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The Lorentz and Poincaré Groups in Relativistic Field Theory

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The Lorentz and Poincaré Groups in Relativistic Field Theory Lorentz Group Poincar´e Group The Lorentz and Poincar´eGroups in Relativistic Field Theory Term Project Nicol´asFern´andezGonz´alez University of California { Santa Cruz June 2015 1 / 14 Using our group theory language the Lorentz group is just the indefinite orthogonal group O(1; 3) Lorentz Group Lorentz Group Lorentz Group Generators of the Lorentz Group Our first encounter with the Lorentz group is in special relativity it Boost and Rotations composed of the transformations that preserve the line element in Lie Algebra 2 µ ν µ 0µ µ ν of the Minkowski space (s = ηµν x x ). In particular, for x ! x = Λν x , Lorentz Group we get: Poincar´e Λµη Λν = η Group ρ µν σ ρσ Or in matrix notation Λ satisfy the following relation: Λ|ηΛ = η =) O|ηO = η Where η is the matrix: η = diag(1; −1; −1; −1) 2 / 14 Lorentz Group Lorentz Group Lorentz Group Generators of the Lorentz Group Our first encounter with the Lorentz group is in special relativity it Boost and Rotations composed of the transformations that preserve the line element in Lie Algebra 2 µ ν µ 0µ µ ν of the Minkowski space (s = ηµν x x ). In particular, for x ! x = Λν x , Lorentz Group we get: Poincar´e Λµη Λν = η Group ρ µν σ ρσ Or in matrix notation Λ satisfy the following relation: Λ|ηΛ = η =) O|ηO = η Where η is the matrix: η = diag(1; −1; −1; −1) Using our group theory language the Lorentz group is just the indefinite orthogonal group O(1; 3) 2 / 14 There are two discrete transformations P and T " " # # L = L+ [L− [L+ [L− " " " " = L+ [ P L+ [ T L+ [ PT L+ " We get the decomposition of L into cosets of the restricted group L+. 4 Islands, 2 Boats Lorentz Group Lorentz Group The Lorentz group consists of four separated components: Generators of the Lorentz Group " 0 Boost and L : det Λ = 1 and Λ ≥ 1. Rotations + 0 Lie Algebra of the " 0 Lorentz L : det Λ = −1 and Λ ≥ 1. Group − 0 Poincar´e Group # 0 L+: det Λ = 1 and Λ0 ≤ −1. # 0 L−: det Λ = −1 and Λ0 ≤ −1. The subgroup of the Lorentz group that exclude spatial reflections and time reversal is called the proper orthochronous Lorentz group SO+(1; 3). 3 / 14 4 Islands, 2 Boats Lorentz Group Lorentz Group The Lorentz group consists of four separated components: Generators of the Lorentz Group " 0 Boost and L : det Λ = 1 and Λ ≥ 1. Rotations + 0 Lie Algebra of the " 0 Lorentz L : det Λ = −1 and Λ ≥ 1. Group − 0 Poincar´e Group # 0 L+: det Λ = 1 and Λ0 ≤ −1. # 0 L−: det Λ = −1 and Λ0 ≤ −1. The subgroup of the Lorentz group that exclude spatial reflections and time reversal is called the proper orthochronous Lorentz group SO+(1; 3). There are two discrete transformations P and T " " # # L = L+ [L− [L+ [L− " " " " = L+ [ P L+ [ T L+ [ PT L+ " We get the decomposition of L into cosets of the restricted group L+. 3 / 14 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." Why not both? UCSC does not discriminate against metrics. Link: http://www.math.columbia.edu/~woit/wordpress/ The West Coast or East Coast? Lorentz Group Lorentz Usually we have two possible choices for the Minkowski metric: Group Generators of the Lorentz Group West Coast metric (1; −1; −1; −1). Feynman's favorite Boost and Rotations Lie Algebra East Coast metric (−1; 1; 1; 1). The choice of Weinberg and of the Lorentz Schwinger (also Ross and Joey?) Group Poincar´e Group 4 / 14 Why not both? UCSC does not discriminate against metrics. Link: http://www.math.columbia.edu/~woit/wordpress/ The West Coast or East Coast? Lorentz Group Lorentz Usually we have two possible choices for the Minkowski metric: Group Generators of the Lorentz Group West Coast metric (1; −1; −1; −1). Feynman's favorite Boost and Rotations Lie Algebra East Coast metric (−1; 1; 1; 1). The choice of Weinberg and of the Lorentz Schwinger (also Ross and Joey?) Group Poincar´e Group 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 Link: http://www.math.columbia.edu/~woit/wordpress/ The West Coast or East Coast? Lorentz Group Lorentz Usually we have two possible choices for the Minkowski metric: Group Generators of the Lorentz Group West Coast metric (1; −1; −1; −1). Feynman's favorite Boost and Rotations Lie Algebra East Coast metric (−1; 1; 1; 1). The choice of Weinberg and of the Lorentz Schwinger (also Ross and Joey?) Group Poincar´e Group 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." Why not both? UCSC does not discriminate against metrics. 