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

Подождите немного. Документ загружается.

on the possibility of mapping its Hamiltonian in a

system of fr ee bosons. Such a mapp ing is not

possible for the Hamiltonian [3], which is not

solvable; however, one can use renormalization

group methods and suitable Ward identities to

show that its behavior is similar to the Luttinger

model (in a sense, one makes perturbation theory

not around the free Fermi gas, but around the

Luttinger model).

If we take into account the interaction with an

external periodic potential with period a, that is,

consider u 6¼ 0, we find that if k

6¼ n=a, then the

Schwinger function behaves essentially like [8].On

the contrary, in the filled-band case, k

= n=a, one

finds that there is still an energy gap which becomes

O(u

1þ

) with 

= O(); this means that the

renormalization of the gap is described by a critical

index; moreover, S( x) ’ O(e

juj

1þ

jxj

). A similar

behavior is also observed in the presence of qua si-

periodic potential. In the attractive case, <0,

u = 0, the behavior is much less understood; it is

believed that the inte raction produces a gap 



the spectrum which is nonanalytic in , and S(x)

shows an exponential decay rather than a power-

law decay, and the interaction converts the system

from a metal to an insulator.

Finally, it is remarkable that a large variety of

models, like Heisenberg spin chains or bidimensional

classical statistical mechanics models, such as the

eight-vertex or the Ashkin–Teller model, can be

mapped into interacting d = 1 fermionic systems, and

consequently their critical behavior can be understood

by using fermionic techniques (see Gentile and

Mastropietro (2001), and references therein).

Superconductors

The theory up to T = 0 for d = 2, 3 systems with

dispersion relation jkj

=2m is based only on

approximate computations, predicting the phenom-

enon of superconductivity. According to the theory

of Bardeen, Cooper, and Schrieffer (BCS theory),

the interaction between fermions leads to the

formation of a gap in the energy spectrum, below

the critical temperature. There are many ways to

derive the BCS theory. One is based on the fact that

one verifies, by perturbative computations, that the

effective interaction is stronger when the four

momenta of the fermions are such that k

’k

and k

’k

. This suggests, heuristically, to

replace in [7]  with



BCS

¼

L

k;k

k;

k;



;



k

;

which is an interaction be tween p airs of electrons

with opposite spin and momenta, which a re called

Cooper pairs. R eplacing  with 

BCS

has the great

advantage that it makes the Schwinger functions

exactly computable and explains the mechanism

of superconductivity in many metals (but not in

the recently discovered high-T

superconductors).

On the other hand, proving that [7] with  or 

BCS

has a similar behavior is still an important open

problem. The two-point Schwinger function in the

model with 

BCS

can be written, after the so-called

Hubbard–Stratonovitch transformation, as

; kðÞ¼L

ik

"ðkÞþ

þ"

ðkÞþu

L

ðuÞ

L

ðuÞ

½11

where (u) is a function with a global minimum in

u = 0 for repulsive interactions <0, whereas for

>0 and sufficiently small temperatures (for

T  T

,withT

= O(e

a=jj

)), it has the form of a

double well with two minima at u =  



with





= O(e

a=jj

); for T greater than T

, there is only

a global minimum at u = 0. By the saddle-point

theorem, we find, for T  T

and <0,

lim

L!1

SðkÞ¼

ik

 "ðkÞþ

þð"ðkÞÞ

þ 



½12

The physical properties predicted by [12] are

completely different with respect to the free case:

the occupation number is continuous, there is an

energy gap in the spectrum , the specific heat is

O(e





) and the phenomenon of superconduc tivity

appears. The fact that the interaction generates a

gap is called mass generation; a similar mechanism

appears in particle theory.

Conclusions

Many other physical phenomena, observed experi-

mentally, can be essentially understood by studying

fermionic systems, but a clear mathematical com-

prehension is still lacking. We mention: the Kondo

effect, that is, the resistance minimum observed in

some metals due to magnetic impurities; Mott

transition, in which a strong interaction produces

an insula ting state in a system which should be

conductors; antiferromagnetism; fractional quantum

Hall effect, and many others. We can say that the

situation in this area of study reminds one of the

classical mechanics at the end of the nineteenth

century; there is agreement on the models to

consider, which are believed to be able to take into

306 Fermionic Systems

account the marvelous properties of the matter

experimentally found, but to extract information

from them requires deeper and compl ex analytical

and mathematical investigations.

See also: Falicov–Kimball Model; Fractional Quantum

Hall Effect; Quantum Statistical Mechanics: Overview;

Renormalization: Statistical Mechanics and Condensed

Matter.

Further Reading

Abrikosov AA, Gorkov LP, and Dzyaloshinskii IE (1965) Methods of

Quantum Field Theory in Statistical Physics. New York: Dover.

Anderson PW (1985) Basic Principles in Solid State Physics.

Menlo Park, CA: Benjamin Cummings.

Anderson PW (1997) The Theory of Superconductivity in High T

Cuprates. Princeton, NJ: Princeton University Press.

Bardeen J, Cooper LN, and Schrieffer JR (1957) Theory of

superconductivity. Physical Review 108: 1175–1204.

Benfatto G and Gallavotti G (1995) Renormalization Group.

Princeton: Princeton University Press.

Berezin FA (1966) The Method of Second Quantization.

