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154-404: The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum gravity , particle and condensed matter physics , cosmology , and pure mathematics . String theory represents an outgrowth of S-matrix theory ,

308-620: A 1 1 − ( 1 − z a ) = 1 z = 1 ( z + a ) − a {\displaystyle {\begin{aligned}f(z)&=\sum _{k=0}^{\infty }a_{k}(z-a)^{k}\\&=\sum _{k=0}^{\infty }(-1)^{k}a^{-k-1}(z-a)^{k}\\&={\frac {1}{a}}\sum _{k=0}^{\infty }\left(1-{\frac {z}{a}}\right)^{k}\\&={\frac {1}{a}}{\frac {1}{1-\left(1-{\frac {z}{a}}\right)}}\\&={\frac {1}{z}}\\&={\frac {1}{(z+a)-a}}\end{aligned}}} which has radius of convergence |

462-591: A ) k + 1 = 1 2 π i ∑ n = 0 ∞ ( − 1 ) n ∫ ∂ D ( ζ − 1 ) n d ζ ( ζ − a ) k + 1 = 1 2 π i ∑ n = 0 ∞ ( − 1 ) n ∫ 0 2 π (

616-1465: A ) m = ( − 1 ) k a − k − 1 . {\displaystyle {\begin{aligned}a_{k}&={\frac {f^{(k)}(a)}{k!}}\\&={\frac {1}{2\pi i}}\int _{\partial D}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}\\&={\frac {1}{2\pi i}}\int _{\partial D}{\frac {\sum _{n=0}^{\infty }(-1)^{n}(\zeta -1)^{n}d\zeta }{(\zeta -a)^{k+1}}}\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }(-1)^{n}\int _{\partial D}{\frac {(\zeta -1)^{n}d\zeta }{(\zeta -a)^{k+1}}}\\&={\frac {1}{2\pi i}}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {(a+re^{i\theta }-1)^{n}rie^{i\theta }d\theta }{(re^{i\theta })^{k+1}}}\\&={\frac {1}{2\pi }}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {(a-1+re^{i\theta })^{n}d\theta }{(re^{i\theta })^{k}}}\\&={\frac {1}{2\pi }}\sum _{n=0}^{\infty }(-1)^{n}\int _{0}^{2\pi }{\frac {\sum _{m=0}^{n}{\binom {n}{m}}(a-1)^{n-m}(re^{i\theta })^{m}d\theta }{(re^{i\theta })^{k}}}\\&={\frac {1}{2\pi }}\sum _{n=0}^{\infty }(-1)^{n}\sum _{m=0}^{n}{\binom {n}{m}}(a-1)^{n-m}r^{m-k}\int _{0}^{2\pi }e^{i(m-k)\theta }d\theta \\&={\frac {1}{2\pi }}\sum _{n=k}^{\infty }(-1)^{n}{\binom {n}{k}}(a-1)^{n-k}\int _{0}^{2\pi }d\theta \\&=\sum _{n=k}^{\infty }(-1)^{n}{\binom {n}{k}}(a-1)^{n-k}\\&=(-1)^{k}\sum _{m=0}^{\infty }{\binom {m+k}{k}}(1-a)^{m}\\&=(-1)^{k}a^{-k-1}\end{aligned}}.} The last summation results from

770-622: A k = f ( k ) ( a ) k ! = 1 2 π i ∫ ∂ D f ( ζ ) d ζ ( ζ − a ) k + 1 = 1 2 π i ∫ ∂ D ∑ n = 0 ∞ ( − 1 ) n ( ζ − 1 ) n d ζ ( ζ −

924-451: A k ( z − a ) k = ∑ k = 0 ∞ ( − 1 ) k a − k − 1 ( z − a ) k = 1 a ∑ k = 0 ∞ ( 1 − z a ) k = 1

1078-503: A k ( z − a ) k . {\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}(z-a)^{k}.} We'll calculate the a k {\displaystyle a_{k}} 's and determine whether this new power series converges in an open set V {\displaystyle V} which is not contained in U {\displaystyle U} . If so, we will have analytically continued f {\displaystyle f} to

1232-532: A | {\displaystyle |a|} around 0 {\displaystyle 0} . If we choose a ∈ U {\displaystyle a\in U} with | a | > 1 {\displaystyle |a|>1} , then V {\displaystyle V} is not a subset of U {\displaystyle U} and is actually larger in area than U {\displaystyle U} . The plot shows

1386-527: A − 1 ) n − k ∫ 0 2 π d θ = ∑ n = k ∞ ( − 1 ) n ( n k ) ( a − 1 ) n − k = ( − 1 ) k ∑ m = 0 ∞ ( m + k k ) ( 1 −

1540-414: A − 1 ) n − m r m − k ∫ 0 2 π e i ( m − k ) θ d θ = 1 2 π ∑ n = k ∞ ( − 1 ) n ( n k ) (

1694-439: A − 1 ) n − m ( r e i θ ) m d θ ( r e i θ ) k = 1 2 π ∑ n = 0 ∞ ( − 1 ) n ∑ m = 0 n ( n m ) (

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1848-457: A − 1 + r e i θ ) n d θ ( r e i θ ) k = 1 2 π ∑ n = 0 ∞ ( − 1 ) n ∫ 0 2 π ∑ m = 0 n ( n m ) (

2002-423: A + r e i θ − 1 ) n r i e i θ d θ ( r e i θ ) k + 1 = 1 2 π ∑ n = 0 ∞ ( − 1 ) n ∫ 0 2 π (

