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The Future of Fundamental Physics

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The Future of Fundamental Physics

Nima Arkani-Hamed

Abstract: Fundamental physics began the twentieth century with the twin revolutions of relativity and quantum mechanics, and much of the second half of the century was devoted to the con- struction of a theoretical structure unifying these radical ideas. But this f o undation has also led us to a number of paradoxes in our understanding of nature. Attempts to make sense of quantum mechanics and gravity at the smallest distance scales lead inexorably to the conclusion that space- time is an approximate notion that must emerge f r om more primitive building blocks. Further- more, violent short-distance quantum f l uctuations in the vacuum seem to make the existence of a macroscopic world wildly implausible, and yet we live com f o rtably in a huge universe. What, if anything, tames these f l uctuations? Why is there a macroscopic universe? These are two of the central theoretical challenges of f u ndamental physics in the twenty-½rst century. In this essay, I describe the circle of ideas surrounding these questions, as well as some of the theoretical and experimental f r onts on which they are being attacked.

Ever since Newton realized that the same force of gravity pulling down on an apple is also responsible for keeping the moon orbiting the Earth, fundamental physics has been driven by the program of uni½cation: the realization that seemingly disparate phenomena are in fact different aspects of the same underlying cause. By the mid-1800s, electricity and magnetism were seen as different aspects of electromagnetism, and a seemingly unrelated phenomenon–light–was understood to be the undulation of electric and magnetic ½elds. Relativity and quantum mechanics pushed the

NIMA ARKANI-HAMED, a Fellow

trend toward uni½cation into territory far removed from ordinary human experience. Einstein taught us that space and time are different aspects of a single entity: space-time. Energy and momentum are united analogously, leading to the famous equiva-

of the American Academy since 2009, is a Professor in the School of Natural Sciences at the Institute for Advanced Study. His interests range from quantum ½eld theory and string theory to cosmology and collider physics. He has published

his work in the Journal of High Ener- gy Physics, the Journal of Cosmology and Astroparticle Physics, and Nucle-

ar Physics, among other places.

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lence between mass and energy, E = mc  , as an immediate consequence. Einstein further realized that space-time is not a static stage on which physics unfolds, but a dynamic entity that can curve and bend. Gravity is understood as a manifestation of

© 2012 by the American Academy of Arts & Sciences

53
The Future  space-time curvature. This new picture ways. The nature of the interactions is

of Funda-

of space-time made it possible to conceive almost completely dictated by the rules

mental

of ideas that were impossible to articulate of quantum mechanics, together with the in the Newtonian picture of the world. requirement that the interactions take Consider the most important fact about place at points in space-time, in complicosmology: we live in an expanding uni- ance with the laws of special relativity. The verse. The distance between two galaxies latter requirement is known as the princigrows with time. But the galaxies are not ple of locality.

Physics

  • rushing apart from each other into some
  • One of the startling general predictions

preexisting space, as though blown out of of quantum ½eld theory is the existence of an explosion from some common center. anti-particles such as the positron, which Rather, more and more space is being gen- has the same properties as the electron but erated between the galaxies all the time, the opposite electric charge. This predicso from the vantage point of any one gal- tion has another striking consequence: axy, the others appear to be rushing away. namely, that even the vacuum has strucThis picture, impossible to imagine in ture and dynamics. Newton’s universe, is an inevitable consequence of Einstein’s theory.
Suppose we attempt to check that some small region of space-time is empty. Be-
Quantum mechanics represented a more cause of the uncertainty principle, we need radical departure from classical physics, higher energies to probe short distances. involving a completely new conceptual Eventually there is enough energy to make framework, both physically and mathe- an electron and a positron, without viomatically. We learned that nature is not lating either the conservation of energy deterministic, and only probabilities can or the conservation of charge. Instead of be predicted. One consequence is the fa- seeing nothing, probing the vacuum at mous uncertainty principle, by which we small distances yields particle/anti-particle cannot simultaneously know the position pairs. It is useful to think of the vacuum and velocity of a particle to perfect accu- as ½lled with quantum fluctuations, with racy. Quantum mechanics also allowed “virtual” particles and anti-particles poppreviously irreconcilable phenomena to ping in and out of existence on faster and be understood in a uni½ed way: particles faster timescales at shorter and shorter and waves came to be seen as limiting as- distances.

