Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Relativistic quantum mechanics of the majorana particle. In particular, three features of wigners work are worthy of mention. I am not sure this is the right answer and this is too long for comments. This representation is reducible, the decomposition into irreducibles is just the decomposition of rn into the various. We are now at a stage to study the irreducible representations of the lorentz group. I am currently working through the method of induced representations in order to calculate the unitary irreducible representations of the poincare group. Wigner representation theory of the poincare group, localization. Most particle physicists will recognise this title immediately, but to nonspecialists it will be just gibberish.
Pdf structural properties of irreducible twoparticle. Calgebras and mackeys theory of group representations. Cossey university of arizona irreducible representations of the symmetric group. The symmetric group representation theory james, mathas, and fayers irreducible representations of the symmetric group j. As a result the irreducible representations can be classi. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. The poincare group must act on the description of this single object. Group theory qmii 2017 1 the proper lorentz group and poincare. Invariant subspaces a linear subspace w v of the carrier space v of a representation d of a group gis an invariant subspace if it is preserved by the action of all operators of the representation.
Character tables list of the complete set of irreducible representations rows and symmetry classes columns of a point group. The lorentz group is a collection of linear transformations of spacetime coordinates x. His main motivation was to understand in intrinsic physical terms the ever increasing \zoo of linear relativistic higher spin eld. Any ray representation of the poincare group can, by a suitable choice of phases, be made into an ordinary representation of the covering group. Why do we say that irreducible representation of poincare group represents the oneparticle state. Recall that in the case of dirac particle one obtains a reducible representation composed of two spin 12 irreducible.
In this chapter we develop the relation between the relativistic wave equations for free fields and the representation theory of the poincare group p. Wigner representation theory of the poincar e group. A nonzero matrix which commutes with all of the matrices of an irreducible representation is a constant multiple of the unit matrix. In physical models, we need unitary representations to make sense of a probabilistic interpretation in a quantum field theory. The role of the irreducible representations of the. This splits into a direct sum or direct integral of irreducible reps.
For massless particles w2 p2 0, and in fact the spin vector w and the energymomentum vector p are parallel. W p 27 it is easy to see that is just the helicity. Another result at the core of representation theory that we will make use of is the following. The poincare group, the full symmetry group of minkowski space translation in space or time, boosts, rotations has as its unitary irreducible representations a two parameter family m. All these steps are covered in many details and with examples. In 1939 wigner 1 classi ed the irreducible rayrepresentations of the poincar e group or what amounts to the same, the irreducible vectorrepresentations of its covering. The representation is unitary because the portion of the wave function depending on the internal variable x is squareintegrable, and all. The poincare group is a continuous group of symmetries, so a lot of the questions that one is required to solve depends on understanding in nitesimal trasnformations of the poincare group, and it boils down to understanding the set of unitary irreducible representations of the lie algebra of the group in question. The use of the irreducible representations corresponds to the expansion of the fields and currents into modes which have the simplest possible.
But even if you need to skip the technical bits, you might find the observations about the sociology of physics interesting. In particular the description includes the tachyonic and infinite spin representations, which do. Stable particles are then identified with irreducible representations of the lorentz group. It turns out that they are related to representations of lorentz group. The theory of induced representations in field theory. We study the real representations of the poincare group and its relation with the complex representations. Chapter 4 properties of irreducible representations. Representations of the poincare group sciencedirect. It explains the method of induced representations for the poincare group representations construction and the complete classification of all unitary irreducible representations. For an irreducible twoparticle representation of the poincare group, the calculated normalization factor matches wylers semiempirical formula for the finestructure constant alpha. It emphasizes group theorys role as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems. These components are the irreducible representations 3. The unitary representations of the poincare group in any spacetime.
The k 0 and k n cases give the trivial representation, but we get fundamental irreducible representations for k 1,n. The classical electromagnetic fieldfrom which the poincare group was originally. Wigners 1939 representation theory of the poincare group. In 1939, eugene wigner classi ed the fundamental particles using the irreducible rep resentations of the double cover of the poincar e group. In 1939 wigner 1 classified the irreducible ray representations of the poincare group or what amounts to the same. Symmetry and particle physics university of surrey. Representations, character tables, and one application of. Why do we say that irreducible representation of poincare. The double cover of the poincare group acts to classify the fundamental particles in. The classic example and one of fundamental importance in physics is the wigner construction of representations of. Tensor representations are given by the direct tensor product of copies of the vector representation.
Including this transformations as well is known as the poincare group. We then give a brief discussion of the description of invariance or covariance in the formalism ofquantum theory thereafter turning to a systematic theory of unitary irreducible representations of p. We show that there is a map from the complex to the real irreducible representations of. Contents 1 lorentz group national tsing hua university. The construction of the principal continuous series of unitary representations of the simply. The determination of all unitary, irreducible representations of the inho. Then the irreducible unitary representations of g classify indivisible intrinsic descriptions of the system and, boldly.
Familiarity with basic group concepts and techniques is. Unitary irreducible representations of the little group. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. Representations of the poincare group, spin and gauge. A 2ajgjgbe the matrices of a ddimensional irreducible representation of a group g, i. Week 1 1 unitary representations of the poincare group.