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History Of String Theory
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. 1943–1959: S-matrix theory String theory represents an outgrowth of S-matrix theory, 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 the strong interactions. The theory presented a radical rethin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 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 conde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley Mandelstam
Stanley Mandelstam (; 12 December 1928 – 23 June 2016) was a South African theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations. The double dispersion relations were a central tool in the bootstrap program which sought to formulate a consistent theory of infinitely many particle types of increasing spin. Early life Mandelstam was born in Johannesburg, South Africa to a Jewish family.William D. Rubinstein, Michael Jolles, Hilary L. Rubinstein, ''The Palgrave Dictionary of Anglo-Jewish History'', Palgrave Macmillan (2011), p. 110 Work Mandelstam, along with Tullio Regge, did the initial development of the Regge theory of strong interaction phenomenology. He reinterpreted the analytic growth rate of the scattering amplitude as a function of the cosine of the scattering angle as the power law for the falloff of scattering amplitudes a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing (physics)
In quantum field theory, a branch of theoretical physics, crossing is the property of scattering amplitudes that allows antiparticles to be interpreted as particles going backwards in time. Crossing states that the same formula that determines the S-matrix elements and scattering amplitudes for particle \mathrm to scatter with \mathrm and produce particle \mathrm and \mathrm will also give the scattering amplitude for \scriptstyle \mathrm+\bar+\mathrm to go into \mathrm, or for \scriptstyle \bar to scatter with \scriptstyle \mathrm to produce \scriptstyle \mathrm+\bar. The only difference is that the value of the energy is negative for the antiparticle. The formal way to state this property is that the antiparticle scattering amplitudes are the analytic continuation of particle scattering amplitudes to negative energies. The interpretation of this statement is that the antiparticle is in every way a particle going backwards in time. History Murray Gell-Mann and Marvin Leonard Gol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marvin Leonard Goldberger
Marvin Leonard "Murph" Goldberger (October 22, 1922 – November 26, 2014) was an American theoretical physicist and former president of the California Institute of Technology. Biography Goldberger was born in Chicago, Illinois. He went on to receive his B.S. at the Carnegie Institute of Technology (now Carnegie Mellon University), and Ph.D. in physics from the University of Chicago in 1948. His advisor on thesis, ''Interaction of High-Energy Neutrons with Heavy Nuclei'', was Enrico Fermi. Goldberger was a postdoc at MIT at least by 1951 where he shared a communal physics office with at least Murray Gell-Mann where they worked together on various projects and he encouraged him to join him at Chicago 1952 onwards, before he became professor of physics at Princeton University from 1957 through 1977. He received the Dannie Heineman Prize for Mathematical Physics in 1961, and in 1963 was elected to the U.S. National Academy of Sciences. In 1965 he was elected a Fellow of the America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. 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 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. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kramers–Kronig Relations
The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform. Formulation Let \chi(\omega) = \chi_1(\omega) + i \chi_2(\omega) be a complex function of the complex variable \omega , where \chi_1(\omega) and \chi_2(\omega) are real. Suppose this function is analytic in the closed upper half-plane of \omega and vanishes faster than 1/, \omega, as , \omega, \to \infty. Slightly weaker conditi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Kronig
Ralph Kronig (10 March 1904 – 16 November 1995) was a German physicist. He is noted for the discovery of particle spin and for his theory of X-ray absorption spectroscopy. His theories include the Kronig–Penney model, the Coster–Kronig transition and the Kramers–Kronig relations. Background Ralph Kronig (later Ralph de Laer Kronig) was born on 10 March 1904 to German parents (Harold Theodor Kronig, Augusta de Laer) in Dresden, Germany. He died in Zeist on 16 November 1995 at the age of 91. Kronig received his primary and high-school education in Dresden and went to New York City to study at Columbia University where he received his PhD in 1925 and subsequently became instructor (1925) and assistant professor (1927). Early in Kronig's career he had encountered Paul Ehrenfest who, while visiting America in 1924, had advised the young physicist Ralph Kronig to revisit Europe. Kronig left for that continent later in 1924 and paid visits to the important centers for theoreti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Kramers
Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistical physics. Background and education Hans Kramers was born on 17 December 1894 in Rotterdam. the son of Hendrik Kramers, a physician, and Jeanne Susanne Breukelman. In 1912 Hans finished secondary education ( HBS) in Rotterdam, and studied mathematics and physics at the University of Leiden, where he obtained a master's degree in 1916. Kramers wanted to obtain foreign experience during his doctoral research, but his first choice of supervisor, Max Born in Göttingen, was not reachable because of the First World War. Because Denmark was neutral in this war, as was the Netherlands, he travelled (by ship, overland was impossible) to Copenhagen, where he visited unannounced the then still relatively unknown Niels Bohr. Bohr took him on as a Ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion Relations
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation. Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media. In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase vel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical Physics Emeritus at the California Institute of Technology, a distinguished fellow and one of the co-founders of the Santa Fe Institute, a professor of physics at the University of New Mexico, and the Presidential Professor of Physics and Medicine at the University of Southern California. Gell-Mann spent several periods at CERN, a nuclear research facility in Switzerland, among others as a John Simon Guggenheim Memorial Foundation fellow in 1972. Early life and education Gell-Mann was born in Lower Manhattan to a family of Jewish immigrants from the Austro-Hungarian Empire, specifically from Czernowitz in present-day Ukraine. His parents were Pauline (née Reichstein) and Arthur Isidore Gell-Mann, who taught English as a second language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Series
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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