  • pects of the underlying description where
  • These quantum fluctuations give rise to

there are no waves at all, only quantum- measurable physical effects. For instance, mechanical particles.

he laws of relativity and quantum the cloud of virtual electrons and positrons surrounding an electron is slightly perturbed by the electron’s electric ½eld. Any physical measurement of the electron’s charge, then, will vary just slightly with distance, growing slowly closer in to the electron as more of the virtual cloud is pierced. These virtual effects can be calculated very precisely; in some circumstances, theoretical predictions and experimental observations can be compared to an astonishing level of precision. The virtual corrections to the magnetic proper-

T

mechanics are the pillars of our current understanding of nature. However, describing physics in a way that is compatible with both of these principles turns out to be extremely challenging; indeed, it is possible only with an extremely constrained theoretical structure, known as

quantum ½eld theory. A quantum ½eld the-

ory is characterized by a menu of particles that interact with each other in various

Dædalus, the Journal of the American Academy of Arts & Sciences

54 ties of the electron, for example, have been same quantum-½eld-theoretic language. theoretically computed to twelve decimal Electromagnetism is associated with interplaces, and they agree with experiment to actions between electrons and photons of

Nima Arkani- Hamed

  • that level of precision.
  • a speci½c sort. Strong interactions arise

The second-half of the twentieth cen- from essentially identical interactions betury saw a flurry of activity, on both ex- tween quarks and gluons, while weak interperimental and theoretical fronts. These actions connect particles like the electron developments culminated in the 1970s and the neutrino in the same way, with with the construction of the Standard massive cousins of the photon known as Model of particle physics, a speci½c quan- the W and Z particles.

  • tum ½eld theory that describes all known
  • Differences appear at long distances for

elementary particles and their interactions subtle reasons. The electromagnetic interdown to the smallest distances we have action was the ½rst to be detected and unprobed so far. There are four basic inter- derstood because the photon is massless actions: gravity and electromagnetism, and the interaction is long-ranged. The W which were familiar even to the ancients, and Z particles are massive, thus mediatas well as the weak and strong interactions ing an interaction with a short range of

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that reveal themselves only on nuclear about 10 cm. The difference with quarks scales. Atomic nuclei consist of neutrons and gluons is more subtle still: the virtual and protons. An isolated neutron is un- effects of the cloud of gluons surroundstable, living for about ½fteen minutes be- ing a quark make the “strong charge” of fore disintegrating into a proton, electron, quarks slowly grow stronger at longer dis-

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and an anti-neutrino. (This process is also tances. At a distance of roughly 10 cm, responsible for radioactivity.) Fifteen min- the interaction is so strong as to permautes is enormously long compared to the nently con½ne quarks inside protons and typical timescales of atoms and nuclei, so neutrons.

  • the interaction responsible for triggering
  • But from a fundamental short-distance

this decay must be very feeble–hence, perspective, these are details: the character weak interaction. The earliest incarnation of the laws is essentially identical. This of the strong interaction was noticed in the fact illustrates the central reason why we attraction keeping protons inside nuclei, probe short distances in fundamental counterbalancing their huge electrical re- physics. It is not so much because we care

  • pulsion.
  • about the “building blocks of matter” and

Some familiar particles, such as elec- the associated set of particles we may distrons and photons, remain as elementary cover, but because we have learned that point-like entities in the Standard Model. the essential unity, simplicity, and beauty Others, like the proton, are understood to of the underlying laws manifest most

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be bound states, around 10 cm in diam- clearly at short distances. eter made of quarks, which are permanently trapped inside the proton through umphs of physics in the twentieth century. their interaction with gluons. It gives us a simple and quantitatively ac-
The Standard Model is one of the tri-
Strong, weak, and electromagnetic inter- curate description of everything we know actions seem completely different from about elementary particles and their intereach other at long distances, but we now actions. Only one element of the theory has know that these differences are a long- yet to be de½nitely con½rmed by experidistance illusion. At short scales, these in- ment. In the fundamental short-distance teractions are described in essentially the theory, where all the interactions are treat-

141 (3)   Summer 2012

55 ed on a symmetrical footing, the particles time is necessarily an approximate notion are massless. The mass of particles, such that must emerge from more primitive as electrons or the Wand Zparticles, arises building blocks.

