Introductory Nuclear Physics

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PHY431 Lecture 9 8 4/30/2000 11.4 The Complex Scalar Field A complex scalar field can be likened to a simultaneous description of two real scalar fields describing particles of precisely the same mass m . We will consider the symmetry that exists in the Lagrangian for such two equal-mass fields: 2 2 2 2 1 1 1 2 2 2 1 1 ( ) ( ) 2 2 L m m µ µ µ µ φ φ φ φ φ φ = + (11.27) We could have started with a different set of fields φ 1 ’ and φ 2 ’, related to φ 1 and φ 2 by a rotation: 1 1 1 1 2 2 2 2 2 2 2 1 1 2 2 1 2 1 2 1 2 1 2 ' ' cos sin cos sin and ' ' sin cos sin cos 1 1 ( ) ( ) 2 2 1 term: ( 'cos 'sin ) ( 'cos 'sin ) ( 'sin 'cos ) st L m µ µ µ µ µ µ µ φ φ φ φ α α α α φ φ φ φ α α α α φ φ φ φ φ φ φ α φ α φ α φ α φ α φ α = = = + ∂ + + ∂ + 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 ( 'sin 'cos ) cos ' ' sin ' ' sin ' ' cos ' ' ' ' ' ' 2 term: ( 'cos 'sin ) ( 'sin 'cos ) ' ' nd µ µ µ µ µ µ µ µ µ µ µ µ µ φ α φ α α φ φ α φ φ α φ φ α φ φ φ φ φ φ φ φ φ α φ α φ α φ α φ φ + = = + + + = ∂ + ∂ + = + + = + (11.28) We can also write the Lagrangian in terms of a complex scalar field φ with the identification: φ = ( φ 1 + i φ 2 )/ 2. The rotation ( φ 1 , φ 2 ) into ( φ 1 ’, φ 2 ’) is equivalent to: ( ) 1 2 1 2 1 2 1 2 1 1 1 ' ( ' ') cos sin sin cos ( ) , 2 2 2 i i i i i e i e α α φ φ φ φ α φ α φ α φ α φ φ φ = + = + + = + = (11.29) where we used the Euler relation e i α = cos α + i sin α . The Lagrangian in terms of φ becomes: ( ) ( ) * * * 1 2 1 2 1 2 1 2 2 2 2 1 1 2 2 1 2 2 2 * * * * * * * * 1 1 1 ( ) with , real ( ) and: ( ), ( ) 2 2 2 2 1 ( ) ( ) 2 1 1 term: ( ) ( ) ( ) ( ) 2 2 1 ( ) ( ) ( ) ( ) 2 st i i i L m i µ µ µ µ µ µ µ µ µ µ µ µ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ = + = = + = = + ∂ + + + + = = + + − ∂ ( ) ( ) * 2 2 2 2 * 2 * 2 * 2 * 2 * 1 2 * 2 * 1 1 2 term: ( ) ( ) ( ) ( ) 2 2 2 1 2 nd i L m µ µ µ µ φ φ φ φ φ φ φ φ φ φ φ φ φφ φ φ φφ = ∂ + = + + = + = = (11.30) and this Lagrangian is invariant under φ φ = e i α φ . Invariance of the Lagrangian under such a sym- metry transformation requires the existence of a conserved Noether current . We will derive this from first principles. For ease of derivation we’ll consider small phase rotations e i α , with | α |<<1. Then:
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PHY431 Lecture 9 9 4/30/2000 * * * * * * * * ' (1 ) , with and: ' ( ) (1 ) , with i i i e i i e e i i α α α φ φ α φ φ δφ δφ αφ φ φ φ α φ φ δφ δφ αφ + = = + ≡ − = = + = + (11.31) Invariance of L requires: * * * * 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ( ) L L L L L L L L L L L L i µ µ µ µ µ µ µ µ µ µ µ µ µ µ δ δφ δ φ φ φ δφ δφ φ φ φ φ φ φ δφ δφ δφ φ φ φ φ φ δφ δφ φ φ φ φ φ α φ φ φ φ = = + + = + + = ∂ ∂ ∂ ∂
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