5 Path Integrals in Quantum Mechanics and Quantum Field Theory

In chapter 4 we discussed the Hilbert space picture of QuantumMechanics and Quantum Field Theory for the case of a free relativistic scalar ﬁelds. Here we will present the Path Integral picture of Quantum Mechanics and of relativistic scalar ﬁeld theories.

The Path Integral picture is important for two reasons. First, it oﬀers an alternative, complementary, picture of Quantum Mechanics in which the role of the classical limit is apparent. Secondly, it gives a direct route to the study regimes where perturbation theory is either inadequate or fails com- pletely. In Quantum mechanics a standard approach to such problems is the WKB approximation, of Wentzel, Kramers and Brillouin. However, as it happens, it is extremely diﬃcult (if not impossible) the generalize the WKB approximation to a Quantum Field Theory. Instead, the non-perturbative treatment of the Feynman path integral, which in Quantum Mechanics is equivalent to WKB, is generalizable to non-perturbative problems in Quan- tum Field Theory. In this chapter we will use path integrals only for bosonic systems, such as scalar ﬁelds. In subsequent chapters we willalsogivea full treatment of the path integral, including its applications to fermionic ﬁelds, abelian and non-abelian gauge ﬁelds, classical statistical mechanics, and non-relativistic many body systems.

There is a huge literature on path integrals, going back to theoriginal papers by Dirac (Dirac, 1933), and particularly Feynman’s 1942 PhD Thesis (Feynman, 2005), and his review paper (Feynman, 1948). Popular textbooks on path integrals include the classic by Feynman and Hibbs (Feynman and Hibbs, 1965), and Schulman’s book (Schulman, 1981), among many others. 5.1 Path Integrals and Quantum Mechanics 115 5.1 Path Integrals and Quantum Mechanics Consider a simple quantum mechanical system whose dynamics can be de- scribed by a generalized coordinate operatorq ˆ.Wewanttocomputethe amplitude

F qf ,tf qi,ti = qf ,tf qi,ti (5.1) known as the Wightman function. This function represents theamplitude ( ∣ ) ⟨ ∣ ⟩ to ﬁnd the system at coordinate qf at the ﬁnal time tf knowing that it was at coordinate qi at the initial time ti.TheamplitudeF qf ,tf qi,ti is just amatrixelementoftheevolutionoperator

ˆ − ( ∣ ) iH ti tf h̵ F qf ,tf qi,ti = qf e qi (5.2) ( )/ q ,t = , q ,t = q,t Let us set, for simplicity,( i∣ i ) 0 ⟨0 and∣ f f ∣ ⟩ .Then,fromthe deﬁnition of this matrix element, we ﬁnd out that it obeys ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ lim F q,t 0, 0 = q 0 = δ q (5.3) t→0

Furthermore, after some algebra( ∣ we) also⟨ ﬁnd∣ ⟩ that( )

∂F ∂ ∂ −iHtˆ h ih =ih q,t 0, 0 = ih q e ̵ 0 ̵ ∂t ̵ ∂t ̵ ∂t − ˆ / = ˆ iHt h̵ q He⟨ ∣ ⟩0 ⟨ ∣ ∣ ⟩ ′ / ′ ′ −iHtˆ h = dq q Hˆ q q e ̵ 0 (5.4) ⟨"∣ ∣ ⟩ / where we have used that, since⟨ ∣q ∣ is⟩⟨ a complete∣ ∣ set⟩ of states, the identity operator I has the expansion, called the resolution of the identity, {∣ ⟩} ′ ′ ′ I = dq q q (5.5) "

Here we have assumed that the states are∣ ⟩⟨ orthonormal,∣ ′ ′ q q = δ q − q (5.6)

Hence, ⟨ ∣ ⟩ ( ) ∂ ′ ′ ′ ih F q,t 0, 0 = dq q Hˆ q F q ,t 0, 0 ≡ Hˆ F q,t 0, 0 (5.7) ̵ ∂t " q

In other words,( ∣F q,t) 0, 0 is⟨ the∣ solution∣ ⟩ ( of∣ the) Schr¨odinger( ∣ Equation) that satisﬁes the initial condition of Eq. (5.3). For this reason,theamplitude F q,t 0, 0 is called( the∣ Schr¨odinger) Propagator.

( ∣ ) 116 Path Integrals in Quantum Mechanics and Quantum Field Theory

q f qf ,tf

( )

q′ q′,t′

q ( ) i qi,ti

′ t t ti( ) t f

Figure 5.1 The amplitude to go from qi,ti to qf ,tf is a sum of products ′ ′ of amplitudes through the intermediate states q ,t . ∣ ⟩ ∣ ⟩ ∣ ⟩ The superposition principle tells us that the amplitude to ﬁnd the system in the ﬁnal state at the ﬁnal time is the sum of amplitudes of theform

′ ′ ′ ′ ′ F q ,t q ,t = dq q ,t q ,t q ,t q ,t (5.8) f f i i " f f i i ′ where the system( is in∣ an arbitrary) set⟨ of states∣ ⟩⟨ at an∣ interme⟩ diate time t . Here we represented this situation by inserting the identityoperatorI at ′ the intermediate time t in the form of the resolution of the identity of Eq. (5.8). Let us next deﬁne a partition of the time interval ti,tf into N sub- intervals each of length ∆t, [ ] tf − ti = N∆t (5.9)

Let tj ,withj = 0,...,N+ 1, denote a set of points in the interval ti,tf , such that { } [ ] ti = t0 ≤ t1 ≤ ...≤ tN ≤ tN+1 = tf (5.10)

Clearly, tk = t0 + k∆t,fork = 1,...,N+ 1. By repeating the procedure used in Eq.(5.8) of inserting the resolution of the identity at theintermediate times tk ,weﬁnd

F q ,t q ,t = dq ...dq q ,t q ,t q ,t q ,t × ... {f }f i i " 1 N f f N N N N N−1 N−1 × ... q ,t q ,t ... q ,t q ,t ( ∣ ) ⟨ ∣ j j ⟩⟨j−1 j−1 ∣ 1 1 ⟩i i (5.11) ⟨ ∣ ⟩ ⟨ ∣ ⟩ 5.1 Path Integrals and Quantum Mechanics 117

Each factor qj,tj qj−1,tj−1 in Eq.(5.11) has the form − ˆ − − ˆ = iH tj tj−1 h̵ ≡ iH∆t h̵ qj,tj qj−⟨1,tj−1∣ qj e ⟩ qj−1 qj e qj−1 (5.12) ( )/ / In the limit N → ∞,with tf − ti ﬁxed and ﬁnite, the interval ∆t becomes ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ inﬁnitesimally small and ∆t → 0. Hence, as N → ∞ we can approximate the expression for qj,tj qj∣−1,tj−1∣ in Eq.(5.12) as follows − q ,t q ,t = q e iH∆t h̵ q j j j−1 ⟨j−1 ∣ j ⟩ j−1 ∆/ t ˆ 2 = qj I − i H + O ∆t qj−1 ⟨ ∣ ⟩ ⟨ ∣ h̵ ∣ ⟩ t = ∆ ˆ 2 δ⟨ q∣$j − qj−1 − i ((qj H)q)j−%∣1 + O⟩ ∆t h̵ (5.13) ( ) ⟨ ∣ ∣ ⟩ (( ) ) which becomes asymptotically exact as N → ∞.

q(t) qf

qi t t ti ∆t f

Figure 5.2 A history q t of the system.

We can also introduce at each intermediate( ) time tj acompletesetof momentum eigenstates p using their resolution of the identity ∞ = {∣ ⟩}I dp p p (5.14) "−∞ Recall that the overlap between the states∣ ⟩⟨q ∣and p is 1 = ipq h̵ q p e ∣ ⟩ ∣ ⟩ (5.15) 2πh̵ / For a typical Hamiltonian of⟨ ∣ the⟩ form& pˆ2 Hˆ = + V qˆ (5.16) 2m

( ) 118 Path Integrals in Quantum Mechanics and Quantum Field Theory its matrix elements are

∞ 2 dpj − pj ˆ ipj qj qj−1 h̵ qj H qj−1 = e + V qj (5.17) "−∞ 2πh 2m ̵ ( )/ Within the⟨ ∣ same∣ level⟩ of approximation we can' also write( )(

dpj pj qj + qj−1 qj,tj qj−1,tj−1 ≈ exp i qj − qj−1 − ∆tH pj, " 2πh̵ h̵ 2 (5.18) where⟨ we∣ have introduce⟩ the “mid-point) * ( rule” which) amounts* tothereplace-++, q → 1 q + q H p, q ment j 2 j j−1 inside the Hamiltonian .Puttingeverything together we ﬁnd that the matrix element qf ,tf qi,ti becomes - . ( ) N ∞ N+1 ⟨ dpj ∣ ⟩ qf ,tf qi,ti = lim dqj N→∞ " # " # 2πh j=1 −∞ j=1 ̵ ⟨ ∣ ⟩ N+1 i qj + qj−1 exp p q − q − ∆tH p , h $ j j j−1 j 2 ⎧ ̵ j=1 ⎫ ⎪ ⎪ ⎨⎪ ) ( ) * +,⎬⎪(5.19) ⎪ ⎪ ⎩⎪ ⎭⎪ Therefore, in the limit N → ∞,holding ti − tf ﬁxed, the amplitude qf ,tf qi,ti is given by the (formal) expression ∣ ∣ i tf ⟨ ∣ ⟩ dt pq˙ − H p, q h " q ,t q ,t = DpDqe̵ ti (5.20) f f i i " [ ( )] where we have⟨ used∣ the⟩ notation N dpjdqj DpDq ≡ lim (5.21) N→∞ # 2πh j=1 ̵ which deﬁnes the integration measure. The functions, or conﬁgurations, q t ,p t must satisfy the initial and ﬁnal conditions = = ( ( ) ( )) q ti qi,qtf qf (5.22)

Thus the matrix element (qf),tf qi,ti can( be) expressed as a sum over his- tories in phase space.Theweightofeachhistoryistheexponentialfactor of Eq. (5.20). Notice that⟨ the quantity∣ ⟩ in brackets it is just the Lagrangian

L = pq˙ − H p, q (5.23)

( ) 5.1 Path Integrals and Quantum Mechanics 119 Thus the matrix element is just

i S q,p q ,t q,t = DpDqeh̵ (5.24) f f " ( ) where S q,p is the action⟨ ∣ of each⟩ history q t ,p t .Alsonoticethat the sum (or integral) runs over independent functions q t and p t which are not required( ) to satisfy any constraint (apart( ( ) from( )) the initial and ﬁnal conditions) and, in particular they are not the solution( ) of the equations( ) of motion. Expressions of these type are known as path-integrals.Theyare also called functional integrals, since the integration measure is a sum over a space of functions,insteadofaﬁeldofnumbersasinaconventionalintegral. Using a Gaussian integral of the form (which involves an analytic contin- uation) 2 p ∆t ∆t 2 ∞ dp i pq˙ − m i q˙ e 2m h = e 2h (5.25) " 2πh ̵ 2πih∆t ̵ −∞ ̵ 6 ̵ ( ) we can integrate out explicitly the momenta in the path-integral and ﬁnd a formula that involves only the histories of the coordinate alone. Notice that there are no initial and ﬁnal conditions on the momenta since the initial and ﬁnal states have well deﬁned positions. The result is

i tf dt L q,q˙ h " q ,t q ,t = Dqe̵ ti (5.26) f f i i " ( ) which is known as⟨ the Feynman∣ ⟩ Path Integral (Feynman, 2005, 1948). Here L q,q˙ is the Lagrangian, 1 L q,q˙ = mq˙2 − V q (5.27) ( ) 2 and the sum over histories(q t ) is restricted( by) the boundary conditions q ti = qi and q tf = qf . → The Feynman path-integral( tells) us that in the correspondence limit, h̵ 0,( the) only history( ) (or possibly histories) that contribute signiﬁcantly to the path integral must be those that leave the action S stationary since, otherwise, the contributions of the rapidly oscillating exponential would add up to zero. In other words, in the classical limit there is onlyonehistoryqc t that contributes. For this history, qc t ,theactionS is stationary, δS = 0, and qc t is the solution of the Classical Equation of Motion ( ) ∂L d( )∂L − = 0(5.28) ( ) ∂q dt ∂q˙ 120 Path Integrals in Quantum Mechanics and Quantum Field Theory

q tf

( ) q

q ti t ti tf ( )

Figure 5.3 Two histories with the same initial and ﬁnal states.