4 / 14 The West Coast or East Coast? Lorentz Group Lorentz Usually we have two possible choices for the Minkowski metric: Group Generators of the Lorentz Group West Coast metric (1; −1; −1; −1). Feynman's favorite Boost and Rotations Lie Algebra East Coast metric (−1; 1; 1; 1). The choice of Weinberg and of the Lorentz Schwinger (also Ross and Joey?) Group Poincar´e Group 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." Why not both? UCSC does not discriminate against metrics. Link: http://www.math.columbia.edu/ woit/wordpress/ 4 / 14 ~ We already know these six parameters: 3 rotations about orthogonal directions in space and 3 boosts along these same directions. Generators of the Lorentz group Lorentz Group Lorentz Group µ Generators of An infinitesimally Lorentz transformation Λν should be of form, the Lorentz Group Boost and µ µ µ Rotations Λν = δν + !ν Lie Algebra of the Lorentz µ Group Where with !ν is a matrix of infinitesimal coefficients. Plugging in our 2 Poincar´e definition of Lorentz group and keeping O(! ), we have: Group µ ν ηρσ = Λρ ηµν Λσ µ µ ν ν ηρσ = δρ + !ρ ηµν (δσ + !σ) ηρσ = ηρσ + !ρσ + !σρ 0 = !ρσ + !σρ We conclude that !ρσ = −!σρ Thus the generators ! are 4 × 4 antisymmetric matrices characterized by six parameters so that there are six generators. 5 / 14 Generators of the Lorentz group Lorentz Group Lorentz Group µ Generators of An infinitesimally Lorentz transformation Λν should be of form, the Lorentz Group Boost and µ µ µ Rotations Λν = δν + !ν Lie Algebra of the Lorentz µ Group Where with !ν is a matrix of infinitesimal coefficients. Plugging in our 2 Poincar´e definition of Lorentz group and keeping O(! ), we have: Group µ ν ηρσ = Λρ ηµν Λσ µ µ ν ν ηρσ = δρ + !ρ ηµν (δσ + !σ) ηρσ = ηρσ + !ρσ + !σρ 0 = !ρσ + !σρ We conclude that !ρσ = −!σρ Thus the generators ! are 4 × 4 antisymmetric matrices characterized by six parameters so that there are six generators. We already know these six parameters: 3 rotations about orthogonal directions in space and 3 boosts along these same directions. 5 / 14 Infinitesimal rotation for x,y and z: 00 1 00 1 00 1 B 0 C B 0 1C B 0 −1 C J1 = i B C J2 = i B C J3 = i B C @ 0 −1A @ 0 A @ 1 0 A 1 0 −1 0 0 Infinitesimal boost: 00 −1 1 0 0 −1 1 0 0 −11 B1 0 C B 0 C B 0 C K1 = i B C K2 = i B C K3 = i B C @ 0 A @−1 0 A @ 0 A 0 0 −1 0 These matrices are the generators of the Lorentz Group (4-vector basis) Boost and Rotations Lorentz Group Lorentz Group Generators of the Lorentz The rotations can be parametrized by a 3-component vector θi with Group Boost and jθij ≤ π, and the boosts by a three component vector βi (rapidity) Rotations Lie Algebra with jβj < 1. Taking a infinitesimal transformation we have that: of the Lorentz Group Poincar´e Group 6 / 14 Infinitesimal boost: 00 −1 1 0 0 −1 1 0 0 −11 B1 0 C B 0 C B 0 C K1 = i B C K2 = i B C K3 = i B C @ 0 A @−1 0 A @ 0 A 0 0 −1 0 These matrices are the generators of the Lorentz Group (4-vector basis) Boost and Rotations Lorentz Group Lorentz Group Generators of the Lorentz The rotations can be parametrized by a 3-component vector θi with Group Boost and jθij ≤ π, and the boosts by a three component vector βi (rapidity) Rotations Lie Algebra with jβj < 1. Taking a infinitesimal transformation we have that: of the Lorentz Group Infinitesimal rotation for x,y and z: Poincar´e Group 00 1 00 1 00 1 B 0 C B 0 1C B 0 −1 C J1 = i B C J2 = i B C J3 = i B C @ 0 −1A @ 0 A @ 1 0 A 1 0 −1 0 0 6 / 14 Boost and Rotations Lorentz Group Lorentz Group Generators of the Lorentz The rotations can be parametrized by a 3-component vector θi with Group Boost and jθij ≤ π, and the boosts by a three component vector βi (rapidity) Rotations Lie Algebra with jβj < 1.
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