Oliver Thewalt

    Oliver Thewalt

    Theoretical Physics | Quantum Biology | Dark Matter Research Cluster

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    Strings and M-Theory by Stephen Hawking

    Strings and M-Theory

     

    by Stephen Hawking

     

    In the 1990's the subject formerly known as `string theory' evolved into something else, which has now

     

    become known as `M-theory.' M-theory is a circle of ideas connecting strings, quantum gravity, unification

     

    of forces, duality, Kaluza-Klein theory, Yang-Mills theory, and supersymmetry. While the fundamental

     

    principles of M-theory are still unclear, our picture of the subject has evolved rapidly in recent years.

     

    M-theory has the distinction of being the only approach to quantum gravity which has succeeded both in

     

    tying itself firmly to our classical understanding of gravity (albeit in 10 or 11 dimensions) and in addressing

     

    non-perturbative quantum issues such as the entropy of black holes. (See Classical and Quantum Gravity

     

    for other approaches to quantum gravity.) To some researchers M-theory is a candidate for a `theory of

     

    everything' which would underlie all of the structures in our universe. W hether or not this is the case, there

     

    is no doubt that M -theory is an active arena for the development of ideas in quantum gravity, cosmology,

     

    and field theory.

     

    M-theory and the physics of p-branes

     

    String theory used to be a theory of, well, strings. In the not so recent past one could hear string theorists

     

    state that the fundamental principle of string theory was that the things we think of as particles (electrons,

     

    photons, gravitons, etc.) are in reality extended objects that look lik e closed vibrating loops of string. All

     

    distinctions between the particles would derive from the association of each particle with a different normal

     

    mode of vibration.

     

    The picture is now quite different. In addition to strings, M-theory contains a zoo of higher dimensional

     

    objects; e.g. 2-dimensional membranes (aka 2-branes), 3-dimensional `3-branes', etc. An object with p

     

    spatial dimensions is known as a p-brane. These branes are now thought to be as fundamental as the

     

    famous `fundamental string.' Indeed, the various branes are related to fundamental strings by powerful

     

    symmetries (known as dualities). Furthermore, under certain conditions the various branes can

     

    dynamically transform into each other as well as into fundamental strings. As a result, the physics of

     

    p-branes has played an increasingly important role in the understanding of M-theory as a whole.

     

    It turns out that p-branes are far more complicated objects than are strings. One therefore uses a variety

     

    of techniques to study them, each of which applies in a different region of parameter space. These

     

    include string perturbation theory, brane effective actions, and supergravity techniques. By splicing

     

    together these pictures, researchers obtain new insights into brane dynamics and the theory in which they

     

    live.

     

    At Syracuse, such studies are pursued mainly using supergravity physics and the related brane effective

     

    actions. The basic idea here is that the branes of M-theory are related to higher-dimensional

     

    generalizations of black holes. A review by Don Marolf provides an introduction for students with a

     

    background in general relativity.

     

    The Maldacena Conjecture (AdS/CFT)

     

    Perhaps the m ost shocking outgrowth of the physics of branes has been the Maldacena conjecture. T his

     

    conjecture states that M-theory subject to particular boundary conditions is in fact equivalent to some

     

    supersymmetric Yang-Mills (i.e., non-gravitational!) theory on a manifold of smaller dimension! One

     

    example is the so-called AdS/CFT correspondence, in which string theory with boundary conditions

     

    matching the ten-dimensional manifold given by the product of 4+1 Anti-DeSitter space and a five-sphere

     

    (AdS5 x S5) is conjectured to be equivalent to 3+1-dimensional super Yang-Mills theory, a

     

    four-dimensional conformal field theory (CFT). This surprising idea follows from certain arguments

     

    involving taking the low energy limit of D-brane physics from both the spacetime (gravitating) point of view

     

    and from the point of view of string perturbation theory. Unfortunately, no version of this conjecture is

     

    currently known which would apply to asym ptotically flat spacetimes (such as Minkowski space)..