→ In other terms, in the correspondence limit h̵ 0, the evaluation of the Feynman path integral reduces to the requirement that the Least Action Principle should hold. This is the classical limit.

5.2 Evaluating path integrals in Quantum Mechanics Let us ﬁrst discuss the following problem. We wish to know how to compute the amplitude qf ,tf qi,ti for a dynamical system whose Lagrangian has the standard form of Eq. (5.27). For simplicity we will begin with a linear harmonic oscillator.⟨ ∣ ⟩ The Hamiltonian for a linear harmonic oscillator is p2 mω2 H = + q2 (5.29) 2m 2 and the associated Lagrangian is m mω2 L = q˙2 − q2 (5.30) 2 2

Let qc t be the classical trajectory. It is the solution of the classical equa- tions of motion ( ) d2q c + ω2q = 0(5.31) dt2 c Let us denote by q t an arbitrary history of the system and by ξ t its deviation from the classical solution qc t .Sinceallthehistories,including the classical trajectory( ) qc t ,obeythesame initial and ﬁnal conditions( ) ( ) q t = q q t = q (5.32) ( ) i i f f

( ) ( ) 5.2 Evaluating path integrals in Quantum Mechanics 121 it follows that ξ t obeys instead vanishing initial and ﬁnal conditions:

ξ t = ξ t = 0(5.33) ( ) i f After some trivial algebra it is easy to show that the action S for an arbitrary ( ) ( ) history q t becomes

2 tf d dq tf d q S q,q˙ =(S) q , q˙ + S ξ,ξ˙ + dt mξ c + dt mξ c + ω2q c c " " 2 c ti dt dt ti dt (5.34) ( ) ( ) ( ) ) , 7 8 The third term vanishes due to the boundary conditions obeyedbythe ﬂuctuations ξ t ,Eq.(5.33).Thelasttermalsovanishessinceqc is a solution of the classical equation of motion Eq. (5.31). These two features hold for all systems, even( ) if they are not harmonic. However, the Lagrangian (and hence the action) for ξ,thesecondterminEq.(5.34),ingeneralisnotthe same as the action for the classical trajectory (the ﬁrst term). Only for the harmonic oscillator S ξ,ξ˙ has the same form as S qc, q˙c . Hence, for a harmonic oscillator, we get the path integral

( i ) i( tf ) ˙ S qc,q˙c t dtL ξ,ξ q ,t q ,t = e h̵ Dξeh̵ ∫ i (5.35) f f i i " ξ ti =ξ tf =0 ( ) ( ) Notice that⟨ the∣ information⟩ on the initial( ) ( and) ﬁnal states enters only through the factor associated with the classical trajectory. For thelinearharmonic oscillator, the quantum mechanical contribution is independent of the initial and ﬁnal states. Thus, we need to do two things: 1) we need an explicit solu- tion qc t of the equation of motion, for which we will compute S qc, q˙c ,and 2) we need to compute the quantum mechanical correction, the last factor in Eq.( (5.35),) which measures the strength of the quantum ﬂuctuations.( ) For a general dynamical system, whose Lagrangian has the formofEq. (5.27), the action of Eq. (5.34) takes the form ˙ S q,q˙ = S qc, q˙c + Seﬀ ξ,ξ; qc 2 tf d dq tf d q ∂V + dt mξ c + dt m c + ξ t ( " ) ( ) (" ) 2 ti dt dt ti dt ∂q qc 9 9 (5.36) ) , 7 9 8 ( ) 9 where the Seﬀ is the eﬀective action for the ﬂuctuations ξ t which has the form

tf tf tf 2 ( ) ˙ 1 ˙2 1 ′ ∂ V ′ 3 Seﬀ ξ,ξ = dt mξ − dt dt ξ t ξ t − O ξ " 2 2 " " ′ q ti ti ti ∂q t ∂q t c 9 (5.37) ( ) 9 ( ) ( ) ( ) ( ) ( )9 122 Path Integrals in Quantum Mechanics and Quantum Field Theory

Once again, the boundary conditions ξ ti = ξ tf = 0andthefactthe qc t is a solution of the equation of motion together imply that thelasttwo terms of Eq. (5.36) vanish identically. ( ) ( ) (Thus,) to the extent that we are allowed to neglect the O ξ3 corrections (and higher), the eﬀective action Seﬀ can be approximated by an action that is quadratic in the ﬂuctuation ξ. In general, this eﬀective action( ) will depend ′′ on the actual classical trajectory, since in general V qc is not constant but is a function of time determined by qc t .However,ifoneisinterestedinthe quantum ﬂuctuations about a minimum of the potential( ) V q ,thenqc t is constant (and equal to the minimum).( ) We will discuss below this case in detail. ( ) ( ) Before we embark in an actual computation it is worthwhile to ask when it should be a good approximation to neglect the terms O ξ3 (and higher). Since we are expanding about the classical path qc,weexpectthatthis → approximation should be correct as we formally take the limi( t) h̵ 0. In the path integral the eﬀective action always appears in the combination Seﬀ h̵ . Hence, for an eﬀective action that is quadratic in ξ,wecaneliminatethe dependence on h̵ by the rescaling / = ˜ ξ h̵ ξ (5.38) & This rescaling leaves the classical contribution S qc h̵ unaﬀected. However, ˜n n 2 terms with powers higher than quadratic in ξ,sayO ξ ,scalelikeh̵ . Thus the action (divided by h̵ )hasanexpansionoftheform( )/ / ∞ ( ) S 1 = 0 + 2 ˜ + n 2 n ˜ S qc S ξ; qc h̵ S ξ; qc (5.39) h h $= ̵ ̵ ( ) ( ) n 3 / ( ) → ( ) ( ) ( ) Thus, in the limit h̵ 0, we can formally expand the weight of the path integral in powers of h̵ .Thematrixelementwearecalculatingthentakes the form 0 iS qc h̵ 2 qf ,tf qi,ti = e Z qc 1 + O h (5.40) ( ) ̵ ( )/ ( ) 2 The quantity Z⟨ qc∣ is the⟩ result of keeping( only)( the( quadratic)) approxi- mation. The higher( ) order terms are a power series expansion in h̵ and are analytic functions of( h̵).Here,Ihaveusedthefactthat,bysymmetry,inmost cases of interest the odd powers in ξ in general do not contribute, although there are some cases where they do. Let us now calculate the eﬀect of the quantum ﬂuctuations to quadratic order. This is equivalent to the WKB approximation. Let us denote this 5.2 Evaluating path integrals in Quantum Mechanics 123 factor by Z, 2 ˜ ˜˙ 2 iSeﬀ ξ,ξ;qc Z qc = Dξe (5.41) " ( ) ξ˜ ti =ξ˜ tf =0 ( ) ( ) It is elementary to show( that,) due( ) to( the) boundary conditions,theaction ˙ Seﬀ ξ,ξ becomes

t 2 1 f d ′′ ( )S ξ˜,ξ˜˙ = dt ξ˜ t −m − V q t ξ˜ t (5.42) eﬀ 2 " 2 c ti dt The diﬀerential( ) operator ( )' ( ( ))( ( ) 2 d ′′ Aˆ = −m − V qc t (5.43) dt2 has the form of a Schr¨odinger operator for a particle on a “coordinate” t ′′ ( ( )) in a potential −V qc t .Letψn t be a complete set of eigenfunctions of Aˆ satisfying the boundary conditions ψ ti = ψ tf = 0. Completeness and orthonormality implies( ( )) that the eigenfunctions( ) ψn t satisfy t ( ) ( ) ∗ ′ ′ f ∗ ψ t ψ t = δ t − t , dt ψ t ψ t = δ (5.44) $ n n " n { m( )} n,m n ti An arbitrary function( ) ( )ξ˜ t (that satisﬁes) the vanishing( ) ( boundary) conditions of Eq.(5.33) can be expanded as a linear combination of the basiseigenfunctions ψn t , ( ) ξ˜ t = c ψ t (5.45) $ n n { ( )} n ( ) ( ) Clearly, we have ξ˜ ti = ξ˜ tf = 0asweshould. For the sp ecial case of qi = qf = q0,whereq0 is a minimum of the potential ′′ V q , V q0 = ωeﬀ( >)0isaconstant,andtheeigenvectorsoftheSchr¨odinger( ) operator are just plane waves. For a linear harmonic oscillator ωeﬀ = ω.Thus, in( this) case( ) the eigenvectors are

ψn t = bn sin kn t − ti (5.46) where ( )πn ( ( )) kn = n = 1, 2, 3,... (5.47) tf − ti and bn = 1 tf − ti.TheeigenvaluesofAˆ are

: π2 / = 2 − 2 = 2 − 2 An kn ωeﬀ 2 n ωeﬀ (5.48) tf − ti

( ) 124 Path Integrals in Quantum Mechanics and Quantum Field Theory

By using the expansion of Eq. (5.45), we ﬁnd that the action S 2 takes the form ( ) 1 tf 1 S 2 = dt ξ˜ t Aˆ ξ˜ t = A c2 (5.49) 2 " 2 $ n n ti ( ) n where we have used the completeness( ) and( orthonormality) of thebasisfunc- tions ψn t . The expansion of Eq.(5.45) is a canonical transformation ξ˜ t → cn.More to the{ point,( )} the expansion is actually a parametrization of the possible histories in terms of a set of orthonormal functions, and it c(an) be used to deﬁne the integration measure to be dc Dξ˜ = N n (5.50) # n 2π with unit Jacobian. Here N is an irrelevant& normalization constant that will be deﬁned below. Finally, the (formal) Gaussian integral, which is deﬁned by asuitable analytic continuation procedure, is ∞ dc i 2 − n 2 Ancn 1 2 e = −iAn (5.51) "−∞ 2π / can be used to write the amplitude& as [ ] − − Z 2 = N A 1 2 ≡ N DetAˆ 1 2 (5.52) # n ( ) n / / where we have used the deﬁnition that the( determinant) of an operator is equal to the product of its eigenvalues. Therefore, up to a normalization constant, we obtained the result − Z 2 = DetAˆ 1 2 (5.53) We have thus reduced the problem( ) of the computation/ of the leading (Gaus- ( ) sian) ﬂuctuations to the path-integral to the computation ofadeterminant of the ﬂuctuation operator, a diﬀerential operator deﬁned by the choice of classical trajectory. Below we will see how this is done.

