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. Notice that the Lagrangian for the coherent-state representation presented in Eq.(8.43) is ﬁrst order in time derivatives. Because of this feature we are not guaranteed that the paths are necessarily diﬀerentiable.Thisproperty leads to all kinds of subtleties that for the most part we will ignore in what follows.

8.4 Path integral for a non-relativistic Bose gas The ﬁeld theoretic description of a gas of (spinless) non-relativistic bosons is given in terms of the creation and annihilation ﬁeld operators φˆ† x and φˆ x ,thatsatisfytheequaltimecommutationrelations(ind space dimensions) ( ) ( ) † d φˆ x , φˆ y = δ x − y (8.44)

Relative to the empty state# ( 0) ,suchthat( )$ ( ) ˆ = ∣ ⟩ φ x 0 0(8.45)

( )∣ ⟩ 8.4 Path integral for a non-relativistic Bose gas 221 the normal ordered Hamiltonian (in the Grand canonical Ensemble) is

h2 Hˆ = ddx φˆ† x − ̵ !2 − µ + V x φˆ x " 2m

1 d d † † + d (x )-d y φˆ x φˆ y U (x −).y (φˆ )y φˆ x (8.46) 2 " " where m is the mass of the bosons,( )µ is( the) ( chemical) ( potential,) ( ) V x is an external potential and U x − y is the interaction potential between pairs of bosons. ( ) Following our discussion( of the) coherent state path integralweseethatit is immediate to write down a path integral for a thermodynamically large system of bosons. The boson coherent states are now labelled by a complex ﬁeld φ x and its complex conjugate φ¯ x .

( ) dx(φ)x φˆ† x φ x = e" 0 (8.47) ( ) ( ) which has the coherent∣{ state( )} property⟩ of being a right∣ ⟩ eigenstate of the ﬁeld operator φˆ x ,

ˆ = ( ) φ x φ φ x φ (8.48) as well as obeying the resolution( )∣{ of}⟩ the identity( )∣{ }⟩ in this spaceofstates

2 ∗ − dx φ x I = DφDφ e " φ φ (8.49) " ∣ ( )∣ The matrix element of the evolution operator∣{ of} this⟩⟨{ system}∣ between an arbitrary initial state i and an arbitrary ﬁnal state f ,separatedbya time lapse T = tf −ti (not to be confused with the temperature!), now takes the form ∣ ⟩ ∣ ⟩ i − HTˆ f e h̵ i = t i f d h DφDφ¯ exp dt d x ̵ φ x,t ∂ φ¯ x,t − φ¯ x,t ∂ φ x,t − H φ, φ¯ ⟨" ∣ ∣ ⟩ h " " i t t ̵ ti 1 9 : dx; (φ x),t (2 + )φ x,t( 2) ( )< [ ]=> 2 " f i × Ψ¯ f φ x,tf Ψi φ¯ x,ti e (8.50) (∣ ( )∣ ∣ ( )∣ ) ( ( )) ( ( )) 222 Coherent State Path Integral Quantization of Quantum Field Theory where the functional H φ, φ¯ is

h2 H φ, φ¯ = [ ddx]φ¯ x − ̵ !2 − µ + V x φ x " 2m 1 [ ] + ddx ( )d-dy φ x 2 φ y 2U( x).− y( ) 2 " " (8.51) ∣ ( )∣ ∣ ( )∣ ( ) It is also possible to write the action S in the less symmetric but simpler form (where dDx ≡ dtddx)

2 h 2 S = dDx φ¯ x ih∂ + ̵ & + µ − V x φ x " ̵ t 2m ⃗ 1 − dD(x)) dDy φ x 2 φ y 2U (x −)*y ( ) 2 " " (8.52) ∣ ( )∣ ∣ ( )∣ ( ) where U x − y = U x − y δ tx − ty . Therefore the path integral for a system of non-relativisticbosons,with chemical( potential) µ(,hasthesameformosthepathintegralofthecharged) ( ) scalar field we discussed before except that the action is firstorderintime derivatives. The fact that the field is complex follows from the requirement that the number of bosons is a globally conserved quantity, which is why one is allowed to introduce a chemical potential. This formulation is useful to study superfluid Helium and similar problems. Suppose for instance that we want to compute the partition function Z for this system of bosons at finite temperature T ,

−βHˆ Z = tre (8.53) where β = 1 T (in units where kB = 1). The coherent-state path integral representation of the partition function is obtained by 1) restricting the initial and ﬁnal/ states to be the same i = f and arbitrary, 2) summing over all possible states, and 3) a Wick rotation to imaginary time t → − → − iτ,withthetime-span T iβh̵ ,withperiodicboundaryconditionsin∣ ⟩ ∣ ⟩ imaginary time. The result is the (imaginary time) path integral

−S φ, φ¯ Z = DφDφ¯ e E (8.54) " ( ) 8.5 Fermion coherent states 223 where SE is the Euclidean action 1 β h2 S φ, φ¯ = dτ dx φ¯ h∂ − µ − ̵ !2 + V x φ E ̵ τ m h̵ "0 " 2 1 β ( ) + dτ dx- dy U x − y φ x 2( φ).y 2 2h̵ "0 " " (8.55) ( )∣ ( )∣ ∣ ( )∣ The fields φ x = φ x,τ satisfy periodic boundary conditions in imaginary time ( ) ( ) = + φ x,τ φ x,τ βh̵ (8.56) This requirement suggests an expansion of the field φ x in Fourier modes ( ) ( ) of the form ∞ iω τ ( ) φ x,τ = e n φ x, ω (8.57) ! ⃗ n n=−∞ where the frequencies ω(n (the) Matsubara frequencies)( ) must be chosen so that φ obeys the required PBCs. We find 2π 2πT ωn = n = n, n∈ Z (8.58) βh̵ h̵ where n is an arbitrary integer. In one of the problems at the end of this chapter you will evaluate this path integral using a semiclassical approxima- tion.

8.5 Fermion coherent states In this section we will develop a formalism for fermions which follows closely what we have done for bosons while taking into account the anti-commuting nature of fermionic operators. † = Let ci be a set of fermion creation operators, with i 1,...,N,and ci the set of their N adjoint operators, the associated annihilation operators. = † The number{ } operator for the i-th fermion is ni ci ci.Letusdeﬁnethebasis{ } states states of the i-th fermion by the kets 0i and 1i ,whichobeythe obvious deﬁnitions: = † = † = ∣ ⟩ † ∣ ⟩ = ci 0i 0,ci 0i 1i ,ci ci 0i 0,ci ci 1i 1i (8.59) For N fermions the Hilbert space is spanned by the anti-symmetrized states ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ n1,...,nN .Let

0 ≡ 01,...,0N (8.60) ∣ ⟩ ∣ ⟩ ∣ ⟩ 224 Coherent State Path Integral Quantization of Quantum Field Theory be the empty state. A general state in this Hilbert space is n n = † 1 † N n1,...,nN c1 ... cN 0 (8.61) n ,...,n As we saw before, the∣ wave function⟩ + ,1 + N ,Ψ ∣is⟩ fully antisymmetric. If the state Ψ is a product state, the wave function is a Slater determinant. ⟨ ∣ ⟩ ∣ ⟩ 8.5.1 Deﬁnition of fermion coherent states

We now define fermion coherent states. Let ξ¯i,ξi ,withi = 1,...,N,bea set of 2N Grassmann variables,asetofsymbolsalsoknownasthegenera- tors of a Grassmann algebra. By definition, Grassmann{ } variables satisfy the following properties ¯ ¯ ¯ 2 ¯2 ξi,ξj = ξi, ξj = ξi, ξj = ξi = ξi = 0(8.62) Therefore, Grassmann variables behave a a set of time-ordered fermion op- ? @ ? @ ? @ erators. We will also require that the Grassmann variables anti-commute with the fermion operators: = ¯ † = ¯ = ¯ † = ξi,cj ξi,cj ξi,c,j ξi,cj 0(8.63) Let us define the fermion coherent states to be ? @ A B ? @ A B −ξc† ξ ≡ e 0 (8.64) ξ¯c ξ ≡ 0 e (8.65) ∣ ⟩ ∣ ⟩ As a consequence of these definitions we have: ⟨ ∣ ⟨ ∣ † −ξc † e = 1 − ξc (8.66) Similarly, if ψ is a Grassmann variable, then ξ¯c −ψc† ξψ¯ ξ ψ = 0 e e 0 = 1 + ξψ¯ = e (8.67) For N fermions we have, ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ N † − ξ c † ! i i N −ξici i=1 ξ ≡ ξ1,...,ξN = Πi=1e 0 ≡ e 0 (8.68) since the following commutator vanishes, ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ † † = ξici ,ξjcj 0(8.69)