2156-457: A germ . The general theory of analytic continuation and its generalizations is known as sheaf theory . Let be a power series converging in the disk D r ( z 0 ), r > 0, defined by Note that without loss of generality, here and below, we will always assume that a maximal such r was chosen, even if that r is ∞. Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that

2310-401: A brane is a physical object that generalizes the notion of a point particle to higher dimensions. For instance, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p , these are called p -branes. The word brane comes from the word "membrane" which refers to

2464-453: A quantum field theory . One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes , which has complicated efforts to develop theories of particle physics based on string theory. These issues have led some in the community to criticize these approaches to physics, and to question

2618-489: A topology on G {\displaystyle {\mathcal {G}}} . Let r > 0, and let The sets U r ( g ), for all r > 0 and g ∈ G {\displaystyle g\in {\mathcal {G}}} define a basis of open sets for the topology on G {\displaystyle {\mathcal {G}}} . A connected component of G {\displaystyle {\mathcal {G}}} (i.e., an equivalence class)

2772-613: A D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition . The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory. Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain categories , such as

2926-448: A collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. Type I string theory turns out to be equivalent by S-duality to the SO (32) heterotic string theory. Similarly, type IIB string theory is related to itself in

3080-529: A connection called supersymmetry between bosons and the class of particles called fermions . Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in eleven dimensions known as M-theory . In late 1997, theorists discovered an important relationship called the anti-de Sitter/conformal field theory correspondence (AdS/CFT correspondence), which relates string theory to another type of physical theory called

3234-488: A fact which follows from the expression of P ( s ) {\displaystyle P(s)} by the logarithms of the Riemann zeta function as Since ζ ( s ) {\displaystyle \zeta (s)} has a simple, non-removable pole at s := 1 {\displaystyle s:=1} , it can then be seen that P ( s ) {\displaystyle P(s)} has

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3388-963: A four-dimensional (4D) spacetime . In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime. In spite of the fact that the Universe is well described by 4D spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily. There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics. Finally, there exist scenarios in which there could actually be more than 4D of spacetime which have nonetheless managed to escape detection. String theories require extra dimensions of spacetime for their mathematical consistency. In bosonic string theory, spacetime

3542-416: A given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle consistent with non-string models of elementary particles, with its mass , charge , and other properties determined by the vibrational state of

3696-749: A link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories. In general relativity, a black hole is defined as a region of spacetime in which the gravitational field is so strong that no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to arise when massive stars undergo gravitational collapse , and many galaxies are thought to contain supermassive black holes at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for theorists attempting to understand

3850-454: A mathematical transformation between bosons and fermions. String theories that include fermionic vibrations are now known as superstring theories . In 1977, the GSO projection (named after Ferdinando Gliozzi , Joël Scherk, and David I. Olive ) led to a family of tachyon-free unitary free string theories, the first consistent superstring theories (see below ). The first superstring revolution

4004-410: A mystery why there was not just one consistent formulation. However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways. They found that a system of strongly interacting strings can, in some cases, be viewed as a system of weakly interacting strings. This phenomenon is known as S-duality. It was studied by Ashoke Sen in

4158-410: A natural boundary for the function L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} for any fixed choice of c ∈ Z c > 1. {\displaystyle c\in \mathbb {Z} \quad c>1.} Hence, there is no analytic continuation for these functions beyond the interior of the unit circle. The monodromy theorem gives

4312-457: A natural explanation for the weakness of gravity compared to the other fundamental forces. A notable fact about string theory is that the different versions of the theory all turn out to be related in highly nontrivial ways. One of the relationships that can exist between different string theories is called S-duality . This is a relationship that says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as

4466-422: A nontrivial way by S-duality. Another relationship between different string theories is T-duality . Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius R is equivalent to a string propagating around a circle of radius 1/ R in the sense that all observable quantities in one description are identified with quantities in

4620-440: A notable step forward in the bootstrap approach was the principle of DHS duality introduced by Richard Dolen , David Horn , and Christoph Schmid in 1967, at Caltech (the original term for it was "average duality" or "finite energy sum rule (FESR) duality"). The three researchers noticed that Regge pole exchange (at high energy) and resonance (at low energy) descriptions offer multiple representations/approximations of one and

4774-410: A number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as M-theory . Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand

History of string theory - Misplaced Pages Continue

4928-455: A particular analytic function f {\displaystyle f} . In this case, it is given by a power series centered at z = 1 {\displaystyle z=1} : f ( z ) = ∑ k = 0 ∞ ( − 1 ) k ( z − 1 ) k . {\displaystyle f(z)=\sum _{k=0}^{\infty }(-1)^{k}(z-1)^{k}.} By

5082-404: A particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle. In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. Duff and his collaborators showed that this construction reproduces exactly

5236-458: A physical interpretation of the Veneziano amplitude by representing nuclear forces as vibrating, one-dimensional strings. However, this string-based description of the strong force made many predictions that directly contradicted experimental findings. In 1971, Pierre Ramond and, independently, John H. Schwarz and André Neveu attempted to implement fermions into the dual model. This led to

5390-422: A power series has radius of convergence r and defines an analytic function f inside that disc. Consider points on the circle of convergence. A point for which there is a neighbourhood on which f has an analytic extension is regular , otherwise singular . The circle is a natural boundary if all its points are singular. More generally, we may apply the definition to any open connected domain on which f

5544-404: A precise definition of entropy as the natural logarithm of the number of different states of the molecules (also called microstates ) that give rise to the same macroscopic features. In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the 1970s, the physicist Jacob Bekenstein suggested that

