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Fujikawa's Method
In physics, Fujikawa's method is a way of deriving the chiral anomaly in quantum field theory. It uses the correspondence between functional determinants and the partition function, effectively making use of the Atiyah–Singer index theorem. Derivation Suppose given a Dirac field \psi which transforms according to a representation \rho of the compact Lie group ''G''; and we have a background connection form of taking values in the Lie algebra \mathfrak\,. The Dirac operator (in Feynman slash notation) is :D\!\!\!\!/\ \stackrel\ \partial\!\!\!/ + i A\!\!\!/ and the fermionic action is given by :\int d^dx\, \overlineiD\!\!\!\!/ \psi The partition function is :Z \int \mathcal\overline\mathcal\psi\,e^. The axial symmetry transformation goes as :\psi\to e^\psi\, :\overline\to \overlinee^ :S\to S + \int d^dx \,\alpha(x)\partial_\mu\left(\overline\gamma^\mu\gamma_\psi\right) Classically, this implies that the chiral current, j_^\mu \equiv \overline\gamma^\mu\gamma_\psi is con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Slash Notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \gamma^1 A_1 + \gamma^2 A_2 + \gamma^3 A_3 + \gamma^4 A_4 where ''γ'' are the gamma matrices. Using the Einstein summation notation, the expression is simply : \ \stackrel\ \gamma^\mu A_\mu. Identities Using the anticommutators of the gamma matrices, one can show that for any a_\mu and b_\mu, :\begin &\equiv a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 \\ + &\equiv 2 a \cdot b \cdot I_4. \end where I_4 is the identity matrix in four dimensions. In particular, :^2 \equiv \partial^2 \cdot I_4. Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example, :\begin \operatorname() &\equiv 4 a \cdot b \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Strength
In physics, field strength means the ''magnitude'' of a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field results in both electric field strength and magnetic field strength. As an application, in radio frequency telecommunications, the signal strength excites a receiving antenna and thereby induces a voltage at a specific frequency and polarization in order to provide an input signal to a radio receiver. Field strength meters are used for such applications as cellular, broadcasting, wi-fi and a wide variety of other radio-related applications. See also * Dipole field strength in free space * Field strength tensor * Signal strength in telecommunications In telecommunications, particularly in radio frequency engineering, signal strength refers to the transmitter power output as received by a reference antenna at a distance from the transmitting antenna. High-powered transmissions, such as those us ... Referenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta Function Regularization
Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label= Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician letter zayin . Letters that arose from zeta include the Roman Z and Cyrillic З. Name Unlike the other Greek letters, this letter did not take its name from the Phoenician letter from which it was derived; it was given a new name on the pattern of beta, eta and theta. The word ''zeta'' is the ancestor of ''zed'', the name of the Latin letter Z in Commonwealth English. Swedish and many Romanic languages (such as Italian and Spanish) do not distinguish between the Greek and Roman forms of the letter; "''zeta''" is used to refer to the Roman letter Z as well as the Greek letter. Uses Letter The letter ζ represents the voiced alveolar fricative in Modern Greek. The sound represented by zeta in Greek before 40 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguistics, linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an Ordination, ordained Minister (Christianity), minister who taught mathematics and physics at the Stettin Gymnasium (Germany), Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classics, classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics. Although lacking university training in mathematics, it was the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axial Symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. glossary of meteorology. Retrieved 2010-04-08. For example, a without trademark or other design, or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric. Axial symmetry can also be |
Partition Function (quantum Field Theory)
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams. Generating functional Scalar theories In a d-dimensional field theory with a real scalar field \phi and action Sphi/math>, the partition function is defined in the path integral formalism as the functional : Z = \int \mathcal D\phi \ e^ where J(x) is a fictitious source current. It acts as a generating functional for arbitrary n-point correlation functions : G_n(x_1, \dots, x_n) = (-1)^n \frac \frac\bigg, _. The derivatives used here are fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If :D^2=\Delta, \, where ∆ is the Laplacian of ''V'', then ''D'' is called a Dirac operator. In high-energy physics, this requirement is often relaxed: only the second-order part of ''D''2 must equal the Laplacian. Examples Example 1 ''D'' = −''i'' ∂''x'' is a Dirac operator on the tangent bundle over a line. Example 2 Consider a simple bundle of notable import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Anomaly
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa. Such events are expected to be prohibited according to classical conservation laws, but it is known there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a ''chiral law'' of this kind. Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly. Research into chiral symmetry breaking laws is a major endeavor in particle physics research at this time. Informal introduction The chiral anomaly originally referred to the anomalous decay rate of the neutral pion, as computed in the current algebra of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
