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59-505: In quantum field theory, S-duality generalizes a well established fact from classical electrodynamics , namely the invariance of Maxwell's equations under the interchange of electric and magnetic fields . One of the earliest known examples of S-duality in quantum field theory is Montonen–Olive duality which relates two versions of a quantum field theory called N = 4 supersymmetric Yang–Mills theory . Recent work of Anton Kapustin and Edward Witten suggests that Montonen–Olive duality

118-478: A continuous distribution of charge is: where ρ ( r ′ ) {\displaystyle \rho (\mathbf {r'} )} is the charge density, and r − r ′ {\displaystyle \mathbf {r} -\mathbf {r'} } is the distance from the volume element d 3 r ′ {\displaystyle \mathrm {d^{3}} \mathbf {r'} } to point in space where φ

177-407: A distribution of point charges, the forces determined from Coulomb's law may be summed. The result after dividing by q 0 is: where n is the number of charges, q i is the amount of charge associated with the i th charge, r i is the position of the i th charge, r is the position where the electric field is being determined, and ε 0 is the electric constant . If the field

236-446: A flurry of work now known as the second superstring revolution . Classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model . It is, therefore, a classical field theory . The theory provides a description of electromagnetic phenomena whenever

295-429: A moving point particle. Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant mathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena. An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and

354-532: A pair of quantum field theories in two-dimensional spacetime . By analyzing what this dimensional reduction does to certain physical objects called D-branes , they showed that one can recover the mathematical ingredients of the geometric Langlands correspondence. Their work shows that the Langlands correspondence is closely related to S-duality in quantum field theory, with possible applications in both subjects. Another realization of S-duality in quantum field theory

413-409: A system becomes larger or more massive the classical dynamics tends to emerge, with some exceptions, such as superfluidity . This is why we can usually ignore quantum mechanics when dealing with everyday objects and the classical description will suffice. However, one of the most vigorous ongoing fields of research in physics is classical-quantum correspondence . This field of research is concerned with

472-409: A test charge and F is the force on that charge. The size of the charge does not really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C ( newtons per coulomb ). This unit is equal to V/m ( volts per meter); see below. In electrostatics, where charges are not moving, around

531-456: A theory with complexified constant − 1 / τ {\displaystyle -1/\tau } . In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory . Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as

590-451: A very special type of gauge theory called N = 4 supersymmetric Yang–Mills theory , and it says that two such theories may be equivalent in a certain precise sense. If one of the theories has a gauge group G {\displaystyle G} , then the dual theory has gauge group L G {\displaystyle {^{L}}G} where L G {\displaystyle {^{L}}G} denotes

649-458: A weakly coupled theory. S-duality is a particular example of a general notion of duality in physics. The term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. 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

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708-532: Is Seiberg duality , first introduced by Nathan Seiberg around 1995. Unlike Montonen–Olive duality, which relates two versions of the maximally supersymmetric gauge theory in four-dimensional spacetime, Seiberg duality relates less symmetric theories called N=1 supersymmetric gauge theories . The two N=1 theories appearing in Seiberg duality are not identical, but they give rise to the same physics at large distances. Like Montonen–Olive duality, Seiberg duality generalizes

767-409: Is a vector representing the magnetic field, t {\displaystyle t} is time, and c {\displaystyle c} is the speed of light . The other symbols in these equations refer to the divergence and curl , which are concepts from vector calculus. An important property of these equations is their invariance under the transformation that simultaneously replaces

826-528: Is being determined. The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down into simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of the potential. Or: From this formula it is clear that E can be expressed in V/m (volts per meter). A changing electromagnetic field propagates away from its origin in

885-477: Is closely related to a research program in mathematics called the geometric Langlands program . Another realization of S-duality in quantum field theory is Seiberg duality , which relates two versions of a theory called N=1 supersymmetric Yang–Mills theory . There are also many examples of S-duality in string theory. The existence of these string dualities implies that seemingly different formulations of string theory are actually physically equivalent. This led to