New York: Academic Press.

Bratelli O and Robinson D (1979) Operator Algebras and

Quantum Statistical Mechanics. Berlin: Springer.

Feldman J, Knorrer H, and Trubowitz E (2002) Fermionic Functional

Integrals and Renormalization Group, CRM Monograph Series,

vol. 16. Providence: American Mathematical Society.

Gallavotti G (1985) Renormalization group and ultraviolet

stability for scalar fields via renormalziation group methods.

Reviews of Modern Physics 57: 471–562.

Gentile G and Mastropietro V (2001) Renormalization group for

one dimensional fermions. A review of mathematical results.

Physics Reports 352: 273–437.

Mahan GD (1990) Many-Particle Physics. New York: Plenum.

Mattis DC and Lieb E (1966) Mathematical Physics in One

Dimension. New York: Academic Press.

Negele JW and Orland H (1988) Quantum Many Particle

Systems. New York: Addison-Wesley.

Pastur L and Figotin A (1991) Spectra of Random and Almost

Periodic Operators. Berlin: Springer.

Pines D (1961) The Many Body Problem. New York: Addison-

Wesley.

Rivasseou V (1994) From Perturbative to Constructive Renorma-

lization. Princeton, NJ: Princeton University Press.

Feynman Path Integrals

S Mazzucchi, Universita

di Trento, Povo, Italy

Introduction

In nonrelativistic quantum mechanics, the state of

a d-dimensional particle is represented by a

unitary vector in the complex separable Hilbert

space L

), the s o-called ‘‘wave function,’’ while

its time evolution is described by the Schro¨dinger

equation:

ih

¼

h

 þ V

ð0; xÞ¼

ðxÞ

½1

where h is the reduced Planck constant, m > 0 is the

mass of the particle, and F = rV is an external

force.

In 1942 R P Feynman, following a suggestion by

Dirac, proposed an alternative (Lagrangian) formu-

lation of quantum mechanics, and a heuristic but

very suggestive representation for the solution of eqn

[1]. According to Feynman, the wave function of the

system at time t evaluated at the point x 2R

given as an ‘‘integral over histories,’’ or as an

integral over all possible paths  in the configuration

space of the system with finite energy passing at the

point x at time t:

ðt ; xÞ¼

fjðtÞ¼xg

ði=hÞS



ðÞ

D

1



fjðtÞ¼xg

ði=hÞS

ðÞ

ðð0ÞÞD ½2

() is the classical action of the system evaluated

along the path 

ðÞS



ðÞ

VððsÞÞds ½3



ðÞ

ðsÞj

ds ½4

D is a heuristic Lebesgue ‘‘flat’’ measure on the

space of paths and (

{j(t) = x}

(i=h)



()D)

1

is a

normalization constant.

Some time later, Feynman himself extended

formula [2] to more general quantum systems,

including the case of quantum fields.

The Feynman path-integral formulation of quan-

tum mechanics is particularly suggestive, as it

provides a spacetime visualization of quantum

dynamics, reintroducing in quantum mechanics the

concept of trajectory (which was banned in the

‘‘orthodox interpretation’’ of the theory) and creat-

ing a connection between the classical description of

Feynman Path Integrals 307

the physical world and the quantum one. Indeed, it

provides a quantization method, allowing, at least

heuristically, to associate a quantum evolution to

each classical Lagrangian. Moreover, the application

of the stationary-phase method for oscillatory

integrals allows the study of the semiclassical limit

of the Schro¨ dinger equation, that is, the study of the

detailed behavior of the solution when the Planck

constant is regarded as a parameter converging to 0.

Indeed, when h is small, the integrand in [2] is

strongly oscillating and the main contributions to

the integral should come from those paths  that

make stationary the phase function S(). These, by

Hamilton’s least action principle, are exactly the

classical orbits of the system.

Feynman path integrals allow also a heuristic

calculus in path space, leading to variational

calculations of quantities of physical and mathe-

matical interest. An interesting application can be

found in topological field theories, as, for

instance, Chern–Simons models. In this case,

heuristic calculations based on the Feynman

path-integral formulation of the theory, where

the integration is performed on a space of

geometrical objects, lead to the computation of

topological invariants.

Even if from a physical point of view, formula [2]

is a source of important results, from a mathema-

tical point of view, it lacks rigor: indeed, neither the

‘‘infinite-dimensional Lebesgue measure,’’ nor the

normalization constant in front of the integral is

well defined. In this article, we shall describe the

main approaches to the rigorous mathematical

realization of Feynman path integrals, as well as

their most important applications.

Possible Mathematical Definitions

of Feynman’s Measure

In the rigorous mathematical definition of Feynman’s

complex measure



:¼

fjðtÞ¼xg

ði=hÞS



ðÞ

D

1

ði=hÞS



ðÞ

D ½5

one has to face mainly two problems. First of all, the

integral is defined on a space of paths, that is, on an

infinite-dimensional space. The implementation of

an integration theory is nontrivial: for instance, it is

well known that a Lebesgue-type measure cannot be

defined on infinite-dimensional Hilbert spaces.

