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8 Coherent State Path Integral Quantization of Quantum Field Theory

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8 Coherent State Path Integral Quantization of Quantum Field Theory 8 Coherent State Path Integral Quantization of Quantum Field Theory 8.1 Coherent states and path integral quantization. The path integral that we have used so far is a powerful tool butitcan only be used in theories based on canonical quantization. Forexample,it cannot be used in theories of fermions (relativistic or not),amongothers. In this chapter we discuss a more general approach based on theconcept of coherent states. Coherent states, and their application to path integrals, have been widely discussed. Excellent references include the 1975 lectures by Faddeev (Faddeev, 1976), and the books by Perelomov (Perelomov, 1986), Klauder (Klauder and Skagerstam, 1985), and Schulman (Schulman, 1981). The related concept of geometric quantization is insightfully presented in the work by Wiegmann (Wiegmann, 1989). 8.2 Coherent states Let us consider a Hilbert space spanned by a complete set of harmonic oscillator states n ,withn = 0,...,∞.Letˆa† anda ˆ be a pair of creation and annihilation operators acting on this Hilbert space, andsatisfyingthe commutation relations{∣ ⟩} † † † a,ˆ aˆ = 1 , aˆ , aˆ = 0 , a,ˆ aˆ = 0(8.1) These operators generate# $ the harmonic# $ oscillators[ states] n in the usual way, {∣ ⟩} 1 † n n = aˆ 0 , aˆ 0 = 0(8.2) n! where 0 is the vacuum∣ ⟩ state% of& the' oscillator.∣ ⟩ ∣ ⟩ ∣ ⟩ 8.2 Coherent states 215 Let us denote by z the coherent state zaˆ† z¯aˆ ∣ ⟩z = e 0 , z = 0 e (8.3) where z is an arbitrary complex number andz ¯ is the complex conjugate. ∣ ⟩ ∣ ⟩ ⟨ ∣ ⟨ ∣ The coherent state z has the defining property of being a wave packet with optimal spread, i.e. the Heisenberg uncertainty inequalityisanequalityfor these coherent states.∣ ⟩ How doesa ˆ act on the coherent state z ? ∞ n z † n aˆ z = aˆ∣ aˆ⟩ 0 (8.4) ! n! n=0 Since ∣ ⟩ & ' ∣ ⟩ † n † n−1 a,ˆ aˆ = n aˆ , (8.5) we find # ∞& n' $ & ' z † n−1 aˆ z = n aˆ 0 ≡ z z (8.6) ! n! n=0 Therefore z is a right∣ ⟩ eigenvector of& a ˆ'and ∣z ⟩is the∣ (right)⟩ eigenvalue. Likewise we get ∣ ⟩ ∞ n ∞ n−1 † † z † n z † n aˆ z = aˆ aˆ 0 = n aˆ 0 (8.7) ! n! ! n! n=0 n=1 Thus, ∣ ⟩ & ' ∣ ⟩ & ' ∣ ⟩ † aˆ z = ∂z z (8.8) Therefore the operators z and ∂ provide a representation of the algebra of z∣ ⟩ ∣ ⟩ creation and annihilation operators. ′ Another quantity of interest is the overlap of two coherent states, z z , ′ ′ z¯aˆ z aˆ† z z = 0 e e 0 ⟨ (8.9)∣ ⟩ We will calculate this matrix element using the Baker-Hausdorffformulas ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ Aˆ Bˆ Aˆ+Bˆ+ 1 A,ˆ Bˆ A,ˆ Bˆ Bˆ Aˆ e e = e 2 = e e e (8.10) # $ # $ which holds provided the commutator A,ˆ Bˆ is a c-number, i.e. it is pro- portional to the identity operator. Since a,ˆ aˆ† = 1, we find # $ ′ ′ ′ zz z a† za z z = e¯ 0 e# ˆ e¯$ˆ 0 (8.11) ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ 216 Coherent State