NO51.51.5CoherentstatesCoherent states aretheeigenstatesof theannihilationoperators(1.94)[) = [)Note that one cannot define the eigenstates of the creation operators.For Fermions,the eigenvalues cannot be ordinary numbers because annihilation operators anti-commute.TheymustbeGRASSMANNNUMBERS:(1.95)+a=01.5.1 Boson coherent statesDefinition[0) = ea baa, J0).(1.96)Overlap() = (o|e)=e (01)=exp(0*),(1.97)Notethatcoherent statesdonotforman orthonormalbasis.Instead.theyform anover-complete basis.Closurerelationdu(0)[0)(] = 1.(1.98)d(Rep)d(Ima)doadoae-Zaoap=Ealoal2(1.99)dμ(Φ)=2元iTProof.Onecan showthattherighthand sideof therelation commuteswithall creation andannihilationoperators.AccordingtoSchur'slemma,itmustbeproportionaltotheidentityoperator.: To prove the commutability, one can showa o)=at lo) = afeZa gaat j0) =(1.100)10),apad0a 10) (0l(1.101)[at, lo) (al] = (-00.then inserts the relation into the integral and integrates by parts.TraceTA=(n|A|n)= / dμ(o)(n/0)([A|n)Z / d(0)(|4|n)(n/0)= / du(0)(|4|0). (1.102)Coherentstaterepresentation[12) = / du() [0) (1) = / dμ(0) 10) (0*),(1.103)where we define a wave function ($*) = (ol). It is an anti-holomorphic function of s*.AllcoherentstatesformaSegal-Bargmannspace[9].9
1.5 Coherent states NO§1.5 Coherent states are the eigenstates of the annihilation operators: aˆα |ϕ⟩ = ϕα |ϕ⟩. (1.94) Note that one cannot define the eigenstates of the creation operators. For Fermions, the eigenvalues cannot be ordinary numbers because annihilation operators anticommute. They must be GRASSMANN NUMBERS: ϕαϕβ + ϕβϕα = 0. (1.95) 1.5.1 Boson coherent states Definition |ϕ⟩ = e ∑ α ϕαa † α |0⟩. (1.96) Overlap ⟨ϕ|ϕ ′ ⟩ = D 0 e ∑ α ϕ ∗ αaˆα ϕ ′ E = e ∑ α ϕ ∗ αϕ ′ α ⟨0 | ϕ ′ ⟩ = exp (ϕ ∗ · ϕ ′ ). (1.97) Note that coherent states do not form an orthonormal basis. Instead, they form an overcomplete basis. Closure relation Z dµ(ϕ)|ϕ⟩⟨ϕ| = 1. (1.98) dµ(ϕ) ≡ Y α dϕαdϕ ∗ α 2πi e − ∑ α|ϕα| 2 ≡ Y α d(Reϕα)d(Imϕα) π e − ∑ α|ϕα| 2 . (1.99) Proof • One can show that the right hand side of the relation commutes with all creation and annihilation operators. According to Schur’s lemma, it must be proportional to the identity operator. • To prove the commutability, one can show aˆ † α |ϕ⟩ = ˆa † αe ∑ α ϕαaˆ † α |0⟩ = ∂ ∂ϕα e ∑ α ϕαaˆ † α |0⟩ = ∂ ∂ϕα |ϕ⟩, (1.100) aˆ † α, |ϕ⟩ ⟨ϕ| = ∂ ∂ϕα − ϕ ∗ α |ϕ⟩ ⟨ϕ| , (1.101) then inserts the relation into the integral and integrates by parts. Trace TrAˆ = X n D n Aˆ n E = X n Z dµ (ϕ)⟨n | ϕ⟩ D ϕ Aˆ n E = X n Z dµ (ϕ) D ϕ Aˆ n E ⟨n | ϕ⟩ = Z dµ(ϕ) D ϕ Aˆ ϕ E . (1.102) Coherent state representation |ψ⟩ = Z dµ(ϕ)|ϕ⟩ ⟨ϕ | ψ⟩ ≡ Z dµ(ϕ)|ϕ⟩ ψ(ϕ ∗ ), (1.103) where we define a wave function ψ(ϕ ∗ ) ≡ ⟨ϕ|ψ⟩. It is an anti-holomorphic function of ϕ ∗ . • All coherent states form a Segal-Bargmann space [9]. 9
