NationMaster - Encyclopedia: Representation theory of the Galilean group

In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows: Wikisource has original text related to this article: Relativity: The Special and General Theory Albert Einsteins theory of relativity, or simply relativity, refers specifically to two theories: special relativity and general relativity. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the...

The spacetime symmetry group of nonrelativistic mechanics is the Galilean group. Since we are interested in projective representations of this group, which is equivalent to unitary representations of the nontrivial central extension of the universal covering group of the Galilean group by the one dimensional Lie group R, refer to the article Galilean group for the central extension of its Lie algebra. We will focus upon the Lie algebra here because it is simpler to analyze and we can always extend the results to the full Lie group thanks to the Frobenius theorem. The symmetry group of an object (e. ... The Galilean transformation is used to transform between the coordinates of two coordinate systems in a constant relative motion in Newtonian physics. ... In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors... In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group... In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... The universal covering space of a topological group is also a topological group. ... The Galilean transformation is used to transform between the coordinates of two coordinate systems in a constant relative motion in Newtonian physics. ... The Galilean transformation is used to transform between the coordinates of two coordinate systems in a constant relative motion in Newtonian physics. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the Frobenius theorem states, in its smooth version of degree 1, the following: Let U be an open set in Rn and F a submodule of Ω1(U) of constant rank r in U. Then F is integrable if and only if for every p ∈ U the stalk...

If you think about how spatial and time translations, rotations and boosts work, these relations are intuitive (except for the central extension).

The central charge M is of course a Casimir invariant. In theoretical physics, a central charge is an operator Z, typically having discrete eigenvalues, that appears on the right-hand side of a commutator that defines an algebra. ... In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra. ...

is another Casimir invariant. Using Schur's lemma, an irreducible unitary representation would have both of these Casimir invariants as multiples of the identity. Let's call these coefficients m and mE₀ respectively. Remember we are talking about unitary representations here, which means these values have to be real. So, m > 0, m = 0 and m < 0. The last case is similar to the first. from a purely representation theoretic point of view, we'd have to study all of them, but we are interested in applications to quantum mechanics here. There, E represents the energy, which has to be bounded from below if we require thermodynamic stability. Consider first the case where m is nonzero. If we look at the $(E,vec{P})$ space with the constraint In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra. ... In mathematics, Schurs lemma is now a generic term applied to theorems on the commutant of a module M that is simple. ... In mathematics, the term irreducible is used in several ways. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

we find the boosts act transitively upon this subsurface. Look at the stabilizer of a point on the orbit, (E₀, 0). Because of transitivity, we know the unitary irrep contains a nontrivial subspace with these energy-momentum eigenvalues. (This subspace only exists in a rigged Hilbert space because the momentum spectrum is continuous, but that is not an essential detail except to mathematicians. Or from another point of view, it is an essential detail except to non-mathematicians.) It is spanned by E, $vec{P}$ , M and L_ij. We already know how the subspace of the irrep transforms under all but the angular momentum. Note that the rotation subgroup is Spin(3). We have to look at its double cover because we're considering projective representations. This is called the little group, a name given by Eugene Wigner. The method of induced representations tells us the irrep is given by the direct sum of all the fibers in a vector bundle over the mE = mE₀ + P²/2 hypersurface whose fibers are a unitary irrep of Spin(3). Spin(3) is none other than SU(2). See representation theory of SU(2). There, it is shown the unitary irreps of SU(2) are labeled by an nonnegative integer multiple of half, s. This is called the spin, due to historical reasons. So, we have shown for m not equal to zero, the unitary irreps are classified by m, E₀ and a spin s. Looking at the spectrum of E, we find that if m, the mass, is negative, the spectrum of E is not bounded from below. So, only the case with a positive mass is physical. In mathematics, a symmetry group describes all symmetries of objects. ... This article is about mathematical concept. ... In mathematics, groups are often used to describe symmetries of objects. ... Screenshot (from SSCX Star Warzone). ... In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ... This article is in need of attention from an expert on the subject. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... Eugene Wigner (left) and Alvin Weinberg Eugene Paul Wigner (Hungarian Wigner PÃ¡l JenÅ‘) (November 17, 1902 â€“ January 1, 1995) was a Hungarian physicist and mathematician. ... In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the (whole) group G itself. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the...

is nonpositive. Suppose it is zero. Here, the boosts and the rotations form the little group. So, any unitary irrep of this little group also gives rise to a projective irrep of the Galilean group. As far as we can tell, only the case which transforms trivially under the little group has any physical interpretation and it corresponds to the no particle state (vacuum). In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ...

The case where the invariant is negative requires additional comment. This corresponds to the representation class for m = 0 and non-zero $vec{P}$ . Extending the tardyon, luxon, tachyon classification from the representation theory of the Poincare' group to an analogous classification, here, one may term these states as synchrons. They represent an instantaneous transfer of non-zero momentum across a (possibly large) distance. Associated with them is a 'time' operator A tardyon or bradyon is a particle that travels slower than light. ... Luxon is a particle that always travels at the speed of light. ... A tachyon (from the Greek Ï„Î±Ï‡ÏÏ‚ takhÃºs, meaning swift) is any hypothetical particle that travels at superluminal velocity. ...

which may be identified the time of transfer. These states are naturally interpreted as the carriers of instantaneous action-at-a-distance forces.

In the 3+1-dimensional Galilei group, the boost generator may be decomposed into

playing a role analogous to helicity. this page is about helicity in fluid mechanics. ...