Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

Feynman devised practical rules for computing

asymptotic expansions of path integrals, order by

order in perturbation theory. The rules are depicted

by graphs, known as the Feynman diagrams

(’t Hooft and Veltmann 1973). Feynman’s first

explicit nontrivial calculation was the Lamb shift.

It earned him the Nobel prize in 1965 (Feynman

1966). The diagram technique is widely used in

quantum mechanics and quantum field theory. The

time ordering provided by the time parameter in

quantum mechanics becomes, in quantum field

theory, a chronological ordering dictated by light

cones.

Another explicit calculation of a path integral [17]

uses the Taylor expansion of the action functional

S(x) around one of its values. It is known as the

background method (DeWitt 2004). It is called a

semiclassical WKB approximation when one expands

around an extremum S(x

)wherex

is a solution of

the Euler–Lagrange equation S

) = 0(Wenzel

1926, Kramers 1927, Brillouin 1926).

Introduced in 1951 (Morette (1951)) semiclassical

approximations are now the subject of a rich

literature reviewed briefly below.

Gaussian Volume Elements

A lesson from Gaussians on R

suggests a definition

of volume elements on infinite-dimensional Banach

spaces X.Let

ðaÞ :¼

dx exp 



jxj



for a > 0 ½19

dx ¼ dx

dx

and jxj

j¼1



¼ 

An elementary calculation gives I

(a) = a

D=2

. There-

fore, when D = 1,

ðaÞ¼

0if0< a < 1

1ifa ¼ 1

1 if 1 < a

(

½20

This is clearly an unsatisfactory situation, but it can

be corrected by introducing a dimensionless volume

element:

x :¼

D=2

dx

½21

The volume element D

x can be defined by the integral

x exp 



jxj

 2ih x

; xi



:¼ exp ajx



½22

where x

is in the dual R

of R

. Equa tion [22]

suggests the following generalization of Gaussians

on R

to Gaussians on a Banach space X:

s; Q

x exp 



QðxÞ



exp 2ihx

; xiðÞ

:¼ exp sWx

ðÞðÞ ½23

where s 2 {1, i}, Q(x) is a qua dratic form on X (see

condition on Q below). W(x

) is a quadratic form on

the dual X

of X, inverse of Q(x) in the following

sense. Set

QðxÞ¼hDx; xi and Wx

ðÞ¼hx

; Gx

i½24

where h, i is a duality product, for example, the

product of x 2 X and Dx 2 X

; then

DG ¼ 1

; GD ¼ 1

½25

Equation [23] defines a Gaussian volume element d

by its Fourier transform

F

s; Q

ðÞ:¼

d

s; Q

ðxÞexp 2ihx

; xiðÞ

:¼ exp sWx

ðÞðÞ ½26

where the Gaussian volume element

d

s;Q

ðxÞ

s;Q

ðxÞexp 



QðxÞ



½27

This is a qualified equality valid upon integration.

The definition of the Gaussian volume element by

its Fourier transform F is valid for s = 1 (Wiener

integral) when Q(x) > 0; it is valid for s = i

(Feynman integral) when ReQ(x) > 0.

Remark Volume elements were introduced with

the notation such as dx; later they were identified

with forms such as ! = dx.In[26] we omit d on the

left-hand side (LHS) for visual clarity.

Example (diagram expansion). The following inte-

grals follow readily (Cartier and DeWitt-Morette

2006) from the definition [26 ]. Let x

be in the dual

of X,

d

s;Q

ðxÞhx

; xi

2nþ1

¼ 0 ½28

d

s;Q

ðxÞhx

; xi

2n!

2



ðÞ

½29

d

s; Q

ðxÞhx

; xihx

; xi

2



; x

ð ÞWx

2n1

; x

ðÞ½30

436 Functional Integration in Quantum Physics

where

is a sum without repetitions of identical

terms.

For instance when n = 1, eqn [30] reads

d

s; Q

ðxÞhx

; xihx

; xi¼

2

; x

ðÞ½31

W(x

, x

) is called the two-point function (a.k.a. the

propagator). In a diagr am it stands for a line from

to x

Feynman diagrams represent Gaussian integrals of

polynomials.

For instance when n = 2, the diagram representa-

tion of [30] is the sum of three terms,

; x



; x



þ Wx

; x



; x



þ Wx

; x



; x



Example (Linear maps). Linear maps on R

are

limited to L : x !Ax, where A is a D  D constant

matrix. Linear maps on a Banach space X offer

many possibilities:

(i) Projections. For example, let x : T !R and

L : x 2 X ! xt

ðÞ; xt

ðÞ; ...; xt

ðÞ

2 R

Þ½32

This projection is a discretization of the path,

useful in particular in numerical calculations of path

integrals. Equation [32] is unambiguous, whereas

the limit of the discretized expres sion [11] is ill-

defined.