The Future of Funda- mental Physics

  • as a dynamic long-distance effect, known
  • Because of the uncertainty principle, we

as the Higgs mechanism because of the have to use high energies to probe short particles’ interactions with the so-called distances. In a world without gravity, we Higgs ½eld. The typical length scale asso- could resolve arbitrarily small distances in ciated with these interactions is around this way, but gravity eventually and dra-

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10 cm, which is, not coincidentally, also matically changes the picture. At ministhe range of weak interactions. As I discuss cule distances, so much energy has to be at greater length below, it is also fortu- concentrated into such a tiny region of itously the distance scale we are now prob- space that the region itself collapses into ing with the Large Hadron Collider (lhc), a black hole, making it impossible to exthe particle accelerator located at the cern tract any information from the experiment. laboratory just outside Geneva, Switzer- This occurs when we attempt to probe

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land. Collisions at the lhc should put rip- distances around 10 cm, the so-called ples in the Higgs ½eld that manifest as the Planck length.

  • Higgs particle with very de½nite proper-
  • The Planck length is a ridiculously tiny

ties and experimental signatures. Indeed, distance scale–sixteen orders of magnilast December, the lhc experiments re- tude smaller than the tiniest distances we ported preliminary evidence for events are probing today at the lhc. Its tininess consistent with the production of the is a direct reflection of the extreme weakHiggs particle, with its expected prop- ness of gravity compared to other forces erties. Analysis of the 2012 data should of nature. The gravitational attraction beeither yield a de½nitive discovery of the tween a pair of electrons is forty-two orders Higgs particle or de½nitively exclude the of magnitude smaller than their electrical simplest realization of the Higgs mecha- repulsion. Classically, both the gravitation-

  • nism within the Standard Model.
  • al and electric forces vary with distance

The success of the Standard Model gives following an inverse-square law; however,

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us a strong indication that we are headed at a distance of around 10 cm, this gets in the right direction in our understand- corrected in an important way: again being of fundamental physics. Yet profound cause of the uncertainty principle, simply mysteries remain, associated with ques- holding two electrons at shorter distances tions that either lie outside the scope of the requires a huge amount of energy. The Standard Model or are addressed by it, but force of gravity increases with increasing in a seemingly absurd way. Two of these mass, or with equivalently increasing enquestions stand out for both their sim- ergy, so the attraction between electrons plicity and urgency, and will drive the de- begins to increase relative to the electri-

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velopment of fundamental physics in the cal repulsion. At around 10 cm, gravity

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  • twenty-½rst century.
  • surpasses the electric force, and at 10

cm, it dominates all interactions. Thus, the combination of gravity and quantum mechanics makes it impossible to operationally probe Planckian distances. Every time we have encountered ideas in physics that cannot even in principle be observed, we have come to see such

T

he principle of locality–the notion that interactions take place at points in spacetime–is one of the two pillars of quantum ½eld theory. It is therefore unsettling to realize that, due to the effects of both gravity and quantum mechanics, space-

Dædalus, the Journal of the American Academy of Arts & Sciences

56 ideas as approximate notions. However, tum fluctuations, and therefore cannot this instance is particularly disturbing be- record the results of the experiment with cause the notion that emerges as approx- perfect accuracy. Without gravity, noth-

Nima Arkani- Hamed

  • imate is that of space-time itself.
  • ing would stop us from conducting the

The description of the situation seems experiment with an in½nitely big apparato relegate all the mysteries to tiny dis- tus to achieve perfect accuracy, but gravtances, and may suggest some sort of gran- ity obstructs this. As the apparatus gets ular structure to space-time near the bigger, it inevitably also gets heavier. If Planck scale. Much as the smooth surface we are making a local measurement in a of a table is resolved into discrete units ½nite-sized room, at some large but ½nite made of molecules and atoms, one might size it becomes so heavy that it collapses imagine that “atoms of space-time” will the entire room into a black hole.