5698-496: A relationship between type IIB string theory and N = 4 supersymmetric Yang–Mills theory , a gauge theory . This conjecture, called the AdS/CFT correspondence , has generated a great deal of interest in high energy physics . It is a realization of the holographic principle , which has far-reaching implications: the AdS/CFT correspondence has helped elucidate the mysteries of black holes suggested by Stephen Hawking 's work and

5852-571: A research program begun by Werner Heisenberg in 1943 following John Archibald Wheeler 's 1937 introduction of the S-matrix. Many prominent theorists picked up and advocated S-matrix theory, starting in the late 1950s and throughout the 1960s. The field became marginalized and discarded in the mid-1970s and disappeared in the 1980s. Physicists neglected it because some of its mathematical methods were alien, and because quantum chromodynamics supplanted it as an experimentally better-qualified approach to

6006-634: A simple pole at s := 1 k , ∀ k ∈ Z + {\displaystyle s:={\tfrac {1}{k}},\forall k\in \mathbb {Z} ^{+}} . Since the set of points has accumulation point 0 (the limit of the sequence as k ↦ ∞ {\displaystyle k\mapsto \infty } ), we can see that zero forms a natural boundary for P ( s ) {\displaystyle P(s)} . This implies that P ( s ) {\displaystyle P(s)} has no analytic continuation for s left of (or at) zero, i.e., there

6160-499: A small group of physicists were examining the possible applications of higher dimensional objects. In 1987, Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes. Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, Michael Duff , Paul Howe, Takeo Inami, and Kellogg Stelle considered

6314-429: A small sphere, not like a point. Heisenberg proposed that the strongly interacting particles were in fact extended objects, and because there are difficulties of principle with extended relativistic particles, he proposed that the notion of a space-time point broke down at nuclear scales. Without space and time, it becomes difficult to formulate a physical theory. Heisenberg proposed a solution to this problem: focusing on

History of string theory - Misplaced Pages Continue

6468-596: A standard argument for the case where c := 2. {\displaystyle c:=2.} Namely, for integers n ≥ 1 {\displaystyle n\geq 1} , let where D {\displaystyle \mathbb {D} } denotes the open unit disk in the complex plane and | R c , n | = c n {\displaystyle |{\mathcal {R}}_{c,n}|=c^{n}} , i.e., there are c n {\displaystyle c^{n}} distinct complex numbers z that lie on or inside

6622-467: A sufficient condition for the existence of a direct analytic continuation (i.e., an extension of an analytic function to an analytic function on a bigger set). Suppose D ⊂ C {\displaystyle D\subset \mathbb {C} } is an open set and f an analytic function on D . If G is a simply connected domain containing D , such that f has an analytic continuation along every path in G , starting from some fixed point

6776-460: A theoretical idea called supersymmetry . In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa. There are several versions of superstring theory: type I , type IIA , type IIB , and two flavors of heterotic string theory ( SO (32) and E 8 × E 8 ). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries . For example,

6930-598: A two-dimensional brane. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A p -brane sweeps out a ( p +1)-dimensional volume in spacetime called its worldvolume . Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane. In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on

7084-415: A unified description of gravity and particle physics, it is a candidate for a theory of everything , a self-contained mathematical model that describes all fundamental forces and forms of matter . Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details. String theory was first studied in

7238-425: A version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S . In that sense, S is the "one true inverse" of the exponential map. In older literature, sheaves of analytic functions were called multi-valued functions . See sheaf for the general concept. Suppose that

7392-458: Is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments. Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of

7546-506: Is a Riemann surface . G {\displaystyle {\mathcal {G}}} is sometimes called the universal analytic function . is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to it. This is the sheaf of the logarithm function. The uniqueness theorem for analytic functions also extends to sheaves of analytic functions: if

7700-457: Is a broad and varied subject that attempts to address a number of deep questions of fundamental physics . String theory has contributed a number of advances to mathematical physics , which have been applied to a variety of problems in black hole physics, early universe cosmology , nuclear physics , and condensed matter physics , and it has stimulated a number of major developments in pure mathematics . Because string theory potentially provides

7854-466: Is a functional equation for L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} for any z satisfying | z | < 1 {\displaystyle |z|<1} given by L c ( z ) = z c + L c ( z c ) {\displaystyle {\mathcal {L}}_{c}(z)=z^{c}+{\mathcal {L}}_{c}(z^{c})} . It

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8008-514: Is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra . In a paper from 1998, Alain Connes , Michael R. Douglas , and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory , a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry. This established

8162-455: Is a larger open subset of C {\displaystyle \mathbb {C} } , containing U , and F is an analytic function defined on V such that then F is called an analytic continuation of f . In other words, the restriction of F to U is the function f we started with. Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F 1 and F 2 such that U

8316-399: Is a period of important discoveries that began in 1984. It was realized that string theory was capable of describing all elementary particles as well as the interactions between them. Hundreds of physicists started to work on string theory as the most promising idea to unify physical theories. The revolution was started by a discovery of anomaly cancellation in type I string theory via

8470-423: Is also not difficult to see that for any integer m ≥ 1 {\displaystyle m\geq 1} , we have another functional equation for L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} given by For any positive natural numbers c , the lacunary series function diverges at z = 1 {\displaystyle z=1} . We consider

8624-465: Is an example of a duality that relates string theory to a quantum field theory. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. In string theory and other related theories,

