arXiv subjects

Stephen Bruce Sontz

Publications and source records attributed to Stephen Bruce Sontz.

24 records · Page 2Linked to original sources

arXiv2009

Relations among various versions of the Segal-Bargmann transform

We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A.

math-ph↗

arXiv2009

La Probabilidad de la Mecanica Cuantica: Una Introduccion en noventa minutos

Referring to quantum mechanics, Einstein used to say "The old one does not play dice." And this is true since the probability of quantum mechanics is not the classical probability of games such as dice. Historically this was the first example of a non-classical probability theory, which we introduce in this expository article using undergraduate linear algebra. There is a short appendix on qubits. Knowledge of quantum mechanics is not required.

math-ph↗

arXiv2009

On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall's results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in $μ$-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal-Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall's Version C generalized Segal-Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal-Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal-Bargmann kernels for Versions A and C.

math-ph↗

arXiv2009

The $μ$-deformed Segal-Bargmann transform is a Hall type transform

We present an explanation of how the $μ$-deformed Segal-Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarria and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal-Bargmann analysis. This explanation relates the $μ$-deformed Segal-Bargmann transforms to the generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the $μ$-deformed Segal-Bargmann transform can be understood as Hall type transforms. In particular, we define a $μ$-deformation of Hall's "Version C" generalized Segal-Bargmann transform which is then shown to be a $μ$-deformed convolution with a $μ$-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall.

math-ph↗

arXiv2008

How the $μ$-deformed Segal-Bargmann space gets two measures

This note explains how the two measures used to define the $μ$-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the $μ$-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the $μ$-deformed Segal-Bargmann space.

math-ph↗

arXiv2007

Direct and reverse log-Sobolev inequalities in $μ$-deformed Segal-Bargmann analysis

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a $μ$-deformed energy, are shown to hold in a family of $μ$-deformed Segal-Bargmann spaces. This shows that the $μ$-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the $μ$-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform.

math-ph↗