(ii) Liouville decomposition. For example, let D be

a second-order differential operator on a space of

paths x :[t

, t

] ! M

vanishing on the boundary,

x(t

) = 0, x(t

) = 0. Let {’

} be a complete, orthogo-

nal set of eigenfunctions of D, then the decomposi-

tion of x into the basis {’



ðtÞ¼

k¼1



ðtÞ½33

is a linear map

L : x 2 X ! u

; ...; u



2 R

It is useful in particular for diagonalizing (see,

e.g., [107]) the Green function G of D [25] (a.k.a.

the covariance in a Gaussian integral [24], or the

two-point function in [31]).

(iii) Volterra maps. For example, let L : X !Y by

yðtÞ¼

ds ðt  sÞxðsÞ

ðt  sÞ¼1 for s < t; 0 otherwise ½34

Let X be the space of square-integrable functions

on T and Y be an L

2, 1

space (square-integrable

function for which the first derivative is also square

integrable) then L maps the canonic al quadratic

form on X into the canonical quadratic form on Y,

hence the canonical Gaussian on X into the

canonical Gaussian on Y. The identity mapping i

from Y into the space C of continuous functions

maps the canonical Gaussian on Y into the Wiener

Gaussian on C (DeWitt-Morette et al. 1979).

The linear maps [32]–[34] and their obvious

generalizations have been used for computing

explicitly many funct ional integrals (see Figure 2).

The basic formula reads

d

ðxÞFðxÞ¼

d

ðyÞf ðyÞ; F ¼ f  L ½ 35

where the Fourier transform F is given by

F

¼F



L ½36

L is the transpose of the linear map L defined by

; xi¼hy

; Lxi½37

Computing F

does not require any calculation. It

can be read off eqn [36]. Com puting d

is easy in a

number of cases such as the following:

1. Y is finite-dimensional. In other words [35] is

a cylindrical integral. Then

d

ðyÞ¼dy

dy

detQ



1=2

 exp 



QðyÞ



½38

where Q(y) is an abbreviation of

ðyÞ¼Q

Yij

½39

its inverseW

ðy

Þin the sense of ½24–½25 is

ðy

Þ¼W

½40

that is given by [36]:

ðy

Þ¼W



L ½41

′

y ′

, Γ

′

, Γ

′

Figure 2 Linear maps. (Published with permission by Elsevier,

North Holland.)

Functional Integration in Quantum Physics 437

is the quadratic form defining 

by [26].

When D is small, say less than 4, and this is not an

unusual situation, it is easy to compute [38].

Example (Wiener Gaussians and Brownian

motion). The Wiener Gaussian on the space P

of pointed paths x : T ! R, T = [t

, t

], x(t

) = 0is

defined by its variance [26]:

Wðx

Þ :¼

ðt

ðsÞinfðt; sÞ½42

Let Y be the Wiener differential space consisting of

the differences of two consecutive values of x on the

n-discretized time interval. The space Y is finite

dimensional,

L : X ! Y

by y

¼ xt

jþ1



 xt



¼h

jþ1

 

; xi

It follows from [37] that



jþ1

 



and

d

ðxÞ¼dy

dy

j¼1

st



1=2

 exp 



x



t

½43

where t

:= t

jþ1

 t

and x

:= x(t

jþ1

)x(t

When s = 1 the Gaussian 

defines the distribu-

tion of a Brownian path. The Gaussian 

covariance inf(t , s) is the Wiener measure.

2. In semiclassical approximations, Q

is the

Hessian (second variation) of an action functional S:

ðhÞ¼

d

S xðÞðÞ



¼0

with h ¼

@

xðÞ



¼0

½44

where {x()} is a one-parameter family of paths [8].

The Jacobi field technology (the Jacobi operator is

defined by [103]; a Jacobi field is a solution of

[102]) yields the inverse of W

and its determinants

(Cartier and DeWitt-Morette 2006); they have been

worked out for a variety of boundary conditions on

classical paths.