Indeed, the assumption of the existence of a

-additive measure  which is invariant under

rotations and translations and assigns a positive

finite measure to all bounded open sets leads to a

contradiction. In fact, by taking an orthon ormal

system {e

}

i 2N

in an infinite-dimensional Hilbert

space H and by considering the open balls

= {x 2H, kx  e

k< 1=2}, one has that they are

pairwise disjoint and their union is contained in the

open ball B(0, 2) = {x 2H, kxk< 2}. By the Euclidean

invariance of the Lebesgue-type measure , one can

deduce that (B

) = a,0< a < 1, for all i 2N. By the

-additivity, one has

ðBð0; 2 ÞÞ  ð[

Þ¼

ðB

Þ¼1

but, on the other hand, (B(0, 2)) should be finite as

B(0, 2) is bounded. As a consequence, we can also

deduce that the term D in [2] does not make sense.

The second problem is the fact that the exponent

in the density e

(i=h)S



()

is imaginary, so that the

exponential oscillates. Even in finite dimensions,

integrals of the form

i(x)

f (x)dx, with

, f : R

!R are continuous functions and f is not

summable, have to be suitably defined, in order to

exploit the cancelations in the integral due to the

oscillatory behavior of the expon ential.

The study of the rigorous foundation of Feynman

path inte grals began in the 1960s, when Cameron

proved that Feynman’s heuristic complex measure

[5] cannot be realized as a complex bounded

variation -additive measure, even on very nice

subsets of the space (R

)

[0, t]

of paths, contrary to

the case of complex measures on R

of the form

(i=2)jxj

dx. In other words, it is not pos sible to

implement an integration theory in the traditional

(Lebesgue) sense. As a consequence, mathemati-

cians tried to realize [5] as a linear continuous

functional on a sufficiently rich Banach a lgebra of

functions, inspired by the fact that a bounded

measure can be regarded as a continuo us functional

on the space of bounded continuous functions.

In order to mirror the features of the heuristic

Feynman’s measure, such a functional should have

some properties:

1. it should behave in a simple way under ‘‘transla-

tions and rotations in path space,’’ as D denotes

a ‘‘flat’’ measure;

2. it should satisfy a Fubini-type theorem, concern-

ing iterated integrations in path space (allowing

the construction, in physical applications, of a

one-parameter group of unitary operators);

3. it should be approximable by finite-dimensional

oscillatory integrals, allowing a sequential approach

in the spirit of Feynman’s original work; and

4. it should be sufficiently flexible to allow a rigorous

mathematical implementation of an infinite-

dimensional version of the stationary-phase

308 Feynman Path Integrals

method and the corresponding study of the

semiclassical limit of quantum mechanics.

Nowadays, several implementations of this program

can be found in the literature of physics and mathe-

matics, for instance, by means of analytic continuation

of Wiener integrals, or as an infinite-dimensional

distribution in the framework of Hida calculus, or via

‘ ‘complex Poisson measures,’ ’ or via nonstandard

analysis, or as an infinite-dimensional oscillatory inte-

gral. The last of these methods is particularly interesting

as it allows the systematic implementation of an infinite-

dimensional version of the stationary-phase method,

whichcanbeappliedtothestudyofthesemiclassical

limit of the solution of the Schro¨dinger equation [1].

Analytic Continuation

In one of the first approaches in the definition of

Feynman path integrals, formula [2] was realized as

the analytic continuation in a suitable complex

parameter of a (nonoscillatory) Gaussian integral

on the space of paths.

In 1949, inspired by Feynman’s work, M Kac

observed that by considering the heat equation



u ¼

u þ VðxÞu

uð0; xÞ¼

ðxÞ

½6

instead of the Schro¨ dinger equation [1] and by

replacing the oscillatory term e

(i=h)S

()

in Feynman

complex measure with the fast decreasing one

(1=h)S

()

, it is possible to give a well-defined

mathematical meaning to Feynman’s heuristic for-

mula [2] in terms of a well-defined integral on the

space of continuous paths W

t, x

= {w 2C(0, t; R

w(0) = x} with respect to the Wiener Gaussian

measure P

t, x

uðt; xÞ¼

t;x



Vð

ﬃﬃﬃﬃﬃﬃﬃ

1=m

wðÞÞd



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1=m

wðtÞÞdP

t;x

ðwÞ½7

The path-integral representation [7] for the solution of

the heat equation [6] is called Feynman–Kac formula.

