The Lorentz and Poincaré Groups & Quantum Field Theory

The Lorentz and Poincar´egroups & Quantum Field Theory

Lorentz Group topology algebra The Lorentz and Poincar´egroups Fields & Quantum Field Theory Poincar´egroup

Angelo Monteux

phys 251: Group Theory and Modern Physics

June 8th 2011 The Lorentz and The Lorentz Group Poincarégroups & Quantum Field Theory Group of linear transformations acting on the 4-dimensional ∼ 4 Minkowski manifold M = R leaving invariant the quadratic Lorentz Group form defined by the metric (−1, 1, 1, 1): topology algebra µ 0µ µ ν Fields x → x = Λν x Poincarégroup 2 02 µ ν T −1 −1 x = x =⇒ Λρ gµνΛσ = gρσ gΛ g = Λ generalized orthogonal group O(3, 1) 0 proper orthocronous det Λ = 1 Λ0 ≥ 1 1 0 proper non-orthocronous det Λ = 1 Λ0 ≤ −1 T 0 improper orthocronous det Λ = −1 Λ0 ≥ 1 P 0 improper non-orthocronous det Λ = −1 Λ0 ≤ −1 PT proper Lorentz transformations → SO(3, 1) proper,orthocronous Lorentz transformations → SO+(3, 1) O(3, 1) = D = {1, P, T , PT } SO+(3, 1) 2 The Lorentz and The Lorentz Group Poincarégroups & Quantum Field Theory Group of linear transformations acting on the 4-dimensional ∼ 4 Minkowski manifold M = R leaving invariant the quadratic Lorentz Group form defined by the metric (−1, 1, 1, 1): topology algebra µ 0µ µ ν Fields x → x = Λν x Poincarégroup 2 02 µ ν T −1 −1 x = x =⇒ Λρ gµνΛσ = gρσ gΛ g = Λ generalized orthogonal group O(3, 1)

0 proper orthocronous det Λ = 1 Λ0 ≥ 1 1 0 proper non-orthocronous det Λ = 1 Λ0 ≤ −1 T 0 improper orthocronous det Λ = −1 Λ0 ≥ 1 P 0 improper non-orthocronous det Λ = −1 Λ0 ≤ −1 PT proper Lorentz transformations → SO(3, 1) proper,orthocronous Lorentz transformations → SO+(3, 1) + O(3, 1) = D2 o SO (3, 1) The Lorentz and Spinorial map Poincar´egroups & Quantum Field Theory We can construct an homomorphism between SL(2, C) and µ SO(3, 1): given a 4-vector v we associate the matrix Lorentz Group topology algebra µ v0 + v3 v1 − iv2 Fields V = vµσ = v1 + iv2 v0 − v3 Poincar´egroup

2 µ det V = −v = −vµv . The Lorentz transform of V will be

0 0 µ ν µ † V = vµσ = Λµvνσ = ΛV Λ det V = det V 0 =⇒ | det Λ|2 = 1 If we take det Λ = 1 we have a proper Lorentz tranformation. ±Λ give the same Lorentz transformation SL(2, ) SO(3, 1) =∼ C Z2

SL(2, C) is simply connected, then it is the universal covering of the Lorentz group The Lorentz and more on SL(2, ) Poincar´egroups C & Quantum Field Theory Λ = ueh, with | det u| = 1 and Trh = 0, h† = h. Lorentz Group topology h3 h1 − ih2 ∼ 3 algebra = R Fields h1 + ih2 −h3 Poincar´egroup a + ib c + id , a2 + b2 + c2 + d2 = 1 =∼ S3 −c + id a − ib

3 3 Topologically, SL(2, C) is homeomorphic to S × R and

S3 3 SO(3, 1) = /Z2 ⊗ R compact, non connected

non compact, connected The Lorentz and The Lorentz algebra Poincar´egroups & Quantum Field Theory

+ Lorentz Group Algebra associated to SO (3, 1): expand Lorentz topology µ µ µ algebra transformations around the identity: Λν = δν + ν Fields 3 rotations 3 boosts Poincar´egroup [K , K ] = −iε J [Ji , Jj ] = iεijk Jk i j ijk k

[Ji , Kj ] = iεijk Kk 6-dimensional algebra Ji0 = Ki Jµν = xµpν − xνpµ, Jij = εijk Jk

[Jµν, Jρσ] = i(gµσJνρ + gνρJµσ − gνσJµρ + gµρJνσ) The Lorentz and Decomposition of the Lorentz algebra Poincar´egroups & Quantum Field Theory

