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

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summed over N:thisrequires(see[24]) an explicit

calculation of [23] pushed at least to order j = 1if

d = 2 or to order j = 3ifd = 3; furthermore it is also

necessary to check that the resulting V

j; N1

can still

be interpreted as low-coupling spin model so that

[22] can be iterated with N  1 replacing N and then

with N  2replacingN  1, ....

The first necessary check towards a proof of the

discussed heuristic ‘‘expectations’’ is that, defining

recursively V

j; h

from V

j, hþ1

for h = N  1, ...,1,0

by [23] with V

replaced by V

j; hþ1

and V

j; N1

replaced by V

j; h

, the couplings between the vari ables

(h)

do not become ‘‘worse’’ than those discussed in

the case h = N. Furthermore, the field ’

(N1)

has a

high probability of satisfying [6], but fluctuations

are possible: hence the R-estimate has to be

combined with another one dealing with the large

fluctuations of X

(N1)

which has to be shown to be

‘‘not worse.’’

For more details, the reader is referred to Gallavotti

(1978, 1985) and Benfatto and Gallavotti (1995).

Effective Potentials and Their

Scale (In)Dependence

To analyze the first problem mentioned at the end of

the previous section, define V

j; h

by [23] with V

replaced by V

j; hþ1

for h = N  1, N  2, ..., 0. The

quantities V

j; h

, which are called ‘‘effective poten-

tials’’ on scale h (and order j), turn out to be in a

natural sense scale independent: this is a con se-

quence of reno rmalizability, realized by Wilson as a

much more general property which can be checked,

in the very special cases considered here with

d = 2, 3, at fixed j by induction, and in the super-

renormalizable models considered here it requires

only an ele mentary computation of a few Gaussian

integrals as the case j = 3 (or even j = 1ifd = 2) is

already sufficient for our purposes .

It can also be (more easily) proved for general j by

a dimensional argument parallel to the one pre-

sented earlier to check finiteness of the renormalized

series. The derivation is elementary but it should be

stressed that, again, it is possible only because of the

special choice of the counter-terms 

, 

.Ifd = 3,

the boundedness and smoothness of the fields ’

(h)

and z

(h)

expressed by the second of [6] and of [10] is

essential; while if d = 2 the smoothness is not

necessary.

The structure of V

j; h

is conveniently expressed

in terms of the fields X

(h)

, as a sum of three terms

(rel)

(standing for ‘‘relevant’’ part), V

(irr)

(standing

for ‘‘irrelevant’’ part), and a ‘‘field independent’’

part E(j, h)jj.

The relevant part in d = 2 is simply of the form

[21] with h replacing N: call it V

(rel, 1)

.Ifd = 3, it is

given by [20] with h replacing N plus, for h < N,a

second ‘‘nonlocal’’ term

ðrel;2Þ

def

2! 2!





ðhÞ3

 C

ðN Þ3







’

ðhÞ

 ’

ðhÞ



dhdh

which is conveniently expressed in terms of a

‘‘nonlocal’’ field

ðhÞ

def

’

ðhÞ

 ’

ðhÞ

ð

jh  h

jÞ

as V

ðrelÞ

¼ V

ðrel;1Þ

þ V

ðrel;2Þ

with

ðrel;2Þ

def





2h

;



ðhÞ2

ðhÞ

 e

c



jhh

dhdh

jjj

½25

where

0 < a 

ðhÞ

ð

jh  h

jÞ

3ð1=2Þ

< a

with a, a

, c

> 0andthesubscriptN means that the

expression in parenthesis ‘‘saturates at scale N’’, i.e.,

its denoninator becomes 

(3(1=2))(hN)

as jh  h

j!0.

The expression [25] is not the full part of the

potential V

j; h

which is of second order in the fields:

there are several other contributions which are

collected below as ‘‘irrelevant.’’

It should be stressed that ‘‘irrelevant’’ is a

traditional technical term: by no means it should

suggest ‘‘negligibility.’’ On the contrary, it could be

maintained that the whole purpose of the theory is

to study the irrelevant terms. The irrelevant part of

the potential can be better designated as the ‘‘driven

part,’’ as its behavior is ‘‘controlled’’ by the relevant

part: although initially V

j; h

, h = N, contains

no irrelevant terms, it eventually contains them for

h < N and they keep getting generated as h

diminishes. Furthermore, the pa rt of the irrelevant

terms generated at scale h

 N becomes very small

at scales h  h

so that the irrelevant part of V

j; h

small h (e.g., at h = 0, i.e., on the ‘‘physical scale’’ of

the observer) only depend on the relevant terms in a

few scales near h.

It also turns out that the Schwinger functions are

simply related to the irrelevant terms.

The irrelevant part of the effective potential can

be expressed as a finite sum of integrals of

Constructive Quantum Field Theory 625

monomials in the fields X

(h)

if d = 2, or in the fields

(h)

and Y

(h)

if d = 3, which can be written as V

(irr)

j; h

given by



k¼1

ðhÞn

¼1

ðhÞn





dðx

;...;h





ht

 Wðx

; ...; h

k¼1

j

¼1

j

jj

½26

with the integral extended to products 





(

 

) of boxes  2Q

,and

d(x

, ..., h

) is the length of the shortest tree

graph that connects all the p þ 2q > 0 points, the

exponents n, t are 2, and t is 3ifq > 0;

the kernel W depends on all coordinates x

, ..., h

and it is bounded ab ove by C

= 1

for some

; the sums

cannot exceed 4j. The

test functions f do not appear in [26] because by

assumption they are bounded by 1: but W depends

on the f ’s as well.