Volume Elements Other than Gaussian

The definition [26]–[27] of Gaussian volume ele-

ments is a particular case of volume elements on a

Banach space  defined by



; Z

’  ð’; JÞ :¼ ZðJÞ½45

for ’ in , and J in the dual 

of . The volume

element D

, Z

is defined by two continuous bounded

functionals

:   

! C and Z:

! C ½46

In quantum field theory, ’ is a field and J is a

source. The functional Z(J) is then the Schwinger

generating functional for the n-point functions. An

axiomatic and applications of functional integrals

on  with volume elements D

, Z

can be found in

(Cartier and DeWitt-Morette (1993)).

Example (Poisson volume elements) (Cartier and

DeWitt-Morette (2006 ) and Collins (1997)). A

Poisson random variable is a random variable N

taking values in the set N of non-negative integers

such that the probability p

that N = n is

:¼ PrðN ¼ nÞ :¼ expðÞ



; 0 ½47

Thanks to the normalizing constant exp (),

n = 0

= 1. The parameter  is the mean value of N:

hNi¼ ½48

A record of fortuito us events occurring at random

times t

< T

 can consist either of the

number N(t) of events occurring at times less than

or equal to t, or of the waiting times

¼ T

 T

k1

½49

between two consecutive events.

When the waiting times are stochastically inde-

pendent and when

Pr t < W

< t þ dtðÞ¼p

ðtÞdt ½50

ðtÞ¼a expðatÞ; t > 0 ½51

the record is a Poisson random variable. It is related

to the number of events N(t) as follows.

Let T be a finite time interval [t

, t

], and

¼ Nðt

ÞNðt

Þ½52

the number of events during T. The random variable

follows a Poisson law [47] with mean value



ðTÞ¼aðt

 t

Þ½53

For mutually disjoint time intervals T

(1)

, T

(2)

, ... the

random variables N

(1)

, N

(2)

, ... are stochastically

independent.

Whereas the parameter  must be real non-

negative, the parameter a can be pure imaginary;

therefore, Poisson processes defined by waiting

438 Functional Integration in Quantum Physics

times can be used in quantum physics as well as in

probability. When a is real, it is called the decay

constant because its physical dimension is [time]

1

A Poisson path x 2 X

is characterized by n jumps

and the jump times during a given time inte rval

T = [t

, t

](Figure 3 illustrates a Poisson path

in X

). The space X of Poisson paths is the union

of all X

X ¼[X

½54

One can define a volume element D

a, T

on X by its

Fourier transform:

a;T

x  exp ihx; f iðÞ:¼ exp

dtae

if ðtÞ



½55

Here a path x 2 X

, characterized by n jump times

, ..., T

, is represented by the sum



þþ

Hence

hx; f i¼fT

ð ÞþþfT

ðÞ ½56

The dimensionless volume element on T is

dvðtÞ¼a d t

Therefore,

volðTÞ¼aT; T ¼ t

 T

volðX

Þ¼a

=n! ½57

volðXÞ¼exp volðTÞðÞ ½58

and it makes sense to write formally

X ¼ exp T

It can be proved that the volume element D

a, T

x is a

measure, in the technical sense of the word (Cartier

and DeWitt-Morette 2006).

Functional integration on spaces of Poisson paths

have been used extensively in solutions of Klein–

Gordon equations, the telegrapher equation and the

Dirac equation (Cartier and DeW itt-Morette 2006).

Other volume elements of interest in quantum

physics include (LaChapelle 2004):



gamma volume elements, which are to gamma

probability distributions what Gaussian volume

elements are to Gaussia n probability distribu-

tions; and



Hermite volume elements convenient for integrat-

ing Wick-ordered polynomials.

ADirac‘‘-function’’ is formally the limit of a

Gaussian integral. Formally, one can introduce a Dirac

functional volume element as the limit of a Gaussian

volume element.

The Koszul Formula

There are several roadblocks on the road from finite

to infinite-dimensional spaces. For instance, a

volume in a D-dimensional space is a top-differential

form, that is, a D-form. There is no top-form in an

infinite-dimensional space – neither on Grassmann

manifolds since Grassmann forms are totally sym-

metric tensors. A D-form in R

has only one strict

component and is equivalent to a scalar density of

weight 1, but scalar densities of weight 1 do not

form an algebra.

For these reasons, volume elements have so far

been defined by integrals [26], [27], [43], and [55].

Short of giving an explicit expression for their

differential forms, one can require them to satisfy

the Koszul formula

! ¼ DivðXÞ! ½59

where ! is a vo lume element on a Banach space X,

X a vector field generating a group of transforma-

tions on X, L

the Lie derivative defined by X, and

Div(X) the standard generalization of div(ergence)

on finite-dimensional spaces (see, e.g., Cartier and

DeWitt-Morette (2006) for the explicit expression of

divergences on Riemannian, symplectic, Grassmann

manifolds). The Koszul formula dictates how a

volume element changes under a group of

transformations.