5.2.1 Analytic continuation to imaginary time It is useful to consider the related problem obtained by an analytic contin- uation to imaginary time, t → −iτ.Wesawbeforethatthereisarelation between this problem and Statistical Physics. We will now work out one example that will be very instructive. 5.2 Evaluating path integrals in Quantum Mechanics 125 Formally, upon the analytic continuation t → −iτ,thematrixelementof the time evolution operator becomes i 1 − H tf − ti − H τf − τi qf e h̵ qi → qf e h̵ qi (5.54) ( ) ( ) Let us choose⟨ ∣ ∣ ⟩ ⟨ ∣ ∣ ⟩ = = τi 0 τf βh̵ (5.55) where β = 1 T ,andT is the temperature (in units of kB = 1). Hence, we ﬁnd that / − = −βH qf , iβ h̵ qi, 0 qf e qi (5.56) ρ The operator ˆ ⟨ / ∣ ⟩ ⟨ ∣ ∣ ⟩ −βH ρˆ = e (5.57) is the Density Matrix in the Canonical Ensemble of Statistical Mechanics for a system with Hamiltonian H in thermal equilibrium at temperature T . It is customary to deﬁne the Partition Function Z,

−βH −βH Z = tre ≡ dq q e q (5.58) " where I inserted a complete set of eigenstates⟨ ∣ of∣q ˆ.Usingtheresultsthat⟩ were derived above, we see that the partition function Z can be written as a(Euclidean)Feynman path integral in imaginary time,oftheform

2 1 βh̵ 1 ∂q Z = Dq τ exp − dτ m + V q " h̵ "0 2 ∂τ [ ] ; β m' ∂*q 2 + ( )(< ≡ Dq τ exp − dτ + V q (5.59) " " 2 ∂τ 0 2h̵ [ ] ; ' *→ + ( )(< where, in the last equality we have rescaled τ τ h̵ . Since the Partition Function is a trace over states, we must use boundary conditions such that the initial and ﬁnal states are/ the same state, and to sum over all such states. In other words, we must have periodic boundary conditions in imaginary time (PBC’s),

q τ = q τ + β (5.60)

T Therefore, a quantum mechanical( ) system( at) ﬁnite temperature can be described in terms of an equivalent system in classical statistical mechanics 126 Path Integrals in Quantum Mechanics and Quantum Field Theory with Hamiltonian (or energy) m ∂q 2 H = + V q (5.61) 2 ∂τ 2h̵ on a segment of length 1 T and obeying* + PBC’s.( ) This eﬀectively means that the segment is actually a ring of length β = 1 T . Alternatively, upon inserting/ a complete set of eigenstatesoftheHamilto- nian, it is easy to see that an arbitrary matrix/ element of the density matrix has the form ∞ ′ − ′ − q e βH q = q n n q e βEn $ n=0 ∞ ⟨ ∣ ∣ ⟩ ⟨ −∣βE⟩⟨ ∗∣ ⟩′ −βE ∗ ′ = e n ψ q ψ q −− − − −→ e 0 ψ q ψ q (5.62) $ n n → ∞ 0 0 n=0 β where En are the eigenvalues( ) of the( ) Hamiltonian, E0 is( the) ground( ) state energy and ψ0 q is the ground state wave function. Therefore,{ } we can calculate both the ground state energy E0 and the ground state wave( ) function from the density matrix and consequently from the (imaginary time) path integral. For example, from the identity

1 −βH E0 = − lim ln tre (5.63) β→∞ β we see that the ground state energy is given by

1 β m ∂q 2 E0 = − lim ln Dq exp − dτ + V q β→∞ β " " 2 ∂τ q 0 =q β 0 2h̵ (5.64) ; ' * + ( )(< Mathematically, the imaginary( ) ( ) time path integral is a betterbehavedobject than its real time counterpart, since it is a sum of positive quantities, the statistical weights. In contrast, the Feynman path integral(inrealtime)isa sum of phases and, as such, is an ill-deﬁned object. It is actually conditionally convergent, and to make sense of it convergence factors (or regulators) will have to be introduced. The eﬀect of these convergence factors is actually an analytic continuation to imaginary time. We will encounter the same problem in the calculation of propagators. Thus, the imaginary time path integral, often referred to as the Euclidean path integral (as opposed to Minkowski), can be used to describe both a quantum system and astatistical mechanics system. Finally, we notice that at low temperatures T → 0, the Euclidean path integral can be approximated using methods similar to the ones we discussed 5.2 Evaluating path integrals in Quantum Mechanics 127 for the (real time) Feynman Path Integral. The main diﬀerence is that we must sum over trajectories which are periodic in imaginary time with period β = 1 T .Inpracticethissumcanonlybedoneexactlyforsimplesystems such as the harmonic oscillator, and for more general systemsonehasto resort/ to some form of perturbation theory. Here we will consider a physical system described by a dynamical variable q and a potential energy V q which has a minimum at q0 = 0. For simplicity we will take V 0 = 0 2 ′′ and we will denote by mω = V 0 (in other words, an eﬀective harmonic( ) oscillator). The partition function is given by the Euclidean path integral( ) ( ) 1 β Z = Dq τ exp − ξ τ AˆEξ τ dτ (5.65) " 2 "0 where AˆE is the imaginary[ time,] * or Euclidean,( ) version( ) of+ the operator Aˆ, and it is given by m d2 ˆ = − + ′′ AE 2 2 V qc τ (5.66) h̵ dτ The functions this operator acts on obey periodic( ( )) boundary conditions with period β.Noticetheimportantchangeinthesignofthetermofthepoten- tial. Hence, once again we will need to compute a functional determinant, although the operator now acts on functions obeying periodicboundarycon- ditions. In a later chapter we will see that in the case of fermionic theories, the boundary conditions become antiperiodic.

5.2.2 The functional determinant We will now do the computation of the determinant in Z 2 .Wewilldo the calculation in imaginary time and then we will carry out( ) the analytic continuation to real time. We will follow closely the method is explained in detail in Sidney Coleman’s book, (Coleman, 1985). We want to compute

m d2 = − + ′′ D Det 2 2 V qc τ (5.67) h̵ dτ subject to the requirement that' the space of( functions( ))( that the operator acts on obeys speciﬁc boundary conditions in (imaginary) time. We will be interested in two cases: (a) Vanishing Boundary Conditions (VBC’s), which are useful to study quantum mechanics at T = 0, and (b) Periodic Bound- ary Conditions (PBC’s) with period β = 1 T .Theapproachissomewhat diﬀerent in the two situations. / 128 Path Integrals in Quantum Mechanics and Quantum Field Theory A: vanishing boundary conditions = h̵ We deﬁne the (real) variable x m τ.Therangeofx is the interval 0,L , with L = hβ m.Letusconsiderthefollowingeigenvalueproblemforthe ̵ 2 Schr¨odinger operator −∂ + W x , [ ] & / 2 −∂ + W x ψ x = λψ x (5.68) ( ) subject to the boundary conditions ψ 0 = ψ L = 0. Formally, the deter- = ( )> ( ) ( ) minant is given by D = ( )λ ( ) (5.69) # n n 2 where λn is the spectrum of eigenvalues of the operator −∂ + W x for aspaceoffunctionssatisfyingagivenboundarycondition. Let us{ deﬁne} an auxiliary function ψλ x ,withλ arealnumbernotneces-( ) sarily in the spectrum of the operator, such that the following requirements are met: ( ) a. ψλ x is a solution of Eq. (5.68), and b. ψλ obeys the initial conditions, ψλ 0 = 0and∂xψλ 0 = 1. ( ) 2 It is easy to see that −∂ + W x has an eigenvalue at λn if and only = ( ) ( ) if ψλn L 0. (Because of this property this procedure is known as the shooting method.) Hence, the determinant( ) D of Eq. (5.69) is equal to the product( of) the zeros of ψλ x at x = L. Consider now two potentials W 1 and W 2 ,andtheassociatedfunctions, 1 2 ( ) ψλ x and ψλ x .Letusshowthat( ) ( ) ( ) ( ) 2 1 Det −∂ + W x − λ ψ 1 L ( ) ( ) = λ (5.70) ( ) Det −∂2 + W 2 x − λ ψ(2) L ? ( ) @ λ ( ) The l. h. s. of Eq. (5.70) is a meromorphic( ) function( ) of λ in the complex ? ( ) @ ( ) plane, which has simple zeros at the eigenvalues of −∂2 +W 1 x and simple 2 2 poles at the eigenvalues of −∂ + W x .Also,thel.h.s.ofEq.(5.70)( ) approaches 1 as λ → ∞,exceptalongthepositiverealaxiswhichiswhere( ) ( ) the spectrum of eigenvalues of both operators( ) is. Here we haveassumedthat the eigenvalues∣ of∣ the operators are non-degenerate, which is the general case. Similarly, the right hand side of Eq. (5.70) is also a meromorphic function of λ,whichhasexactly the same zeros and the same poles as the left hand side. It also goes to 1 as λ → ∞ (again, except along the positive real axis), since the wave-functions ψλ are asymptotically plane waves in this limit . Therefore, the function∣ formed∣ by taking the ratio r. h. s. / l. h. 5.2 Evaluating path integrals in Quantum Mechanics 129 s. is an analytic function on the entire complex plane and it approaches 1 as λ → ∞.Then,generaltheoremsofthetheoryoffunctionsofacomplex variable tell us that this function is equal to 1 everywhere. ∣ ∣From these considerations we conclude that the following ratio is inde- pendent of W x , Det −∂2 + W x − λ ( ) (5.71) ψλ L = ( ) > We now deﬁne a constant N such that ( ) Det −∂2 + W x = N 2 πh̵ (5.72) ψ0 L = ( )> Then, we can write ( ) −1 2 N − 2 + = −1 2 Det ∂ W πh̵ ψ0 L (5.73) / / Thus we reduced the computation of the determinant, including the normal- ? = >@ [ ( )] ization constant, to ﬁnding the function ψ0 L .Forthecaseofthelinear harmonic oscillator, this function is the solution of ( ) ∂2 − + mω2 ψ x = 0(5.74) ∂x2 0 ' ( (′ ) with the initial conditions, ψ0 0 = 0andψ0 0 = 1. The solution is 1 ψ x = sinh mωx (5.75) 0 ( ) mω ( ) & Hence, ( ) & ( ) 2 −1 2 ∂ − Z = N Det − + mω2 = πhψ L 1 2 (5.76) ∂x2 / ̵ 0 / and we ﬁnd ' 7 8( [ ( )] πh −1 2 Z = ̵ sinh βω (5.77) mω / == where we have used L h̵ β )&m.Fromthisresultweﬁndthattheground( ), state energy is & − / 1 h̵ ω E0 = lim ln Z = (5.78) β→∞ β 2 as it should be. Finally, by means of an analytic continuation back to real time, we can 130 Path Integrals in Quantum Mechanics and Quantum Field Theory use these results to ﬁnd, for instance, the amplitude to return to the origin after some time T .Thus,fortf − ti = T and qf = qi = 0, we get

iπh −1 2 0,T 0, 0 = ̵ sin ωT (5.79) mω /

⟨B: periodic∣ ⟩ boundary)& conditions( ), Periodic boundary conditions imply that the histories satisfy q τ = q τ +β . Hence, these functions can be expanded in a Fourier series of the form ∞ ( ) ( ) iω τ q τ = e n q (5.80) $ n n=−∞ ∗ where ωn = 2πn β.Sinceq τ( )is real, we have the constraint q−n = qn.For such conﬁgurations (or histories) the action becomes

/ β ( ) 2 m ∂q 1 ′′ S = dτ + V 0 q2 " 2 ∂τ 2 0 2h̵ β ′′ '2 * + m 2 ( )′′ ( 2 = V 0 q + β ω + V 0 qn (5.81) 2 0 $ h2 n n≥1 ̵ The integration measure( now) is ) ( ), ∣ ∣ dq dReq dImq Dq τ = N 0 n n (5.82) # 2π 2π n≥1 where N is a normalization[ ] constant& that will be discussed below. After doing the Gaussian integrals, the partition function becomes,

∞ 1 2 1 1 1 Z = N = N ′′ βm 2 ′′ βm 2 ′′ / βV 0 #≥ ω + βV 0 #=−∞ ω + βV 0 n 1 h2 n ⎡n h2 n ⎤ ̵ ⎢ ̵ ⎥ : ⎢ ⎥(5.83) ⎢ ⎥ Formally, the inﬁnite( ) product that enters( ) in this⎣⎢ equation isdivergent.The( )⎦⎥ normalization constant N eliminates this divergence. This is an example of what is called a regularization.Theregularizedpartitionfunctionis

′′ −1 m 1 h2V 0 Z = 1 + ̵ (5.84) h2β ′′ # mω2 6 ̵ βV 0 n≥1 n ( ) Using the identity : ' ( ( ) a2 sinh a 1 + = (5.85) # 2 2 a n≥1 n π 7 8 5.3 Path integrals for a scalar ﬁeld theory 131 we ﬁnd 1 Z = (5.86) ′′ 1 2 βh V 0 2sinh ̵ 2 m / ⎛ ( ) ⎞ which is the partition function for⎜ a linear7 harmonic8 ⎟ oscillator, see L. D. ⎝ ⎠ Landau and E. M. Lifshitz, Statistical Physics,(LandauandLifshitz,1959b).

