On the Regularization of Phase-Space Path Integral in ...ricerca.mat.uniroma3.it/users/mfalconi/other/ms_thesis.pdf · tum ﬁeld theory and its formulation through path integrals

On the Regularization of Phase-Space Path Integral

in Curved Manifolds

Marco Falconi

June 1, 2010

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

I n this work we discuss path integrals both in flat and curved space-time: in particularwe describe how to write the integral kernel of the time evolution operator (in a curvedspace time we describe the evolution with respect to a geodesic affine parameter, such as theproper time if the particle has a non-zero mass) in a path integral form, both in configurationand phase space. We briefly introduce a path integral formulation of Quantum Field Theo-ries, however we focus our attention mainly on quantum mechanical models. We developthe idea, originally due to Schwinger, to describe effective actions in quantum field theoryas quantum mechanical path integrals of a fictitious particle with evolution dictated by asuitable Hamiltonian function. We will see that one could have obtained the same result asin field theory by first quantizing the particle which actually makes the loop of the Feynmangraph corresponding to the effective action.

In the second part we focus on a particle of mass m ≥ 0 that classically moves alonga geodesic of curved space-time. Quantizing this model we will see that even if it is stillpossible to describe its evolution in the affine parameter by means of a path integral, theresults obtained would be ambiguous unless we introduce suitable regularization schemesand related counterterms. This is, in fact, the key point of path integrals used in physics.

Configuration space path integral needs different counterterms, depending on the reg-ularization scheme used, in order to give the same results at any perturbative order for theintegral kernel of the evolution operator. These counterterms are finite and univocally fixedby renormalization conditions arising in two-loop calculations. No higher loop calculationare needed since if we consider quantum mechanics in curved space as a field theory, itwould be super-renormalizable, which means that possible divergences and ambiguitiescan appear only up to a finite number of loops.

Different is the case of phase space path integral, as we show in this work: in such func-tional integral perturbative calculations are not ambiguous nor divergent in the continuumlimit, so as far as we limit ourselves to phase space perturbative calculations the introductionof regularization schemes is not particularly useful: they just serve the purpose of explicitlydefining the path integral measure, but in many perturbative calculations the regulariza-tion can be removed at once. Nevertheless we cannot avoid the regularization procedurecompletely, for example if we want to integrate out momenta, because in the continuumlimit such measure has not a definite meaning. Introducing a cutoff in the Fourier modes ofmomentum and coordinate paths we calculate explicitly the “counterterms” of phase spacepath integral in mode regularization.

i

Contents

Introduction i

I FLAT SPACE-TIME 1

1 Path Integrals 51.1: Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1: Probability Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2: Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3: Equivalence of Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2: A modern formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1: Time-independent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2: Time-dependent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3: Transition to Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . 16

1.3: Applications: Free Theories, Harmonic Oscillators & Perturbative Theory . . 171.3.1: Systems of free particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.2: The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.3: Perturbative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.4: Some Perturbative Calculations . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Quantum Fields or Quantum Particles? 372.1: QFT Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1: Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.2: Q-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1.3: Derivation of Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2: Worldline Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.1: The Quantum Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.2: Fictitious Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 48

A Trotter formula and other systematic approximants 53A.1: Trotter product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2: Zassenhaus formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

iii

B the Dyson expansion 55B.1: Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C Gaussian integrals 59

II CURVED SPACE-TIME 61

3 Path Integrals 653.1: Particles in curved manifolds of space-time . . . . . . . . . . . . . . . . . . . . 65

3.1.1: Relativistic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.2: Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2: Derivation of path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.1: Time slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.2: Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3: Continuum Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.3.1: Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.3.2: Analysis of divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3.3: Perturbative Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4: Regularization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4.1: Mode Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4.2: Dimensional regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4.3: Time slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Regularizations in phase space 914.1: Worldline methods on curved space . . . . . . . . . . . . . . . . . . . . . . . . 924.2: Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.1: Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2.2: Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2.3: Perturbative expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.4: Fixing A and Vph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2.5: Regularization is not needed for perturbative calculations . . . . . . . . . . 104

D Weyl Ordering 107

E Distributional integration in phase space 109

Conclusions 111

Bibliography 113

iv

IPart

FLAT SPACE-TIME

1

T he aim of this first part is to introduce the reader to the concept of “Path-integral”,providing historical remarks as well as its modern derivation; and then to illustratesome of its application, both in Quantum Mechanics and Quantum Field Theory. We willconfine ourselves here to flat space-time, for it is easier to develop basic concepts in thisenvironment, and we will see in detail the generalization to an arbitrary curved space-timein the second part of this work.

outline of first part

We start exposing briefly the formulation of R.P. Feynman, who first introduced in 1949the idea of Path integral inspired by some remarks of P.A.M. Dirac concerning the relationof classical action to quantum mechanics; then we derive Feynman formulas from operatorQuantum Mechanics, following a more modern and more general point of view. Techni-cal discussions about functional integration, Trotter formula, Feynman-Kac formula, time-dependent Hamiltonians and Gaussian integrals are developed in the appendices.

After the theoretical derivation of path integrals we develop some applications: we ana-lyze the analytically solvable free and harmonic theories, as well as some perturbative meth-ods for a quite general class of systems.

In the following chapter we develop the mathematical tools necessary to introduce quan-tum field theory and its formulation through path integrals. We show that the introductionof a particular space, called Q-space, allows us to derive quantum fields path integral in away perfectly analogous to the one followed in quantum mechanics. Some mathematicaltechnicalities are relegated to the appendices.

Finally, we introduce the effective action for a field interacting with an external back-ground source, and discuss briefly the worldline approach that leads to a description of itsone-loop part by means of a quantum mechanical path integral. However we will give adeeper insight of this topic in the second part of this work.

3

Ch

ap

ter

1Path Integrals

“one feels as Cavalieri must have feltcalculating the volume of a pyramid

before the invention of calculus”

R.P.Feynman

1.1: Some History

A s an historical note, we review here Feynman approach to Non-Relativistic QuantumMechanics, based on Space-Time paths, as he presented it on a famous work[1] in 1948.It provides a “third formulation” of quantum theory, equivalent to both the Schrödinger andHeißenberg one, and inspired by a 1933 Dirac work[2].

1.1.1: Probability Amplitudes

We start presenting its first postulate:

Feynman’s first

postulate

Postulate 1. If an ideal measurement is performed to determine whether a particle has a path lying ina region R of space-time, then the probability that the result will be affirmative is the absolute squareof a sum of complex distributions, the probability amplitudes, one for each path in the region:

P (R) = |ϕ(R)|2 , (probability for region R)

with

ϕ(R) = lime→0

∫R

φ(. . . , xi , xi+1 , . . . ) · · · dxi dxi+1 · · · , (probability amplitude for region R)

5

Part I: FLAT SPACE-TIME

and

xi+1 particle’s position at time ti+1 = ti + e .

This postulate prescribes the type of mathematical framework required by Heißenberguncertainty principle for the calculation of probabilities. Now we can imagine that, as eapproaches zero, the probability amplitude φ(. . . , xi , xi+1 , . . . ) essentially depends on theentire path x(t) rather than on the discrete points xi of the path. We might call Φ[x(t)] theprobability amplitude functional of paths x(t). We can now introduce the second postulate,that gives a prescription to compute this functional for each path:

Feynman second

postulate

Postulate 2. The paths contribute equally in magnitude, but the phase of their contribution is theclassical action (in units of }h ); i.e., the time integral of the Lagrangian taken along the path:

Φ[x(t)] ∝ ei}h S[x(t)] ,

where

S[x(t)] =∫L (ẋ(t), x(t))dt ,

and the proportionality is intended up to a normalization global factor.

The Lagrangian is a function of position and velocity, and if we suppose it to be aquadratic function of the velocities we can show the equivalence of these postulates andusual quantum mechanics formulations.

But we have first to interpret this postulate for a discontinuous path, i.e. to calculateϕ(R): we shall assume that x(t) in the interval between ti and ti+1 is the classical pathgiven by Lagrangian L , which starting from xi at ti, reaches xi+1 at ti+1, at least in the limitti

e→0−−→ ti+1. Lagrangian function must therefore depend on no higher time derivatives of theposition than the first, in order to be sufficient the specification of start-point and end-pointto determine classical path. So for a discontinuous path we can write S = ∑i S(xi+1, xi),where

S(xi+1, xi) = minpaths

∫ ti+1tiL (ẋ(t), x(t))dt , (1.1.1)

and the minimum is taken to evaluate action on classical trajectory. So now we can combinethe two postulates and find:

ϕ(R) = lime→0

∫R

ei}h ∑i S(xi+1 , xi) · · · dxi+1A

dxiA · · · , (1.1.2)

where we have split the normalization factor into 1/A factors, one for each time slice.

6

1.1. SOME HISTORY CHAPTER 1. PATH INTEGRALS

1.1.2: Wave Function

In order to check equivalence we have to define the wave function, and show that its timedevelopment is given by Schrödinger equation.

The wave function is a quantity having a well defined (by Schrödinger equation) timedevelopment, so we have to define it by means of successive sections of the path, withrespect to time. In order to do that we choose a particular time, say t, and split region R of(1.1.2) into pieces, future and past relative to t. The split is done in this way:

a) a region R′, restricted arbitrarily in space, but lying entirely earlier than some time t′,such that t′ < t;

b) a region R′′ defined in the same way as in a), but lying later than time t′′, such that t′′ > t;

c) the region between t′ and t′′, in which all values of x coordinates are unrestricted, i.e. allof space-time between t′ and t′′.

Region c) is not necessary at all, for it can be taken as narrow in time as desired. Neverthelessit is convenient in letting us vary t a little without having to redefine a) and b) .So we can define the probability |ϕ(R′, R′′)|2 that if a path had been in region R′, then will befound in region R′′. This is the crucial measured quantity in most experiments: we preparea system in a certain way (e.g., it was in region R′) and measure some other property (e.g.,will it be found in region R′′?).

Now we can use a (1.1.2)-like equation to compute the quantity ϕ(R′, R′′). Assuming thetime t to be one particular point k of the subdivision into e time steps, i.e. t = tk, we can splitthe action exponential in the product of two exponentials:

ei}h ∑i S(xi+1 , xi) ≡ e

i}h

∞∑

i=kS(xi+1 , xi)

· ei}h

k−1∑

i=−∞S(xi+1 , xi)

.