944-504: Is described by a special type of quantum field theory called a gauge theory or Yang–Mills theory . In a gauge theory, the physical fields have a high degree of symmetry which can be understood mathematically using the notion of a Lie group . This Lie group is known as the gauge group . The electromagnetic field is described by a very simple gauge theory corresponding to the abelian gauge group U(1) , but there are other gauge theories with more complicated non-abelian gauge groups . It

1003-401: Is generally characterized by the principle of complete determinism , although deterministic interpretations of quantum mechanics do exist. From the point of view of classical physics as being non-relativistic physics, the predictions of general and special relativity are significantly different from those of classical theories, particularly concerning the passage of time, the geometry of space,

1062-520: Is instead produced by a continuous distribution of charge, the summation becomes an integral: where ρ ( r ′ ) {\displaystyle \rho (\mathbf {r'} )} is the charge density and r − r ′ {\displaystyle \mathbf {r} -\mathbf {r'} } is the vector that points from the volume element d 3 r ′ {\displaystyle \mathrm {d^{3}} \mathbf {r'} } to

1121-414: Is much greater than 1) to a weakly coupled theory (where the coupling constant 1 / g {\displaystyle 1/g} is much less than 1 and computations are possible). For this reason, S-duality is called a strong-weak duality . In classical physics , the behavior of the electric and magnetic field is described by a system of equations known as Maxwell's equations . Working in

1180-462: Is natural to ask whether there is an analog in gauge theory of the symmetry interchanging the electric and magnetic fields in Maxwell's equations. The answer was given in the late 1970s by Claus Montonen and David Olive , building on earlier work of Peter Goddard , Jean Nuyts , and Olive. Their work provides an example of S-duality now known as Montonen–Olive duality . Montonen–Olive duality applies to

1239-412: Is possible to describe a new physical setup in which these electric and magnetic fields are essentially interchanged, and the new fields will again give a solution of Maxwell's equations. This situation is the most basic manifestation of S-duality in a field theory. In quantum field theory, the electric and magnetic fields are unified into a single entity called the electromagnetic field , and this field

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1298-528: Is the coupling constant. For example, in the Yang–Mills theory that describes the electromagnetic field, this number g {\displaystyle g} is simply the elementary charge e {\displaystyle e} carried by a single proton. In addition to exchanging the gauge groups of the two theories, Montonen–Olive duality transforms a theory with complexified coupling constant τ {\displaystyle \tau } to

1357-408: Is the electric potential, and C is the path over which the integral is being taken. Unfortunately, this definition has a caveat. From Maxwell's equations , it is clear that ∇ × E is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of

1416-410: Is the manifestation of the electromagnetic interaction between charged particles. As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity). For the fields of general charge distributions,

1475-405: Is the position. r q {\displaystyle {\textbf {r}}_{q}} and v q {\displaystyle {\textbf {v}}_{q}} are the position and velocity of the charge, respectively, as a function of retarded time . The vector potential is similar: These can then be differentiated accordingly to obtain the complete field equations for

1534-451: Is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force. Although the equation appears to suggest that

1593-466: Is the velocity of the object and c is the speed of light. For velocities much smaller than that of light, one can neglect the terms with c and higher that appear. These formulas then reduce to the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities. Computer modeling has to be as real as possible. Classical physics would introduce an error as in

1652-434: The A vector potential described below. Whenever the charges are quasistatic, however, this condition will be essentially met. From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is: where q is the point charge's charge, r is the position at which the potential is being determined, and r i is the position of each point charge. The potential for

1711-542: The Langlands dual group which is in general different from G {\displaystyle G} . An important quantity in quantum field theory is complexified coupling constant. This is a complex number defined by the formula where θ {\displaystyle \theta } is the theta angle , a quantity appearing in the Lagrangian that defines the theory, and g {\displaystyle g}

1770-460: The Taniyama–Shimura conjecture , which includes Fermat's Last Theorem as a special case. In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult. As a result, some mathematicians have worked on a related conjecture known as the geometric Langlands correspondence . This is a geometric reformulation of

1829-469: The superfluidity case. In order to produce reliable models of the world, one can not use classical physics. It is true that quantum theories consume time and computer resources, and the equations of classical physics could be resorted to in order to provide a quick solution, but such a solution would lack reliability. Computer modeling would use only the energy criteria to determine which theory to use: relativity or quantum theory, when attempting to describe