The underlying idea of the analytic continuation

approach comes from the fact that by introducing in

[6] a suitable parameter , proportional, for

instance, to the time t as in the case  = 



h

u ¼

h

u þVðxÞu

uðt; xÞ¼

t;x

ð1=

hÞ

Vð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h=ðm

wðÞÞd





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h=ðm

wðtÞ



t;x

ðwÞ

or to the Planck constant, as in the case  = 



u ¼



u þVðxÞu

uðt; xÞ¼

t;x

ð1=

Vð

ﬃﬃﬃﬃﬃﬃﬃﬃ



wðÞÞd



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



wðtÞÞdP

t;x

ðwÞ

or to the mass, as in the case  = 

u ¼

2

u  iVðxÞu

uðt; xÞ¼

t;x

i

ﬃﬃﬃﬃﬃﬃﬃﬃ

1=

wðÞ



d



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1=

wðtÞÞdP

t;x

ðwÞ

and by allowing  to assume complex values, then one

gets, at least heuristically, Schro¨ dinger equation and its

solution by substituting, respectively, 

= i, 

= ih,

or 

= im. These procedures can be made com-

pletely rigorous under suitable conditions on the

potential V and initial datum

The Approach via Fourier Transform

This approach has its roots in a couple of papers

by K Ito in the 1960s and was extensively

developed by S Albeverio and R Høegh-Krohn in

the 1970s. The main idea is the definition of

oscillatory integrals with quadratic phase function

on a real separable Hilbert space (H, h, i), the

Fresnel integrals,

ði=2hÞkxk

f ðxÞdx ½8

as the distributional pairing between e

(i=2h)kxk

and

a complex-valued function f belonging to the

space F(H) of functions that are Fourier trans-

forms of complex bounded variation measures on

H,thatis,

f ¼



; f ðxÞ¼

ihx;yi

d

ðyÞ

F(H) is a Banach algebra, wher e the product is the

pointwise one and the identity is the function

f (x) = 1 8x 2H. The norm of an element f is the

total variation of the corresponding measure 

, that

is, k

k= sup

j

)j, where the supremum is

taken over all sequences { E

} of pairwise-disjoint

Borel subsets of H, such that [

= H.

Given a function f 2F(H), f =



, its Fr esnel

integral is defined by the Parseval formula:

ði=2hÞkxk

f ðxÞdx :¼

ðih=2Þkxk

d

ðxÞ½9

Feynman Path Integrals 309

where the right-hand side is a well-defined absolutely

convergent integral with respect to a -additive

measure on H.

It is important to recall that this approach

provides the implementation of a method of

stationary phase for the expansion of the integral

in powers of the small parameter h occurring in the

integrand. We postpone the discussion of these

results, as well as the application to the solution

of the Schro¨ dinger equation, to the next section

where a generalization of the present approach is

described.

Infinite-Dimensional Oscillatory Integrals

The main idea of this approach is the extension of

the definition of oscillatory integrals with quadratic

phase function [8] to infinite-dimensional Hilbert

spaces by means of a twofold limiting procedure.

The study of integrals of the form

IðhÞ:¼

ði=hÞðxÞ

f ðxÞdx ½10

where (x):R

!R is the phase function and

f : R

!C a complex-valued continuous function,

is a classical topic, largely developed in connection

with various problems in mathematics (such as the

theory of pseudodifferential operators) and physics

(such as optics). Particular effort has been devoted

to the study of the detailed be havior of the above

integral in the limit of ‘‘strong oscillations,’’ that is,

when h !0, by means of the method of stationary

phase.

Thanks to the cancellations due to the oscillatory

term e

(i=2h)(x)

, the integral can still be defined, even if

the function f is not summable, as the limit of a

sequence of regularized, hence absolutely convergent,

integrals. According to a Ho¨ rmander’s proposal,

the oscillatory integral of a function f : R

!C is

well defined if, for each test function  2S(R

), such

that (0) = 1, the limit

lim

!0

ði=2hÞðxÞ

ðxÞf ðxÞdx

exists and is independent of .

This definition has been generalized in the 1980s

by D Elworthy and A Truman to the case where the

underlying space R

is replaced by a real separable

infinite-dimensional Hilbert space (H, h, i), under

the assumption that the phase function is quadratic,

that is, (x) = kxk

=2. The ‘‘infinite-dimensional

oscillatory integral’’



ði=2hÞkxk

f ðxÞdx

is defined as the limit of a sequence of finite-

dimensional approximations. More precisely, a

function f : H!C is ‘‘integrable’’ if, for each

increasing sequence {P

}

n 2N

of finite-dimensional

projector operators in H converging strongly to the

identity operator as n !1, the limit

lim

n!1

ði=2hÞkP



1



ði=2hÞkP

f ðP

xÞdP

x ½11

exists and is independent of the sequence {P

}

n 2N

In this case, the limit is denoted by



ði=2hÞkxk

f ðxÞdx

The description of the largest class of integrable

functions is still an open problem, even in finite

dimension, but it is possible to find some interesting

subsets of it. In particular, any function belonging

to F(H), the Banach algebra considered in the

approach by Fourier transform, is integrable. Indeed,

by assuming that the function f in [11] is of the type

f ðxÞ¼e

ði=hÞhx;Lxi

gðxÞ

where L : H!H is a linear self-adjoint trace-class

operator on H such that (I  L) is invertible and

g 2F(H), that is, g(x) =

hx, yi

d

(y), then it is

possible to prove that f is integrable in the sense of

definition [11] and the corresponding infinite-

dimensional oscillatory integral can be explicitly

computed in terms of a well-defined integral with

respect to a bounded variation measure 

by means

of the following Parseval’s type equality:



ði=2hÞkxk

ði=hÞhx;Lxi

gðxÞdx

¼detðI  LÞ

1=2

ðih=2Þhx;ðILÞ

1

d

ðxÞ½12

det (I  L) being the Fredholm determinant of the

operator I  L, that is, the product of its eigenvalues,

counted with their multiplicity. If L = 0, then we

obtain eqn [9], so that we can look at the infinite-

dimensional oscillatory integrals approach as a gen-

eralization of the Fourier transform approach, since it

allows at least in principle to integrate a class of

function larger than F(H). In fact, recently this feature

has been used by S Albeverio and S Mazzucchi in the

proof of a Parseval’s type equality similar to [12] for

infinite-dimensional oscillatory integrals with poly-

nomially growing phase functions.