.One can setupa“Schrodingerequation"for ($*).An application of the representationcanbefound inRef.[8]Representation of operators:a(1.104)aα0(1.105)at→a.Proof(1.106)( ) =*() =*(*),dμ($)中e$*'($**)(0[aa/)=/dμ(0)(0aal0)(0)=(1.107)ob(o*)[ dμ()e* (o*) =(1.108)00*d0aUnit-operator:One can prove theidentity[ d(0)e** (0*)d(Φ*) =(1.109)by appending ($|toboth sides of Eq. (1.103)Matrix-elementsofnormal-orderedoperators:(|A(at,a)[0)=A(o,)e*(1.110)Averageandvarianceoftheparticlenumber(0~[0)N=loa?(1.111)(0[0)(0[(n-n) 0)(AN)?=(1.112)(010)In the limit of N → oo, △N/N -→ 0, a coherent state can be interpreted as a classical field.Forinstances,thecoherentstatesof phononscorrespondtoclassical sound waves,andthecoherentstatesofphotonscorrespondtoclassicalelectromagneticfields.1.5.2 Grassmann algebraGrassmannalgebra isdefined bya setof generators(s.J,ar=1...nwhichanti-commute:(1.113)SS+=0.AmatrixrepresentationofGrassmannnumbersrequiresmatricesofdimensionatleast 2nx2n[1]. It is obvious: = 0.(1.114)Number intheGrassmann algebrais a linear combination with coefficientsof thegenerators:(1.115)[1,Sa,SalSa2...,SalSa...San].Note that a complex number could also be regarded as a linear combination of generators[1,i].10
• One can set up a“Schrödinger equation”for ψ(ϕ ∗ ). An application of the representation can be found in Ref. [8]. Representation of operators: aˆα → ∂ ∂ϕ∗ α , (1.104) aˆ † α → ϕ ∗ α. (1.105) Proof ϕ aˆ † α ψ = ϕ ∗ ⟨ϕ | ψ⟩ ≡ ϕ ∗ψ (ϕ ∗ ), (1.106) ⟨ϕ | aˆα | ψ⟩ = Z dµ(ϕ ′ )⟨ϕ | aˆα | ϕ ′ ⟩ ⟨ϕ ′ | ψ⟩ = Z dµ(ϕ ′ )ϕ ′ αe ϕ ∗ ·ϕ ′ ψ (ϕ ′∗) (1.107) = ∂ ∂ϕ∗ α Z dµ(ϕ ′ )e ϕ ∗ ·ϕ ′ ψ (ϕ ′∗) = ∂ψ (ϕ ∗ ) ∂ϕ∗ α . (1.108) Unit-operator: One can prove the identity ψ (ϕ ′∗) = Z dµ(ϕ)e ϕ ′∗ ·ϕψ (ϕ ∗ ) (1.109) by appending ⟨ϕ ′ | to both sides of Eq. (1.103). Matrix-elements of normal-ordered operators: D ϕ Aˆ aˆ † , aˆ ϕ ′ E = A (ϕ ∗ , ϕ′ ) e ϕ ∗ ·ϕ ′ . (1.110) Average and variance of the particle number N¯ ≡ D ϕ Nˆ ϕ E ⟨ϕ | ϕ⟩ = X α |ϕα| 2 , (1.111) (∆N) 2 ≡ ϕ Nˆ − N¯ 2 ϕ ⟨ϕ | ϕ⟩ = N. ¯ (1.112) In the limit of N¯ → ∞, ∆N/N¯ → 0, a coherent state can be interpreted as a classical field. For instances, the coherent states of phonons correspond to classical sound waves, and the coherent states of photons correspond to classical electromagnetic fields. 