It often happens that an object cannot be defined

per se, but that it is sufficient to define its variation. For

example, one does not define potentials, but potential

differences; the ratio of infinite-dimensional determi-

nants can be defined without defining each determi-

nant; the work of Wiener on ‘‘differential-spaces,’’

which is a landmark in functional integration, is based

N(t )

(0)

Figure 3 A Poisson path in X

Functional Integration in Quantum Physics 439

on differences between two consecutive values of a

function, etc. Similarly, the Koszul formula does not

define ! but gives its variation L

The Operator Formalism of Quantum

Physics

Functional integrals can be used to represent

operator matrix elements, and solutions of the

Schro¨ dinger equation.

1. Matrix elements of operators on Hilbert spaces.

Symbolically,

hjexp iHt=hðÞji¼



Dx exp iSðxÞ=hðÞ½60

The domain of integration X



is a space of paths

x on [t, 0] satisfying initial conditions that

characterize the quantum state , and final

conditions that characterize the quantum state .

The action functional S yields the Hamiltonian H.

A key property of path integrals is their

representations of matrix elements of time-

ordered operators. The path parameter (time,

scale, or any other parameter) provides the

operator ordering [11]. A simp le example is the

two-point function of the Wiener measure [42]:

dðxÞxðtÞxðsÞ¼infðt; sÞ½61

The function integral orders the time, that is, the

argument of the variable of integration. In

quantum field theory, time ordering becomes a

chronological ordering dictated by light cones.

2. Schro¨ dinger equation and other parabolic equa-

tions (Cartier and DeWitt-Morette 2006).

The following theorem provides the mathematical

underpinning for a great variety of functional inte-

grals. It also provides a construction of functional

integrals, which begins with the symmetries of a given

physical system rather than its action functional. The

theorem consists of two parts: the definition of a

functional integral, and the partial differential equa-

tion satisfied by the value of the functional integral, as

a function of a set of parameters.

Given a manifold M, consider the contractible

space P

M of pointed L

2, 1

paths over T = [t

, t

x : T ! M; e:g:, xðt

Þ¼x

;

i:e:, x 2P

M ½62

Given D þ 1 vector field Y,{X

()

}, generators of

group of transformations on M, define a map

P : P

M by z ! x ½63

explicitly

dxðt; zÞ¼X

ðÞ

xðt; zÞðÞdz



þ Yxðt; zÞðÞdt ½64

xðt

; zÞ¼x

; zðt

Þ¼0 ½65

In general, the vector fields do not commute and the

solution of [64]–[65] is of the form

xðt; zÞ¼x



ðt; zÞ½66

where (t, z) is an element of a group of right

actions on M, defined by the D þ 1 generators Y,

()



t þ t

; z  z

ðÞ¼x



ðt; zÞ

ðt

; z

The path z defined on [t

, t] is followed by the path

on [t, t

Consider the following functional integral over

of a functional of paths on P

ðÞðx

Þ :¼

s;Q

z exp 



QðzÞ



  x



ðt; zÞ



½67

where

QðzÞ¼

dth





ðtÞ



ðtÞ½68

The functional (U

)atx

is a function (T, x

). It

is a solution of the generalized Schro¨dinger equation,

@





ðÞ

ðÞ

 þL

 ½69

This equation is valid on manifolds M (e.g., frame

bundles, U(N) bundles, multiply connected spaces,

symplectic manifold phase space) in arbitrary sys-

tems of coordinates.

Example (Polar coordinates on R

). Let us

abbreviate z



(t)toz



, x

(t)tor,andx

(t)to.It

follows from

¼ r cos ; z

¼ r sin  ½70

that

dr ¼ cos   dz

þ sin   dz

¼: X

ð1Þ

þ X

ð2Þ

d ¼

sin 

cos 

¼: X

ð1Þ

þ X

ð2Þ

½71

440 Functional Integration in Quantum Physics

The dynamical vector fields are, therefore,

ð1Þ

¼ cos 



sin 

@

½72

ð2Þ

¼ sin 

cos 

@

½73

Here h



= 



and eqn [69] reads

@

4

@



 ½74

This example is trivial because x(t, z) is not a

functional of z but a function of z(t) given by [70].

In the following example, x(t, z) is a functional of z.