5.3 Path integrals for a scalar ﬁeld theory We will now develop the path-integral quantization picture for a scalar ﬁeld theory. Our starting point will be the canonically quantizedscalarﬁeld.As we saw before, in canonical quantization the scalar ﬁeld φˆ x is an operator that acts on a Hilbert space of states. We will use the ﬁeld representation, which is the analog of the conventional coordinate represen( tation) in Quan- tum Mechanics. Thus, the basis states are labelled by the ﬁeld conﬁguration at some ﬁxed time x0,asetofstatesoftheform φ x,x0 .Theﬁeldoperator ˆ φ x,x0 acts trivially on these states, {∣{( )}⟩} φˆ x,x φ x,x = φ x,x φ x,x (5.87) ( ) 0 0 0 0 The set of states φ x,x is both complete and orthonormal. Com- ( )∣{ (0 )}⟩ ( )∣{ ( )}⟩ pleteness here means that these states span the entire Hilbert space. Con- sequently the identity{∣{ operator( )}⟩Iˆ} in the full Hilbert space can be expanded in a complete basis in the usual manner, which for this basis itmeans

Iˆ = Dφ x,x φ x,x φ x,x (5.88) " 0 0 0

Since the completeness condition( ) is∣{ a sum( over)}⟩⟨ all{ the( states)}∣ in the basis and since this basis is the set of ﬁeld conﬁgurations at a given time x0,wewill need to give a deﬁnition for integration measure which represents the sums over the ﬁeld conﬁgurations. In this case, the deﬁnition of the integration measure is trivial, Dφ x,x = dφ x,x (5.89) 0 # 0 x Likewise, orthonormality of( the basis) states( is the) condition ′ ′ φ x,x φ x,x = δ φ x,x − φ x,x (5.90) 0 0 # 0 0 x Thus, we have⟨ a( working)∣ deﬁnition( )⟩ of the= Hilbert( space) ( for a re)>al scalar ﬁeld. 132 Path Integrals in Quantum Mechanics and Quantum Field Theory

In canonical quantization, the classical canonical momentum Π x,x0 , deﬁned as δL ( ) Π x,x0 = = ∂0φ x,x0 (5.91) δ∂0φ x,x0 becomes an operator( that acts) on the same Hilbert( space) as the ﬁeld itself φ does. The ﬁeld operator φˆ x and the( canonical) momentum operator Πˆ x satisfy equal-time canonical commutation relations ˆ ( ) ˆ = 3 − ( ) φ x,x0 , Π y,x0 ih̵ δ x y (5.92) Here we will consider a real scalar ﬁeld whose Lagrangian density is ? ( ) ( )@ ( ) 1 2 L = ∂ φ − V φ (5.93) 2 µ It is a simple matter to generalize what follows below to more general cases, - . ( ) such as complex ﬁelds and/or several components. Let us also recall that the Hamiltonian for a scalar ﬁeld is given by 1 1 2 Hˆ = d3x Πˆ 2 x + &φˆ x + V φˆ x (5.94) " 2 2

For reasons that will become) ( clear) soon,= it( is)> convenient( ( )) to, add an extra term to the Lagrangian density of the scalar ﬁeld, Eq. (5.93),oftheform

Lsource = J x φ x (5.95) The ﬁeld J x is called an external source.TheﬁeldJ x is the analog of ( ) ( ) external forces acting on a system of classical particles. Here we will always assume that( the) sources J x vanish both at spacial inﬁnity( ) (at all times) and everywhere in both the remote past and in the remote future, ( ) lim J x,x0 = 0limJ x,x0 = 0(5.96) x →∞ x0→±∞

The total Lagrangian∣ ∣ density( ) is ( )

L φ,J = L + Lsource (5.97) Since the source J x is in general a function of space and time, the Hamil- ( ) tonian that follows from this Lagrangian is formally time-dependent. We will derive the( ) path integral for this quantum ﬁeld theory by following the same procedure we used for the case of a ﬁnite quantum mechanical system. Hence we begin by considering the Wightman function deﬁned as the amplitude ′ J φ x,x0 φ y,y0 J (5.98) In other words, we want the transition amplitude in the background of the ⟨{ ( )}∣{ ( )}⟩ 5.3 Path integrals for a scalar ﬁeld theory 133 sources J x .Wewillbeinterestedinsituationsinwhichx0 is in the re- mote future and y0 is in the remote past. It turns out that this amplitude is intimately( related) to the computation of ground state (or vacuum)expec- tation values of time ordered products of ﬁeld operators in the Heisenberg representation

N ˆ ˆ G x1,...,xN ≡ 0 T φ x1 ...φ xN 0 (5.99) ( ) which are known as( the N-point) functions⟨ ∣ [ ( (or) correlators).( )]∣ In⟩ particular the 2-point function

2 ˆ ˆ G x1 − x2 ≡ −i 0 T φ x1 φ x2 0 (5.100) ( ) is called the Feynman( Propagator) for⟨ this∣ [ theory.( ) ( We)] will∣ ⟩ see later on that all quantities of physical interest can be obtained from a suitable correlation function of the type of Eq. (5.99). ˆ ˆ In Eq. (5.99) we have use the notation T φ x1 ...φ xN for the time- ordered product of Heisenberg ﬁeld operators. For any pair Heisenberg of operators Aˆ x and Bˆ y ,thatcommuteforspace-likeseparations,their[ ( ) ( )] time-ordered product is deﬁned to be ( ) ( ) T Aˆ x Bˆ y = θ x0 − y0 Aˆ x Bˆ y + θ y0 − x0 Bˆ y Aˆ x (5.101) where[θ (x )is( the)] step( (or Heaviside)) ( ) function( ) ( ) ( ) ( )

( ) 1ifx ≥ 0, θ x = (5.102) 0otherwise

This deﬁnition is generalized( ) by induction; to to the product of any num- ber of operators. Notice that inside a time-ordered product the Heisenberg operators behave as if they were c-numbers. Let us now recall the structure of the derivation that we gave of the path integral in Quantum Mechanics. We will paraphrase that derivation for this ﬁeld theory. We considered an amplitude equivalent to Eq. (5.98), and realized that this amplitude is actually a matrix elementoftheevolution operator,

x i 0 ′ ′ − dx0 H x0 ′ h "y & ′ J φ x,x0 φ y,y0 J = φ x Te ̵ 0 φ y ( ) (5.103) ′ where⟨{T(stands)}∣ for{ the( time)}⟩ ordering⟨{ ( symbol)}∣ (not temperature!),∣{ and( H&)}⟩x0

( ) 134 Path Integrals in Quantum Mechanics and Quantum Field Theory is the time-dependent Hamiltonian whose Hamiltonian density is 1 1 2 H x = Π2 x,x + &φˆ x,x + V φˆ x,x − J x,x φˆ x,x & 0 2 ̂ 0 2 0 0 0 0 (5.104) Paraphrasing( ) ( the construction) = ( used)> in the( case( of Quantum)) ( Me)chanics( ) of aparticle,weﬁrstpartitionthetimeintervalinalargenumber of steps N, each of width ∆t,andtheninsertacompletesetofeigenstatesoftheﬁeld operator φˆ,sincetheﬁeldplaystheroleofthecoordinate.Asitturnedout, we also had to insert complete sets of eigenstates of the canonical momentum operator, which here means the canonical ﬁeld operator Πˆ x .Uponformally taking the time-continuum limit, N → ∞ and ∆t → 0whilekeepingN∆t ﬁxed, we obtain the result that the phase-space path integral( ) of the ﬁeld theory is i d4x φ˙Π − H φ, Π + Jφ ′ h " J φ x,x0 φ y,y0 J = DφDΠ e ̵ "b. c. ? ( ) (5.105)@ where⟨{ ( b.c.)} indicates∣{ ( the)}⟩ boundary conditions speciﬁed by therequirement ′ that the initial and ﬁnal states be φ x,x0 and φ y,y0 ,respec- tively. Exactly as in the case of the path∣{ integral( )} for⟩ a particle,∣{ ( the)}⟩Hamilto- nian of this theory is quadratic in the canonical momenta Π x .Hence,we can further integrate out the ﬁeld Π x ,andobtaintheFeynmanpathin- tegral for the scalar ﬁeld theory in the form of a sum over histories( ) of ﬁeld conﬁgurations: ( ) i S φ,∂ φ,J ′ = h µ J φ x,x0 φ y,y0 J N Dφe̵ (5.106) "b. c. ( ) where N⟨{ is( an (unimportant))}∣{ ( )}⟩ normalization constant, and S φ,∂µφ,J is the action for a real scalar ﬁeld φ x coupled to a source J x , ( ) 1 2 S φ,∂ φ,J = d4x ∂ φ − V φ + Jφ (5.107) µ " ( ) 2 µ ( )

( ) ) - . ( ) , 5.4 Path integrals and propagators In Quantum Field Theory we will be interested in calculating vacuum (ground state) expectation values of ﬁeld operators at various space-time locations. ′ Thus, instead of the amplitude J φ x,x0 φ y,y0 J we may be in- terested in a transition between an initial state, at y0 → −∞ which is the vacuum state 0 ,i.e.thegroundstateofthescalarﬁeldintheabsenceof⟨{ ( )}∣{ ( )}⟩

∣ ⟩ 5.4 Path integrals and propagators 135 the source J x ,andaﬁnalstateatx0 → ∞ which is also the vacuum state of the theory in the absence of sources. We will denote this matrix element by ( )

Z J = J 0 0 J (5.108)

This matrix element is called the[ vacuum] ⟨ ∣ persistence⟩ amplitude. Let us see now how the vacuum persistence amplitude is relatedtothe Feynman path integral for a scalar ﬁeld of Eq. (5.106). In order to do that ′ we will assume that the source J x is “on” between times t < t and that ′ ′ we watch the system on a much longer time interval T < t < t < T .For this interval, we can now use the Superposition( ) Principle to insert complete ′ sets of states at intermediate times t and t ,andwritetheamplitudeinthe form

′ ′ J Φ x,T Φ x,T J = (5.109) ′ ′ ′ ′ ′ ′ Dφ x,t Dφ x,t Φ x,T φ x,t ⟨{ "( )}∣{ ( )}⟩ × ′ ′ ( ) ( )J⟨{ φ( x,t)}∣{φ (x,t )}⟩J φ x,t Φ x,T

′ ⟨{′ ( ′ )}∣{′ ( )}⟩ ⟨{ ( )}∣{ ( )}⟩ The matrix elements Φ x,T φ x,t and φ x,t Φ x,T are given by ⟨{ ( )}∣{ ( )}⟩ ⟨{ ( )}∣{ ( )}⟩ ∗ −iE t − T h φ x,t Φ x,T = Ψ φ x Ψ Φ x e n ̵ $ n n n ′ ′ ′ ′ ′ ′ ′ ∗ ′ −iE( T −)/t h Φ⟨{ x(,T )}∣{φ(x,t)}⟩ = Ψ [{ Φ( x)}] Ψ[{ (φ )}]x e m ̵ $ m m m ( )/ ⟨{ ( )}∣{ ( )}⟩ [{ ( )}] [{ ( )}] (5.110) where we have introduced complete sets of eigenstates Ψn of the Hamil- tonian of the scalar ﬁeld (without sources) and the corresponding wave func- tions, Ψn Φ x . ∣{ }⟩ ′ At long times T and T these series expansions oscillate very rapidly and a deﬁnition{ must[ ( )]} be provided to make sense on these expressions. To this end, we will now analytically continue T along the positive imaginary time axis, ′ and T along the negative imaginary time axis, as shown in ﬁgure 5.4. After carrying out the analytic continuation, we ﬁnd that the following identities 136 Path Integrals in Quantum Mechanics and Quantum Field Theory

Im t

T

t t′

Re t

T ′

Figure 5.4 Analytic continuation.

hold,

−iE0T h ∗ −iE0t h lim e ̵ φ x,t Φ x,T = Ψ0 φ Ψ0 Φ e ̵ T →+i∞ ′ ′ iE0T h/ ′ ′ ′ ′ ∗ ′ iE0t /h lim e ̵ ⟨Φ{ (x,T)}∣{ φ(x,t)}⟩ = Ψ0[{Φ}] Ψ0[{ φ}] e ̵ T ′→−i∞ / (5.111)/ ⟨{ ( )}∣{ ( )}⟩ [{ }] [{ }]

