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Poincare Symmetry




Poincaré symmetry is the full symmetry of Special Relativity and includes
  • Translations (ie, displacements) in time and space (these form the Abelian Lie Group of translations on space-time)

  • Rotation s in space (this forms the non-Abelian Lie Group of 3-dimensional rotations)

  • Boosts , ie, transformations connecting two uniformly moving bodies.

  • The last two symmetries together make up the Lorentz Group (see Lorentz Invariance ). These are generators of a Lie Group called the ''' Poincare Group ''' which is a Semi-direct Product of the group of translations and the Lorentz group. Things which are invariant under this group are said to have '''Poincaré invariance''' or '''relativistic invariance'''.


The Poincaré group is the full symmetry group of any Relativistic Field Theory . As a result, all Elementary Particle s fall in representations of this group. These are usually specified by the ''four-momentum'' of each particle (i.e. its mass) and the intrinsic Quantum Numbers JPC, where J is the Spin quantum number, P is the Parity and C is the Charge Conjugation quantum number. Many quantum field theories do violate parity and charge conjugation. In those case, we drop the P and the C. Since CPT is an invariance of every Quantum Field Theory , a time reversal quantum number could easily be constructed out of those given.


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