Example (Paths with values on a Riemannian

manifold (M

, g)). Consider the frame bundle over

and a connection  defining the horizontal lift

(t) of a vector

x(t),

ðtÞ¼ððtÞÞ 

xðtÞ½75

In order to bring eqn [75] in the form [64], we think

of a frame u(t) as a linear map from R

into the

tangent space T

x(t)

uðtÞ: R

! T

xðtÞ

½76

Let

zðtÞ :¼ uðtÞ

1

xðtÞ½77

Choose a basis {e

(A)

}inR

and {e

()

}inT

x(t)

such that

zðtÞ¼

ðtÞe

ðAÞ

¼ uðtÞ

1



ðtÞe

ðÞ

Þ½78

Insert u(t)  u(t)

1

into [75], then

ðtÞ¼X

ðAÞ

ðtÞðÞ

ðtÞ½79

where the dynamical vector fields are

ðAÞ

ððt ÞÞ ¼ ðððtÞÞ  uðtÞÞ  e

ðAÞ

½80

The construction [64]–[69] gives a parabolic equation

on the bundle. If the connection  is the metric

connection, then the parabolic equation on the bundle

gives, by projection on the base space, the parabolic

equation with the Laplace–Beltrami operator. Expli-

citly, the projection on the base space of [67] is

; x

ðÞ:¼

s; Q

ðzÞexp 



QðzÞ



  Dev zðÞðt

ÞðÞ ½81

where Dev is the Cartan development map, namely

the bijection, defined by [82], from the space of

pointed paths z on T

(identified to R

via the

frame u

) into the space of pointed paths x on M

(paths such that x(t

) = x

  ðÞðtÞ¼: Dev zðÞðtÞ½82

 is the projection on the base space. The path

integral [81] is the solution of the equations

ðt

; x

Þ¼

4

 ðt

; x

Þ½83

ðt

; xÞ¼ðxÞ½84

where  is the Laplace–Beltrami operator on (M

, g),

 ¼ g

½85

and D

is the covariant derivative defined by the

Riemann connection .

Semiclassical Expansions

Classical mechanics is a limit of quantum

mechanics; therefore, it is natural to expand the

action functional S of a given system around, or

near, its classical value – namely its minimum S(q),

where q is a solution of the Euler–Lagrange

equation,

ðqÞ¼0 ½86

Set

SðxÞ¼SðqÞþS

ðqÞ þ

ðqÞ

000

ðqÞ þ ½87

where x 2 X is a path

x : T ! M

and ,  2 T

X is a vector field at q 2 X. The second

variation of S is called its Hessian

ðqÞ ¼: Hess q; ; ðÞ ½88

The arena of semiclassical expansions of a func-

tional integral schematically written as

I ¼

a;b

Dx exp iSðxÞ=hðÞð xt

ðÞðÞÞ½89

consists of the intersection U

a, b

of two spaces

a, b

 X the space of paths satisfying D initial

conditions (a) and D final conditions (b), and

(S) the space of critical points of S

q 2 U

ðSÞ; S

ðqÞ¼0 ½90

a;b

:¼ X

a;b

\ U

ðSÞ ½91

Functional Integration in Quantum Physics 441

The nature of the intersection U

a, b

determines the

behavior of the system S. Figure 4 shows the

intersection of the space X

a, b

of paths on M

with

fixed points. It also shows the space U

(S)of

critical points of S.

We consider first the case in which U

a, b

consists

of a single point q, or several isolated points q

(i)

. The

semiclassical expansion consists in dropping the

terms beyond the Hessian:

WKB

:¼

D exp

2i



SðqÞþ

ðqÞ



 ð xðt

ÞÞ ½92

where the initial wave function  accounts for the D

initial conditions of the system, and X

is the space

of pointed paths

xðt

Þ¼x

; and ðt

Þ¼0 for every x 2 X

½93

WKB Approximations

The integral I

WKB

is the Gaussian defined by the

Hessian. Explicit calculations of I

WKB

exploit the

power of Jacobi fields of S at q.