8778-608: Is analogous to the summatory form of the Riemann zeta function when ℜ ( s ) > 1 {\displaystyle \Re (s)>1} in so much as it is the same summatory function as ζ ( s ) {\displaystyle \zeta (s)} , except with indices restricted only to the prime numbers instead of taking the sum over all positive natural numbers . The prime zeta function has an analytic continuation to all complex s such that 0 < ℜ ( s ) < 1 {\displaystyle 0<\Re (s)<1} ,

8932-454: Is analytic, and classify the points of the boundary of the domain as regular or singular: the domain boundary is then a natural boundary if all points are singular, in which case the domain is a domain of holomorphy . For ℜ ( s ) > 1 {\displaystyle \Re (s)>1} we define the so-called prime zeta function , P ( s ) {\displaystyle P(s)} , to be This function

9086-455: Is assumed to be on the order of the Planck length , or 10 meters, the scale at which the effects of quantum gravity are believed to become significant. On much larger length scales, such as the scales visible in physics laboratories, such objects would be indistinguishable from zero-dimensional point particles, and the vibrational state of the string would determine the type of particle. One of

9240-515: Is believed to provide a resolution of the black hole information paradox . In 2003, Michael R. Douglas 's discovery of the string theory landscape , which suggests that string theory has a large number of inequivalent false vacua , led to much discussion of what string theory might eventually be expected to predict, and how cosmology can be incorporated into the theory. A possible mechanism of string theory vacuum stabilization (the KKLT mechanism )

9394-510: Is called a sheaf . We also note that the map defined by ϕ g ( h ) = h 0 : U r ( g ) → C , {\displaystyle \phi _{g}(h)=h_{0}:U_{r}(g)\to \mathbb {C} ,} where r is the radius of convergence of g , is a chart . The set of such charts forms an atlas for G {\displaystyle {\mathcal {G}}} , hence G {\displaystyle {\mathcal {G}}}

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9548-399: Is contained in V and for all z in U then on all of V . This is because F 1  −  F 2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions . A common way to define functions in complex analysis proceeds by first specifying

9702-561: Is equal to a given mass and charge for the resulting black hole. Their calculation reproduced the Bekenstein–Hawking formula exactly, including the factor of 1/4 . Subsequent work by Strominger, Vafa, and others refined the original calculations and gave the precise values of the "quantum corrections" needed to describe very small black holes. The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference

9856-488: Is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality. In general, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, Montonen–Olive duality is an example of an S-duality relationship between quantum field theories. The AdS/CFT correspondence

10010-408: Is formulated within the framework of classical physics , whereas the other fundamental forces are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity. In addition to

10164-820: Is no continuation possible for P ( s ) {\displaystyle P(s)} when 0 ≥ ℜ ( s ) {\displaystyle 0\geq \Re (s)} . As a remark, this fact can be problematic if we are performing a complex contour integral over an interval whose real parts are symmetric about zero, say I F ⊆ C   such that   ℜ ( s ) ∈ ( − C , C ) , ∀ s ∈ I F {\displaystyle I_{F}\subseteq \mathbb {C} \ {\text{such that}}\ \Re (s)\in (-C,C),\forall s\in I_{F}} for some C > 0 {\displaystyle C>0} , where

10318-415: Is not enough to determine the scattering, and the proposal was ignored for many years. Heisenberg's proposal was revived in 1956 when Murray Gell-Mann recognized that dispersion relations —like those discovered by Hendrik Kramers and Ralph Kronig in the 1920s (see Kramers–Kronig relations )—allow the formulation of a notion of causality, a notion that events in the future would not influence events in

10472-420: Is plotted against mass squared on a so-called Chew–Frautschi plot ), which implied that the scattering of these particles would have very strange behavior—it should fall off exponentially quickly at large angles. With this realization, theorists hoped to construct a theory of composite particles on Regge trajectories, whose scattering amplitudes had the asymptotic form demanded by Regge theory. In 1967,

10626-461: Is the speed of light , k is the Boltzmann constant , ħ is the reduced Planck constant , G is Newton's constant , and A is the surface area of the event horizon. Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features. The Bekenstein–Hawking entropy formula gives the expected value of

10780-405: Is to find a solution of the theory that reproduces the observed spectrum of elementary particles, with a small cosmological constant , containing dark matter and a plausible mechanism for cosmic inflation . While there has been progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of details. One of

10934-402: The c n {\displaystyle c^{n}} -th roots of unity. Hence, since the set formed by all such roots is dense on the boundary of the unit circle, there is no analytic continuation of L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} to complex z whose modulus exceeds one. The proof of this fact is generalized from

11088-518: The Cauchy–Hadamard theorem , its radius of convergence is 1. That is, f {\displaystyle f} is defined and analytic on the open set U = { | z − 1 | < 1 } {\displaystyle U=\{|z-1|<1\}} which has boundary ∂ U = { | z − 1 | = 1 } {\displaystyle \partial U=\{|z-1|=1\}} . Indeed,

11242-485: The Green–Schwarz mechanism (named after Michael Green and John H. Schwarz) in 1984. The ground-breaking discovery of the heterotic string was made by David Gross , Jeffrey Harvey , Emil Martinec , and Ryan Rohm in 1985. It was also realized by Philip Candelas , Gary Horowitz , Andrew Strominger , and Edward Witten in 1985 that to obtain N = 1 {\displaystyle N=1} supersymmetry ,