This result is known as the Gell-Mann-Low Theorem.(Gell-Mann and Low, 1951) In this limit, the contributions from excited states drop out provided the vacuum state 0 is non-degenerate. This procedure is equivalent to the standard adiabatic turning on and oﬀof the external sources.Therestriction to a non-degenerate∣ ⟩ vacuum state can be done by lifting a possible degen- eracy by means of an inﬁnitesimally weak external perturbation, which is switched oﬀafter the inﬁnite time limit is taken. We will encounter similar issues in our discussion of spontaneous symmetry breaking inlaterchapters. 5.4 Path integrals and propagators 137 Hence, in the same limit, we also ﬁnd that the following relation holds

′ ′ Φ x,T Φ x,T lim lim ′ ∗ ′ T →+i∞ ′→− ∞ − − T i exp iE0 T T h̵ Ψ0 Φ Ψ0 Φ ′ ∗ ⟨′ { (′ )}∣ ( )}⟩ ′ ′ = DΦDΦ Ψ φ x,t Ψ φ x,t φ x,t φ x,t " [0 ( )/ 0] [{ }] J [{ }] J ≡ 0 0 (5.112) J J [{ ( )}] [{ ( )}] ⟨{ ( )}∣{ ( )}⟩

Eq.(5.112)⟨ gives∣ ⟩ us a direct relation between the Feynman Path Integral and the vacuum persistence amplitude of the form

′ i T d4x L φ,∂ φ + Jφ h " µ Z J = J 0 0 J = N lim lim Dφe̵ T T →+i∞ T ′→−i∞ " M ( ) (5.113)N In other[ ] words,⟨ ∣ ⟩ in this asymptotically long-time limit, the amplitude of Eq. (5.98) becomes identical to the vacuum persistence amplitude J 0 0 J , regardless of the choice of the initial and ﬁnal states. Hence we ﬁnd a direct relation between the vacuum persistencefunction⟨ ∣ ⟩ Z J and the Feynman path integral, given by Eq. (5.113). Notice that, in this limit, we can ignore the “hard” boundary condition and work instead with[ ] free boundary conditions. Or equivalently, physical properties become independent of the initial and ﬁnal conditions placed. For these reasons, from now on we will work with the simpler expression

i 4 d x L φ,∂µφ + Jφ Z J = 0 0 = N Dφeh " (5.114) J J " ̵ M ( ) N This is a[ very] useful⟨ ∣ ⟩ relation. We will see now that Z J is the generating function(al) of all the vacuum expectation values of time ordered products of ﬁelds, i.e. the correlators of the theory. [ ] In particular, let us compute the expression

2 2 2 1 δ Z J 1 δ 0 0 i ′ = J J = 0 T φ x φ x 0 Z 0 δJ x δJ x′ 0 0 δJ x δJ x′ h 9J=0 9J=0 ̵ [ ] 9 ⟨ ∣ ⟩ 9 (5.115) 9 9 * + ⟨ ∣ [ ( ) ( )]∣ ⟩ Thus, the 2-point function,9 i.e. the Feynman9 propagator or propagator of the [ ] ( ) ( )9 ⟨ ∣ ⟩ ( ) ( )9 scalar ﬁeld φ x ,becomes

′ 1 ′ i 0 T φ x( φ) x 0 = −i Dφφ x φ x exp S φ,∂µφ 0 0 " h̵ (5.116) ⟨ ∣ [ ( ) ( )]∣ ⟩ ( ) ( ) * [ ]+ ⟨ ∣ ⟩ 138 Path Integrals in Quantum Mechanics and Quantum Field Theory

Similarly, the N-point function 0 T φ x1 ...φ xN 0 becomes 1 δN 0 0 = − ⟨ N∣ [ ( ) (J )]∣J⟩ 0 T φ x1 ...φ xN 0 ih̵ 0 0 δJ x1 ...δJ xN 9J=0 1 ⟨ ∣ ⟩ 9 i ⟨ ∣ [ ( ) ( )]∣ ⟩ = ( ) D 9 ⟨ φφ∣ ⟩ x1( ...) φ xN ( exp)9 S φ,∂µφ 0 0 " 9 h̵ (5.117) ( ) ( ) * [ ]+ ⟨ ∣ ⟩ where i Z 0 = 0 0 = Dφ exp S φ,∂µφ (5.118) " h̵ Therefore, we ﬁnd that[ ] the⟨ ∣ path⟩ integral always* [ yields vacuu]+ mexpectation values of time-ordered products of operators. The quantity Z J can thus be viewed as the generating functional of the correlation functions of this theory. These are actually general results that hold for the path[ integrals] of all theories.

5.5 Path integrals in Euclidean space-time and Statistical Physics In the last section we saw how to relate the computation of transition am- plitudes to path integrals in Minkowski space-time with speciﬁc boundary conditions dictated by the nature of the initial and ﬁnal states. In particular we derived explicit expressions for the case of ﬁxed boundaryconditions. However we could have chosen other boundary conditions. For instance, for the amplitude to begin in any state at the initial time and to go back to the same state at the ﬁnal time, but summing over all states. This is the same as to ask for the trace

′ ′ Z J = DΦ Φ x,t Φ x,t " J J i 4 [ ] −⟨{ ( d x)}H∣{−(Jφ )}⟩ ≡Tr Te h̵ " i d(4x L +)Jφ ≡ Dφeh̵ " (5.119) "PBC ( ) where PBC stands for periodic boundary conditions on some generally ﬁnite ′ time interval t − t,andT is the time-ordering symbol. Let us now carry the analytic continuation to imaginary time t → −iτ,i.e. aWickrotation.UponaWickrotationthetheoryhasEuclideaninvariance, 5.5 Path integrals in Euclidean space-time and Statistical Physics 139 i.e. rotations and translations in D = d + 1-dimensional space. Imaginary time plays the same role as the other d spacial dimensions. Hereafter we will denote imaginary time by xD,andallvectorswillhaveindicesµ that run from 1 to D. We will consider two cases: inﬁnite imaginary time interval,andﬁnite imaginary time interval. imaginary time

0 β

space

Figure 5.5 Periodic boundary conditions wraps space-time into a cylinder.

5.5.1 Inﬁnite imaginary time interval In this case the path integral becomes

D L ′ − d x E − Jφ Z J = Dφe " (5.120) " ( ) where D is the total number[ ] of space-time dimensions. For the sake of deﬁ- niteness here we discuss the four-dimensional case but the results are obvi- ously valid more generally. Here LE is the Euclidean Lagrangian 1 1 L = ∂ φ 2 + &φ 2 + V φ (5.121) E 2 0 2

The path integral of Eq. (5.120)( has) two( interpretations.) ( ) One is simply the inﬁnite time limit (in imaginary time) and therefore it 140 Path Integrals in Quantum Mechanics and Quantum Field Theory must be identical to the vacuum persistence amplitude J 0 0 J .Theonly diﬀerence is that from here we get all the N-point functions in Euclidean space-time (imaginary time). Therefore, the relativistic interval⟨ ∣ ⟩ is 2 2 2 2 x0 − x → −τ − x < 0(5.122) which is always space-like. Hence, with this procedure we will get the correla- tion functions for space-like separations of its arguments.Togettotime-like separations we will need to do an analytic continuation back to real time. This we will do later on. The second interpretation is that thepathintegral of Eq. (5.120) is the partition function of a system in Classical Statistical Mechanics in D dimensions with energy density (divided by T )equalto LE − Jφ.Thiswillturnouttobeaveryusefulconnection(bothways!).

5.5.2 Finite imaginary time interval In this case we have

0 ≤ x0 = τ ≤ β = 1 T (5.123) where T will be interpreted as the temperature. Indeed, in this case the path / integral is ′ −βH Z 0 = Tr e (5.124) and we are eﬀectively looking at a problem of the same Quantum Field [ ] Theory but at ﬁnite temperature T = 1 β.Thepathintegralisonceagain the partition function but of a system in Quantum StatisticalPhysics!The = partition function thus is (after setting h/̵ 1) β − L − ′ dτ E Jφ Z J = Dφe"0 (5.125) " ( ) where the ﬁeld φ x,τ[ obeys] periodic boundary conditions in imaginary time, ( ) φ x,τ = φ x,τ + β (5.126)

This boundary condition will hold for all bosonic theories. We will see later ( ) ( ) on that theories with fermions obey instead anti-periodic boundary condi- tions in imaginary time. Hence, Quantum Field Theory at ﬁnite temperature T is just Quantum Field Theory on an Euclidean space-time which is periodic (and ﬁnite) in one direction, imaginary time. In other words, we have wrapped (or compactiﬁed) 5.6 Path integrals for the free scalar ﬁeld 141 Euclidean space-time into a cylinder with perimeter (circumference) β = 1 T = = (in units of h̵ kB 1). The correlation functions in imaginary time (which we will call the Eu-/ clidean correlation functions) are given by ′ δN Z J 1 = ′ φ x1 ...φ xN (5.127) Z J δJ x1 ...J xN 9J=0 [ ] 9 which are just the correlation functions9 in⟨ the( ) equivalent( )⟩ problem in Statis- [ ] ( ) ( )9 tical Mechanics. Upon analytic continuation9 the Euclidean correlation func- tions φ x1 ...φ xN and the N-point functions of the QFT are related by ⟨ ( ) ( )⟩ ↔ N φ x1 ...φ xN ih̵ 0 Tφ x1 ...φ xN 0 (5.128) For the case of a quantum ﬁeld theory at ﬁnite temperature T ,thepath ⟨ ( ) ( )⟩ ( ) ⟨ ∣ ( ) ( )∣ ⟩ integral yields the correlation functions of the Heisenbergﬁeldoperators in imaginary time. These correlation functions are often called the thermal correlation functions (or propagators). They are functions of the spatial po- sitions of the ﬁelds, x1,...,xN and of their imaginary time coordinates, xD1,...,xDN (here xD ≡ τ). To obtain the correlation functions as a func- tion of the real time coordinates x01,...,x0N at ﬁnite temperature T it is necessary to do an analytic continuation. We will discuss howthisisdone later on.

5.6 Path integrals for the free scalar ﬁeld We will consider now the case of a free scalar ﬁeld.Wewillcarryourdis- cussion in Euclidean space-time (i.e. in imaginary time), and we will do the relevant analytic continuation back to real time at the end ofthecalculation. The Euclidean Lagrangian LE for a free ﬁeld φ coupled to a source J is

1 2 1 L = ∂ φ + m2φ2 − Jφ (5.129) E 2 µ 2 where we are using the notation - . 2 ∂µφ = ∂µφ∂µφ (5.130) Here the index is µ = 1,...,D for an Euclidean space-time of D = d + 1 - . dimensions. For the most part (but not always) we will be interested in the case of d = 3andEuclideanspacehasfourdimensions.Noticetheway the Euclidean space-time indices are placed in Eq. (5.130). This is not a misprint! 142 Path Integrals in Quantum Mechanics and Quantum Field Theory We will compute the Euclidean Path Integral (or Partition Function) ZE J exactly. The Euclidean Path Integral for a free ﬁeld has the form

D 1 2 1 2 2 [ ] − d x ∂µφ + m φ − Jφ Z J = N Dφe" 2 2 (5.131) E " ) - . , In Classical[ ] Statistical Mechanics this theory is known as the Gaussian model. In what follows I will assume that the boundary conditions of the ﬁeld φ (and the source J)atinﬁnityareeithervanishingorperiodic,andthat the source J also either vanishes at spatial inﬁnity or is periodic. With these assumptions all terms which are total derivatives dropoutidentically. Therefore, upon an integration by parts and after dropping boundary terms, the Euclidean Lagrangian becomes 1 L = φ −∂2 + m2 φ − Jφ (5.132) E 2 φ Since this action is a quadratic form? of the ﬁ[email protected] this path integral can be calculated exactly. It has terms which are quadratic (or, rather bilinear) in φ and a term linear in φ,thesourceterm.Bymeansofthefollowingshift of the ﬁeld φ φ x = φ¯ x + ξ x (5.133) the Lagrangian becomes ( ) ( ) ( ) 1 L = φ −∂2 + m2 φ − Jφ E 2 1 1 = φ¯ −∂2 + m2 φ¯ − Jφ¯ + ξ −∂2 + m2 ξ + ξ −∂2 + m2 φ¯ − Jξ 2 ? @ 2 (5.134) ? @ ? @ ? @ Hence, we can decouple the source J x by requiring that the shift φ¯ be such that the terms linear in ξ cancel each other exactly. This requirement leads to the condition that the classical ﬁeld( ) φ¯ be the solution of the following inhomogeneous partial diﬀerential equation