Feynman’s heuristic formula [2] for the representa-

tion of the solution of the Schro¨dinger equation [1] can

310 Feynman Path Integrals

be realized as an infinite-dimensional oscillatory integral

on the Hilbert space H

of absolutely continuous paths

 : [0, t] 2R

with fixed endpoint (t) = 0 and finite

kinetic energy



()d<1, endowed with the inner

product h

, 



()



()d. One has to take

an initial datum

) that is the Fourier trans-

form of a complex bounded variation measure on R

that is,

(x) =

ikx

d

(k). Moreover, one has to

assume that the potential V in [1] is the sum of a

harmonic oscillator part plus a bounded perturbation

that is the Fourier transform of a complex bounded

variation measure 

on R

VðxÞ¼

x

x þ V

ðxÞ

ðxÞ¼

ikx

d

ðkÞ

(

being a symmetric positive d  d matrix).

In this case, it is possible to prove that the linear

operator L on H

defined by

ð; LÞ

ðÞ

ðÞd

is self-adjoint and trace class, and (I  L)isinvertible.

Moreover, by considering the function v : H

vðÞ

ððÞþxÞ d

þ 2x

ðÞd; 2H

it is possible to prove that the function f : H

given by

f ðÞ¼e

ði=hÞvðÞ

ðð0ÞþxÞ

is the Fourier transform of a complex bounded

variation measure 

on H

and the infinite-

dimensional Fresnel integral of the function

g() = e

(i=2h)(, L)

f (), that is,:

ðtÞ¼0

ði=2hÞ

_

ðÞd

ði=hÞ

VððÞþxÞd

ðð0ÞþxÞd

ði=2hÞð;ðILÞÞ

ði=hÞvðÞ

ðð0ÞþxÞd ½13

is well defined and it is equal to

detðI  LÞ

1=2

ðih=2Þð;ðILÞ

1

Þ

d

ðÞ

Moreover, it is a representation of the solution of

equation [1] evaluated at x 2R

at time t. Recently,

solutions of the Schro¨ dinger equation with quartic

anharmonic potential via infinite-dimensional oscil-

latory integrals have been provided by S Albeverio

and S Mazzucchi using a combination of Parseval

formula and a new analytic method (the inclusion of

such potentials had been a stumbling block for many

years).

In this framework, it is possible to implement an

infinite-dimensional version of the stationary-phase

method and study the asymptotic behavior of the

oscillatory integrals in the limit h !0.

The method of stationary phase was originally

proposed by Stokes, who noted that when h !0 the

oscillatory integral [10] is O(h

) for any n 2N,

provided that there are no critical points of the

phase function  in the support of the function f.As

a consequence, one can deduce that the leading

contribution to the integral [10] should come from a

neighborhood of those points c 2R

, such that

r(c) = 0. More precisely, by assuming that the set

C of critical points is finite, that is, C = {c

, ..., c

}

and that every criti cal point is nondegenerate, that

is, det D

(c

) 6¼ 0 8c

2C, then one has

IðhÞ

ði=hÞðc



ðhÞ½14

where I



: R !C are C

functions of R, such that



ð0Þ¼f ðc

Þð2ihÞ

N=2

ðdet D

ðc

ÞÞ

1=2

If some critical point is degenerate, the situation is

more complicated: one has to take into account the

type of degeneracy and apply the theory of unfold-

ings of singularities.

These results can be generalized to infinite-

dimensional oscillatory integrals of the form

IðhÞ¼

ði=2hÞhx;ðILÞxi

ði=hÞvðxÞ

gðxÞdx ½15

with v(x) =

ihx, yi

d(y), g(x) =

ihx, yi

d(y), , 

being complex bounded variation measures on H

satisfying suitable assumptions and L : H!H is a

self-adjoint and trace-class linear operator, such that

(I  L) is invertible. Under suitable growth condition on

the moments of the measures ,  and by assuming

that the phase function (x) = hx,(I  L)xiv(x)

has a finite number of nondegenerate critical points

, ..., c

, it is possible to prove that the integral I(h)

in [15] is equal to

IðhÞ¼

k¼1

ði=hÞðc



ðhÞþI

ðhÞ

for some C

functions I



satisfying:



ð0Þ¼½detðI  L  D

Vðc

ÞÞ

1=2

gðc

k ¼ 1; ...; s

ðjÞ

ð0Þ¼0; j ¼ 0; 1; 2; ...

Feynman Path Integrals 311

Moreover, under some additional smallness assump-

tions on v, it has been proved that the phase function

 has a unique stationary point c and as h !0

IðhÞe

ði=hÞðcÞ



ðhÞ

for some C

function I



. Each term of the

asymptotic expansion in powers of h of the function



can be explicitly computed, and it is possible to

prove that such an asymptotic expansion is Borel-

summable and determines I



uniquely.

