different directions, of the work of Segal and Bargmann, notably (1) Perelomov's notion of generalized coherent states and (2) work, beginning with Berezin and Rawnsley, on the geometric quantization of Kahler manifolds. (The geometric quantization program does intersect directly with what I am working on; see below.) For example, a search in MathSciNet with "coherent state" in the title line gives (as of November 2003) 1086 hits.

Connections with stochastic analysis

The motivation for my thesis work was work of Leonard Gross in stochastic analysis. Gross proved a result now known as the Gross ergodicity theorem (see for example, Publication 17 on my publications page). As an outgrowth of his proof, Gross established an analog on a compact Lie group of the Fock (symmetric tensor) decomposition. This led him to suggest to me to look for a version of the Segal-Bargmann transform on a compact Lie group. Although I was motivated by Gross's work in stochastic analysis, my thesis itself was purely "finite dimensional" and had no stochastic analysis in it.
Later on, Gross and Paul Malliavin found a direct connection between the Gross ergodicity theorem and the generalized Segal-Bargmann transform for a compact group. Ambar Sengupta and I then expanded on this connection--see Publications 5 and 8.

Connections with quantized Yang-Mills theory

Meanwhile, Ken Wren, then a student of Klaas Landsman at Cambridge University, was considering the problem of the canonical quantization of Yang-Mills theory on a spacetime cylinder. This problem has been studied many times in many different ways over the years, but Wren used the "Rieffel induction" approach proposed by Landsman. In this problem, one has an infinite-dimensional configuration space of "connections" over the spatial circle.One then "reduces" by the group of (based) gauge tranformations. A remarkable feature of this system is that the reduced system, consisting of connections modulo gauge transformations is finite dimensional. Specifically, the space of connections modulo gauge transformations can be identified with a single copy of the compact structure group K.

In considering the quantum theory, Wren uses coherent states for the space of connections and then attempts to "project" these into the (non-existent) "gauge-invariant subspace." This "projection" is supposed to to be accomplished by integration with respect to the (also non-existent) "Haar measure" on the infinite-dimensional group of gauge transformations. Although this appears very ill defined, Wren was able to combine a "Gaussian" factor in the relevant integrals with the fictitious Haar measure to obtain a well-defined integral involving the Wiener measure on the group of gauge transformations and thus to obtain a well-defined "projected" family of coherent states. Remarkably, these projected coherent states obtained by Wren turn out to coincide with the coherent states on K coming from my generalized Segal-Bargmann transform. Thus the coherent states that I introduced from a purely finite-dimensional point of view turn out to be interpretable as the coherent states for Yang-Mills theory over a spacetime cylinder. See my review of Wren's paper in Mathematical Reviews [99g:58019] for further information.

Bruce Driver and I elaborated on the ideas of Wren, using a somewhat different "projection," in order to obtain not just the coherent states themselves but also the associated Segal--Bargmann transform out of the projection method. See Publications 6 and 12 on my publications page.

More on coherent states and connections with quantum gravity and particle physics

In another direction, there have been several attempts to apply the generalized Segal-Bargmann transform in physics. This began in a paper of A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourão, and T. Thiemann [J. Funct. Anal. 135 (1996), 519-551], in which the authors glue together the generalized Segal-Bargmann transform for multiple copies of SU(2) in order to provide a Segal-Bargmann-type transform for the Ashtekar approach to quantum gravity. This was designed to deal with certain "reality conditions" in the Ashtekar approach.

In a related development, T. Thiemann has developed a pure real-variables approach to quantum gravity, called "quantum spin dynamics" described in a series of papers in Classical and Quantum Gravity. Since then, Thiemann and co-authors have been trying to establish whether that theory reduces to ordinary general relativity in the classical limit. In order to do this, they have been trying to build coherent states that approximate a state of classical general relativity by piecing together "my" coherent states for various copies of SU(2) associated to spatial graphs. This has involved, among other things, investigating the semiclassical properties of my coherent states in the SU(2) case. [See papers by Thiemann and co-authors in concerning "gauge field theory coherent states."]

Along the way, Thiemann developed a new way of thinking about these generalized coherent states, which he calls the "complexifier" method. This method has the potential to produce many other examples of generalized coherent states (although this possibility has not yet been explored in great detail) and even in the cases where the coherent states are already known it gives an interesting alternative point of view. From another direction, the Polish physicists Kowalski and Rembielinski independently discovered the Hall-Stenzel type coherent states for the case of a 2-sphere, using a "polar decomposition" method. [See J. Phys. A 33 (2000), 6035--6048 and J. Math. Phys. 42 (2001), 4138--4147.] Jeff Mitchell and I set out to understand better both the complexifier method and the polar decomposition method. We considered this for the case of an n-sphere and showed how the Thiemann and the Kowalski and Rembieli_ski approaches fit together in this case. We also gave self-contained elementary proofs of all the relevant results in this case and investigated the large-radius limit of the coherent states. See Publications 14, 16 and 30 on my publications page. There have been quite a few other papers along the lines of Thiemann's work, which make use of the heat-kernel coherent states on SU(2), in the quantum gravity literature.

