Representation theory of the Poincaré group

Representation theory of the Poincaré group

In mathematics, the representation theory of the double cover of the Poincaré group is an example of the theory for a Lie group, in a case that is neither a compact group nor a semisimple group. It is important in relation with theoretical physics.

For unitary representations, see Wigner's classification.

For more general (linear) representations of the double cover, see below.

A rep of the Poincaré is called a covariant field rep if there is an injective intertwiner from it to a rep consisting of the space of sections of a fiber bundle over Minkowski spacetime where if a is mapped to b under the action of an element of the double cover, then the base point of a maps to the base point of b as the (affine) Minkowski rep of the same group.

In general, single particle reps are unitary irreps while operators form covariant field reps (see operator valued distribution). This is because single particle reps form a Hilbert space while if we have any operator A let's say, looking at the operator valued function over the (double cover of the) Poincaré group defined by g [A] for each element g of the double cover of the Poincaré group, we can decompose them under harmonic analysis and each component of the decomposition would correspond to a linear covariant field rep. More precisely, Minkowski space is a homogeneous space of the Poincaré group, with the Lorentz group as the stabilizer, which means that under harmonic analysis, the operator valued function over the Poincaré group can be rewritten as operator valued rep of the Lorentz group function over Minkowski space.

In general, the injective intertwiner needn't be surjective. In other words, the space of all sections (forget about topology for the moment) could be a reducible representation. A covariant intertwiner is an intertwiner mapping one covariant field rep to another. The derivative (since Minkowski space is flat, we don't need a covariant derivative) for example is a nonzero intertwiner from a covariant field rep to another. Also, if the fiber is a reducible rep, we could have a (pointwise) intertwiner projecting out one of its components for example.

Such an analysis doesn't explain the spacelike commutativity/anticommutativity properties of the fields though. This form of locality is an additional assumption.

Things like resonances are actually more complicated.

This injective intertwiner needn't necessarily be unique. Ignoring topological issues like continuity, smoothness, boundary conditions, topology, etc, for example, the rep of all scalar field configurations satisfying the Klein-Gordon equation for mass m is equivalent to a subrep of all smooth closed 1-forms satisfying the Klein-Gordon equation for mass m.

Also, the reps, and hence the fibers of the fiber bundle needn't necessarily be linear. It could be affine or even nonlinear, like for example nonlinear sigma models.

A rep is called a quotient covariant field rep if it is equivalent to the quotient representation of a covariant field rep by a subrep of it which is also a covariant field rep (with respect to the same fiber bundle). This is the case, for example in free Abelian Yang-Mills theories.

ee also

*representation of the diffeomorphism group
*Wigner's classification
*representation theory of the Galilean group
*Particle physics and representation theory


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Representation theory of the Galilean group — In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group. In 3+1 dimensions, this is the subgroup of the… …   Wikipedia

  • Representation theory of the Lorentz group — The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half integer spins in quantum field theory. These representations are normally constructed out of spinors.The group may also be …   Wikipedia

  • Poincaré group — In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. It is a 10 dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz… …   Wikipedia

  • List of representation theory topics — This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. Contents 1 General representation theory 2… …   Wikipedia

  • Representation theory — This article is about the theory of representations of algebraic structures by linear transformations and matrices. For the more general notion of representations throughout mathematics, see representation (mathematics). Representation theory is… …   Wikipedia

  • Particle physics and representation theory — There is a natural connection, first discovered by Eugene Wigner, between the properties of particles, the representation theory of Lie groups and Lie algebras, and the symmetries of the universe. According to it, the different states of an… …   Wikipedia

  • Lorentz group — Group theory Group theory …   Wikipedia

  • General linear group — Group theory Group theory …   Wikipedia

  • Orthogonal group — Group theory Group theory …   Wikipedia

  • Lie group — Lie groups …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”