The application of these results to the infinite-

dimensional oscillatory integral representation [13]

for the solution of the Schro¨ dinger equation allows

the study of its semiclassical limit. One has to

consider a potential V that is the Fourier transform

of a complex bounded variation measure  on (R

such that

jj

djj() < 1 for some >0, and a

particular form for the initial wave function

(x) = e

(i=h)(x)

(x), where  is real and

,  2C

) are independent of h. This initial

datum corresponds to an initial particle distribution



(x) = jj

(x) and to a limiting value of the

probability current J

h = 0

= r(x) 

(x)=m, giving an

initial particle flux associated to the velocity field

r(x)=m. One also has to assume that the Lagrange

manifold L

 (y, rf) intersects transversally the

subset 

of the phase space made of all points (y, p)

such that p is the momentum at y of a classi cal

particle that starts at time zero from x, moves under

the action of V, and ends at y at time t. In this case,

the Feynman path integral [13] has an asymptotic

expansion in powers of h for h !0, whose leading

term is the sum of the values of the function

det



ðjÞ

ðy

ðjÞ

; tÞ

! !



1=2

ði=2Þm

ðjÞ

ði=hÞS

ði=hÞ



taken at the points y

(j)

such that a classical particle

starting at y

(j)

at time zero with momentum r(y

(j)

)

is at x at time t. S is the classical action along this

classical path



(j)

and m

(j)

is the Maslov index of the

path



(j)

, that is, m

(j)

is the number of zeros of

det



ðjÞ

ðy

ðjÞ

;Þ

! !

as  varies on the interval (0, t).

White-Noise Calculus

The leading idea of the present approach, which was

originally proposed by C DeWitt-Morette and

P Kre´e and presently realized in the framework of

white-noise calculus by T Hida, L Streit, and many

other authors, is the realization of the Feynman

integrand e

(i=h)S



()

as an infinite-dimensional distri-

bution. This idea is similar to the one of the

approach via Fourier transform, where the expres-

sion (2i)

d=2

(i=2)(x, x)

f (x)dx is realized as a

distributional pairing be tween e

(i=2)(x, x)

=(2i)

d=2

and

the function f 2F(R

) by means of the Parseval-type

equality [9] and generalized to infinite-dimensional

spaces. In white-noise calculus, the pairing is

realized in a different measure space. Indeed, by

manipulating the integrand in

ð2iÞ

d=2

ði=2Þðx; xÞ

f ðxÞdx

one has

ði=2Þðx;xÞ

ð2iÞ

d=2

f ðxÞdx

ði=2Þðx;xÞþð1=2Þðx;xÞ

d=2

f ðxÞ

ð1=2Þðx;xÞ

ð2Þ

d=2

dx ½16

where the latter line can be interpreted as the

distributional pairing of

ði=2Þðx;xÞþð1=2Þðx;xÞ

d=2

and f not with respect to Lebesgue measure but

rather with respect to the standard Gaussian

measure

ð1=2Þðx;xÞ

ð2Þ

d=2

on R

. The RHS of [16] can be generalized to the

case in which R

is replaced by a path space, thanks

to the fact that on infinite-dimensional spaces, even

if Lebesgue measure is meaningless, Gaussia n

measures are well defined and can be used as

reference measures. The detailed realization of this

idea as well as its application to the mathematical

realization of the Feynman integrand are rather

technical and we certainly do not provide details

here. We recall that this approach has been success-

fully applied to the rigorous realization of Feynman

path-integral formulation of Chern–Simons models.

Other Possible Approaches

Another possible mathematical definition of Feyn-

man path integrals is based on Poisson measures. It

was originally proposed by A M Chebotarev and

V P Maslov and further developed by several

authors such as S Albeverio, Ph Blanchard,

Ph Com be, R Høegh-Krohn, M Sirugue, and

V Kolokol’tsov. It can be applied to ‘‘phase-space

integrals,’’ to the Dirac equation and in particular

algebraic settings, as well as to the Schro¨ dinger

312 Feynman Path Integrals

equation, with potentials of the same type ‘‘Fourier

transform of bounded measure’’ discussed in the

subsection ‘‘Infinite-dimensional oscillatory integrals.’’

Another possible definition of Feynman path

integrals is based on a ‘‘time-slicing’’ approximation

and a limiting procedure, rather closed to Feynman’s

original work based on Trotter product formula.

The ‘‘sequential approach’’ was proposed originally

by A Truman and further extensively developed by

D Fujiwara and N Kumano-go. The paths  in

formula [2] are approximated by piecewise linear

paths and the Feynman path integral is correspond-

ingly approximated by a finite-dimensional integral.

In particular, D Fujiwara and N Kumano-go proved

that the integrals defined in this way have some

important properties, such as invariance under

translations and orthogonal transformations. It is

also possible to interchange the ord er of integration

with Riemann–Stieltjies integrals and study the

semiclassical approxim ation.

Finally, it is worthwhile to recall a very interesting

and intuitive approach to the Feynman integration

which is based on nonstandard analysis. It was

introduced by S Albeverio, J E Fenstad, R Høegh-

Krohn, and T Linstrøm in the 1980s, but it has not

been systematically developed yet.

Abbreviations

D Heuristic Lebesgue-type measure on the space

of paths

t, x

Wiener Gaussian measure on W

t, x

Action functional



Action functional for the free particle

V Potential

t, x

Space of continuous paths with fixed initial

point W

t, x

= {w 2C(0, t; R

):w(0) = x}

h Reduced Planck constant

 Phase function

 Path,  : [0, t] !R

 Fourier transform of the measure 

Wave function, solution of the Schro¨ dinger

equation

H Hilbert space

Fresnel integral on the Hilbert space H



Infinite-dimensional oscillatory integral on the

Hilbert space H

h,i inner product

kk norm

See also: Chern–Simons Models: Rigorous Results;

Euclidean Field Theory; Functional Integration in

Quantum Physics; Path Integrals in Noncommutative

Geometry; Quillen Determinant; Singularity and

Bifurcation Theory; Stationary Phase Approximation.

