a r X i v :g r -q c /9701042v 1 18 J a n 1997Abstracts of Seminars given at the Workshop on
Mathematical Problems of Quantum Gravity
held at the Erwin Schr¨o dinger Institute,Vienna
Peter Aichelburg 1and Abhay Ashtekar 2Organizers 1Institute of Theoretical Physics University of Vienna,Boltzamanngas 5,A-1090Vienna 2
Center for Gravitational Physics and Geometry Physics Department,Penn State,University Park,PA 16802This pre-print contains the abstracts of minars (including key references)prented at the ESI workshop on mathematical problems in quantum gravity held during July and August of 1996.Contributors include A.Ashtekar,J.Baez,F.Barbero,A.Barvinsky,F.Embacher,R.Gambini,D.Giulini,J.Halliwell,T.Jacobson,R.Loll,D.Marolf,K.Meissner,R.Myers,J.Pullin,M.Reinberger,C.Rovelli,T.Strobl and T.Thiemann.While the contributions cover most of the talks given during the workshop,there were also a few additional speakers who contributions were not received in time.
Contributors Ashtekar,Abhay:Ashtekar@phys.psu.edu
Baez,John:Baez@.math.ucr.edu
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Fernando Barbero:Barbero@LAEFF.ESA.Es
A.O.Barvinsky:grg@ibrae.msk.edu
Franz Embacher:fe@pap.univie.ac.at政审个人总结
Rodolfo Gambini:Rgambini@fisica.edu.uy
Domenico Giulini:DGiulini@esi.ac.at
J.J.Halliwell:J.halliwell@ic.ac.uk
T.Jacobson:Jacobson@umdhep.umd.edu
Renate Loll:Loll@aei-potsdam.mpg.de
Don Marolf:Marolf@suhep.phy.syr.edu
Krzysztof A.Meissner:h
Robert Myers:RCM@hep.Physics.McGill.CA
剥莲蓬
Jorge Pullin:Pullin@phys.psu.edu
Michael P.Reinberger:Mreis@medb.physics.utoronto.ca Carlo Rovelli:Rovelli@vms.cis.pitt.edu
Thomas Strobl:Tstrobl@wth-aachen.de Thomas Thiemann:Thiemann@abel.math.harvard.edu
Quantum theory of geometry
Abhay Ashtekar
This was primarily a review talk,bad largely on joint work with Jerzy Lewandowski.
Over the last three years,a new functional calculus has been developed on the quantum configuration space of general relativity without any reference to a background geometrical structure in space-time(such as a metric).The purpo of this talk was to indicate how this machinery can be applied to systematically construct a quantum theory of geometry.The kinematical Hilbert space of quantum gravity was prented.States reprent polymer-like,1-dimensional excitations of geometry.Regulated Operators corresponding to areas of2-surfaces were introduced on the kinematical Hilbert space of quantum gravity and shown to be lf-adjoint.Their full spectrum was pr
ented.It is purely discrete and contains some physically interesting information.First,the“area gap”,i.e.,the value of the smallest non-zero excitation,contains information about the global topology of the surface.Second,in the large eigenvalue limit,the eigenvalues become clor and clor to each other such that|a n+1−a n|≤[(l P/2√
One of the implications of the work reported in thefirst talk is that area operators as-sociated with different operators do not always commute.This is atfirst surprising becau the classical formula for areas involves only triads,without any reference to connections and from the basic Poisson bracket relations one expects the triads to commute among themlves.It turns out,however,that the naive expectation is incorrect.The reason is that the formula for areas involves triads which are smeared only on2-surfaces rather than in3dimensions and the Poisson brackets between such objects are,strictly speaking, singular.(Furthermore,in our framework bad on holonomies,triad operators which are smeared in3dimensions are not likely to be well-defined!)
To analyze this issue in detail,we examine the Poisson algebra between the following pha space functions:cylindrical functions of(smooth)connections and triads smeared on 2-surfaces.(Incidentally,since the triads have density weight one,they are in fact2-forms and it is thus geometrically natural to smear them on2-surfaces.)If we assume naively that the smeared triads com
mute,we run into a problem with the Jacobi identity;the naive Poisson algebra is not a Lie algebra and is therefore incorrect.One can regulate the naive algebra carefully to obtain a Lie algebra.Then,onefinds that the(2-dimensionally) smeared triads fail to Poisson commute.The commutators between the quantum triad operators just mirror this correct Poisson algebra and this is why the area operators fail to commute.
A.Ashtekar,A.Corichi,J.Lewandowski and J.A.Zapata,CGPG pre-print.
Geometry of quantum mechanics
Abhay Ashtekar
This talk summarized joint work with Troy Schilling which constitutes his1996Ph.D. thesis at Penn State.