This operation is possible essentially because the Lagrangian is a function only of positionsand velocities, and so we could split the action into a sum of discretized parts. Now, callingxk ≡ x and tk ≡ t we obtain:

Probability

amplitudes and

wave functions

ϕ(R′, R′′) =∫

χ∗(x, t)ψ(x, t)dx , (1.1.3)

where

ψ(x, t) = lime→0

∫R′

ei}h

k−1∑

i=−∞S(xi+1 , xi) dxk−1

Adxk−2

A · · · , (1.1.4)

and

χ∗(x, t) = lime→0

∫R′′

ei}h

∞∑

i=kS(xi+1 , xi) 1

Adxk+1

Adxk+2

A · · · . (1.1.5)

7


The quantity ψ depends only on past information on the system, with respect to t, and itis completely defined when the region R′ is known. The information on the future lies en-tirely on the quantity χ. We have thus defined the concept of wave functions ψ(x, t) , χ(x, t)defining the state of a system.

The quantity |∫

χ∗(x, t)ψ(x, t)dx|2 is the probability that a system in state ψ will befound by an experiment whose characteristic state is χ, in agreement with ordinary quantummechanics.

1.1.3: Equivalence of Formulations

If the wave functions just defined satisfy Schrödinger equation we have completed our taskto show the equivalence between Feynman formulations and the other ones. In order to dothat we have to ask thatL is a quadratic, but perhaps inhomogeneous, form in the velocitiesẋ(t).

If we compute ψ in (1.1.4) at the next time slice we obtain:

ψ(xk+1, t + e) =∫R′

ei}h

k∑

i=−∞S(xi+1 , xi) dxk

Adxk−1

A · · · =∫

ei}h S(xk+1,xk)ψ(xk, t)dxkA . (1.1.6)

This equation is not exact, it is only true in the limit e → 0 and we shall derive Schrödingerequation by assuming it is valid to first order in e. In order to do this we need first of allto give an approximate value of (1.1.1), provided the error of the approximation be of anorder smaller than the first in e. The Lagrangian is a quadratic form in the velocities, andwe will see that the paths important in the calculation are those for which xi+1 − xi ∼ e

12 .

Under these circumstances, it is sufficient to calculate the integral in (1.1.1) over the classicalpath taken by a free particle. If we chose the coordinates of the system to be Cartesian1,2, thepath of a free particle is a straight line. So in this case it is sufficiently accurate to replace theintegral by3:

S(xi+1, xi) = eL(

xi+1 − xie

, xi+1

). (1.1.7)

1more generally, coordinates for which the terms quadratic in the velocities in the Lagrangian functionappear with constant coefficients.

2if one seeks the differential equation in a different coordinate system, the easiest way is to find the equationin Cartesian coordinates and then to transform the coordinate system to the one desired.

3this approximation is not valid if we have a vector potential or other terms linear in the velocity in theLagrangian, if those are the cases we need to use an approximation of the type

S(xi+1, xi) =e

2L(

xi+1 − xie

, xi+1

)+

e

2L(

xi+1 − xie

, xi

)or

S(xi+1, xi) = eL(

xi+1 − xie

,xi+1 + xi

2

).

8

1.2. A MODERN FORMULATION CHAPTER 1. PATH INTEGRALS

Thus, in the simple unidimensional case of a particle of mass m moving under a potentialV(x), we have:

S(xi+1, xi) =me2

(xi+1 − xi

e

)2− eV(xi+1) .

So if we use this last equation and we call xk+1 ≡ x , (xk+1− xk) ≡ ξ so that xk = x− ξ, then(1.1.6) becomes:

ψ(x, t + e) = e−i}h eV(x)

1A

∫e

im2e}h ξ

2ψ(x− ξ, t)dξ , (1.1.8)

so if we Taylor expand around ξ the wave function, perform Gaussian integrations, andconsider terms up to first order in e1, we obtain:

ψ(x, t) + e∂tψ =(

2π}heim

) 12 1

A

(1− ie}h

V(x))(

ψ(x, y) +}hei2m

∂2

∂x2ψ

).

In order that both sides may agree both to zero and to first order in e, we must have:

A =(

2π}heim

) 12

and of course

i}h∂tψ =1

2m(−i}h∂x)2ψ + V(x)ψ ,

which is the Schrödinger equation we sought. The generalization to arbitrary degrees offreedom is straightforward. For what it concerns χ∗, a differential equation can be devel-oped in the same way, but it appears to have the time reversed. However taking the complexconjugates we can see that χ satisfies the same equation as ψ. We have finally shown thatFeynman formulation is equivalent2 to the usual formulation of quantum mechanics pro-vided by Schrödinger or Heißenberg.

We shall conclude here our historic note, and begin to deal with modern formulations ofpath integral.

1.2: A modern formulation

P ath integral emerges naturally in usual non-relativistic quantum mechanics as onedeals with the dynamics, i.e. as one considers evolution of a state through time. Inorder to know such an evolution in wave functions space, one has to seek the kernel of thefollowing integral equation:

ψ(x f , t f ) =∫

dxi K(x f , t f ; xi, ti) ψ(xi, ti) , (1.2.1)1Taylor expanding the first member of (1.1.8) and the exponential of the potential up to first order in e.2at least if the system is described by a Lagrangian that is function only of time, position, and velocities and

is a quadratic form of the latter (perhaps inhomogeneous).

9


where x is intended to be a vector of the system’s configuration space, and dx is the infinites-imal volume element of the same space.

If the Hamiltonian is time independent is straightforward to show that K is a solutionof the Schrödinger equation, i.e.

(i}h∂t − H(x,−i}h∇)

)K(x, t; x′, t′) = 01. Let’s see how

this kernel can be calculated when we deal with a generic vector of the Hilbert space2:suppose that we have a state |ψ(t)〉 ∈ H , its time evolution is governed by the evolu-tion operator U (t, t′) in such a way that |ψ(t f )〉 = U (t f , ti)|ψ(ti)〉. We know that positioneigenvectors form a complete set of commuting observables, so we can use the relation1 =

∫dxi |xi〉〈xi| to obtain |ψ(t f )〉 =

∫dxi U (t f , ti)|xi〉〈xi|ψ(ti)〉; now finally projecting

each member on |x f 〉 and remembering the usual definition 〈x|ψ(t)〉 ≡ ψ(x, t) we obtainψ(x f , t f ) =

∫dxi 〈x f |U (t f , ti)|xi〉ψ(xi, ti), so from (1.2.1) we immediately identify:

K(x f , t f ; xi, ti) = 〈x f |U (t f , ti)|xi〉 . (1.2.2)

Now U is a unitary operator (in order to conserve probabilities), and can be written in the in-finitesimal form U (t + e, t) = 1− ieH(t), providedH, called the infinitesimal generator of thetransformation, to be a self-adjoint operator. In fact we can identifyH(t) as the total Hamilto-nian function of the system. We can also obtain a finite transformation iterating infinitesimalones, the result being the exponentiation of the generator, provided to be time-ordered topreserve the causality condition3:

Hamiltonian-

induced time

evolution U (t f , ti) = T{

exp[−i∫ t f

tidtH(t)

]}where T stands for time-ordered product ; (1.2.3)

such a relation is known as Dyson series, and has to be intended as an integral series:

U (t f , ti) = 1 +∞

∑n=1

(−i)n∫ t f

ti

∫ t1ti· · ·

∫ tn−1ti

dt1 · · ·dtnH(t1) · · · H(tn) . (1.2.4)

If, as it happens in most cases, the Hamiltonian function does not depend explicitly on time,

equation (1.2.3) takes the well known form U (t f , ti) ≡ U (t f − ti) = exp[−i(t f − ti)H

]4, as

the time ordering of operators is no longer necessary and the integration is straightforward5.Now that the background is set, we can see how to put the propagator K(x f , t f ; xi, ti) in

a path integral form. We will start analyzing the case of a time independent HamiltonianH,for it happens to be the case in a very wide range of physical situations, in particular thosewe will treat in greater detail in the rest of the work; nevertheless afterwards we will showhow to obtain a path integral propagator even in the case of a time dependent Hamiltonian,although not in the very detail.

1for a prove see [3]; for an analogous prove in relativistic quantum mechanics, i.e. for Klein-Gordon andDirac equations, see [4].

2from now on we will set }h = 1.3See [5, 6] for ulterior details.4the two-parameter family of unitary operators U (s, t) reduces in that case to the unitary one-parameter group

U (t), usual representation of time-translations group on a Hilbert spaceH .5H does not bear any time dependence and could be carried out.

10


1.2.1: Time-independent Hamiltonian

In that case relation (1.2.2) becomes:

K(x f , t f ; xi, ti) = 〈x f |e−i(t f−ti)H|xi〉 . (1.2.5)

For the sake of simplicity we will consider an usual Hamiltonian H(x, p) = T0(p) + V(x),where T0 is a quadratic function of only the momenta and V is an arbitrary function of thecoordinates1.

Now we can use a formula introduced by H.F.Trotter [7]2:

Trotter Product

Formula

eA+B = limn→∞

(e1n Ae

1n B)n (1.2.6)

that in our case becomes

e−i(t f−ti)[T0(p)+V(x)] = lime→0

(e−ieT0(p)e−ieV(x)) · · · (e−ieT0(p)e−ieV(x))︸︷︷︸n−1 times

, (1.2.7)

with e = (t f − ti)/(n − 1). Now using the two decompositions of identity operator 1x =∫dx |x〉〈x| and 1p =

∫ dp(2π)D |p〉〈p| (the subscript x or p has no meaning than to distinguish

between different decompositions), and defining xn ≡ x f , x0 ≡ xi , we obtain:

K(x f , t f ; xi, ti) = lime→0〈xn|

↓1pn

e−ieT0(p)e−ieV(x)↓

1xn−1 · 1pn−1

· · ·↓

1x1 · 1p1

e−ieT0(p)e−ieV(x)|x0〉 (1.2.8)

acting with exp[−ieT0(p)] on 〈p| and with exp[−ieV(x)] on |x〉

= lime→0

∫ ( n∏k=1

dpk(2π)D

)(n−1∏k=1

dxk

)e−ie

n∑

k=1

(T0(pk)+V(xk−1)

)〈xn|pn〉 · · · 〈p1|x0〉

finally recalling that 〈x|p〉 = 〈p|x〉∗ = exp[i(p · x)

]= lim

e→0

∫ ( n∏k=1

dpk(2π)D

)(n−1∏k=1

dxk

)e

ien∑

k=1

[pk ·(xk−xk−1)

e −(T0(pk)+V(xk−1)

)]. (1.2.9)

We have obtained a result for the integral kernelK(x f , t f ; xi, ti) which is formally equivalentto the one formulated by Feynman in (1.1.2), despite the fact that it is defined in phase spacerather than in configuration space. However we will see that in most cases3 the integral inthe momenta can be easily performed reducing effectively (1.2.9) to (1.1.2).