# $ 8.5 Fermion coherent states 225 8.5.2 Analytic functions of Grassmann variables We will deﬁne ψ ξ to be an analytic function of the Grassmann variable if it has a power series expansion in ξ, ( ) 2 ψ ξ = ψ0 + ψ1ξ + ψ2ξ + ... (8.70) where ψn ∈ C.Since ( ) n ξ = 0, ∀n ≥ 2(8.71) then, all analytic functions of a Grassmann variable reduce to a ﬁrst degree polynomial,

ψ ξ ≡ ψ0 + ψ1ξ (8.72) Similarly, we deﬁne complex conjugation by ( ) ¯ ¯ ¯ ψ ξ ≡ ψ0 + ψ1ξ (8.73) where ψ¯ and ψ¯ are the complex conjugates of ψ and ψ respectively. 0 1 ( ) 0 1 We can also deﬁne functions of two Grassmann variables ξ and ξ¯, ¯ ¯ ¯ A ξ,ξ = a0 + a1ξ + a¯1ξ + a12ξξ (8.74) where a ,¯a and a are complex numbers; a anda ¯ are not necessarily 1 1 12 ( ) 1 1 complex conjugates of each other.

8.5.3 Differentiation over Grassmann variables Since analytic functions of Grassmann variables have such a simple struc- ture, differentiation is just as simple. Indeed, we define the derivative as the coefficient of the linear term

∂ξψ ξ ≡ ψ1 (8.75) Likewise we also have ( ) ¯ ∂ξ¯ψ ξ ≡ ψ1 (8.76) Clearly, using this rule we can write ( ) ∂ξ ξξ¯ = −∂ξ ξξ¯ = −ξ¯ (8.77) Asimilarargumentshowsthat ( ) ( ) ¯ ¯ ∂ξA ξ,ξ = a1 − a12 ξ (8.78) ¯ ∂ξ¯A ξ,ξ = a¯1 + a12 ξ (8.79) (¯ ) ¯ ∂ξ¯∂ξA ξ,ξ = −a12 = −∂ξ∂ξ¯A ξ,ξ (8.80) ( ) ( ) ( ) 226 Coherent State Path Integral Quantization of Quantum Field Theory from where we conclude that ∂ξ and ∂ξ¯ anti-commute,

∂ξ¯,∂ξ = 0, and ∂ξ∂ξ = ∂ξ¯∂ξ¯ = 0(8.81)

? @ 8.5.4 Integration over Grassmann variables The basic diﬀerentiation rule of Eq. (8.75) implies that

1 = ∂ξξ (8.82) which suggests the following deﬁnitions:

dξ 1 = 0, dξ ∂ ξ = 0, dξ ξ = 1(8.83) " " ξ " Analogous rules also apply for the conjugate variables ξ¯. It is instructive to compare the diﬀerentiation and integration rules: dξ 1 = 0 ↔ ∂ 1 = 0 ∫ ξ (8.84) dξ ξ = 1 ↔ ∂ ξ = 1 ∫ ξ Thus, for Grassmann variables diﬀerentiation and integration are exactly equivalent

∂ ⟺ dξ (8.85) ξ " These rules imply that the integral of an analytic function f ξ is

dξf ξ = dξ f + f ξ = f (8.86) " " 0 1 1 ( ) and ( ) ( ) dξA ξ¯,ξ = dξ a + a ξ + a¯ ξ¯+ a ξξ¯ = a − a ξ¯ " " 0 1 1 12 1 12 dξ¯A ξ¯,ξ = dξ a + a ξ + a¯ ξ¯+ a ξξ¯ = a¯ + a ξ " ( ) " C 0 1 1 12 D 1 12 dξ¯dξA ξ¯,ξ = − dξdξ¯A ξ¯,ξ = −a " ( ) " C 12 D (8.87) ( ) ( ) It is straightforward to show that with these definitions, thefollowingex- pression is a consistent definition of a delta-function: ′ ′ −η ξ − ξ δ ξ ,ξ = dηe (8.88) " ′ ( ) where ξ, ξ and η are Grassmann( ) variables. 8.5 Fermion coherent states 227 Finally, given that we have a vector space of analytic functions we can define an inner product as follows: −ξξ¯ f g = dξ¯dξe f¯ ξ g ξ¯ = f¯ g + f¯ g (8.89) " 0 0 1 1 as expected. ⟨ ∣ ⟩ ( ) ( )

8.5.5 Properties of fermion coherent states We deﬁned above the fermion bra and ket coherent states † − ξ c ξ¯ c ! j j ! j j j j ξj = e 0 , ξj = 0 e (8.90) After a little algebra, using the rules deﬁned above, it is easy to see that the ∣{ }⟩ ∣ ⟩ ⟨{ }∣ ⟨ ∣ following identities hold: = † = − ci ξj ξi ξj ci ξj ∂ξi ξj (8.91) † ξj ci =∂ξ¯ ξj ξj c = ξ¯i ξj (8.92) ∣{ }⟩ i∣{ }⟩ ∣{ }⟩i ∣{ }⟩ ′ The inner product of two coherent states, ξ and ξ ,is ⟨{ }∣ ⟨{ }∣ ⟨{ }∣ j ⟨{ }j ∣ ξ¯ ξ !∣{ j}⟩j ∣{ }⟩ ′ j ξj ξj = e (8.93) Similarly, we also have the Resolution of the Identity (which is easy to prove) ⟨{ }∣{ }⟩ N − ξ¯iξi N != I = Π = dξ¯ dξ e i 1 ξ ξ (8.94) " i 1 i i i i

Let Ψ be some state. Then,& we can' use Eq. (8.94)∣{ }⟩⟨ to{ expand}∣ the state Ψ in fermion coherent states ξ , ∣ ⟩ N ∣ ⟩ ∣ ⟩ − ξ¯iξi N != Ψ = Π = dξ¯ dξ e i 1 Ψ ξ ξ (8.95) " i 1 i i i where ∣ ⟩ & ' ( )∣{ }⟩ ¯ ¯ ¯ Ψ ξ ≡ Ψ ξ1,...,ξN (8.96) We can use the rules derived above to compute the following matrix elements ( ) ( ) = ¯ † = ¯ ¯ ξ cj Ψ ∂ξ¯j Ψ ξ , ξ cj Ψ ξj Ψ ξ (8.97) which is consistent with what we concluded above. ⟨ ∣ ∣ ⟩ ( ) ⟨ ∣ ∣ ⟩ ( ) 228 Coherent State Path Integral Quantization of Quantum Field Theory Let 0 be the “empty state”. We will not call it the “vacuum” since it is not in the sector of the ground state of the systems of interest. Let † A cj ∣, ⟩cj be a normal ordered operator (with respect to the state 0 ). By using the formalism worked out above one can show without diﬃculty ′ that&{ its} matrix{ }' elements in the coherent states ξ and ξ are ∣ ⟩ ′ ξ¯ ξ ! i i ∣ ⟩ ∣ ⟩ † ′ = i ¯ ′ ξ A cj , cj ξ e A ξj , ξj (8.98) ˆ For example, the⟨ ∣ expectation&{ } { } value' ∣ ⟩ of the fermion&{ number} { }' operator N, † Nˆ = c c (8.99) ! j j j in the coherent state ξ is ξ Nˆ ξ ∣ ⟩ = ξ¯ ξ (8.100) ξ ξ ! j j j ⟨ ∣ ∣ ⟩ ⟨ ∣ ⟩ 8.5.6 Grassmann gaussian integrals Let us consider a Gaussian integral over Grassmann variablesoftheform