  • replace space-time near the Planck length.
  • This means that there is no way, not

This naive idea is very likely wrong. Any even in principle, to make perfectly accusort of granular structure to space-time rate local measurements, and thus local obpicks a preferred frame of reference, where servables cannot have a precise meaning. the size of the granularity is “small,” in There is an irreducible error associated sharp conflict with the laws of relativity. with any local measurement that is made But there is a deeper reason to suspect in a ½nite room. While this error is signi½- that something much more interesting and cant close to the Planck scale, it is negligisubtle than “atoms of space-time” is at ble in ordinary circumstances. But this play. The problems with space-time are not does not diminish the importance of this only localized to small distances; in a pre- observation. The fact that quantum mecise sense, “inside” regions of space-time chanics makes it impossible to determine cannot appear in any fundamental descrip- precisely the position and velocity of a

  • tion of physics at all.
  • baseball is also irrelevant to a baseball

The slogan is that due to quantum me- player. However, it is of fundamental imchanics and gravity, there are no “local portance to physics that we cannot speak observables.” Indeed, before worrying precisely of position and momentum, but about what a correct theory combining only position or momentum. Similarly, the quantum mechanics and gravity ought to fact that gravity makes it impossible to look like, it is worth thinking about what have precise local observables has the draperfectly precise measurements can ever matic consequence that the “inside” of be made by experiments. These (in prin- any region of space-time does not have a ciple) exact observables provide a target sharp meaning, and is likely an approxi-

  • for what the theory should predict.
  • mate notion that cannot appear in a deeper

Imagine trying to perform any sort of underlying theory. local measurement, by which I mean an experiment that can be done in a ½nite-
If we cannot speak precisely of local obsized room. To extract a perfectly precise measurement, we need (among other things) to use an in½nitely large apparatus in order to avoid inaccuracies arising from the quantum fluctuations of the apparatus. If the apparatus has a large but ½nite number of components, on a huge but ½nite timescale, it suffers its own quanservables, what observables can we talk about? Instead of performing observations inside some region of space-time, we can push our detectors out to in½nite distances, at the boundary of space-time, where we can make them in½nitely big. We can then throw particles into the interior, where they interact and scatter with

141 (3)   Summer 2012

57 each other in some way and emerge back into the interior of the box and watch them out to in½nity where they are measured. come back out to the walls at some ½nite The results of these scattering experiments time in the future. Because these experican be the perfectly precise observables ments start and end on the walls, it is natthat one might hope to calculate from a ural to wonder whether there is a way of

The Future of Funda- mental Physics

fundamental underlying theory. String theory is our best attempt to make of the box makes no appearance. sense of the mysteries of quantum grav- Amazingly, such a description exists, and describing the physics where the interior ity, and it perfectly exempli½es this basic is given in terms of a completely ordinary ideology. In its earliest incarnation, string quantum ½eld theory living on the walls theory computed the results of scattering of the box, made from particles very much processes and was thought of as a gener- like the quarks and gluons of the strong alization of quantum ½eld theory, with interactions. When the interactions bepoint-like particles replaced by extended tween the “gluons” are made very strong, loops of string. This idea miraculously the physics is completely equivalent to passed several physical and mathematical that of string theory living on the inside of consistency checks and spawned a huge the box. In a speci½c sense, gravity, strings, amount of theoretical activity. The 1990s and an extra direction of space emerge brought a steady stream of surprises re- from the strong interactions of a perfectly vealing that string theory is not in fact a ordinary quantum ½eld theory in one lowtheory of strings, but contains both point- er dimension, much like an apparently like particles as well as higher-dimensional three-dimensional image can be encoded objects as important ingredients. By the late 1990s, these developments led in a two-dimensional hologram. At ½rst sight, this holographic equivato an amazing realization, widely consid- lence seems impossible. If we had a ball ered to be the most important theoretical in the middle of the box, how could its advance in the ½eld in the past two de- position in the interior be encoded only cades. Early work in string theory focused on the walls? The presence of the ball in on understanding scattering processes in the interior is represented as some lump flat space-time, where time marches uni- of energy in the description on the walls; formly from the in½nite past to the in½- as the ball moves around the interior, this nite future and where space is not curved. lump correspondingly shrinks and grows But it is also possible to consider a differ- in size. What about the force of gravity ent kind of geometry on very large scales, between two balls in the interior? The two known as anti-de Sitter space. Here, time corresponding lumps of energy modify still marches uniformly from the in½nite the virtual cloud of gluons surrounding past to the in½nite future, but space is them, which in turn induces a net attraccurved. While the distance from a point tion between the lumps, precisely reproon the interior to the boundary of space ducing the correct gravitational force. In is in½nite, due to the curvature, a light every physical sense, gravity and the extra beam takes a ½nite amount of time to direction of space making up the inside make it to the boundary. Thus, this geom- of the box do indeed emerge “holographetry can be usefully thought of as the ically,” from the dynamics of the theory