     

    Although the conjecture has not yet been proven, an impressive variety of supporting evidence has been

     

    obtained. These range from the classification of linearized perturbations to calculations of black hole

     

    entropy (see below). Another piece of such evidence stems from the studies of gravitating branes

     

    mentioned above. Marolf and Sumati Surya (a past Syracuse student, now at UBC) used supergravity

     

    techniques to uncover certain links between brane physics and black hole no-hair theorems. This work

     

    was then extended by Marolf and Amanda Peet (Toronto) and the Maldacena conjecture was used to

     

    suggest a `dual version' of the effect in the super Yang-Mills quantum field theory description. By showing

     

    that quantitative information governing the no-hair phenomenon was reproduced by the appropriate

     

    quantum field theory calculation, they added a new piece of evidence in support of the Maldacena

     

    conjecture and refined the `dictionary' that translates between the gravitating and non-gravitating sides of

     

    the correspondence.

     

    The correspondence can also be used in the other direction. As an example, Marolf and Peet turned

     

    their arguments around to predict certain gravitational features of branes. Supporting evidence for these

     

    predictions was then found by Marolf, Andres Gomberoff (then a postdoc at Syracuse, now at CECS),

     

    David Kastor (U. Mass) and Jennie Traschen (U. Mass). A more detailed analysis using numerical

     

    techniques is now being pursued in conjunction with Pablo Laguna (Penn State).

     

    However, this phenomenon may yet have more more to teach us. Marolf and Pedro Silva are exploring

     

    this possibility by investigating the relationship between the above no-hair results and non-abelian D-brane

     

    effective actions, which is another story in itself.

     

    Field Theory and Non-Commutative Geom etry

     

    Recently, it has been shown that field theories on so-called non-commutative spaces also play a role in

     

    M-theory and shed light on interesting questions of brane dynamics. A non-commutative geometry is an

     

    algebraic generalization of a manifold (with metric) in which the coordinates do not commute. As an

     

    example, one could roughly refer to a quantum mechanical Hilbert space as a non-commutative phase

     

    space. At Syracuse, the study of non-commutative geometry has been pursued for some time by A. P.

     

    Balachandran and by Kamesh W ali. Be sure to read the corresponding entry under Elementary Particles

     

    and Fields for a description of this work.

     

    Black Holes and Quantum Mechanics in M-theory

     

    Black holes have long been a focal point for studies of quantum gravity. In part, this stems from

     

    dimensional analysis which suggests that the fundamental physics of quantum gravity takes place at the

     

    Plank scale, roughly 10-35 meters. The fact that quantum fluctuations in vacuum energy can create black

     

    holes at this scale suggests that the fundamental structure m ay be a `soup of virtual black holes,'

     

    sometimes known as `spacetime foam.' The other reason for the focus on black holes is the intriguing

     

    phenomenon of Hawking radiation, first uncovered by Stephen Hawking in the early 1970's. Although it is

     

    not possible for any energy to escape from a black hole in classical physics, quantum effects cause black

     

    holes to radiate like black bodies. The corresponding temperature is tiny for everyday black holes, but is

     

    large for tiny Plank scale Schwarzschild black holes. Since black holes have a temperature, they also

     

    have an entropy, which turns out to be enormous but finite and an intense point of discussion. The

     

    tension between the classical notion of causality (which is, after all, what determines that nothing can

     

    escape from a black hole) and Hawking radiation also suggests that quantum gravity effects may cause a

     

    fundamental shift in our understanding of space and time. The study of such issues sometimes goes

     

    under the heading of `the information paradox,' which refers to the issue of whether information that

     

    enters a black hole can in fact leave again through quantum processes.

     

    String (or M-) theory provides a number of tools that can be used to study the quantum physics of black

     

    holes. (Be sure to also read the discussion of black holes and quantum mechanics under Classical and

     

    Quantum Gravity.) One of the most powerful has been the use of D-brane techniques. D-branes are

     

    non-perturbative objects around which string perturbation theory can still describe physics. In this context

     

    they are well known as places where strings can end. Placing enough D-branes together can create a

     

    black hole. As first described by Andrew Strominger and Cumrun Vafa, string techniques then predict

     

    certain properties of this black hole. In particular, such m ethods have been used to successfully calculate

     

    both Hawking radiation from the hole and the entropy of these black holes. These are the only known

     

    techniques through which one can precisely predict the entropy of a black hole by counting m icroscopic

     

    states. Interestingly, such calculations are done in a regime in which no horizon exists -- supersymmetry

     

    is used to extrapolate the res ult to honest black holes. As a res ult, m any fundamental questions rem ain

     

    and are the subject of on-going research. Marolf has participated [1,2,3] in the use of D-brane techniques

     

    to probe black hole entropy and inform ation and continues to address such issues, e.g. recent work with

     

    Jorma Louko (Nottingham ) and Simon Ross (Durham).

