1 Coherent states and path integral quantiza- tion. 1.1 Coherent States Let q and p be the coordinate and momentum operators. They satisfy the Heisenberg algebra, [q,p] = i~. Let us introduce the creation and annihilation operators a† and a, by their standard relations ~ q = a† + a (1) r2mω m~ω p = i a† a (2) r 2 − Let us consider a Hilbert space spanned by a complete set of harmonic os- cillator states n , with n =0,..., . Leta ˆ† anda ˆ be a pair of creation and annihilation operators{| i} acting on that∞ Hilbert space, and satisfying the commu- tation relations a,ˆ aˆ† =1 , aˆ†, aˆ† =0 , [ˆa, aˆ] = 0 (3) These operators generate the harmonic oscillators states n in the usual way, {| i} 1 n n = aˆ† 0 (4) | i √n! | i aˆ 0 = 0 (5) | i where 0 is the vacuum state of the oscillator. Let| usi denote by z the coherent state | i † z = ezaˆ 0 (6) | i | i z = 0 ez¯aˆ (7) h | h | 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 opti- mal spread, i.e.,| ithe Heisenberg uncertainty inequality is an equality for these coherent states. How doesa ˆ act on the coherent state z ? | i ∞ n z n aˆ z = aˆ aˆ† 0 (8) | i n! | i n=0 X Since n n−1 a,ˆ aˆ† = n aˆ† (9) we get h i ∞ n z n n aˆ z = a,ˆ aˆ† + aˆ† aˆ 0 (10) | i n! | i n=0 X h i 1 Thus, we find ∞ n z n−1 aˆ z = n aˆ† 0 z z (11) | i n! | i≡ | i n=0 X Therefore z is a right eigenvector ofa ˆ and z is the (right) eigenvalue. Likewise| i we get ∞ n z n aˆ† z =a ˆ† aˆ† 0 | i n! | i n=0 X ∞ n z n+1 = aˆ† 0 n! | i n=0 X ∞ n z n+1 = (n + 1) aˆ† 0 (n + 1)! | i n=0 X ∞ n−1 z n = n aˆ† 0 n! | i n=1 X (12) Thus, ∂ aˆ† z = z (13) | i ∂z | i Another quantity of interest is the overlap of two coherent states, z z′ , h | i ′ † z z′ = 0 ez¯aˆ ez aˆ 0 (14) h | i h | | i We will calculate this matrix element using the Baker-Hausdorff formulas 1 ˆ ˆ Aˆ + Bˆ + A,ˆ Bˆ A,ˆ Bˆ ˆ ˆ eA eB = e 2 = e eB eA (15) h i h i which holds provided the commutator A,ˆ Bˆ is a c-number, i.e., it is propor- tional to the identity operator. Since a,ˆh aˆ† =i 1, we find ′ ′ † z z′ = ezz¯ 0 ez aˆ ez¯aˆ 0 (16) h | i h | | i But ez¯aˆ 0 = 0 (17) | i | i and ′ † 0 ez aˆ = 0 (18) h | h | Hence we get ′ z z′ = ezz¯ (19) h | i 2 An arbitrary state ψ of this Hilbert space can be expanded in the harmonic oscillator basis states| i n , {| i} ∞ ∞ ψ ψ n ψ = n n = n aˆ† 0 (20) | i √ | i n! | i n=0 n! n=0 X X The projection of the state ψ onto the coherent state z is | i | i ∞ ψ n z ψ = n z aˆ† 0 (21) h | i n! h | | i n=0 X Since z aˆ† =z ¯ z (22) h | h | we find ∞ ψ z ψ = n z¯n ψ(¯z) (23) h | i n! ≡ n=0 X Therefore the projection of ψ onto z is the anti-holomorphic (i.e., anti- analytic) function ψ(¯z). In other| i words,| i in this representation, the space of states ψ are in one-to-one correspondence with the space of anti-analytic functions.{| i} In summary, the coherent states z satisfy {| i} aˆ z = z z z aˆ = ∂z¯ z †| i | i h | † h | aˆ z = ∂z z z aˆ =z ¯ z (24) z|ψi = ψ(¯|z)i hψ|z = ψ¯h(z|) h | i h | i Next we will prove the resolution of identity dzdz¯ Iˆ = e zz¯ z z (25) 2πi − | ih | Z Let ψ and φ be two arbitrary states | i | i ∞ ψ ψ = n n | i √ | i n=0 n! X∞ ψ ψ = n n | i √ | i n=0 n! X∞ φ ψ φ ψ = n n (26) h | i n! n=0 X Let us compute the matrix element φ¯ ψ φ Iˆ ψ = n n n Iˆ m (27) n! h | | i m.n h | | i X 3 Thus we need to find dzdz¯ 2 n Iˆ m = e z n z z m (28) h | | i 2πi −| | h | ih | i Z Recall that the integration measure is defined to be given by dzdz¯ dRezdImz = (29) 2πi π where 1 zn n z = 0 (ˆa)n z = 0 z (30) h | i √n!h | | i √n!h | i and m 1 m z¯ z m = z aˆ† 0 = z 0 (31) h | i √m!h | | i √m!h | i Now, since 0 z 2 = 1, we get |h | i| z 2 ∞ 2π ρ2 ˆ dzdz¯ e−| | n m dϕ e− n+m i(n m)ϕ n I m = z z¯ = ρdρ ρ e − h | | i 2πi √n!m! 0 0 2π √n!m! Z Z Z (32) Thus, δ ∞ n Iˆ m = n,m dx xne x = n m (33) h | | i n! − h | i Z0 Hence, we have found that φ Iˆ ψ = φ ψ (34) h | | i h | i for any pair of states ψ and φ . Therefore Iˆ is the identity operator in that space. We conclude that| i the set| ofi coherent states z is an over-complete set of states. {| i} Furthermore, since n ′ z aˆ† (ˆa)m z′ =z ¯nz′m z z′ =z ¯nz′mezz¯ (35) h | | i h | i we conclude that the matrix elements of any arbitrary normal ordered operator of the form † n m Aˆ = An,m aˆ (ˆa) (36) n,m X are equal to ′ ′ n ′m zz¯ z Aˆ z = An,mz¯ z e (37) h | | i n,m ! X Therefore, if Aˆ(ˆa, aˆ†) is an arbitrary normal ordered operator (relative to the state 0 ), its matrix elements are given by | i ′ z Aˆ(ˆa, aˆ†) z′ = A(¯z,z′)ezz¯ (38) h | | i 4 where A(¯z,z′) is a function of two complex variablesz ¯ and z′, obtained from Aˆ by the formal replacement aˆ z′ , aˆ† z¯ (39) ↔ ↔ For example, the matrix elements of the the operator Nˆ =ˆa†aˆ, which measures the number of excitations, is ′ z Nˆ z′ = z aˆ†aˆ z′ =zz ¯ ′ ezz¯ (40) h | | i h | | i 1.2 Path Integrals and Coherent States As usual we will want to compute the matrix elements of the evolution operator , U T i Hˆ (ˆa†, aˆ) = e− ~ (41) U where Hˆ (ˆa†, aˆ) is a normal ordered operator. Thus, if i and f denote two arbitrary initial and final states, we can write the matrix| i element| i of as U T i Hˆ (ˆa†, aˆ) ǫ N f e− ~ i = lim f 1 i Hˆ (ˆa†, aˆ) i (42) h | | i ǫ→0,N→∞ h | −− ~ | i However now, instead of inserting a complete set of states at each intermediate time tj (with j = 1,...,N), we will insert an over-complete set zj at each {| i} time tj through the insertion of the resolution of the identity, ǫ N f 1 i Hˆ (ˆa†, aˆ) i = h | − ~ | i N 2 zj N − | | dzj dz¯j j=1 ǫ † = e f 1 i Hˆ (ˆa , aˆ) zN 2πi X h | − ~ | i j=1 Z Y ǫ † ǫ † zN 1 i Hˆ (ˆa , aˆ) zN− ... z 1 i Hˆ (ˆa , aˆ) i = ×h | − ~ | 1i h 1| − ~ | i N 2 zj N − | | N−1 dzj dz¯j j=1 ǫ † e zk 1 i Hˆ (ˆa , aˆ) zk ≡ 2πi X h +1| − ~ | i j=1 " k # Z Y Y=1 ǫ † ǫ † f 1 i Hˆ (ˆa , aˆ) zN z 1 i Hˆ (ˆa , aˆ) zi ×h | − ~ | ih 1| − ~ | i (43) In the limit ǫ 0 these matrix elements are → ǫ † ǫ † zk 1 i Hˆ (ˆa , aˆ) zk = zk zk i zk Hˆ (ˆa , aˆ) zk h +1| − ~ | i h +1| i− ~h +1| | i ǫ = zk zk 1 i H(¯zk ,zk) h +1| i − ~ +1 h i (44) 5 where H(¯zk+1,zk) is a function which is obtained from the normal ordered † Hamiltonian by the substitutionsa ˆ z¯k+1 anda ˆ zk. Hence, we can write the following expression for the matrix→ element → T i Hˆ (ˆa†, aˆ) f e− ~ i = h | | i N N−1 2 zj z¯j zj N − | | +1 N−1 dzj dz¯j j=1 j=1 ǫ = lim e X e X 1 i H(¯zk+1,zk) ǫ→0,N→∞ 2πi − ~ j=1 j=1 Z Y Y h i ǫ f Hˆ zN ǫ z1 Hˆ i f zN z1 i 1 i h | | i 1 i h | | i ×h | ih | i − ~ f zN − ~ z i " h | i # " h 1| i # (45) By further expanding the initial and final states in coherent states 2 dzf dz¯f zf f = e ψ¯f (zf ) zf h | 2πi −| | h | Z 2 dzidz¯i zi i = e ψi(¯zi) zi | i 2πi −| | | i Z (46) we find T i Hˆ (ˆa†, aˆ) f e− ~ i = h | | i i tf ~ dt (z∂ z¯ z∂¯ z) H(z, z¯) 1 2 2 ~ t t ( zi + zf ) ti 2i − − ¯ = z z¯ e e2 | | | | ψf (zf )ψi(¯zi) D D Z Z (47) This is the coherent-state form of the path integral.