− ξ¯ M ξ + ξ¯ ζ + ζ¯ ξ N ! i ij j i i i i Z ζ¯,ζ = dξ¯ dξ e i,j (8.101) " $ i i i=1

where ζi [and] ζ¯i are) a set of* 2N Grassmann variables, and the matrix Mij is a complex Hermitian matrix. We will now show that { } { } − ζ¯ M 1 ζ ! i ij j Z ζ¯,ζ = det M e ij (8.102) & ' Before showing that Eq. (8.102) is correct, let us make a few observations: [ ] ( ) 1. Eq. (8.102) looks like the familiar expression for Gaussian integrals for − bosons except that, instead of a factor of det M 1 2,wegetafactorof det M in the numerator. Except for the absence of a/ square root, thatthe ﬂuctuation determinant appears in the numerator( ) is the main eﬀect of the Fermi statistics!. 2. Moreover, if we had considered a system of N Grassmann variables (instead of 2N)wewouldhaveobtainedinsteadafactorof

det M 1 2 = Pf M (8.103) / ( ) ( ) 8.5 Fermion coherent states 229 where M would now be an N ×N real anti-symmetric matrix, and Pf M denotes the Pfaﬃan of the matrix M. ( ) To prove that Eq. (8.102) is correct it will be suﬃcient to consider the case ζi = ζ¯i = 0, since the contribution from these sources is identical to the bosonic case (except for the ordering of the Grassmann variables). Using the Grassmann identities we can write the exponential factoras

− ξ¯ M ξ ! i ij j e i,j = 1 − ξ¯ M ξ (8.104) $ i ij j ij The integral that we need to do is C D

N Z 0, 0 = dξ¯ ξ 1 − ξ¯ M ξ (8.105) " $ i i $ i ij j i=1 ij From the integration[ rules,] we) can easily* seeC that the onlyD non-vanishing terms in this expression are those that have the just one ξi and one ξ¯i (for each i). Hence we can write

N N Z 0, 0 = −1 dξ¯ dξ ξ¯ M ξ ξ¯ M ξ ...+ permutations " $ i i 1 12 2 2 23 3 i=1 N [ ] ( )N ) * = −1 M M M ... dξ¯ dξ ξ¯ ξ ξ¯ ξ ξ¯ ...ξ¯ ξ 12 23 34 " $ i i 1 2 2 3 3 N 1 i=1 (+ permutations) ) * 2N = −1 M12M23M34 ...MN,1 + permutations (8.106) What is the contribution of the terms labeled “permutations”? It is easy ( ) to see that if we permute any pair of labels, say 2 and 3, we will get a contribution of the form 2N −1 −1 M13M32M24 ... (8.107) Hence we conclude that the Gaussian Grassmann integral is just the deter- ( ) ( ) minant of the matrix M,

− ξ¯ M ξ N ! i ij j Z 0, 0 = dξ¯ dξ e ij = det M (8.108) " $ i i i=1 Alternatively,[ we] can diagonalize) * the quadratic form and notice that the Jacobian is “upside-down” compared with the result we found for bosons. 230 Coherent State Path Integral Quantization of Quantum Field Theory 8.6 Path integrals for fermions We are now ready to give a prescription for the construction ofafermion path integral in a general system. Let H the a normal-ordered Hamiltonian, with respect to some reference state 0 ,ofasystemoffermions.Let Ψi be the ket at the initial time ti and Ψf be the ﬁnal state at time tf .The matrix element of the evolution operator∣ ⟩ can be written as a Grassmann∣ ⟩ path integral ∣ ⟩ i − H t − t = f i Ψf ,tf Ψi,ti Ψf e h̵ Ψi i ( S ψ¯,)ψ ⟨ ∣ ⟩ ≡⟨ D∣ψ¯Dψeh ∣ ×⟩ projection operators " ̵ ( ) (8.109) where we have not written down the explicit form of the projection operators onto the initial and ﬁnal states. The action S ψ¯,ψ is

tf S ψ¯,ψ = dt ihψ∂¯ ψ −( H ψ)¯,ψ (8.110) " ̵ t ti This expression of the( fermion) path integral; holds for( any)< theory of fermions, relativistic or not. Notice that it has the same form as the bosonic path integral. The only change is that for fermions the determinant appears in the numerator while for bosons it is in the denominator!

8.7 Path integral quantization of the Dirac field We will now apply the methods we just developed to the case of the Dirac theory. Let ψα x ,withα = 1,...,4beafreemassiveDiracfieldin3+1 space-time dimensions. This field satisfies the Dirac Equation as an equation of motion, ( ) i∂ − m ψ = 0(8.111)

ψ ∂ = γµ∂ γ where is a 4-spinor and C /µ.RecallthattheDiracD -matrices satisfy the algebra / µ ν µν γ ,γ = 2g (8.112) where gµν is the Minkowski space metric tensor (in the Bjorken-Drell form). ? @ We saw before that in the quantum field theory description of the Dirac theory, ψ is an operator acting on the Fock space of (fermionic) states.We 8.7 Path integral quantization of the Dirac field 231 also saw that the Dirac equation can be regarded as the classical equation of motion of the Lagrangian density L = ψ¯ i∂ − m ψ (8.113) where ψ¯ = ψ†γ0.Wealsonotedthatthemomentumcanonicallyconjugate C / D to the field ψ is iψ†,fromwherethestandardfermionicequaltimeanti- commutation relations follow † = 3 ψα x,x0 ,ψβ y,x0 δαβ δ x − y (8.114)

The Lagrangian density L for a Dirac fermion coupled to sources ηα andη ¯α A ( ) ( )B ( ) is L = ψ¯ i∂ − m ψ + ψη¯ + ηψ¯ (8.115) where the sources are “classical” (Grassmann) anticommuting fields. C / D We can follow the same steps described in the preceding sections to find the following expression for the path-integral of the Dirac field in terms of Grassmann fields ψ x and ψ¯ x (which here are independent variables!)

4 ¯ ( ) 1 ( ) i d x ψη + ηψ¯ Z η¯,η = 0 Te " 0 0 0 C D [ ] ⟨ ∣ iS + i d4x ψη¯ + ηψ¯∣ ⟩ ≡⟨ ∣D⟩ψ¯Dψe " (8.116) " ( ) where S = d4xL. ∫ From this result it follows that the Dirac propagator is givenby

iSαβ x − y = 0 Tψα x ψ¯β y 0 −i 2 δ2Z η¯,η ( ) =⟨ ∣ ( ) ( )∣ ⟩ Z 0, 0 δη¯α x δηβ y Eη¯=η=0 ( ) [ ] E 1 E = x, α y,β E (8.117) [ i]∂ − m( ) ( )E Similarly, the partition function⟨ for∣ a theory∣ of⟩ free Dirac fermions is / ZDirac = Det i∂ − m (8.118) In contrast, in the case of a free real scalar field we found C / D − 2 2 1 2 Zscalar = Det ∂ + m (8.119) / Therefore for the case of the Dirac field the partition function is a determi- / & '0 nant whereas for the scalar field is the inverse of a determinant (actually, 232 Coherent State Path Integral Quantization of Quantum Field Theory of its square root). The fact that one result is the inverse of the other is aconsequencethatDiracfieldsarequantizedasfermionswhereas scalar fields are quantized as bosons. As we saw, this is a general result for Fermi and Bose fields regardless of whether they are relativistic ornot.Moreover, from this result it follows that the vacuum (ground state) energy of a Dirac field is negative whereas the vacuum energy for a scalar field ispositive. We will see that in interacting field theories the result of Eq.(8.118) leads to the Feynman diagram rule that each fermion loop carries a minus sign which reflects the Fermi-Dirac statistics, and hence this rule holds even in the absence of relativistic invariance. We now note that, due to the charge-conjugation symmetry of the Dirac theory, the spectrum of the Dirac operator is symmetric, i.e for every positive eigenvalue of the Dirac operator there is a negative eigenvalue of equal magnitude. More formally, since γ5,γµ = 0,

− = − − γ5 i∂ m{ γ5 } i∂ m (8.120) 2 = and γ5 I (the 4 × 4identitymatrix),itfollowsthatC / D C / D Det i∂ − m = Det i∂ + m (8.121)

Hence C / D C / D 1 2 Det i∂ − m = Det i∂ − m Det i∂ + m / 2 = 2 2 C / D /Det C∂/ + mD C / D0 (8.122) where we used that the “square”/ of& the Dirac'0 operator is the Klein-Gordon operator multiplied by the 4 × 4identitymatrix,whichiswhytheexponent = × 1 of the r.h.s of the equation above is 2 4 2 .Itiseasytoseethatthis Dirac result implies that the vacuum energy for the Dirac fermion E0 and the scalar vacuum energy of a scalar ﬁeld E0 ,withthesamemassm,arerelated by Dirac scalar E0 = −4E0 (8.123)

