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Particle Physics And Representation Theory
There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their energy spectrum, spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe. General picture Symmetries of a quantum system In quantum mechanics, any particular one-particle state is represented as a vector space, vector in a Hilbert space \mathcal H. To help understand what types of particles can exist, it is important to classify the possibilities for \mathcal H allowed by Wigner's theorem#Symmetry transformations, symmetries, and their properties. Let \mathcal H be a Hilbert space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three generations of fermions, but ordinary matter is made only from the first fermion generation. The first generation consists of up and down quarks which form protons and neutrons, and electrons and electron neutrinos. The three fundamental interactions known to be mediated by bosons are electromagnetism, the weak interaction, and the strong interaction. Quarks cannot exist on their own but form hadrons. Hadrons that contain an odd number of quarks are called baryons and those that contain an even number are called mesons. Two baryons, the proton and the neutron, make up most of the mass of ordinary matter. Mesons are unstable and the longest-lived last for only a few hundredths of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Hilbert Space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by :w \sim v if and only if v = \lambda w for some non-zero complex number \lambda. The equivalence classes of v for the relation \sim are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. Overview The physical significance of the projective Hilbert space is that in quantum theory, the wave functions \psi and \lambda \psi represent the same ''physical state'', for any \lambda \ne 0. It is conventional to choose a \psi from the ray so that it has unit norm, \langle\psi, \psi\rangle = 1, in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine \psi within the ray, since \psi could be multiplied by any \lambda with absolute value 1 (the U(1) action ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of The Poincaré Group
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.) In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts on the space of sections. Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Space
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Transformations
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the origins coinciding at 0, where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \left ( \sqrt\right )^ is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. The three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by: :\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra Extension
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges. Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra. Central extensions are needed in physi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Representation
In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(''V'') is the general linear group of invertible linear transformations of ''V'' over ''F'', and ''F''∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of G is a collection of operators \rho(g)\in\mathrm(V),\, g\in G satisfying the homomorphism property up to a constant: :\rho(g)\rho(h) = c(g, h)\rho(gh), for some constant c(g, h)\in F. Equivalently, a projective representation of G is a collection of operators \tilde\rho(g)\in\mathrm(V), g\in G, such that \tilde\rho(gh)=\tilde\rho(g)\tilde\rho(h). Note that, in this notation, \tilde\rho(g) is a ''set'' of linear operators related by multiplication with some nonze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the future, in general, the macroscopic universe does not show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantum noninvasive measurements are predicted to violate time symmetry even in equilibrium, contrary to their classical counterparts, although this has not yet been experimentally confirmed. Time ''asymmetries'' generally are caused by one of three categories: # intrinsic to the dynamic physical law (e.g., for the weak force) # due to the initial conditions of the universe (e.g., for the second law of thermodynamics) # due to measurements (e.g., for the noninvasive measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
