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Unformatted text preview: 8 Coherent State Path Integral Quantization of Quantum Field Theory 8.1 Coherent states and path integral quantization. 8.1.1 Coherent States Let us consider a Hilbert space spanned by a complete set of harmonic oscillator states { n )} , with n = 0 ,..., . Let a and a be a pair of creation and annihi lation operators acting on that Hilbert space, and satisfying the commutation relations bracketleftbig a, a bracketrightbig = 1 , bracketleftbig a , a bracketrightbig = 0 , [ a, a ] = 0 (1) These operators generate the harmonic oscillators states { n )} in the usual way,  n ) = 1 n ! ( a ) n  ) (2) a  ) = 0 (3) where  ) is the vacuum state of the oscillator. Let us denote by  z ) the coherent state  z ) = e z a  ) (4) ( z  = (  e z a (5) where z is an arbitrary complex number and z is the complex conjugate. The coherent state  z ) has the defining property of being a wave packet with opti mal spread, i.e., the Heisenberg uncertainty inequality is an equality for these coherent states. How does a act on the coherent state  z ) ? a  z ) = summationdisplay n =0 z n n ! a ( a ) n  ) (6) Since bracketleftBig a, ( a ) n bracketrightBig = n ( a ) n 1 (7) we get a  z ) = summationdisplay n =0 z n n ! parenleftBigbracketleftBig a, ( a ) n bracketrightBig + ( a ) n a parenrightBig  ) (8) Thus, we find a  z ) = summationdisplay n =0 z n n ! n ( a ) n 1  ) z  z ) (9) Therefore  z ) is a right eigenvector of a and z is the (right) eigenvalue. 1 Likewise we get a  z ) = a summationdisplay n =0 z n n ! ( a ) n  ) = summationdisplay n =0 z n n ! ( a ) n +1  ) = summationdisplay n =0 ( n + 1) z n ( n + 1)! ( a ) n +1  ) = summationdisplay n =1 n z n 1 n ! ( a ) n  ) (10) Thus, a  z ) = z  z ) (11) Another quantity of interest is the overlap of two coherent states, ( z  z ) , ( z  z ) = (  e z a e z a  ) (12) We will calculate this matrix element using the BakerHausdorff formulas e A e B = e A + B + 1 2 bracketleftBig A, B bracketrightBig = e bracketleftBig A, B bracketrightBig e B e A (13) which holds provided the commutator bracketleftBig A, B bracketrightBig is a cnumber, i.e., it is propor tional to the identity operator. Since bracketleftbig a, a bracketrightbig = 1, we find ( z  z ) = e zz (  e z a e z a  ) (14) But e z a  ) =  ) (15) and (  e z a = (  (16) Hence we get ( z  z ) = e zz (17) An arbitrary state  ) of this Hilbert space can be expanded in the harmonic oscillator basis states { n )} ,  ) = summationdisplay n =0 n n !...
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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
 Quantum Field Theory

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