1we will analyze later the case of more general Hamiltonian functions.2this formula is discussed in Appendix A. However, as better explained in the appendix, it gives us a valid

approximant only to first order, and for our particular needs it turns out to be sufficient; however if we wanteda more precise approximation (with respect to some parameter, perhaps the commutator itself), we had to usethe Zassenhaus formula[8, 9]:

eλ(A+B) = eλAeλBeλ2 1

2 [B,A]eλ3 1

6 [[B,A],A+2B] · · · .

See Appendix A of this part for further details.3obviously, even in the particular case we are analyzing.

11


Our result, eq. (1.2.9), can be put in a much more elegant form effectively taking thelimit e → 0. If we consider the paths in phase space x(t) and p(t), such that x(tk) ≡ xk andp(tk) ≡ pk, they define a functional space, and in the limit the argument of the exponentialin (1.2.9) becomes just an integral over t1:

en

∑k=1

[ pk · (xk − xk−1)e

−(T0(pk) + V(xk−1)

)]= e

n

∑k=1

[p(tk) · ẋ(tk)−H

(x(tk), p(tk)

)]+ O(e2)

→∫ t f

tidt(

p(t) · ẋ(t)−H(x(t), p(t)

)). (1.2.10)

Furthermore, the integration over all the points x(tk) and p(tk) in the limit leads to a func-

Functional

Integration

tional integration∫D[x(t)]D[p(t)] over all the possible paths in phase space2,3 that start from

xi at ti and end at x f at t f . Eq. (1.2.9) then becomes an integral over all constrained pathsx(t), as well as over all unconstrained p(t)4:

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)]D[p(t)] exp{

i∫ t f

tidt(

p(t)ẋ(t)−H(x(t), p(t)

))}. (1.2.11)

We discuss now the case of a general Hamiltonian operator of two variables x and p5 (thatcan possibly contain mixed xi pj terms)6. We would like to repeat the same procedure wehave just done through Eq.s (1.2.5)– (1.2.11), but we cannot use Trotter formula (1.2.6) any-more, since it is difficult to treat exponentials of mixed terms xi pj, because we don’t knowhow to act on them with |p〉 and |x〉, in order to extract the Hamiltonian function from theHamiltonian operator. If it would be possible to manage exp

{ieH

}to have all the ps to the

left of all the xs7, then we could repeat the procedure done in (1.2.8), and then continue inthe same previous way to obtain (1.2.11)8.The big problem is that eO (x,p) 6= : eO (x,p) : , in

1about O(e2) in the second line, refer to Appendix A for a more detailed discussion on errors and approxi-mations in path integrals.

2this integration is intended to be defined by a regularization process, e.g.:∫D[x(t)]D[p(t)] ≡ lim

e→0

∫ n∏k=1

dpk(2π)D

n−1∏k=1

dxk (time-slicing).

However time slicing is not the only regularization possible for this functional measure, as we will see in thefollowing.

3There is in fact a mathematical notion of integration over paths due to Wiener[10, 11], but unfortunatelythis notion cannot be used in this case (we need to perform an analytic continuation in time, the Wick rotation, inorder to use Wiener measure).

4that is why it is called path integral.5in this general formulation, the variables x and p are two complete sets of independent observables, maybe

not the ones related to position and momentum coordinates.6that doesn’t represent a big deal as far as we limit ourselves to order e in the approximations, as it is done

in most cases, however it becomes tougher to use more precise approximations (see Appendix A).7we call such an operator ordering normal ordering, and it will be denoted by : . . . : , i.e. given an arbitrary

operator O (x, p), we will denote with : O (x, p) : its normal ordered form, with all ps to the left of all xs.8considering the whole : exp

{ieH

}: instead of exp

{ieT0

}exp

{ieV}

.

12


particular either e:O (x,p): 6= : e:O (x,p): : ! But in the latter case, we are able to write a relationbetween the two different exponentials. In particular if the exponential depends on someparameter (e.g. e in exp{−ieH} ), the relation will be a power series of such parameter.

So when we are dealing with a general time-independent Hamiltonian, all we have todo is to put it in normal ordered formHN (x, p), and use the following relation[12]:

e−ieHN (x,p) =: e−ieHN (x,p) : −e2∞

∑n=0

(−ie)n(n + 2)!

(Hn+2N (x, p)− : Hn+2N (x, p) :

); (1.2.12)

finally, since we need to consider only terms up to order e in the limit e→ 0, one can proceedthrough Eq.s (1.2.8)–(1.2.9) using : e−ieHN (x,p) : to obtain

Path Integral for

time-independent

Hamiltonians

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)]D[p(t)] exp{

i∫ t f

tidt(

p(t)ẋ(t)−HN(x(t), p(t)

))}. (1.2.13)

1.2.2: Time-dependent Hamiltonian

The most general Hamiltonian function could bear an explicit time dependence, so it wouldbe not a constant of the motion anymore, since dHdt =

∂H∂t . Nevertheless we know that evo-

lution operator obeys Dyson series (1.2.3), and we will show that a path integral expressionof the propagator is still possible1. Consider an Hamiltonian function in normal orderingH(x, p, t) bearing an explicit time dependence. The integral kernel we have to calculate hasthe form:

K(x f , t f ; xi, ti) = 〈x f |U (t f , ti)|xi〉 . (1.2.14)

Up to this point we need to describe the two-parameter family of unitary operators U (t, s)a little more in depth. We know that for a time-independent Hamiltonian it reduces to aone-parameter family of unitary operators that satisfies the properties of a group, and beingtime its parameter it turns out to be the unitary representation of time translations group. Wewould like to know the properties of the two-parameter family U (t, s) as well.

Definition 1.2.1 (Unitary Propagator). A two-parameter family of unitary operatorsA(s, t),s , t ∈ R which satisfies:

(i) A(r, s)A(s, t) = A(r, t)

(ii) A(t, t) = 1

(iii) A(s, t) is jointly strongly continuous in s and t

is called a unitary propagator.

1this derivation follows the one given in [5].

13


Theorem 1.2.1 (the Dyson expansion1). Let t → H(t) be a strongly continuous map of R intothe bounded self-adjoint operators on a Hilbert spaceH . Then there is a unitary propagator U (t, s)onH so that, for all ψ ∈ H ,

ϕs(t) = U (t, s)ψ

satisfies

ddt

ϕs(t) = −iH(t)ϕs(t) , ϕs(s) = ψ .

Unfortunately in most cases the HamiltonianH(x, p, t) does not satisfy the conditions of

Evolution in a

time-dependent

system

Theorem 1.2.1, since it often has an unbounded kinetic part. Nevertheless time dependentHamiltonians physically relevant are usually of the form H(t) = H0 + V(t), where both H0and V(t) are normal ordered operators depending on coordinates and momenta, and H0 ispossibly unbounded, while V(t) satisfies the conditions of Theorem 1.2.1. In such a case weare able to express K(x f , t f ; xi, ti) in a functional integral form2. In fact we can use the lasttheorem, passing to the “interaction representation”. We define

Ṽ(t) = eiH0tV(t)e−iH0t ;

then t→ Ṽ(t) also satisfies the hypotheses of Theorem 1.2.1, and we call the correspondingpropagator Ũ (t, s). We recall that it is determined by a Dyson series (see Appendix B for aproof):

Ũ (t, s) = 1 +∞

∑n=1

(−i)n∫ t

s

∫ t1s· · ·

∫ tn−1s

dt1 · · ·dtn eiH0t1V(t1) · · · e−iH0(tn−1−tn)V(tn)e−iH0tn .

(1.2.15)

Now setting

U (t, s) = e−itH0 Ũ (t, s)eisH0 , (1.2.16)

we see that, at least formally, U (t, s) satisfies

ddtU (t, s) = −iH0e−itH0 Ũ (t, s)eisH0 + e−itH0

(−iṼ(t)

)Ũ (t, s)eisH0

= −i(

H0 + V(t))U (t, s) ,

so ϕs(t) = U (t, s)ψ should be a strong solution of

ddt

ϕs(t) = −i(

H0 + V(t))

ϕs(t) , ϕs(s) = ψ .

1for a proof, and a discussion on its implications see Appendix B.2see Appendix B for the most general case.

14


The problem arises from the fact that H0 U (t, s)ψ may not take sense, since Ũ (t, s)ψ may notbe in the domain of H0 even if ψ is. Now a particular case of the general Theorem B.1.2 ofAppendix B assures us that if t → [H0, V(t)] is strongly continuous, then ϕs(t) is a strongsolution; however it is always a “weak” solution, in the sense that for any η ∈ D(H0),(η, ϕs(t))1 is differentiable and

ddt

(η, ϕs(t)) = −i(H0η, ϕs(t))− i(V(t)η, ϕs(t)) .

Now the evolution operator U (t f , ti), if exists, is indeed a unitary propagator, since Ũ (t, s) isa unitary propagator. Property (i) in particular turns out to be very useful since it allows usto write, defining t f ≡ tn , ti ≡ t0 :

U (tn, t0) = U (tn, tn−1) · · · U (t1, t0) (1.2.17)

with tk ≡ tk+1 − e , e =t f − ti

nso obviously tn > tn−1 > . . . > t0 .

So Eq. (1.2.14) becomes:

K(xn, tn; x0, t0) = 〈xn|U (tn, tn−1) · · · U (t1, t0)|x0〉 . (1.2.18)

Now using the decompositions of the identity operator 1x and 1p defined in section 1.2.1 wehave:

K(xn, tn; x0, t0) = 〈xn|↓

1pn

U (tn, tn−1)↓

1xn−1 · 1pn−1

· · ·↓

1x1 · 1p1

U (t1, t0)|x0〉

=∫ ( n

∏k=1

dpk(2π)D

)(n−1∏k=1

dxk

)〈xn|pn〉〈pn|U (tn, tn−1)|xn−1〉 · · · 〈p1|U (t1, t0)|x0〉 .