The field-independent part is simply the value

of log Z

(, f ) computed by the perturbation

analys is in the sect ion ‘‘Per turbation theory ’’ up to

order j in  but using as propagator (C

(N)

 C

(h)

thus, E(j, h) is a constant depending on N but

unifor mly bounde d as N !1 (beca use of the

renorm alizability proved in the section ‘‘Perturba-

tion theory ’’).

If d = 2, there is no need to introduce the nonlocal

fields Y

(h)

and in [26] one can simply take q = 0,

and the relevant part also can be expressed by

omitting the term V

(rel, 2)

in [25]: unlike the d = 3

case, the estimate on the kernels W by an

N-independent C

holds uniformly in h without

having to introduce Y. For d = 2, it will there fore be

supposed that V

(rel, 2)

 0in[25] and q = 0in[26].

It is not necessary to have more information on

the structure of V

j; h

even though one can find simple

graphical rules, closely related to the ones in the

section ‘‘Perturbat ion theory ,’’ to constr uct the

coefficients W in full detail. The W depend, of

course, on h but the uniformity of the bound on W

is the only relevant property and in this sense the

effective potentials are said to be (almost) ‘‘scale

independent.’’

The above bounds on the irrelevant part can

be checked by an elementary direct computation if

j  3: in spite of its ‘‘elementary character,’’ the

uniformity in h  N is a result ultimately playing an

essential role in the theory together with the

dominance of the relevant part over the irrelevant

one which, once the fields are properly scaled, is

‘‘much smaller’’ (by a factor of order 

h

, see [26]),

at least if h is large.

Remarks

(i) Checking scale independence for j = 1 is just

checking that

P(dz

(h)

1; h

= V

1; h1

. Note that

1;h

def





’

ðhÞ4

 6C

ðhÞ

’

ðhÞ2

þ 3C

ðhÞ2



hence, calling :’

(h)4

: the polynomial in the integral

(Wick’s monomial of order 4), we have here an

elementary Gaussian integral (‘‘martingale property

of Wick monomials’’). Note the essential role of the

counter-terms. For j > 1, the computation is similar

but it involves higher-order polynomials (up to 4j)

and the distinction between d = 2andd = 3

becomes important.

(ii) V

j;0

contains only the field-independent part

E(j,0)jj (see above) which is just a number (as

there are no fields of scale 0): by the above

definitions, it is identical to the perturbative

expansion truncated to jth order in  of

log Z

(, f ), well defined as discussed earlier.

Nonperturbative Renormalization:

Small Fields

Having introduced the notion of effective potential

j; h

, of order j and scale h, satisfying the bounds

(described after [26]) on the kernels W representing

it, the problem is to estimate the remainder in [22]

and find its relation with the value [24] given by the

heuristic Taylor expansion. Assume <1 to avoid

distinguishing this case from that with   1 which

would lead to very similar estimates but to different

-dependence on some constants.

Define 

(h)

) = 1ifkz

(h)



 Bh

for all  2Q

see [8], and 0 otherwise; then the following lemma

holds:

Lemma 1 Let kX

(h)



be defined as [8] with z

replaced by X and suppose kX

(h)



 Bh

for all 

then, for all j  1, it is

j;hþ1



ðz

ðhþ1Þ

ÞdPðz

ðhþ1Þ

¼ e

j;h

þR

ðj;hþ1Þjj

½27

with, for suitable constants c



, c



ðj; h þ 1Þj  R



ðj; h þ 1Þ

def

Rðj; h þ 1Þþc



c



ðhþ1Þ

and R(j; h þ 1) given by [24] with h þ 1 in place

of N.

Since Z

(, f ) 

h = 1



(h)

)P(dz

(h)

) this

immediately gives a lower bound on E = (1=jj)

log Z

(, f ): in fact if 

(kz

)

k) = 1 for

626 Constructive Quantum Field Theory

= 1, ..., h, then kX

(h)



 cBh

for some c so

that, by recursive application of Lemma 1,

(, f )  e

j,0



h = 1



(j, h)jj

. By the remark at the

end of the previous section, given j the lower bound

on E just described agrees with the perturbation

expansion of E = (1=jj) log Z

(, f ) truncated to

order j (in ) up to an error bounded by

h = 1



(j, h).

Remark The prob lem solved by Lemma 1 is

usually referred to as the small-field problem, to

contrast it with the large-field problem discussed

later. The proof of the lemma is a simple Taylor

expansion in 

h

if d = 3orinh



2h

if d = 2to

order j (in ). The constraint on z

(hþ1)

makes the

integrations over z

(hþ1)

, necessary to compute V

j; h

from V

j; hþ1

, not Gaussian. But the tail estimates [9],

together with the Markov property of the distribu-

tion of z

(h)

can be used to estimate the difference

with respect to the Gaussian unconstrained integra-

tions of z

(hþ1)

: and the result is the addition of the

small ‘‘tail error’’ changing R into R



in [27]. The

estimate of the main part of the remainder R would

be obvious if the fields z

(h)

were independent on

boxes of scale 

h

: they are not independent but

they are Markovian and the estimate can be done by

taking into account the Markov property.

For more details, the reader is referred to Wilson

(1970, 1972), Gallavotti (1978, 1981, 1985), and

Benfatto et al. (1978).

Nonperturbative Renormalization: Large

Fields, Ultraviolet Stability

The small-field estimates are not sufficient to obtain

ultraviolet stability: to control the cases in which

(h)

j> Bh

for some x or some h,orjY

(h)

j> Bh

for

some jx  hj <

h

, a further idea is necessary and it

rests on making use of the assumption that >0

which, in a sense to be determined, should suppress

the contribution to the integral defining Z

(, f )

coming from very large values of the field. Assume

also <1 for the same reasons advanced in the

section ‘‘Effective potentials and their scale

(in)dependence.’’

