The Lorentz and Poincaré Groups in Relativistic Field Theory

The and s in Relativistic Field Theory Term Project Nicolás Fernández González University of California Santa Cruz June 2015 1 / 14

the Our first encounter with the group is in special relativity it composed transformations that preserve the line element in Minkowski space (s 2 = η µν x µ x ν ). In particular, for x µ x µ = Λ µ ν x ν, we get: Λ µ ρη µν Λ ν σ = η ρσ Or in matrix notation Λ satisfy the following relation: Where η is the matrix: Λ ηλ = η = O ηo = η η = diag(1, 1, 1, 1) 2 / 14

4 Islands, 2 Boats the Poincare The group consists of four separated components: L + : det Λ = 1 and Λ00 1. L : det Λ = 1 and Λ00 1. L + : det Λ = 1 and Λ00 1. L : det Λ = 1 and Λ00 1. The subgroup group that exclude spatial reflections and time reversal is called the proper orthochronous group SO+ (1, 3). There are two discrete transformations P and T L = L + L L + L = L + P L + T L + P T L + We get the decomposition of L into cosets restricted group L +. 3 / 14

The West Coast or East Coast? the Usually we have two possible choices for the Minkowski metric: West Coast metric (1, 1, 1, 1). Feynman s favorite East Coast metric ( 1, 1, 1, 1). The choice of Weinberg and Schwinger (also Ross and Joey?) Peter Woit has a nice blog entry about this:...physicists dont appreciate the mathematical concept of real structure or complexification. and...pseudo-majorana spinors a nebulous concept best kept undisturbed. 4 / 14

Generators group the An infinitesimally transformation Λ µ ν should be of form, Λ µ ν = δ µ ν + ω µ ν Where with ω µ ν is a matrix of infinitesimal coefficients. Plugging in our definition of group and keeping O(ω 2 ), we have: η ρσ = Λ µ ρη µν Λ ν σ η ρσ = ( δ µ ρ + ω µ ρ ) ηµν (δ ν σ + ω ν σ) η ρσ = η ρσ + ω ρσ + ω σρ 0 = ω ρσ + ω σρ We conclude that ω ρσ = ω σρ Thus the generators ω are 4 4 antisymmetric matrices characterized by six parameters so that there are six generators. 5 / 14

the The rotations can be parametrized by a 3-component vector θ i with θ i π, and the boosts by a three component vector β i (rapidity) with β <. Taking a infinitesimal transformation we have that: Infinitesimal rotation for x,y and z: 0 J 1 = i 0 0 1 1 0 0 J 2 = i 0 1 0 1 0 0 J 3 = i 0 1 1 0 0 Infinitesimal boost: 0 1 K 1 = i 1 0 0 0 0 1 K 2 = i 0 1 0 0 0 1 K 3 = i 0 0 1 0 These matrices are the generators (4-vector basis) 6 / 14

the Having the generation of any representation we can know the Lie algebra. [J i, J j ] = iɛ ijk J k [J i, K j ] = iɛ ijk K k [K i, K j ] = iɛ ijk J k 7 / 14

the Having the generation of any representation we can know the Lie algebra. [J i, J j ] = iɛ ijk J k [J i, K j ] = iɛ ijk K k [K i, K j ] = iɛ ijk J k We can be even more efficient writing as a rank-2 tensor representation Where V µν is [V µν, V ρσ ] = i (V µσ η νρ + V νρ η µσ V µρ η νσ V νσ η µρ ). 0 K 1 K 2 K 3 V µν = K 1 0 J 3 J 2 K 2 J 3 0 J 1 K 3 J 2 J 1 0 So now we can define the group as the set of transformations generated by these generators. 7 / 14

Generators the We can define a suspicious combinations se two sets of generators as J + i = 1 2 (J i + ik i ) J i = 1 2 (J i ik i ) 8 / 14

Generators the We can define a suspicious combinations se two sets of generators as J + i = 1 2 (J i + ik i ) J i = 1 2 (J i ik i ) Which satisfy the commutation relations, [ J + i, J + ] j = iɛijk J + k [ J i, J ] j = iɛijk J k [ J + i, J ] j = 0 Black magic! We now see that the algebra se generators decouple into two SU(2) algebras. 8 / 14

Representations group the Representations Lie algebra group j + j Reps Dim Field 0 0 (0, 0) 1 Scalar or Singlet 1 2 0 ( 1 2, 0) 2 Left handed Weyl spinor 1 0 ) 2 Right handed Weyl spinor 1 2 2 (0, 1 2 1 2 ( 1 2, 1 2 ) 4 Vector 1 0 (1, 0) 3 Self-dual 2-form field 0 1 (0, 1) 3 Anti-self-dual 2-form field 1 1 (1, 1) 9 Traceless symmetric tensor field. ( 1 2, 0) (0, 1 2 ) is a Dirac spinor (1, 0) (0, 1) is the electromagnetic field tensor F µν 9 / 14 We can label the representations of so(3, 1) by the weights (quantum numbers) of su(2) su(2)

We have a problem the group are not unitary and that is a deal braking for Physics because we really really like unitary for Quantum Field Theory. Where we were wrong? 10 / 14

We have a problem the group are not unitary and that is a deal braking for Physics because we really really like unitary for Quantum Field Theory. Where we were wrong? It all comes from the factor of i associated with boosts. Or in other words, The precise relationship between the two groups are that the complex linear combinations (complexification) generators algebra to SU(2) SU(2). 10 / 14

Casimir Operators The group, as the group, preserve the line element in Minkowski space but now with the translations, : x µ x µ = Λ µ ν x ν + a µ (Λ 1, a 1 ) (Λ 2, a 2 ) = (Λ 2 Λ 1, Λ 2 a 1 + a 2 ) The group has 10 parameters: 6 from a and 4 from translations a 0 a 1 η(a, Λ) = Λ a 2 a 3 0 0 0 0 1 11 / 14

Generators Casimir Operators The group is represented by the algebra [V µν, V ρσ ] = i (V µσ η νρ + V νρ η µσ V µρ η νσ V νσ η µρ ) [P µ, P ν ] = 0 [V µν, P σ ] = i (P µ η νσ P ν η µσ ) 12 / 14

Representations group Casimir Operators Wigner, on a paper 1939, gave us a method to classify all reducible representations group by considering the little group of a quantum eigenstate with definite momentum P 2. In the literature is called: Classification of representations by orbits. P 0 P 2 P µ Little m m 2 (m, 0, 0, 0) SO(3) m m 2 ( m, 0, 0, 0) SO(3) P 0 ( P, 0, 0, P ) E(2) P 0 ( P, 0, 0, P ) E(2) 0 m 2 (0, 0, 0, m) SO(2, 1) 0 0 (0, 0, 0, 0) SO(3, 1) 13 / 14

Casimir Operators Casimir Operators Finally, there are only two invariants in the group that commute with all generators. The first Casimir invariant C 1 is associated with the mass invariance, and the second Casimir invariant C 2 refers to spin invariance. C 1 = P µ P µ C 2 = W µ W µ Where W µ is the Pauli-Lubanski (pseudo) vector defined by W µ = 1 2 ɛ µνρσp ν V ρσ A particle with fix energy m, s; P µ, σ 14 / 14