−∂2 + m2 φ¯ = J x (5.135) ¯ Equivalently, we can write? the [email protected] ﬁeld(φ )is terms of the source J x through the action of the inverse of the operator −∂2 + m2, 1 ( ) φ¯ = J (5.136) −∂2 + m2 5.6 Path integrals for the free scalar ﬁeld 143 The solution of Eq. (5.135) is

D ′ E ′ ′ φ¯ x = d x G x − x J x (5.137) " 0 where ( ) ( ) ( ) E ′ 1 ′ G x − x = x x (5.138) 0 −∂2 + m2 2 2 is the correlation function( of the linear) ⟨ partial∣ diﬀerentia∣ ⟩ loperator−∂ +m . E ′ Thus, G0 x − x is the solution of E ′ D ′ −∂2 + m2 G x − x = δ x − x (5.139) ( ) x 0 E − ′ In terms of G0 x? x ,thetermsoftheshiftedactionbecome,@ ( ) ( ) 1 dDx φ¯ x −∂2 + m2 φ¯ x − Jφ¯ x " 2 ( ) 1 = − dDxφ¯ x J x O ( )? @ ( 2)" ( )P 1 ′ ′ ′ = − dDxdDx J x GE x − x J x 2 " ( ) ( ) 0 (5.140) ( ) ( ) ( ) Therefore the path integral for the generating function of the free Euclidean scalar ﬁeld ZE J ,deﬁnedinEq.(5.131),isgivenby

1 ′ ′ ′ dDx dDx J x GE x − x J x [ ] 2 " " 0 ZE J = ZE 0 e (5.141) where ZE 0 is ( ) ( ) ( ) [ ] [ ] 1 D 2 2 [ ] − d xξ x −∂ + m ξ x Z 0 = Dξe 2 " (5.142) E " ( )? @ ( ) Eq. (5.141)[ shows] that, after the decoupling, ZE J is a product of two factors: (a) a factor that is function of a bilinear form in thesourceJ,and (b) a path integral, ZE 0 ,thatisindependentofthesources.[ ]

[ ] 5.6.1 Calculation of ZE 0

The path integral ZE 0 is analogous to the ﬂuctuation factor that we found in the path integral for a harmonic oscillator in[ elementary] quantum me- chanics. There we saw[ that] the analogous factor could be written as a deter- minant of a diﬀerential operator, the kernel of the bilinear form that entered in the action. The same result holds here as well. The only diﬀerence is that 144 Path Integrals in Quantum Mechanics and Quantum Field Theory the kernel is now the partial diﬀerential operator Aˆ = −∂2 + m2 whereas in Quantum Mechanics is an ordinary diﬀerential operator. Here too, the operator Aˆ has a set of eigenstates Ψn x which, once the boundary con- ditions in space-time are speciﬁed, are both complete and orthonormal, and the associated spectrum of eigenvalues{ A( n)}is 2 2 −∂ + m Ψn x =AnΨn x

D d x Ψ x Ψ x =δ " ? n @ m( ) n,m ( ) ′ ′ Ψ x Ψ x =δ x − x (5.143) $ n( ) n ( ) n Hence, once again we can expand( the) ﬁeld( )φ x( in the) complete set of states Ψn x φ x = c Ψ ( x) (5.144) $ n n { ( )} n

Hence, the ﬁeld conﬁgurations( are) thus parametrized( ) by the coeﬃcients cn . The action now becomes, { } D 1 S = d x L φ,∂φ = A c2 (5.145) " E 2 $ n n n

Thus, up to a normalization factor,( we ﬁnd) that ZE 0 is given by

− −1 2 Z 0 = A 1 2 ≡ Det −∂2 + m2 , (5.146) E # n [ ] n / / and we have reduced[ ] the calculation= of Z?E 0 to [email protected]> computation of the determinant of a diﬀerential operator, Det −∂2 + m2 . In chapter 8 we will discuss eﬃcient methods[ ] to compute such determi- nants. For the moment, it will be suﬃcient? to notice [email protected] thereisasimple, but formal, way to compute this determinant. First, we noticethatifweare interested in the behavior of an inﬁnite system at T = 0, the eigenstates of the operator −∂2 + m2 are simply suitably normalized plane waves. Let L be the linear size of the system, with L → ∞.Then,theeigenfunctionsare labeled by a D-dimensional momentum pµ (with µ = 0, 1,...,d)

1 ipµ xµ Ψp x = e (5.147) 2πL D 2 with eigenvalues, ( ) / ( ) 2 2 Ap = p + m (5.148) 5.6 Path integrals for the free scalar ﬁeld 145 Hence the logarithm of determinant is

ln Det −∂2 + m2 =Tr ln −∂2 + m2 = ln p2 + m2 $ ? @ p ? @ (dDp ) =V ln p2 + m2 (5.149) " 2π D where V = LD is the volume of Euclidean space-time.( Hence,) ( ) D V d p 2 2 ln ZE 0 = − ln p + m (5.150) 2 " 2π D This expression is has two[ ] singularities: an infrared( divergence) and an ultra- violet divergence.lnZ 0 ,Eq(5.150),divergesas( ) V → ∞.Thisinfrared (IR) singularity actually is not a problem since ln ZE 0 should be an extensive quantity which must[ scale] with the volume of space-time. In other words, this is how it should behave. However, the integral[ ] in Eq.(5.150) diverges at large momenta unless there is an upper bound (or cutoﬀ)fortheallowed momenta. This is an ultraviolet (UV) singularity. It has the same origin of the UV divergence of the ground state energy. In fact ZE 0 is closely related to the ground state (vacuum) energy since [ ] −βEn −βE0 ZE 0 = lim e ∼ e + ... (5.151) β→∞ $ n Thus, [ ] D 1 1 d d p 2 2 E0 = − lim ln ZE 0 = L ln p + m (5.152) β→∞ β 2 " 2π D where Ld is the volume of space[ ] ,andV = Ldβ.NoticethatEq.(5.152)isUV( ) ( ) divergent. Later in this chapter we will discuss how to compute expressions of the form of Eq. (5.152).

5.6.2 Propagators and correlators Anumberofinterestingresultsarefoundimmediatelybydirect inspection of Eq. (5.141). We can easily see that, once we set J = 0, the correlation 0 ′ function GE x − x ( ) 0 ′ 1 ′ ( ) G x − x = x x (5.153) E −∂2 + m2 ( ) ( ) ⟨ ∣ ∣ ⟩ 146 Path Integrals in Quantum Mechanics and Quantum Field Theory is equal to the 2-point correlation function for this theory (at J = 0),

δ2Z J ′ = 1 E = 0 ′ φ x φ x ′ GE x − x (5.154) ZE 0 δJ x δJ x = 9J 0 ( ) [ ] 9 ⟨ ( ) ( )⟩ 9 ( ) Likewise we ﬁnd that, for a free ﬁeld theory,9 the N-point correlation function [ ] ( ) ( )9 φ x1 ...φ xN is equal to

1 δN Z J ⟨ ( φ)x ...( φ )x⟩ = E 1 N = ZE 0 δJ x1 ...δJ xN J 0 = [ ] 9 φ x1 φ x2 ... φ xN−19φ xN + permutations ⟨ ( ) ( )⟩ 9 [ ] ( ) ( )9 (5.155) ⟨ ( ) ( )⟩ ⟨ ( ) ( )⟩ Therefore, for a free ﬁeld, up to permutations of the coordinates x1,...,xN , the N-point functions reduces to a sum of products of 2-point functions. Hence, N must be a positive even integer. This result, Eq. (5.155), which we derived in the context of a theory for a free scalar ﬁeld, is actually much more general. It is known as Wick’s Theorem.Itappliestoallfreetheories, theories whose Lagrangians are bilinear in the ﬁelds, it is independent of the statistics and on whether there is relativistic invariance or not. The only caveat is that, as we will see later on, for the case of fermionic theories there is a sign associated with each term of this sum. It is easy to see that, for N = 2k,thetotalnumberoftermsinthesumis 2k ! k − k − ...... = 2 1 2 3 k (5.156) 2 k! ( ) Each factor of a 2-point( function)( φ x)1 φ x2 ,afreepropagator,anditis also called a contraction.Italsocommontousethenotation ⟨ ( ) ( )⟩ φ x1 φ x2 = φ x1 φ x2 (5.157) to denote a contraction⟨ or( propagator.) ( )⟩ ( ) ( )

5.6.3 Calculation of the propagator 0 ′ We will now calculate the 2-point function, or propagator, GE x − x for inﬁnite Euclidean space. This is the case of interest in QFT at( T) = 0. Later on we will do the calculation of the propagator at ﬁnite temperature.( ) 0 ′ Eq. (5.139) tells us that GE x − x is the Green function of the operator 0 ′ 2 2 ( ) −∂ + m .WewilluseFouriertransformmethodsandwriteGE x − x in ( ) ( ) ( ) 5.6 Path integrals for the free scalar ﬁeld 147 the form D ′ 0 ′ d p E i pµ xµ − xµ G x − x = G0 p e (5.158) E " 2π D ( ) ( ) which is a solution( of Eq.) (5.139) if ( ) ( ) 0 1 G p = (5.159) E p2 + m2 ( ) Therefore the correlation function( in) real (Euclidean!) space is the integral ′ D i pµ xµ − xµ 0 ′ d p e G x − x = (5.160) E " 2π D p2 + m2 ( ) ( ) We will often encounter( integrals) of this type and for that reason we will do this one in some detail. We begin by( using) the identity ∞ 1 1 − A α = dαe 2 (5.161) A 2 "0 where A > 0isapositiverealnumber.Thevariableα is called a Feynman- Schwinger parameter. We now choose A = p2 + m2,andsubstitutethisexpressionbackinEq. (5.160), which takes the form

∞ D α 2 2 ′ ′ 1 d p − p + m + ip x − x 0 − = 2 µ µ µ GE x x dα D e (5.162) 2 "0 " 2π ( ) ( ) ( ) The integrand( ) is a Gaussian, and the integral can be calculated by a shift ( ) of the integration variables pµ,i.e.bycompletingsquares

′ 2 ′ 2 α ′ 1 xµ − xµ 1 xµ − xµ p2 +m2 −ip x −x = αp − i − (5.163) 2 µ µ µ 2 µ α 2 α & and( by using) the( Gaussian) integral7 & 8 7 & 8

′ 2 1 xµ − xµ D − αpµ − i d p 2 α −D 2 e = 2πα (5.164) " 2π D & 7 & 8 / After all of this is done, we ﬁnd the formula ( ) ( ) ′ 2 ∞ x − x 1 2 0 ′ 1 −D − − m α G x − x = dα α 2 e 2α 2 (5.165) E D 2 2π 2 "0 ( ) / ∣ ∣ ( ) / ( ) 148 Path Integrals in Quantum Mechanics and Quantum Field Theory Let us now deﬁne a rescaling of the variable α, α = λt (5.166) by which ′ ′ x − x 2 1 x − x 2 1 + m2α = + m2λt (5.167) 2α 2 2λt 2 If we choose ∣ ∣ ∣ ∣ x − x′ = λ m , (5.168) the exponent becomes ∣ ∣ ′ ′ x − x 2 1 m x − x 1 + m2α = t + (5.169) 2α 2 2 t ∣ ∣ ∣ ∣ After this ﬁnal change of variables, we ﬁnd that the* correlat+ ion function is D − m 1 0 ′ = 1 2 ′ GE x − x ′ K D −1 m x − x (5.170) 2π D 2 x − x 2 ( ) where Kν( z is) the Modiﬁed/ * Bessel function,+ which( has∣ the integral∣) repre- sentation ( ) ∣ ∣ ( ) z 1 ∞ − t + 1 ν − 1 2 t Kν z = dt t e (5.171) 2 "0 * + ν = D − z = m x − x′ where 2 1, and ( ) .

power-law decay∣ ∣ exponential decay

a ξ =1/m distance

Figure 5.6 Behaviors of the Euclidean propagator.