(1.2.19)

All we have to do is evaluate 〈pk|U (tk, tk−1)|xk−1〉 , and put it back in (1.2.19). But expandingtk around tk−1 in Eq. (1.2.15) we obtain:

Ũ (tk, tk−1) = 1− ie eiH0tk−1V(tk−1)e−iH0tk−1 + O(e2) ;

then

U (tk, tk−1) = e−itk H0 Ũ (tk, tk−1)eitk−1 H0

= e−i(tk−tk−1)H0(

1− ieV(tk−1))

+ O(e2)

= e−ieH0 e−ieV(tk−1) + O(e2) = e−ieH(tk−1) + O(e2)

=: e−ieH(tk−1) : +O(e2) ,

recalling Eq. (1.2.12) in the last equality. So if we take the limit e → 0 we can forget aboutsecond-order e-terms, and acting with 〈pk| and |xk−1〉 on the normal ordered exponentialwe get:

lime→0〈pk|U (tk, tk−1)|xk−1〉 = lim

e→0e−ieH(xk−1,pk ,tk−1)〈pk|xk−1〉 ; (1.2.20)

1D(H0) is the domain of H0 and (η, ϕs(t)) is the scalar product between η, ϕs(t) ∈ H .

15


finally inserting this result in (1.2.19) we obtain the discretized path integral:

K(xn, tn; x0, t0) = lime→0

∫ ( n∏k=1

dpk(2π)D

)(n−1∏k=1

dxk

)e

ien∑

k=1

[pk ·(xk−xk−1)

e −H(xk−1,pk ,tk−1)]

. (1.2.21)

Following the same reasoning of (1.2.10), we obtain the final path integral formula:

Functional Integral

for time-dependent

Hamiltonians

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)]D[p(t)] exp{

i∫ t f

tidt(

p(t)ẋ(t)−H(x(t), p(t), t

))}. (1.2.22)

We see that this formula is perfectly analogous to Eq. (1.2.11), provided with the right timedependence of the Hamiltonian function.

1.2.3: Transition to Lagrangian formulation

We see that in Equations (1.2.11), (1.2.13) and (1.2.22) the integrand in the exponential lookslike the Lagrangian function L associated with the Hamiltonian H; path integral thus be-coming the integral over paths of the exponential of i1 times the classical action associatedwith the system. But this appearance is misleading, since the momenta p(t) are indepen-dent variables, not yet related to x(t) or their derivatives. However, if the Hamiltonian isa quadratic function of the p(t), then the integral over this variables can be performed justsubstituting them with their values dictated by canonical formalism, thus the integrand inthe exponential really is the Lagrangian function[13].

Let’s see how this passage is performed: we deal with a quadratic form of the p(t), i.e.2

H(x(t), p(t), t) = 12

Aij[x(t), t]pi(t)pj(t) + Bi[x(t), t]pi(t) + C[x(t), t] ;

so the exponential becomes

exp{−i(1

2A ij pi pj +B i pi +C

)},

where we have used a shorthand notation where an index like i stands for a multi-index,both discrete and continuous, and the contraction of equal indices stands for summationover discrete indices and integration over continuous ones3 and

A ij ≡ Aij[x(t), t]δ(t− t′) ,B i ≡ Bi[x(t), t]− ẋi(t) ,

C ≡∫

dt C[x(t), t] .

1(times }h−1).2making explicit the components of the x(t) and p(t) vectors, and using Einstein summation convention.3see Appendix C for further informations.

16

1.3. APPLICATIONS: FREE THEORIES, HARMONIC OSCILLATORS & PERTURBATIVE THEORY CHAPTER 1. PATH INTEGRALS

Now we know how to solve Gaussian integrals like∫D[p(t)] exp

{−i(

12A

ij pi pj +B i pi +

C)}

1 : it is sufficient to substitute to the variables pi their value p̄i ≡ −A −1ij B j , i.e. the sta-

tionary point of the exponent2. But the stationary point of∫

dt(

p(t)ẋ(t)−H(x(t), p(t), t

))obeys the equation:

ẋi(t) =δH(

x(t), p(t), t)

δpi(t)

∣∣∣∣p= p̄

,

i.e. the usual formula dictated by canonical formalism. So performing the integration overmomentum paths we see that the exponent of Equations (1.2.11) , (1.2.13) and (1.2.22) isindeed i times the classical action of the system; therefore finally we can write the pathintegral in configuration space3:

Path Integral in

Lagrangian

formulation

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)] exp{

i∫ t f

tidtL

(x(t), ẋ(t), t

)}. (1.2.23)

However this derivation is just formal and does not take into account the factor

1√Det

[iA [x(t)]

2π

] ,that contains a functional determinant that has to be suitably managed. In particular if A

Integrating out

momenta is a

touchy business

does not depend on x(t), such a determinant could be adsorbed in the definition of the mea-sureD[x(t)], however if it depends on x(t) could give rise to modifications to the LagrangianL that one has to take into account, in particular we will see that in curved space-time theintegration of momenta modifiesL introducing a counterterm that depends on the regular-ization scheme we have chosen to properly define the functional measure D[x(t)]D[p(t)].

1.3: Applications: Free Theories, Harmonic Oscillators &Perturbative Theory

I t is now time to explicit our general results in some useful cases. Let’s start from thesimplest case: a system consisting of n free4 particles of equal mass m in a N dimensional1see Appendix C2the value p̄ obeying:

δ

δpi

(−i(1

2A ij pi pj +B i pi +C

))∣∣∣∣∣p= p̄

= 0 ;

where the derivative is a functional derivative.3we present here only the most general case of a time-dependent Lagrangian function, the other are partic-

ular cases.4as free particles we mean distinguishable particles that don’t interact neither mutually, neither with some

external potential.

17


space. In that case we can describe our system with D ≡ nN coordinates and momenta i.e.with a D-dimensional configuration space described by a Lagrangian function

L f ree(

x(t), ẋ(t))

=m2

ẋ(t)2 (1.3.1)

where we use a shorthand notation for a D-dimensional vector

x(t)2 ≡D

∑i=1

xi(t)2 ,

and the dot stands for a derivative with respect to time as usual.Before dealing with the path integral of such a system we will see that if the Lagrangian

function is a quadratic form, the kernel K(x f , t f ; xi, ti) is proportional to the exponentialof i times the action evaluated on the classical path, i.e. on the path satisfying the classicalequations of motion and the boundary conditions1: consider a general quadratic Lagrangianfunction

L =12(aij ẋi ẋj + bijxixj) + cijxi ẋj + di ẋi + eixi ,

where the implicit time dependence of coordinates and their time derivatives has been omit-ted, and a, b, c are D× D dimensional time-dependent matrices and d, e are D-dimensionaltime-dependent vectors, in the usual component notation with summation over repeated in-dices. Now it is immediate to see that aij = aji and bij = bji , i.e. both a and b are symmetricmatrices2. With matrix c we have to be more careful: a priori it has not a definite symmetry,but as usual we can split it in a symmetric and an antisymmetric part defining

cij = csij + caij ,

with

csij ≡cij + cji

2,

caij ≡cij − cji

2;

but the term csijxi ẋj can be written as a total derivative (that we can cancel out of the La-

grangian, not contributing to the equations of motion) plus a term with the structure αijxixj

that can be adsorbed redefining bij in a suitable way:

csijxi ẋj =

ddt

(12

csijxixj)− 1

2ċsijx

ixj .

1the action evaluated on the classical path is not to be confused with the classical action: the latter is theaction of a classical system corresponding to the quantum mechanical Hamiltonian function of the system underexamination; the former is the classical action evaluated on the path dictated by classical equations of motionand boundary conditions.

2in fact if they have not a definite symmetry, they can be split in a symmetric and antisymmetric part, butthe antisymmetric part contracted with the product of two coordinates (that is symmetric) vanishes.

18


In an analogous way we can see that the term di ẋi equals a total derivative plus a termadsorbed in redefining vector e; so finally a general quadratic Lagrangian can be written as

L =12(asij ẋ

i ẋj + bsijxixj) + caijx

i ẋj + eixi , (1.3.2)

where the symmetry or antisymmetry of the matrices has been made explicit with indices s

and a that will be henceforward omitted. Equations of motion

ddt

(∂L∂ẋi

)=

∂L∂xi

read in that case

ȧijxj + aij ẍj − ċijxj − 2cij ẋj − bijxj − di = 0 . (1.3.3)

If we now perform a split x(t) = xcl(t) + q(t), where xcl(t) is a vector satisfying equationsof motion and the boundary conditions, i.e.1:

Classical Path &

Quantum

Fluctuations split

xcl(ti) = xi , xcl(t f ) = x f ,

ȧijxjcl + aij ẍ

jcl − ċijx

jcl − 2cij ẋ

jcl − bijx

jcl − d

i = 0 :

and q(t) has vanishing boundary conditions, i.e.2:

q(ti) = 0 , q(t f ) = 0 ;

we immediately see that the Lagrangian can be split as:

L (x, ẋ) ≡ L (xcl , ẋcl) +L2(q, q̇) +(

aijq̇i ẋjcl + bijq

ixjcl − cijq̇ixjcl + cijq

i ẋjcl + diqi)

,

whereL2(q, q̇) ≡ L (q, q̇)− diqi (it is the purely quadratic part of the Lagrangian). In order toevaluate the kernelK(x f , t f ; xi, ti) we need the action S[x, t f , ti], i.e. the time integral betweenthe initial and final time of the Lagrangian function. But S[x, t f , ti] ≡ S[xcl , t f , ti] + S2[q, t f , ti]where

S2[q, t f , ti] =∫ t f

tidtL2(q, q̇) :

in fact the time integral

∫ t fti

dt(

aijq̇i ẋjcl + bijq

ixjcl − cijq̇ixjcl + cijq

i ẋjcl + diqi)

vanishes. In order to prove the last assertion is sufficient to integrate by parts the q̇-terms,remembering that the q function vanishes at the boundaries and xcl satisfies equations of

1xcl(t) is often called the “classical path”.2q(t) is often called the “quantum fluctuation”.

19


motion. Finally since we usually define the functional measureD[x(t)] to be invariant undertranslations, so that D[x(t)] ≡ D[q(t)], formula (1.2.23) becomes in that case:

K(x f , t f ; xi, ti) = eiS[xcl ,t f ,ti ]∫

VBC

D[q(t)] exp{

i∫ t f

tidtL2

(q(t), q̇(t), t

)}, (1.3.4)

where VBC stands for vanishing boundary conditions. Now defining∫VBC

D[q(t)] exp{

i∫ t f

tidtL2

(q(t), q̇(t), t

)}≡ F (t f , ti) ,

we see explicitly that the kernel is proportional to the exponential of the action on the clas-sical path:

K(x f , t f ; xi, ti) = F (t f , ti) eiS[xcl ,t f ,ti ] . (1.3.5)

1.3.1: Systems of free particles

We will now calculate explicitly K(x f , t f ; xi, ti) in the case of a system of free particles. Wehave seen at the beginning of this section that such a system has in general a Lagrangianfunction:

L f ree(

x(t), ẋ(t))

=m2

ẋ(t)2 (1.3.6)

where x(t) is a D-dimensional vector. We have to make a sense out of the functional measureD[x(t)], and in that simple case it is useful to remember the time discretization that led toour path integral formulation, so we will write

D[x(t)] ≡ lime→0

( m2πie

) D2 N N−1

∏i=1

dxi ,

where dxi is the D dimensional Lebesgue measure, e ≡t f−ti

N and the constant multiplyingthe measures is due to the integration over momenta (see section 1.2.3). The time discretizedaction becomes e ∑ m2 (

xi−xi−1e )

2, so we have to evaluate the following expression1:

K(x f , t f ; xi, ti) = lime→0

∫ N−1∏i=1

dxi( m

2πie

) D2 N

exp{

ieN

∑i=1

m2

( xi − xi−1e

)2}. (1.3.7)

But if we define

Fe(x) ≡( m

2πie

) D2

exp{

imx2

2e

},

we see that with the usual convolution product2(Fe ∗ Fe′

)(x) ≡

∫ ∞−∞

dy Fe(y)Fe′(x− y)1identifying xN ≡ x f and x0 ≡ xi.2being the Lebesgue measure translationally invariant, this definition is equivalent to:(

Fe ∗ Fe′)(x) ≡

∫ ∞−∞

dy Fe(y− a)Fe′ (x− y) ,

where a is an arbitrary factor.

20


holds the following property:(Fe ∗ Fe′

)(x) = Fe+e′(x) . (first convolution property)

The demonstration of the (first convolution property) involves some boring calculationscompleting squares and performing a Gaussian integration on y1. The usefulness of thisproperty is seen considering the integration over dxi in Eq. (1.3.7), isolating xi-dependingterms we see that this integration reduces to:

( m2πie

) D2( m

2πie

) D2∫

dxi exp{

im(xi − xi−1)22e

}exp

{im(xi+1 − xi)2

2e

}=

remembering the definition of convolution product

=(

Fe ∗ Fe)(xi+1 − xi−1) = F2e(xi+1 − xi−1) .

Now the integration over dxi+1 is perfectly analogous but involves the convolution product(F2e ∗ Fe

)(xi+2− xi−1); so it is straightforward to perform all consecutive N− 1 integrations,

starting from∫

dx1 , obtaining2:

FNe(xN − x0) ≡( m

2πi(t f − ti)

) D2

exp{

im(x f − xi)22(t f − ti)

};

so finally

K f ree(x f , t f ; xi, ti) =( m

2πi(t f − ti)

) D2

exp{

im(x f − xi)22(t f − ti)

}. (1.3.8)

As we expected, the Kernel in Eq. (1.3.8) has the structure predicted in (1.3.5), being thefree Lagrangian effectively a quadratic form3. Furthermore, we could have found K f reestarting from Eq. (1.3.5): we know that K f ree(x f , t f ; xi, ti) = F (t f , ti) exp

{iS[xcl , t f , ti]

}, so

all we need to evaluate is F (t f , ti); but since Eq. (1.2.1) tells us that K obeys Schrödinger1for examples on such a method as “completing squares” in Gaussian integration see Appendix C.2remembering that xN ≡ x f , x0 ≡ xi and Ne ≡ t f − ti.3the solution of equations of motion

ẍ = 0

respecting the boundary conditions x(ti) = xi and x(t f ) = x f is

x(t) =x f − xit f − ti

t +xit f − x f ti

t f − ti,

so the action on the classical path is effectively

S[xcl , t f , ti] =m2

(x f − xi)2

(t f − ti).

21


Equation1, we can fix F (t f , ti) to be the value for which such equation is satisfied by K f ree.A straightforward calculation fixes obviously:

F (t f , ti) =( m

2πi(t f − ti)

) D2

,

as we expected.

1.3.2: The Harmonic Oscillator

Another system of physical interest is a system of n uncoupled and distinguishable Har-monic Oscillators of equal mass m, in N dimensions, that is characterized in a D = nNdimensional space by a Lagrangian function

Losc(x(t), ẋ(t)) =m2

(ẋ(t)2 −ω2x(t)2

),

with usual vectorial notations. It is a discomforting task to evaluate directly K(x f , t f ; xi, ti),since the path satisfying equations of motion and these general boundary conditions is:

xcl(t) =x f

cos(ωt f )tan(ωti) + xicos(ωti) tan(ωt f )

tan(ωt f )− tan(ωti)cos(ωt) +

x fcos(ωt f )

− xicos(ωti)tan(ωt f )− tan(ωti)

sin(ωt) ;

but since m and ω are constants, the Lagrangian function doesn’t depend explicitly ontime, so time translations are a symmetry of the system, and therefore K(x f , t f ; xi, ti) ≡K(

x f , (t f − ti); xi, 0), with the advantage that with these boundaries xcl(t) reduces to2:

xcl(t) = xi cos(ωt) +x f − xi cos

(ω(t f − ti)

)sin(ω(t f − ti)

) sin(ωt) . (1.3.9)The action evaluated on Eq. (1.3.9) is

Sosc[xcl , (t f − ti)] =mω

2 sin(ω(t f − ti)

)((x2f + x2i ) cos(ω(t f − ti))− 2x f · xi) . (1.3.10)Since as the free Lagrangian function, Losc is a quadratic form, we can use formula (1.3.5)and evaluate only

F (t f − ti) =∫

VBC

D[q(t)] exp{

im2

∫ (t f−ti)0

dt(q̇(t)2 −ω2q(t)2

)}.

In order to do that we will use a particular technique, called “mode expansion”: the idea isquite simple, we will expand paths q(t) in a Fourier series, and define the integration overall possible paths as the integration of the Fourier coefficients over all their possible values,

1since K(x f , t f ; xi, ti) is the only x f and t f -dependent object on the right member of Eq. (1.2.1), andSchrödinger Equation has to be satisfied for ψ(x f , t f ).

2we will limit ourselves to the case 0 < ω(t f − ti) < π in order to determine xcl(t) univocally.

22


since every path vanishing at the boundaries could be written as a series of sines with realFourier coefficients1. So we perform the expansion

q(t) =∞

∑n=1

Cnφn(t) ,

where Cn ∈ RD and φn(t) is a complete set of orthonormal functions on [0, (t f − ti)], van-ishing at the boundaries, i.e. the sine functions

φn(t) =

√2

t f − tisin(

nπtt f − ti

).

A simple integration by parts allow us to rewrite the action S2[q, (t f − ti)] as a function ofthe Fourier coefficients, precisely

S2[q, (t f − ti)] ≡m2

∫ (t f−ti)0

dt(q̇(t)2 −ω2q(t)2

)=

m2

∞

∑n=1

C2n

[( nπt f − ti

)2−ω2

].

So, setting

D[q(t)] =M limN→∞

N

∏n=1

dCn ,

we need to calculate

F (t f − ti) =M limN→∞

∫ N∏n=1

dCn exp{

im2

N

∑n=1

C2n

[( nπt f − ti

)2−ω2

]},

after performing N Gaussian integrations we obtain

F (t f − ti) =M′ limN→∞

N

∏n=1

[1−

(ω(t f − ti)nπ

)2]− D2=M′

(ω(t f − ti)

sin(ω(t f − ti)

)) D2 ;whereM′ ≡ t f−tinπ

(2πim

) ND2 M and we have used the formula

sin xx

= limN→∞

N

∏n=1

[1−

( xnπ

)2].

In order to fix M′, we remember that in the limit ω → 0 the system under considerationreduces to a system of free particles, that has a well known kernel K f ree that we calculatedin the previous subsection. But since

Sosc[xcl , t f , ti]ω→0−−→ S f ree[xcl , t f , ti] ,

it has to be

Fosc(t f − ti)ω→0−−→ F f ree(t f , ti) ;

1more precisely we will limit to say that the functional measure D[q(t)] is proportional to the measure of allFourier coefficients, the proportionality factor being fixed later to satisfy the free particles limit ω → 0.

23


so being

Fosc(t f − ti)ω→0−−→M′ ,

we can identifyM′ = F f ree(t f , ti), so finally we can write1

Kosc(x f , t f ; xi, ti) =(

mω2πi sin

(ω(t f − ti)

)) D2 exp{i mω2 sin

(ω(t f − ti)

)((x2f + x2i )· cos

(ω(t f − ti)

)− 2x f · xi

)}.

(1.3.11)

1.3.3: Perturbative Methods

Unfortunately it is quite difficult to explicitly solve path integrals for more complicated sys-tems, but we will see soon that we can develop a quite powerful tool to calculate perturba-tive expansions with path integrals introducing the so-called “correlation functions”.

We recall that there are two equivalent realizations of time evolution on a Hilbert SpaceH : one is often referred to as Schrödinger picture, the other as Heißenberg picture. The for-mer makes the states, i.e. the elements of Hilbert Space, carry any implicit time dependence(in fact they have to obey Schrödinger equation) while operators might have only explicittime dependence2; in the latter elements of H are constant in time, and operators carry alltime dependence3. Now, since they must describe the same physical system, they have to berelated by a unitary transformation, soHS andHH would be isomorphic. This is possible iftime evolution is unitary, i.e. realized by a unitary operator U (t, s) ∈ L(H ). As explained inAppendix B, this happens in most cases of physical interest. Furthermore, if time evolution

Heißenberg &

Schrödinger

pictures

is not unitary, the entire Schrödinger picture would be almost unmanageable, since in factit consists in a mapping t → Ht, where t ∈ R is time, and Ht is the Space of states associ-ated with the system at time t; now since the evolution is not unitary Ht and Ht′ could benon-isomorphic, or worse even ifHt is a Hilbert Space,Ht′ could not!