1.5.2 Grassmann algebra Grassmann algebra is defined by a set of generators {ξα}, α = 1 . . . n which anti-commute: ξαξβ + ξβξα = 0. (1.113) A matrix representation of Grassmann numbers requires matrices of dimension at least 2 n × 2 n [1]. It is obvious: ξ 2 α = 0. (1.114) Number in the Grassmann algebra is a linear combination with coefficients of the generators: {1, ξα, ξα1 ξα2 , . . . , ξα1 ξα2 . . . ξαn }. (1.115) Note that a complex number could also be regarded as a linear combination of generators {1, i}. 10
Conjugate has properties:(1.116)(a)* = Sa,(1.117)()* = a,(1.118)(a)* = *a,(1.119)(Sa...an)* -- Sa...SaFunction(1.120)f() = fo + fis,(1.121)A($*,s)=ao+ai+ais*+a12*.Derivative is defined to be identical to the complex derivative, except that g has to be anti-commutedthrough until it reaches to act on :oα() -(1.122)Note that Og and Oe- also anti-commute.Integral is defined by the rules:*1 = 0.(1.123)1** = 1(1.124)1Note that de*is not defined! Justtreatand * as two independent variables.:Anti-commute between an integral and anotherintegral, or an integral and a Grassmannvariable.. The integral coincides with the derivative.Reproducingkernel(Diracfunction)(1.125)8(5,E) = -(E - E) ,Id'8(s,s)f(),(1.126)f(6) =Scalarproduct ofGrassmannfunctions:(1.127)(f Ig) = / de*dee-* f*(E)g($*)(1.128)/de*de(1-*)(fo+ff)(go+gis*)d'defogo+/edtfigi(1.129)(1.130)=fo90+f91.1.5.3Fermion coherent statesDefinition[5) =e-2 a 0) = II(1 - aat) 10) .(1.131)a(1.132)a (S) = [S),(1.133)(5at =(Ss)aRat ls) = -15) ,(1.134)aaa(51(1.135)(slaα =O11
Conjugate has properties: (ξα) ∗ = ξ ∗ α, (1.116) (ξ ∗ α) ∗ = ξα, (1.117) (λξα) ∗ = λ ∗ ξ ∗ α, (1.118) (ξα1 . . . ξαn ) ∗ = ξ ∗ αn . . . ξ∗ α1 . (1.119) Function f(ξ) = f0 + f1ξ, (1.120) A(ξ ∗ , ξ) = a0 + a1ξ + ¯a1ξ ∗ + a12ξ ∗ ξ. (1.121) Derivative is defined to be identical to the complex derivative, except that ∂ξ has to be anti-commuted through until it reaches to act on ξ: ∂ ∂ξ (ξ ∗ ξ) = −ξ ∗ . (1.122) Note that ∂ξ and ∂ξ ∗ also anti-commute. Integral is defined by the rules: Z dξ 1 = Z dξ ∗ 1 = 0, (1.123) Z dξ ξ = Z dξ ∗ ξ ∗ = 1. (1.124) • Note that R dξ ∗ ξ is not defined! Just treat ξ and ξ ∗ as two independent variables. • Anti-commute between an integral and another integral, or an integral and a Grassmann variable. • The integral coincides with the derivative. Reproducing kernel (Dirac function) δ(ξ, ξ′ ) = − (ξ − ξ ′ ), (1.125) f(ξ) = Z dξ ′ δ(ξ, ξ′ )f(ξ ′ ). (1.126) Scalar product of Grassmann functions: ⟨f | g⟩ ≡ Z dξ ∗dξ e−ξ ∗ ξ f ∗ (ξ)g(ξ ∗ ) (1.127) = Z dξ ∗dξ (1 − ξ ∗ ξ) (f ∗ 0 + f ∗ 1 ξ) (g0 + g1ξ ∗ ) (1.128) = − Z dξ ∗dξ ξ∗ ξf ∗ 0 g0 + Z dξ ∗dξ ξξ∗ f ∗ 1 g1 (1.129) = f ∗ 0 g0 + f ∗ 1 g1. (1.130) 1.5.3 Fermion coherent states Definition |ξ⟩ = e − ∑ α ξαaˆ † α |0⟩ = Y α 1 − ξαaˆ † α |0⟩. (1.131) aˆα |ξ⟩ = ξα |ξ⟩, (1.132) ⟨ξ| aˆ † α = ⟨ξ| ξ ∗ α, (1.133) aˆ † α |ξ⟩ = − ∂ ∂ξα |ξ⟩, (1.134) ⟨ξ| aˆα = + ∂ ∂ξ∗ α ⟨ξ| . (1.135) 11