Further Reading

Albeverio S (1997) Wiener and Feynman – path integrals and

their applications. Proceedings of the Norbert Wiener Cen-

tenary Congress 1994 (East Lansing, MI, 1994), 163–194,

Proc. Sympos. Appl. Math., 52. Providence, RI: American

Mathematical Society.

Albeverio S and Høegh-Krohn R (1977) Oscillatory integrals and

the method of stationary phase in infinitely many dimensions,

with applications to the classical limit of quantum mechanics.

Inventiones Mathematicae 40(1): 59–106.

Albeverio S and Høegh-Krohn R (2005) Mathematical Theory of

Feynman Path Integrals, 2nd edn., with Mazzucchi S Lecture

Notes in Mathematics, vol. 523. Berlin: Springer.

Elworthy D and Truman A (1984) Feynman maps, Cameron–

Martin formulae and anharmonic oscillators. Annales de

l’Institut Henri Poincare Physique Theorique 41(2): 115–142.

Hida T, Kuo HH, Potthoff J, and Streit L (1995) White Noise.

Dordrecht: Kluwer.

Johnson GW and Lapidus ML (2000) The Feynman Integral and

Feynman’s Operational Calculus. New York: Oxford University

Press.

Special issue of Journal of Mathematical Physics on functional

integration, vol. 36, no. 5 (1995).

Finite-Dimensional Algebras and Quivers

A Savage, University of Toronto, Toronto, ON, Canada

ª 2006 A Savage. Published by Elsevier Ltd.

Introduction

Algebras and their representations are ubiquitous in

mathematics. It turns out that representations of

finite-dimensional algebras are intimately related to

quivers, which are simply oriented graphs. Quivers

arise naturally in many areas of mathematics,

including representation theory, algebraic and dif-

ferential geometry, Kac–Moody algebras, and quan-

tum groups. In this article, we give a brief overview

of some of these topics. We start by giving the basic

definitions of associative algebras and their repre-

sentations. We then introduce quivers and their

representation theory, mentioning the connection to

the representation theory of associative algebras. We

also discuss in some detail the relationship between

quivers and the theory of Lie algebras.

Finite-Dimensional Algebras and Quivers 313

Associative Algebras

An ‘‘algebra’’ is a vector space A over a field k

equipped with a multiplication which is distributive

and such that

aðxyÞ¼ðaxÞy ¼xðayÞ; 8a 2k; x; y 2A

When we wish to make the field explicit, we call A a

k-algebra. An algebra is ‘‘associative’’ if (xy)z = x(yz)

for all x, y, z 2A. A has a ‘‘unit,’’ or ‘‘multiplicative

identity,’’ if it contains an element 1

such that

x = x1

= x for all x 2A.Fromnowon,wewill

assume all algebras are associative with unit. A is said

to be ‘‘commutative’’ if xy = yx for all x, y 2A and

finite dimensional if the underlying vector space of A

is finite dimensional.

A vector subspace I of A is called a ‘‘left (resp.

right) ideal’’ if xy 2I for all x 2A, y 2I (resp.

x 2I, y 2A). If I is both a right and a left ideal, it

is called a two-sided ideal of A.IfI is a two-sided

ideal of A, then the factor space A=I is again an

algebra.

An algebra homomorphism is a linear map

f : A

between two algebras such that

f ð1

Þ¼1

f ðxyÞ¼f ðxÞf ðyÞ; 8x; y 2A

A representation of an algebra A is an algebra

homomorphism  : A !End

(V)forak-vector

space V.HereEnd

(V) is the space of endomorph-

isms of the vector space V with multiplication

given by composition. Given a representation of

an algebra A on a vector space V,wemayviewV

as an A-module with the action of A on V given

a v ¼ ðaÞv; a 2A; v 2V

A morphism : V !W of two A-modules (or

equivalently, representations of A) is a linear map

commuting with the action of A. That is, it is a

linear map satisfying

a  ðvÞ¼ ða vÞ; 8a 2A; v 2V

Let G be a commutative monoid (a set with an

associative multiplication and a u nit element). A

G-graded k-algebra is a k-algebra which can be

expressed as a direct sum A = 

g 2G

such that

 A

for all a 2k and A

 A

þg

for all

, g

2G.Amorphism : A !B of G-graded

algebras is a k-algebra morphism respecting the

grading, that is, satisfying (A

)  B

for all

g 2G.

Quivers and Path Algebras

A ‘‘quiver’’ is simply an oriented graph. More

precisely, a quiver is a pair Q = (Q

, Q

)whereQ

is a finite set of vertices and Q

is a finite set of

arrows (oriented edges) between them. For a 2Q

we let h(a) denote the ‘‘head’’ of a and t(a)denotethe

‘‘tail’’ of a.ApathinQ is a sequence x = 



...

of arrows such that h(

iþ1

) = t(

)for1i m  1.

We let t(x) = t(

)andh(x) = h(

) denote the initial

and final vertices of the path x. For each vertex

i 2Q

,welete

denote the trivial path which starts

and ends at the vertex i.