Here we have ignored the fact that both the l.h.s. and the r.h.s. of this equation are divergent, as we saw before. However since they have the same divergence, or what is the same is they are regularized in the same way, the comparison is meaningful. On the other hand, if instead of Dirac fermions, which are charged (and hence complex) fields, we consider Majorana fermions, which are charge 8.7 Path integral quantization of the Dirac field 233 neutral, and hence are real fields, we would have obtained instead the results 1 2 ZMajorana = Pf i∂ − m = Det i∂ − m (8.124) / where Pf is the Pfaffian or, what is the same, the square root of the deter- C / D / C / D0 minant. Thus the vacuum energy of a Majorana fermion is half the vacuum energy of a Dirac fermion. Finally, since a massless Dirac fermion is equivalent to two Weyl fermions, one for each chirality, it follows that the vacuum energy of a Majorana Weyl fermion is equal and opposite to the vacuum energy of a scalar field. Moreover this relation holds for all the states of their spectra which are identical. This observation is the origin of the concept of supersymmetry in which the fermionic states and the bosonic states are precisely matched. As aresult,thevacuumenergyofasupersymmetrictheoryiszero. We end by discussing briefly the theory of Dirac fermions in Euclidean space-time. We will focus on the theory in four dimensions, but this can be done in any dimension. There are two equivalent ways to do thisanalytic continuation. One option is to define a set of four anti-hermitian Dirac gamma matrices j 0 γj = γ ,γ4 = −iγ (8.125) with j = 1, 2, 3, that satisfy the algebra

γµ,γν = −2δµν (8.126) with µ = 1,...,4. Similarly, we deﬁne the hermitian γ matrix { } 5 5 γ5 = γ = γ1γ2γ3γ4 (8.127) In this notation the partition function is

Z = DψDψ¯ exp −S ψ, ψ¯ (8.128) " E where SE is the Euclidean for a Dirac spinor( ( which)) coupled to an abelian gauge field becomes S = d4xψ¯ iD + m ψ (8.129) E " = = = where Dµ ∂µ + iAµ,againwithµ( / 1, 2, 3), 4, with A4 −iA0.The Euclidean Dirac propagator in momentum space is given by (with p0 = −ip4) 1 p + m S p = = (8.130) −p + m p2 + m2 / Alternatively, we can define( ) the four gamma matrices to be hermitian and / 234 Coherent State Path Integral Quantization of Quantum Field Theory satisfy the Clifford algebra γµ,γν = 2δµnu.InthisnotationtheEuclidean action is S{ = d}4xψ¯ D + m ψ (8.131) E " where the operator ∂ is anti-hermitian.( In/ this) notation the propagator is 1 ip + m / S p = = (8.132) −ip + m p2 + m2 / ( ) 8.8 Functional/ determinants We will now face the problem of how to compute functional determinants. We have discussed before how to do that for path-integrals with a few de- grees of freedom (i.e. in Quantum Mechanics). We will now generalize these ideas to Quantum Field Theory. We will begin by discussing some simple determinants that show up in systems of fermions and bosons atfinitetem- perature and density.

8.8.1 Functional determinants for coherent states Consider a system of fermions (or bosons) with one-body Hamiltonian hˆ at non-zero temperature T and chemical potential µ.Thepartitionfunction −β Hˆ − µNˆ Z = tr e (8.133) where β = 1 kBT , & ' † Hˆ = dx ψˆ x hˆ ψˆ x (8.134) / " and ( ) ( ) † Nˆ = dx ψˆ x ψˆ x (8.135) " is the number operator. Here x denotes( ) both( ) spacial and internal (spin) labels. The functional (or path) integral expression for the partition function is

i ∗ dt ψ ih∂ − hˆ + µ ψ ∗ ̵ t Z = Dψ Dψeh " (8.136) " ̵ → − ≤ ≤& ' In imaginary time set t iτ,with0 τ βh̵ : βh ̵ ∗ − + ˆ − ∗ dτ ψ ∂τ h µ ψ Z = Dψ Dψe "0 (8.137) " & ' 8.8 Functional determinants 235 The fields ψ τ can represent either be bosons,inwhichcasetheyarejust complex functions of x and τ,orfermions,inwhichcasetheyarecomplex Grassmann functions( ) of x and τ.Theonlysubtletyresidesinthechoiceof boundary conditions 1. Bosons: Since the partition function is a trace, in this case the fields(becomplex or real) must obey the usual periodic boundary conditions in imaginary time, = + ψ τ ψ τ βh̵ (8.138) 2. Fermions: ( ) ( ) In the case of fermions the fields are complex Grassmann variables. How- ever, if we want to compute a trace it turns out that, due to the anti- commutation rules, it is necessary to require the fields to obey instead anti-periodic boundary conditions, = − + ψ τ ψ τ βh̵ (8.139) Let λ be a complete set of eigenstates of the one-body Hamiltonian hˆ, ( ) ( ) ελ be its eigenvalue spectrum with λ aspectralparameter.Hence,we have{ a∣ suitable⟩} set of quantum numbers spanning the spectrum of hˆ,andlet {φλ}τ be the associated complete set of eigenfunctions. We now expand the field configurations in the basis of eigenfunctions of hˆ, { ( )} ψ τ = ψ φ τ (8.140) ! λ λ λ The eigenfunctions of hˆ are complete( ) and orthonormal.( ) Thus, if we expand the fields, the path-integral of Eq. (8.136)becomes = (with h̵ 1) ∗ − dτ ψλ −∂τ − ελ + µ ψλ ∗ " ! Z = dψ dψ e λ (8.141) " $ λ λ λ ( ) which becomes) (after absorbing* all uninteresting constant factors in the integration measure) σ Z = Det −∂ − ε + µ (8.142) $ τ λ λ where σ = +1forfermions and[σ = (−1forbosons.)] λ Let ψn τ be the solution of the equation λ λ −∂τ − ελ + µ ψ τ = αnψ τ (8.143) ( ) n n ( ) ( ) ( ) 236 Coherent State Path Integral Quantization of Quantum Field Theory λ where αn is the (generally complex) eigenvalue. The eigenfunctions ψn τ will be required to satisfy either periodic or anti-periodicboundarycondi- tions, ( ) λ λ ψn τ = −σψn τ + β (8.144) where, once again, σ = ±1. ( ) ( ) The eigenvalue condition, Eq. (8.143) is solved by

iαnτ ψn τ = ψn e (8.145) provided α satisﬁes n ( ) αn = −iωn + µ − ελ (8.146) where the Matsubara frequencies are given by (with kB = 1)

πT n + 1 , 2 2 for fermions ωn = (8.147) 2πTn, for bosons & ' Let us consider now the functionF ϕα τ which is an eigenfunction of −∂τ − ελ + µ, ( ) −∂τ − ελ + µ ϕα τ = αϕα τ (8.148) which satisfies only an initial condition for ϕλ 0 ,suchas ( ) ( ) α ( ) λ ϕ 0 = 1(8.149) α ( ) Notice that since the operator is linear in ∂ we cannot impose additional ( ) τ conditions on the derivative of ϕα. The solution of λ ∂τ ln ϕα τ = µ − ε − α (8.150) is ( ) λ λ µ − ελ − α τ ϕα τ = ϕα 0 e (8.151) After imposing the initial condition of( Eq. (8.149),) we find ( ) ( ) λ − α + ελ − µ τ ϕα τ = e (8.152) λ But, although this function ϕ τ satisfies( all the) requirements, it does not ( α) have the same zeros as the determinant Det −∂τ + µ − ελ − α .However, the function ( ) λ λ( ) σFα τ = 1 + σϕα τ (8.153)

( ) ( ) 8.8 Functional determinants 237 does satisfy all the requirements. Indeed,

