1. UNITARY REPRESENTATIONS OF THE POINCARE GROUP

2. SPECIAL RELATIVITY + QUANTUM MECHANICS
Simple foundations
Einstein’s theory
Lorentz group
No roadblocks to development
Dodgy foundations
States are rays
g(h(x)) = eigh(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)

3. 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 ( ) 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
Feynman 32
Heisenberg 28
Schwinger 24
Pauli
Wigner 21

4. 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

5. 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

6. 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-

7. 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

8. The Induced Representation

9. 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

10. 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,…