Fix a field k. The path algebra kQ associated to a

quiver Q is the k-algebra whose underlying vector

space has basis the set of paths in Q, and with the

product of paths given by concatenation. Thus, if

x = 

...

and y = 

...

are two paths, then

xy = 

...



...

if h(y) = t(x) and xy = 0 other-

wise. We also have

if i ¼ j

0ifi 6¼j

(

x ¼

x if hðxÞ¼i

0ifhðxÞ 6¼ i

(

x if tðxÞ¼i

0iftðxÞ 6¼ i

(

for x 2kQ. This mul tiplication is associative. Note

that e

A and Ae

have bases given by the set of paths

ending and starting at i, respectively. The path

algebra has a unit given by

i 2Q

Example 1 Let Q be the following quiver:

1234

ρλσ

then kQ has a basis given by the set of paths

, e

, , , , }. Some sample products are

 = 0,  = 0,  = 0, e

 = e

= , e

 = 0.

Example 2 Let Q be the following quiver (the

so-called ‘‘Jordan quiver’’).

Then kQ ﬃk[t], the algebra of polynomials in one

variable.

Note that the path algebra kQ is finite dimen-

sional if and only if Q has no oriented cycles (paths

with the same head and tail vertex).

314 Finite-Dimensional Algebras and Quivers

Example 3 Let Q be the following quiver:

123 n – 2 n – 1 n

Then for every 1 i j  n, there is a uniqu e path

from i to j. Let f : kQ !M

(k) be the linear map from

the path algebra to the n n matrices with entries in

the field k that sends the unique path from i to j to the

matrix E

with (j, i) entry 1 and all other entries zero.

Then one can show that f is an isomorphism onto the

algebra of lower triangular matrices.

Representations of Quivers

Fix a field k. A representation of a quiver Q is an

assignment of a vector space to each vertex and to

each arrow a linear map between the vector spaces

assigned to its tail and head. More precisely, a

representation V of Q is a collection

ji 2Q

of finite-dimensional k-vector spaces together with a

collection



: V

tðÞ

hðÞ

j 2Q

of k-linear maps. Note that a representation V of a

quiver Q is equivalent to a represent ation of the

path algebra kQ. The dimension of V is the map

: Q

0

given by d

(i) = dim V

for i 2Q

If V and W are two representations of a quiver Q,

then a morphism : V !W is a collection of k-linear

maps

: V

ji 2Q

such that



tðÞ

hðÞ



; 8 2Q

Proposition 1 Let A be a finite-dimensional

k-algebra. Then the category of representations of

A is equivalent to the category of representations of

the algebra kQ/I for some quiver Q and some two-

sided ideal I of kQ.

It is for this reason that the study of finite-

dimensional associative algebras is intimately related

to the study of quivers.

We define the direct sum V W of two repre-

sentations V and W of a quiver Q by

ðV WÞ

¼ V

W

; i 2Q

and (V W)



: V

t()

W

t()

h()

W

h()

ðV WÞ



ððv; wÞÞ ¼ ðV



ðvÞ; W



ðwÞÞ

for v 2V

t()

, w 2W

t()

,  2Q

. A representation V is

‘‘trivial’’ if V

= 0 for all i 2Q

and ‘‘simple’’ if its

only subrepresentations are the zero repre sentation

and V itself. We say that V is ‘‘decomposable’’ if it is

isomorphic to W U for some nontrivial represen-

tations W and U. Otherwise, we call V ‘‘indecom-

posable.’’ Every representation of a quiver has a

decomposition into indecomposable representations

that is unique up to isomorphism and permutation

of the components. Thus, to classify all representa-

tions of a quiver, it suffices to classify the indecom-

posable representations.

Example 4 Let Q be the following quiver:

Then Q has three indecomposable representations

U, V, and W given by:

¼k; U

¼0; U



¼0

¼0; V

¼k; V



¼0

¼k; W



¼1

Then any representation Z of Q is isomo rphic to

Z ﬃU

r

V

r

W

where d

= dim Z

, d

= dim Z

, r = rank Z



Example 5 Let Q be the Jordan quiver. Then

representations V of Q are classified up to iso-

morphism by the Jordan normal form of V



where 

is the single arrow of the quiver. Indecomposable

representations correspond to single Jordan blocks.

These are parametrized by a discrete parameter n

(the size of the block) and a continuous parameter 

(the eigenvalue of the block).

A quiver is said to be of ‘‘finite type’’ if it has only

finitely many indecomposable representations (up to

isomorphism). If a quiver has infinitely many

isomorphism classes but they can be split into

families, each parametrize d by a single continuous

parameter, then we say the quiver is of ‘‘tame’’ (or

‘‘affine’’) type. If a quiver is of neither finite nor

tame type, it is of ‘‘wild type.’’ It turns out that there

is a rather remarkable relationship between the

classification of quivers and their representations

and the theory of Kac–Moody algebras.

The ‘‘Euler form’’ or ‘‘Ringel form’’ of a quiver Q

is defined to be the asymmetric bilinear form on Z

given by

h; i¼

i 2Q

ðiÞðiÞ

 2Q

ðtðÞÞðhðÞÞ

Finite-Dimensional Algebras and Quivers 315