λ − α + ελ − µ β σFα β = 1 + σe (8.154) which vanishes for α = α .Then,aversionofColeman’sargument,discussed( ) n( ) in Sec.5.2.2, tells us that −∂ + µ − ε − α Det τ λ = λ constant (8.155) σFα β ( ) where the right hand side is a constant in the sense that it doesnotdepend ( ) on the choice of the eigenvalues ελ . Hence, { } λ Det −∂τ + µ − ελ = const. σF0 β (8.156) The partition function is ( ) ( ) −βF σ Z = e = Det −∂ + µ − ε (8.157) $ τ λ λ where F is the free energy, which we[ ﬁnd( it is given by)]

F = −σT ln Det −∂ + µ − ε − α ! τ λ λ β µ − ε = −σT ln 1 +(σe λ + f) βµ (8.158) ! λ ( ) which is the correct result for non-interacting: fermions= ( an) dbosons.Here, we have set 0fermions f βµ = (8.159) N βµ ⎧−2T ln 1 − e bosons ⎪ ( ) ⎨⎪ where N is the number of⎪ states in the+ spectrum, λ . ⎩⎪ In some cases the spectrum has the symmetry ελ = −ε−λ, e. g. the Dirac 2 2 theory whose spectrum is ε± = ± p + m ,andtheseexpressionscanbe{ } simplified further, K Det −∂ + iε = Det −∂ + iε Det −∂ − iε $ t λ $ t λ t λ λ λ>0 ( ) = [ Det( ∂2 + ε2 ) ( )] $ t λ λ>0 ≡ Det &−∂2 + ε'2 (8.160) $ τ λ λ>0 & ' 238 Coherent State Path Integral Quantization of Quantum Field Theory where, in the last step, we performed a Wick rotation. This last expression we have encountered before. The result is Det −∂2 + ε2 = const.ψ β (8.161) $ τ λ 0 λ>0 where ψ0 τ is the solution of& the differential' equation( ) −∂2 + ε2 ψ τ = 0(8.162) ( ) τ λ 0 which satisfies the initial conditions & ' ( ) ψ0 0 = 0 ∂τ ψ0 0 = 1(8.163) The solution is ( ) ( ) sinh ελ τ ψ0 τ = (8.164) ελ (∣ ∣ ) Hence ( ) ∣ ∣ ελ β sinh ελ β e ψ0 β = ⟶ as β → ∞, (8.165) ελ 2 ελ ∣ ∣ (∣ ∣ ) In particular,( since) ∣ ∣ ∣ ∣ ε β β ελ −β ελ e λ ! ! = e λ>0 = e λ<0 (8.166) $ 2 ελ λ>0 ∣ ∣ ∣ ∣ we get that the ground state energy E is the sum of the single particle ∣ ∣ G energies of the occupied (negative energy) states: E = ε (8.167) G ! λ λ<0

8.8.2 Functional determinants, heat kernels and ζ-function regularization We have seen before that the evaluation of the effects of quantumfluctua- tions involves the calculation of the determinant of a differential operator. In the case of non-relativistic single particle Quantum Mechanics, is Section 5.2.2 we discussed in detail how to calculate a functional determinant of 2 the form Det −∂t + W t .Howeverthemethodweusedforthatpurpose becomes unmanageably cumbersome if applied to the calculation of determi- 2 nants of partial# differential( )$ operators of the form Det −D + W x ,where x ≡ xµ and D is some differential operator. Fortunately there are better and more efficient ways of doing such calculations. # ( )$ 8.8 Functional determinants 239

Let Aˆ be an operator, and fn x be a complete set of eigenstates of Aˆ, with the eigenvalue spectrum an ,suchthat { ( )} Afˆ x = a f x (8.168) { n } n n We will assume that Aˆ has a discrete spectrum of real positive eigenvalues, ( ) ( ) and hence it is bounded from below. For the case of a continuousspectrum we will put the system in a finite box, which makes the spectrum discrete, and take limits at the end of the calculation. The function ζ s , ∞ 1 ( ) ζ s = , for Re s > 1(8.169) ! ns n=1 is the well known Riemann( ) ζ-function. We will now use the eigenvalue spectrum of the operator Aˆ to define the generalized ζ-function 1 ζ s = (8.170) A ! as n n where the sum runs over the labels( ) (here denoted by n)ofthespectrumof the operator Â.Wewillassumethatthesum(infiniteseries)isconvergent which, in practice, will require that we introduce some sort of regularization at the high end (high energies) of the spectrum. Upon differentiation we find

dζA d −s ln a ln an = e n = − (8.171) ds ! ds ! as n n n + where we have assumed convergence. Then, in the limit s → 0 we ﬁnd that the following identity holds

dζA lim = − ln an = − ln an ≡ − ln DetA (8.172) → + ds ! $ s 0 n n

Hence, we can formally relate the generalized ζ-function, ζA,tothefunc- tional determinant of the operator A dζ A = − ln DetA (8.173) ds s→0+ E We have thus reduced the computationE of a determinant to the computation E of a function with speciﬁc properties.E Let us deﬁne now the generalized Heat Kernel

−a τ ∗ −τAˆ G x, y; τ = e n f x f y ≡ x e y (8.174) A ! n n n ( ) ( ) ( ) ⟨ ∣ ∣ ⟩ 240 Coherent State Path Integral Quantization of Quantum Field Theory where τ > 0. The Heat Kernel GA x, y; τ clearly obeys the diﬀerential equation

−∂τ GA x, y; τ (= AGˆ A) x, y; τ (8.175) which can be regarded as a generalized Heat Equation. Indeed,forAˆ = 2 ( ) ( ) −D ! ,thisistheregularHeatEquation(whereD is the diffusion constant) and, in this case τ represents time. In general we will refer to τ as proper time. The Heat Kernel GA x, y; τ satisfies the initial condition ∗ lim GA x, y; τ = fn x fn y = δ x − y (8.176) → + ! τ 0 ( ) n where we have used the( completeness) relation( ) of( ) the eigenfun( )ctions fn x . Hence, GA x, y; τ is the solution of a generalized Heat Equation with kernel Aˆ.ItdefinesageneralizedrandomwalkorMarkovprocess. { ( )} We will now( show) that GA x, y; τ is related to the generalized ζ-function ζA s .Indeed,letusconsidertheHeatKernelGA x, y; τ at short distances, y → x,andcomputetheintegral(belowwedenoteby( ) D is the dimensionality of( space-time)) ( )

D −anτ D ∗ d x lim GA x, y; τ = e d xfn x fn x " y→x ! " n −a τ −τAˆ ( ) = e n ≡ tr e ( ) ( ) (8.177) ! n where we assumed that the eigenfunctions are normalized to unity

D d x f x 2 = 1(8.178) " n i.e. normalized inside a box. ∣ ( )∣ We will now use that, for s > 0andan > 0(oratleastthatithasa positive real part) we can write ∞ s−1 −anτ Γ s dτ τ e = s (8.179) "0 an where Γ s is the Euler Gamma function: ( ) ∞ s−1 −τ ( ) Γ s = dτ τ e (8.180) "0 Then, we obtain the identity( ) ∞ s−1 D Γ s dτ τ d x lim GA x, y; τ = (8.181) " " y→x ! as 0 n n ( ) ( ) 8.8 Functional determinants 241

Therefore, we ﬁnd that the generalized ζ-function, ζA s ,canbeobtained from the generalized Heat Kernel GA x, y; τ : ∞ ( ) 1 s−1 D ζA s = dτ τ (d x lim) GA x, y; τ (8.182) Γ s "0 " y→x This result( suggests) the following strategy for the( computa) tion of deter- ( ) minants. Given the hermitian operator Aˆ,wesolvethegeneralizedHeat Equation

AGˆ A = −∂τ GA (8.183) subject to the initial condition

D lim GA x, y; τ = δ x − y (8.184) τ→0+

Next we ﬁnd the associated ζ-function,( ) ζA s(,usingtheexpression) ∞ 1 s−1 D ζA s = dτ τ d x( lim) GA x, y; τ (8.185) Γ s "0 " y→x where we recognize( ) the identity ( ) ( ) −τAˆ D tr e = d x lim GA x, y; τ (8.186) " y→x

+ We next take the limit s → 0 to relate the generalized( ) ζ-function to the determinant: dζ s lim A = − ln det Aˆ (8.187) s→0+ ds ( ) In practice we will have to exercise some care in this step since we will ﬁnd singularities as we take this limit. Most often we will keep the points x and y apart by a small but ﬁnite distance a,whichwewilleventuallyattemptto take to zero. Hence, we will need to understand in detail the short distance behavior of the Heat Kernel. Furthermore, the propagator of the theory