Example (Momentum-to-position transitions) (e.g.,

Cartier and DeWitt-Moret te 2006). We have

WKB

ðx

; t

; p

; t

Þ¼exp

2i

Sqðt

Þ; pðt

ÞðÞ

 det

ðt

Þ@p

ðt



1=2

½94

where S is the action function (a.k.a. Hamilton’s

principal function)

Sqðt

Þ; pðt

ÞðÞ¼SðqÞþhp

; xðt

Þi ½95

where the classical path q is characterized by its

initial momentum p

and its final position x

. The

proof of [94] rests on the following property of

quadratic forms Q. Let L : X ! Y linearly and

¼ Q

 L ½96

According to the notations used in [26 ], [27] ,

ðxÞexp 



ðxÞ



¼ 1 ½97

According to [35], [27],

1 ¼

ðxÞexp 



ðxÞ



ðLxÞexp 



ðLxÞ



½98

¼jdetLj

ðxÞexp 



ðxÞ



½99

If s = 1, that is, if Q

and Q

are positive definite,

then

ðxÞ exp Q

ðxÞ

ðÞ

¼¼ det Q

ðÞ

1=2

½100

If s = i, that is, for Feynman integrals

ðxÞexp Q

ðxÞðÞ

¼ det Q

ðÞ

1=2

IndðQ

½101

where ‘‘Ind(Q

)’’ is the ratio of the numbers of

negative eigenvalues of Q

and Q

respectively, and

i =

ﬃﬃﬃﬃﬃﬃ

1

= e

i=2

Equation [100] is a key equation for semiclassical

expansions where it is convenient to break up the

second variation S

(q) into two quadratic forms:

ðqÞ ¼ Q

ðÞþQð Þ½102

where Q

is the kinetic energy. The quadratic form

is a convenient Gaussian volume element for

computing [92]. Moreover, splitting the Hessian

into Q

þ Q corresponds to splitting the system into

a ‘‘free’’ system and a perturbation.

In eqns [100] and [101] the determinant of the

ratios of the infinite-dimensional quadratic forms

have been shown (Cartier and DeWit t-

Morette 2006) to be a finite-dimensional determi-

nant, thanks to Jacobi field technology.

Degenerate Hessians; Beyond WKB

When U

a, b

consists of isolated points, the Hessian is

not degenerate, and the semiclassical expansion is

usually called the (strict) WKB approximation.

When the Hessian is degenerate,

ðqÞ ¼ 0 for  6¼ 0 ½103

(S )

a, b

: U

(S )

a, b

Figure 4 Intersection of the space P

a, b

(abbreviated to

a, b

) of paths on M

with fixed points, and the 2D-dimensional

space U

(S) of critical points of the system S. (Adapted from a

Plenum Press publication with permission by Springer-Verlag.)

442 Functional Integration in Quantum Physics

there is at least one nonzero Jacobi fie ld h along q,

ðqÞh ¼ 0; h 2 T

ðSÞ ½ 104

with D vanishing initial conditions (a)andD

vanishing final conditions (b). Equation [104] is

the defining equation of Jacobi fields. The vanishing

boundary conditions imply that h 2 T

a, b

as well

as being a Jacobi field.

For understanding the intersections U

a, b

when the

Hessian is degenerate, one can construct the follow-

ing basis for the intersecting tangent spaces

(S) and T

a, b



Basis for T

(S): a complete set (if it exists) of

linearly independent Jacobi fields. It can be

constructed by varying the 2D conditions (a), (b)

satisfied by q 2 X

a, b



Basis for T

a, b

: a complete set of orthonormal

eigenvectors {

} of the Jacobi operator J(q)

defined by the Hessian

ðqÞ ¼: hhJðqÞ;i;i½105

JðqÞ

¼ 



; k 2f0; 1; ...g½106

The basis {

} diagonalizes the Hessian. When

the Hessian is degenerate, there is at least one

eigenvector of J(q) with zero eigenvalue.

1. The intersection U

a, b

is of dimension l > 0. Let

} be the coordinates of  in the {

} basis of

a, b

. Then the diagonalized Hessian is

ðqÞ ¼

k¼0



ðu

½107

There are l zero eigenvalues {

} when the system of

Euler–Lagrange equations decouples (possibly after

a change of variable in X

a, b

) into two sets: l

constraint equations, and D  l equations determin-

ing D  l coordinates {q

}ofq. Say l = 1, for

simplicity. Then

SðxÞ¼Sðq Þþc

k¼1



ðu

þ Oðjuj

Þ½108

where

S

q

ðtÞ



ðtÞ½109

The change of variable  ! {u

} is a linear change

of variable of type [33]. The integral [92]

decomposes into the product of an ordinary integral

over u

and a Gaussian functional integral defined

by a nondegenerate quadratic form. The integral

over u

yields a Dirac -function, (c

=h). The

propagator vanishes unless the conservation law

= 0 is satisfied.

Conservation laws appear in the classical limit of

quantum physics. The quantum system may have

less symmetry than its classical limit.