  • inside of a box.
  • that lives fundamentally on the walls. This

There is a rich set of observables that correspondence gives us our ½rst concrete we can talk about in this geometry: start- clue as to how space-time may emerge ing on the walls, we can throw particles from more primitive building blocks.

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    UCB-PTH-05/19 LBNL-57959 Supersymmetry Then and Now Bruno Zumino Department of Physics, University of California, and Theoretical Physics Group, 50A-5104, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA ABSTRACT A brief description of some salient aspects of four-dimensional super- symmetry: early history, supermanifolds, the MSSM, cold dark matter, the cosmological constant and the string landscape. arXiv:hep-th/0508127v1 17 Aug 2005 1 A brief history of the beginning of supersymmetry Four dimensional supersymmetry (SUSY) has been discovered independently three times: first in Moscow, by Golfand and Likhtman, then in Kharkov, by Volkov and Akulov, and Volkov and Soroka, and finally by Julius Wess and me, who collaborated at CERN in Geneva and in Karlsruhe. It is re- markable that Volkov and his collaborators didn’t know about the work of Golfand and Likhtman, since all of them were writing papers in Russian in Soviet journals. Julius and I were totally unaware of the earlier work. For information on the life and work of Golfand and Likhtman, I refer to the Yuri Golfand Memorial Volume [1]. For information on Volkov’s life and work, I refer to the Proceedings of the 1997 Volkov Memorial Seminar in Kharkov [2]. Supersymmetry is a symmetry which relates the properties of integral- spin bosons to those of half-integral-spin fermions. The generators of the symmetry form what has come to be called a superalgebra, which is a su- per extension of the Poincar´eLie algebra of quantum field theory (Lorentz transformations and space-time translations) by fermionic generators.
  • Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen

    Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen

    Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen Basic Theoretical Physics AConciseOverview With 31 Figures 123 Prof. Dr. Uwe Krey University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Dr. rer nat habil Anthony Owen University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Library of Congress Control Number: 2007930646 ISBN 978-3-540-36804-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper SPIN 11492665 57/3180/YL - 5 4 3 2 1 0 Preface This textbook on theoretical physics (I-IV) is based on lectures held by one of the authors at the University of Regensburg in Germany.
  • Quantum Gravity: a Primer for Philosophers∗

    Quantum Gravity: a Primer for Philosophers∗

    Quantum Gravity: A Primer for Philosophers∗ Dean Rickles ‘Quantum Gravity’ does not denote any existing theory: the field of quantum gravity is very much a ‘work in progress’. As you will see in this chapter, there are multiple lines of attack each with the same core goal: to find a theory that unifies, in some sense, general relativity (Einstein’s classical field theory of gravitation) and quantum field theory (the theoretical framework through which we understand the behaviour of particles in non-gravitational fields). Quantum field theory and general relativity seem to be like oil and water, they don’t like to mix—it is fair to say that combining them to produce a theory of quantum gravity constitutes the greatest unresolved puzzle in physics. Our goal in this chapter is to give the reader an impression of what the problem of quantum gravity is; why it is an important problem; the ways that have been suggested to resolve it; and what philosophical issues these approaches, and the problem itself, generate. This review is extremely selective, as it has to be to remain a manageable size: generally, rather than going into great detail in some area, we highlight the key features and the options, in the hope that readers may take up the problem for themselves—however, some of the basic formalism will be introduced so that the reader is able to enter the physics and (what little there is of) the philosophy of physics literature prepared.1 I have also supplied references for those cases where I have omitted some important facts.
  • The Goal of Theoretical Physics - 2