In the way we normally formulate the theories,classical mechanics has deep roots in(symplectic)geometry while quantum mechanics is esntially algebraic.However,one can recast quantum mechanics in a geometric language which brings out the similarities and differences between the two theories.The idea is to pass from the Hilbert space to the space of the“tru
e”space of states of quantum mechanics.The space of rays –or the projective Hilbert space,is in particular,a symplectic manifold,which happens to be equipped with a further K¨a hler structure.Regarding it as a symplectic manifold, one can repeat the familiar constructions from classical mechanics.For example,given any function,one can construct its Hamiltonian vectorfield.If one us the expectation value of the Hamiltonian operator as the function,it turns out that the resulting“clas-sical”symplectic evolution is precily the(projection of the)Schr¨o dinger evolution on the Hilbert space.Roughly,properties of quantum mechanics which it“shares”with clas-sical mechanics u only the symplectic structure on the projective Hilbert space.The
“genuinely”quantum properties such as uncertainties and probabilities refer to the K¨a hler metric.Thus,purely in mathematical physics terms,one can regard quantum mechanics as a special ca of classical mechanics,one in which the pha space happens to have a K¨a hler structure(which then enables one to do more.)This geometrical formulation of quantum mechanics sheds considerable light on the cond quantization procedure and on mi-classical states and dynamics.
After the work was completed,we found that many of our results were discovered independently by a number of authors,most notably L.Hughstone and by R.Cirelli,A. Mani´a and L.Pizzochero.
A.Ashtekar and T.Schilling;in:The Proceedings of the First Canadian-Mexican-American Physical Societies’Conference,edited by A.Zapeda(American Institute of Physics,NY1995.)
T.Schilling;Geometry of Quantum Mechanics,Ph.D.Thesis,Penn State,1996.
Probing quantum gravity through
exactly soluble midi-superspaces
Abhay Ashtekar
This talk summarized joint work with Monica Pierri which constituted part of her Ph.D.thesis.
The idea was to consider midi-superspaces which are simple enough to be exactly soluble both classically and quantum mechanically and u the solution to probe various nagging issues of quantum gravity such as the issue of time and the nature of the vacuum. The specific example prented was the midi-superspace of ,cylindrical) gravitational waves.This model was analyzed by Karel Kuchˇa r already in the early ven-ties and by Michel Allen in the mid-eighties.However certain issues concerning boundary conditions,surface terms and functional analytic subtleties could not be discusd then. Using results on asymptotics and techniques of regul
arization that have been developed since then,their discussion can be completed to construct a complete quantum theory. We ud this solution to construct a regulated,quantum space-time metric operator and address issues such as“light conefluctuations”.That this is possible within the canonical framework is noteworthy since concerns are often expresd that the canonical quantiza-tion procedure may not be able to handle such“space-time”issues.The model has a well-defined,non-trivial Hamiltonian operator and a stable vacuum state(the eigenstate of the Hamiltonian with zero eigenvalue.)On general states,one can write the Schr¨o dinger equation.However,the mathematical parameter in this equation has the physical inter-pretation of time only on mi-classical states.Finally,this solution can also be ud to probe a key issue in our non-perturbative quantum gravity program:existence of operators
corresponding to traces of holonomies around clod loops.The operators do exist in spite of the fact that they involve smearing of the connection only in one dimension.
More recently,this model has been ud to show the existence of certain unforeen quantum gravity effects which can be large even when the space-time curvature is small.
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A.Ashtekar and M.Pierri,gr-qc/9606085,J.Math.Phys.,37,6250-70,(1996).
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A.Ashtekar,gr-qc/9610008,Phys.Rev.Lett.77,4864-67(1996).
历史的作用
Topological Quantum Field Theory
John Baez
The simplest sort of topological quantumfield theory is BF theory,where the La-grangian is of the form tr(BF),with F being the curvature of a connection and B being a Lie-algebra valued(n−2)-form in n dimensions.When n is3or4one can also add a”cosmological constant term”of form tr(BBB)or tr(BB),respectively.In this talk,I summarized what is known about BF theory in dimensions2,3,and4,as well as the equivalent state sum models.In particular,I described how state sum models of BF-like theories in2dimensions ari from certain monoids,while in3dimensions they ari from certain monoidal categories and in4dimensions from certain monoidal2-categories (most notably the category of reprentations of a quantum group,which may be en as a monoidal2-category with one object).I also sketched how4-dimensional BF theory underlies Chern-Simons theory in3dimensions.
References:
John Baez and James Dolan,Higher-dimensional algebra and topological quantum field theory,Jour.Math.Phys..36(1995),6073-6105.
谢延信John Baez,Four-dimensional BF theory as a topological quantumfield theory,to appear in Lett.Math.Phys.,preprint available as q-alg/9507006.
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