Infinite-dimensional Heisenberg Groups Research Articles

On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modeled on an abstract Wiener space.

LT-equivariant index from the viewpoint of KK-theory: A global analysis on the infinite-dimensional Heisenberg group

Let T be a circle group, and LT be its loop group. We hope to establish an LT-equivariant index theory for Hamiltonian LT-spaces, from the viewpoint of KK-theory. We have already constructed several objects in the previous paper (Takata, arXiv:1701.06055), including a Hilbert space H which can be regarded as a “Hilbert space consisting of L2-sections of a Spinor bundle on the infinite-dimensional manifold”, an “LT-equivariant Dirac operator D” acting on H, and the “twisted group C∗-algebra of LT”, in an ad hoc way.In this paper, we will construct more sophisticated objects for the index problem, in terms of KK-theory. Concretely, we will define analytic index of D as a K-theory element of the “twisted group C∗-algebra of LT” mentioned above. Then, we will also define the “latter half” of the assembly map at the “index element” (H,D). Finally, we will prove that the value of the “latter half” of the assembly map at the “index element”, coincides with the above analytic index of D. This is an infinite-dimensional version of a part of the index theorem for complete Riemannian manifolds with proper cocompact actions of Kasparov.We are mainly interested in the LT-equivariant index problem of infinite-dimensional manifolds, and so we use the language of representation of LT. However, our theory can be easily generalized to the setting of infinite-dimensional abstract Heisenberg groups equipped with some additional structures. This is potentially useful for researchers of other areas.

Paley-Wiener Isomorphism Over Infinite-Dimensional Unitary Groups

An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl–Schrödinger irreducible representations of complexified infinite-dimensional Heisenberg groups.

On Wigner transforms in infinite dimensions

We investigate the Schrödinger representations of certain infinite-dimensional Heisenberg groups, using their corresponding Wigner transforms.

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron–Martin subgroup, and that the Radon–Nikodym derivative is Malliavin smooth.

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in \cite{BaudoinGarofalo2011}.

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, { ν t } t > 0 , are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron–Martin type quasi-invariance results, (3) good L p -bounds on the corresponding Radon–Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.

Integrability of One-Parameter Groups Generated by the Virasoro Operators

Local one-parameter groups generated by the Virasoro operators were constructed via Feynman path integrals on a coadjoint of the infinite-dimensional Heisenberg group in the previous paper [T. Hashimoto, 1996.J. Funct. Anal.137, 191–218]. The main purpose of this paper is to prove that the one-parameter groups satisfy the integrability condition, i.e., the actions of the one-parameter subgroups are compatible with the action of the Virasoro algebra.

Construction of Local One-Parameter Subgroups Generated by the Virasoro Operators via Feynman Path Integrals

It is well known that the Virasoro algebra acts on the representations of an infinite-dimensional Heisenberg algebra, called the Fock models. We calculate Feynman path integrals on an coadjoint orbit of the infinite-dimensional Heisenberg group to give explicit formulae of actions of local one-parameters subgroups generated by the Virasoro operators.

Nonlinear quantum fields in ≥4 dimensions and cohomology of the infinite Heisenberg group

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if ϕ \phi denotes the free massless scalar field in d-dimensional Minkowski space ( d ≥ 4 d \geq 4 , even) and if q is an even integer greater than or equal to 4, then ∫ M 0 : ϕ ( X ) q : d X {\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : ϕ ( X ) q \phi {(X)^q} : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case d = q = 4 d = q = 4 and to corresponding scattering theory.