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Unformatted text preview: 2 Classical Field Theory In what follows we will consider rather general field theories. The only guiding principles that we will use in constructing these theories are (a) symmetries and (b) a generalized Least Action Principle. 2.1 Relativistic Invariance Before we saw three examples of relativistic wave equations. They are Maxwell’s equations for classical electromagnetism, the KleinGordon and Dirac equations. Maxwell’s equations govern the dynamics of a vector field, the vector potentials A μ ( x ) = ( A , vector A ), whereas the KleinGordon equation describes excitations of a scalar field φ ( x ) and the Dirac equation governs the behavior of the four component spinor field ψ α ( x )( α = 0 , 1 , 2 , 3). Each one of these fields transforms in a very definite way under the group of Lorentz transformations or Lorentz group. The Lorentz group is defined as a group of linear transformations Λ of Minkowski spacetime M onto itself Λ : M → M such that x ′ μ = Λ μ ν x ν (1) The spacetime components of Λ are the Lorentz boosts which relate inertial reference frames moving at relative velocity vectorv . Thus, Lorentz boosts along the x 1axis have the familiar form x ′ = x + vx 1 /c radicalbig 1 − v 2 /c 2 x 1 ′ = x 1 + vx /c radicalbig 1 − v 2 /c 2 x 2 ′ = x 2 x 3 ′ = x 3 (2) where x = ct , x 1 = x , x 2 = y and x 3 = z (note: these are components, not powers!). If we use the notation γ = (1 − v 2 /c 2 ) − 1 / 2 ≡ cosh α , we can write the Lorentz boost as a matrix: x ′ x 1 ′ x 2 ′ x 3 ′ = cosh α sinh α sinh α cosh α 1 1 x x 1 x 2 x 3 (3) The space components of Λ are conventional threedimensional rotations R . Infinitesimal Lorentz transformations are generated by the hermitian oper ators L μν = i ( x μ ∂ ν − x ν ∂ μ ) (4) 1 where ∂ μ = ∂ ∂x μ . They satisfy the algebra [ L μν ,L ρσ ] = ig νρ L μσ − ig μρ L νσ − ig νσ L μρ + ig μσ L νρ (5) where g μν is the metric tensor for Minkowski spacetime (see below). This is the algebra of the group SO (3 , 1). Actually any operator of the form M μν = L μν + S μν (6) where S μν are 4 × 4 matrices satisfying the algebra of Eq. 5 satisfies the same algebra. Below we will discuss explicit examples. Lorentz transformations form a group , since (a) the product of two Lorentz transformations is a Lorentz transformation, (b) there exists an identity trans formation, and (c) Lorentz transformations are invertible. Notice, however, that in general two transformations do not commute with each other. Hence, the lorentz group is nonAbelian....
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This note was uploaded on 09/20/2010 for the course PHYS 582 taught by Professor Leigh during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Leigh

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