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1888-420: The branches of theory sometimes included in classical physics are variably: In contrast to classical physics, " modern physics " is a slightly looser term that may refer to just quantum physics or to 20th- and 21st-century physics in general. Modern physics includes quantum theory and relativity, when applicable. A physical system can be described by classical physics when it satisfies conditions such that

1947-489: The classical Langlands correspondence which is obtained by replacing the number fields appearing in the original version by function fields and applying techniques from algebraic geometry . In a paper from 2007, Anton Kapustin and Edward Witten suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen–Olive duality. Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct

2006-483: The context of quantum mechanics , classical theory refers to theories of physics that do not use the quantisation paradigm , which includes classical mechanics and relativity . Likewise, classical field theories , such as general relativity and classical electromagnetism , are those that do not use quantum mechanics. In the context of general and special relativity, classical theories are those that obey Galilean relativity . Depending on point of view, among

2065-497: The coupling constant 1 / g {\displaystyle 1/g} . Similarly, type I string theory with the coupling g {\displaystyle g} is equivalent to the SO(32) heterotic string theory with the coupling constant 1 / g {\displaystyle 1/g} . The existence of these dualities showed that the five string theories were in fact not all distinct theories. In 1995, at

2124-399: The discoverer of that particular equation. Computer modeling is essential for quantum and relativistic physics. Classical physics is considered the limit of quantum mechanics for a large number of particles. On the other hand, classic mechanics is derived from relativistic mechanics . For example, in many formulations from special relativity, a correction factor ( v / c ) appears, where v

2183-408: The discovery of how the laws of quantum physics give rise to classical physics found at the limit of the large scales of the classical level. Today, a computer performs millions of arithmetic operations in seconds to solve a classical differential equation , while Newton (one of the fathers of the differential calculus) would take hours to solve the same equation by manual calculation, even if he were

2242-402: The electric and magnetic fields are independent, the equation can be rewritten in term of four-current (instead of charge) and a single electromagnetic tensor that represents the combined field ( F μ ν {\displaystyle F^{\mu \nu }} ): The electric field E is defined such that, on a stationary charge: where q 0 is what is known as

2301-454: The electric field E {\displaystyle \mathbf {E} } by the magnetic field B {\displaystyle \mathbf {B} } and replaces B {\displaystyle \mathbf {B} } by − 1 / c 2 E {\displaystyle -1/c^{2}\mathbf {E} } : In other words, given a pair of electric and magnetic fields that solve Maxwell's equations, it

2360-447: The equations of Albert Einstein 's general theory of relativity . Similarly, the strength of the electromagnetic force is described by a coupling constant, which is related to the charge carried by a single proton . To compute observable quantities in quantum field theory or string theory, physicists typically apply the methods of perturbation theory . In perturbation theory, quantities called probability amplitudes , which determine

2419-409: The following force (often called the Lorentz force) on charged particles: where all boldfaced quantities are vectors : F is the force that a particle with charge q experiences, E is the electric field at the location of the particle, v is the velocity of the particle, B is the magnetic field at the location of the particle. The above equation illustrates that the Lorentz force

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2478-400: The form of a wave . These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths . Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves , microwaves , light ( infrared , visible light and ultraviolet ), x-rays and gamma rays . In the field of particle physics this electromagnetic radiation

2537-433: The graduate level, textbooks like Classical Electricity and Magnetism , Classical Electrodynamics , and Course of Theoretical Physics are considered as classic references. The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be an electromagnetic wave. However,

2596-406: The language of vector calculus and assuming that no electric charges or currents are present, these equations can be written Here E {\displaystyle \mathbf {E} } is a vector (or more precisely a vector field whose magnitude and direction may vary from point to point in space) representing the electric field, B {\displaystyle \mathbf {B} }

2655-626: The laws of classical physics are approximately valid. In practice, physical objects ranging from those larger than atoms and molecules , to objects in the macroscopic and astronomical realm, can be well-described (understood) with classical mechanics. Beginning at the atomic level and lower, the laws of classical physics break down and generally do not provide a correct description of nature. Electromagnetic fields and forces can be described well by classical electrodynamics at length scales and field strengths large enough that quantum mechanical effects are negligible. Unlike quantum physics, classical physics

2714-418: The mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities. One of these dualities is S-duality. The existence of S-duality in string theory was first proposed by Ashoke Sen in 1994. It was shown that type IIB string theory with the coupling constant g {\displaystyle g} is equivalent via S-duality to the same string theory with

2773-495: The motion of bodies in free fall, and the propagation of light. Traditionally, light was reconciled with classical mechanics by assuming the existence of a stationary medium through which light propagated, the luminiferous aether , which was later shown not to exist. Mathematically, classical physics equations are those in which the Planck constant does not appear. According to the correspondence principle and Ehrenfest's theorem , as

2832-419: The particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative Classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift , then

2891-432: The point in space where E is being determined. Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral where φ ( r ) {\displaystyle \varphi ({\textbf {r}})}

2950-581: The previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the area of "classical physics". As such, the definition of a classical theory depends on context. Classical physical concepts are often used when modern theories are unnecessarily complex for a particular situation. Most often, classical physics refers to pre-1900 physics, while modern physics refers to post-1900 physics, which incorporates elements of quantum mechanics and relativity . Classical theory has at least two distinct meanings in physics. In

3009-417: The probability for various physical processes to occur, are expressed as sums of infinitely many terms , where each term is proportional to a power of the coupling constant g {\displaystyle g} : In order for such an expression to make sense, the coupling constant must be less than 1 so that the higher powers of g {\displaystyle g} become negligibly small and

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3068-512: The realization, in the mid-1990s, that all of the five consistent superstring theories are just different limiting cases of a single eleven-dimensional theory called M-theory . In quantum field theory and string theory, a coupling constant is a number that controls the strength of interactions in the theory. For example, the strength of gravity is described by a number called Newton's constant , which appears in Newton's law of gravity and also in

3127-553: The relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics which is a quantum field theory . Fundamental physical aspects of classical electrodynamics are presented in many textbooks. For the undergraduate level, textbooks like The Feynman Lectures on Physics , Electricity and Magnetism , and Introduction to Electrodynamics are considered as classic references and for

3186-458: The retarded potentials can be computed and differentiated accordingly to yield Jefimenko's equations. Retarded potentials can also be derived for point charges, and the equations are known as the Liénard–Wiechert potentials. The scalar potential is: where q {\displaystyle q} is the point charge's charge and r {\displaystyle {\textbf {r}}}

3245-433: The string theory conference at University of Southern California , Edward Witten made the surprising suggestion that all five of these theories were just different limits of a single theory now known as M-theory . Witten's proposal was based on the observation that type IIA and E 8 ×E 8 heterotic string theories are closely related to a gravitational theory called eleven-dimensional supergravity . His announcement led to

3304-448: The sum is finite. If the coupling constant is not less than 1, then the terms of this sum will grow larger and larger, and the expression gives a meaningless infinite answer. In this case the theory is said to be strongly coupled , and one cannot use perturbation theory to make predictions. For certain theories, S-duality provides a way of doing computations at strong coupling by translating these computations into different computations in

3363-457: The symmetry of Maxwell's equations that interchanges electric and magnetic fields. Up until the mid 1990s, physicists working on string theory believed there were five distinct versions of the theory: type I , type IIA , type IIB , and the 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. In

3422-647: The theory of electromagnetism , as it is currently understood, grew out of Michael Faraday 's experiments suggesting the existence of an electromagnetic field and James Clerk Maxwell 's use of differential equations to describe it in his A Treatise on Electricity and Magnetism (1873). The development of electromagnetism in Europe included the development of methods to measure voltage , current , capacitance , and resistance . Detailed historical accounts are given by Wolfgang Pauli , E. T. Whittaker , Abraham Pais , and Bruce J. Hunt. The electromagnetic field exerts

3481-456: The transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. S-duality is useful because it relates a theory with coupling constant g {\displaystyle g} to an equivalent theory with coupling constant 1 / g {\displaystyle 1/g} . Thus it relates a strongly coupled theory (where the coupling constant g {\displaystyle g}

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