introduction tothe feynman path integralscienze-como.uninsubria.it/phil/corsi/ln-fpi.pdf · a ket...

Introdu tion to

the Feynman Path Integral

(student handout version)

Philip G. Rat lie

(philip.rat lieuninsubria.it)

Dipartimento di S ienza e Alta Te nologia

Università degli Studi dell'Insubria in Como

via Valleggio 11, 22100 Como (CO), Italy

(last revised 6th February 2018)

Prefa e

The present is a written version of le ture notes for a short introdu tory ourse

on the Feynman path integral , held within the framework of the Physi s Ph.D.

programme at Insubria University in Como. The le tures were rst delivered in

the a ademi year 2015/16.

i

ii PREFACE

Contents

Prefa e i

Contents iii

1 Introdu tion 1

1.1 Supplementary reading . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Lagrangian versus Hamiltonian . . . . . . . . . . . . . . . . 4

1.4.2 Dira 's seed . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Derivation of the Path Integral 7

2.1 Huygens' prin iple . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The transition amplitude . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Paths in lassi al me hani s . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Quantum temporal evolution . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 The S hrödinger equation . . . . . . . . . . . . . . . . . . . 13

2.5 The path-integral formulation of NRQM . . . . . . . . . . . . . . . 16

2.6 Rederiving the S hrödinger equation . . . . . . . . . . . . . . . . . 20

2.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Quantum Field Theory 23

3.1 The path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 A real s alar eld . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 External sour e terms . . . . . . . . . . . . . . . . . . . . . 26

3.1.3 A omplex ( harged) s alar eld . . . . . . . . . . . . . . . . 27

3.1.4 The generalisation to N degrees of freedom . . . . . . . . . . 28

3.1.5 Transition matrix elements for produ ts of elds . . . . . . . 28

3.1.6 External sour e terms . . . . . . . . . . . . . . . . . . . . . 29

iii

iv CONTENTS

3.2 S alar quantum eld theory . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 The generating fun tional Z [J ] . . . . . . . . . . . . . . . . 35

3.2.2 The free-eld ase . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.3 Free-eld Green fun tions . . . . . . . . . . . . . . . . . . . 39

3.2.4 Conne ted Green fun tions . . . . . . . . . . . . . . . . . . . 40

3.2.5 The ee tive a tion . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A Theoreti al Ba kground 49

A.1 Wi k rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.2 A simplied approa h . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.3 The Legendre transform . . . . . . . . . . . . . . . . . . . . . . . . 53

A.3.1 Denition and basi s . . . . . . . . . . . . . . . . . . . . . . 53

A.3.2 Relationship to the path integral . . . . . . . . . . . . . . . 55

B Glossary of A ronyms 57

Chapter 1

Introdu tion

1.1 Supplementary reading

A short list of suggested supplementary reading material follows. Note, however,

that the intention of these few le tures is in any ase to be as far as possible a self-

ontained introdu tion. The referen e material listed here is thus merely provided

as an indi ation for those who might wish to study in more detail the various topi s

presented.

The book by Kleinert provides a very omprehensive treatment of the subje t,

as do the two volumes by Chai hian and Demi hev, while the lessons by Ma Kenzie

represent a more on ise but still rather omplete presentation of the Feynman

path integral and some of its appli ations. There is, of ourse, also the ex ellent

book by Feynman and Hibbs, whi h remains a very useful sour e of material and

ideas.

On the other hand, the short note by Derbes provides a brief a ount of the

manner in whi h Feynman apparently arrived at su h a formulation. The book by

Brown ontains Feynman's original thesis (whi h is, of ourse, wonderfully lear

and full of insight) and other related works su h as the seminal paper by Dira ,

whi h provided the intuition for Feynman's great step forward.

In the rst part of these notes we shall largely follow the derivation as given

in Abers and Lee (1973) and also in Bailin and Love (1993), where it is presented

in the ontext of se ond quantisation or quantum eld theory. Finally, my own

le ture notes on quantum eld theory (Rat lie, 2009) over all aspe ts of the

appli ation to eld quantisation, in luding derivation of Feynman rules for su h

omplete theories as quantum ele trodynami s and quantum hromodynami s.

1

2 CHAPTER 1. INTRODUCTION

Reading list

Abers, E.S. and Lee, B.W. (1973), Phys. Rep. 9, 1.

Bailin, D. and Love, A. (1993), Introdu tion to Gauge Field Theory (IOP Pub.),

revised edition.

Brown, L.M., ed. (2005), Feynman's Thesis: A New Approa h to Quantum Theory

(World S i.).

Chai hian, M. and Demi hev, A. (2001), Path Integrals in Physi s. Vol. 1: Sto has-

ti Pro esses and Quantum Me hani s (IOP Pub.); Path Integrals in Physi s.

Vol. 2: Quantum Field Theory, Statisti al Physi s and other Modern Appli a-

tions (IOP Pub., 2001).

Derbes, D. (1995), Am. J. Phys. 64, 881.

Dira , P.A.M. (1933), Phys. Z. Sowjetunion 3, 64; reprinted in, in Feynman's

Thesis: A New Approa h to Quantum Theory, Brown, L.M., ed. (World S i.,

2005), p. 111.

Dira , P.A.M. (1945), Rev. Mod. Phys. 17, 195.

Feynman, R.P. and Hibbs, A.R. (1965), Quantum Me hani s and Path Integrals

(M GrawHill).

Kleinert, H. (2006), Path Integrals in Quantum Me hani s, Statisti s, Polymer

Physi s, and Finan ial Markets (World S i.), 4th. edition.

Ma Kenzie, R. (2000), le tures in pro . of Ren ontres du Vietnam: VI Vietnam

S hool of Physi s (Vung-Tau, Vietnam, De . 1999), eds. N. Van Hieu and J. Trân

Thanh Vân, to appear; quant-ph/0004090.

Rat lie, P.G. (2012), m.s . le ture ourse presented at Università degli Studi

dell'Insubria (Como, 20092012).

S hulman, L.S. (2005), Te hniques and Appli ations of Path Integration (Dover),

2nd. edition.

http://www.arXiv.org/abs/quant-ph/0004090

1.2. AIMS AND PHILOSOPHY 3

1.2 Aims and philosophy

The spe i topi of this short set of le ture notes is pre isely the Feynman path

integral (FPI). This approa h was developed within the framework of quantum

me hani s and is ertainly of great importan e in se ond or eld quantisation.

However, it nds useful appli ations in many other elds: as diverse as statisti al

me hani s and even, for example, nan ial analysis.

Sin e the time available for this spe i ourse is limited (about eight hours of

front-on le tures), there will be little or no room for stri t mathemati al rigour,

whi h in any ase is somewhat orthogonal to my personal vision of physi s. In

other words, I shall attempt to provide a lear and motivated des ription of the

physi al signi an e and some of the interesting appli ations of the FPI approa h.

Indeed, the great advantage of this formulation is pre isely the insight it provides

on, for example, the pro ess of se ond quantisation and the derivation of the

so- alled Feynman rules (Feynman, 1949a,b) for al ulating physi al quantities

su h as ross-se tions, de ay rates and quantum orre tions to magneti moments,

oupling onstants, masses et .

1.3 Conventions and notation

We give here a brief forewarning of the onventions and notational devi es to be

use in the following notes. While some may leave the mathemati al expressions

a little hermeti on rst en ounter, the driving motivation is always to simplify

them by removing as mu h unne essary ornamentation as possible.

It is ommon pra ti e to indi ate operators via a ir umex, e.g. H . In the

following hapters, however, to avoid unne essary luttering of formulæ, unless

there is a real ambiguity, we shall avoid su h notation and rely on the ontext as

indi ator of the nature (operator or lassi al variable) of the various symbols used.

Indeed, the ir umex will usually indi ate a unit ve tor: thus, x≡x/|x|. Note

also that ve tors in three-dimensional spa e will be set in bold typefa e whereas

four-ve tors will be set in normal typefa ethere should be no ause for onfusion

however. In writing s alar produ ts, in either three- or four-dimensional spa e,

it is onvenient to adopt the Einstein summation onvention for repeated indi es:

thus, for example, x·y≡xiyi and x·y≡xµyµ.The braket notation, introdu ed by Dira in 1939, has long been universally

adopted as a onvenient and powerful short-hand in quantum theory; it will thus

be the main representation used here from the very start of the ourse. Students

unfamiliar with su h usage would be well advised to prepare by onsulting any of

the less advan ed texts available.


1.4 Motivation

1.4.1 Lagrangian versus Hamiltonian

In this the losing se tion of this hapter we shall briey examine the paper by

Dira (1933) that apparently motivated Feynman in his quest for a new formulation

of quantum me hani s. Dira opens by noting that whereas

Quantum me hani s was built up on a foundation of analogy with the Hamil-

tonian theory of lassi al me hani s.

Now there is an alternative formulation for lassi al dynami s, provided by

the Lagrangian.

This is evidently quite in ontrast with his own approa h to deriving the relativisti

quantum equation (Dira , 1928), whi h adopts the Hamiltonian formulation as the

starting point.

He then goes on to state that

The two formulations are, of ourse, losely related, but there are reasons

for believing that the Lagrangian one is the more fundamental.

Su h a view is indeed iterated by Feynman (1949b):

By forsaking the Hamiltonian method, the wedding of relativity and quantum

me hani s an be a omplished most naturally.

Although a posteriori we shall indeed see that the path-integral approa h leads

naturally to an intrinsi ally Lagrangian formulation, the motivation a priori for

su h a preferen e is manifold: all the equations of motion may be derived from a

single a tion prin iple; moreover, the a tion fun tion is naturally relativisti ally

invariant whereas

. . . the Hamiltonian method is essentially non-relativisti in form, sin e it

marks out a parti ular time variable as the anoni al onjugate of the Hamil-

tonian fun tion.

It might be added that in the Lagrangian approa h symmetries are more manifest,

with related onservation laws being immediately derivable. In this regard see, in

parti ular, the work of Noether (1918).

1.4.2 Dira 's seed

Now, the observation that sparked Feynman's imagination is the following: given

two omplete sets of generalised oordinates qi and Qi,

1.4. MOTIVATION 5

There will be a transformation fun tion

∗ (q′|Q′) onne ting the two rep-

resentations. We shall now show that this transformation fun tion is the

quantum analogue of eiS/h

.

Indeed, Feynman (1972), in his Nobel le ture, re ounts his personal rea tion to

this statement:

Professor Jehle showed me this, I read it, he explained it to me, and I

said, what does he mean, they are analogous; what does that mean,

analogous? What is the use of that? He said, you Ameri ans! You

always want to nd a use for everything! I said, that I thought that

Dira must mean that they were equal. No, he explained, he doesn't

mean they are equal. Well, I said, let's see what happens if we make

them equal.

And this then is essentially what he did, or rather, he a tually demonstrated that

they are (more-or-less) equal, whi h is what we shall do in the next hapter.

∗By transformation, Dira means what today we usually all a transition amplitude, whi h

moreover we now usually represent (in Dira notation) as 〈q′|Q′〉.


1.5 Bibliography

Dira , P.A.M. (1928), Pro . Royal So . (London) A117, 610.



2005), p. 111.

Dira , P.A.M. (1939), Math. Pro . Cambridge Phil. So . 35, 416.

Einstein, A. (1916), Annalen Phys. 49, 769; reprinted in, Annalen Phys. 14, 517.

Feynman, R.P. (1949a), Phys. Rev. 76, 749.

Feynman, R.P. (1949b), Phys. Rev. 76, 769.

Feynman, R.P. (1972), in Nobel Le tures, Physi s 19631970, ed. N. Foundation

(Elsevier), p. 155.

Noether, E. (1918), Na hr. v. d. Ges. d. Wiss. zu Göttingen, 235; transl., Transport

Theory and Statisti al Me hani s 183.

Chapter 2

Derivation of the Path Integral

In this hapter we introdu e the on ept of the Feynman path integral (FPI). We

shall see that although the mathemati al formulation due to Feynman is absolutely

original, in some sense the physi al on epts had already been at least partially

introdu ed in the study of lassi al opti s in the nineteenth entury and even

earlier. Indeed, it will always be of help to bear in mind that quantum me hani s

is none other than wave me hani s applied to what we usually onsider as the

propagation and intera tion of matter elds.

The on eptual di ulty, as always, lies in the wave nature of all dynami s: if it

were as easy to a ept su h a des ription for what are normally onsidered parti les

as it is to a ept that light propagates as individual photons, i.e. dis rete pa kets

of quantised energy and momentum, then there would indeed be no mystery.

2.1 Huygens' prin iple

Let us begin these lessons by re alling a well known view of lassi al wave me h-

ani s: namely, Huygens' prin iple (1690). Put briey, this states that given a

wavefront at some initial instant, a method for onstru ting the orresponding

wavefront at some later instant is provided by viewing ea h innitesimal portion

of the initial wavefront as a sour e giving rise to a later innitesimal disturban e

the only aveat is that the nal disturban es should then be summed as amp-

litudes and may thus interfere onstru tively or destru tively a ording to the

relative phases a rued during propagation (Fresnel, 1816); see Fig. 2.1. This is

then held to be a true physi al des ription of wave propagation.

Quantum me hani s des ribes all motion as wave propagation; it is thus nat-

ural to translate this pi ture to systems governed by, say, the S hrödinger (wave)

equation. Mathemati ally, the solutions to wave equations may des ribed via the

7

8 CHAPTER 2. DERIVATION OF THE PATH INTEGRAL

Figure 2.1: A pi torial representation of the HuygensFresnel prin iple.

use of a Green's fun tion or, as physi ists prefer to say, a propagator:

ψα(x, t) =

∫

d3x′K(x, t;x′, t′)ψα(x

′, t′), (2.1.1)

whi h is just the pre ise mathemati al formulation of Huygens' prin iple des ribed

above. Here, of ourse, impli itly t′<t, but we shall omment on this later.

Moving over to the more general Dira notation and exploiting the ompleteness

of position eigenstates |x′;t〉,∫

d3x′ |x′;t〉〈x′;t| = 1, (2.1.2)

we may write

〈x;t| =∫

d3x′ 〈x;t|x′;t′〉〈x′;t′| =

∫

d3x′K(x, t;x′, t′)〈x′;t′|. (2.1.3)

That is, the kernel of Eq. (2.1.1) is just the transition amplitude:

∗

K(x, t;x′, t′) = 〈x;t|x′;t′〉. (2.1.4)

We should mention here that there is a te hni al problem in this des ription

with ba kward propagation, whi h is usually solved by the use of so- alled ad-

van ed and retarded potentials; one then dis ards the advan ed solutions (i.e. those

that propagate ba kwards) as being unphysi al. We shall not dwell on this here

although we should note in passing that it was again Feynman who a tually solved

the problem te hni ally.

∗Re all that in Dira notation the wave-fun tion is given by ψα(x;t)= 〈x;t|α〉.

2.2. THE TRANSITION AMPLITUDE 9

2.2 The transition amplitude

The transition amplitude 〈xf

;tf

|xi

;ti

〉 is, in fa t, a entral obje t in quantum me h-

ani s (or wave me hani s). The meaning of this expression is simple: it represents

the probability amplitude for a parti le that was lo ated at the point xi

at time

ti

to then nd itself at the point xf

at some later time tf

; as already note, it is

also known to physi ists as the propagator or to mathemati ians as the Green's

fun tion. In the sense that it ompletely embodies the dynami s of the system,

it may be onsidered as being as fundamental as, say, the S hrödinger equation

itself. Indeed, as we shall see, a general expression for the transition amplitude

may be taken as the obje t dening the quantum theory.

Now, the ompleteness property of position eigenstates, Eq. (2.1.2), also allows

us to onsider the evolution from ti

to tf

, say, divided into two periods: rst from

ti

to some intermediate t′ and then from t′ to tf

. Thus,

〈xf

;tf

|xi

;ti

〉 =

∫

d3x′ 〈x

f

;tf

|x′;t′〉〈x′;t′|xi

;ti

〉 (tf

> t′ > ti

). (2.2.1)

This is the starting point for the FPI and somehow already embodies the on ept

of a sum over all paths.

One usually onsiders a nite interval between the initial and nal times; it is,

however, quite natural to onsider iteration over a large number of innitesimal

time steps. In a quantum des ription of motion the transition amplitude (modulus

squared) gives the probability that the parti le under study may be found at some

given point in spa e at some given instant in time. From the previous expression,

we already see that the nal amplitude may be seen as arising from onsidering

passage through all possible intermediate positions at the hosen intermediate

time.

A possible (seemingly uninteresting) exer ise is then to divide the time interval

between the initial and nal instants ti

and tf

into n+1 equal subintervals:

tf

≥ tn ≥ tn−1 ≥ · · · ≥ t2 ≥ t1 ≥ ti

, (2.2.2)

with, say, tm− tm−1= δt a xed interval for m=1,... ,n (t0≡ ti and tn+1≡ tf). Us-ing on e again the ompleteness relation (2.1.2), we an rewrite the transition

amplitude as:

K(xf

, tf

; xi

, ti

) =

∫

dxn

∫

dxn−1 · · ·∫

dx1

× 〈xf

;tf

|xn;tn〉〈xn;tn|xn−1;tn−1〉 · · · 〈x1;t1|xi;ti〉. (2.2.3)

The physi al meaning is perhaps best rendered pi torially, as in Fig. 2.2. The


ti

t1 t2...

.

.

.

tn−1 tn tf

t

x

Figure 2.2: A pi torial representation of the FPI or equivalently of wave or quantum

propagation.

probability amplitude for propagation from the point xi

at time ti

to the point xf

at time tf

may be al ulated as the sum over all paths with all possible intermediate

positions. The HuygensFresnel prin iple translates into summing over all possible

traje tories. Again, the physi al interpretation is as in Fig. 2.2.

2.3 Paths in lassi al me hani s

Let us briey omment on the on ept of paths in lassi al me hani s before on-

tinuing with their signi an e in quantum physi s. The lassi al Lagrangian for a

parti le of mass m, subje t to an external potential V (x) is

L l

(x, x) = 12mx2 − V (x). (2.3.1)

Fixing the boundary onditions in one of the many possible ways (e.g. by on-

straining the two end points xi

and xf

), there is usually just one path (or at

least only a nite number) orresponding to the lassi ally allowed motion of the

parti le. Consider, for example, the simple ase of a parti le moving in the Earth's

gravitational eld, V (x)=mgx, with the following boundary onditions:

(ti

, xi

) = (0, h) and (tf

, xf

) = (√

2h/g, 0). (2.3.2)

This leads to a lassi al path, whi h an then be nothing other than

x = h− 12gt2. (2.3.3)

2.4. QUANTUM TEMPORAL EVOLUTION 11

The general treatment follows Hamilton's prin iple: the lassi al path is that for

whi h the lassi al a tion is stationary; in other words, for whi h

δ

∫ tf

ti

dt L l

(x, x) = 0. (2.3.4)

Solving this ondition leads to Lagrange's equation of motion.

2.4 Quantum temporal evolution

The problem we are posed is the des ription of a given state, |α〉 say, at some time

t0 and its subsequent temporal evolution to a later instant t. We may express this

ompa tly as:

|α,t0;t〉 (t > t0), (2.4.1)

whi h, in words, is the state at time t that was |α〉 at an earlier time t0. The

standard pro edure, already met, is that of introdu ing the so- alled temporal-

evolution operator U(t,t0). We shall thus write

|α,t0;t〉 = U(t, t0)|α,t0〉, (2.4.2)

where, temporarily, we indi ate expli itly that |α〉 is the state at time t0.Clearly, there are natural restri tions to be pla ed on the operator U. First of

all, the requirement that probability be onserved in quantum me hani s (states

annot simply appear from or disappear into the va uum) implies that the norm

be preserved. In other words, U must be unitary. To see this, onsider a generi

state expanded over a basis set of eigenstates |a〉 of some operator A:

|α,t0;t〉 =∑

a

ca(t)|a〉. (2.4.3)

Assuming the eigenstates to be properly normalised, the generalised Fourier oef-

ient ca(t) simply represents the probability amplitude for obtaining result a for

a measurement A performed at time t; in other words, |ca(t)|2 is the probability

of obtaining result a. We must therefore have

∑

a

c2a(t) = 1 (∀ t), (2.4.4)

or, equivalently,

〈α,t0;t|α,t0;t〉 = 〈α,t0|α,t0〉. (2.4.5)


Then, sin e

〈α,t0;t|α,t0;t〉 = 〈α,t0|U†(t,t0)U(t,t0)|α,t0〉 (2.4.6)

must hold for any state |α,t0〉, we require

U†(t, t0)U(t, t0) = 1, (2.4.7)

where 1 is just the identity operator. This fundamental ondition is known as

unitarity and is intimately related to the probabilisti interpretation of quantum

me hani s.

A somewhat trivial, but ne essary, requirement is the property of omposition:

U(t, t0) = U(t, t′)U(t′, t0) (t0 < t′ < t). (2.4.8)

With this property in hand, it makes sense to work with innitesimal temporal

translations:

|α,t0;t0+δt〉 = U(t0 + δt, t0)|α,t0〉. (2.4.9)

Finally, ontinuity of the solutions requires that

U(t0, t0) = 1. (2.4.10)

It is not di ult to see then that the required operator must take the (innitesimal)

form

U(t0 + δt, t0) = 1− iΩ(t0)δt, (2.4.11)

where the unitarity ondition (2.4.7) requires that Ω be an Hermitian operator

(Ω†=Ω). It is immediately obvious that Ω has dimensions of frequen y. Re all

now that in lassi al me hani s the Hamiltonian H is the generator of temporal

translations and that if our system is to be invariant under time-reversal, H too

must be Hermitian. A natural identi ation is therefore

Ω = 1~H (2.4.12)

and we shall thus take

U(t0 + δt, t0) = 1− i~H δt. (2.4.13)

If Ehrenfest's theorem is to hold and thus guarantee the orre t lassi al limit, the

onstant of proportionality ~ introdu ed must learly be the same as that used

in the relationship involving spatial translations or, equivalently, in the de Broglie

relationship. As we shall now see, the hypothesis of a universal onstant ~ is

su ient to derive the S hrödinger equation.


2.4.1 The S hrödinger equation

As noted, the pre eding assumptions are su ient to derive the dierential equa-

tions governing the temporal evolution of physi al systems. Let us begin with the

evolution operator itself: we have

U(t+ δt, t0) = U(t+ δt, t)U(t, t0) =(

1− i~H δt

)

U(t, t0). (2.4.14)

The dieren e in U between t and t+δt is thus

δU(t, t0) ≡ U(t+ δt, t0)− U(t, t0) = − i~H δtU(t, t0), (2.4.15)

from whi h we immediately obtain the desired dierential, or S hrödinger, equation

for U:

i~d

dtU(t, t0) = HU(t, t0). (2.4.16)

On e we know how U evolves, the evolution of the states is also determined:

i~d

dt

[

U(t, t0)|α,t0〉]

=

[

i~d

dtU(t, t0)

]

|α,t0〉 = HU(t, t0)|α,t0〉, (2.4.17)

sin e |α,t0〉 is independent of t. We nally rewrite this as

i~d

dt|α,t0;t〉 = H |α,t0;t〉. (2.4.18)

It is, however, lear from the foregoing that on e we have U, we may abandon

the S hrödinger equation as superuous; all information regarding the evolution

of the states is unambiguously ontained in U. It is important to underline the

simpli ation this engenders: we need only obtain the solution of Eq. (2.4.16) and

apply it to any given starting ket.

Where did we get that [the S hrödinger equation from? It's not possible to

derive it from anything you know. It ame out of the mind of S hrödinger.

The Feynman Le tures on Physi s

In general, one needs to onsider three dierent possible ases of in reasing

omplexity:

1. H is time independent,

2. H is time dependent, with [H(t),H(t′)]=0 (∀ t and t′),

3. H is time dependent, with [H(t),H(t′)] 6=0 (for t′ 6= t).


Although it will turn out that the rst (and simplest) ase is su ient for our

purpose, it is important to be aware of the general problem.

Case 1: In this ase H does not vary with time, in other words, the sour e of

the for e elds is onstant (e.g. a spinning, harged parti le in a onstant magneti

eld). The solution of Eq. (2.4.16) is then simply

∗

U(t, t0) = exp[

− i~H ·(t− t0)

]

. (2.4.19)

To verify that this really is a solution, it is su ient to expand

†in powers of

H and dierentiate. Alternatively, one an divide the full interval [t0,t] into Nsubintervals of equal length δt=(t− t0)/N . One then has

U(t, t0) = U(t, t− δt) U(t− δt, t− 2δt) · · ·U(t0 + δt, t0)

=[

1− i~H ·(t− t0)/N

]N. (2.4.20)

Taking the limit N→∞ leads to Eq. (2.4.19).

Exer ise 2.4.1. Che k result Eq. (2.4.19) by expansion in powers of H.

Case 2: If H depends on t but ommutes with itself at dierent times, then the

solution may be written formally as

U(t, t0) = exp

[

− i

~

∫ t

t0

dt′ H(t′)

]

. (2.4.21)

This an be proved in a similar manner to the previous ase. An example might

be a spinning, harged parti le in a magneti eld of onstant dire tion but time-

varying intensity.

Exer ise 2.4.2. Prove result (2.4.21). Use both approa hes outlined above.

‡

Case 3: This is the most general ase: H may depend on t and does not ne es-

sarily ommute with itself at dierent times. The example might now be ome a

spinning, harged parti le in a time-dependent magneti eld of varying dire tion

∗The dot pla ed between H and (t− t0) is used here merely to larify that (t− t0) multiplies H

and is not its argument.

†Indeed, a fun tion of an operator su h as H is to be understood pre isely in terms of its

power-series expansion.

‡Feynman de lared himself never satised unless he had to dierent methods to solve a problem.


(re all that the operators sx and sy, e.g. do not ommute). The solution is a little

more di ult here. At least formally, we may integrate Eq. (2.4.16) to obtain

U(t, t0) = 1− i

~

∫ t

t0

dt′ H(t′)U(t′, t0), (2.4.22)

where the 1 (the integration onstant) is determined by the boundary ondition

U(t0,t0)=1. This is learly not mu h use sin e U still appears on the right-hand

side. We may, however, substitute the entire right-hand side, evaluated at t= t′,for U in the integrand:

U(t, t0) = 1− i

~

∫ t

t0

dt′ H(t′)+

(

i

~

)2∫ t

t0

dt′ H(t′)

∫ t′

t0

dt′′ H(t′′)U(t′′, t0). (2.4.23)

Iterating the substitution of U using Eq. (2.4.22) in the last integrand we obtain

the (formal) innite power-series (in H), or Neumann series, solution

U(t, t0) = 1+

∞∑

n=1

(

− i

~

)n∫ t

t0

dt1

∫ t1

t0

dt2 · · ·∫ tn−1

t0

dtn H(t1)H(t2) · · ·H(tn).

(2.4.24)

This is known as a Dyson series (although the original treatment by Dyson was

a tually performed within the framework of quantum eld theory).

Without proof, we express the Dyson series in a more symmetri fashion:

U(t, t0) = 1+

∞∑

n=1

1

n!

(

− i

~

)n∫ t

t0

dt1

∫ t

t0

dt2 · · ·∫ t

t0

dtn T(

H(t1)H(t2) · · ·H(tn))

,

(2.4.25)

where the operator T stands for time ordering. It is dened by

T(

O(t) O(t′))

≡

O(t) O(t′) for t > t′,

O(t′) O(t) for t < t′,(2.4.26)

with the obvious extension to a multiple produ t of operators. This all permits a

rather ompa t nal expression:

U(t, t0) = T

exp

[

− i

~

∫ t

t0

dt′ H(t′)

]

. (2.4.27)

Exer ise 2.4.3. Derive Eq. (2.4.25) expli itly, in parti ular the fa tor 1/n!, start-ing from the less symmetri Eq. (2.4.24).

Hint: onsider the ee t of reordering the integralsthe indi es 1,... ,n are merely


labels that may be freely reordered; it may help to start by onsidering the two-

dimensional ase, where there are only two possible orderings.

Clearly, we may make no statements a priori as to the onvergen e properties

of this or indeed the previous series solutions. Note that it is now trivial to obtain

formal solutions although a general expli it solution in anything but the rst ase

is not easily a essible; the ases of interest here will however always involve a

time-independent H.

2.5 The path-integral formulation of NRQM

In this se tion we shall present the path-integral formulation (Feynman, 1948) of

non-relativisti quantum me hani s, born of a suggestion by Dira (1933). The

approa h we shall follow is lose to that found in Abers and Lee (1973).

Given a time-independent operator QS

(in the S hrödinger pi ture) measur-

ing the generalised oordinate q, the (therefore time-independent) eigenstates are

dened, as usual, through

QS

|q〉S

≡ q|q〉S

. (2.5.1)

Moving over to the Heisenberg pi ture, re all that we have

QH

(t) := e i H t QS

e− i H tand |q,t〉

H

= e i H t |q〉S

, (2.5.2)

where H is the Hamiltonian (from now on assumed time independent) and we may

thus write

QH

|q,t〉H

≡ q(t)|q,t〉H

. (2.5.3)

Re all that in the Heisenberg pi ture the physi al states are frozen in time and

all time dependen e is absorbed into the operators, whi h onsequently then have

time-dependent eigenstates. Note further that here obviously HH

=HS

.

Now, the so- alled transition matrix element in the Heisenberg pi ture

M(qf

, tf

; qi

, ti

) = 〈qf

;tf

|qi

;ti

〉H H

= 〈qf

|e− iH(tf

−ti

) |qi

〉S S

(2.5.4)

des ribes the probability amplitude for a state initially having eigenvalue qi

at some

instant ti

to nd itself with qf

at a later instant tf

. It en odes all the important

information (dynami s) of the quantum theory. The idea then will be to express

this obje t as a path integral.

The rst step is to divide the nite interval [ti

,tf

] into n+1 subintervals of

innitesimal duration ∆t=(tf

− ti

)/(n+1) by inserting n omplete sets of states

2.5. THE PATH-INTEGRAL FORMULATION OF NRQM 17

(ti

<t1< · · ·<tn<tf):

M(qf

, tf

; qi

, ti

) =

∫

dq1 dq2 . . .dqn 〈qf,tf|qn,tn〉 · · · 〈q2,t2|q1,t1〉〈q1,t1|qi,ti〉, (2.5.5)

where (as hen eforth) we have omitted the sux H, on the understanding that

su h time-dependent eigenstates belong ex lusively to the Heisenberg pi ture.

Consider a generi innitesimal time sli e:

〈q′,t+∆t|q,t〉 = 〈q′|e− iH∆t|q〉

= 〈q′|1− iH∆t+O(∆t2)|q〉

= δ(q′ − q)− i∆t〈q′|H|q〉+O(∆t2). (2.5.6)

The HamiltonianH=H(Q,P ) is a fun tion of the generalised position and onjug-ate momentum operators Q and P .∗ The simplest, but most useful and ommon,

ase is a massive parti le (i.e. with kineti energy quadrati in P ) subje t to a

potential depending only on position Q:

H =P 2

2m+ V (Q). (2.5.7)

Using the fa t that 〈q|p〉=e ipq and 〈p′|p〉= δ(p′−p), we have

〈q′|H(Q,P )|q〉 =

∫

dp′ dp

[〈q′|p′〉〈p′|P 2|p〉〈p|q〉2m

]

+ 〈q′|V (Q)|q〉

=

∫

dp e ip(q′−q)

[

p2

2m+ V (q)

]

=

∫

dp e ip(q′−q)

H(q, p), (2.5.8)

where H(q,p) is now simply the lassi al Hamiltonian fun tion.

†That is, all

referen e to operators has been entirely eliminated.

To rst order in ∆t, we may thus rewrite (2.5.6) as

〈q′,t+∆t|q,t〉 ≃∫

dp e i [p(q′−q)−H(q,p)∆t] . (2.5.9)

∗Capital letters will generally be used here for operators whereas lower ase will indi ate ordinary

variables.

†In some texts one nds the more symmetri

1

2(q+q′) in pla e of q as the argument of H, but

this is quite unne essary, in view of the limit to be taken.


Now, to rst order in ∆t, (q′−q)≃ q∆t and so we have

〈q′,t+∆t|q,t〉 ≃∫

dp e i [pq−H(q,p)]∆t . (2.5.10)

Substituting this expression into that above for the matrix element, we have

M(qf

, tf

; qi

, ti

) = limn→∞

∫ n∏

k=1

dqk

n+1∏

k=1

dpk

× exp

i

n+1∑

j=1

[

pj qj −H(qj , pj)]

∆t

, (2.5.11)

with the identi ation q0≡ qi and qn+1≡ qf. In the limit ∆t→0, the sum be omes

an integral and we may rewrite the (now innite) produ ts as

∗

M(qf

, tf

; qi

, ti

) =

∫

DqDp exp

i

∫ tf

ti

dt[

p q −H(q, p)]

. (2.5.12)

This passage is then to be taken as dening the FPI.

With the above hoi e of Hamiltonian, the omplex integrals in p are Gaussianand may therefore be performed via a Wi k rotation (see App. A.1):

∫ ∞

−∞

dp e i (p q−p2/2m)∆t =

[ m

2π i∆t

]1/2e i

1

2mq

2∆t . (2.5.13)

For the transition matrix element, we thus nally obtain

M(qf

, tf

; qi

, ti

) = limn→∞

∫ n∏

k=1

dqk[

2π i∆t/m]1/2

× exp

i

n+1∑

j=1

[

12mq2j − V (qj)

]

∆t

= limn→∞

∫ n∏

k=1

dqk[

2π i∆t/m]1/2

exp

i

∫ tf

ti

dt L(q, q)

. (2.5.14)

∗Note that, in general, any numeri al fa tors a ompanying the measures dqk and dpk may

simply be absorbed into irrelevant overall normalisation onstants.

2.5. THE PATH-INTEGRAL FORMULATION OF NRQM 19

We may rewrite this as

M(qf

, tf

; qi

, ti

) =

∫

Dq

i

∫ tf

ti

dt L(q, q)

, (2.5.15)

where L(q,q)= 12mq2−V (q) is simply the lassi al Lagrangian and we thus see

that the exponent is pre isely the lassi al a tion

S(tf

, ti

) :=

∫ tf

ti

dt L(q, q), (2.5.16)

whi h then determines the temporal evolution in quantum me hani s. The nal,

rather ompa t, expression is thus

M(qf

, tf

; qi

, ti

) =

∫

Dq e iS(tf,ti) . (2.5.17)

The important point is that, whereas the nal expression for the transition

amplitude presented above has been derived starting from the anoni al formula-

tion of quantum me hani s, we may now equally take the FPI as the premise from

whi h to derive the S hrödinger equation.

It is important to stress, however, that the nal simple form shown above is a

dire t onsequen e of the parti ular hoi e of Hamiltonian, whi h is, fortunately,

generally su ient for our purposes. If, on the other hand, the Hamiltonian were

to involve produ ts of P and Q, then the nal form would not depend simply on

the lassi al a tion, but on what might be alled an ee tive a tion, whi h ould

then be al ulated from Eq. (2.5.11).

Exer ise 2.5.1. Starting from the non-linear Lagrangian

L(q, q) = 12q2 f(q),

where f(q) is some non-singular fun tion of q, show that the transition amplitude

is as above but with an ee tive a tion (Lee and Yang, 1962)

Se

=

∫ tf

ti

dt[

L(q, q)− 12iδ(0) ln f(q)

]

.

One then nds that the s attering matrix al ulated using su h a transition

amplitude generates another innite ontribution, whi h an els the δ(0) term.

In any ase, the evident advantage of this approa h is that we express quantum-

me hani al quantities in terms of the lassi al Lagrangian. In parti ular, this per-


mits a dire t study of the importan e of the known symmetries at the lassi al

level of su h quantities and their properties under various symmetry transforma-

tions. Perhaps, we need only re all here the importan e of Noether's seminal work

(1918) on the relationship between ontinuous symmetries and onserved quantit-

ies, whi h stands at the very foundation of all modern theory, both lassi al and

quantum.

2.6 Rederiving the S hrödinger equation

We lose this se tion by showing expli itly that the transition amplitude dened

by the FPI does indeed obey the time-dependent S hrödinger wave equation in

the variables xf

and tf

(for xi

and ti

xed). Let us split the amplitude into just

two regions, [ti

,t] and [t,tf

], with t= tf

−δt:

〈xf

,tf

|xi

,ti

〉 =

∫ +∞

−∞

dx′ 〈xf

,tf

|x′,t〉〈x′,t|xi

,ti

〉

=

√

m

2π i~ δt

∫ +∞

−∞

dx′ exp

[

im

2~

(xf

− x′)2

δt−

iV(

12(x

f

+ x′))

δt

~

]

〈x′,t|xi

,ti

〉.

(2.6.1)

Sin e we are assuming xf

−x′ innitesimal, it is onvenient to shift the integration

variable to ξ≡xf

−x′ and rewrite xf

=x and tf

= t+δt:

〈x,t+δt|xi

,ti

〉 =

√

m

2π i~ δt

∫ +∞

−∞

dξ exp

[

im

2~

ξ2

δt− iV (x− 1

2ξ) δt

~

]

〈x−ξ,t|xi

,ti

〉.(2.6.2)

Now, one of the many denitions of the Dira δ-fun tion is

limδt→0

√

m

2π i~ δtexp

[

imξ2

2~δt

]

= δ(ξ), (2.6.3)

from whi h we see that the dominant region for δt→0 is ξ∼0. That being the

ase, we may now expand 〈x−ξ,t|xi

,ti

〉 on the right-hand side of Eq. (2.6.2) aroundthe point x (i.e. as a Taylor series in ξ). Note that expansions in δt and ξ2 are

equivalent sin e the integral over ξ adds a power of

√δt for every power of ξ

present; the term in V is already O(δt). It is thus also ne essary to expand the

left-hand side as a power series in δt. Keeping terms only up to and in luding

2.6. REDERIVING THE SCHRÖDINGER EQUATION 21

O(δt), we have

〈x,t|xi

,ti

〉+ δt∂

∂t〈x,t|x

i

,ti

〉

=

√

m

2π i~ δt

∫ +∞

−∞

dξ exp

(

im

2~

ξ2

δt

)[

1− iV (x) δt

~+ · · ·

]

×[

〈x,t|xi

,ti

〉+ ξ2

2

∂2

∂x2〈x,t|x

i

,ti

〉+O(ξ4)

]

, (2.6.4)

where terms linear (and ubi ) in ξ have already been dropped, as they vanish on

integration from −∞ to +∞. The leading 〈x,t|xi

,ti

〉 terms an el between the two

sides and, using

∫ +∞

−∞

dξ ξ2 exp

(

im

2~

ξ2

δt

)

=√2π

(

i~δt

m

)3/2

, (2.6.5)

in the limit δt→0 we are left with

δt∂

∂t〈x,t|x

i

,ti

〉 =

√

m

2π i~ δt

√2π

(

i~δt

m

)3/21

2

∂2

∂x2〈x,t|x

i

,ti

〉

− i

~δt V (x)〈x,t|x

i

,ti

〉. (2.6.6)

Finally, rearranging and gathering fa tors (δt drops out), we obtain the S h-

rödinger equation for 〈x,t|xi

,ti

〉:

i~∂

∂t〈x,t|x

i

,ti

〉 =

[

− ~2

2m

∂2

∂x2+ V (x)

]

〈x,t|xi

,ti

〉. (2.6.7)

We may thus on lude that the transition amplitude onstru ted following the

Feynman formulation is the same as that of the S hrödinger formulation in wave

me hani s. Unfortunately, the integrals to be evaluated even for the simplest

problem in ordinary quantum me hani s are quite ompli ated and thus there is

little to be gained over the usual formulation. In the ase of quantum eld theory

(or se ond quantisation), however, the Feynman approa h is de idedly easier and

more intuitive. The parallel with the partition fun tion Z(β) also makes this

approa h parti ularly suited to ertain problems in statisti al quantum me hani s

and, in parti ular, the study of quantum me hani s at nite temperature.


2.7 Bibliography

Abers, E.S. and Lee, B.W. (1973), Phys. Rep. 9, 1.



2005), p. 111.

Feynman, R.P. (1948), Rev. Mod. Phys. 20, 367.

Fresnel, A. (1816), Ann. Chim. et Phys. 1, 339; eds. H.H. de Sénarmont, É. Verdet

and L.F. Fresnel, Ouvres Complètes d'Augustin Fresnel (in Fren h) (Imprimerie

Impériale, Paris, 1866).

Huygens, Chr. (1690), Traité de la Lumiere (Pieter van der Aa); ompleted in

1678.

Lee, T.-D. and Yang, C.-N. (1962), Phys. Rev. 128, 885.

Noether, E. (1918), Na hr. v. d. Ges. d. Wiss. zu Göttingen, 235; transl., Transport

Theory and Statisti al Me hani s 183.

Chapter 3

Quantum Field Theory

3.1 The path integral

We shall now examine the appli ation of the FPI to the problem of se ond or

eld quantisation. As in the previous hapter, the basi denition of the path or

fun tional integral starts with a dis retised version of the theory; we shall present

the approa h for a simple real s alar eld. For the more ompli ations ases of

spinor and gauge elds, the reader is referred to my online notes on quantum eld

theory.

3.1.1 A real s alar eld

The starting point is the following simple one-dimensional Gaussian integral:

∫ ∞

−∞

dq e−12aq

2

=√2π a−

1/2 , (3.1.1)

where q is intended to represent a generalised real oordinate. The natural exten-

sion to a real n-dimensional spa e leads to

∫ ∞

−∞

dq1 dq2 . . . dqn e−12qTA q =

(√2π)n

(detA)−1/2 , (3.1.2)

where q is an n- omponent ve tor: q=(q1,... ,qn) and A a real, symmetri n×n-matrix; onvergen e of the integral then requires A to be also positive-denite.

∗

Exer ise 3.1.1. Prove the last equation above.

Hint: This is most easily performed by diagonalising the matrix A.

∗We shall normally be dealing with Hermitian operators su h as H and L, whi h will thus be

naturally represented by Hermitian matri es.

23

24 CHAPTER 3. QUANTUM FIELD THEORY

It is onvenient, for the path-integral quantisation pro edure, to reformulate

su h expressions as exponentials and we thus rewrite this as

∫ ∞

−∞

dq1 dq2 . . .dqn e−12qTA q =

(√2π)n

e−12Tr lnA, (3.1.3)

where we have exploited the following identity:

ln detA ≡ Tr lnA. (3.1.4)

Exer ise 3.1.2. Prove this last equality.

Hint: The simplest proof is again obtained by diagonalising A. Re all that both the

tra e and determinant are invariant under unitary transformation.

The appli ation to eld theory now pro eeds by taking the limit n→∞, when e

the ve tor qi

:= q(xi)→ q(x) and the matrix Aij :=Axi,yj→A(x,y), with xi and yj

representing spa etime oordinates. One may think of the dis retised quantit-

ies as lo al averages over innitesimal spa etime hyper ubes entred around the

points xi, yj et . We shall thus take the ontinuum limit of (3.1.3) to dene a

so- alled fun tional integral :

∫ ∞

−∞

Dq e−12

∫dx dy q(x)A(x,y) q(y) = e−

12Tr lnA . (3.1.5)

In this denition the numeri al

√2π fa tors have been absorbed into an overall

(innite) normalisation onstant (whi h, as we shall soon see, is generally irrelev-

ant). For larity, we have limited the spatial integrals to one dimension; however,

the extension to three-dimensional spa e or four-dimensional spa etime is trivial.

We must now learn how to evaluate the expression on the right-hand side.

To do this, let us rst dene a little more arefully what is meant by a fun tion

of an operator, su h as A(x,y). By onsidering the passage from the dis rete ase

to the ontinuum, it is evident that multipli ation may now be naturally dened

as

C = A·B, meaning C(x, y) =

∫

dz A(x, z)B(z, y), (3.1.6)

where the integration is understood to be over the entire x-spa e. Moreover, the

tra e operation thus be omes trivially

TrA :=

∫

dxA(x, x). (3.1.7)

Now, in all ases of interest here we shall be dealing with lo al intera tions and

3.1. THE PATH INTEGRAL 25

thus the most general form of operator to o ur will a tually be

A(x, y) = δ(x− y)A(x, y), (3.1.8)

where A(x,y) is an analyti fun tion of x, y and their derivatives. It is useful then

to dene the Dira δ-fun tion via a Fourier-transform representation

δ(x− y) :=

∫ ∞

−∞

dk e ik(x−y) . (3.1.9)

We may thus write

A(x, y) =

∫

dk e ik(x−y)A(k). (3.1.10)

Consider, for example, the following operator:

A(x, y) = δ(x− y)

[

∂

∂x

∂

∂y+ c

]

, (3.1.11a)

with c a onstant, when e we have

A(x, y) =

∫

dk e ik(x−y) [k2 + c]. (3.1.11b)

It is left as an exer ise to show that su h an operator raised to a power is given

by the following expression:

∗

An(x, y) ≡∫

dz1 . . .dzn−1A(x, z1)A(z1, z2) . . . A(zn−1, y)

=

∫

dk e ik(x−y)A

n. (3.1.12)

In other words, inasmu h as a power-series representation exists, a fun tion, or

rather fun tional, of an operator may be represented as (note the use of square

bra kets)

F [A(x, y)] =

∫

dk e ik(x−y) F(

A(k))

. (3.1.13)

In parti ular, for the inverse (whi h ertainly exists if, as required, the orrespond-

∗From now on we shall omit the tilde in Fourier transforms, as the argument su es to indi ate

the spa e over whi h the obje t is dened.


ing matrix A is positive denite or Hermitian) we have

A−1(x, y) =

∫

dk e ik(x−y)A

−1(k). (3.1.14)

And nally, for example, for the operator A dened in (3.1.11a), we then have

lnA(x, y) =

∫

dk e ik(x−y) ln[k2 + c]. (3.1.15)

3.1.2 External sour e terms

A further very useful generalisation of the previous formulæ is obtained by adding

a term linear in q to the exponent; thus,

∫ ∞

−∞

dq1 dq2 . . .dqn e−12qTA q+j·q =

(√2π)n

e−12Tr lnA e

12jTA−1

j , (3.1.16)

where j=(j1,... ,jn) is some arbitrary external ( urrent or sour e) ve tor.

Exer ise 3.1.3. Prove this last result.

Hint: Complete the square in the exponent, then shift the integration variables;

note that su h an operation is always possible sin e the integrals onverge.

In path-integral or fun tional language this be omes

∫ ∞

−∞

Dq exp

−12

∫

dx dy q(x)A(x, y) q(y) +

∫

dy j(y) q(y)

= exp

−12Tr lnA+ 1

2

∫

dx dy j(x)A−1(x, y) j(y)

. (3.1.17)

With this expression, we an now evaluate rather more ompli ated integrals.

Dierentiating Eq. (3.1.16) with respe t to the dis rete sour es ji1 , ji2 , . . . jim and

then setting all of the j1= · · ·= jn=0, we obtain

∫ ∞

−∞

dq1 dq2 . . .dqn qi1 qi2 . . . qim e−12qTA q

=[

A−1i1i2

· · ·A−1im−1im

+ perms] (√

2π)n

e−12Tr lnA, (3.1.18)

The natural extension to the ontinuum of the previous derivatives, the fun tional

derivative, may now be intuitively dened via

δ

δj(x)

∫

dy j(y) q(y) := q(x). (3.1.19)


And the fun tional-integral version (or ontinuum) of (3.1.18) is then simply (for

m even)

∫ ∞

−∞

Dq q(x1) q(x2) . . . q(xm) e− 1

2

∫dy dz q(y)A(y,z) q(z)

=[

A−1(x1, x2) · · ·A−1(xm−1, xm) + perms]

e−12Tr lnA . (3.1.20)

3.1.3 A omplex ( harged) s alar eld

The only immediately useful extension remaining is to omplex variables qi

∈C and

thus to omplex elds q(x)∈C.∗In order that the integrals remain well-dened

and onvergent, we now require that the omplex-valued matrix A be Hermitian

†

(or self-adjoint) and positive denite. The omplex measure is

dq dq∗ ≡ 2 d(Re q) d(Im q) . (3.1.21)

And, starting again with the simple Gaussian integral (a real and positive)

∫

dq dq∗ e−12aq

∗q =

√2π a−1, (3.1.22)

through to the multi-dimensional version

∫

dq1 dq∗1 . . . dqn dq

∗n e−

12q†A q =

(√2π)n

e−Tr lnA, (3.1.23)

we obtain,

∫

DqDq∗ exp

−12

∫

dx dy q∗(x)A(x, y) q(y) +

∫

dx[

j∗(x) q(x) + j(x) q∗(x)]

= exp

−Tr lnA+ 12

∫

dx dy j∗(x)A−1(x, y) j(y)

. (3.1.24)

Exer ise 3.1.4. Using the methods illustrated, prove the nal result given here.

∗Su h a ase naturally represents a harged s alar (spin-zero) eld.

†This will usually be the ase for physi al systems of interest here.


3.1.4 The generalisation to N degrees of freedom

It is now straightforward to generalise Eq. (2.5.12) to a system with N ontinuous

degrees of freedom:

M(qf

, tf

; qi

, ti

) =

∫

DqDp exp

i

∫ tf

ti

dt[

p·q −H(p, q)]

, (3.1.25)

where q :=(q1,q2,...qN) and p := (p1,p2,...pN ). This then will be the starting pointfor a quantum eld theory . However, for the rest of this se tion a single ontinuous

degree of freedom will su e.

3.1.5 Transition matrix elements for produ ts of elds

Let us now try to al ulate more ompli ated matrix elements: inserting the oper-

ator Q(t) into the transition matrix element (2.5.4), for some instant t su h that

ti

<t<tf

, leads to

〈qf

;tf

|Q(t)|qi

;ti

〉 =

∫

dq1 dq2 . . .dqn

× 〈qf

,tf

|qn,tn〉 · · · 〈qk,tk|Q(t)|qk−1,tk−1〉 · · · 〈q2,t2|q1,t1〉〈q1,t1|qi,ti〉, (3.1.26)

where we have identied tk and tk−1 su h that tk−1<t<tk. To rst order in ∆t we an onsider |qk−1,tk−1〉 to be an approximate eigenstate of Q(t) and thus repla e

Q(t) with q(t). It is then easy to see that we may pro eed as before, with simply

an extra fa tor q(t) inside the integral, to obtain

〈qf

;tf

|Q(t)|qi

;ti

〉 =

∫

DqDp q(t) exp

i

∫ tf

ti

dt′[

pq −H(q, p)]

. (3.1.27)

Consider now a produ t of two su h operators Q(ta)Q(tb): if ta>tb, we have

〈qf

;tf

|Q(ta)Q(tb)|qi;ti〉 =

∫

dq1 dq2 . . .dqn

× 〈qf

,tf

|qn,tn〉 · · · 〈qka,tka |Q(ta)|qka−1,tka−1〉 · · ·

· · · 〈qkb ,tkb|Q(tb)|qkb−1,tkb−1〉 · · · 〈q1,t1|qi,ti〉, (3.1.28a)


whi h eventually leads us to

〈qf

;tf

|Q(ta)Q(tb)|qi;ti〉 =

∫

DqDp q(ta) q(tb) exp

i

∫ tf

ti

dt[

pq −H(q, p)]

.

(3.1.28b)

The derivation rests on the ordering ta>tb, i.e. the operator on the right a ts

temporally before that on the left. Indeed, it is not generally possible to derive

su h a formula with the opposite ordering. We thus see that, to avoid spe i

restri tions on ta and tb, we should write

〈qf

;tf

|T[

Q(ta)Q(tb)]

|qi

;ti

〉

=

∫

DqDp q(ta) q(tb) exp

i

∫ tf

ti

dt[

pq −H(q, p)]

, (3.1.29)

where we have introdu ed the standard time-ordered produ t T[

Q(ta)Q(tb)]

.

T[

Q(ta)Q(tb)]

:=

Q(ta)Q(tb) for ta > tb,

Q(tb)Q(ta) for tb > ta.(3.1.30)

The extension to a time-ordered produ t of M ≥2 su h operators is obvious

〈qf

;tf

|T[

Q(t1)Q(t2)...Q(tM)]

|qi

;ti

〉

=

∫

DqDp q(t1) q(t2) . . . q(tM) exp

i

∫ tf

ti

dt[

pq −H(q, p)]

. (3.1.31)

3.1.6 External sour e terms

It will also be useful to onsider the ee t of adding external sour e terms (or,

in lassi al language, driving for es) into the Hamiltonian, whi h, as always, we

assume does not itself depend expli itly on time. We shall require the unperturbed

energy eigenstates |n〉 and their orresponding wave-fun tions:

ϕn(q, t) = 〈q,t|n〉 = e− iEnt 〈q|n〉 with ϕn(q) = 〈q|n〉. (3.1.32)

Let ϕ0(q,t) be the lowest-energy or ground state of the system. We now aim

to al ulate the transition amplitude for a system in the ground state at some

initial instant ti

in the distant past to be in the same state at some nal instant

tf

in the distant future. In parti ular, we onsider a sour e term J(t)Q(t), with


J(t) 6=0 only during the interval ta<t<tb, added to the Hamiltonian.

∗That is,

the arbitrary sour e term is initially swit hed o and remains zero until t= ta andthen remains a tive only until t= tb, when it is again set to zero. We thus write

〈qf

;tf

|qi

;ti

〉J =

∫

DqDp exp

i

∫ tf

ti

dt[

p q −H(q, p) + J q]

. (3.1.33)

Inserting omplete sets of states, we an now split the amplitude into a produ t

of pie es running from ti

to ta, to tb, to tf:

〈qf

;tf

|qi

;ti

〉J =

∫

dqa dqb 〈qf;tf|qb;tb〉〈qb;tb|qa;ta〉J〈qa;ta|qi;ti〉. (3.1.34)

The rst and last elements on the right-hand side are given by expressions su h as

Eq. (3.1.33) without the sour e term. Consider, for example, the last:

〈qa;ta|qi;ti〉 = 〈qa|e− iH(ta−ti

) |qi

〉

=∑

n

〈qa|e− iH(ta−ti

) |n〉〈n|qi

〉

=∑

n

〈qa|n〉〈n|qi〉 e− iEn(ta−ti

)

=∑

n

ϕn(qa)ϕ∗n(qi) e

− iEn(ta−ti

) . (3.1.35)

Sin e the dependen e on ti

is expli it, we may make the analyti ontinuation

ti

→ i∞ (or Wi k rotation). The rapidly os illating imaginary phases then turn

into de aying exponentials, whi h kill all ontributions for n>0, leaving

limti

→ i∞〈qa;ta|qi;ti〉 = ϕ0(qa, ta)ϕ

∗0(qi), (3.1.36a)

assuming qi

to be some onstant asymptoti value. Likewise, the rst term be omes

limtf

→− i∞〈q

f

;tf

|qb;tb〉 = ϕ0(qf)ϕ∗0(qb, tb), (3.1.36b)

again assuming a onstant asymptoti value for qf

. Substituting these ba k into

∗The external sour e J(t) is not, of ourse, really an operator, but we use a apital letter for

larity of notation.


the original expression, we obtain

limti

→+i∞

tf

→− i∞

〈qf

;tf

|qi

;ti

〉J

exp[− iE0(tf − ti

)]ϕ∗0(qi)ϕ0(qf)

=

∫

dqa dqb ϕ∗0(qb, tb) 〈qb;tb|qa;ta〉J ϕ0(qa, ta). (3.1.37)

On examining this equation, we nd that the right-hand side is just the ground-

state to ground-state amplitudenote that ta,b may be taken as large as we like.

This obje t is what we shall all Z [J ].∗ The left-hand side above then simply

provides the method to al ulate it.

The generating fun tional Z [J ] also allows the derivation of ground-state to

ground-state expe tations values of produ ts of elds, as in Eq. (3.1.31), simply

by taking fun tional derivatives with respe t to J(t), whi h pull down fa tors of

iϕ(t). Taking n su h derivatives and then setting J(t)=0 leads to

δnZ [J ]

δJ(t1)δJ(t2) . . . δJ(tn)

∣

∣

∣

∣

J(t)=0

= in∫

dqa dqb ϕ∗0(qb, tb)ϕ0(qa, ta)

∫

DqDp q(tn) . . . q(t2) q(t1) exp

i

∫ tb

ta

dt[

pq −H(q, p)]

, (3.1.38)

where the time variables are ordered ta<t1, t2, ... , tn<tb. This then is just the

expe tation value of the time-ordered produ t T[

Q(t1)Q(t2) ...Q(tn)]

evaluated

between ground states at instants ti

and tf

:

⟨

0, tf

∣

∣T[

Q(t1)Q(t2) . . . Q(tn)]∣

∣ 0, ti

⟩

(3.1.39)

These will be the Green fun tions of the quantised eld theory.

Up to onstant fa tors, whi h may, as always, be absorbed into the overall

normalisation, on performing the p integral we have

Z [J ] ∼ limti

→+i∞

tf

→− i∞

〈qf

;tf

|qi

;ti

〉J

∼ limti

→+i∞

tf

→− i∞

∫

Dq exp

i

∫ tf

ti

dt[

Le

(q, q) + J(t) q(t)]

, (3.1.40)

∗The reader should be warned that in some texts, the symbol W [J ] is used. However, in order

to underline the lose ties between the obje t dened here and the partition fun tion Z(β) instatisti al me hani s, we shall use Z [J ].


with boundary onditions

limti

→−∞q(t

i

) = qi

and limtf

→+∞q(t

f

) = qf

, (3.1.41)

where qi

and qf

are some xed onstant values. We an then write

〈T[

Q(t1)Q(t2)...Q(tn)]

〉0

∼ limti

→+i∞

tf

→− i∞

∫

dq1 . . .dqn 〈qf;tf|qn;tn〉 qn . . . q2 〈q2;t2|q1;t1〉 q1 〈q1;t1|qi;ti〉, (3.1.42)

where qi

:= q(ti

), t1<t2< · · ·<tn and 〈···〉0 indi ates a ground-state expe tation

value.

We shall again need the Wi k rotation or analyti ontinuation to imaginary

time ti

→+i∞ and tf

→− i∞. To perform the ontinuation, note that

〈qf

;tf

|qi

;ti

〉 = limn→∞

∫ n∏

k=1

dqk[

2π i∆t/m]1/2

× exp

i

n+1∑

j=1

Le

(

qj + qj−1

2,qj − qj−1

∆t

)

∆t

, (3.1.43)

only depends on time through ∆t. We may therefore write

〈qf

;tf

|qi

;ti

〉∣

∣

∣

ti

=− iτi

tf

=− iτf

= limn→∞

∫ n∏

k=1

dqk[

2π∆τ/m]1/2

× exp

n+1∑

j=1

Le

(

qj + qj−1

2,qj − qj−1

− i∆τ

)

∆τ

, (3.1.44)

where ∆τ := (τf

−τi

)/(n+1). The analyti ontinuation of the ground-state ex-

pe tation value of the time-ordered produ t of operators thus be omes

〈T[

Q(t1)Q(t2)...Q(tn)]

〉0

∣

∣

∣

tj=− iτj

∼ limτi

→−∞

τf

→+∞

∫

Dq q(τn) . . . q(τ2) q(τ1) exp

∫ τf

τi

dτ Le

(

q, idq

dτ

)

. (3.1.45)

3.2. SCALAR QUANTUM FIELD THEORY 33

Su h a move to imaginary time is none other than a Eu lidean formulation.

∗We

shall thus also dene a Eu lidean version of the fun tional generator:

ZE

[J ] ∼∫

Dq exp

∫ ∞

∞

dτ

[

Le

(

q, idq

dτ

)

+ J(τ) q(τ)

]

, (3.1.46)

with the boundary onditions that q(τ) approa hes some onstant values for τ→±∞. Note that the problem of overall normalisation of these expressions is expli-

itly avoided when relating the Minkowski and Eu lidean versions:

1

Z [J ]

δnZ [J ]


∣

∣

∣

∣

∣

J=0

=in

ZE

[J ]

δnZE

[J ]

δJ(τ1)δJ(τ2) . . . δJ(τn)

∣

∣

∣

∣

∣ J=0τi

=i ti

.

(3.1.47)

For an example of the Eu lidean formulation of the approa h presented here,

see the treatment of the harmoni os illator presented by Feynman (1950) and also

Feynman and Hibbs (1965).

It may be worth stressing that we an go ba k one step to the more general

formula, before performing the fun tional p integration:

Z [J ] ∼ limti

→+i∞

tf

→− i∞

∫

DqDp exp

i

∫ tf

ti

dt[

pq −H(q, p) + J(t) q(t)]

. (3.1.48)

In any ase, we then nally have

〈T[

Q(t1)Q(t2)...Q(tn)]

〉0= − in

δnZ [J ]


∣

∣

∣

∣

J(t)=0

. (3.1.49)

3.2 S alar quantum eld theory

We now apply the path-integral formulation of quantum theory to the simplest ase

of a s alar eld theory. The idea is that the eld ϕ(t,x) will be ome an operator

(in the Heisenberg pi ture). The operator ϕ(t,x) should a t on the o upation-

number spa ein anoni al quantisation the b(k) and b∗(k) reate and destroy

parti les (seen as o upation levels), in analogy with, e.g. the ladder operators

a± :=p± iq (or a and a∗) for the harmoni os illator.

However, the important point to bear in mind is that the approa h outlined

so far is an alternative to the anoni al quantisation pro edure via substitution of

∗Note that for t→− iτ , we have xµxµ→−(τ2+x

2). In other words, gµν →−diag[1,1,1,1].


operators: indeed, the pro edure is far more dire t and on eptually simpler. One

merely needs to identify the Lagrangian governing the lassi al dynami s and use

this in the FPI, whi h is then unambiguously dened.

The extension of the path integral to N degrees of freedom was given in

Eq. (3.1.25). This an now be applied to a s alar eld theory as follows. We

divide the spatial volume into ubi sub-volumes of dimension (∆x)3 and use the

index a, the index to the ve tors in Eq. (3.1.25), to label them: Va. The a-th omponent of the ve tor q(t) thus be omes ϕa(t), whi h we dene by

ϕa(t) := (∆x)−3

∫

Va

d3xϕ(t,x). (3.2.1)

It is thus the average of ϕ over the ell volume (we might equally have simply

taken the value of ϕ at the ell entre). The Lagrangian then be omes

L(t) ≡∫

d3x L(x, t) →

∑

a

(∆x)3 L(

ϕa(t), ϕa(t))

. (3.2.2)

As usual, we may now dene the anoni al momenta

pa(t) :=∂L

∂ϕa(t)= (∆x)3

∂L

∂ϕa(t)=: (∆x)3 πa(t). (3.2.3)

The Hamiltonian is then

H =∑

a

pa ϕa(t)− L :=∑

a

(∆x)3 Ha, (3.2.4)

where, naturally, the Hamiltonian density is just

H(

πa(t), ϕa(t))

= πa(t) ϕa(t)− L(

ϕa(t), ϕa(t))

. (3.2.5)

With these denitions, the path integral (3.1.25) be omes

limn→∞∆x→0

∫

∏

a

(

n∏

k=1

dϕa(tk)

n+1∏

k=1

(∆x)3dπa(tk)

)

× exp

i

n+1∑

j=1

∆t∑

b

(∆x)3[

πb(tj)ϕb(tj)− ϕb(tj−1)

∆t−H

(

πb(tj), ϕb(tj))

]

=

∫

DϕDπ exp

i

∫ tf

ti

dt d3x[

π(t,x) ϕ(t,x)−H(t,x)]

, (3.2.6)


where, following our earlier formulation, we have dened π(t,x), the eld onjugateto ϕ(t,x), by

π(t,x) :=∂L

∂ϕ(t,x). (3.2.7)

In analogy with the quantum-me hani al ase, the path integral runs over all eld

ongurations ϕ(t,x) and π(t,x), subje t to boundary onditions of the form

ϕ(ti

,x) = ϕi

(x) and ϕ(tf

,x) = ϕf

(x). (3.2.8)

Let us stress on e more, the quantities involved here, ϕ(t,x) and π(t,x), remain

lassi al eld ongurations and not operators.

3.2.1 The generating fun tional Z [J ]

The natural extension of Eq. (3.1.33) to a eld is simply

Z [J ] ∝ 〈ϕf

(x),tf

|ϕi

(x),ti

〉J

=

∫

DϕDπ exp

i

∫ tf

ti

d4x[

π(x) ϕ(x)−H(

π(x), ϕ(x))

+ J(x)ϕ(x)]

.

(3.2.9)

The normalisation of Z [J ] is hosen su h that Z [J ]∣

∣

J=0=1 and, as before, the

auxiliary, external, sour e term J(x)ϕ(x) serves to study the generi ground-state

transition amplitude. Sin e we are dealing here with the des ription of parti le

dynami s in a relativisti ontext and, in parti ular, we expe t reation and de-

stru tion to o ur, it is natural to take the ground state as oin iding with the

va uum; that is, the empty state.

∗Following the example from non-relativisti

quantum me hani s, i.e. Eq. (3.1.49), we shall thus dene the n-point Green fun -

tion:

G(n)(x1, . . . , xn) := (− i )n

δnZ [J ]

δJ(x1) . . . δJ(xn)

∣

∣

∣

∣

J=0

= 〈0|T[

ϕ(x1)...ϕ(xn)]

|0〉, (3.2.10)

∗That the va uum state should be the lowest-energy state is not guaranteed a priori ; it is indeed

possible that non-trivial intera tions lead to ongurations with non-zero expe tation value for

some elds and yet with lower energy than the naïve va uum (as in the ase of the phenomenon

known as spontaneous symmetry breaking, whi h we shall study later). For the moment we

shall simply ignore su h possible ompli ations.


whi h is none other than the va uum expe tation value of the time-ordered produ t

of n elds.

On a histori al note, the relationship between G(n)

and the expe tation value

of the time-ordered produ t of elds was originally derived by S hwinger (1951)

within the ontext of his sour e theory, but without the auxiliary of the path integ-

ral. The important point of the path-integral formulation, apart from the elegant

proof it permits, is that it also provides an expli it (albeit formal) expression for

al ulation.

Re all that all quantities are lassi al and therefore ommute; we may thus

rewrite Z [J ] as an expansion in powers of J , for whi h the Green fun tions are

then just the oe ients:

Z [J ] =

∞∑

n=0

in

n!

∫

d4x1 . . .d4xn G

(n)(x1, . . . , xn) J(x1) . . . J(xn), (3.2.11)

where the rst term (n=0) is just 1, representing the trivial zero-point fun tion.

We an again perform the analyti ontinuation to Eu lidean spa etime (i.e.

Wi k rotate to imaginary time). However, sin e the pro ess should be well un-

derstood by now, we shall remain in Minkowski spa e and simply appeal, when

ne essary, to the possibility of a Eu lidean formulation in order to justify the vari-

ous operations. To render the transition from one to the other unambiguous, we

may introdu e an iε term (equivalent to a omplex mass or nite lifetime):

H → H−12iεϕ2 (ε = 0+). (3.2.12)

As before, the spe ial ase, in whi h the Lagrangian density has the simple

separated form (i.e. the ϕ's and ϕ's are not mixed)

L(ϕ, ∂µϕ) = 12ϕ2 + F(ϕ,∇ϕ), (3.2.13)

allows us to perform the π integral expli itly. We start from, f. (3.2.9),

Z [J ] =

∫

DϕDπ exp

i

∫ tf

ti

d4x[

π ϕ− 12π2 − F(ϕ,∇ϕ) + Jϕ

]

. (3.2.14)

Completing the square in π and performing the Gaussian fun tional integral, leads


dire tly to

Z [J ] ∝∫

Dϕ exp

i

∫ tf

ti

d4x[

12ϕ2 + F(ϕ,∇ϕ) + Jϕ

]

=

∫

Dϕ exp

i

∫ tf

ti

d4x[

L(ϕ, ∂µϕ) + Jϕ]

. (3.2.15)

As usual, the normalisation is xed by requiring Z [J ]∣

∣

J=0=1. Again, this simple

result is obtained only by virtue of the parti ular form of L. Otherwise, the integral

in π must be performed expli itly rst and, as dis ussed earlier, the nal result

will then depend on some Le

6=L.

3.2.2 The free-eld ase

Let us rst examine the simple free-eld ase, in whi h F does not ontain terms

higher than quadrati in ϕ. That is, the Lagrangian takes the form

L → L0 = 12[(∂µϕ)(∂µϕ)−m2ϕ2], (3.2.16)

where the subs ript 0 stands for free. For larity, we return to the dis rete

formulation and for simpli ity we shall hoose ∆t=∆=∆x. Thus,

Z0[J ] ∝∫

Dϕ exp

i

∫ tf

ti

d4x[

L0(ϕ, ∂µϕ) + Jϕ]

→ lim∆→0

∫

∏

a

dϕa exp

i

[

∆8∑

a,b

12ϕaKab ϕb +∆4

∑

a

Jaϕa

]

, (3.2.17)

The matrix operator Kab is su h that in the limit we have

lim∆→0

Kab = −[

+m2 − iε]

δ4(x− y) =: K(x, y), (3.2.18)

with a→x and b→ y as ∆→0; we shall all this the KleinGordon operator. The

ϕa integrations an be performed by exploiting the formulæ derived earlier:

Z0[J ] = lim∆→0

1√detKab

∏

c

√

2π

i∆8 exp

−12i∑

a,b

Ja(K−1)abJb

, (3.2.19)


where the inverse of Kab (the Green fun tion) is naturally dened by

∑

c

(K−1)acKcb = δab. (3.2.20)

Note that the ontinuum limits for the delta and sum are:

∆−4δab → δ4(x− y) and

∑

a

∆4 →∫

d4x . (3.2.21)

In the ontinuum then (with the usual normalisation) we have just

Z0[J ] = exp

−12i

∫

d4x d4y J(x) K−1(x, y) J(y)

. (3.2.22)

We now rewrite the inverse of K(x,y) in the following suggestive form:

K−1(x, y) = ∆

F

(x− y). (3.2.23)

This is known as the Feynman propagator and we shall now show that it really

does only depend on the dieren e (x−y). From the denition of K, we an write

K(x, y) =

∫

d4p e ip·(x−y)[

p2 −m2 + iε]

, (3.2.24)

from whi h it immediately follows that

K−1(x, y) =

∫

d4p e ip·(x−y)[

p2 −m2 + iε]−1

. (3.2.25)

This is indeed the Feynman propagator (or Green fun tion), as may be seen by

onsidering that from the above we have

[

x+m2]

∆F

(x− y) = −∫

d4p e ip·(x−y) = −δ4(x− y). (3.2.26)

And thus ∆F

is asso iated with the propagation of solutions to the KleinGordon

equation

[

+m2]

ϕ(x) = 0. (3.2.27)

One should note that the iε pres ription, as applied here (via the additional

iεϕ2term in the Lagrangian), annot be readily generalised to the fermion or

gauge-eld ases (owing to the presen e of negative-energy solutions in the former

ase and the pre lusion of a mass term in the latter). We shall thus, in general,


simply insert an iε into the propagator by hand. The nal form of the free-eld

generating fun tional is then

Z0[J ] = exp

− i

∫

d4x d4y 12J(x)∆

F

(x− y) J(y)

, (3.2.28)

with ∆F

dened as above.

3.2.3 Free-eld Green fun tions

As already seen, fun tional derivatives of Z [J ] with respe t to J generate the

n-point Green fun tions of the theory. The formula just derived then provides

an expli it method to al ulate them. First of all, note that an odd number of

derivatives always leaves one fa tor of J , whi h auses the entire term to vanish

on setting J =0; we need thus only onsider n even. We shall present n=2 and 4,the ontinuation to n≥6 being straightforward and G

(0)0 is, of ourse, unity.

G(2)0 (x1, x2) = i∆

F

(x1 − x2), (3.2.29a)

G(4)0 (x1, x2, x3, x4) = i

[

∆F

(x1 − x2)∆F

(x3 − x4)

+ ∆F

(x1 − x3)∆F

(x2 − x4)

+ ∆F

(x1 − x4)∆F

(x2 − x3)]

. (3.2.29b)

Note that a full intera ting theory may also generate terms for n odd (even n=1may o ur). This simple mathemati al stru ture leads to a natural diagrammati al

representation, in whi h, rst of all, the free-eld propagator or two-point (Green)

fun tion is indi ated symboli ally by a line:

G(2)0 (x1, x2) = x1 x2 (3.2.30a)

and the free-eld four-point fun tion is then

G(4)0 (x1, x2, x3, x4) =

x2

x1

x4

x3

+ (x2 ↔ x3) + (x3 ↔ x4). (3.2.30b)

Given that G(2)(x1,x2) is asso iated with the propagation of one eld between

the spa etime points x1 and x2, then it is natural to assume that G(4)(x1,x2,x3,x4)

will have to do with the propagation of two elds between the spa etime points

x1, x2, x3 and x4. We shall later show how this omes about.


It turns out that al ulations are more easily performed in momentum spa e;

it is therefore useful to apply a Fourier transform:

G(n)(p1, . . . , pn) δ

4(p1 + · · ·+ pn)

:=

∫

d4x1 . . .d4xn e i (p1·x1+···+pn·xn) G

(n)(x1, . . . , xn). (3.2.31)

The energymomentum onserving δ-fun tion has been written expli itly sin e

translation invarian e implies that Green fun tions only depend on spa etime

dieren es. Moreover, we do not distinguish symboli ally between G(n)

for ordinary

spa etime and momentum spa e sin e, given the arguments, there is evidently no

ambiguity. For the free-eld two-point fun tion, we thus write

G(2)0 (p,−p) = i∆

F

(p) =i

p2 −m2 + iε. (3.2.32)

The orresponding diagrammati representation is just

G(2)0 (p,−p) = i∆

F

(p) =p

. (3.2.33)

Note that the energymomentum here has a spe i diagrammati dire tion of ow

and so sometimes an expli it arrow is atta hed to the propagator line (although this

is only stri tly ne essary when there is transport of onserved quantum numbers).

3.2.4 Conne ted Green fun tions

As shown in Eq. (3.2.29b), the Green fun tions generated by dierentiating Z [J ]are dis onne ted. However, there an evidently be no interesting dynami s on-

tained in su h pro esses; e.g. G(4)0 merely des ribes the independent propagation of

two non-intera ting elds. We shall now show how to ex lude su h ontributions

and leave only onne ted pro esses.

To this end, we dene a new generating fun tional Zc[J ] by

exp

iZc[J ]

:= Z [J ], (3.2.34a)

or equivalently

iZc[J ] := lnZ [J ]. (3.2.34b)

We then see that

iδnZc[J ]

δJ1δJ2 . . . δJn=

1

Z [J ]

δnZ [J ]

δJ1δJ2 . . . δJn


−

1

Z [J ]

δZ [J ]

δJ1

1

Z [J ]

δn−1Z [J ]

δJ2 . . . δJn+ perms

−

1

Z [J ]

δ2Z [J ]

δJ1δJ2

1

Z [J ]

δn−2Z [J ]


− . . .

+

1

Z [J ]

δZ [J ]

δJ1

1

Z [J ]

δZ [J ]

δJ2

1

Z [J ]

δn−2Z [J ]


+ . . . , (3.2.35)

where we have adopted the shorthand Jn :=J(xn). It is almost immediate that

in a free-eld theory the only onne ted Green fun tion is the two-point fun tion.

To see this, let us express Zc[J ] as a power series, of whi h the oe ients should

now be onne ted Green fun tions:

iZc[J ] =∞∑

n=0

in

n!

∫

d4x1 d4x2 . . . d

4xn G(n)c (x1, x2, . . . , xn) J(x1)J(x2) . . . J(xn).

(3.2.36)

Note that, sin e we require e iZc[J ] |J=0=1 for a free-eld theory, the n=0 term

(va uum diagrams) vanishes, as too the n=1 term (so- alled tadpole diagrams).

They have been retained, however, for omplete generality.

Comparison of Eq. (3.2.34a) and Eq. (3.2.28) immediately leads to

Zc 0[J ] = −12

∫

d4x d4y J(x)∆F

(x− y) J(y) (3.2.37)

and thus

G(2)0 c (x1 − x2) = G

(2)0 (x1 − x2) = i∆

F

(x1 − x2). (3.2.38)

On e we move over to the full intera ting theory, there will also be onne ted

diagrams with n 6=2 external elds. Moreover, even the simple two-point fun tion

will have higher-order ontributions (quantum orre tions) and removal of the

dis onne ted terms will then be ome important. We shall, however, postpone

further examination of this question until later.

What may not be so obvious is that the terms following the rst on the right-

hand side of Eq. (3.2.35) indeed have the fun tion of removing the dis onne ted

ontributions. We shall now demonstrate this. First let us omment on the Z [J ]fa tors in the denominators. Although the free-eld version of Z [J ] for J =0 is

unity, the intera ting theory generates higher-order (in perturbation theory) on-

tributions. These are purely va uum diagrams (bubbles), whi h are ee tively an

overall renormalisation with no physi al importan e. Pre isely the same sum of

bubble graphs appears multiplying all higher-order Green fun tions. The denom-

inator Z [J ] asso iated with ea h set of derivatives of Z [J ] in Eq. (3.2.35) then

simply an els this irrelevant overall renormalisation fa tor in all ases.


The simplest proof of the onne tedness of G(n)c pro eeds via indu tion. We

shall therefore rst examine the lowest-order derivatives:

iδZc[J ]

δJ1=

1

Z [J ]

δZ [J ]

δJ1, (3.2.39a)

iδ2Zc[J ]

δJ1δJ2=

1

Z [J ]

δ2Z [J ]

δJ1δJ2− 1

Z [J ]

δZ [J ]

δJ1

1

Z [J ]

δZ [J ]

δJ2

=1

Z [J ]

δ2Z [J ]

δJ1δJ2− iδZc[J ]

δJ1

iδZc[J ]

δJ2. (3.2.39b)

The rst of these shows that, apart from the renormalisation dis ussed above, the

onne ted and dis onne ted one-point fun tions are (rather obviously) identi al.

The se ond expli itly displays an ellation of the possible dis onne ted two-point

ontribution formed by the produ t of two ( onne ted) one-point diagrams.

Exer ise 3.2.1. Evaluate the third-order derivative, show expli itly by regrouping

of terms and substitution that the onne ted three-point fun tion is the dis onne ted

three-point fun tion with the three possible produ ts of onne ted one- and two-point

fun tions together with the produ t of three one-point fun tions subtra ted.

Consider now the n-th derivative of Eq. (3.2.34a): dividing out a fa tor of Z [J ],we obtain

1

Z [J ]

δnZ [J ]

δJ1δJ2 . . . δJn=

iδnZc[J ]

δJ1δJ2 . . . δJn+

iδZc[J ]

δJ1

iδn−1Zc[J ]


+

iδ2Zc[J ]

δJ1δJ2

iδn−2Zc[J ]


+ . . .

+

iδZc[J ]

δJ1

iδZc[J ]

δJ2

iδn−2Zc[J ]


+ . . . (3.2.40)

By hypothesis, all of the rst n−1 orders of derivatives appearing on the right-handside generate onne ted fun tions. The terms after the rst on the right-hand side

thus generate all possible dis onne ted ontributions to the n-point fun tion. It

therefore follows that the rst term generates the onne ted n-point ontribution.Sin e G

(1)c and G

(2)c have been shown expli itly to be onne ted, the indu tive proof

is omplete.


3.2.5 The ee tive a tion

We have

Z [J ] = eiZc[J ] =

∫

Dϕ exp

i

∫

d4y[

L(ϕ, ∂µϕ) + J(y)ϕ(y)]

. (3.2.41)

Taking now the rst fun tional derivative of Zc[J ], we obtain

δZc[J ]

δJ(x)=

1

Z [J ]

∫

Dϕϕ(x) exp

i

∫

d4y[

L(ϕ, ∂µϕ) + J(y)ϕ(y)]

. (3.2.42)

This obje t is just the ground-state expe tation value of ϕ(x) in the presen e of thesour e J(x). In other words, it is just the lassi al eld (but more on this shortly).

The lassi al eld so-dened has some importan e, as we shall see later, when

dis ussing spontaneous symmetry breaking. Note though that it is a fun tional of

J(x). We therefore dene

ϕc(x) :=δZc[J ]

δJ(x), (3.2.43)

where the index c stands for lassi al on the left-hand side but onne ted on the

right-hand side. And writing Zc[J ]= 〈0|0〉J , we thus have

ϕc(x) =〈0|ϕ(x)|0〉J

〈0|0〉J, (3.2.44)

where |0〉 is used to indi ate the va uum or ground state.

Let us now perform a Legendre transform on Zc[J ]:

Γ[ϕc] := Zc[J ]−∫

d4x J(x)ϕc(x). (3.2.45)

Su h a transform is well-known in lassi al me hani s and statisti al thermody-

nami s; it is analogous here, e.g., to the relation F =E−TS relating the Helmholtz

free energy F to the entropy S.

Exer ise 3.2.2. Take the fun tional derivative of Γ[ϕc] with respe t to J(x) andthus show that Γ[ϕc] indeed only depends on ϕc(x) and not on J(x).

The symmetry of the equations is evident and J(x) an now be obtained by

taking a fun tional derivative of Γ[ϕc] with respe t to ϕc(x):

J(x) = − δΓ[ϕc]

δϕc(x). (3.2.46)


The signi an e of the dual pair of equations (3.2.43 & 46) is that in the absen e

of J (i.e. J =0) Eq. (3.2.43) gives pre isely the true lassi al eld, whi h in turn

is seen to extremise (usually minimise) the fun tional Γ[ϕc]. We shall thus be led

to interpret Γ[ϕc] as the ee tive a tion.From the denition of ϕc, in the free-eld ase we have

ϕc(x) = −∫

d4y∆F

(x− y) J(y). (3.2.47)

Now, sin e ∆F

(x−y) is pre isely the Green fun tion for the KleinGordon equa-

tion, ϕc(x) then satises

[

+m2]

ϕc(x) = J(x), (3.2.48)

whi h, indeed, is none other than the lassi al eld equation in the presen e of an

external sour e J(x).With these denitions in hand, we may now pro eed with the al ulation of the

ee tive a tion Γ0[ϕc] for the free-eld ase. Let us rst add one more denition

or shorthand for the KleinGordon operator (i.e. the same symbol as used earlier

but now having only one argument; so there should be no onfusion):

K(x) := x +m2. (3.2.49)

We may start from Eq. (3.2.37) and use (3.2.48) to eliminate J(x):

Zc[J ] = −12

∫

d4x d4y J(x)∆F

(x− y) J(y)

= −12

∫

d4x d4y [K(x)ϕc(x)]∆F

(x− y) [K(y)ϕc(y)] . (3.2.50)

where the square bra kets are used to delimit the a tion of the operator. Thus,

Γ0[ϕc] = −12

∫

d4x d4y [K(x)ϕc(x)]∆F

(x− y) [K(y)ϕc(y)]

−∫

d4x [K(x)ϕc(x)]ϕc(x)

= +12

∫

d4xϕc(x) [K(x)ϕc(x)]−∫

d4x [K(x)ϕc(x)]ϕc(x)

= −∫

d4x 12

[

∂µ∂µϕc(x) +m2ϕc(x)]

ϕc(x)


=

∫

d4x 12

[

∂µϕc(x) ∂µϕc(x)−m2ϕ2c(x)

]

=

∫

d4x L0(ϕc, ∂µϕc), (3.2.51)

where in the various steps we have integrated by parts and exploited the relation

K−1(x)=∆

F

(x). The nal expression is just the lassi al a tion orresponding to

the lassi al free-eld; hen e the previous hoi e of the term ee tive a tion.

Su h an interpretation takes on a parti ularly useful meaning in the full inter-

a ting theory. While intera tions generally render the theory insoluble in losed

form (owing to the indu ed quantum orre tions), we may still perform a formal

fun tional expansion in powers of the eld ϕc(x):

Γ[ϕc] =

∞∑

n=1

in

n!

∫

d4x1 . . .d4xn Γ

(n)(x1, . . . , xn)ϕc(x1) . . . ϕc(xn). (3.2.52)

It will turn out that the oe ients Γ(n) orrespond not only to n-point fun tions

that are onne ted, but that are also what are termed one-parti le irredu ible

(1PI); that is, they annot be rendered dis onne ted by utting just a single in-

ternal line. Some examples of 1PI and non-1PI fun tions are shown in Fig. 3.1.

(a) (b) ( ) (d)

Figure 3.1: Examples of (a ) 1PI and (d) non-1PI or one-parti le redu ible diagrams.

In the free-eld ase, as already shown, there is only one su h non-vanishing

n-point 1PI fun tion: that for n=2,

Γ(2)0 (x, y) = K(x) δ4(x− y). (3.2.53)

Just as for G(n), we perform a Fourier transform and dene the momentum-spa e

versions:

Γ(n)(p1, p2, . . . , pn) δ4(p1 + p2 + · · ·+ pn)

:=

∫

d4x1 . . .d4xn e i (p1·x1+···+pn·xn) Γ(n)(x1, x2, . . . , xn). (3.2.54)


The free-eld two-point momentum-spa e 1PI fun tion is then

Γ(2)0 (p,−p) = −(p2 −m2). (3.2.55)

One an also make a low-energy or Taylor expansion, whi h is equivalent to an

expansion in powers of momenta:

Γ[ϕc] =

∫

d4x[

−V(ϕc) +12A(ϕc) ∂

µϕc ∂µϕc + . . .]

, (3.2.56)

where now the oe ients V(ϕc), A(ϕc), . . . are simple fun tions of ϕc and hen e

of x. The rst term V(ϕc) is then alled the ee tive potential. Note that in the

ase where the lassi al eld is spa etime independent (so that only this term

survives), from Eq. (3.2.46) we have

∂V(ϕc)

∂ϕc(x)= J(x). (3.2.57)

Setting J =0, we then see that the lassi al eld onguration ϕc is that whi h

extremises

∗the ee tive potential. In other words, on e we know V(ϕc), then we

an immediately obtain the va uum expe tation value of the eld ϕc. Of ourse,

if ϕc is not onstant, then we must minimise the full expression for Γ[ϕc].We should perhaps remark here that one an show that higher derivatives of

Zc[J ] for J =0 are a tually the n-point onne ted Green fun tions for the true

minimum; i.e. for the shifted eld ϕ :=ϕ−ν, where

ν :=δZc[J ]

δJ(x)

∣

∣

∣

∣

J=0

. (3.2.58)

That is, for ϕ with vanishing va uum expe tation value. Note that, sin e the right-

hand side is translationally invariant, ν is ne essarily spa etime independent.

Exer ise 3.2.3. By dierentiating (3.2.42) with respe t to J (remembering to

dierentiate the denominator too) and then setting J =0, show, for example, that

δ2Zc[J ]

δJ(x1) δJ(x2)

∣

∣

∣

∣

J=0

= i

∫

Dϕ ϕ(x1)ϕ(x2) ei∫d4x L(x)

∫

Dϕ e i∫d4x L(x)

.

By indu tion, this an be extended to the general n-point fun tion.

In general, as would appear natural from the foregoing observation, we shall use

∗Sin e one an show that V(ϕc) orresponds to an energy density, then the ground state (lowest-

energy state) must, in fa t, evidently be a minimum.


the shifted elds to onstru t the quantum eld theory. That is, we shall always

onsider quantum u tuations (or perturbations) around the ground state of the

theory. This be omes espe ially important in the presen e of spontaneously broken

symmetries, when e elds may a quire non-vanishing va uum expe tation values

owing to the parti ular form of the potential o urring in the Lagrangian. It would

then be in orre t to perturb around the va uum (zero-eld) onguration as this

is unstable by onstru tion. In su h ases the shift must, however, be performed

by hand before the quantisation pro edure.


3.3 Bibliography

Feynman, R.P. (1950), Phys. Rev. 80, 440.

Feynman, R.P. and Hibbs, A.R. (1965), Quantum Me hani s and Path Integrals

(M GrawHill).

S hwinger, J.S. (1951), Pro . Nat. A ad. S i. 37, 452.

Appendix A

Theoreti al Ba kground

A.1 Wi k rotation

In order to exemplify the pro edure of Wi k rotation (?), ne essary for orre t

evaluation of the highly os illatory integrals typi al of quantum me hani s, let us

onsider the integral to be performed in Eq. (2.5.13):

∫ ∞

−∞

dp e i (p q−p2/2m)∆t . (A.1.1)

To evaluate this integral, we need to regularise the highly os illatory integrand at

innity. This may be onveniently performed by attributing the parti le a omplex

mass m+ iε (with ε>0 for ∆t>0).∗ We may then use Cau hy's theorem to shift

and rotate

†the path of integration in the omplex p′ plane to run along a path at

45to the original real axis (see Fig. A.1), this is then what is generally known as

a Wi k rotation.

In other words, we wish to evaluate the integral

I =

∫ +∞

−∞

dz e− ia(z2+2bz), (A.1.2a)

where a and b are real and a > 0. We rst rewrite this as

= e iab2

∫ +∞

−∞

dz e− ia(z+b)2

, (A.1.2b)

∗Note that this is equivalent to assuming a nite lifetime (indeed the imaginary part of the mass

or energy of a state is none other than the de ay rate or inverse lifetime).

†Note that the sign of the iε term determines the sense of the rotation.

49

50 APPENDIX A. THEORETICAL BACKGROUND

Imz′′

Rez′′

Figure A.1: A Wi k rotation in the omplex z′′plane; here the rotated integration of

Eq. (A.1.2 ) lies along the path at 45.

after whi h we may shift to a new variable z′ = z+b:

= e iab2

∫ +∞

−∞

dz′ e− iaz′2

. (A.1.2 )

We nally hange variable to z′′=e iπ/4z′ (this is equivalent to rotating the path of

integration as shown in Fig. A.1). To use Cau hy's theorem, the ontour must be

losed and we must ensure that there are no singularities en losed by the ontour

dened by the new and old paths. To guarantee that the integral vanishes along

the ar s in Fig. A.1 for |z′′|→∞, we also temporarily shift a→a− iε with ε>0 (asalready noted, this orresponds to attributing the parti le with a omplex mass):

I = e− iπ/4 e iab2

∫ +∞

−∞

dz′′ e−az′′2

. (A.1.2d)

The desired integral is now trivial and the result (for ε vanishing) is

√

m

2π i ∆te i

12mq

2∆t . (A.1.3)

A.2 A simplied approa h

Here we shall attempt a simplied derivation of the path integral dire tly in the

onguration-spa e representation. We thus wish to al ulate the innitesimal

A.2. A SIMPLIFIED APPROACH 51

transition amplitude

M(x2, t2; x1, t1) =

∫

dxϕ∗(x; x2, t2)U(t2, t1)ϕ(x; x1, t1), (A.2.1)

where t2= t1+δt and, for deniteness, we might take

ϕ(x; x1, t1) = limε→0

1√

ε√πe−(x−x1)

2/2ε

2

. (A.2.2)

The wave-fun tion, hosen to represent a position eigenstate, is su h that |ϕ|2 (aGaussian) redu es to a δ-fun tion (with the orre t normalisation) in the limit.

It will be ne essary at some point to move over to the momentum-spa e rep-

resentation ϕ(p,ξ):

ϕ(p, ξ) =

∫

dx e− ipx ϕ(x, ξ), (A.2.3a)

ϕ(x, ξ) =

∫

dp e ipx ϕ(p, ξ), (A.2.3b)

where

ϕ(p, ξ) = limε→0

∫

dx e− ipx 1√

ε√πe−(x−ξ)

2/2ε

2

,

= limε→0

e− ipξ ε√2π

1√

ε√πe−p

2ε2/2 .

= limε→0

√

2ε√π e− ipξ e−p

2ε2/2 . (A.2.4)

As in the main text, the innitesimal version of the temporal translation oper-

ator may be written here as

U(t + δt, t) ≃ 1− i H(x, p) δt. (A.2.5)

Taking now the ase H(x,p)= p2

2m+V (x), this be omes

1− i

[

p2

2m+ V (x)

]

δt. (A.2.6)

Inserting this into the above expression for the transition matrix element, we

immediately see that x will be repla ed by the position x, whi h in turn be omes

x1 in the δt→0 limit under the integral with position eigenstates.


The only ompli ation then is the term in p2:

∫

dxϕ∗(x; x2, t2) p2 ϕ(x; x1, t1)

=

∫

dx dp2 dp1 e− ip2x ϕ∗(p2, x2) p2 e ip1x ϕ(p1, x1). (A.2.7)

The operator p2 may now be repla ed by p21, after whi h the x integral may be per-

formed to give a δ-fun tion that sets p2 = p1 (and we may then drop the superua e

index on p):

=

∫

dp ϕ∗(p, x2) p2 ϕ(p, x1). (A.2.8)

We may now substitute ba k for ϕ(p,ξ) with ϕ(x,ξ):

=

∫

dp dx′ dx e ipx′

ϕ∗(x′; x2) p2 e− ipx ϕ(x; x1). (A.2.9)

(A.2.10)

The remaining terms are

∫

dxϕ∗(x; x2) [1− iV (x1)δt]ϕ(x; x1)

=

∫

dx dx ′dp e ip(x′−x) ϕ∗(x′; x2) [1− iV (x1)δt]ϕ(x; x1)

(A.2.11)

In the exponentials we may now set x=x1 and x′=x2 and ombining the terms

we have

∫

dp dx′ dxϕ∗(x′; x2) eip(x2−x1)

1− i

[

p2

2m+ V (x)

]

δt

ϕ(x; x1) (A.2.12)

≃∫

dp dx′ dxϕ∗(x′; x2) ei [px−H(p,x)]δt ϕ(x; x1). (A.2.13)

As in the main text, for the given Hamiltonian, the p integral now transforms

px−H(p,x) in the exponent into L(x,x)= 12mx2−V (x).

A.3. THE LEGENDRE TRANSFORM 53

A.3 The Legendre transform

A.3.1 Denition and basi s

The Legendre transform appears in many diverse areas of physi s: perhaps the best

known is the onne tion in lassi al me hani s between the Lagrangian L(q,q) andthe Hamiltonian H(q,p), but it also relates the Helmholtz free energy F to the

entropy S in statisti al me hani s; in the present notes it is also seen to provide

the relation between the generating fun tional for onne ted Green fun tions Zc[J ]and the ee tive a tion Γ[ϕ]. For a more omplete and re ent dis ussion, see ?.

Just as the Fourier and Lapla e transforms, the Legendre transform provides a

method of re-en oding the dynami al information on a system. Consider a fun -

tion f(x) whose gradient is stri tly monotoni (i.e., it is either stri tly onvex or

stri tly on ave). In su h a ase there exists a unique one-to-one relation between

the variable x and the derivative f ′(x)=df(x)/dx. We may therefore hange vari-

ables from x to y(x)=f ′(x) without loss of information. Note that the onvexity

ondition guarantees that the fun tion y(x) may also be inverted to obtain x(y).The usefulness in general of su h a hange lies in the possibility that y may

be more a essible than x. And in parti ular, while the Hamiltonian may be a

more immediately determined fun tion, the Lagrangian lends itself better to a

relativisti treatment and the appli ation of Noether's theorem.

Of ourse, the Legendre transform is more than just a simple hange of vari-

ables. The stri t mathemati al denition is as follows:

g(y) := maxx

[

xy − f(x)]

, (A.3.1)

where the maximum is taken with respe t to variations in x for y held xed.

Setting the x derivative of the square bra kets to zero gives

0 = y − f ′(x), or y = f ′(x). (A.3.2)

A simple pi torial representation of this denition is shown in Fig. A.2. The

fun tion −g(y)=−[xy−f(x)] is given by the inter ept with the verti al axis of the

straight line rossing the urve f(x) at the point x with slope y. From the diagram,

it is lear that, for a given slope y, the maximum of [xy−f(x)] with respe t to

variations in x is attained when the straight line is pre isely tangent to f(x) andthus at the point x for whi h y=f ′(x). The need for onvexity is also obvious from

the diagram. Note, of ourse, that there may also be other independent variables

involved but these are mere spe tators in the transform.

With the understanding that su h a pi torial des ription provides, we an shift


f(x)

g(y)

xg

f

xy

x

Figure A.2: A diagrammati illustration of the Legendre transformation g(y) of the

fun tion f(x), as dened in the text.

to a simpler denition, more ommonly found in physi s:

g(y) := xy − f(x) for y =df(x)

dx, (A.3.3)

where it is impli it that to evaluate g(y) we must write x as a fun tion of y by

inverting the equation y=f ′(x). From this denition the symmetry of the Legendre

transform is immediately manifest as we may thus write

xy = f(x) + g(y), (A.3.4)

or the inverse transform

f(x) := xy − g(y) for x =dg(y)

dy. (A.3.5)

Note that in all ases the variables x and y are not independent as ea h should be

viewed as a fun tion of the other.

Some simple properties of the transform are immediately obvious from Fig. A.2.

Consider the minimum of the fun tion f(x) with respe t to x: fmin

:=f(xmin

). Bydenition, the derivative vanishes at the point x

min

; that is, y=0. We therefore

have

fmin

= −g(0) and gmin

= −f(0). (A.3.6)

Further, by taking derivatives of the relations dening the variables x and y, weobtain

dy

dx=

d2f

dx2and

dx

dy=

d2g

dy2, (A.3.7)

A.3. THE LEGENDRE TRANSFORM 55

from whi h we see that

d2f

dx2d2g

dy2= 1. (A.3.8)

The above re ipro al relation between the urvatures highlights the onvexity re-

quirement: neither of the se ond derivatives may vanish. Moreover, it is somewhat

reminis ent of the un ertainty relations ∆x∆p, ∆θ∆L≥~ et .

A.3.2 Relationship to the path integral

The question now arises as to the relationship with the path integral. We have seen

in the main text, for the ase of separated p and q dependen e and a quadrati formin p for the Hamiltonian, that the integration over p transforms the Hamiltonian

into the Lagrangian, see the passage from Eq. (2.5.12) to Eq. (2.5.15).

Let us examine a little more losely the path integral to be performed:

M(qf

, tf

; qi

, ti

) =

∫

DqDp exp

i

∫ tf

ti

dt[

p q −H(q, p)]

. (2.5.12)

The dominant ontribution learly arises from the region of stationary phase; in

this ase that will orrespond to the maximum of the exponent as a fun tion of

p, whi h is just the denition of the Legendre transform. For H(x,p)= p2

2m+V (x),

this is just the point p=mq, whi h suggests we make the following shift of variables:

p=mq+p′. This then gives

p q −H(q, p) = (mq + p′) q − (mq + p′)2

2m− V (p)

= L(q, q)− p′2

2m. (A.3.9)

We thus see that the lassi al transformation from the Hamiltonian to Lagrangian

formalism is determined by a stationary phase ondition in the quantum theory.

Appendix B

Glossary of A ronyms

1PI: one-parti le irredu ible

FPI: Feynman path integral

57

58 APPENDIX B. GLOSSARY OF ACRONYMS

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