The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics.
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👉 Poincaré group in the context of Exotic meson
In particle physics, exotic mesons are mesons that have quantum numbers not possible in the quark model; some proposals for non-standard quark model mesons could be:
All exotic mesons are classed as mesons because they are hadrons and carry zero baryon number. Of these, glueballs must be flavor singlets – that is, must have zero isospin, strangeness, charm, bottomness, and topness. Like all particle states, exotic mesons are specified by the quantum numbers which label representations of the Poincaré symmetry, q.e., by the mass (enclosed in parentheses), and by J, where J is the angular momentum, P is the intrinsic parity, and C is the charge conjugation parity; One also often specifies the isospin I of the meson. Typically, every quark model meson comes in SU(3) flavor nonet: an octet and an associated flavor singlet. A glueball shows up as an extra (supernumerary) particle outside the nonet.
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Poincaré group in the context of Relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré-covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high-energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.
Key features common to all RQMs include: the prediction of antimatter, spin magnetic moments of elementary spin-1/2 fermions, fine structure, and quantum dynamics of charged particles in electromagnetic fields. The key result is the Dirac equation, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the Hamiltonian operator to achieve agreement with experimental observations.
Poincaré group in the context of Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non-commuting symmetry operators or that the non-degenerate states are also eigenvectors of symmetry operators.
This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators, and relates them to the Lie groups, and relativistic transformations in the Lorentz group and Poincaré group.
Poincaré group in the context of Conformal symmetry
Conformal symmetry is a property of spacetime that ensures angles remain unchanged even when distances are altered. If you stretch, compress, or otherwise distort spacetime, the local angular relationships between lines or curves stay the same. This idea extends the familiar Poincaré group —which accounts for rotations, translations, and boosts—into the more comprehensive conformal group.
Conformal symmetry encompasses special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.