−1 SA x, y = x Aˆ y (8.188) can also be related to the Heat( Kernel.) ⟨ ∣ Indeed,∣ ⟩ by expending Eq.(8.188) in the eigenstates of Aˆ,weﬁnd ∗ x n n y fn x fn y SA x, y = = (8.189) ! an ! an n n ⟨ ∣ ⟩⟨ ∣ ⟩ ( ) ( ) ( ) 242 Coherent State Path Integral Quantization of Quantum Field Theory We can now write the following integral of the Heat Kernel as

∞ ∞ ∗ ∗ −anτ fn x fn y dτGA x, y; τ = fn x fn y dτe = " ! " ! an 0 n 0 n ( ) (8.190)( ) ( ) ( ) ( ) Hence, the propagator SA x, y can be expressed as an integral of the Heat Kernel ( ) ∞ SA x, y = dτGA x, y; τ (8.191) "0

Equivalently, we can say that( ) since GA x, y; τ( satisﬁes) the Heat Equation −τAˆ AGˆ A = −∂τ GA, then GA (x, y; τ )= x e y (8.192)

Thus, ( ) ⟨ ∣ ∣ ⟩ ∞ −τAˆ −1 SA x, y = dτ x e y = x Aˆ y (8.193) "0 and that it is indeed( the) Green function⟨ ∣ of ∣Aˆ,⟩ ⟨ ∣ ∣ ⟩

AˆxS x, y = δ x − y (8.194)

G x, y τ It is worth to note that the Heat( Kernel) ( A ) ; ,ascanbeseenfrom Eq. (8.192), is also the Gibbs density matrix of the bounded Hermitian operator Aˆ.Assuchithasanimaginarytime(τ!)( path-integral) representation. Here, to actually insure convergence, we must also require that the spectrum of Aˆ be positive. In that picture we view GA x, y; τ as the amplitude for the imaginary-time (proper time) evolution from the initialstate y to the ﬁnal state x .Inotherwords,wepictureSA(x, y as) the amplitude to go from y to x in an arbitrary time. ∣ ⟩ ∣ ⟩ ( )

8.9 The determinant of the Euclidean Klein-Gordon operator As an example of the use of the Heat Kernel method we will use it to compute the determinant of the Euclidean Klein-Gordon operator. Thus, we will take the hermitian operator Aˆ to be

Aˆ = −&2 +m2 (8.195) in D Euclidean space-time dimensions. This operator has a bounded positive spectrum. Here we will be interested in a system with infinite size L → ∞, and a large volume V = LD.Wewillfollowthestepsoutlinedabove. 8.9 The determinant of the Euclidean Klein-Gordon operator 243 We begin by constructing the Heat Kernel G x, y; τ .Bydefinitionitis the solution of the partial differential equation

2 2 ( ) −& +m G x, y; τ = −∂τ G x, y; τ (8.196) satisfying the initial& condition' ( ) ( ) D lim G x, y; τ = δ x − y (8.197) τ→0+ We will ﬁnd G x, y; τ by Fourier( transforms,) ( )

D d p ip ⋅ x − y (G x, y); τ = G p,τ e (8.198) " 2π D ( ) We ﬁnd that in order( for) G x, y; τ to satisfy( ) Eq.(8.196), its Fourier trans- ( ) form G p; τ must satisfy the diﬀerential equation ( ) − = 2 + 2 ( ) ∂τ G p; τ p m G p; τ (8.199)

The solution of this equation,( consistent) & with the' initial( ) condition of Eq.(8.197) is − p 2 + m2 τ G p; τ = e (8.200)

& ' We can now easily ﬁnd G(x, y;)τ by simply ﬁnding the anti-transform of G p; τ :

( D ) d p − p 2 + m2 τ + ip ⋅ x − y ( ) G x, y; τ = e " 2π D & 'x − y 2 ( ) ( ) − m2τ + ( 1 ) 4τ = e (8.201) D 2 ∣ ∣ 4πτ ) * Notice that for m → 0, G x, y; τ/ reduces to the usual diffusion kernel (with ( ) unit diffusion constant) ( ) x − y 2 1 − lim G x, y; τ = e 4τ (8.202) m→0 D 2 4πτ ∣ ∣ Next we construct the(ζ-function) / ( ) ∞ 1 s−1 D ζ−&2+m2 s = dτ τ d x lim G x, y; τ (8.203) Γ s "0 " y→x ( ) ( ) ( ) 244 Coherent State Path Integral Quantization of Quantum Field Theory We first do the integral ∞ s− D dτ τ 1 d xG x, y; τ "0 " 2 ( ) 2 R ∞ − m τ + V s− −D 4τ = dτ τ 1 2 e (8.204) D 4π 2 "0 / ) * where R = x − y .Uponscalingthevariable/ τ = λt,withλ = R 2m,we find that ( ) ∞ ∣ ∣ s− D / s−1 2 R 2 dτ τ G x, y; τ = K D mR (8.205) D −s "0 4π 2 2m 2 / where Kν z , ( ) : = ( ) ( ) z 1 ∞ − t + ( ) 1 ν−1 2 t Kν z = dt t e (8.206) 2 "0 : = is a modified Bessel function.( ) Its short argument behavior is Γ ν 2 ν K z ∼ + ... (8.207) ν 2 z ( ) As a check, we notice that for( s)= 1theintegralofEq.(8.204)doesreproduce: = the Euclidean Klein-Gordon propagator that we discussed earlier in these lectures. The next step is to take the short distance limit ∞ 1−s D−2s s−1 2 m = D lim dτ τ G x, y; τ lim D K −s mR R→0 " R→0 D 2 −s 2 0 2π mR 2 ( ) s − D/ ( ) Γ 2 = ( ) ( ) (8.208) 4π D 2m2s−D & ' Notice that we have exchanged the order/ of the limit and the integration. Also, after we took the short distance( ) limit R → 0, the expression above acquired a factor of Γ s − D 2 ,whichissingularass−D 2approacheszero (or any negative integer). Thus, a small but finite R smears this singularity. Finally we find the (ζ-function/ ) by doing the (trivial) integration/ over space ∞ 1 D ζ s = dτ d x lim G x, y; τ Γ s "0 " y→x D D s − − s ( ) −2s m Γ 2 (m 2 ) = Vµ( ) (8.209) 4π D 2 Γ s µ & ' / + , ( ) ( ) 8.10 Path integral for spin 245 where µ = 1 R plays the role of a cutoffmass (or momentum) scale that we will need to make some quantities dimensionless. The appearance of this quantity is also/ a consequence of the singularities. We will now consider the specific case of D = 4dimensions.ForD = 4 the ζ-function is − m4 µ 2s m −2s ζ s = V (8.210) 16π2 s − 1 s − 2 µ We can now compute( the) desired (logarithm of the)+ , determinantforD = 4 ( )( ) dimensions: dζ m4 m 3 ln Det −&2 +m2 = − lim = ln − V (8.211) s→0+ ds 16π2 µ 4 where V = L4#.Asimilarcalculationfor$ D = 2yieldstheresult7 8

m2 m 1 ln Det −&2 +m2 = − ln − V (8.212) 2π µ 2 where V = L2. # $ 7 8

8.10 Path integral for spin We will now discuss the use of path integral methods to describe a quantum mechanical spin. Consider a quantum mechanical system whichconsistsof aspininthespin-S representation of the group SU 2 .Thespaceofstates of the spin-S representation is 2S + 1-dimensional, and it is spanned by the 2 basis S, M which are the eigenstates of the operators( ) S and S3,i.e. S2 S, M =S S + 1 S, M {∣ ⟩} S3 S, M =M S, M (8.213) ∣ ⟩ ( )∣ ⟩ with M ≤ S (in integer-spaced intervals). This set of states is complete ad ∣ ⟩ ∣ ⟩ it forms a basis of this Hilbert space. The operators S1, S2 and S3 obey the SU 2∣ algebra,∣

Sa,Sb = i(abcSc (8.214) ( ) where a, b, c = 1, 2, 3. [ ] The simplest physical problem involving spin is the couplingtoanexternal magnetic ﬁeld B through the Zeeman interaction

HZeeman = µ B ⋅ S (8.215) 246 Coherent State Path Integral Quantization of Quantum Field Theory where µ is the Zeeman coupling constant ( i.e. the product of the Bohr magneton and the gyromagnetic factor). Let us denote by 0 the highest weight state S, S .Letusdeﬁnethespin ± raising and lowering operators S , ∣ ⟩ ± ∣ ⟩ S = S1 ± iS2 (8.216) + The highest weight state 0 is annihilated by S , + + S 0 = S S, S = 0(8.217) ∣ ⟩ Clearly, we also have ∣ ⟩ ∣ ⟩ S2 0 =S S + 1 0

S3 0 =S 0 (8.218) ∣ ⟩ ( )∣ ⟩ ∣ ⟩ ∣ ⟩

n0 θ n n0 × n

Figure 8.1 Geometry for a spin coherent state n (see text).

