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.
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]
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
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]
Pseudo-Goldstone boson Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken (hidden), and would thus engender massless Nambu-Goldstone bosons. The additional explicit symmetry breaking gives these bosons a small mass. The properties of these pseudo-Goldstone bosons can normally be found by an expansion around the (exactly) symmetric theory in terms of the explicit symmetry-breaking parameters. In QCD, this is interpreted as a consequence of spontaneous symmetry breaking of chiral symmetry in a sector of QCD with 3 flavours of light quarks.[1] Such a theory, for idealized massless quarks, has global In actual full QCD, the small quark masses further break the chiral symmetry explicitly as well.[2] The masses of the actual pseudoscalar meson octet are found by an expansion in the quark masses,[3] which goes by the name of chiral perturbation theory. See also[edit]
General covariant transformations In physics, general covariant transformations are symmetries of gravitation theory on a world manifold . From the physical viewpoint, general covariant transformations are treated as particular (holonomic) reference frame transformations in general relativity. Mathematical definition[edit] Let be a fiber bundle coordinated by . is projected onto a diffeomorphism of its base . need not give rise to an automorphism of In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of is a projectable vector field on . , whose flow is a one-parameter group of diffeomorphisms of . be a vector field on . projected onto , every vector field gives rise to the horizontal vector field . yields a monomorphism of the -module of vector fields on to the , but this monomorphisms is not a Lie algebra morphism, unless is flat. However, there is a category of above mentioned natural bundles which admit the functorial lift onto of any vector field such that is a Lie algebra monomorphism . . of . .
Goldstone boson In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in the context of the BCS superconductivity mechanism,[1] and subsequently elucidated by Jeffrey Goldstone,[2] and systematically generalized in the context of quantum field theory.[3] Goldstone's theorem[edit] Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken; i.e., its currents are conserved, but the ground state (vacuum) is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) scalar particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the ground state. Examples[edit] Natural[edit] Theory[edit]