Poincaré group Basic explanation[edit] If one ignores the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation (by a fixed angle) around any of the three spatial axes, or a boost in any of the three spatial directions, altogether 1 + 3 + 3 + 3 = 10. Technical explanation[edit] Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the "inhomogeneous Lorentz group". In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group. The Poincaré algebra is the Lie algebra of the Poincaré group. where P is the generator of translations, M is the generator of Lorentz transformations, and η is the Minkowski metric (see sign convention). Poincaré symmetry[edit] See also[edit]
Symmetrie (Physik) Dieser Artikel wurde den Mitarbeitern der Redaktion Physik zur Qualitätssicherung aufgetragen. Wenn Du Dich mit dem Thema auskennst, bist Du herzlich eingeladen, Dich an der Prüfung und möglichen Verbesserung des Artikels zu beteiligen. Der Meinungsaustausch darüber findet derzeit nicht auf der Artikeldiskussionsseite, sondern auf der Qualitätssicherungs-Seite der Physik statt. Die mathematische Beschreibung von Symmetrien erfolgt durch die Gruppentheorie. Das sog. Wichtig sind nicht nur die Symmetrien selbst, sondern auch Symmetriebrechungen: So wird in der Theorie der elektroschwachen Wechselwirkung die Eichsymmetrie durch den Higgs-Mechanismus gebrochen, wozu das einzige bisher noch nicht nachgewiesene Teilchen des Standardmodells der Elementarteilchenphysik, das Higgs-Boson, benötigt wird. Folgende Tabelle gibt einen Überblick über wichtige Symmetrien und ihre Erhaltungsgrößen. Transformationen oder Symmetrieoperationen können wie die Symmetrien selbst stetig oder diskret sein.
Symmetric group Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on G. Definition and first properties[edit] The symmetric group on a finite set X is the group whose elements are all bijective functions from X to X and whose group operation is that of function composition.[1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. Applications[edit] Elements[edit] Sym(2)
Lorentz covariance In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings: This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. Examples[edit] In general, the nature of a Lorentz tensor can be identified by its tensor order, which is the number of indices it has. Scalars[edit] Spacetime interval: Proper time (for timelike intervals):
Noether's theorem Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law. Basic illustrations and background[edit] As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion. Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. Informal statement of the theorem[edit]
CP violation It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present Universe, and in the study of weak interactions in particle physics. CP-symmetry[edit] The idea behind parity symmetry is that the equations of particle physics are invariant under mirror inversion. This leads to the prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay) occurs at the same rate as the original reaction. Parity symmetry appears to be valid for all reactions involving electromagnetism and strong interactions. Overall, the symmetry of a quantum mechanical system can be restored if another symmetry S can be found such that the combined symmetry PS remains unbroken. Simply speaking, charge conjugation is a simple symmetry between particles and antiparticles, and so CP-symmetry was proposed in 1957 by Lev Landau as the true symmetry between matter and antimatter. CP violation in the Standard Model[edit] and . .
Gauge covariant derivative The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations. Fluid dynamics[edit] In fluid dynamics, the gauge covariant derivative of a fluid may be defined as where is a velocity vector field of a fluid. Gauge theory[edit] is the electromagnetic vector potential. What happens to the covariant derivative under a gauge transformation[edit] If a gauge transformation is given by and for the gauge potential then transforms as and so that in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.
Noether-Theorem Eine Erhaltungsgröße eines Systems von Teilchen ist eine Funktion der Zeit , des Ortes der Teilchen und ihrer Geschwindigkeit , deren Wert sich auf jeder physikalisch durchlaufenen Bahn nicht mit der Zeit ändert. eines Teilchens der Masse bewegt, eine Erhaltungsgröße, d. h. für alle Zeiten gilt Beispiele für Symmetrien und zugehörige Erhaltungsgrößen[Bearbeiten] Aus der Homogenität der Zeit (Wahl der Startzeit spielt keine Rolle) folgt die Erhaltung der Energie (Energieerhaltungssatz). Mathematische Formulierung[Bearbeiten] Wirkung[Bearbeiten] Der im Noether-Theorem formulierte Zusammenhang von Symmetrien und Erhaltungsgrößen gilt für solche physikalischen Systeme, deren Bewegungs- oder Feldgleichungen aus einem Variationsprinzip abgeleitet werden können. Bei der Bewegung von Massepunkten ist dieses Wirkungsfunktional durch eine Lagrangefunktion der Zeit und der Geschwindigkeit charakterisiert und ordnet jeder differenzierbaren Bahnkurve das Zeitintegral zu. durch den Startpunkt und zur Endzeit Sei mit
The Revolutionary Galois Theory On May 31, 1832, a French Republican revolutionary called Évariste Galois died from a gunshot. He was 20 years old. The night before, fearing his own death, Galois had written many letters, frenetically scribbling “I have no time; I have no time“. Had he? The revolution Galois initiated turned out to be bigger and more profound than he could have possibly envisioned it! To romanticize even a bit more the story-telling of Galois’ dramatic life, here’s an abstract of a documentary I made about algebra: Waw! He definitely is! How did Galois theory still managed to be known? In 1843, 10 years after Galois’ death, finally, a brilliant French mathematician named Joseph Liouville managed to grasp some of Galois’ ideas. Haha! It was definitely a huge boost. This article is going to increasingly push you intellectually. Imaginary and Complex NumbersBy Lê Nguyên Hoang | Updated:2014-12 | Views: 3357My first reaction to imaginary numbers was... Fields in Pure Algebra What do you mean? Oh yeah! Haha!
Galilean invariance Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. The fact that the Earth orbits around the sun at approximately 30 km/s offers a somewhat more dramatic example, though it is technically not an inertial reference frame. Formulation[edit] Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames. Among the axioms from Newton's theory are: There exists an absolute space, in which Newton's laws are true. Galilean relativity can be shown as follows. Newton's theory versus special relativity[edit] See also[edit]