Let us consider now the spin coherent state n ,(Perelomov,1986)∣ ⟩

iθn0 × n ⋅ S n = e ∣0 ⟩ (8.219) where n is a three-dimensional unit vector (n2 = 1), n is a unit vector ∣ ⟩ ∣ ⟩ 0 pointing along the direction of the quantization axis (i.e. the “North Pole” of the unit sphere) and θ is the colatitude,(seeFig.8.1)

n ⋅ n0 = cos θ (8.220) As we will see the state n is a coherent spin state which represents a

∣ ⟩ 8.10 Path integral for spin 247 spin polarized along the n axis. The state n can be expanded in the basis S, M , S ∣ ⟩ S ∣ ⟩ n = D n S, M (8.221) ! MS M=−S ( ) S ∣ ⟩ ( )∣ ⟩ Here DMS n are the representation matrices in the spin-S representation. It is( important) to note that there are many rotations that leadtothe same state( n) from the highest weight 0 .Forexampleanyrotationalong the direction n results only in a change in the phase of the state n .These rotations are∣ equivalent⟩ to a multiplication∣ ⟩ on the right by arotationabout the z axis. However, in Quantum Mechanics this phase has no∣ physic⟩ ally observable consequence. Hence we will regard all of these states as being physically equivalent. In other words, the states form equivalence classes (or rays)andwemust pick one and only one state from each class. These rotations are generated by S3,the(only)diagonalgeneratorofSU 2 .Hence,thephysicalstatesare not in one-to-one correspondence with the elements of SU 2 but instead with the elements of the right coset SU 2( )U 1 ,withtheU 1 generated by S3.InthecaseofofcoherentstateofamoregeneralLiegroup,t( ) he coset is obtained by dividing out the maximal( torus)/ ( generated) by al(lthediagonal) generators of the group. In mathematical language, if we consider all the rotations at once, the spin coherent states are said to form a hermitian line bundle.

n1 n2

Figure 8.2 Spherical triangle with vertices at the unit vectors, n1, n2 and n3.

AconsequenceoftheseobservationsisthattheD matrices do not form 248 Coherent State Path Integral Quantization of Quantum Field Theory agroupundermatrixmultiplication.Insteadtheysatisfy

S S S iΦ n1, n2, n3 S3 D n1 D n2 = D n3 e (8.222) ( ) ( ) ( ) where the phase factor is usually called a cocycle( .Here)Φn1, n2, n3 is the ( ) ( ) ( ) (oriented) area of the spherical triangle with vertices at n1, n2, n3. However, since the sphere is a closed surface, which area( do weactually) mean? “Inside” or “outside”? Thus, the phase factor is ambiguous by an amount determined by 4π,thetotalareaofthesphere, i4πM e (8.223) However, since M is either an integer or a half-integer this ambiguity in Φ has no consequence whatsoever, i4πM e = 1(8.224) We can also regard this result as a requirement that M be quantized to be an integer or a half-integer, i.e. the representations of SU 2 . The states n are coherent states which satisfy the following properties (Perelomov, 1986). The overlap of two coherent states n and n is ⃗1( ) 2 ∣ ⟩ S † S n n = 0 D n D n 0 1 2 1 2 ∣ ⟩ ∣ ⟩ (S) iΦ( n) , n , n S = 0 D n e 1 2 ⃗0 3 0 ⟨ ∣ ⟩ ⟨ ∣ ( 0) ( )∣ ⟩ ( ) S 1 + n1 ⋅ n2 iΦ n , n , n S = (e 1 2) 0 (8.225) ⟨ ∣ 2( ) ∣ ⟩ ( ) The (diagonal) matrix element: of the spin= operator is n S n = S n (8.226) Finally, the (over-complete) set of coherent states n have a resolution ⟨ ∣ ∣ ⟩ of the identity of the form {∣ ⟩} Iˆ = dµ n n n (8.227) " where the integration measure dµ n( is)∣ ⟩⟨ ∣ 2S + 1 dµ n = δ n2 − 1 d3n (8.228) (4π)

Let us now use the coherent( ) : states = n( to ﬁnd) the path integral for a spin. In imaginary time τ (and with periodic boundary conditions) the path integral is simply the partition function{∣ ⟩} −βH Z = tre (8.229) 8.10 Path integral for spin 249 where β = 1 T (T is the temperature) and H is the Hamiltonian. As usual the path integral form of the partition function is found by splitting up the imaginary time/ interval 0 ≤ τ ≤ β in Nτ steps each of length δτ such that Nτ δτ = β.Hencewehave −δτH Nτ Z = lim tr e (8.230) Nτ →∞,δτ→0 and insert the resolution of the identity at+ every, intermediate time step, N N τ τ −δτH Z = lim dµ nj n τj e n τj+1 N →∞,δτ→0 $ " $ τ j=1 j=1 ⎛ ⎞ ⎛ ⎞ N⎜τ ( )⎟N⎜τ ⟨ ( )∣ ∣ ( )⟩⎟ ≃ lim ⎝ dµ nj ⎠ ⎝ n τj n τj+1 − δτ n τj⎠ H n τj+1 N →∞,δτ→0 $ " ⃗ $ ⃗ τ j=1 j=1 ⎛ ⎞ ⎛ ⎞ ⎜ ( )⎟ ⎜ ;⟨ ( )∣ ( )⟩ ⟨ ( )∣ ∣ ((8.231))⟩<⎟ ⎝ ⎠ ⎝ ⎠ However, since

n τj H n τj+1 ≃ n τj H n τj = µSB ⋅ n τj (8.232) n τj n τj+1 ⟨ ( )∣ ∣ ( )⟩ and ⟨ ( )∣ ∣ ( )⟩ ( ) ⟨ ( )∣ ( )⟩ S 1 + n τj ⋅ n τj+1 iΦ n τ , n τ , n S n τ n τ = e j j+1 0 j j+1 2 ( ) ( ) ( ( ) ( ) (8.233)) we can⟨ ( write)∣ ( the partition)⟩ : function in the form= −S n Z = lim Dn e E (8.234) Nτ →∞,δτ→0 " [ ] where SE n is given by

Nτ −S n[ =] iS Φ n τ , n τ , n E ! j ⃗ j+1 0 j=1

Nτ Nτ [ ] (1 +( n) τj( ⋅ n )τj+1 ) +S ln ⃗ − δτ µS n τ ⋅ B (8.235) ! 2 ! j j=1 j=1 ( ) ( ) The ﬁrst term of the) right hand side of Eq.* (8.235)( ) contains( ) theexpression Φ n τ , n τ , n which has a simple geometric interpretation: it is the j ⃗ j+1 0 sum of the areas of the Nτ contiguous spherical triangles. These triangles have( ( the) pole( n0) as a) common vertex, and their other pairs of vertices trace asphericalpolygonwithverticesat n τj .Inthetimecontinuumlimit this spherical polygon becomes the history of the spin, which traces a closed { ( )} 250 Coherent State Path Integral Quantization of Quantum Field Theory + oriented curve Γ = n τ (with 0 ≤ τ ≤ β). Let us denote by Ω the region of the sphere whose boundary is Γand which contains thepolen0. − The complement of this{ ( region)} is Ω and it contains the opposite pole −n0. Hence we ﬁnd that + − lim Φ n τj , n τj+1 , n0 = A Ω = 4π − A Ω (8.236) Nτ →∞,δτ→0 where A Ω is the( area( ) of the( region) ) Ω. Once[ ] again, the[ ambiguity] of the area leads to the requirement that S should be an integer or a half-integer. [ ] Ω+ n0

