STATEMENT RESARCH PHILIP TILLMAN
The general questions driving my research are What are the foundations of quantum eld theories (QFT) in curved spacetimes? and What is the correct theory that encompasses both quantum theory and gravity? The interplay between gravity and quantum theory is one of the most debated subjects in physics. My approach to the topic is to investigate the structure of QFT on curved space-times and incorporate new s developments such as from deformation quantization (DQ), algebraic quantum eld theory, geometrical algebras, and others. In other words, gravity here is treated as a classical background for quantum systems.
Ph.D. Dissertation
Canonical quantization seems to break the dieomorphism invariance of general relativity (GR) because of the need to choose the coordinates of phase-space (x and p) before quantization is performed. In fact, the theorem of Groenewold and van Howe states that the rules of canonical quantization are an inconsistent framework. Yet, canonical quantization is enormously successful and so this raises the question of how we understand both its successes and its shortcomings. It is a conventional belief that quantum theory is independent of the choice of coordinates, which implies that this independence is hidden in the current mathematical formulation of canonical quantization. There are two main competing alternatives to canonical quantization: deformation quantization (DQ) [HiHe] and geometric quantization, both of which attempt to solve this problem as well as other conceptual problems of canonical quantization. In [Til] I argue that DQ provides a very reasonable modication of those rules that make it consistent. This is the main reason why I chose to study DQ. An example of DQ is Fedosov quantization [Fed],[GdT]. The Fedosov quantization map is a map from C 1 phase-space functions to linear Hilbert space operators. The power of the Fedosov quantization map is that it can be written at least perturbatively, by means of Fedosov algorithm, for all nite-dimensional s symplectic manifolds, thought as generalized phase-spaces. However, establishing that the series converges needs to be done on a case-by-case basis, because of topological obstructions. In my dissertation I constructed the Fedosov quantization map on constant curvature manifolds of codimension one exactly. This class includes the two-sphere (also in [TiSp2]) and the de Sitter/Anti-de Sitter (dS/AdS) space-times of GR (also in [TiSp1]). I chose this class of manifolds because if Fedosov quantization procedure is a good choice to generalize, it certainly should be constructed for these manifolds, which are relatively simple yet nontrivial. I observed that Fedosov construction is very powerful and general, because s it gives us a guide to possibly generalize all quantum theories on space-times to curved space-times. at In essence, my dissertation validated Fedosov quantization of DQ by showing that it agrees with previous quantizations for constant curvature manifolds of codimension one. In addition, the insights of Fedosov s algorithm were rened, and the physical motivations for key formulas and steps were claried.
Current and Future Research
A longstanding problem in QFT on curved space-times is the axiomatic construction of general QFT The s. free eld is well understood (see [Wald]), but the interacting and gauge eld cases need to be developed. Most of the literature on interacting QFT deal only with Riemannian space-times, but, as is known, the s connection between the Riemannian and Lorentzian case is in general not valid. The Osterwalder-Schrader reconstruction theorem [OsSc] gives the precise conditions for the connection to hold. To this end, I am currently investigating the work Brunetti and Fredenhagen [BrFr] in which they presented a perturbative of construction '4 theory on all smooth globally hyperbolic space-times. It relies on a new method of renormalization which is a local version of a renormalization program developed by Epstein and Glaser [Sch]. The goal of my work here is to rst understand and then simplify this renormalization program which is highly technical. One important outstanding issue in [BrFr] is the existence of a full quantum theory associated to the series expansion of the perturbation. Using this techique of local renormalization combined with the insights of Fedosov quantization we may be able to extend these constructions to more general QFT (e.g., gauge QFT and possibly non-perturbative s s
1
QFT in curved space-times. Further in the future, these insights may be instrumental in the development s) and understanding of quantum gravity.
References
[BrFr] Brunetti R. and Fredenhagen K., Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208, 623 (2000). Brunetti R. and Fredenhagen K., Interacting Quantum Fields in Curved Space: Renormalizability of '4 , proceedings of the conference Operator Algebras and Quantum Field Theory held at Accademia Nazionale dei Lincei, Rome, Italy, July 1996, Preprint gr-qc/9701048. [Buc] Buchholz, D. Proc. XIIIth Int. Congress on Mathematical Physics (London) (2000) Preprint mathph/0011044. [DFr] Dtsch M. and Fredenhagen K., (Proc. Conf. Math. Physics in Mathematics and Physics (Siena) (2000), Preprint hep-th/0101079. [Fed] Fedosov B., Deformation Quantization and Index Theory, Akademie, Berlin 1996. [Fre] Fredenhagen K. et al, Quantum Field Theory: Where We Are, contribution to: An Assessment of Current Paradigms in the Physics of Fundamental Phenomena, Springer-Verlang, Preprint hep-th/0603155. [GdT] Gadella M., del Olmo M.A., Tosiek J., Geometrical Origin of the product in the Fedosov Formalism, (2005) J. Geom. Phys. 55, 316. [Haag] Haag R., Local Quantum Physics, Springer-Verlag (1992). [HiHe] Hirshfeld A., Henselder P., Deformation quantization in the teaching of quantum mechanics, Am. J. Phys. 70 (5), May 537 547 (2002), quant-ph/0208163; Hirshfeld A., Henselder P., Deformation Quantization for Systems with Fermions, Annals Phys. 302 (2002) 59 77.; Hirshfeld A., Henselder P.,...