Path Integral Quantization of Quantum Field Theory But ′ za z a† e¯ˆ 0 = 0 , 0 e ˆ = 0 (8.12) Hence we get ∣ ⟩ ∣ ⟩ ⟨ ∣ ⟨ ∣ ′ ′ zz z z = e¯ (8.13) An arbitrary state ψ of this Hilbert space can be expanded in the harmonic ⟨ ∣ ⟩ oscillator basis states n , ∣ ⟩ ∞ ∞ ψ ψ † n ψ {=∣ ⟩} n n = n aˆ 0 (8.14) ! ! n! n=0 n! n=0 The projection of the∣ ⟩ state ψ% onto∣ ⟩ the coherent& state' ∣ ⟩z is ∞ ψ † n z ψ∣ =⟩ n z aˆ 0 ∣ ⟩ (8.15) ! n! n=0 Since ⟨ ∣ ⟩ ⟨ ∣& ' ∣ ⟩ † z aˆ = z¯ z (8.16) we find ⟨ ∣∞ ⟨ ∣ ψ n z ψ = n z¯ ≡ ψ z¯ (8.17) ! n! n=0 Therefore the projection⟨ of∣ ψ⟩ onto z is the(anti-holomorphic) (i.e. anti- analytic)functionψ z¯ .Inotherwords,inthisrepresentation,thespaceof states ψ are in one-to-one∣ correspondence⟩ ∣ ⟩ with the space of anti-analytic functions. ( ) In summary,{∣ ⟩} the coherent states z satisfy the following properties aˆ z = z z z aˆ = ∂z¯ z † {∣ ⟩} † aˆ z = ∂z z z aˆ = z¯ z (8.18) z∣ψ⟩ = ψ∣ z¯⟩⟨ψ∣z = ψ¯⟨z∣ ∣ ⟩ ∣ ⟩⟨∣ ⟨ ∣ Next we will prove the resolution of identity ⟨ ∣ ⟩ ( )⟨∣ ⟩ ( ) dzdz¯ −zz¯ Iˆ = e z z (8.19) " 2πi Let ψ and φ be two arbitrary states ∣ ⟩⟨ ∣ ∞ ∞ ψ φ ∣ ⟩ ∣ ⟩ ψ = n n , φ = n n (8.20) ! ! n=0 n! n=0 n! ∣ ⟩ % ∣ ⟩ ∣ ⟩ % ∣ ⟩ 8.2 Coherent states 217 such that their inner product is ∞ 1 φ ψ = φ¯ ψ (8.21) ! n! n n n=0 Let us compute the matrix⟨ element∣ ⟩ of the operator Iˆ given in Eq.(8.19), φ¯ ψ φ Iˆ ψ = n n n Iˆ m (8.22) ! n! m.n Thus we need to compute⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ dzdz¯ − z 2 n Iˆ m = e n z z m (8.23) " 2πi ∣ ∣ Recall that the integration⟨ ∣ ∣ ⟩ measure is defined⟨ to∣ ⟩⟨ be∣ given⟩ by dzdz¯ dRezdImz = (8.24) 2πi π The overlaps are given by n 1 n z n z = 0 aˆ z = 0 z (8.25) n! n! and ⟨ ∣ ⟩ % ⟨ ∣( ) ∣ ⟩ % ⟨ ∣ ⟩ m 1 † m z¯ z m = z aˆ 0 = z 0 (8.26) m! m! Now, since 0 z 2⟨ =∣ 1,⟩ we% get ⟨ ∣& ' ∣ ⟩ % ⟨ ∣ ⟩ − z 2 ∣⟨ ∣ ⟩∣ dzdz¯ e n m n Iˆ m = z z¯ " 2πi n!m! ∣ ∣ 2 ∞ 2π −ρ ⟨ ∣ ∣ ⟩ % dϕ e n+m i n − m ϕ = ρdρ ρ e (8.27) "0 "0 2π n!m! ( ) Thus, % ∞ δn,m n −x n Iˆ m = dx x e = n m (8.28) n! "0 Hence, we have found⟨ ∣ ∣ that⟩ ⟨ ∣ ⟩ φ Iˆ ψ = φ ψ (8.29) for any pair of states ψ and φ .ThereforeIˆ is the identity operator in ⟨ ∣ ∣ ⟩ ⟨ ∣ ⟩ this Hilbert space. We conclude that the set of coherent states z is an over-complete set of states.∣ ⟩ ∣ ⟩ {∣ ⟩} 218 Coherent State Path Integral Quantization of Quantum Field Theory Furthermore, since ′ † n m ′ n ′m ′ n ′m zz¯ z aˆ aˆ z = z¯ z z z = z¯ z e (8.30) ′ we conclude that⟨ the∣& matrix' ( ) elements∣ ⟩ in generic⟨ ∣ ⟩ coherent states z and z of any arbitrary normal ordered operator of the form † n m ∣ ⟩ ∣ ⟩ Aˆ = A aˆ aˆ (8.31) ! n,m n,m are equal to & ' ( ) ′ ′ n ′m zz¯ z Aˆ z = A z¯ z e (8.32) ! n,m n,m ⟨ ∣ ∣ ⟩ ) * Therefore, if Aˆ a,ˆ aˆ† is an arbitrary normal ordered operator (relative to the state 0 ), its matrix elements are given by ( ) ′ † ′ ′ zz¯ ∣ ⟩ z Aˆ a,ˆ aˆ z = A z,z¯ e (8.33) ′ ′ where A z,z¯ is a function of two complex variablesz ¯ and z obtained from ⟨ ∣ ( )∣ ⟩ ( ) Aˆ by the formal replacement ( ) ′ † aˆ ↔ z , aˆ ↔ z¯ (8.34) ′ The complex function A z,z¯ is