11396-654: The derived category of coherent sheaves on a complex algebraic variety , or the Fukaya category of a symplectic manifold . The connection between the physical notion of a brane and the mathematical notion of a category has led to important mathematical insights in the fields of algebraic and symplectic geometry and representation theory . Prior to 1995, theorists believed that there were five consistent versions of superstring theory (type I, type IIA, type IIB, and two versions of heterotic string theory). This understanding changed in 1995 when Edward Witten suggested that

11550-408: The first superstring revolution in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects. Another feature of string theory that many physicists were drawn to in the 1980s and 1990s

11704-492: The k th derivation of the geometric series , which gives the formula 1 ( 1 − x ) k + 1 = ∑ m = 0 ∞ ( m + k k ) x m . {\displaystyle {\frac {1}{(1-x)^{k+1}}}=\sum _{m=0}^{\infty }{\binom {m+k}{k}}x^{m}.} Then, f ( z ) = ∑ k = 0 ∞

11858-440: The strong interactions . The theory presented a radical rethinking of the foundations of physical laws. By the 1940s it had become clear that the proton and the neutron were not pointlike particles like the electron. Their magnetic moment differed greatly from that of a pointlike spin-½ charged particle, too much to attribute the difference to a small perturbation . Their interactions were so strong that they scattered like

12012-460: The Austrian physicist Ludwig Boltzmann , who showed that the thermodynamic properties of a gas could be derived from the combined properties of its many constituent molecules . Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give

12166-503: The BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting. The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry . This subject

12320-524: The November 1986 issue (vol. 7, #11) featured a cover story written by Gary Taubes , "Everything's Now Tied to Strings", which explained string theory for a popular audience. In 1987, Eric Bergshoeff  [ de ] , Ergin Sezgin  [ de ] and Paul Townsend showed that there are no superstrings in eleven dimensions (the largest number of dimensions consistent with a single graviton in supergravity theories), but supermembranes . In

12474-483: The S-matrix directly, without any assumptions about space-time structure. But when transitions from the far-past to the far-future occur in one step with no intermediate steps, it becomes difficult to calculate anything. In quantum field theory , the intermediate steps are the fluctuations of fields or equivalently the fluctuations of virtual particles. In this proposed S-matrix theory, there are no local quantities at all. Heisenberg proposed to use unitarity to determine

12628-584: The S-matrix had to fulfil. Prominent advocates of the new "dispersion relations" approach included Stanley Mandelstam and Geoffrey Chew , both at UC Berkeley at the time. Mandelstam discovered the double dispersion relations , a new and powerful analytic form, in 1958, and believed that it would provide the key to progress in the intractable strong interactions. By the late 1950s, many strongly interacting particles of ever higher spins had been discovered, and it became clear that they were not all fundamental. While Japanese physicist Shoichi Sakata proposed that

12782-426: The S-matrix. In all conceivable situations, the sum of the squares of the amplitudes must equal 1. This property can determine the amplitude in a quantum field theory order by order in a perturbation series once the basic interactions are given, and in many quantum field theories the amplitudes grow too fast at high energies to make a unitary S-matrix. But without extra assumptions on the high-energy behavior, unitarity

12936-480: The analytic continuation of an analytic function . The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces . Analytic continuation is used in Riemannian manifolds , in the context of solutions of Einstein's equations . For example, Schwarzschild coordinates can be analytically continued into Kruskal–Szekeres coordinates . Begin with

13090-493: The basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. In quantum field theory, one typically computes the probabilities of various physical events using the techniques of perturbation theory . Developed by Richard Feynman and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagrams to organize computations. One imagines that these diagrams depict

13244-426: The boundary of the unit circle, there are an infinite number of points z within this arc such that L c ( z ) = ∞ {\displaystyle {\mathcal {L}}_{c}(z)=\infty } . This condition is equivalent to saying that the circle C 1 := { z : | z | = 1 } {\displaystyle C_{1}:=\{z:|z|=1\}} forms

13398-408: The challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. The scattering of strings is most straightforwardly defined using the techniques of perturbation theory , but it is not known in general how to define string theory nonperturbatively . It is also not clear whether there is any principle by which string theory selects its vacuum state ,

13552-409: The compact extra dimensions must be shaped like a Calabi–Yau manifold . A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians Eugenio Calabi and Shing-Tung Yau . Another approach to reducing the number of dimensions is the so-called brane-world scenario. In this approach, physicists assume that

13706-432: The concept of "spinning strings", and pointed the way to a method for removing the problematic tachyon (see RNS formalism ). Dual resonance models for strong interactions were a relatively popular subject of study between 1968 and 1973. The scientific community lost interest in string theory as a theory of strong interactions in 1973 when quantum chromodynamics became the main focus of theoretical research (mainly due to

13860-419: The context of heterotic strings in four dimensions and by Chris Hull and Paul Townsend in the context of the type IIB theory. Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent. At around the same time, as many physicists were studying the properties of strings,

14014-468: The development of bosonic string theory . String theory is formulated in terms of the Polyakov action , which describes how strings move through space and time. Like springs, the strings tend to contract to minimize their potential energy, but conservation of energy prevents them from disappearing, and instead they oscillate. By applying the ideas of quantum mechanics to strings it is possible to deduce

14168-499: The different vibrational modes of strings, and that each vibrational state appears to be a different particle. The mass of each particle, and the fashion with which it can interact, are determined by the way the string vibrates—in essence, by the " note " the string "sounds." The scale of notes, each corresponding to a different kind of particle, is termed the " spectrum " of the theory. Early models included both open strings, which have two distinct endpoints, and closed strings, where