∼ Lorentz Group as complex algebras, so(3, 1) = sl(2, C) ⊕ sl(2, C) topology algebra so(4) =∼ su(2) ⊕ su(2) Fields Poincar´egroup  1  [Ai , Aj ] = iεijk Ak Ai = 2 (Ji + iKi ) 1 =⇒ [Bi , Bj ] = iεijk Bk Bi = 2 (Ji − iKi )  [Ai , Bj ] = 0 The Lorentz and Decomposition of the Lorentz algebra Poincar´egroups & Quantum Field Theory

as complex algebras, so(3, 1) =∼ sl(2, ) ⊕ sl(2, ) Lorentz Group C C topology  algebra [A , A ] = iε A Fields A = 1 (J + iK )  i j ijk k i 2 i i Poincarégroup 1 =⇒ [Bi , Bj ] = iεijk Bk Bi = 2 (Ji − iKi )  [Ai , Bj ] = 0 • SO(3, 1) is a non-compact group, then there is no finite-dimensional unitary representation. • still, the SO(3) (so(3) =∼ su(2)) subgroup of rotations is compact Ji = Ai + Bi The irreducible representations of the Lorentz algebra are given by irreducible representations of the two su(2)’s: a field in the (JA, JB ) representation of SO(3) has spin

j = JA + JB , JA + JB − 1,..., |JA − JB |

(2JA + 1)(2JB + 1)-dimensional The Lorentz and Representations and Fields Poincar´egroups & Quantum Field Theory Given an arbitrary ﬁeld Φ(x), after a Lorentz transformation

it will be: Lorentz Group topology 0 0 i ab ab Φ (x ) = D[ω]Φ(x), D[ω] = exp( ωabΣ ), Σ ∈ so(3, 1) algebra 2 Fields Poincarégroup • one-dimensional (trivial) representation: (0, 0) φ0(x0) = φ(x) scalar field • two-dimensional representations: 1 (0, 1 ) ( 2 , 0) 2 Weyl spinors ψα = ψL(x) ψβ˙ = ψR (x) A two-dimensional representation of su(2) is given by the ~σ/2: the rotations and boosts are represented by  σi ( − i ~σ·(θ~−iξ~)  R[Ji ] = ψ → e 2 ψ 2 =⇒ L L σi − i ~σ·(θ~+iξ~) R[K ] = −i ψ → e 2 ψ  i 2 R R The Lorentz and Representations and Fields Poincarégroups & Quantum Field Theory • three-dimensional representations: (1, 0) and (0, 1):

antisymmetric 4x4 tensor =⇒ Fµν Lorentz Group topology algebra Fields 1 ˜ 1 ˜ Fµν = 2 (Fµν + Fµν) + 2 (Fµν − Fµν) = (1, 0) ⊕ (0, 1) Poincar´egroup self-dual anti-self-dual 1 1 • four-dimensional reducible representation: ( 2 , 0) ⊕ (0, 2 ) ψ(x) Dirac spinor 1 1 • four-dimensional irreducible representation: ( 2 , 2 ) given a vector Aµ, ν Aµ → ΛµAν vector ﬁeld µ which can also be written as Aαα˙ = Aµσαα˙

α α˙ i µν i ρσ e 2 ωµν σ στ e− 2 ωρσσ¯ = Λτ σσ αα˙ σ ββ˙ β β˙ The Lorentz and Representations and Fields Poincar´egroups & Quantum Field Theory • three-dimensional representations: (1, 0) and (0, 1):

α α˙ i µν i ρσ e 2 ωµν σ στ e− 2 ωρσσ¯ = Λτ σσ αα˙ σ ββ˙ β β˙ The Lorentz and The PoincaréGroup Poincarégroups & Quantum Field Theory Group of ALL the transformations leaving invariant the quadratic form defined by the metric (−1, 1, 1, 1): Lorentz Group topology algebra µ µ ν µ Fields x → Λν x + a Poincarégroup

[Jµν, Jρσ] = i(gµσJνρ + gνρJµσ − gνσJµρ + gµρJνσ)

[Pµ, Pν] = 0, [Jµν, Pρ] = i(gµρPν − gνρPµ)

• The Poincarégroup is the semidirect product of the group of translations and the Lorentz group. • Any relativistic field is in a representation of the Poincaré group, defined by its 4-momentum and the quantum numbers (JA, JB ) defined for the Lorentz group. The Lorentz and Casimir operators Poincarégroups & Quantum Field Theory

Lorentz Group topology algebra 2 2 Fields Pµ, Wµ Poincar´egroup µ 1 µνρσ where W = 2 ε PνJρσ is the Pauli-Lubanski vector.