    The Goal of Theoretical Physics - 2

    The goal of theoretical physics - 2 Werner Heisenberg, discoverer of the famous Uncertainty Principle. Physics becomes even more uncertain when black holes are involved. (Photo Pfeiffer) Last month we published the first part prediction of singularities presuma­ of Stephen Hawking's inaugural lec­ bly indicates that the Classical The­ ture * as Lucasian Professor of Mathe­ ory will break down. However, there matics in the University of Cambridge. seems to be no reason why it should In the second part of the lecture, break down until the gravitational which follows here. Hawking turned field becomes strong enough so that to the subject of gravity. After Ein­ quantum gravitational effects are stein's monumental work on gravita­ important. Thus a quantum theory of tion earlier this century, efforts to gravity is essential if we are to des­ mite this force with the other cribe the early Universe and to give mechanisms found in Nature have some explanation for the initial con­ been consistently unfruitful. But there ditions beyond merely appealing to are indications that this is beginning to the Anthropic Principle, which can change. be paraphrased as 'things are as they are because we are'. Such a theory is also required if we 'So far most of the effort has been are to answer the question 'Does devoted to unifying the first three time really have a beginning and, categories of physical interactions, possibly, an end as is predicted by the strong and weak nuclear forces Classical General Relativity or are and electromagnetism. The fourth the singularities in the Big Bang and and last, gravity, has been neglected.
  • CURRICULUM VITAE June, 2016 Hu, Bei-Lok Bernard Professor Of

    CURRICULUM VITAE June, 2016 Hu, Bei-Lok Bernard Professor Of

    CURRICULUM VITAE June, 2016 Hu, Bei-Lok Bernard Professor of Physics, University of Maryland, College Park 胡悲樂 Founding Fellow, Joint Quantum Institute, Univ. Maryland and NIST Founding Member, Maryland Center for Fundamental Physics, UMD. I. PERSONAL DATA Date and Place of Birth: October 4, 1947, Chungking, China. Citizenship: U.S.A. Permanent Address: 3153 Physical Sciences Complex Department of Physics, University of Maryland, College Park, Maryland 20742-4111 Telephone: (301) 405-6029 E-mail: [email protected] Fax: MCFP: (301) 314-5649 Physics Dept: (301) 314-9525 UMd Physics webpage: http://umdphysics.umd.edu/people/faculty/153-hu.html Research Groups: - Gravitation Theory (GRT) Group: http://umdphysics.umd.edu/research/theoretical/87gravitationaltheory.html - Quantum Coherence and Information (QCI) Theory Group: http://www.physics.umd.edu/qcoh/index.html II. EDUCATION Date School Location Major Degree 1958-64 Pui Ching Middle School Hong Kong Science High School 1964-67 University of California Berkeley Physics A.B. 1967-69 Princeton University Princeton Physics M.A. 1969-72 Princeton University Princeton Physics Ph.D. III. ACADEMIC EXPERIENCE Date Institution Position June 1972- Princeton University Research Associate Jan. 1973 Princeton, N.J. 08540 Physics Department Jan. 1973- Institute for Advanced Study Member Aug. 1973 Princeton, N.J. 08540 School of Natural Science Sept.1973- Stanford University Research Associate Aug. 1974 Stanford, Calif. 94305 Physics Department Sept.1974- University of Maryland Postdoctoral Fellow Jan. 1975 College Park, Md. 20742 Physics & Astronomy Jan. 1975- University of California Research Mathematician Sept.1976 Berkeley, Calif. 94720 Mathematics Department Oct. 1976- Institute for Space Studies Research Associate May 1977 NASA, New York, N.Y.