University:
Massachusetts Institute of Technology
Course:
8.322 | Quantum Theory II
Academic year:

2021
Views:

342

Pages:

34
Author:

Accinkiquicpqw

Rel. QM 7.1 Basics of 𝝰 Space-time coordinates: x^μ, μ = 0, 1, 2, 3 x^μ = (ct, x, y, z) Minkowski metric: S² = η_μνx^μx^ν = c²t² - |x|² η_μν = (1 0 0 0) (0 -1 0 0) (0 0 -1 0) (0 0 0 -1) Lorentz x-forms: x'^μ = Λ^μ_νx^ν where Λ preserves S² η_μνΛ^μ_αΛ^ν_β = η_αβ Satisfy Group of Λ's with (ΛΛ' = Λ'') Lorentz group for Λ ∈ 𝔏 (det Λ = ±1) proper (det Λ = +1) improper (det Λ = -1) Includes: Rotations Ex. not around x? (cosθ sinθ 0 0) (-sinθ cosθ 0 0) (0 0 1 0) (0 0 0 1) Boosts Ex. boost along x: Λ = (γ -βγ 0 0) (-βγ γ 0 0) (0 0 1 0) (0 0 0 1) with γ = 1/√(1 - β²) 23.2 Improper: spatial inversion 4-vectors: 4-component objects transform under Λ^μ_ν as (U')^μ = Λ^μ_νU^ν Ex. momentum: P^μ = (E/c, P) P^μP_μ = E²/c² - |P|² = m²c² P'^μ = Λ^μ_νP^ν P'^μP'_μ = Λ^μ_αΛ^ν_βP^αP^β = η_αβP^αP^β = P^μP_μ = m²c² Tensors: Ex. F^μν = ∂_μA_ν - ∂_νA_μ F^μν = (0 E¹ E² E³) (-E¹ 0 B³ -B²) (-E² -B³ 0 B¹) (-E³ B² -B¹ 0 ) Products of γ matrices Can form 2^4 = 16 matrices from products of the γ's 1, γ^μ, γ^μγ^ν, γ^μγ^νγ^λ, γ^μγ^νγ^λγ^ρ 1, 4, 6, 4, 1 Denote Γ_n, n = 1, ..., 16 Any 4x4 matrix can be written as M = Σc_nΓ_n (sum i = 1 to 16) Pauli's Fundamental Theorem Given any 2 sets of 4x4 matrices, Γ_μ, Γ'_μ, satisfying Σ_μΓ_μΓ'_μ = c₁γ^μΓ_μ + c₂I Σ_μΓ_μΓ'_μ = c₁γ^μΓ_μ - c₂I ∃ non-singular 4x4 matrix S so that Γ'_μ = SΓ_μS^-1 So, only one Dirac representation. Above is "Pauli-Dirac" representation. Above is "Pauli-Dirac" representation. Most useful when kinetic energy is low. (Weyl rep: 1/2(1 ± γ5)γ^μ)

Relativistic Quantum Mechanics

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock