582-chapter2 - 2 Classical Field Theory In what follows we...

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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 Klein-Gordon and Dirac equations. Maxwell’s equations govern the dynamics of a vector field, the vector potentials A μ ( x ) = ( A , vector A ), whereas the Klein-Gordon 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 space-time M onto itself Λ : M → M such that x ′ μ = Λ μ ν x ν (1) The space-time components of Λ are the Lorentz boosts which relate inertial reference frames moving at relative velocity vectorv . Thus, Lorentz boosts along the x 1-axis 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 three-dimensional 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 space-time (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 non-Abelian....
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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.

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582-chapter2 - 2 Classical Field Theory In what follows we...

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