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Quantum Theory of Geometry I: Area Operators

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Quantum Theory of Geometry I: Area Operators The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Quantum Theory of Geometry I Area Op erators Abhay Ashtekar Jerzy Lewandowski Vienna Preprint ESI August Supp orted by Federal Ministry of Science and Research Austria Available via httpwwwesiacat Quantum Theory of Geometry I Area Op erators Abhay Ashtekar and Jerzy Lewandowski August Center for Gravitational Physics and Geometry Physics Department Penn State University Park PA USA Institute of Theoretical Physics Warsaw University ul Hoza Warsaw Poland Max Planck Institut fur Gravitationphysik Schlaatzweg Potsdam Germany Abstract A new functional calculus develop ed recently for a fully nonp erturbative treatment of quantum gravity is used to b egin a systematic construction of a quantum theory of geometry Regulated op erators corresp onding to areas of surfaces are intro duced and shown to b e selfadjoint on the underlying kinematical Hilb ert space of states It is shown that their sp ectra are purely discrete indicating that the underlying quantum geometry is far from what the continuum picture might suggest Indeed the fundamental excitations of quantum geometry are dimensional rather like p olymers and the dimensional continuum geometry emerges only on coarse graining The full Hilb ert space admits an orthonormal decomp osition into nite dimensional subspaces which can b e interpreted as the spaces of states of spin systems Using this prop erty the complete sp ectrum of the area op erators is evaluated The general framework constructed here will b e used in a subsequent pap er to discuss dimensional geometric op erators eg the ones corresp onding to volumes of regions It is a pleasure to dedicate this article to Professor Andrzej Trautman who was one of the rst to recognize the deep relation b etween geometry and the physics of gauge elds which lies at the heart of this investigation Intro duction In his celebrated inaugural address Riemann suggested that geometry of space may b e more than just a ducial mathematical entity serving as a passive stage for physical phenomena and may in fact have a direct physical meaning in its own right General relativity proved this vision to b e correct Einsteins equations put geometry on the same fo oting as matter Now the physics of this century has shown us that matter has constituents and the dimensional ob jects we p erceive as solids in fact have a discrete underlying structure The continuum description of matter is an approximation which succeeds brilliantly in the macroscopic regime but fails hop elessly at the atomic scale It is therefore natural to ask if the same is true of geometry Do es geometry also have constituents at the Planck scale What are its atoms Its elementary excitations Is the spacetime continuum only a coarsegrained approximation If so what is the nature of the underlying quantum geometry To prob e such issues it is natural to lo ok for hints in the pro cedures that have b een successful in describing matter Let us b egin by asking what we mean by quantization of physical quantities Let us take a simple example the hydrogen atom In this case the answer is clear while the basic observables energy and angular momentum take on a continuous range of values classically in quantum mechanics their sp ectra are discrete So we can ask if the same is true of geometry Classical geometrical observables such as areas of surfaces and volumes of regions can take on continuous values on the phase space of general relativity Are the sp ectra of corresp onding quantum op erators discrete If so we would say that geometry is quantized Thus it is rather easy to p ose the basic questions in a precise fashion Indeed they could have b een formulated so on after the advent of quantum mechanics An swering them on the other hand has proved to b e surprisingly dicult The main reason it seems is the inadequacy of the standard techniques More precisely the traditional approach to quantum eld theory has b een p erturbative where one begins with a continuum background geometry It is then dicult to see how discreteness would arise in the sp ectra of geometric op erators To analyze such issues one needs a fully nonp erturbative approach geometric op erators have to b e constructed ab initio without assuming any background geometry To prob e the nature of quantum geometry we can not b egin by assuming the validity of the continuum picture We must let quantum gravity itself decide whether this picture is adequate at the Planck scale the theory itself should lead us to the correct microscopic picture of geometry In this pap er we will use the nonp erturbative canonical approach to quantum gravity based on connections to prob e these issues Over the past three years this approach has b een put on a rm mathematical fo oting through the development of a new functional calculus on the space of gauge equivalent connections This calculus do es not use any background elds such as a metric and is therefore wellsuited to a fully nonp erturbative treatment The purp ose of this pap er is to use this framework to explore the nature of quantum geometry In section we recall the relevant results from the new functional calculus and outline the general strategy In section we present a regularization of the area op erator Its prop erties are discussed in section in particular we exhibit its entire sp ectrum Our analysis is carried out in the connection representation and the discussion is selfcontained However at a nontechnical level there is a close similarity b etween the basic ideas used here and those used in discussions based on the lo op representation Indeed the development of the functional calculus which underlies this analysis itself was motivated in a large measure by the pioneering work on lo op representation by Rovelli and Smolin The relation b etween various approaches will discussed in section The main result of this pap er should have ramications on the statistical me chanical origin of the entropy of black holes along the lines of This issue is b eing investigated Preliminaries This section is divided into three parts In the rst we will recall the basic structure of the quantum conguration space and in the second that of the Hilb ert space of kinematic quantum states The overall strategy will b e summarized in the third part Quantum conguration space In general relativity one can regard the space AG of SU connections mo dulo gauge transformations on a spatial manifold as the classical conguration space For systems with only a nite numb er of degrees of freedom the classical conguration space also serves as the domain space of quantum wave functions ie as the quantum conguration space For systems with an innite numb er of degrees of freedom on the other hand this is not true generically the quantum conguration space is an enlargement of the classical In free eld theory in Minkowski space as well as exactly solvable mo dels in low spacetime dimensions for example while the classical conguration space can b e built from suitably smo oth elds the quantum conguration space includes all temp ered distributions This is an imp ortant p oint b ecause typically the classical conguration spaces are of zero measure wave functions with supp ort only on smo oth congurations have zero norm The overall situation is the same in general relativity The quantum conguration AG is a certain completion of AG space AG inherits the quotient structure of AG ie AG is the quotient The space of the space A of generalized connections by the space G of generalized gauge trans formations To see the nature of the generalization involved recall rst that each R A smo oth connection denes a holonomy along paths in h A P exp p p Generalized connections capture this notion That is each A in A can b e dened as a map which assigns to each oriented path p in an element Ap of SU such that i Ap Ap and ii Ap p Ap Ap where p is obtained from p by simply reversing the orientation p p denotes the comp osition of the two paths obtained by connecting the end of p with the b eginning of p and Ap Ap is the comp osition in SU A generalized gauge transformation is a map g which assigns to each p oint v of an SU element g x in an arbitrary p ossibly discontinuous fashion It acts on A in the exp ected manner at the end p oints of paths Ap g v Ap g v where v and v are resp ectively the b eginning and the end p oint of p If A happ ens to b e a smo oth connections say A we have Ap h A However in general Ap can not b e expressed as a path p ordered exp onential of a smo oth form with values in the Lie algebra of SU Similarly in general a generalized gauge transformation can not b e represented by a smo oth group valued function on A G and AG seem to o large to b e mathematically At rst sight the spaces controllable However they admit three characterizations which enables one to in tro duce dierential and integral calculus on them We will conclude this subsection by summarizing the characterization as suitable limits of the corre sp onding spaces in lattice gauge theory which will b e most useful for the main b o dy of this pap er We b egin with some denitions An edge is an oriented dimensional submanifold of with two b oundary p oints called vertices which is analytic everywhere including the vertices A graph in is a collection of edges such that if two distinct edges meet they do so only at vertices In the physics
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