oftentimes called the symbol of the (normal- ordered) operator Aˆ. † For example, the matrix( elements) of the number operator Nˆ = aˆ aˆ,which measures the number of excitations, is ′ ′ † ′ ′ zz¯ z Nˆ z = z aˆ aˆ z = zz¯ e (8.35) ⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ 8.3 Path integrals and coherent states The concept of coherent states have been applied to broad areas of Quantum Mechanics (Perelomov, 1986; Klauder and Skagerstam, 1985).Herewewill focus on its application to path integrals (Faddeev, 1976). We want to compute the matrix elements of the evolution operator U, T † −i Hˆ aˆ , aˆ U = e h̵ (8.36) † where Hˆ aˆ , aˆ is a normal ordered operator( and) T = tf −ti is the total time ( ) 8.3 Path integrals and coherent states 219 lapse. Thus, if i and f denote two arbitrary initial and final states, we can write the matrix element of U as ∣ ⟩ ∣ ⟩ T † −i Hˆ a , a N ˆ ˆ ( ˆ † f e h̵ i = lim f 1 − i H aˆ , aˆ i (8.37) "→0,N→∞ h̵ ( ) However,⟨ ∣ instead of inserting∣ ⟩ a complete⟨ ∣+ set of states( at), eachintermediate∣ ⟩ time tj (with j = 1,...,N), we will now insert an over-complete set of coherent states zj at each time tj through the insertion of the resolution of the identity, {∣ ⟩} ( † N f 1 − i Hˆ aˆ , aˆ i = h̵ N ⟨ ∣+ ( ), ∣ ⟩ 2 N − zj N−1 dzj dz¯j ! ( † = e j=1 z 1 − i Hˆ aˆ , aˆ z " $ 2πi $ k+1 h k j=1 ∣ ∣ k=1 ̵ + , ( ˆ †- ⟨ ∣+ ((ˆ †), ∣ ⟩. × f 1 − i H aˆ , aˆ zN z1 1 − i H aˆ , aˆ zi h̵ h̵ (8.38) ⟨ ∣+ ( ), ∣ ⟩⟨ ∣+ ( ), ∣ ⟩ In the limit ( → 0thesematrixelementsbecome ( † ( † z 1 − i Hˆ aˆ , aˆ z = z z − i z Hˆ aˆ , aˆ z k+1 h k k+1 k h k+1 k ̵ ̵ ( = z z 1 − i H z ,z ⟨ ∣+ ( ), ∣ ⟩ ⟨ k+1∣ k⟩ ⟨ ∣¯k+(1 k )∣ ⟩ (8.39) h̵ where H z¯k+1,zk is a function obtained⟨ ∣ from⟩/ the normal-ordered( )0 Hamilto- † nian by performing the substitutionsa ˆ → z¯k+1 anda ˆ → zk. Hence,( we can write) the following expression for the matrix element of the evolution operator of the form T † −i Hˆ aˆ , aˆ f e h̵ i = N N−1 ( ) 2 ⟨ ∣ ∣ ⟩ − z z¯ + z N dz dz ! j ! j 1 j N−1 j ¯j j=1 j=1 ( = lim e e 1 − i H z¯k+1,zk "→0,N→∞ " $ 2πi $ h j=1 ∣ ∣ j=1 ̵ ⎛ ⎞ ⎜ ( f ⎟Hˆ zN ( z1 Hˆ i / ( )0 × f zN z1 i⎝ 1 − i ⎠ 1 − i (8.40) h̵ f zN h̵ z1 i ⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ ⟨ ∣ ⟩⟨ ∣ ⟩- .- . ⟨ ∣ ⟩ ⟨ ∣ ⟩ 220 Coherent State Path Integral Quantization of Quantum Field Theory By further expanding the initial and final states in coherent states 2 dzf dz¯f − z f = e f ψ¯ z z " 2πi f f f 2 dzidz¯i − z∣ ∣ ⟨ i∣ = e i ψ z¯( z)⟨ ∣ (8.41) " 2πi i i i ∣ ∣ we find the (formal) result∣ ⟩ ( )∣ ⟩ T † −i Hˆ aˆ , aˆ f e h̵ i = i tf h ( ) dt ̵ z∂ z¯ − z¯∂ z − H z,z¯ ⟨ ∣ ∣ ⟩ h " 2i t t = DzDze¯ ̵ ti " 7 ( ) ( )8 1 2 2 zi + zf × e2 ψ¯f zf ψi z¯i (8.42) (∣ ∣ ∣ ∣ ) This is the coherent-state form of the path integral.( We) ( can) identify in this expression the Lagrangian L as the quantity h L = ̵ z∂ z¯ − z¯∂ z − H z,z¯ (8.43) 2i t t It is easy to check that this expression( is equivalent) ( to) the phase-space path integral derived in Section 5.1.
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