Consider first d = 2. Let D

be the ‘‘large-field

region’’ where jX

(N)

j> BN

and let V

(=D

)be

the integral defining the potential in [21] extended

to the regio n =D

, complement of D

. This regi on

is typically very irregular (and random as X itself is

random with distribution P

An upper bound on the integral defining Z

(, f )

is obtained by simply replacing e

by e

(=D

)

because in D

the first term in the integrand in [21]

is N



(BN

) < 0 and it overwhelmingly

dominates on the remaining terms whose value is

bounded by a similar expression with a smaller

power of N. Then if E

def

=E denotes the comple-

ment in  of a set E:

Lemma 2 Let d = 2. Define V

) to be given by

the expression [22] with the integrals extending over



and define R(j, h þ 1) by [24]. Then

hþ1

ðD

hþ1



ðhþ1Þ



¼ e

ðD

ÞþR

ðj;hþ1Þjj

½28

where j

(j, h þ 1jR

(j, h þ 1 =

def

R(j; h þ 1) þ

c

(hþ1)

with suitable c

, c

Remark Lemma 2 is genuinely not perturbative

and making essential use of the positivity of .

Below the analysis of the proof of the lemma, which

consists essentially in its reduction to Lemma 1,is

described in detail. It is perhaps the most interesting

part and the core of the theory of the proof that

truncating the expansion in  of (1=jj) log Z

(, f )

to order j gives as a result an estimate exact to order



jþ1

of (1=jj) log Z

(, f ).

Let R

be the cubes  2Q

in which there is at

least one point x where jz

(N)

jBN

. By definit ion,

the region D

N1

is covered by R

Remark that in the region D

N1

the field

(N1)

is large but z

is not large so that X

(N)

is still

very large: this is so because the bounds set to define

the regions D and R are quite different being BN

and BN

, respectively. Hence, if a point is in D

N1

and not in R

, then the field X

(N)

must be of the

order BN

. Therefore, by positivity of the ’

(N)4

term (which dominates all other terms so that

(N)

(’

(N)

) < 0 for x 2D

[ (D

N1

)) we can

replace V

)byV((D

[ (D

N1

))

), for the

purpose of obtaining an upper bound.

Furthermore, modulo a suitable correction, it is

possible to replace V((D

[ (D

N 1

))

)by

V((D

N1

[ R

)

): because the integrand in V

bounded below by

b

2N

if d = 2 (by b

N

if d = 3), for some b, so that the

points in R

can at most lower V(( D

[

N1

))

)bybN



(4d)N

#(R

)if#R

the number of boxes of Q

in R

and V(’

)is

bounded below by its minimum: thus,

VððD

N1

[ R

ÞþbN



ð4dÞN

#ðR

is an upper bound to V((D

[ (D

N1

))

In the complem ent of D

N1

[ R

, all fields are

‘‘small’’; if X

(N1)

and R

are fixed this region is not

random (as a function of z

(N)

) any more. Therefore,

Constructive Quantum Field Theory 627

if X

(N1)

, R

are fixed the integration over z

(N)

conditioned to having z

(N)

fixed (and large) in the

region R

, is performed by means of the same

argument necessary to prove Lemma 1 (essentially a

Taylor expansion in 

(4d)N

). The large size of

(N)

in R

does not affect too much the result

because on the boundary of R

the field z

(N)

BN

(recalling that z

(N)

is continuous) and since

the variable z

(N)

is Markovian, the boundary effect

decays exponentially from the boundary @R

:it

adds a quantity that can be shown to be bounded by

the number of boxes in R

on the boundary of R

hence by #R

, times b

(N  1)



(4d)

(B(N  1)

)

for some b

The result of the integration over z

(N)

((D

[(D

N1

))

)

conditioned to the large-field

values of z

(N)

in R

leads to an upper bound on

P(dz

(N)

)as

j;N1

ðD

N1

ÞþRðj;NÞjj



2R

c e

c

ðBN

þc



ð4dÞN

ðBN



½29

where c, c

, c

are suitable constants: this is

explained as follows.

1. Taylor expansion (in ) of the integral

((D

N1

)

)þbN



(4d )N

#(R

)

(which, by cons-

truction, is an upper bound on e

)

)with

respect to the field z

(N)

, conditioned to be fixed

and large in R

, would lead to an upper bound as

j; N1

ððD

N1

ÞþR

ðj;NÞjjþb

ðBN



ð4dÞN

#ðR

with R

equal to [24] possibly with some C

replacing C

. The second exponential on the RHS

of [29] arises partly from the above correction

(BN

)



(4d)N

#(R

) and partly from a

contribution of similar form explained in (3)

below.

2. Integration over the large conditioning fields

fixed in R

is controlled by the second estimate

in [9] (the tail estimate): the first factors in

parentheses in [29] is the tail estimate just

mentioned, i.e., the probability that z

(N)

is large

in the region R

. The second factor is only partly

explained in (1) above.

3. Without further estimates, the bound [29] would

contain V

j; N1

((D

N1

[ R

)

)ratherthan

j; N1

N1

). Hence, there is the need to change

the potential V

j;N1

((D

N1

[ R

)

) by ‘ ‘reintrodu-

cing’ ’ the contribution due to the fields in

N1

in order to reconstruct V

j; N1

N1

Reintroducing this part of the potential costs a

quantity like b

N



(4d)N

(BN

)

#(R

) (because

the reintroduction occurs in the region R

N1

which is covered by R

and in such points the field

(N1)

is not large, being bounded by B(N  1)

);

so that their contribution to the effective potential

is still dominated by the ’

-term and therefore by



(4d)N

times a power of BN

times the volume of

(in units 

N

, i.e., #R

).Allthisistakencare

of by suitably fixing c

Note that the sum over R

of [29] is

ð1 þ c e

c

þc



ð4dÞN

ðBN



jj

(because  contains jj

cubes of Q

); hence, it is

bounded above by e

c

for suitably defined

, c

The same argument can be repeated for V

j; h

)

with any h if V

j; h

) is defined by the sum over  ’s

in Q

of the same integrals as those in [25] and [26]

with 

replacing 

in the integration domains.

