String theory: Difference between revisions

Revision as of 01:55, 7 October 2024

Revision as of 19:17, 17 November 2024

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{{main|String (physics)}}

[[Image:World lines and world sheet.svg|left|thumb|250pxupright=1.2|Interaction in the quantum world: [[worldline]]s of point-like [[particles]] or a [[worldsheet]] swept up by closed [[string (physics)|strings]] in string theory]]

The application of quantum mechanics to physical objects such as the [[electromagnetic field]], which are extended in space and time, is known as [[quantum field theory]]. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields.<ref name="Zee 2010"/>

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=== Dualities ===

[[File:Dualities in String Theory.svg|right|thumb|alt=A diagram indicating the relationships between M-theory and the five superstring theories.|450pxthumb|upright=2|A diagram of string theory dualities. Blue edges indicate [[S-duality]]. Red edges indicate [[T-duality]].]]

{{main|S-duality|T-duality}}

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=== Unification of superstring theories ===

[[File:Limits of M-theory.svg|right|350pxupright=1.6|thumb|alt=A star-shaped diagram with the various limits of M-theory labeled at its six vertices.|A schematic illustration of the relationship between [[M-theory]], the five [[superstring theory|superstring theories]], and eleven-dimensional [[supergravity]]. The shaded region represents a family of different physical scenarios that are possible in M-theory. In certain limiting cases corresponding to the cusps, it is natural to describe the physics using one of the six theories labeled there.]]

In the 1970s, many physicists became interested in [[supergravity]] theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit on the number of dimensions.<ref>[[#Duff1998|Duff]], p. 64</ref> In 1978, work by [[Werner Nahm]] showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.<ref name=Nahm/> In the same year, [[Eugene Cremmer]], [[Bernard Julia]], and [[Joël Scherk]] of the [[École Normale Supérieure]] showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.<ref name=Cremmer/><ref name="Duff 1998, p. 65">[[#Duff1998|Duff]], p. 65</ref>

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One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensional anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/> It looks like a solid [[cylinder (geometry)|cylinder]] in which any [[cross section (geometry)|cross section]] is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is [[curvature|curved]] in such a way that any point in the interior is actually infinitely far from this boundary surface.<ref name="Maldacena 2005, p. 61"/>

[[File:AdS3.svg|thumb|right|alt=A cylinder formed by stacking copies of the disk illustrated in the previous figure.|350pxthumb|upright=1.6|Three-dimensional [[anti-de Sitter space]] is like a stack of [[Poincaré disk model|hyperbolic disks]], each one representing the state of the universe at a given time. The resulting [[spacetime]] looks like a solid [[cylinder (geometry)|cylinder]].]]

This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.<ref name="Maldacena 2005, p. 60"/>

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{{main|String cosmology}}

[[File:WMAP 2012.png|thumb|right|300pxupright=1.4|A map of the [[cosmic microwave background]] produced by the [[Wilkinson Microwave Anisotropy Probe]]]]

The Big Bang theory is the prevailing [[physical cosmology|cosmological]] model for the universe from the earliest known periods through its subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including galactic [[redshift]]s, the relative abundance of light elements such as [[hydrogen]] and [[helium]], and the existence of a [[cosmic microwave background]], there are several questions that remain unanswered. For example, the standard Big Bang model does not explain why the universe appears to be the same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as [[magnetic monopoles]] are not observed in experiments.<ref>[[#Becker|Becker, Becker and Schwarz]], pp. 530–531</ref>

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{{main|Monstrous moonshine}}

[[File:Labeled Triangle Reflections.svg|left|thumb|200pxupright=1|alt=An equilateral triangle with a line joining each vertex to the midpoint of the opposite side|An equilateral triangle can be rotated through 120°, 240°, or 360°, or reflected in any of the three lines pictured without changing its shape.]]

[[Group theory]] is the branch of mathematics that studies the concept of [[symmetry]]. For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. One can rotate it through 120°, 240°, or 360°, or one can reflect in any of the lines labeled {{math|''S''₀}}, {{math|''S''₁}}, or {{math|''S''₂}} in the picture. Each of these operations is called a ''symmetry'', and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a [[group (mathematics)|group]]. In this particular example, the group is known as the [[dihedral group]] of [[order (group theory)|order]] 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a [[finite group]].<ref name=Dummit/>

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This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called [[monster group]], has over {{math|10⁵³}} elements, more than a thousand times the number of atoms in the Earth.<ref name="Klarreich 2015"/>

[[Image:KleinInvariantJ.jpg|right|thumb|300pxupright=1.4|A graph of the [[j-invariant|{{math|''j''}}-function]] in the complex plane]]