14322-421: The disk of radius r {\displaystyle r} around a {\displaystyle a} ; and let ∂ D {\displaystyle \partial D} be its boundary. Then D ∪ ∂ D ⊂ U {\displaystyle D\cup \partial D\subset U} . Using Cauchy's differentiation formula to calculate the new coefficients, one has

14476-425: The dual description. For example, a string has momentum as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the winding number . If a string has momentum p and winding number n in one description, it will have momentum n and winding number p in the dual description. For example, type IIA string theory

14630-663: The early 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of an 11-dimensional theory that became known as M-theory (for details, see Introduction to M-theory ). These discoveries sparked the second superstring revolution that took place approximately between 1994 and 1995. The different versions of superstring theory were unified, as long hoped, by new equivalences. These are known as S-duality , T-duality , U-duality , mirror symmetry , and conifold transitions. The different theories of strings were also related to M-theory. In 1995, Joseph Polchinski discovered that

14784-461: The eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes." In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the M should stand for "magic", "mystery", or "membrane" according to taste, and

14938-423: The endpoints are joined to make a complete loop. The two types of string behave in slightly different ways, yielding two spectra. Not all modern string theories use both types; some incorporate only the closed variety. The earliest string model has several problems: it has a critical dimension D = 26, a feature that was originally discovered by Claud Lovelace in 1971; the theory has a fundamental instability,

15092-774: The entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity. Indeed, in 1998, Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry. In collaboration with several other authors in 2010, he showed that some results on black hole entropy could be extended to non-extremal astrophysical black holes. Analytic continuation The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with

15246-402: The entropy of a black hole is instead proportional to the surface area of its event horizon , the boundary beyond which matter and radiation are lost to its gravitational attraction. When combined with ideas of the physicist Stephen Hawking , Bekenstein's work yielded a precise formula for the entropy of a black hole. The Bekenstein–Hawking formula expresses the entropy S as where c

15400-479: The entropy of a black hole, but by the 1990s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory. In a paper from 1996, Andrew Strominger and Cumrun Vafa showed how to derive the Bekenstein–Hawking formula for certain black holes in string theory. Their calculation

15554-449: The extra dimensions are assumed to "close up" on themselves to form circles. In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches

15708-591: The five theories were just special limiting cases of an eleven-dimensional theory called M-theory. Witten's conjecture was based on the work of a number of other physicists, including Ashoke Sen , Chris Hull , Paul Townsend , and Michael Duff . His announcement led to a flurry of research activity now known as the second superstring revolution . 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

15862-472: The framework of quantum mechanics. One important example of a matrix model is the BFSS matrix model proposed by Tom Banks , Willy Fischler , Stephen Shenker , and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that

16016-406: The function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann zeta function and the gamma function . The concept of a universal cover was first developed to define a natural domain for

16170-424: The hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions. Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics,

16324-550: The idea that they were observed particles. At the time, Chew's approach was considered more mainstream because it did not introduce fractional charge values and because it focused on experimentally measurable S-matrix elements, not on hypothetical pointlike constituents. In 1959, Tullio Regge , a young theorist in Italy, discovered that bound states in quantum mechanics can be organized into families known as Regge trajectories , each family having distinctive angular momenta . This idea

16478-458: The integrand is a function with denominator that depends on P ( s ) {\displaystyle P(s)} in an essential way. For integers c ≥ 2 {\displaystyle c\geq 2} , we define the lacunary series of order c by the power series expansion Clearly, since c n + 1 = c ⋅ c n {\displaystyle c^{n+1}=c\cdot c^{n}} there

16632-487: The late 1960s as a theory of the strong nuclear force , before being abandoned in favor of quantum chromodynamics . Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory , incorporated only the class of particles known as bosons . It later developed into superstring theory , which posits

16786-443: The micro-level. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the universe , from elementary particles to atoms to the evolution of stars and the universe as a whole. In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of quantum gravity . The general theory of relativity

16940-505: The months following Witten's announcement, hundreds of new papers appeared on the Internet confirming different parts of his proposal. Today this flurry of work is known as the second superstring revolution. Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wrote "As it has been proposed that

17094-466: The notion of momentum state, which is defined far away from the experimental chamber, still works. The physical quantity he proposed as fundamental is the quantum mechanical amplitude for a group of incoming particles to turn into a group of outgoing particles, and he did not admit that there were any steps in between. The S-matrix is the quantity that describes how a collection of incoming particles turn into outgoing ones. Heisenberg proposed to study

17248-597: The number of dimensions. In 1978, work by Werner Nahm showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven. 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. Initially, many physicists hoped that by compactifying eleven-dimensional supergravity , it might be possible to construct realistic models of our four-dimensional world. The hope

17402-409: The observable quantities—those things measurable by experiments. An experiment only sees a microscopic quantity if it can be transferred by a series of events to the classical devices that surround the experimental chamber. The objects that fly to infinity are stable particles, in quantum superpositions of different momentum states. Heisenberg proposed that even when space and time are unreliable,

17556-442: The observable universe is a four-dimensional subspace of a higher dimensional space. In such models, the force-carrying bosons of particle physics arise from open strings with endpoints attached to the four-dimensional subspace, while gravity arises from closed strings propagating through the larger ambient space. This idea plays an important role in attempts to develop models of real-world physics based on string theory, and it provides

17710-402: The obtained values of f ( − 1 ) {\displaystyle f(-1)} are the same when the successive centers have a positive imaginary part or a negative imaginary part. This is not always the case; in particular this is not the case for the complex logarithm , the antiderivative of the above function. The power series defined below is generalized by the idea of