Applying Lemma 1 recursiv ely with j  1 (if

d = 3 it woul d become necessary to take j  3), it

follows that there exist N-indep endent upper and

lower bounds E



jj on log Z(, f ) of the form

j;0



h = 1

(R(j, h) þ c



c



)jj for c



, c



> 0

suitably chosen and -independent for <1.

By the remark at the end of Sec.6, given j, the

bounds just described agree with the perturbation

expansion E(j,0)jjV

j;0

of log Z(, f ) truncated

to order j (in ) up to the remainders



h = 1



(j, h). Hence, if B is chosen proportional

to log



1

def

log (e þ 

1

), the upper and lower

bounds coincide to order j in  with the value

obtained by truncating to order j the perturbative

series.

The latter remark is important as it implies

not only that the bounds are finite (by the

section ‘‘Perturbat ion theory ’’) but also that the

function (1=j j) log Z(, f ) is not quadratic in f:

already to order 1 in  it is quartic in f (containing a

term equal to (

x,0

dx)

The latter prop erty is important as it excludes

that the result is a ‘‘Gaussian’’ generating function.

Thus, the outline of the proof of Lemma 2, which

together with Lemma 1 forms the core of the

analysis of the ultraviolet stability for d = 2, is

completed.

If d = 3, more care is needed because (very mild)

smoothness, like the considered Ho¨ lder continuity

with exponent 1/4, of z, X is necessary to obtain the

key scale independence property discussed in earlier:

therefore, the natural measure of the size of z

(h)

and

(h)

in a box  2Q

is no longer the maximum of

(h)

j or of jX

(h)

j. The region D

becomes more

628 Constructive Quantum Field Theory

involved as it has to consist of the points x

where jX

(h)

j> Bh

and of the pairs h, h

where

h;h

j

ðhÞ

 X

ðhÞ



ð

jh  h

jÞ

> Bh

i.e., it is not just a subset of .

However, if d = 3, the relevant part also contains

the negative term V

(rel, 2)

, see [25] : and since it

dominates over all other terms which contain a

Y-field (because their couplings [25] are smaller by

about 

h

), the argument given for d = 2 can be

adapted to the new situation. Two regions D

, D

will be defined: the first consists of all the points x

where jX

(h)

j> Bh

and the second of all the pairs

h, h

where jY

(h)

h, h

j> Bh

. The region R

will be

the collection of all  2Q

, where kz

(h)



> Bh

see [8] with  = 0. Then V(D

) will be defined as the

sum of the integrals in [25] and [26] with the integrals

over x

further restricted to x

62D

and those over the

pairs h

, h

are further restricted to ( h

, h

) 62D

. With

the new settings, Lemma 2 can be proved also for

d = 3 along the same lines as in the d = 2 case.

For more details, the reader is referred to Wilson

(1970, 1972), Benfatto et al. (1978), and Gallavotti

(1981).

Ultraviolet Limit, Infrared Behavior, and

Other Applications

The results on the ultraviolet stability are nonper-

turbative, as no assumption is made on the size of 

(the assumption <1 has been imposed in the last

two sections only to obtain simpler expressions for

the -dependence of various constants): nevertheless

the multiscale analysis has allowed us to use

perturbative techniques (i.e., the Taylor expansion

in Lemmata 1, 2) to find the solution. The latter

procedure is the essence of the renormalization

group methods: they aim at reducing a difficult

multiscale problem to a sequence of simple single-

scale problems. Of course, in most cases, it is

difficult to implement the approach and the scalar

quantum fields in dimensions 2, 3 are among the

simplest examples. The analysis of the beta function

and of the running couplings, which appear in

essentially all renormalization group applications,

does not play a role here (or, better, their role is so

inessential that it has even been possible to avoid

mentioning them). This makes the models somewhat

special from the renormalization group viewpoint:

the running couplings at length scale h, if intro-

duced, would tend exponentially to 0 as h !1;

unlike what happens in the most interesting

renormalization group applications in which they

either tend to zero only as powers of h or do not

tend to zero at all.

The multiscale analysis method, i.e., the renorma-

lization group method, in a form close to the one

discussed here has been appli ed very often since its

introduction in physics and it has led to the solution

of several important problems. The following is not

an exhaustive list and includes a few open questions.

1. The arguments just discussed imply, with minor

extra work that Z

(, f )asN !1not only admit

uniform upper and lower bounds but also that the

limit as N !1actually exists and it is a C

function

of  , f . Its  and f-derivatives at  = 0andf = 0are

given by the formal perturbation calculation. In some

cases, it is even possible to show that the formal series

for Z

(, f )inpowersof is Borel summable.

2. The problem of removing the infrared cutof f (i.e.,

 !1) is in a sense more a problem of statistical

mechanics. In fact, it can be solved for d = 2, 3 by a

typical technique used in statistical mechanics, the

‘‘cluster expansion.’’ This is not intended to mean

that it is technically an easy task: understanding its

connection with the low-density expansions and

the possibility of using such techniques has been a

major achievement that is not discussed here.

3. The third problem mentioned in the introduction,

that is, checking the axioms so that the theory could

be interpreted as a quantum field theory is a difficult

problem which required important efforts to con-

trol and which is not analyzed here. An introduction

to it can be its analysis in the d = 2case.

4. Also the problem of keeping the ultraviolet cutoff

and removing the infrared cutoff while the para-

meter m

in the propagator approaches 0 is a very

interesting problem related to many questions in

statistical mechanics at the critical point.