In fact in cases where we are not able to tell whether time evolution is unitary or notwe must rely on Heißenberg picture, as in the case of interacting Quantum Field Theories,since in that picture the Hilbert Space describing the system is constant through time. If timeevolution is unitary, we can use U (t, s) to associate HH and HS(t)4, with arbitrary t ∈ R;

1being, as we have already seen, Kosc(x f , t f ; xi, ti) ≡ Kosc(x f , (t f − ti); xi, 0).2in fact

dAS(t)dt

=∂AS(t)

∂t,

where AS is an arbitrary operator in L(HS).3we recall that their time evolution is given by

dAH(t)dt

=(

∂AS(t)∂t

)H− i[AH(t),HH(t)] ,

where AH is an arbitrary operator in L(HH), andHH is the Hamiltonian function.4since time evolution is unitary, HS(t′) ≡ U (t′, t)HS(t) is isomorphic to HS(t) for any t′, t ∈ R, so HS has

the same structure at any time.

24


in particular HH ≡ U †(t, t0)HS(t), where t0 is a fixed parameter. So HS(t0) ≡ HH, whileany other HS(t) is isomorphic to it. L(HH), the family of linear operators in Heißenbergpicture is related to L(HS) as usually done when we perform a unitary transformation, i.e.L(HH) ≡ U †(t, t0)L(HS)U (t, t0).

So we immediately see that K(x f , t f ; xi, ti) ≡ 〈x f (t f )|U (t f , ti)|xi(ti)〉S is the transitionelement between position eigenstates in Heißenberg picture:

K(x f , t f ; xi, ti) = 〈x f (t f )|U (t f , t0)U †(ti, t0)|xi(ti)〉S = 〈x f , t f |xi, ti〉H .

Now on we will use Heißenberg picture, omitting index H, so K(x f , t f ; xi, ti) = 〈x f , t f |xi, ti〉.Up to this point, we would like to be able to calculate 〈x f , t f |O(t1)|xi, ti〉, where O(t1) ∈L(H ) and t1 ∈ [ti, t f ], with a path integral. This is easily done:

〈x f , t f |O(t1)|xi, ti〉 = 〈x f (t f )|U (t f , t1)O(t1)U (t1, ti)|xi(ti)〉S

=∫

dxdx′K(x f , t f ; x, t)〈x(t)|O(t1)|x′(t)〉SK(x′, t; xi, ti) ;

if O(t)S is diagonal in the coordinates representation, i.e.

〈x(t)|O(t1)|x′(t)〉S = O(x(t1), t1)δD(x− x′) ,

then above expression reduces to a single path integral1:

〈x f , t f |O(t1)|xi, ti〉 =∫

x(ti)=xix(t f )=x f

D[x(t)] eiS[x,t f ,ti ]O[x(t1)] . (1.3.12)

This construction can be generalized to a time-ordered product of operators

T{

O1(t1)O2(t2) · · ·}

with t1, t2, . . . ∈ [ti, t f ] ,

obtaining, if the operators are all diagonal in the coordinates representation,calculation of

transition elements

with path integrals

〈x f , t f |T{

O1(t1)O2(t2) · · ·}|xi, ti〉 =

∫x(ti)=xix(t f )=x f

D[x(t)] eiS[x,t f ,ti ]O1[x(t1)]O2[x(t2)] · · · . (1.3.13)

If O1(t1) ≡ xi1(t1), and so on, formula (1.3.13) reduces to:

〈x f , t f |T{

xi1(t1)xi2(t2) · · ·}|xi, ti〉 =

∫x(ti)=xix(t f )=x f

D[x(t)] eiS[x,t f ,ti ]xi1(t1)xi2(t2) · · · ;

so we can define the n-point propagator, or n-point correlation function2

〈 xi1(t1) · · · xin(tn) 〉 ≡〈x f , t f |T

{xi1(t1) · · · xin(tn)

}|xi, ti〉

〈x f , t f |xi, ti〉(1.3.14)

1to make this possible it is necessary that t ∈ [ti, t f ], in order to identify the integration over x as one of theslices of the path at an intermediate time t.

2with this definition the zero point correlation function, i.e. 〈 1 〉 = 1; but we will use a different notationhere, we define 〈 1 〉 ≡ 〈x f , t f |xi, ti〉: so 〈x f , t f |T

{xi1 (t1) · · · xin (tn)

}|xi, ti〉 = 〈 1 〉〈 xi1 (t1) · · · xin (tn) 〉.

25


through a path integral. We will see in a moment that there is a beautiful way to calculatesuch correlations functions with n arbitrary knowing only the 2-point propagator. Define a“generating functional”

Generating

Functional

Z[J] =∫

x(ti)=xix(t f )=x f

D[x(t)] ei(

S[x,t f ,ti ]+∫

dt Ji(t)xi(t))

, (1.3.15)

where the “sources” J(t) are D-dimensional vectors. It is immediate to see that the n-pointpropagator can be written in terms of functional derivatives of generating functional1:

〈 xi1(t1) · · · xin(tn) 〉 =1

Z[J](−i)n δ

δJi1(t1)· · · δ

δJin(tn)Z[J]

∣∣∣∣J=0

. (1.3.16)

Now if S[x, t f , ti] reduces to S2[x, t f , ti] = −∫ t f

ti

∫ t fti dtdt

′ 12 Kij(t, t

′)xi(t)xj(t′), where Kij(t, t′)is a differential operator, Z[J] reduces to a Gaussian Integral2,3

Z[J] =∫

x(ti)=xix(t f )=x f

D[x(t)] e−i(

12 Kijx

ixj−Jixi)

= Det−12

[ iK2π

]Det−

12

[ iA2π

]e

i2 (K

−1)ij Ji Jj .

So we immediately deduce by derivation the so called “Wick Theorem”, i.e. that any odd-point propagator must vanish, and any even-point propagator is calculated by means of allpossible products of 2-point propagators4:

〈 xi 〉2 = 0 ,〈 xi1 xi2 〉2 = −i(K−1)ij ,

〈 xi1 · · · xin+1 〉2 = 0 ,

〈 xi1 · · · xi2n 〉2 = ∑permutations of

i1...i2n

(−i(K−1)ip1ip2

)· · ·(−i(K−1)ip(2n−1)ip2n

).

(Wick Theorem)

Unfortunately only when we deal with quadratic Lagrangian functions we can write theaction as S2[q, t f , ti] = −

∫ t fti dt

12 Kij(t)q

i(t)qj(t), following the method described at the be-ginning of Section 1.3. Nevertheless we can use these tools in the case of a general actionS[x, t f , ti] if we are able to write it as S2[x, t f , ti] + Sint[x, t f , ti]: in fact K(x f , t f ; xi, ti) thus

1the functional derivative is defined (in the unidimensional case, the generalization is straightforward) asthe following: given a functional F[ f (x)] its functional derivative is the fraction

δF[ f (x)]δ f (y)

≡ lime→0

F[ f (x) + eδ(x− y)]− F[ f (x)]e

.

2using the usual compact multi-index notation.3see Appendix C and Section 1.2.3 for an explanation of the factor Det−1/2

[iA2π

].

4we have used 2 to make explicit that this Theorem holds if the averages are calculated with an action S2.Since now on we will always calculate averages of that kind, the index 2 will be omitted.

26


becomes

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)] eiS2[x,t f ,ti ]eiSint[x,t f ,ti ] = 〈 1 〉〈 eiSint[x,t f ,ti ] 〉 ;

and since the exponential can be expanded in a power series, and the average of a sum isthe sum of the averages we obtain:

K(x f , t f ; xi, ti) = 〈 1 〉(

1 +∞

∑n=1

in

n!〈 Snint 〉

).

If this series depends on a small parameter, this expression could be seen as the perturbativeexpansion we sought1. We stress that with this construction all we need to know to performperturbative calculations is the 2-point propagator, i.e. the inverse of the kinetic operatorKij(t, t′) appearing in S2[x, t f , ti].

1.3.4: Some Perturbative Calculations

Let’s see how to apply the powerful tools of the previous section to some simple case. Infact we will use generating functionals, propagators and perturbative expansions almosteverywhere throughout this work, but some simple example will help us get acquaintedwith such techniques.

Let’s start with a well known problem, a system of uncoupled harmonic oscillators: wewill calculate the 2-point propagator for such a theory (recall that Sosc is a S2-type action), in aparticular case. It is particularly useful to analyze this propagator even if we already solvedthe path integral for such a system, because a quite crucial problem will emerge quite natu-rally in performing that calculation, related with the impossibility to define coherently pathintegrals in real time without recurring to analytic continuation of calculations in imaginarytime. We already encountered a first problem: path integrals in real time involve “imagi-nary” Gaussian integrals, like

∫D[p]eip2/2, also known as Fresnel integrals, that are analytic

continuations of well-known real Gaussian integrals, like∫D[p]e−p2/2, see Appendix C for

further details. This underlies a deeper problem: we are not able to define a functional Euclidean time &Wick rotationmeasure in the space of paths in real time due to oscillatory factors eiS, but we can do that

if we continue time analytically to purely imaginary values; this procedure is called “WickRotation”, and consists in a redefinition of time t→ −it: with that procedure the oscillatoryfactor becomes a real exponential e−SE , where SE is the so-called Euclidean action, and wecan define Wiener measures in path space in such a case. So if we want to be more rigorousand solve ambiguities, we have to perform calculations in Euclidean (imaginary) time, andthen “rotate back” to real time through analytic continuation.

Returning to the propagator for a theory of harmonic oscillators, we will see that in realtime we have poles on the integration path, and we have an ambiguity on how to “pass by”

1recall that such a parameter could be }h that we set to 1 throughout our derivation but else would haveappeared as a factor }h−n in the perturbative expansion; also a suitable choice of t f − ti could serve as the smallparameter as it appears in the power series (only if the Hamiltonian is time-independent).

