Quantum field theory

Standard Model The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, as well as classifying all the subatomic particles known. It was developed throughout the latter half of the 20th century, as a collaborative effort of scientists around the world.[1] The current formulation was finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, discoveries of the top quark (1995), the tau neutrino (2000), and more recently the Higgs boson (2013), have given further credence to the Standard Model. Because of its success in explaining a wide variety of experimental results, the Standard Model is sometimes regarded as a "theory of almost everything". Historical background The Higgs mechanism is believed to give rise to the masses of all the elementary particles in the Standard Model. Overview Particle content Fermions Gauge bosons Higgs boson Main article: Higgs boson E.S.

Complex logarithm In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the real natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of a complex number z is a complex number w such that ew = z.[1] The notation for such a w is ln z or log z. Since every nonzero complex number z has infinitely many logarithms,[1] care is required to give such notation an unambiguous meaning. If z = reiθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2πi gives all the others.[1] Problems with inverting the complex exponential function For a function to have an inverse, it must map distinct values to distinct values, i.e., be injective. forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. There are two solutions to this problem. Definition of principal value For example, Log(-3i) = ln 3 − πi/2. .

Photon polarization Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photon eigenstates have either right or left circular polarization. A photon that is in a superposition of eigenstates can have linear, circular, or elliptical polarization. The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well, and forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. Polarization of classical electromagnetic waves Polarization states Linear polarization Effect of a polarizer on reflection from mud flats. .

Quantization (physics) The first method to be developed for quantization of field theories was canonical quantization. While this is extremely easy to implement on sufficiently simple theories, there are many situations where other methods of quantization yield more efficient procedures for computing quantum amplitudes. However, the use of canonical quantization has left its mark on the language and interpretation of quantum field theory. The method does not apply to all possible actions (like for instance actions with a noncausal structure or actions with gauge "flows"). -deformed in the same way as in canonical quantization. Actually, there is a way to quantize actions with gauge "flows". Main article: Weyl quantization. Also see the phase space formulation of quantum mechanics, Moyal bracket, Star product, and Wigner quasi-probability distribution. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. Jump up ^ H.J.

Higgs boson The Higgs boson is named after Peter Higgs, one of six physicists who, in 1964, proposed the mechanism that suggested the existence of such a particle. Although Higgs's name has come to be associated with this theory, several researchers between about 1960 and 1972 each independently developed different parts of it. In mainstream media the Higgs boson has often been called the "God particle", from a 1993 book on the topic; the nickname is strongly disliked by many physicists, including Higgs, who regard it as inappropriate sensationalism.[17][18] In 2013 two of the original researchers, Peter Higgs and François Englert, were awarded the Nobel Prize in Physics for their work and prediction[19] (Englert's co-researcher Robert Brout had died in 2011). A non-technical summary "Higgs" terminology Overview If this field did exist, this would be a monumental discovery for science and human knowledge, and is expected to open doorways to new knowledge in many fields. History

Abstract algebra The permutations of Rubik's Cube have a group structure; the group is a fundamental concept within abstract algebra. History As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Major themes include: Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. Early group theory There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, in his generalization of Fermat's little theorem. Modern algebra Basic concepts More complicated examples include:

Quantum field theory in curved spacetime In particle physics, quantum field theory in curved spacetime is an extension of standard, Minkowski-space quantum field theory to curved spacetime. A general prediction of this theory is that particles can be created by time-dependent gravitational fields (multigraviton pair production), or by time-independent gravitational fields that contain horizons. Description Interesting new phenomena occur; owing to the equivalence principle the quantization procedure locally resembles that of normal coordinates where the affine connection at the origin is set to zero and a nonzero Riemann tensor in general once the proper (covariant) formalism is chosen; however, even in flat spacetime quantum field theory, the number of particles is not well-defined locally. For non-zero cosmological constants, on curved spacetimes quantum fields lose their interpretation as asymptotic particles. Applications Approximation to quantum gravity See also References Notes N.D.

Strange Physical Theory Proved After Nearly 40 Years When physicist Vitaly Efimov heard his theory had finally been proven, he ran up to the younger scientist who had verified it and gave him a high five. Efimov had predicted a quantum-mechanical version of Borromean rings, a symbol that first showed up in Afghan Buddhist art from around the second century. The symbol depicts three rings linked together; if any ring were removed, they would all come apart. Efimov theorized an analog to the rings using particles: Three particles (such as atoms or protons or even quarks) could be bound together in a stable state, even though any two of them could not bind without the third. The physicist first proposed the idea, based on a mathematical proof, in 1970. A team of physicists led by Randy Hulet of Rice University in Houston finally achieved the trio of particles, and published their findings in the online journal Science Express. Hulet presented his work at a meeting in Rome in October that Efimov also attended.

Complex analysis Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics". Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). History Complex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Complex functions For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts: and where are real-valued functions. In other words, the components of the function f(z), can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponential functions, logarithmic functions, and trigonometric functions) into the complex domain. Holomorphic functions See also: analytic function, holomorphic sheaf and vector bundles.