5. Field theory methods can be applied to various

statistical mechanics problems away from criti-

cality: particularly interes ting is the theory of the

neutral Coulomb gas and of the dipole gas in two

dimensions.

6. The methods can be applied to Fermi systems in

field theory as well as in equilibrium statistical

mechanics. The understanding of the ground state

in not exactly soluble models of spinless fermions

in one dimension at small coupling is one of the

results. And via the transfer matrix theory it has

led to the understanding of nontrivial critical

behavior in two-dimensional models that are not

exactly soluble (like Ising next-nearest-neighbor or

Ashkin–Teller model). Fermi systems are of

particular interest also because in their analysis

the large-fields problem is absent, but this great

Constructive Quantum Field Theory 629

technical advantage is somewhat offset by the

anticommutation properties of the fermionic

fields, which do not allow us to employ

probabilistic techniques in the estimates.

7. An outstanding open problem is whether the scalar

’

-theory is possible and nontrivial in dimension

d = 4: this is a case of a renormalizable not

asymptotically free theory. The conjecture that

many support is that the theory is necessarily trivial

(i.e., the function Z

(, f ) becomes necessarily a

Gaussian in the limit N !1). One of the main

problems is the choice of the ultraviolet cut-off;

unlike the d = 2, 3 cases in which the choice is a

matter of convenience it does not seem that the

issue of triviality can be settled without a careful

analysis of the choice and of the role of the

ultraviolet cut-off.

8. Very interesting problems can be found in the

study of highly symmetric quantum fields: gauge

invariance presents serious difficulties to be

studied (rigorously or even heuristically) because

in its naive forms it is incompatible with

regularizations. Rigorous treatments have been

in some cases possible and in few cases it has been

shown that the naive treatment is not only not

rigorous but it leads to incorrect results.

9. In connection with item (8) an outstanding problem

is to understand relativistic pure gauge Higgs fields

in dimension d = 4: the latter have been shown to be

ultraviolet stable but the result has not been

followed by the study of the infrared limit.

10. The classical gauge theory problem is quantum

electrodynamics, QED, in dimension 4: it is a

renormalizable theory (taking into account gauge

invariance) and its perturbative series truncated

after the first few orders give results that can be

directly confronted with experience, giving very

accurate predictions. Nevertheless, the model is

widely believed to be incomplete: in the sense that,

if treated rigorously, the result would be a field

describing free noninteracting assemblies of

photons and electrons. It is believed that QED

can make sense only if embedded in a model with

more fields, representing other particles (e.g., the

standard model), which would influence the

behavior of the electromagnetic field by providing

an effective ultraviolet cutoff high enough for not

altering the predictions on the observations on the

time and energy scales on which present (and,

possibly, future over a long time span) experi-

ments are performed. In dimension d = 3, QED is

super-renormalizable, once the gauge symmetry is

properly taken into account, and it can be studied

with the techniques described above for the scalar

fields in the corresponding dimension.

In general, constru ctive quantum field theory

seems to be in a deep crisis: the few solutions that

have been found concern very special problems and

are very demanding technically; the results obtained

have often not been considered to contribute

appreciably to any ‘‘progress.’’ And many consider

that the work dedicated to the subject is not worth

the results that one can even hope to obtain.

Therefore, in recent years, attempts have been

made to follow other paths: an attitude that in the

past usually did not lead, in general to great

achievements but that is always tempting and

worth pursuing because the rare major progresses

made in physics resulted precisely by such changes

of attitude, leaving aside developments requiring

work whi ch was too technical and possibly hopeless:

just to mention an important case, one can recall

quantum mechanics which disposed of all attempts

at understanding the observed atomic levels quanti-

zation on the basis of refined developments of

classical electromagnetism.

For more details, the reader is referred to Nelson

(1966), Guerra (1972), Glimm et al. (1973), Glimm

and Jaffe (1981), Simon (1974), Benfatto et al.

(1978, 2003), Aizenman (1982), Gawedzky and

Kupiainen (1983, 1985a, b), Balaban (1983), and

Giuliani and Mastropietro (2005).

See also: Algebraic Approach to Quantum Field Theory;

Axiomatic Quantum Field Theory; Euclidean Field

Theory; Integrability and Quantum Field Theory;

Perturbation Theory and its Techniques; Quantum Field

Theory: A Brief Introduction; Scattering, Asymptotic

Completeness and Bound States.

Further Reading

Aizenman M (1982) Geometric analysis of ’

-fields and Ising

models. Communications in Mathematical Physics 86: 1–48.

Balaban T (1983) (Higgs)

3, 2

quantum fields in a finite volume. III.

Renormalization. Communications in Mathematical Physics

88: 411–445.

Benfatto G, Cassandro M, Gallavotti G et al. (1978) Some

probabilistic techniques in field theory. Communications in

Mathematical Physics 59: 143–166.

Benfatto G, Cassandro M, Gallavotti G et al. (1980) Ultraviolet

stability in Euclidean scalar field theories. Communications in

Mathematical Physics 71: 95–130.

Benfatto G and Gallavotti G (1995) Renormalization Group,

pp. 1–143. Princeton: Princeton University Press .

Benfatto G, Giuliani A, and Mastropietro V (2003) Low

temperature analysis of two dimensional Fermi systems with

symmetric Fermi surface. Annales Henry Poincare´ 4: 137–193.

De Calan C and Rivasseau V (1981) Local existence of the Borel

transform in euclidean 

. Communications in Mathematical

Physics 82: 69–100.

Fro¨ hlich J (1982) On the triviality of 

theories and the

approach to the critical point in d  4 dimensions. Nuclear

Physics B 200: 281–296.

630 Constructive Quantum Field Theory

Gallavotti G (1978) Some aspects of renormalization problems in

statistical mechanics. Memorie dell’ Accademia dei Lincei

15: 23–59.