n τj

n τj+1 ( ) (

Ω−

Figure 8.3 The function n τ,s as arbitrary extension of the of the history n τ to the upper cap Ω+ of the sphere S2. ( ) There is a simple an elegant way to write the area enclosed by Γ.Letn τ ( ) ⃗ be a history and Γbe the set of points o the 2-sphere traced by n τ for ⃗ 0 ≤ τ ≤ β.Letusdeﬁnen τ,s (with 0 ≤ s ≤ 1) to be an arbitrary extension( ) + of n τ from the curve Γto the interior of the upper cap Ω ,asshownin( ) Fig.8.3, such that ( ) ( ) n τ,0 = n τ , n τ,1 = n0, n τ,0 = n τ + β,0 (8.237) Then the area can be written in the compact form ( ) ( ) ( ) ( ) ( ) 1 β + A Ω = ds dτ n τ,s ⋅ ∂τ n τ,s × ∂sn τ,s ≡ SWZ n (8.238) "0 "0 In Mathematics[ ] this expression( ) for the( area) is called( the) (simplectic)[ ] 2-form, and in the literature is usually called a Wess-Zumino action (Witten, 1984), SWZ,oraBerry phase.(Berry,1984;Simon,1983)Thecoherentstatepath integral for spin is a special case of the method of geometric quantization (Wiegmann, 1989). 8.10 Path integral for spin 251

Total Flux Φ = 4πS

Figure 8.4 A hairy ball or monopole

Thus, in the (formal) time continuum limit, the action SE becomes (Frad- kin and Stone, 1988)

β β Sδτ 2 SE = −iS SWZ n + dτ ∂τ n τ + dτµS B ⋅ n τ (8.239) 2 "0 "0 Notice that we have[ ] kept (temporarily)( ( a)) term of order δτ,whichwewill( ) drop shortly. How do we interpret Eq. (8.239)? Since n τ is constrained to be a point 2 on the surface of the unit sphere, i.e. n = 1, the action SE n can be interpreted as the action of a particle of mass M = Sδτ → 0andn τ is ( ) ⃗ the position vector of the particle at (imaginary) time τ.Thus,thesecond[ ] term is a (vanishingly small) kinetic energy term, and the last term of( ) Eq. (8.239) is a potential energy term. What is the meaning of the ﬁrst term? In Eq. (8.238) we saw that SWZ n , the the so-called Wess-Zumino or Berry phase term in the action, is the area of the (positively oriented) region A Ω+ “enclosed” by the “path” n τ [.In] fact, 1 [ β ] ( ) SWZ n = ds dτn ⋅ ∂τ n × ∂sn (8.240) "0 "0 + is the area of the oriented[ ] surface Ω whose boundary is the oriented path + Γ = ∂Ω (see Fig. 8.3). Using Stokes theorem we can write the the expression SA n as the circulation of a vector ﬁeld A n ,

[ ] dn ⋅ A n τ = dS ⋅[!] n × A n τ (8.241) (∂Ω )Ω+ [ ( )] [ ( )] 252 Coherent State Path Integral Quantization of Quantum Field Theory provided the “magnetic ﬁeld” !n × A is “constant”, namely

B = !n × A n τ = S n τ (8.242)

In other words, this is the magnetic field[ ( of)] a magnetic( ) monopole located at the center of the sphere. What is the total flux Φof this magnetic field?

Φ = dS ⋅ !n × A n = S dS ⋅ n ≡ 4πS (8.243) "sphere "

Thus, the total number of ﬂux quanta[ ] Nφ piercing the unit sphere is Φ N = = 2S = magnetic charge (8.244) φ 2π We reach the condition that the magnetic charge is quantized,aresultknown as the Dirac quantization condition. Is this result consistent with what we know about charged particles in magnetic ﬁelds? In particular, how is this result related to the physics of spin? To answer these questions we will go back to real time andwritethe action

T M dn 2 dn S n = dt + A n t ⋅ − µSn t ⋅ B (8.245) "0 2 dt dt

[ ] - 2 : = [ ( )] ( ) . with the constraint n = 1andwherethelimitM → 0isimplied. The classical hamiltonian associated to the action of Eq. (8.245) is 1 2 H = n × p − A n + µSn ⋅ B ≡ H + µSn ⋅ B (8.246) 2M 0

It is easy to check/ that( the[ vector])0 Λ, Λ = n × p − A (8.247) satisﬁes the algebra ( ) = − Λa, Λb ih̵ (abc Λc hSn̵ c (8.248) = where a, b, c 1, 2, 3, (abc[ is the] (third rank)( Levi-Civita) tensor, and with Λ ⋅ n = n ⋅ Λ = 0(8.249) the generators of rotations for this system are

= + L Λ hS̵ n (8.250) 8.10 Path integral for spin 253 The operators L and Λ satisfy the (joint) algebra

= − 2 = La,Lb ih̵ (abcLs La, L 0 (8.251) = = [ La,nb] ih̵ (abcnc La#, Λb $ih̵ (abcΛc Hence [ ] [ ] 2 La, Λ = 0 ⇒ La,H = 0(8.252) since the operators La satisfy the angular momentum algebra, we can diago- 2 # $ [ ] nalize L and L3 simultaneously. Let m, 6 be the simultaneous eigenstates 2 of L and L3, ∣ ⟩ 2 = 2 + L m, 6 h̵ 6 6 1 m, 6 (8.253) = L3 m, 6 hm̵ m, 6 (8.254) ∣ ⟩ (2 )∣ + ⟩ − h̵ 6 6 1 S H0 m, 6 = m, 6 (8.255) ∣ ⟩ 2MR∣ 2 ⟩ 2S ( ) where R = 1istheradiusofthesphere.Theeigenvalues∣ ⟩ : = ∣ ⟩ 6 are of the form + + 6 = S + n, m ≤ 6,withn ∈ Z ∪ 0 and 2S ∈ Z ∪ 0 .Henceeachlevel is 26 + 1-fold degenerate, or what is equivalent, 2n + 1 + 2S-fold degenerate. Then, we get∣ ∣ { } { } 2 = 2 − 2 2 2 = 2 − 2 2 Λ L n h̵ S L h̵ S (8.256)

Since M = Sδt → 0, the lowest energy in the spectrum of H0 are those with the smallest value of 6, i. e. states with n = 0and6 = S.Thedegeneracyof this “Landau” level is 2S +1, and the gap to the next excited states diverges as M → 0. Thus, in the M → 0limit,thelowest energy states have the same degeneracy as the spin-S representation. Moreover, the operators L 2 and L3 become the corresponding spin operators. Thus, the equivalency found is indeed correct. Thus, we have shown that the quantum states of a scalar (non-relativistic) particle bound to a magnetic monopole of magnetic charge 2S,obeyingthe Dirac quantization condition, are identical to those of those of a spinning particle! (Wu and Yang, 1976) We close this section with some observations on the semi-classical motion. From the (real time) action (already in the M → 0limit)

T T 1 S = − dt µS n ⋅ B + S dt ds n ⋅ ∂tn × ∂sn (8.257) "0 "0 "0 we can derive a Classical Equation of Motion by looking at the stationary 254 Coherent State Path Integral Quantization of Quantum Field Theory conﬁgurations. The variation of the second term in Eq. (8.257) is T 1 T δS = Sδ dt ds n ⋅ ∂tn × ∂sn = S dtδn t ⋅ n t × ∂tn t (8.258) "0 "0 "0 the variation of the ﬁrst term in Eq. (8.257) is ( ) ( ) ( ) T T δ dt µSn t ⋅ B = dt δn t ⋅ µSB (8.259) "0 "0 Hence, ( ) ( ) T δS = dt δn t ⋅ − µSB + Sn t × ∂tn t (8.260) "0 which implies that the classical( ) + trajectories must( ) satisfy( )the, equation of motion

µB = n × ∂tn (8.261) If we now use the vector identity 2 n × n × ∂tn = n ⋅ ∂tn n − n ∂tn (8.262) and ( ) 2 n ⋅ ∂tn = 0, and n = 1(8.263) we get the classical equation of motion

∂tn = µB × n (8.264)

Therefore, the classical motion is precessional with an angular velocity Ωpr = µB.