27


them, but in Euclidean time there are no poles on the integration path so the ambiguity issolved, and analytic continuation of the result gives us back the real time propagator. Weare not able to perform calculations in the case of an arbitrary Kosc(x f , t f ; xi, ti)1, so we willrestrict to the calculation of Z(t f , ti) ≡

∫dx 〈x, t f |x, ti〉: 〈x, t f |x, ti〉 is a path integral, with a

periodic boundary condition x(t f ) = x(ti) = x, and such boundary value is then integratedover all its possible values; this integration over the boundary value is equivalent to modifythe space of paths we are considering in defining the functional measure: we pass fromintegration over all possible paths with fixed endpoints to integration over all possible pathswith the initial point equivalent to the final one, and we will call this constraint “PeriodicBoundary Conditions”, or PBC. So we can write

Z(t f , ti) =∫

PBC

D[x(t)] eiS[x,t f ,ti ] . (1.3.17)

It is straightforward that all the considerations about generating functionals, propagatorsetc. we did for Kosc(x f , t f ; xi, ti) remain valid for Z(t f , ti): it is sufficient to substitute inany formula fixed endpoints conditions with PBC. We will see the usefulness of Z imme-diately as we will consider the Euclidean case. By now we will limit ourselves to find the2-point propagator for Z in the case of Harmonic Lagrangian function. First of all since timetranslation is a symmetry of Harmonic systems, we can write:

Z(t f , ti) ≡ Z(β) =∫

PBC

D[x(t)] eiS[x,β2 ,−

β2 ] with β ≡ t f − ti .

Then we note that periodic boundary conditions allow us to integrate by parts in the ac-tion, since the boundary term vanishes, so we can write Harmonic action, using multi-indexnotation as usual:

Sosc =12

Kijxixj ,

with Kij(t, t′) = m( d2

dt2 + ω2)δ(t− t′)δij. So in order to find the propagator all we have to do

is find the inverse of Kij(t, t′); we have to solve the distributional equation∫dt′′ Kij(t, t′′)(K−1)jk(t′′, t′) = δ ki δ(t− t′) ,

i.e.:∫dt′′ mδijδD(t− t′′)(

d2

dt2+ ω2)(K−1)jk(t′′, t′) = mδij(

d2

dt2+ ω2)(K−1)jk(t, t′) = δ ki δ(t− t′) ,

⇒ (K−1)jk(t, t′) = δjk

mK−1(t− t′) ,

1because even if the Lagrangian function of such a system is a quadratic function, we are not able to finda solution for the inverse of the Harmonic kinetic operator that satisfies either general boundary conditions orvanishing ones, as we would have to if we wanted to solve the general case. We will see that for a free systemwe know a solution of the inverse equation that satisfies vanishing boundary conditions, so we are able to findthe propagator for a general Kernel.

28


here we guess that since K(t, t′) ≡ K(t− t′) the same property holds for K−1(t, t′); if we canfind a solution we prove that guess is right, since the solution is unique. So we have to seeka solution of:

(d2

dx2+ ω2)K−1(x) = δ(x) ,

so if we consider the limit β → ∞ we can apply Fourier transform, so the equation forK̂−1(k) reads

−(k2 −ω2)K̂−1(k) = 1 ,

⇒ K̂−1(k) = − 1k2 −ω2 ,

⇒ K−1(x) = −∫ dk

2π1

k2 −ω2 eikx ;

here we see that there are poles on the integration path, as we anticipated, and we don’tknow what is the right prescription to manage these singularities. We are thus forced toperform calculations in Euclidean time and recover the real time result through analyticcontinuation. But what is the difference between real and Euclidean time calculations? SinceS[x, t f , ti] =

∫ t fti dtL (x(t), ẋ(t), t) we have, performing the transformation t

′ = it:

iS[x, t f , ti] = i∫ t′f

t′id(−it′)L (x(t′), iẋ(t′),−it′) = −SE[x, t f , ti]

where SE[x, t f , ti] = −∫ t f

ti dtL (x(t), iẋ(t),−it), recalling t′ = t. So Euclidean time path

integral becomes

K(x f , t f ; xi, ti) =∫

x(ti)=xix(t f )=x f

D[x(t)] e−SE[x,t f ,ti ] , (1.3.18)

Z(t f , ti) =∫

PBC

D[x(t)] e−SE[x,t f ,ti ] ; (1.3.19)

and it is straightforward to generalize generating functionals, correlation functions and

29


Wick Theorem1 in Euclidean time:

Z[J] =∫

boundaries

D[x(t)] e−(

SE[x,t f ,ti ]−∫

dt Ji(t)xi(t))

, (1.3.20)

〈 xi1(t1) · · · xin(tn) 〉 =1

Z[J]δ

δJi1(t1)· · · δ

δJin(tn)Z[J]

∣∣∣∣J=0

, (1.3.21)

〈 xi1 xi2 〉2 = (K−1)ij ,〈 xi1 · · · xin+1 〉2 = 0 ,〈 xi1 · · · xi2n 〉2 = ∑

permutations ofi1 ...i2n

(K−1)ip1ip2 · · · (K−1)ip(2n−1)ip2n .(Eucl. Wick Th.)

In the case of Harmonic oscillators Euclidean action becomes:

SE,osc[x, t f − ti] =m2

∫ β2

− β2dt (ẋ(t)2 + ω2x(t)2) ,

so integrating by parts2 we obtain the usual form S2 = 12 Kijxixj with Kij(t, t′) = m(− d

2

dt2 +ω2)δ(t− t′)δij. Now repeating the same procedure we did in real time for the propagator ofZ(β) we find that

(K−1)jk(t, t′) =δjk

mK−1(t− t′)

with

K−1(x) =∫ dk

2π1

k2 + ω2eikx ;

it’s clear that we don’t have any pole in the integration path, so a usual integration in thecomplex plane gives us

(K−1)ij(t, t′) =δij

2mωe−ω|t−t

′| .

Clearly the Euclidean propagator is

〈 xi(t)xj(t′) 〉E =δij

2mωe−ω|t−t

′| ,

the real time propagator is given by analytic continuation of the Euclidean one3:

〈 xi(t)xj(t′) 〉R =δij

2mωe−iω|t−t

′| .

1we recall that in Euclidean time we have SE2 =∫

dtdt′ 12 Kij(t, t′)xi(t)xj(t′).

2in the calculation of Z , where the integration by parts is allowed without boundary terms.3in that case absolute value is used to write the propagator in a compact way, but when making analytic

continuation we have to consider separately the cases t− t′ < 0 and t− t′ > 0, and then restore the compactnotation.

30


We note that the result obtained through analytic continuation is the same we would haveobtained with the so-called Feynman prescription

K−1(x) = −∫ dk

2π1

k2 −ω2 + ie eikx

for the treatment of poles on integration path. Let’s see now the physical meaning of Z(β)in Euclidean time: we have seen in Eq. (1.3.19) that it can be written in a path integral form,but if we remember its own definition1 and perform Wick rotation we obtain:

Z(β) ≡∫

dx 〈x|e−βH|x〉 = Tr e−βH ;

it is clear now that Z(β) is the partition function of a statistical system with an HamiltonianH at temperature T = (kβ)−1, where k is the Boltzmann constant.

So, at least in the asymptotic limit β → ∞, we know the 2-point propagator for thepartition function of Harmonic systems, and also 〈 1 〉: calculating

∫dxKosc(x, β2 ; x,−

β2 ),

we obtain (performing Wick rotation in Eq. (1.3.11) and then integrating),

〈 1 〉 =(

12(cosh(ωβ)− 1)

) D2

.

We have thus all we need to make perturbative calculations for a system with action SE =Sosc,E + Sint,E : consider for example a system of Harmonic oscillators subjected to a pertur-bation potential

Vint(x) = gijkxixjxk + λijklxixjxkxl ;

we can see the effect of that perturbation on the partition function for large β by means ofa perturbative series in g and λ: in that case Sint,E =

∫dt (gijkxixjxk + λijklxixjxkxl), so we

have2

Zint(β)= 〈 1 〉(

1−∫ β

2

− β2dt(

gijk〈 xi(t)xj(t)xk(t) 〉+ λijkl〈 xi(t)xj(t)xk(t)xl(t) 〉))

+O(g2 + λ2)

but since odd-point propagators vanish we see that the only first-order contribution to per-turbation is given by the quartic potential, in particular recalling that 〈 xi(t)xj(t) 〉E =δij/2mω, that λijkl has to be completely symmetric in the exchange of two indices (the even-tual antisymmetric part would vanish when contracted with four xs) and that 〈 1 〉 ≡Zosc(β), we obtain:

Zint(β) = Zosc(β)(

1− 3βλi k

i k4m2ω2

)+ O(g2 + λ2) = Zosc(β) e−β

3λ i ki k4m2ω2 + O(g2 + λ2) ,

1in the case of a time-independent Hamiltonian, i.e. when time translation is a symmetry of the system andZ(t f , ti) ≡ Z(t f − ti, 0).

2remembering that in Euclidean time:∫boundaries

D[x(t)] e−(

SE2+Sint,E) = 〈 1 〉〈 e−Sint,E 〉 = 〈 1 〉(

1 +∞

∑n=1

(−)nn!〈 Snint,E 〉

).

31


with λ i ki k ≡ λijklδijδkl .A useful application of this result is to calculate the first order effect of the anharmonic

perturbation on the ground-state energy of the system. In fact if we calculate the trace in thepartition function using a basis of Hamiltonian eigenfunctions we obtain:

Z(β) = ∑n〈n|e−βH|n〉 = ∑n e−βEn = e−βE0(

1 + ∑n e−β(En−E0))

β→∞∼ e−βE0 ;

so the ground-state energy shift is:

∆E0 = Eint0 − Eosc0 = limβ→∞− 1

βln(Zint/Zosc) =

34

λ i ki km2ω2

.

As a last simple example we will find propagators for a free theory, and use them tomake perturbative calculations in the case of a system of particles with the same masssubjected to an arbitrary external potential (using evolution time as the small parameter).In order to find propagators we will use directly Euclidean time, and in that case the ac-tion reads S f ree,E[x, β] =

∫ β0 dt

m2 ẋ(t)

2, with β ≡ (t f − ti) and recalling that since free La-grangian does not depend explicitly on time, time translation is a symmetry of the systemso K(x f , t f , xi, ti) ≡ K(x f , β; xi, 0). Since the free Lagrangian is also a quadratic function, wecan do the usual xcl-q split, remembering that in the Euclidean case we obtain:

K(x f , t f ; xi, ti) = e−S[xcl,E ,t f ,ti ]∫

VBC

D[q(t)] e−SE2[q,t f ,ti ] ; (1.3.22)

that reads in this case:

S[xcl,E , β] =m2

(x f − xi)2β

,

SE2[q, β] =∫ β

0dtdt′

12

mδij(−d2

dt2)δ(t− t′)qi(t)qj(t′) .