′ There are two interesting regimes: (a) long distances, m x −x ≫ 1,and ′ (b) short distances, m x − x ≪ 1. ∣ ∣ ∣ A: long∣ distance behavior ′ In this regime, z = m x − x ≫ 1, a saddle-point calculation shows that the Bessel Function Kν z has the asymptotic behavior, ∣ ∣ π −z ( )K z = e 1 + O 1 z (5.172) ν 2z 6 ( ) [ ( / )] 5.6 Path integrals for the free scalar ﬁeld 149 Thus, in this regime the Euclidean propagator (or correlation function) be- haves like ′ D − 2 −m x − x 0 ′ π 2 m e 1 G x − x = 1 + O E D − 1 m x − x′ ( ) : ∣ ∣ 2π/ D 2 m x − x′ 2 ( ) ) * +, / ∣ ∣ (5.173) ( ) ( ∣ ∣) Therefore, at long distances, the Euclidean (or imaginary time) propagator has an exponential decay with distance (and imaginary time).Thelength scale for this decay is 1 m,whichisnaturalsinceitistheonlyquantitywith units of length in the theory. In real time, and in conventional units, this length scale is just the/ Compton wavelength, h̵ mc.InStatisticalPhysics this length scale is known as the correlation length ξ. / B: short distance behavior In this regime we must use the behavior of the Bessel function for small values of the argument,

Γ ν ν−2 Kν z = ν + O 1 z (5.174) 2 z (2 ) The correlation function now( ) behaves instead( / like, ) = > Γ D − 1 0 ′ = 2 GE x − x + ... (5.175) 4πD 2 x − x′ D−2 ( ) ( ′ ) where ... are terms that( vanish) as m/x − x → 0. Notice that the leading ∣ ∣ term is independent of the mass m.Thisisthebehaviorofthefreemassless theory. ∣ ∣

5.6.4 Behavior of the propagator in Minkowski space-time We will now ﬁnd the behavior of the propagator in real time.Thismeans that now we must do the analytic continuation back to real time. Let us recall that in going from Minkowski to Euclidean space we con- tinued x0 → −ix4.Inaddition,thereisalsofactorofi diﬀerence in the deﬁnition of the propagator. Thus, the propagator in Minkowski space-time ′ G 0 x − x is the expression that results from the analytic continaution ( ) ′ 0 ′ G 0 x − x = iG x − x (5.176) ( ) E x4→ix0 ( ) We can also obtain( ) this result from the path integral formulation in ( ) ( )Q 150 Path Integrals in Quantum Mechanics and Quantum Field Theory Minkowski space-time. Indeed, the generating functional for a free real mas- sive scalar ﬁeld Z J in D = d + 1dimensionalMinkowskispace-timeis

2 [ ] D 1 2 m 2 i d x ∂φ − φ + Jφ " 2 2 Z J = Dφe (5.177) " ' ( ) ( Hence, the expectation[ ] value to the time-ordered product of two ﬁeld is

1 δZ J 0 Tφ x φ y 0 = − (5.178) Z J δJ x δJ y J=0 [ ] 9 On the other hand,⟨ ∣ for( a) free( )∣ ﬁeld⟩ the generating function9 is given by (up [ ] ( ) ( )9 to a normalization constant N )

i D D 0 −1 2 d x d yJ x G x − y J y Z J = N ∂2 + m2 e 2 " " Det ( ) / ( ) ( ) (5.179)( ) where[ ]G0 x −? y is= the Green>@ function of the Klein-Gordon operator and satisﬁes ( ) D ∂2 + m2 G 0 x − y = δ x − y (5.180) ( ) Hence, we obtain the= expected> result( ) ( )

0 Tφ x φ y 0 = −iG 0 x − y (5.181) ( ) Let us compute the⟨ propagator∣ ( ) ( in)∣D⟩ = 4Minkowskispace-timebyanalytic( ) continuation from the D = 4Euclideanpropagator.Therelativisticinterval s is given by

2 ′ 2 ′ 2 s = x0 − x0 − x − x (5.182)

′ The Euclidean interval (length)( x −)x ,andtherelativisticinterval( ) s are related by

′ ∣ ∣ x − x = x − x′ 2 → −s2 (5.183) R : Therefore, in D = 4space-timedimensions,theMinkowskispacepropagator∣ ∣ ( ) is

0 ′ i m 2 G x − x = K1 m −s (5.184) 4π2 −s2 ( ) : ( ) & ( ) 5.6 Path integrals for the free scalar ﬁeld 151

We will need the asymptotic behavior of the Bessel function K1 z , π −z 3 K z = e 1 + + ... , for z ≫ 1 ( ) 1 2z 8z 6 1 z 1 K (z) = + ln z)+ C − +,... , for z ≪ 1(5.185) 1 z 2 2 where C =(0.)577215 ...* is the Euler-Mascheroni+ constant. Let us examine

Oscillatory Behavior

time-like separations light cone

Power law

Exponential Decay Exponential Decay Decay

space-like separations

Oscillatory Behavior

Figure 5.7 Behaviors of the propagator in Minkowski space-time. the behavior of Eq. (5.184) in the regimes: (a) space-like, s2 < 0, and (b) time-like, s2 > 0, intervals.

′ A: space-like intervals: x − x 2 = s2 < 0 This is the space-like domain. By inspecting Eq. (5.184) we see that for space-like separations, the factor −s2 (is a positive) real number. Conse- quently the argument of the Bessel& function is real (and positive), and the propagator is pure imaginary. In particular we see that, for s2 < 0the Minkowski propagator is essentially the Euclidean correlation function, 0 ′ = 0 ′ 2 < G x − x iGE x − x , for s 0(5.186) ( ) ( ) ( ) ( ) 152 Path Integrals in Quantum Mechanics and Quantum Field Theory

Hence, for s2 < 0wehavetheasymptoticbehaviors,

2 2 0 ′ π 2 m −m −s G x − x =i e , for m −s2 ≫ 1 2 3 2 :4π m −s2 : ( ) / : ( ′) i / G 0 x − x = ,& for m −s2 ≪ 1(5.187) 4π2 −s=2 > ( ) : ( ) ′ 2 = 2 > B:( time-like) intervals: x − x s 0 This is the time-like domain. The analytic continuation yields ( ) 0 ′ m 2 G x − x = K1 im s (5.188) 4π2 s2 ( ) : For pure imaginary arguments,( ) the Bessel& function( K)1 iz is the analytic K iz = − π H 1 −z continuation of the Hankel function, 1 2 1 .Thisfunctionis oscillatory for large values of its argument. Indeed, we( ) now( get) the behaviors

2 ( 2 ) ( ) 0 ′ π 2 m im s G x − x = e , for m s2 ≫ 1 2 3 2 :4π m s2 : ( ) / : ( ′) 1 / G 0 x − x = , & for m s2 ≪ 1(5.189) 4π2s2 = > ( ) : Notice( that, up) to a factor of i,theshortdistancebehavioristhesame for both time-like and space like separations. The main diﬀerence is that at large time-like separations we get an oscillatory behavior instead of an exponential decay. The length scale of the oscillations is, once again, set by the only scale in the theory, the Compton wavelength.

5.7 Exponential decays and mass gaps The exponential decay at long space-like separations (and the oscillatory behavior at long time-like separations) is not a peculiarityofthefreeﬁeld theory. It is a general consequence of the existence of a mass gap in the spectrum. We can see that by considering the 2-point functionofageneric theory, for simplicity in imaginary time. The 2-point function is ′ ′ ′ ′ G 2 x − x ,τ − τ = 0 T φˆ x,τ φˆ x ,τ 0 (5.190) where T is the imaginary( ) time-ordering operator. ( ) ⟨ ∣ ( ) ( )∣ ⟩ The Heisenberg representation of the operator φˆ in imaginary time is = (h̵ 1) Hτ −Hτ φˆ x,τ = e φˆ x, 0 e (5.191)

( ) ( ) 5.7 Exponential decays and mass gaps 153 Hence, we can write the 2-point function as ′ ′ G 2 x − x ,τ − τ = ( ) ′ ′ = ′ Hτ ˆ −H τ − τ ˆ ′ −Hτ ( θ τ −)τ 0 e φ x, 0 e φ x , 0 e 0 ′ ′ ′ Hτ ˆ ′ −H( τ −)τ ˆ −Hτ + θ(τ − τ)⟨0∣e φ( x ,)0 e (φ x, 0) e ∣0⟩ ′ ′ = ′ E0 τ − τ ˆ ( −H )τ − τ ˆ ′ θ(τ − τ )⟨e ∣ ( 0 )φ x, 0 e ( )φ x , 0 ∣ 0⟩ ′ ′ ′ E0(τ − τ) ˆ ′ −H( τ − τ) ˆ + θ(τ − τ) e ⟨0∣φ(x , 0) e φ( x, 0)∣0⟩ (5.192) ( ) ( ) ( ) ⟨ ∣ ( ) ( )∣ ⟩ We now insert a complete set of eigenstates n of the Hamiltonian Hˆ , with eigenvalues En .The2-pointfunctionnowreads, ′ ′ {∣ ⟩} G 2 x − x ,τ − τ = { } ′ ( ) ′ ′ − E − E τ − τ = θ τ − τ 0 φˆ x, 0 n n φˆ x , 0 0 e n 0 ( ) $ n ( )( ′ ) ′ ′ − E − E τ − τ + θ(τ − τ) ⟨0∣φˆ(x , 0)∣ n⟩⟨ n∣ φˆ( x, 0)∣0⟩ e n 0 $ n ( )( ) ( ) ⟨ ∣ ( )∣ ⟩⟨ ∣ ( )∣ ⟩ (5.193) Since iPˆ ⋅ x −iPˆ ⋅ x φˆ x, 0 = e φˆ 0, 0 e (5.194) and, in a translation invariant system, the eigenstates of the Hamiltonian ( ) ( ) are also eigenstates of the total momentum P

Pˆ 0 = 0, Pˆ n = Pn n , (5.195) where Pn is the linear momentum of state n ,wecanwrite ∣ ⟩ ∣ ⟩ ∣ ⟩ ′ ′ 2 −iPn ⋅ x − x 0 φˆ x, 0 n n φˆ x , 0 0 = 0 φˆ 0, 0 ∣ n⟩ e (5.196) Using the above expressions we can rewrite Eq. (5.193) in( the form) ⟨ ∣ ( )∣ ⟩⟨ ∣ ( )∣ ⟩ ∣⟨ ∣ ( )∣ ⟩∣ ′ ′ G 2 x − x ,τ − τ ( ) ′ ′ 2 ′ −iP ⋅ x − x − E − E τ − τ = 0 φˆ 0, 0 n θ τ − τ e n e n 0 $ ( ) n ′ ′ ′ −iP ⋅( x − x) −( E − E)( τ − τ) ∣⟨ ∣ ( )∣ ⟩∣ +S θ( τ − τ) e n e n 0

( ) ( )( (5.197)) ( ) T 154 Path Integrals in Quantum Mechanics and Quantum Field Theory ′ Thus, at equal positions, x = x ,weobtainthefollowingsimplerexpression ′ in the imaginary time interval τ − τ

′ 2 ′ 2 − E − E τ − τ G 0,τ − τ = 0 φˆ 0, 0 n × e n 0 (5.198) $ ( ) n ( )∣ ∣ ( ) ∣⟨ ∣ ( )∣ ⟩∣ ′ In the limit of large imaginary time separation, τ − τ → ∞,thereis always a largest non-vanishing term in the sums. This is the term for the state n0 that mixes with the vacuum state 0 through∣ the∣ ﬁeld operator ˆ − φ,andwiththelowest excitation energy, the mass gap En0 E0.Hence,for ′ large imaginary∣ ⟩ time separations, τ − τ → ∞∣ ,the2-pointfunctiondecays⟩ exponentially, ∣ ∣ ′ 2 ′ 2 − − − ˆ En0 E0 τ τ G 0,τ − τ ≃ 0 φ 0, 0 n0 × e (5.199) ( ) ( )∣ ∣ aresultthatwealreadyderivedforafreeﬁeldinEq.(5.173)( ) ∣⟨ ∣ ( )∣ ⟩∣ .Therefore, if the spectrum has a gap, the correlation functions (or propagators) de- cay exponentially in imaginary time. In real time we will get,instead,an oscillatory behavior. This is a very general result. Finally, notice that Lorentz invariance in Minkowski space-time (real time) implies rotational (Euclidean) invariance in imaginary time. Hence, exponen- tial decay in imaginary times, at equal positions, must imply(ingeneral) exponential decay in real space at equal imaginary times. Thus, in a Lorentz invariant system the propagator at space-like separations is always equal to the propagator in imaginary time.

