Reflection positivity has several applications in both mathematics and physics. For example, reflection positivity induces a duality between group representations.
In this thesis, we coin a new definition for a new kind of reflection positivity, namely, twisted reflection positive representation on a vector space. We show that all of the non-compactly causal symmetric spaces give rise to twisted reflection positive representations. We discover examples of twisted reflection positive representations on the sphere and on the Grassmannian manifold which are not unitary, namely, the generalized principle series with the Cosine transform as an intertwining operator. We give a direct proof for the reflection positivity of the Cosine transform on SO(n).
On the other hand, we generalize an integrability theorem to the case of non-positive definite distribution. As a result, we give a relation between the non-compactly causal symmetric spaces and the reflection positive distributions. Cocycle conditions are also treated.
We construct a general method to generate twisted reflection positive representations and then we apply it to get twisted reflection positive representations on the Euclidean space.
Finally, we introduce a reflection positive cyclic distribution vector for the circle case. Then we prove that this distribution vector generates a well known reflection positive function.
| Identifer | oai:union.ndltd.org:LSU/oai:etd.lsu.edu:etd-04042016-205851 |
| Date | 21 April 2016 |
| Creators | Hayajneh, Mostafa Ahmad |
| Contributors | Davidson, Mark, Litherland, Richard A., Shipman, Stephen, Yakimov, Milen, Rau, Ravi, Olafsson, Gestur |
| Publisher | LSU |
| Source Sets | Louisiana State University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lsu.edu/docs/available/etd-04042016-205851/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached herein a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to LSU or its agents the non-exclusive license to archive and make accessible, under the conditions specified below and in appropriate University policies, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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