UNITARY REPRESENTATIONS OF THE POINCARE GROUP
SPECIAL RELATIVITY + QUANTUM MECHANICS Simple foundations Einstein’s theory Lorentz group No roadblocks to development Dodgy foundations States are rays g(h(x)) = eigh(x) Many contributors State space The Gruppenpest Group theory applied to QM in late 20s Chemists adopted relatively quickly Physicists slow to utilize (finally in early 60s with Gell-Mann)
Eugene Wigner Symmetries must be unitary or anti-unitary operators State space is a representation of the Lorentz group Any ray representation of the Lorentz group can be made into an ordinary representation of its covering group Determined the unitary representations of the Lorentz group Wigner (1902-1995) an early advocate Unitary/anti-unitary theorem 1931 Representations of the Lorentz group 1939 Nobel prize 1963 “for his contribution to the theory of the atomic spectra, particularly through the discovery and applications of fundamental symmetry principles” Weinberg author index Dirac 51 Feynman 32 Heisenberg 28 Schwinger 24 Pauli 23 Wigner 21
Poincare Group Proper (det = 1) orthochronous (00 1) Lorentz group SO+(1,3) Translations (a, )(b, ) = (a + b, ) Minkowski space R1,3 with metric = diag(1,-1,-1,-1) (a, )x = a + x Defining representation Semi-direct product – translation the maximal abelian subgroup is invariant (normal) Non-compact so either non-unitary or not finite dimensional
Poincare Group II R1,3 in 1-1 correspondence with 2 by 2 complex matrices with real diagonal entries M(x) = x = x0 + x3 x1 - ix2 det M(x) = |x| x1 + ix2 x0-x3 SL(2,C) acts by AM(x)A† since det AM(x)A† = det M(x) = |x| S: SL(2,C) SO+(1,3) by M(S(A)x) = AM(x)A† S is a homomorphism, onto, but not 1-1: S(A) = S(-A) SL(2,C) is the universal cover of SO+(1,3) Poincare group has a lower degree representation Any path between two pints can be continuously deformed into any other – simply connected
State space as a representation All particles are partially characterized by their four momentum p Possible states may need additional parameters denoted by Representations are concentrated on orbits m+ 0+ 0- m-
Wigner’s Method Choose a representative point in each orbit For each point in an orbit, pick a Lorentz transformation that takes the representative point to that point Find the Lorentz transformations that fix the representative point – the “little” group A representation of the little group induces a representation of the Poincare group
The Induced Representation
Massive representations Choose (m,0,0,0) as the representative point The little group is SU(2) Its irreducible representations are spin representations of dimension 2j + 1 with states and matrices Eigenvalues half an integer for entirely algebraic reasons Irreducible Unitary Infinite dimensional in p
Massless representations Choose the representative point (k,0,0,k) The little group is a subgroup of the Lorentz group consisting of rotations around the 3-axis and translations in the plane The Lie algebra is generated by J3, L1 = J1 + K2, L2 = J2 - K1 [L1 , L2] = 0 [J3 , Li] = iεijLj Define These ladder operators generate an infinite tower of eigenstates from each eigenstate of J3, spaced apart by integers. The topology of the Lorentz group restricts eigenvalues of J3 to 0, ½, 1,…