Connections with geometric quantization

Geometric quantization [e.g., the book of N. M. J. Woodhouse] gives a method of building quantum Hilbert spaces from classical systems. A by-now standard example is geometric quantization is the construction of the Segal--Bargmann space for C^n using geometric quantization with a "complex polarization." One can also construct the ordinary Segal--Bargmann transform by means of the "pairing map" of geometric quantization. On the surface of it, the method I use to construct the generalized Segal--Bargmann transform for a compact Lie group seems completely unrelated to the geometric quantization. I use heat kernel methods and geometric quantization seemingly knows nothing about heat kernels.

Nevertheless, it makes sense to apply geometric quantization in the setting of a compact Lie group and see how the results compare to my heat kernel approach. I carried out this computation in Publication 15, and amazingly the two approaches give precisely the same results! This mysterious result has been understood better thanks to work of Florentino, Matias, Mourao, and Nunes [J. Funct. Anal., 2005 and 2006] and recent work of Kirwin, Mourao, and Nunes [arXiv:1203.4767 mathDG].

This result, in conjunction with the results described previously concerning the quantization of two-dimensional Yang--Mills theory, also raises interesting questions about the relationship of quantization to reduction. See Publications 11 and 12 for further discussion of this point.

Holomorphic Sobolev spaces

In a recent paper, Wicharn Lewkeeratiyutkul and I have considered how smoothness of a function f on a compact Lie group K affects the behavior of the Segal-Bargmann transform of f. We have found that each derivative that f possesses gives a specific amount of improvent in the behavior at infinity of the transform. This leads to a necessary-and-sufficient pointwise condition characterizing the image under the Segal--Bargmann transform of the space of smooth functions on K. The proofs involve a notion of "holomorphic Sobolev spaces" on the complexification of K, along with a holomorphic version of the Sobolev embedding theorem. See Publication 19.

The noncompact case

Most recently, I have been working with Jeff Mitchell to construct something like a Segal-Bargmann transform for noncompact symmetric spaces. In the work of Stenzel on Segal-Bargmann for compact symmetric spaces, each fiber in the cotangent bundle of a compact symmetric space gets identified with the "dual" noncompact symmetric space. For example, in the transform for the n-sphere, each fiber in the cotangent bundle gets identified with n-dimensional hyperbolic space. It is thus very natural to try to reverse the roles of the compact and noncompact symmetric spaces in order to get a transform that starts with a function on a noncompact symmetric space such as hyperbolic space.

Unfortunately, this nice idea runs into trouble almost immediately--one gets all sorts of singularities in the noncompact case that do not arise in the compact case. If one could simply close one's eyes and pretend these singularities do not exist, then one could formally argue precisely as in the compact case. What goes wrong is not so much that the proofs from the compact case do not go through, but rather that the statements from the compact case do not make sense, because of these singularities. I have been looking for years now for ways to get these singularities to cancel out or otherwise go away. Jeff Mitchell and I finally made some headway on this problem, as described in Publications 21, 23, 24 and 35. A complementary approach to this problem was taken by B. Krötz, G. Ólafsson, and R. Stanton [The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type. Int. Math. Res. Not. 2005, no. 22, 1307–1329].

Complex structures

A key part of the construction of a holomorphic type Hilbert space is the construction of the appropriate complex structure on the classical phase space. In the case of the cotangent bundle of a Riemannian manifold, one beautiful construction of such complex structures is the "adapted complex structure," introduced independently by Guillemin and Stenzel and by Lempert and Szoke. Will Kirwin and I have given a new way of thinking about these structures, using the "imaginary time geodesic flow". (See Publication 28.) We have then used the idea of imaginary-time flows to generalize the notion of adapted complex structure to flows that incorporate a magnetic field on the base manifold. (See Publication 31.)

The large-N limit

It is natural to attempt to take the limit of the Segal-Bargmann transform on some compact Lie group as the dimension tends to infinity. We may consider, for example, the nested family of unitary groups U(N) and try to let N tend to infinity. Work of Maria Gordina [Potential Analysis and J. Functional Analysis, 2000] shows that the most obvious approach to this limit does not work. Nevertheless, work of Ph. Biane [J. Funct. Analysis 1997] suggests a way that one can obtain something interesting in the limit, by rescaling the metric on U(N) as a function of N in an appropriate way. Recent work with B. Driver and T. Kemp shows how this works in detail. See Publications 32 and 34.

 The Makeenko-Migdal equation

In work with Bruce Driver, Franck Gabriel, and Todd Kemp, I have explored the Makeenko-Migdal equation for Yang-Mills theory over the plane (Publication 36) and over an arbitrary compact surface (Publication 37). In the plane case, we give new, short proofs of the Makeenko-Migdal equation, building on ideas in the original proof by Thierry Levy. In Publication 37, we provide the first rigorous proof of the Makeenko-Migdal equation for general surfaces.

 

 

 

Department of Mathematics | University of Notre Dame
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