17864-434: The one-dimensional diagram representing the path of a point particle by a two-dimensional (2D) surface representing the motion of a string. Unlike in quantum field theory, string theory does not have a full non-perturbative definition, so many of the theoretical questions that physicists would like to answer remain out of reach. In theories of particle physics based on string theory, the characteristic length scale of strings

18018-487: The particles could be understood as bound states of just three of them (the proton, the neutron and the Lambda ; see Sakata model ), Geoffrey Chew believed that none of these particles are fundamental (for details, see Bootstrap model ). Sakata's approach was reworked in the 1960s into the quark model by Murray Gell-Mann and George Zweig by making the charges of the hypothetical constituents fractional and rejecting

18172-601: The past, even when the microscopic notion of past and future are not clearly defined. He also recognized that these relations might be useful in computing observables for the case of strong interaction physics. The dispersion relations were analytic properties of the S-matrix, and they imposed more stringent conditions than those that follow from unitarity alone. This development in S-matrix theory stemmed from Murray Gell-Mann and Marvin Leonard Goldberger 's (1954) discovery of crossing symmetry , another condition that

18326-422: The paths of point-like particles and their interactions. The starting point for string theory is the idea that the point-like particles of quantum field theory can also be modeled as one-dimensional objects called strings. The interaction of strings is most straightforwardly defined by generalizing the perturbation theory used in ordinary quantum field theory. At the level of Feynman diagrams, this means replacing

18480-457: The physical state that determines the properties of our universe. These problems have led some in the community to criticize these approaches to the unification of physics and question the value of continued research on these problems. 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

18634-405: The presence of singularities . The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology . Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C {\displaystyle \mathbb {C} } . If V

18788-469: The presence of tachyons (see tachyon condensation ); additionally, the spectrum of particles contains only bosons , particles like the photon that obey particular rules of behavior. While bosons are a critical ingredient of the Universe, they are not its only constituents. Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry (in the West ) in 1971,

18942-546: The problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei , black holes , and the early universe. String theory is a theoretical framework that attempts to address these questions and many others. The starting point for string theory is the idea that the point-like particles of particle physics can also be modeled as one-dimensional objects called strings . String theory describes how strings propagate through space and interact with each other. In

19096-408: The quantum aspects of black holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated in the discovery of the anti-de Sitter/conformal field theory correspondence or AdS/CFT. This is a theoretical result that relates string theory to other physical theories which are better understood theoretically. The AdS/CFT correspondence has implications for

19250-401: The quantum aspects of gravity. String theory has proved to be an important tool for investigating the theoretical properties of black holes because it provides a framework in which theorists can study their thermodynamics . In the branch of physics called statistical mechanics , entropy is a measure of the randomness or disorder of a physical system. This concept was studied in the 1870s by

19404-482: The question of analytic continuation of L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} to other complex z such that | z | > 1. {\displaystyle |z|>1.} As we shall see, for any n ≥ 1 {\displaystyle n\geq 1} , the function L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} diverges at

19558-599: The region U ∪ V {\displaystyle U\cup V} which is strictly larger than U {\displaystyle U} . The distance from a {\displaystyle a} to ∂ U {\displaystyle \partial U} is ρ = 1 − | a − 1 | > 0 {\displaystyle \rho =1-|a-1|>0} . Take 0 < r < ρ {\displaystyle 0<r<\rho } ; let D {\displaystyle D} be

19712-470: The region contains points not in U ∪ V {\displaystyle U\cup V} , then we will have analytically continued f {\displaystyle f} even further. This particular f {\displaystyle f} can be analytically continued to the whole punctured complex plane C ∖ { 0 } . {\displaystyle \mathbb {C} \setminus \{0\}.} In this particular case

19866-457: The result for a = 1 2 ( 3 + i ) . {\displaystyle a={\tfrac {1}{2}}(3+i).} We can continue the process: select b ∈ U ∪ V {\displaystyle b\in U\cup V} , recenter the power series at b {\displaystyle b} , and determine where the new power series converges. If

20020-609: The same physically observable process. The first model in which hadronic particles essentially follow the Regge trajectories was the dual resonance model that was constructed by Gabriele Veneziano in 1968, who noted that the Euler beta function could be used to describe 4-particle scattering amplitude data for such particles. The Veneziano scattering amplitude (or Veneziano model) was quickly generalized to an N -particle amplitude by Ziro Koba and Holger Bech Nielsen (their approach

20174-464: The series diverges at z = 0 ∈ ∂ U {\displaystyle z=0\in \partial U} . Pretend we don't know that f ( z ) = 1 / z {\displaystyle f(z)=1/z} , and focus on recentering the power series at a different point a ∈ U {\displaystyle a\in U} : f ( z ) = ∑ k = 0 ∞

20328-399: The set of germs G {\displaystyle {\mathcal {G}}} . Let g and h be germs . If | h 0 − g 0 | < r {\displaystyle |h_{0}-g_{0}|<r} where r is the radius of convergence of g and if the power series defined by g and h specify identical functions on the intersection of

20482-406: The sheaf of an analytic function contains the zero germ (i.e., the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero. Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f ( z ) then this function will have the property that exp( f ( z )) = z . If we had decided to use