2. The intersection U

a, b

is a multiple root of the

Euler–Lagrange equation. The flow of classical

solutions has an envelope, known as a caustic.

Caustics abound in physics: the soap bubble

problem, scattering of particles by a repulsive

Coulomb potential (see Figure 5), rainbow scatter-

ing from a source at infinity, glory scattering etc.

(Cartier and DeWitt-Morette 2006).

Let us consider a specific example for simplicity.

For instance, the scattering of particles of given

momenta p

by a repulsive Coulomb potential.

Let q and q



be two solutions of the Euler–

Lagrange equation with slightly different boundary

conditions at t

.ComputeI(x



, t

; p

, t

)byexpand-

ing the action functional not around q



but around q.

The path q



is not in X

a, b

and the expansion of the

action functional has to be carried up to and including

the third variation. As before, let {u

}bethe

coordinates of  in the base {

}, k 2 {0, 1 ...}. The

integral over u

is an Airy integral



1=3

Ai 

1=3



exp i cu







½110

where

 ¼



h



q



ðrÞq



ðsÞq



ðtÞ

 



ðrÞ



ðsÞ



ðtÞ

½111

c ¼

2

S

qðtÞ

 

ðtÞ x



 x



½112

Figure 5 A flow of particles scattered by a repulsive Coulomb

potential. (Reprinted from Physical Review D with permission by

the American Physical Society.)

Functional Integration in Quantum Physics 443

The leading contribution of the Airy function when

h tends to zero can be computed by the stationary

phase method. When x



is in the ‘‘illuminated’’

region, the probability amplitude I(x



, t

; p

, t

)

oscillates rapidly as h tends to zero. When x



is in

the ‘‘dark’’ region, the probability amplitude decays

exponentially. Quantum mechanics softens up the

caustics.

The two kinds of degeneracies described in

sections (1) and (2) may occur simultaneously. This

happens, for instance, in glory scattering for which

the cross section, to leading terms in the semiclassi-

cal expansions, has been obtained by functional

integration in closed form in terms of Bessel

functions (Cartier and DeWitt-Morette 2005).

3. The intersection U

a, b

is the empty set. There is

no classical solution corresponding to the quantum

transition. This phenomenon, called ‘‘tunneling’’ or

‘‘barrier penetration,’’ is a rich chapter of quantum

physics which can be found in most of the books

listed under ‘‘Further reading.’’

A Multipurpose Tool

Functional integration provides insight and techni-

ques to quantum physics not available from the

operator formalism. Just as an example, one can

quote the section ‘‘Beyond WKB’’ which has often

been dismissed in the operator formalism by stating

that ‘‘WKB breaks down’’ in such cases.

The power of functional integration stems from

the power of infinite-dimensional spaces. For

instance, compare the Lagrangian of a system with

its action functional

SðxÞ¼

dtL

xðtÞ; xðtÞðÞ; x 2 X

a;b

x : T ! M

; S : X

a;b

! R ½113

A classical solution q of the system can be defined

either by a solution of the Euler–Lagrange equation,

together with the boundary conditions dictated by

q 2 X

a, b

or by an extremum of the action func-

tional, S

(q) = 0. The path q is a significant point in

a, b

but it is not isolated and the Hessian S

(q)

gives much information on q, such as conservation

laws, caustics, tunneling.

A list of applications is beyond the scope of this

article. We treat only two applications, then give in

the ‘‘Further reading’’ section a short list of books

that develop such applications as polarons, phase

transitions, properties of quantum gases, scattering

processes, many-body theory of bosons and fer-

mions, knot invariants, quantum crystals, quantum

field theory, anomalies, etc.

The Homotopy Theorem for Paths Taking Their

Values in a Multiply-Connected Space

The space X

a, b

of paths x

x: T ! M

; x 2 X

a;b

probes the global properties of their ranges M

When M

is multiply connected, X

a, b

is the sum of

distinct homotopy classes of paths. The integral over

a, b

is a linear combination of integrals over each

homotopy class of paths. The coefficients of this

linear combinati ons are provided by the homotopy

theorem.