     

    A related topic is the idea of `holography,' which suggests that a fundamental description of an n+1

     

    dimensional spacetime may in fact be through an n-dimensional theory (or, more properly, and (n-1)+1

     

    dimensional theory). T his idea was originally suggested by Lenny Susskind, W illy Fischler, Gerard t'Hooft,

     

    and others motivated by the fact that the entropy of black holes scales with their surface area instead of

     

    their volume. Assuming that the Maldacena conjecture is correct, it provides a striking implementation of

     

    this idea.

     

    A particular version of holography is known as the Bousso conjecture. W hile less sweeping (and less

     

    precise) than the Maldacena conjecture, it has the advantage that it can in fact apply to general

     

    spacetimes which need not satisfy special boundary conditions. A rough statement of Bousso's

     

    conjecture is that the entropy flux through any null surface is bounded by the area of this null surface. In a

     

    recent paper, Marolf, Eanna Flanagan (Cornell), and Robert W ald (Chicago) were able to prove that this

     

    bound in fact follows from conventional Einstein gravity in the appropriate semi-classical setting.

     

    String Cosmology

     

    Mark Bowick, Mark Trodden, Joel Rozowsky and Salah Nasri are studying elements of superstring

     

    cosmology. In particular they are interested in the issue of the dimensionality of spacetime.

     

    Nonperturbative effects from geometry

     

    An important feature of M-theory is that, at least in certain regimes, it is properly described as an

     

    eleven-dim ensional theory. This is in contrast to the original string theory which lives in ten dimensions.

     

    These descriptions of the theory are related through the process of Kaluza-Klein reduction, where a higher

     

    dimensional theory can be made to seem like a lower dimensional theory containing extra fields. The ten

     

    dimensional description arises when one of the eleven dimensions is a circle whose size is small enough

     

    to be ignored.

     

    The original formulation of string theory in terms of the scattering of quantum strings m akes use of a small

     

    parameter known as the string coupling, g. This description is inherently tied to a perturbative expansion

     

    in powers of g. Now, the string coupling turns out to be related to the size of the tiny circle that constitutes

     

    the eleventh dimension. Small g arises for small circles while large g arises for large circles.

     

    For large g, one may consider situations in which quantum effects are small so that one can use classical

     

    eleven-dimensional gravity to accurately describe the physics. While the description in terms of string

     

    scattering is inherently perturbative, eleven-dimensional gravity is not. Thus, one can use properties of

     

    eleven-dimensional gravity to obtain non-perturbative information about M-theory. In some cases, one

     

    can use supersymmetry to argue that classically derived conclusions also remain valid when quantum

     

    mechanics is tak en into account.

     

    An excellent example of this kind of result is the Kaluza-Klein monopole, discovered by Rafael Sorkin long

     

    before the days of M-theory. This is a stable solution to the 4+1-dimensional Einstein equations whose

     

    3+1-dimensional description is as a magnetic monopole in gravity coupled to an electromagnetic field (and

     

    a scalar field). While magnetic monopoles are singular, in this case the singularity is merely an artifact of

     

    the 3+1-dimensional description. The 4+1 description is a perfectly smooth spacetime. Thus, the higher

     

    dimensional geometry implies that such a theory does in fact contain magnetic monopoles.

     

    Kaluza-Klein monopoles (generalized to 9+1 and 10+1 dimensions) continue to be of im portance in

     

    M-theory, and in fact they have the sam e status as the p-branes described above. The monopoles are

     

    related to various branes by the duality symmetries of M-theory, and in fact one D-brane can described as

     

    a Kaluza-Klein monopole in eleven dimensions. An example of how these monopoles can be used to

     

    derive non-perturbative effects in string theory can be found in a recent paper by M arolf which uses their

     

    eleven-dim ensional geometry to resolve certain issues involving charge quantization. The monopole

     

    geometry makes a single brane (known as a M2-brane) in eleven dimensions appear as a pair of

     

    D-branes in ten dimensions. Not surprisingly, these two branes must always remain attached to each

     

    other. This leads to a phenomenon in which certain external fields cause D-branes to be confined in

     

    pairs. Further studies of Kaluza-Klein monopoles and other aspects of eleven-dimensional geom etry are

     

    certain to uncover additional effects that are invisible to string perturbation theory