So we recognize again the usual form S2 = 12 Kijqiqj. But now in order to find (K−1)ij we will

proceed in a slightly different way: we perform the mode expansion we did in Section 1.3.2,so we can perform the integration over t in SE2, obtaining a form1

S2 = limN,P→∞

12

KijCiCj ,

where Kij depends only on discrete indices (it is a real matrix), and so it is much easier toinvert.

If one defines the Fourier expansion:

q(t) = limN→∞

N

∑n=1

√2β

Cn sin(nπt

β

)where Cn ∈ RD , (1.3.23)

1Ci ≡ Cin are Fourier coefficients of expansion, recalling notation of Section 1.3.2.

32


the integration over t in SE2[q, β] is straightforward, obtaining the usual form1

S2 = limN,P→∞

12

KijCiCj

with C ∈ RDN and

Knpij =mπ2n2

β2δijδ

np ,

so the inverse is immediately found as:

(K−1)ijnp =β2

mπ2n2δijδnp ,

and since integration over all possible paths vanishing at the boundaries is equivalent to anintegration over all Fourier coefficients Cin, we can find propagators of the Cs in a standardway obtaining:

〈 CinCjp 〉E =

β2

mπ2n2δijδnp ,

so using Eq. (1.3.23) we find

〈 qi(t)qj(t′) 〉E = limN,P→∞

N

∑n=1

P

∑p=1

2β

sin(πnt

β

)sin(πpt′

β

)〈 CinC

jp 〉E =

δij

m∆(t, t′) , (1.3.24)

with

∆(t, s) =∞

∑n=1

[ 2βπ2n2

sin(πnt

β

)sin(πns

β

)]. (1.3.25)

In order to perform calculations, we need to know the distributional meaning of ∆(t, s) inthe space of continuous functions on [0, β] vanishing at the boundaries, i.e. the algebra ofcontinuous functions with compact support κ([0, β]). Since the propagator (1.3.24) has tobe the inverse of the kinetic operator mδijδ(t− t′)(− d

2

dt2 ) we expect that ∆(t, s) ∈ κ([0, β])∗ 2

satisfies the equation

− d2

dt2∆(t, s) = δ(t− s) . (1.3.26)

Let’s see directly if it is the case: first of all we note that ∆(t, s) is symmetric in t and s, so ifEq. (1.3.26) holds, it holds even

− d2

ds2∆(t, s) = δ(t− s) ;

denoting a derivation of ∆ with a •, to the left of ∆ if we derive with respect to the firstargument, to the right if we derive with respect to the second, we have:

−••∆(t, s) = −∆••(t, s) =∞

∑n=1

2β

sin(πnt

β

)sin(πns

β

), (1.3.27)

1the cutoff N in the Fourier expansion is used in order to make K a finite DN × DN dimensional matrix.2κ([0, β])∗ is the dual space of κ([0, β]).

33


is it δ(t− s)? Yes, indeed: the definition of δ(t− s) as a functional in κ([0, β])∗ is

δ(t− s)[ f (t)] ≡∫ β

0dt δ(t− s) f (t) = f (s) with f (t) ∈ κ([0, β]) ;

so if we search the coefficients Dn(s) of a Fourier expansion

δ(t− s) =∞

∑n=1

√2β

Dn(s) sin(πnt

β

)we have to solve1

∑n,p2β

Dn(s)Fp∫ β

0dt sin

(πntβ

)sin(πpt

β

)= ∑n,p Dn(s)Fpδnp

⇒ Dn(s) =√

2β

sin(πns

β

),

⇒ δ(t− s) =∞

∑n=1

2β

sin(πnt

β

)sin(πns

β

)= −••∆(t, s) .

So a general solution of (1.3.26), symmetric in t and s and in κ([0, β])∗ is:

Analysis of

Distributions in

κ([0, β])∗ bymeans of Fourier

series

∆(t, s) =1β[(β− t)sθ(t− s) + (β− s)tθ(s− t)] ,

where θ(t− s) ∈ κ([0, β])∗ is defined as

θ(t− s)[ f (t)] ≡∫ β

0dt θ(t− s) f (t) =

∫ βs

dt f (t) with f (t) ∈ κ([0, β]) .

From its own definition θ(t− s) has a Fourier expansion

θ(t− s) = 2( ∞

∑n=1

1π2n

[cos(π2ns

β

)−1]

sin(π2nt

β

)+

∞

∑n=0

1π(2n + 1)

[cos(π(2n + 1)s

β

)+1]

sin(π(2n + 1)t

β

));

so it is directly checked that θ(0) ≡ θ(t− t) satisfies for f (t) ∈ κ([0, β]) :

θ(0)[ f (t)] ≡∫ β

0dt θ(0) f (t) =

12

∫ β0

dt f (t) ≡ 12[ f (t)] ,

where 1/2 is an element of κ([0, β])∗, so we can say that in κ([0, β])∗, θ(0) = 1/2 ; thereforewe obtain that

∆(t, t) = t− 1β

t2 (1.3.28)

as an element of κ([0, β])∗. Now that we know the behavior of ∆ in time integrals we canperform perturbative calculations for a wide class of physical systems. In fact in most cases

1expanding f (t) in a Fourier series with coefficients Fn.

34


of physical interest the quantum Hamiltonian function of the system isH = −m2∇2 + V(x),where ∇2 is the Laplace operator in D dimensions and V(x) ∈ C∞(RD), the infinitely dif-ferentiable functions in RD. The Euclidean action corresponding to such a system is simply:

SE(x, t f , ti) = S f ree,E(x, β) +∫ β

0dt V

(x(t)

),

where as usual β = t f − ti; so we are able to write

K(x f , t f ; xi, ti)= 〈 1 〉(

1 +∞

∑n=1

(−)nn!

∫ β0

dt1 · · ·dtn 〈 V(q(t1)+ xcl(t1)

)· · ·V

(q(tn)+ xcl(tn)

)〉)

with 〈 qi(t)qj(t′) 〉 = (δij/m)∆(t, t′) and xcl(t) = [(x f − xi)/β]t + xi . This is not yet aperturbative series, because we don’t have defined a controlling parameter that permits toapproximate the series when it is small. If we use β as a perturbative parameter we areinterested to write a series

K(x f , t f ; xi, ti) = a0(x f , xi) + a1(x f , xi)β + a2(x f , xi)β2 + . . . ;

we will not perform all the boring calculations here, we will limit to describe the methodol-ogy: first of all we expand the potential V(x) in a Taylor series around the initial point xi,obtaining a series1

V(x) = V(xi) + (qi(t) +ξ i

βt)∂iV(xi) +

12(qi(t) +

ξ i

βt)(qj(t) +

ξ j

βt)∂i∂jV(xi) + . . . ,

then we perform averages using the 2-point propagator 〈 qi(t)qj(t′) 〉 and finally solve in-tegrals remembering the distributional meaning of ∆(t, s); for example one has to solve theintegral∫ β

0dt ∆(t, t) =

∫ β0

dt(

t− 1β

t2)

=16

β2 .

As a mention we write here the values of the first three coefficients if x f = xi = x, see[14]for further details:

a0(x, x) = 〈 1 〉 =( m

2πβ

) D2

,

a1(x, x) = −( m

2πβ

) D2

V(x) ,

a2(x, x) =( m

2πβ

) D2(1

2V(x)2 − 1

12m∇2V(x)

).

1remembering that

x(t) = xcl(t) + q(t) = q(t) +x f − xi

βt + xi ;

and defining ξ = x f − xi.

35

Ch

ap

ter

2Quantum Fields or Quantum Particles?

“The vacuum current of a chargedDirac field [. . . ] can be related to the

dynamical properties of a ‘particle’with space-time coordinates that

depend upon a proper-timeparameter.”

Julian Schwinger

P revious chapter analyzed the usefulness of functional integration describing the timeevolution of a system of Quantum mechanical particles. We will see in this chapter thatpath integrals can be extended quite easily to Quantum Field Theories and provide an alter-native method to calculate propagators and derive Feynman rules in such an environment.Then we see that using the Schwinger proper-time representation we can perform Quan-tum Field Theory calculations1 through the path integral of a Quantum mechanical particlemoving in space-time2.

2.1: QFT Path Integrals

I n order to construct a functional integral formulation of Quantum Field Theory we needto have a deeper insight on the mathematical framework describing a relativistic system,the so-called “second quantization”, so we will introduce it briefly.

1in particular the calculation of Effective Actions.2these methods are often called “world-line path integral representations” for Effective Actions.

37


2.1.1: Fock Spaces

The Hilbert space in the Quantum mechanical description of a single particle depends onthe system we are considering: as an example a Schrödinger particle of spin one half hasH = L2(R3, dx; C2) ≡ L2(R3) ⊗ C2, where ≡ means natural isomorphism; i.e. the set ofpairs {ψ1(x), ψ2(x)} of square-integrable functions (dx is the Lebesgue measure). If wewant to deal with a relativistic system, in which the number of particles is not fixed, weneed to define a different framework, called Fock space[15]:

Definition 2.1.1 (Fock Space overH ). LetH be a Hilbert space andH n = H ⊗ · · · ⊗H ann-fold tensor product. SettingH 0 = C we define the Fock Space overH as:

F (H ) =∞⊕

n=0H n .

However F (H ) is too large, since quantum particles are indistinguishable, so we haveto restrict to two of its subspaces, the symmetric and antisymmetric Fock spaces. Let’s seehow to construct such subspaces: let Pn be the permutation group on n elements and {φk}be a basis for H . For each σ ∈ Pn, we define an operator (also denoted by σ) on basis

Fock Spaceselements ofH n by

σ(φk1 ⊗ · · · ⊗ φkn) = φkσ(1) ⊗ · · · ⊗ φkσ(n) ;

σ extends by linearity to a bounded operator of norm one on H n so we can define Sn =(1/n!) ∑σ∈Pn σ . Sn is an orthogonal projection and its range is called the n-fold symmetrictensor product of H . If H = L2(R3), SnH n is just the subspace of L2(R3n) of all functionsleft invariant under any permutation of the variables. So we can define

Definition 2.1.2 (Symmetric (or Boson) Fock Space overH ).

Fs(H ) =∞⊕

n=0SnH n .

Let e : Pn → {1,−1} which is one on even permutations and minus one on odd per-mutations. Define An = (1/n!) ∑σ∈Pn e(σ)σ ; then An is an orthogonal projection on H

n.AnH n is called the n-fold antisymmetric tensor product ofH . So we define

Definition 2.1.3 (Antisymmetric (or Fermion

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