The principle of superposition of quantum states

requires the probability amplitude for a given

transition to be a linear combination of probability

amplitudes. It follows that the absolute value of the

probability amplitude for a transition from the state

a at t

to the state b at t

has the form

Kb; t

; a; t

ðÞ



ðÞK



b; t

; a; t

ðÞ



½114

where K



is the interval over paths in the same

homotopy class. The homotopy theorem (Laidlaw

and Morette-DeWitt 1971) and (Schulman 1971)in

Cartier and DeWitt-Morette (2006)) states that the

set {()} forms a representation of the fundamental

group of the multiply connected space M

. One

cannot label a homotopy class by an element of the

fundamental group unless one has chosen a point

c 2 M

and a homotopy class for paths going from

c to a an d for paths going from c to b – in brief,

unless one has chosen a homotopy mesh on M

The fundamental group based at c is isomorphic to

the fundamental grou p based at any other point of

but not canonically so. Therefore, eqn [114] is

only an equality between absolute values of prob-

ability amplitudes. The proof of the homotopy

theorem consists in requiring [114] to be indepen-

dent of the chosen homotopy mesh.

Application: Systems of n-Indistinguishable

Particles in R

In order that there be a one-to-one correspondence

between the system and its configuration space,

x : T ! R

Dn

¼: R

D;n

where S

is the symmetric group for n permutations;

the coincidence points in R

D, n

are excluded so that

acts effectively on R

D, n

. Note that R

1, n

is not

connected, but R

2, n

is multiply connected. When

D  3, R

D, n

is simply connected and the fundamen-

tal group on R

D, n

is isomorphic to S

444 Functional Integration in Quantum Physics

There are only two scalar unitary representations

of S



:  2 S

! 1 for all permutations 



:  2 S

1 for even permutations

1 for odd permutations



Therefore, in R

there are two different propagators

of indistinguishable particles:

bose





ðÞK



½115

is a symmetric propagator

fermi





ðÞK



½116

is an antisymmetric propagator.

The arguments leading to the existence of (scalar)

bosons and fermions in R

fails in R

. Stat istics

cannot be assigned to particles in R

; particles

‘‘without’’ statistics have been called anyons.

Application: a Spinning Top

Schulman’s analysis of the Schro¨ dinger equation for

a spinning top (Schulmann 1968) motivated the

formulation of the homotopy theorem. Therefore,

Schulman’s results can easily be form ulated as an

application of [114].

Application: Instantons (DeWitt 2004)

The homotopy theorem reformulated for functional

integrals applies to the total houtjini amplitude of

instantons in Minkowski spacetime.

Scaling Properties of Gaussians

We rewrite the definition [26] of Gaussian volume

elements as

d

ðxÞexp 2ihx

; xiðÞ:¼ exp iWðx

ÞðÞ½117

where the covariance G is defined by the variance W,

Wðx

Þ¼hx

; Gx

In quantum field theory the definition [26] reads



d

ð’Þexp 2ihJ;’iðÞ:¼ exp iWðJÞðÞ½118

where ’ is a field on spacetime (Minkowski, or

Euclidean) and J is called the source. A Gaussian 

can be decomposed into the convolution of any

number of Gaussians. For example, if

W ¼ W

þ W

! G ¼ G

þ G

½119

then



¼ 

 

½120

Explicitly, in QFT



d

ð’Þexp 2ihJ;’iðÞ



d

’

ðÞ

d

’

ðÞ

 exp 2ihJ;’

þ ’

iðÞ½121

where

’ ¼ ’

þ ’

½122

The additive property [119] makes it possible to

express a covariance G as an integral over an

independent scale variable.

Let  2 [0, 1] be an independent scale variable.

(some authors use  2 [1, 1[ and 

1

2 [0, 1[). A

scale variable has no physical dimension:

½¼0 ½123

The scaling operator S



acting on a function f of

length dimension [f ] is by definition



f ðxÞ :¼ 

½f 

fx=ðÞ ½124

the scaling of an interval [a, b[ is given by



[a, b[ = {s=js 2 [a, b[}, that is,



½a; b½¼½a=; b=½½125

The scaling of a functional F is



FðÞð’Þ¼FS



’ðÞ ½126

In order to decompose a covariance into an integral

of scale-dependent contributions we note that a

covariance G is a two-point function [31].In

quantum field theory [118], the engineering lengt h

dimension of G is twice the field dimension

½G¼2½’½127

Let x, y 2 spacetime and G be a Laplacian Green

function. One can introduce a scaled (truncated)

Green function

½l

;l½

ðx; yÞ :¼



s=l

u jx  yjðÞ½128

where

½l¼1; ½s¼1; d



s ¼ ds=s

 l; ½u¼½G½129

such that

lim

¼0;l¼1

½l

;l ½

ðx; yÞ¼Gðx; yÞ½130

Functional Integration in Quantum Physics 445