5.8 Scalar ﬁelds at ﬁnite temperature We will now discuss brieﬂy the behavior of free scalar ﬁelds inthermal equilibrium at ﬁnite temperature T .Wewillgiveamoredetaileddiscussion in Chapter 10 where we will discuss more extensively the relation between observables and propagators. We saw in Section 5.5 that the ﬁeld theory is now deﬁned on an Euclidean cylindrical space-time which is ﬁnite and periodic along theimaginarytime direction with circumference β = 1 T ,whereT is the temperature (where we set the Boltzmann constant kB = 1). Hence the imaginary time dimension has been compactiﬁed. / 5.8 Scalar ﬁelds at ﬁnite temperature 155 5.8.1 The free energy Let us begin by computing the free energy. We will work in D = d + 1 Euclidean space-time dimensions. The partition function Z T is computed by the result of Eq.(5.146) except that the diﬀerential operator now is 2 2 2 ( ) Aˆ = −∂τ − ∂ + m , (5.200) with the caveat that now ∂2 denotes the the Laplacian operator that acts only on the spacial coordinates, x,andthattheimaginarytimeisperiodic. The mode expansion for the ﬁeld in this Euclidean (cylinder) space is ∞ d d p iω τ+ip⋅x φ x,τ = φ ω , p e n (5.201) $ " d n n=−∞ 2π where ωn = 2πTn( are) the Matsubara frequencies( ) and n ∈ Z.Theﬁeld ( ) operator is periodic in the imaginary time τ with period β = 1 T .The Euclidean action now is β / d 1 2 1 2 1 2 2 S = dτ d x ∂τ φ + ∂φ + m φ "0 " 2 2 2 d β d p 2) ( 2 ) (2 ) , = p + m φ0 p 2 " 2π d ddp +β ( ω2 +)p∣2 +( m)∣2 φ ω , p 2 (5.202) " ( )d $ n n 2π n≥1 where we split the action into= the sum of the>∣ contribution( )∣ fromthezero- ( ) frequency Matsubara mode, denoted by φ0 p = φ 0, p ,andthecontribu- tions of the modes for the rest of the frequencies. Since the free energy is given by F T = −( T)ln Z (T ,weneedtocompute) (again, up to the usual UV divergent normalization constant) T ( ) ( ) F T = ln Det − ∂2 − ∂2 + m2 (5.203) 2 τ We can now expand the determinant in the eigenvalues of the op erator 2 2 2 ( ) M N −∂τ − ∂ + m ,andobtaintheformallydivergentexpression ∞ 1 ddp F T = VT ln β ω2 + p2 + m2 (5.204) 2 " d $ n 2π n=−∞ where V is( the) spatial volume. This expression= M is formallyN> divergentbothin the momentum integrals and( in) the frequency sum and needs to beregular- ized. We already encountered this problem in our discussion of path integrals in Quantum Mechanics. As in that case we will recall that we have a formally 156 Path Integrals in Quantum Mechanics and Quantum Field Theory divergent normalization constant, N ,whichwehavenotmadeexplicithere and that can be deﬁned as to cancel the divergence of the frequency sum (as we did in Eq.(5.84)). The regularized frequency sum can now be computed d ∞ 2 2 d p 1 2 p + m F T = VT ln β p2 + m2 1 + (5.205) " d # ω2 2π / n=1 n Using( the) identity of Eq.(5.85)S * = the free energy> + F 7T becomes 8T ( ) ddp p2 + m2 F T = VT ( ) d ln 2sinh (5.206) " 2π : 2T which can be written( ) in the form ' 7 8( ( ) 2 2 d p + m d p − = T F T Vε0 + VT ln 1 − e : (5.207) " 2π d ⎛ ⎞ ⎜ ⎟ ( ) ⎜ ⎟ where ( ) ⎜ ⎟ d ⎝ ⎠ 1 d p 2 2 ε0 = p + m (5.208) 2 " 2π d R is the (ultraviolet divergent) vacuum (ground state) energydensity.Notice that the ultraviolet divergence is absent( ) in the ﬁnite temperature contribu- tion.

5.8.2 The thermal propagator The thermal propagator is the time-ordered propagator in imaginary time. It is equivalent to the Euclidean correlation function on thecylindricalge- ometry. We will denote the thermal propagator by 0 = GT x,τ φ x,τ φ 0, 0 T (5.209) ( ) It has the Fourier expansion ( ) ⟨ ( ) ( )⟩ ∞ + ⋅ 1 ddp eiωnτ ip x φ x,τ φ 0, 0 = (5.210) T β $ " d 2 2 2 n=−∞ 2π ωn + p + m where, once⟨ ( again,) (ωn =)⟩2πTn are the Matsubara frequencies. We will now obtain two useful expressions( ) for the thermal propagator. The expression follows from doing the momentum integrals ﬁrst. In fact, by observing that the Matsubara frequencies act as mass terms ofaﬁeldinone dimension lower, which allows us to identify the integrals inEq.(5.210)with 5.8 Scalar ﬁelds at ﬁnite temperature 157 the Euclidean propagators of an inﬁnite number of ﬁelds, eachlabeledby an integer n,ind Euclidean dimensions with mass squared equal to 2 2 2 mn = m + ωn (5.211) We can now use the result of Eq.(5.170) for the Euclidean correlator (now in d Euclidean dimensions) and write the thermal propagator as the following series d ∞ iωnτ − 1 0 1 e mn 2 G x,τ = K d m x (5.212) T d 2 ′ −1 n β =$ 2π x − x 2 ( ) n −∞ where m( is given) in Eq.(5.211)./ * Since the+ thermal propagator( ∣ ∣) is expressed n ( ) ∣ ∣ as an inﬁnite series of massive propagators, each with increasing masses, it −1 implies that at distances large compared with the length scale λT = 2πT , known as the thermal wavelength, all the terms of the series become negli- gible compared with the term with vanishing Matsubara frequency.( In) this limit, the thermal propagator reduces to the correlator of the classical theory in d (spatial) Euclidean dimensions,

0 ≃ ≫ GT x,τ φ x φ 0 , for x λT (5.213) ( ) In other terms, at distances large compared with the circumference β of ( ) ⟨ ( ) ( )⟩ ∣ ∣ the the cylindrical Euclidean space-time, the theory becomes asymptotically equivalent to the Euclidean theory in one space-time dimension less.

C+

+i!m2 + p2

C+

C−

−i!m2 + p2 C−

Figure 5.8 158 Path Integrals in Quantum Mechanics and Quantum Field Theory We will now ﬁnd an alternative expression for the thermal propagator by doing the sum over Matsubara frequencies shown in Eq.(5.210). We will use the residue theorem to represent the sum as a contour integral on the complex plane, as shown in Fig.5.8,

∞ 1 eiωnτ 1 dz eizτ z = cot (5.214) β $ 2 2 2 2 ( 2πi 2 2 2 2T n=−∞ ωn + p + m C+∪C− z + m + p O P where the (positively oriented) contour C = C− ∪ C+ in the complex z plane is shown in Fig.5.8. The black dots on the real axis of represent the integers z = n,whiletheblackdotsontheimaginaryaxisrepresentthepoles 2 2 at ±i m + p .UpondistortingthecontourC+ to the negatively oriented + contour C of the upper half-plane, and the contour C− to the negatively : − oriented contour C of the lower half-plane, we can evaluate the integrals by using the residue theorem once again, but now at the poles on the imaginary axis. This computation yields the following result for the thermalpropagator

p2 + m2 coth d T d p : 2 − τ p2 + m2 ip ⋅ x G x,τ = e e T d (5.215) " 2π 27p2 + m2 8 R ∣ ∣ ( ) : This expression applies( ) to the regime τ ≪ β = 1 T in which quantum ﬂuctuations play a dominant role. After a little algebra, we can now write the thermal propagator as /

d − τ p2 + m2 ip ⋅ x d p e e = GT x,τ R " 2π d p2 + 2 ∣ ∣2 m ( ) d : − τ p2 + m2 ip ⋅ x (d p) 1 e e + R (5.216) " 2π d p2 + m2 p2 + m2 exp − 1 ∣ ∣ : T : ( ) 7 8 By inspection of Eq.(5.216), we see that the ﬁrst term on the r.h.s is the T → 0limit,andthatitisjust(asitshouldbe!)thepropagatorin D = d+1 0 Euclidean space-time dimensions GE x,τ ,afteranintegrationoverfre- quencies. The second term in the r.h.s.( ) of Eq.(5.216) describes the contri- butions to the thermal propagator from( thermal) ﬂuctuations,showninthe 5.8 Scalar ﬁelds at ﬁnite temperature 159 form of the Bose occupation numbers, the Bose-Einstein distribution, 1 n p,T = (5.217) p2 + m2 exp − 1 ( ) : T The appearance of the Bose-Einstein7 distribution8 was to be expected (and, in fact, required) since the excitations of the scalar ﬁeld are bosons. Finally, we can ﬁnd the time-ordered propagator, in real time x0,atﬁnite 0 temperature T ,thatwewilldenotebyG x,x0; T .Bymeansofthean- alytic continuation τ → ix0 of the of the thermal( ) propagator of Eq.(5.216), we ﬁnd, ( ) 0 = 0 G x,x0; T GM x ( ) ( ) d −i x p2 + m2 ip ⋅ x ( d )p ( ) 1 e 0 e + R (5.218) " 2π d p2 + m2 p2 + m2 exp − 1 ∣ ∣ : T : ( ) 0 7 8 where GM x is the (Lorentz-invariant) Minkowski space-time propagator in D dimensions( ) (i.e. at zero temperature), given in Section 5.6.4. Notice that the ﬁnite temperature( ) contribution is not Lorentz-invariant. This result was expected since at ﬁnite temperature space and time do not playequivalent roles. 160 Path Integrals in Quantum Mechanics and Quantum Field Theory Exercises 5.1 Path Integral for a particle in a double well potential Consider a particle with coordinate q,massm moving in the one- dimensional double well potential V q

= 2 2 2 V q λ q( )− q0 (5.219) In this problem you will use the path integral methods, in imaginary ( ) = > time, that were discussed in class to calculate the matrix element, 1 T −T − HT q , − q , = q e h − q (5.220) 0 2 0 2 0 ̵ 0 to leading order in the semiclassical expansion, in the limit T → ∞. ⟨ ∣ ⟩ ⟨ ∣ ∣ ⟩ 1Writedowntheexpressionoftheimaginarytimepathintegral that is appropriate for this problem. Write an explicit expression for the Euclidean Lagrangian (the Lagrangian in imaginary time). How does it diﬀer from the Lagrangian in real time? Make sure that you specify the initial and ﬁnal conditions. Do not calculate anything yet! 2DerivetheEuler-Lagrangeequationforthisproblem(always in imag- inary time). Compare it with the equation of motion in real time. Find the explicit solution for the trajectory (in imaginary time) that satisﬁes the initial and ﬁnal conditions. Is the solution unique? Ex- plain. What is the physical interpretation of this trajectory and of the amplitude? Hint:Yourequationofmotioninimaginarytimelookslikeafunny. Asimplewaytosolveforthetrajectorythatyouneedinthisprob- lem is to think of this equation of motion as if imaginary time was real time, then to ﬁnd the analog of the classical energy and touse the conservation of energy to ﬁnd the trajectory. 3Computetheimaginarytimeactionforthetrajectoryyoufound above. 4Expandaroundthesolutionyoufoundabove.Writeaformalexpres- sion of the amplitude to leading order. Find an explicit expression for the operator that enters in the ﬂuctuation determinant. 5.2 Path Integral for a charged particle moving on a plane in the presence of a perpendicular magnetic ﬁeld. Consider a particle of mass m and charge −e moving on a plane in the presence of an external uniform magnetic ﬁeld perpendicular to the plane and with strength B.Letr = x1,x2 and p = p1,p2 represent

( ) ( )