A seemingly unrelated construction is the [[j-invariant|{{math|''j''}}-function]] of [[number theory]]. This object belongs to a special class of functions called [[modular function]]s, whose graphs form a certain kind of repeating pattern.<ref>[[#Gannon|Gannon]], p. 2</ref> Although this function appears in a branch of mathematics that seems very different from the theory of finite groups, the two subjects turn out to be intimately related. In the late 1970s, mathematicians [[John McKay (mathematician)|John McKay]] and [[John G. Thompson|John Thompson]] noticed that certain numbers arising in the analysis of the monster group (namely, the dimensions of its [[irreducible representation]]s) are related to numbers that appear in a formula for the {{math|''j''}}-function (namely, the coefficients of its [[Fourier series]]).<ref>[[#Gannon|Gannon]], p. 4</ref> This relationship was further developed by [[John Horton Conway]] and [[Simon P. Norton|Simon Norton]]<ref name=Conway/> who called it [[monstrous moonshine]] because it seemed so far fetched.<ref>[[#Gannon|Gannon]], p. 5</ref>

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[[File:GabrieleVeneziano.jpg|right|thumb|upright|[[Gabriele Veneziano]]]]

The result was widely advertised by [[Murray Gell-Mann]], leading [[Gabriele Veneziano]] to construct a [[Veneziano scattering amplitude|scattering amplitude]] that had the property of Dolen–Horn–Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight-line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line—the [[gamma function]]— which—which was widely used in Regge theory. By manipulating combinations of gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits and had a suggestive integral representation that could be used for generalization.

Over the next years, hundreds of physicists worked to complete the [[Bootstrap model|bootstrap program]] for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a [[tachyon]]. [[Miguel Ángel Virasoro (physicist)|Miguel Virasoro]] and Joel Shapiro found a different amplitude now understood to be that of closed strings, while [[Ziro Koba]] and [[Holger Bech Nielsen|Holger Nielsen]] generalized Veneziano's integral representation to multiparticle scattering. Veneziano and [[Sergio Fubini]] introduced an operator formalism for computing the scattering amplitudes that was a forerunner of [[world-sheet conformal theory]], while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states. [[Claud Lovelace]] calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is 26. [[Charles Thorn]], [[Peter Goddard (physicist)|Peter Goddard]] and [[Richard Brower]] went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26.

In 1969–701969–1970, [[Yoichiro Nambu]], [[Holger Bech Nielsen]], and [[Leonard Susskind]] recognized that the theory could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the action principle by [[Peter Goddard (physicist)|Peter Goddard]], [[Jeffrey Goldstone]], [[Claudio Rebbi]], and [[Charles Thorn]], giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the [[Virasoro algebra|Virasoro conditions]].

In 1971, [[Pierre Ramond]] added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. [[John Henry Schwarz|John Schwarz]] and [[André Neveu]] added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. [[Stanley Mandelstam]] formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. [[Michio Kaku]] and [[Keiji Kikkawa]] gave a different formulation of the bosonic string, as a [[string field theory]], with infinitely many particle types and with fields taking values not on points, but on loops and curves.

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* {{Cite book| first = Roger | last = Penrose | year = 2005 | title = The Road to Reality: A Complete Guide to the Laws of the Universe | publisher = Knopf | isbn = 978-0-679-45443-4 | title-link = The Road to Reality: A Complete Guide to the Laws of the Universe }}

* {{Cite book| first = Lee | last = Smolin | year = 2006 | title = The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next | publisher = Houghton Mifflin Co. | ___location = New York | isbn = 978-0-618-55105-7 | title-link = The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next }}

* {{Cite book | first = Peter | last = Woit | year = 2006 | title = Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law | publisher = New York: Basic Books | ___location =London, LondonEngland: Jonathan Cape & | isbn = 978-0-465-09275-8 <!-- both are correct -->}}

=== Textbooks ===

* {{cite book |last1=Becker |first1=K. |last2=Becker |first2=M. |last3=Schwarz |first3=J. H. |title=String Theory and M-Theory: A Modern Introduction |publisher=Cambridge University Press |date=2006 |isbn=978-0521860697 }}

* {{cite book |last1=Blumenhagen |first1=R. |last2=Lüst |first2=D. |last3=Theisen |first3=S. |title=Basic Concepts of String Theory |publisher=Springer |date=2012 |isbn=978-3642294969 }}

* {{cite book |last1=Green |first1=Michael |last2=Schwarz |first2=John |last3=Witten |first3=Edward |title=Superstring theory. Vol. 1: Introduction |publisher=Cambridge University Press |date=2012 |isbn=978-1107029118 }}

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; Video :

* [https://www.bbc.co.uk/science/horizon/2001/paralleluni.shtml bbc-horizon: parallel-uni] — 2002—2002 feature documentary by [[Horizon (British TV series)|BBC Horizon]], episode [[Parallel Universes (film)|Parallel Universes]] focus on history and emergence of M-theory, and scientists involved.

* [https://www.pbs.org/wgbh/nova/physics/elegant-universe.html pbs.org-nova: elegant-uni] — 2003—2003 [[Emmy Award]]-winning, three-hour miniseries by [[Nova (American TV program)|Nova]] with [[Brian Greene]], adapted from his [[The Elegant Universe]] (original [[PBS]] broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003).

{{String theory topics |state=collapsed}}