20636-496: The six small extra dimensions (the D = 10 critical dimension of superstring theory had been originally discovered by John H. Schwarz in 1972) need to be compactified on a Calabi–Yau manifold . (In string theory, compactification is a generalization of Kaluza–Klein theory , which was first proposed in the 1920s.) By 1985, five separate superstring theories had been described: type I, type II (IIA and IIB) , and heterotic (SO(32) and E 8 × E 8 ) . Discover magazine in

20790-405: The string scale, a string looks just like an ordinary particle, with its mass , charge , and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton , a quantum mechanical particle that carries the gravitational force . Thus, string theory is a theory of quantum gravity . String theory

20944-455: The string. String theory's application as a form of quantum gravity proposes a vibrational state responsible for the graviton , a yet unproven quantum particle that is theorized to carry gravitational force. One of the main developments of the past several decades in string theory was the discovery of certain 'dualities', mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered

21098-432: The strings appearing in type IIA superstring theory. Speaking at a string theory conference in 1995, Edward Witten made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of higher-dimensional branes in string theory. In

21252-434: The study of black holes and quantum gravity, and it has been applied to other subjects, including nuclear and condensed matter physics . Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it will eventually be developed to the point where it fully describes our universe, making it a theory of everything . One of the goals of current research in string theory

21406-453: The theoretical appeal of its asymptotic freedom ). In 1974, John H. Schwarz and Joël Scherk , and independently Tamiaki Yoneya , studied the boson -like patterns of string vibration and found that their properties exactly matched those of the graviton , the gravitational force's hypothetical messenger particle . Schwarz and Scherk argued that string theory had failed to catch on because physicists had underestimated its scope. This led to

21560-428: The theory requires the inclusion of higher-dimensional objects, called D-branes : these are the sources of electric and magnetic Ramond–Ramond fields that are required by string duality . D-branes added additional rich mathematical structure to the theory, and opened possibilities for constructing realistic cosmological models in the theory (for details, see Brane cosmology ). In 1997–98, Juan Maldacena conjectured

21714-405: The true meaning of the title should be decided when a more fundamental formulation of the theory is known. In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within

21868-537: The two domains, then we say that h is generated by (or compatible with) g , and we write g ≥ h . This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity , we obtain a symmetric relation, which is therefore also an equivalence relation on germs (but not an ordering). This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted ≅ {\displaystyle \cong } . We can define

22022-499: The type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA, IIB and heterotic include only closed strings. In everyday life, there are three familiar dimensions (3D) of space: height, width and length. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to

22176-412: The unit circle such that z c n = 1 {\displaystyle z^{c^{n}}=1} . Now the key part of the proof is to use the functional equation for L c ( z ) {\displaystyle {\mathcal {L}}_{c}(z)} when | z | < 1 {\displaystyle |z|<1} to show that Thus for any arc on

22330-466: The value of continued research on string theory unification. In the 20th century, two theoretical frameworks emerged for formulating the laws of physics. The first is Albert Einstein 's general theory of relativity , a theory that explains the force of gravity and the structure of spacetime at the macro-level. The other is quantum mechanics , a completely different formulation, which uses known probability principles to describe physical phenomena at

22484-425: The vector is a germ of f . The base g 0 of g is z 0 , the stem of g is (α 0 , α 1 , α 2 , ...) and the top g 1 of g is α 0 . The top of g is the value of f at z 0 . Any vector g = ( z 0 , α 0 , α 1 , ...) is a germ if it represents a power series of an analytic function around z 0 with some radius of convergence r > 0. Therefore, we can safely speak of

22638-492: The vibrational states of a string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. The original version of string theory was bosonic string theory , but this version described only bosons , a class of particles that transmit forces between the matter particles, or fermions . Bosonic string theory was eventually superseded by theories called superstring theories . These theories describe both bosons and fermions, and they incorporate

22792-478: Was based on the observation that D-branes—which look like fluctuating membranes when they are weakly interacting—become dense, massive objects with event horizons when the interactions are strong. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge

22946-588: Was dubbed the Koba–Nielsen formalism ), and to what are now recognized as closed strings by Miguel Virasoro and Joel A. Shapiro (their approach was dubbed the Shapiro–Virasoro model ). In 1969, the Chan–Paton rules (proposed by Jack E. Paton and Hong-Mo Chan ) enabled isospin factors to be added to the Veneziano model. In 1969–70, Yoichiro Nambu , Holger Bech Nielsen , and Leonard Susskind presented

23100-566: Was generalized to relativistic quantum mechanics by Stanley Mandelstam , Vladimir Gribov and Marcel Froissart , using a mathematical method (the Sommerfeld–Watson representation ) discovered decades earlier by Arnold Sommerfeld and Kenneth M. Watson : the result was dubbed the Froissart–Gribov formula . In 1961, Geoffrey Chew and Steven Frautschi recognized that mesons had straight line Regge trajectories (in their scheme, spin

23254-450: Was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian . In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory. Although there were only a handful of consistent superstring theories, it remained

23408-555: Was proposed in 2003 by Shamit Kachru , Renata Kallosh , Andrei Linde , and Sandip Trivedi . Much of the present-day research is focused on characterizing the " swampland " of theories incompatible with quantum gravity . String theory In physics , string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings . String theory describes how these strings propagate through space and interact with each other. On distance scales larger than

23562-446: Was that Strominger and Vafa considered only extremal black holes in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge. Strominger and Vafa also restricted attention to black holes in five-dimensional spacetime with unphysical supersymmetry. Although it was originally developed in this very particular and physically unrealistic context in string theory,

23716-550: Was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces , and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as chirality . Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions. In

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