Gallavotti G (1981) Elliptic operators and Gaussian processes. In:

Aspects Statistiques et Aspects Physiques des Processus Gaus-

siens, pp. 349–360. Colloques Internat. C.N.R.S, St. Flour.

Publications du CNRS, Paris.

Gallavotti G (1985) Renormalization theory and ultraviolet

stability via renormalization group methods. Reviews of

Modern Physics 57: 471–569.

Gawedzky K and Kupiainen A (1983) Block spin renormalization

group for dipole gas and (@)

. Annals of Physics 147: 198–243.

Gawedzky K and Kupiainen A (1985a) Gross–Neveu model

through convergent perturbation expansion. Communications

in Mathematical Physics 102: 1–30.

Gawedzky K and Kupiainen A (1985b) Massless lattice 

theory:

rigorous control of a renormalizable asymptotically free model.

Communications in Mathematical Physics 99: 197–252.

Giuliani A and Mastropietro V (2005) Anomalous universality in

the anisotropic Ashkin–Teller model. Communications in

Mathematical Physics 256: 681–735.

Glimm J, Jaffe A, and Spencer T (1973) Velo G and Wightman A

(eds.) Constructive Field theory, Lecture Notes in Physics,

vol. 25, pp. 132–242. New York: Springer.

Glimm J and Jaffe A (1981) Quantum Physics. Springer.

Guerra F (1972) Uniqueness of the vacuum energy density and Van

Hove phenomena in the infinite volume limit for two-dimensional

self-coupled Bose fields. Physical Review Letters 28: 1213–1215.

Hepp K (1966) The´orie de la re´normalization. Lecture Notes in

Physics, vol. 2. Heidelberg: Springer.

Nelson E (1966) A quartic interaction in two dimensions. In:

Goodman R and Segal I (eds.) Mathematical Theory of

Elementary Particles, pp. 69–73. Cambridge: M.I.T.

Osterwalder K and Schrader R (1973) Axioms for Euclidean

Green’s functions. Communications in Mathematical Physics

31: 83–112.

Simon B (1974) The P(’)

Euclidean (Quantum) Field Theory.

Princeton: Princeton University Press.

Streater RF and Wightman AS (1964) PCT, Spin, Statistics and

All That. Benjamin-Cummings (reprinted Princeton University

Press, 2000).

Wightman AS and Ga¨rding L (1965) Fields as operator-valued

distributions in relativistic quantum theory. Arkiv fo¨ r Fysik

28: 129–189.

Wilson KG (1970) Model of coupling constant renormalization.

Physical Review D 2: 1438–1472.

Wilson KG (1972) Renormalization of a scalar field in strong

coupling. Physical Review D 6: 419–426.

Contact Manifolds

J B Etnyre, University of Pennsylvania,

Philadelphia, PA, USA

Introduction

Contact geometry has been seen to underly many

physical phenomena and is related to many other

mathematical structures. Contact structures first

appeared in the work of Sophus Lie on partial

differential equations. They reappeared in Gibbs’

work on thermodynamics, Huygens’ work on

geometric optics, and in Hamiltonian dynamics.

More recently, contact structures have been seen to

have relations with fluid mechanics, Riemannian

geometry, and low-dimensional topology, and these

structures provide an interesting class of subelliptic

operators.

After summarizing the basic definitions, exam-

ples, and facts concerning contact geometry, this

article discusses the connections between contact

geometry and symplectic geometry, Riemannian

geometry, complex geometry, analysis, and

dynamics. The article ends by discussing two of

the most-studied connections with physics: Hamil-

tonian dynamics and geometric optics. References

for other important topics in contact geometry

(e.g., thermodynamics, fluid dynamics, holo-

morphic c urves, and open book decompositions)

are provided in the ‘‘Further reading’’ section.

Basic Definitions and Examples

A hyperplane field  on a manifold M is a codimen-

sion-1 sub-bundle of the tangent bundle TM. Locally,

a hyperplane field can always be described as the

kernel of a 1-form. In other words, for every point in

M there is a neighborhood U and a 1-form  defined

on U such that the kernel of the linear map



: T

M !R is 

for all x in U.Theform is called

a local defining form for . A contact structure on a

(2n þ 1)-dimensional manifold M is a ‘‘maximally

nonintegrable hyperplane field’’ . The hyperplane

field  is maximally nonintegrable if for any (and hence

every) locally defining 1-form  for  the following

equation holds:

 ^ðdÞ

6¼ 0 ½1

(this means that the form is, pointwise, never equal

to 0). Geometrically, the nonintegrability of  means

that no hypersurface in M can be tangent to  along

an open subset of the hypersurface. Intuitively, this

means that the hyperplanes ‘‘twist too much’’ to be

tangent to hypersurfaces (Figure 1). The pair (M, )

Contact Manifolds 631

is called a contac t manifold and any locally defining

form  for  is called a contact form for .

Example 1 The most basic example of a contact

structure can be seen on R

2nþ1

as the kernel of the

1-form  = dz 

i = 1

, where the coordinates

on R

2nþ1

are (x

, y

, ..., x

, y

, z). This example is

shown in Figure 1 when n = 1.

Example 2 Recall that on the cotangent space of

any n-manifold M, there is a canonical 1-form ,

called the Liouville form. If (q

, ..., q

) are local

coordinates on M, then any 1-form can be expressed

i¼1

,so(q

, p

, ..., q

, p

) are local coor-

dinates on T



M. In these coordinates,

 ¼

i¼1





½2

where  : T



M !M is the natural projecti on

map. The 1-jet space of M is the manifold

(M) = T



M  R and can be considered as a bundle

over M. The 1-jet space has a natural contact

structure given as the kernel of  = dz  , where z

is the coordinate on R. Note that if M = R

then we

recover the previous example.