: Sa is a Grassmann number-The Fermion Fock space must be enlarged to define a coher-entstate.. s, $t, a, and at anti-commute, and (sa)t = ate*.Proofa 5)=II (1-t) (aa + t) 0)=II (1-d) a 10)±= II (1- Ep) a (1 - Et) (0) = a (5) (1.136)B+at 15) = II (1-Spat)(at + tat) 0)= II (1 - pa) at 10)是15) (1.137) (1 - t) 0) = = II (1-gat) (--aB+oOverlap(515)=e5"s"(1.138)Closure relation[II dsdae-$* [5) (] = 1.(1.139)Trace of an operator:A=(n|4|n)= /IIddae-s*(n5)(||n[1Iddae-s* (-|4|n) (n15)= /Idae-ss(|A|s),(1.140)The extra minus sign is due to[ d*dee-s"g()f($") = / de*de-$"f(-*)g()(1.141)Coherent state representation1b) = [Iddae-* 15)《E)= /Idde-* [5) (s*),(1.142)(1.143)(51a10)-是(),(1.144)(s[a /) =(c*),Matrix element of a normal-ordered operator(s|A(at,a) [e') = es*s'A($,s").(1.145)Caveats. There are no classical interpretation of the coherent states of Fermions..Noviableapproximation(e.g.,stationary-phaseapproximation)exists.12
• ξα is a Grassmann number –The Fermion Fock space must be enlarged to define a coherent state. • ξ, ξ ∗ , aˆ, and aˆ † anti-commute, and (ξaˆ) † = ˆa † ξ ∗ . Proof aˆα |ξ⟩ = Y β̸=α 1 − ξβaˆ † β aˆα + ξαaˆαaˆ † α |0⟩ = Y β̸=α 1 − ξβaˆ † β ξα |0⟩ = Y β̸=α 1 − ξβaˆ † β ξα 1 − ξαaˆ † α |0⟩ = ξα |ξ⟩ (1.136) aˆ † α |ξ⟩ = Y β̸=α 1 − ξβaˆ † β aˆ † α + ξαaˆ † αaˆ † α |0⟩ = Y β̸=α 1 − ξβaˆ † β aˆ † α |0⟩ = Y β̸=α 1 − ξβaˆ † β − ∂ ∂ξα 1 − ξαaˆ † α |0⟩ = − ∂ ∂ξα |ξ⟩ (1.137) Overlap ⟨ξ | ξ ′ ⟩ = e ξ ∗ ·ξ ′ (1.138) Closure relation Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ |ξ⟩ ⟨ξ| = 1. (1.139) Trace of an operator: TrAˆ = X n D n Aˆ n E = Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ X n ⟨n | ξ⟩ D ξ Aˆ n E = Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ X n D −ξ Aˆ n E ⟨n | ξ⟩ ≡ Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ D −ξ Aˆ ξ E . (1.140) The extra minus sign is due to Z dξ ∗dξ e−ξ ∗ ·ξ g(ξ)f(ξ ∗ ) = Z dξ ∗dξ e−ξ ∗ ·ξ f(−ξ ∗ )g(ξ). (1.141) Coherent state representation |ψ⟩ = Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ |ξ⟩ ⟨ξ | ψ⟩ ≡ Z Y α dξ ∗ αdξαe −ξ ∗ ·ξ |ξ⟩ ψ(ξ ∗ ). (1.142) ⟨ξ | aˆα | ψ⟩ = ∂ ∂ξ∗ α ψ(ξ ∗ ), (1.143) ξ aˆ † α ψ = ξ ∗ αψ(ξ ∗ ), (1.144) Matrix element of a normal-ordered operator D ξ Aˆ aˆ † , aˆ ξ ′ E = e ξ ∗ ·ξ ′ A (ξ ∗ , ξ′ ). (1.145) Caveats • There are no classical interpretation of the coherent states of Fermions. • No viable approximation (e.g., stationary-phase approximation) exists. 12
