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: 22 11 1 2 2 2 () ( ) Lm m µµ φφ φ =∂∂− + ∂∂ − (11.27) We could have started with a different set of fields 1 ’ and 2 ’, related to 1 and 2 by a rotation: 1 1 2 2 2 1 2 12 '' cos sin cos sin and sin cos sin cos ( ) 1 term: ( 'cos 'sin ) ( 'cos 'sin ) ( 'sin 'cos ) st µ αα α φ φ φα    ==       =∂∂+ ∂∂ − + ∂− + ∂+ 222 2 2 2 1 2 1 2 1 2 (' s i n ' c o s) c o s ' 's i n ' i n ' 'c o s ' ' ' ' ' ' 2 term: ( 'cos 'sin ) ( 'sin 'cos ) ' ' nd µµµ µ += = ∂∂+ ∂∂= ∂∂ + + =+ (11.28) We can also write the Lagrangian in terms of a complex scalar field with the identification: = ( 1 + i 2 )/ 2. The rotation ( 1 , 2 ) into ( 1 ’, 2 ’) is equivalent to: 1 2 1 2 1 '( ' ' ) c o s s i n s i n c o s ( ) , 2 ii i e i e −− = + + = + = (11.29) where we used the Euler relation e i = cos + i sin . The Lagrangian in terms of becomes: ** * 1 2 1 2 2 1 2 1 ( ) with , real ( ) and: ( ), ( ) 2 2 1 ( ) 2 1 1 t e r m : 1 2 st i i =− = =∂ + + ∂++ ∂−= =∂ +∂ + − ∂ −∂ − * *2 * * 2 t e r m : ()() 2 1 2 nd i + = ++ −= +− = =∂∂ − (11.30) and this Lagrangian is invariant under = e i . 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 , with | |<<1. Then:
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PHY431 Lecture 9 9 4/30/2000 ** * * * * * * ' (1 ) , with and: ' ( ) ) , with i ii ei i ee i i α αα φφ φ δ δφ αφ −+ =≈ = + == + = + (11.31) Invariance of L requires: * * 0 ( )( ) ) () 0( LL L L L L i µµ µ ∂∂ + ∂+ →= + +  =+ + =    =− + + =   + * )0 ( ) i =∂ = (11.32) The final quantity in square brackets in equation (11.32) is a covariant fourvector, and is conserved: * * 0, with ( ) jj i q ∂= (11.33) where we added some constant “charge” in the definition of the current. 11.4.1 Current Conservation The four-current for the complex scalar field Lagrangian (11.30) becomes: j = iq [ * ( µ ) ( µ * ) ]. (11.34) This current has interesting properties: if the field represents a particle with mass m , then the field * represents another particle with exactly the same mass: it is natural to consider * to represent the anti- particle. Indeed, under the interchange ( * ), the four-current changes sign: j j indicating that the charge of the antiparticle must be opposite that of the particle. This true for generalized charge: i.e. for every additive quantum number: strangeness, charm, etc. as well as for electric charge.
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This note was uploaded on 02/05/2008 for the course PHY 431 taught by Professor Rijssenbeek during the Spring '01 term at SUNY Stony Brook.

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Lecture 09 - PHY431 Lecture 9 8 4/30/2000 11.4 The Complex...

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