Example 3 The (oriented) projectivized cotangent

space of a manifold M is the set P



M of nonzero

covectors in T



M where two covectors are identified

if they differ by a positive real number, that is,



M ¼ðT



M nf0gÞ=R

½3

where {0} is the zero section of T



M and R

denotes

the positive real numbers. If M has a metric then P



can be easily identified with the space of unit

covectors. Considering P



M as unit covectors, we can

restrict the canonical 1-form  to P



M to get a 1-form

 whose kernel defines a contact structure  on P



(Although there is no canonical contact form on P



the contact structure  is still well defined.) Note that if

M is compact then so is P



M; so this gives examples of

contact structures on compact manifolds.

If  and 

are two locally defining 1-forms for ,then

there is a nonzero function f such that 

= f .Thus,



^ (d

)

= f

nþ1

 ^ (d)

is a nonzero top dimen-

sional form on M and if n is odd then the orientation

defined by the local defining form is independent of the

actual form. Hence, when n is odd, a contact structure

defines an orientation on M (this is independent of

whether or not  is orientable!). If M had a preassigned

orientation (and n is odd), then the contact structure is

called ‘‘positive’’ if it induces the given orientation and

‘‘negative’’ oth erwise. One should be careful when

reading the literature, as some authors build

positive into their definition of contact structure,

especially when n = 1. If there is a globally defined

1-form  whose kernel defines , then  is called

transversally orientable or co-orientable. This is

equivalent to the bundle  being orientable when n

is odd or when n is even and M is orientable. In

this article the discussion is restricted to transver-

sely orientable contact structures.

Suppose that  is a contact form for ,theneqn [1]

implies that d j



is a symplectic form on .This

is one sense in which a contact structure is like an

odd-dimensional analog of a symplectic structure.

AsubmanifoldL of a contact manifold (M, )is

called Legendrian if dimM=2dimL þ1andT

L 

Example 4 A fiber in the unit cotangent bundle

with the contact structure from Example 3 is a

Legendrian sphere.

Example 5 Let f : M !R be a function. Then

(f )(q) = (q,df

, f (q)) is a section of the 1-jet space

(M)ofM; it is called the 1-jet of f.Ifs is any

section of the 1-jet space, then it is Legendrian if and

only if it is the 1-jet of a function.

This observation is the basis for Lie’s study of

partial differential equations. More specifically, a

first-order partial differential equation on M can be

considered as giving an algebraic equation on J

(M).

Then, a section of J

(M) satisfying this algebraic

equation corresponds to the 1-jet of a solution to the

original partial differential equation if and only if it

is Legendrian.

Recently, Legendrian submanifolds have been

much studi ed. The re are various classification results

in three dimensions and several striking existence

results in higher dimensions.

Local Theory

The natural equivalence between contact structures

is contactomorphism. Two contact structures 

and

Figure 1 The standard contact structure on R

given as the

kernel of dz  y dx : Courtesy of Stephan Scho

nenberger.

632 Contact Manifolds



on manifolds M

and M

, respectively, are

contactomorphic if there is a diffeomorphism

f : M

such that f



(

) = 

. All contact struc-

tures are locally contactomorphic. In particular, we

have the following theorem.

Theorem 1 (Darboux’s Theorem). Suppose 

is a

contact structure on the manifold M

, i = 0, 1, and

and M

have the same dimension. Given any

points p

and p

in M

and M

, respectively, there

are neighborhoods N

of p

in M

and a contacto-

morphism from (N

, 

) to (N

, 

). Moreov er,

if 

is a contact form for 

near p

, then the

contactomorphism can be chosen to pull 

back to 

Thus, locally all contact structure s (and contact

forms!) look like the one given in Example 1 above.

Furthermore, contact structures are ‘‘local in

time.’’ That is, compact deformations of contact

structures do not produce new contact structures.

Theorem 2 (Gray’s theorem). Let M be an oriented

(2n þ 1)-dimensional manifold and 

, t 2 (0, 1), a

family of contact structures on M that agree off of

some compact subset of M. Then there is a family of

diffeomorphisms 

: M !M such that (

)





= 

In particular, on a compact manifold, all

deformations of contact structures come from

diffeomorphisms of the underlying manifold. The

theorem is not true if the contac t structures do not

agree off of a compact set. For example, there is a

one-parameter family of nonco ntactomorphic

contact structures on S

 R

Existence and Classification

The existence of contact structures on closed odd-

dimensional manifolds is quite difficult. However,

Gromov has shown that contact structures on

open manifolds obey an h-principle. To explain

this, w e note that if (M

2nþ1

, ) is a co-oriented

contact manifold then the tangent bundle of M can

be written as   R and thus the structure group

of TM can be reduced to U(n)(since has

a conformal symplectic structure on it). Such

a reduction of the structure group is called an

almost contact structure on M.Clearly,acontact

structure on M induces an almost contact struc-

ture. If M is an open manifold, Gromov proved

that the inclusion of the space of co-oriented

contact structures o n M into the space of almost

contact structures on M is a weak homotopy

equivalence. In particular, if an open manifold

meets the necessary algebraic condition for the

existence of an almost contact structure, then the

manifold has a co-oriented contact structure.

Lutz and Martinet proved a similar, but weaker,

result for oriented closed 3-manifolds. More

specifically, every closed oriented 3-manifold admits

a co-oriented contact structure and in fact has at least

one for every homotopy class of plane field. There has

been much progress on classifying contact structures

on 3-manifolds and here an interesting dichotomy has

appeared. Contact structures break into one of two

types: tight or overtwisted. Overtwisted contact

structures obey an h-principle and are in general easy

to understand. Tight contact structures have a more

subtle, geometric nature. In higher dimensions there is

much less known about the existence (or classification)

of contact structures.