• massive particle: Pµ = (m, 0)

2 2 Wµ = (0, −mJi ) =⇒ Wµ = m s(s + 1) : spin

• massless particle: P2 = 0 J · P W |pi = hP |pi, h = ± : helicity µ µ |P| The Lorentz and Coleman-Mandula theorem Poincar´egroups & Quantum Field Theory

A symmetry transformation of the S-matrix is a unitary operator Lorentz Group U commuting with S. topology algebra An internal symmetry is a transformation that commutes with the Fields Poincarégroup P. Poincarégroup Theorem: Let G be a connected symmetry group of the S matrix such that 1. G has a subgroup locally isomorphic to P 2. all particles have positive energy and there is only one zero-particle vacuum separated from one-particle states 3. elastic scattering amplitudes are analytic functions of s and t (except at thresholds) 4. T |p, p0i= 6 0 then G is locally isomorphic to the direct product of an internal symmetry group and P. The Lorentz and Coleman-Mandula theorem Poincarégroups & Quantum Field Theory A symmetry transformation of the S-matrix is a unitary operator

U commuting with S. Lorentz Group An internal symmetry is a transformation that commutes with the topology algebra Poincarégroup P. Fields Theorem: Let G be a connected symmetry group of the S matrix Poincarégroup such that 1. G has a subgroup locally isomorphic to P 2. all particles have positive energy and there is only one zero-particle vacuum separated from one-particle states 3. elastic scattering amplitudes are analytic functions of s and t (except at thresholds) 4. T |p, p0i= 6 0 then G is locally isomorphic to the direct product of an internal symmetry group and P. The internal symmetry group is the direct product of a compact semisimple group with U(1) factors. The Lorentz and Coleman-Mandula theorem Poincarégroups & Quantum Field Theory A symmetry transformation of the S-matrix is a unitary operator

U commuting with S. Lorentz Group An internal symmetry is a transformation that commutes with the topology algebra Poincarégroup P. Fields Theorem: Let G be a connected symmetry group of the S matrix Poincarégroup such that 1. G has a subgroup locally isomorphic to P 2. all particles have positive energy and there is only one zero-particle vacuum separated from one-particle states 3. elastic scattering amplitudes are analytic functions of s and t (except at thresholds) 4. T |p, p0i= 6 0 then G is locally isomorphic to the direct product of an internal symmetry group and P. Derived for the Lie algebra so discrete symmetries are allowed The Lorentz and Coleman-Mandula theorem Poincarégroups & Quantum Field Theory A symmetry transformation of the S-matrix is a unitary operator

U commuting with S. Lorentz Group An internal symmetry is a transformation that commutes with the topology algebra Poincarégroup P. Fields Theorem: Let G be a connected symmetry group of the S matrix Poincarégroup such that 1. G has a subgroup locally isomorphic to P 2. all particles have positive energy and there is only one zero-particle vacuum separated from one-particle states 3. elastic scattering amplitudes are analytic functions of s and t (except at thresholds) 4. T |p, p0i= 6 0 then G is locally isomorphic to the direct product of an internal symmetry group and P. If condition2 does not hold P is replaced by the conformal group The Lorentz and Coleman-Mandula theorem Poincarégroups & Quantum Field Theory A symmetry transformation of the S-matrix is a unitary operator

U commuting with S. Lorentz Group An internal symmetry is a transformation that commutes with the topology algebra Poincarégroup P. Fields Theorem: Let G be a connected symmetry group of the S matrix Poincarégroup such that 1. G has a subgroup locally isomorphic to P 2. all particles have positive energy and there is only one zero-particle vacuum separated from one-particle states 3. elastic scattering amplitudes are analytic functions of s and t (except at thresholds) 4. T |p, p0i= 6 0 then G is locally isomorphic to the direct product of an internal symmetry group and P. Supersymmetry is excluded from this theorem because it involves a graded Lie algebra instead of a normal Lie algebra The Lorentz and Essential Bibliography Poincarégroups & Quantum Field Theory

Lorentz Group topology algebra S. Weinberg, The Quantum Theory of Fields, vol. Fields Poincar´egroup 1. M. Kaku, Quantum Field Theory: A Modern Introduction. Wu Ki Tung, Group theory in Physics. http://en.wikipedia.org/wiki/Lorentz_group S. Coleman, J. Mandula - Physical Review 159(5): 1251-1256 (1967)