1.5.4 Gaussian integralsForcomplexvariables dz, dziea H=+Jt+t] = [detH]-1 Jt H- ](1.146)2元iH is a matrix with a positive-definite Hermitianpart.ForGrassmannvariables[IIdn;dnen' H+J++'] = [detH]ejH-].(1.147)Both (ni, nt) and [J,J) are Grassmann variables. H is not necessary to be positive definite..ThelawforlineartransformationsofGrassmannvariables:0(n.n)(1.148)II dct dc;P(ct,c) =IIdn dn;P(c*(n*, n), c(nt,n)0(c)Note that the Jacobian is inverted.. For more general cases involving nin and ni n quadratic terms, a generalized GaussianintegralformulacanbefoundinRef.[20]1.6SummaryCommutationrelationaat= aB, [aa,ap]-=at,a]= (Coherent state ([s) = exp (.Saat) (0)Operations[5)=1s),at [s)=(0g Is),(at =(S1,(Sla=O(5lMatrix element (|A(at,a)le') = es*-'A($*,")Closure relation 1 =Jdμ($)(s)(slTrace TrA= dμ() (C|A|E)Representation ) = J d(s)[E) ($*), ($*) =(5), (sat) =(s*), (5a) = ($*)Gaussian integraldedsa-aptHapEa+Da(nEa+na)=[detH-eDnaHang11NBosonsFermionsdadaedμ($) = IINn[2iBosonsN=1FermionsC?.Csaar13
1.5.4 Gaussian integrals For complex variables Z Y i dz ∗ i dzi 2πi e −z †Hz+J † z+z †J = [detH] −1 e J †H−1J . (1.146) H is a matrix with a positive-definite Hermitian part. For Grassmann variables Z Ydη ∗ i dηie −η †Hη+J †η+η †J = [detH] e J †H−1J . (1.147) Both {ηi , η∗ i } and {Ji,J ∗ i } are Grassmann variables. H is not necessary to be positive definite. • The law for linear transformations of Grassmann variables: Z Ydζ ∗ i dζiP (ζ ∗ , ζ) = ∂(η ∗ , η) ∂(ζ ∗, ζ) Z Ydη ∗ i dηiP (ζ ∗ (η ∗ , η), ζ(η ∗ , η)). (1.148) Note that the Jacobian is inverted. • For more general cases involving ηiηj and η ∗ i η ∗ j quadratic terms, a generalized Gaussian integral formula can be found in Ref. [20]. 1.6 Summary Commutation relation h aˆα, aˆ † β i −ζ = δαβ, [ˆaα, aˆβ]−ζ = h aˆ † α, aˆ † β i −ζ = 0. Coherent state |ξ⟩ = exp ζ P α ξαaˆ † α |0⟩ Operations aˆα |ξ⟩ = ξα |ξ⟩, aˆ † α |ξ⟩ = ζ∂ξα |ξ⟩, ⟨ξ| aˆ † α = ⟨ξ| ξ ∗ α, ⟨ξ| aˆα = ∂ξ ∗ α ⟨ξ| Matrix element ⟨ξ|Aˆ(ˆa † , aˆ)|ξ ′ ⟩ = e ξ ∗ ·ξ ′ A(ξ ∗ , ξ′ ) Closure relation 1 = R dµ(ξ)|ξ⟩ ⟨ξ| Trace TrAˆ = R dµ(ξ)⟨ζξ|Aˆ|ξ⟩ Representation |ψ⟩ = R dµ(ξ)|ξ⟩ ψ(ξ ∗ ), ψ(ξ ∗ ) = ⟨ξ|ψ⟩, ⟨ξ|aˆ † α|ψ⟩ = ξ ∗ αψ(ξ ∗ ), ⟨ξ|aˆα|ψ⟩ = ∂ξ ∗ α ψ(ξ ∗ ) Gaussian integral Z Y α dξ ∗ αdξα N e − ∑ αβ ξ ∗ αHαβξβ+ ∑ α(η ∗ αξα+ξ ∗ αηα) = [detH] −ζ e ∑ αβ η ∗ αH−1 αβ ηβ ζ = ( 1 Bosons −1 Fermions dµ(ξ) = Y α dξ ∗ αdξα N e −ξ ∗ ·ξ N = ( 2πi Bosons 1 Fermions ξ ∗ · ξ ′ = X α ξ ∗ αξ ′ α 13