Relations with Symplectic Geometry

Let (X, !) be a symplectic manifold. A vector field v

satisfying

! ¼ ! ½4

(where L

! is the Lie derivative of ! in the direction

of v) is called a symplectic dilation. A compact

hypersurface M in (X, !) is said to have ‘‘contact

type’’ if there exists a symplectic dilation v in a

neighborhood of M that is transverse to M.Givena

hypersurface M in (X, !), the characteristic line field

LM in the tangent bundle of M is the symplectic

complement of TM in TX.(SinceM is codimension 1,

it is coisotropic; thus, the symplectic complement lies

in TM and is one dimensional.)

Theorem 3 Let M be a compact hy persurface in a

symplectic manifold (X, !) and denote the inclusion

map i : M !X. Then M has contact type if and only

if there exists a 1-form  on M such that d = i



and the form  is never zero on the characteristic

line field.

If M is a hypersurface of contact type, then the

1-form  is obtained by contracting the symplectic

dilation v into the symplectic form:  = 

!.Itis

easy to verify that the 1-form  is a contact form

on M. Thus, a hypersurface of contact type in a

symplectic m anifold inherits a co-oriented contact

structure.

Given a co-orientable contact manifold (M, ), its

symplectization Symp(M, ) = (X, !) is constructed

as follows. The manifold X = M  (0, 1), and given

a global contact form  for  the symplectic

form is ! = d(t), where t is the coordinate on R.

(The symplec tization is also equivalently defined as

(M  R, d(e

)).)

Example 6 The symplectization of the standard

contact structure on the unit cotangent bundle

Contact Manifolds 633

(see Example 3) is the standard symplectic structure

on the complement of the zero section in the

cotangent bundle.

The symplectization is independent of the choice

of contact from . To see this, fix a co-orientation

for  and note the manifold X which can be

identified (in many ways) with the sub-bundle of



M whose fiber over x 2 M is

f 2 T



M : ð

Þ¼0and

>0 on vectors positively transverse to 

g½5

and restricting d to this subspace yields a symplec-

tic form !, where  is the Liouville form on T



defined in Example 2. A choice of contact form 

fixes an identification of X with the sub-bundle of



M under which d(t) is taken to d.

The vector field v = @=@t on (X, !) is a symplectic

dilation that is transverse to M  {1}  X. Clearly,



M{1}

= . Thus, we see that any co-orientable

contact manifold can be realized as a hypersurface

of contact type in a symplectic manifold. In

summary, we have the following theorem.

Theorem 4 If (M, ) is a co-oriented contact

manifold, then there is a symplectic manifold

Symp(M, ) in which M sits as a hypersurface of

contact type. Moreover, any contact form  for 

gives an embedding of M into Symp(M, ) that

realizes M as a hypersurface of contact type.

We also note that all the hypersurfaces of contact

type in (X, !) look locally, in X, like a contact

manifold sitting inside its symplec tification.

Theorem 5 Given a compact hypersurface M of

contact type in a symplectic manifold (X, !) with the

symplectic dilation given by v, there is a neighbor-

hood of M in X symplectomorphic to a neighbor-

hood of M  {1} in Symp(M, ) where the

symplectization is identified with M  (0, 1) using

the contact form  = 

and  = ker .

The Reeb Vector Field and Riemannian

Geometry

Let (M, ) be a contact manifold. Associated to a

contact form  for  is the Reeb vector field v



This is the unique vector field satisfying





 ¼ 1 and 



d ¼ 0 ½6

One may readily check that v



is transverse to the

contact hyperplanes and the flow of v



preserves 

(in fact, it preserves ). These two conditions

characterize Reeb vector fields; that is, a vector

field v is the Reeb vect or field for some contact form

for  if and only if it is transverse to  and its flow

preserves .

The fundamental question concerning Reeb vector

fields asks if its flow has a (contractible) periodic

orbit. A paraphrazing of the Weinstein conjecture

asserts a positive answer to this question. Most

progress on this conjecture has been made in

dimension 3 where H Hofer has proved the

existence of periodic orbits for all Reeb fields on S

and on 3-manifolds with essential spheres

(i.e., embedded S

’s that do not bound a 3-ball in

the manifol d). Relations with Hamiltonian dynamics

are discussed below.

Recall, from Example 3, that a Riemannian metric

g on a manifold M provides an identification of the

(oriented) projectivized cotangent bundle P



M with

the unit cotangent bundle. Considered as a subset of



M, P



M inherits not only a contact structure but

also a contact form  (by restricting the Liouville

form). Let v



be the associated Reeb vector field.

The metric g also provides an identification of the

tangent and cotangent bundles of M. Thus, P



may be considered as the unit tangent bundle of M.

Let w

be the vector field on the unit tangent bundle

generating the geodesic flow on M.

Theorem 6 The Reeb vector field v



is identified

with geodesic flow field w

when P



M is identified

with the unit tangent space using the metric g.

Relations with Complex Geometry

and Analysis

Let X be a complex manifold with boundary and

denote the induced complex structure on TX by J.

The complex tangencies  to M = @X are described

by the equation d  J = 0, where  is a function

defined in a neighborhood of the boundary such that

0 is a regular value and 

1

(0) = M. The form

L(v, w) = d(d  J)(v, Jw), for v, w 2 , is called

the Levi form, and when L(v, w) is positive

(negative) defi nite, then X is said to have strictly

pseudoconvex (pseudoconcave) boundary. The

hyperplane field  will be a contact structure if and

only if d(d  J) is a nondegenerate 2-form on  (if

and only if L(v, w) is definite). A well-studied source

of examples comes from Stein manifolds.

Example 7 Let X be a complex manifold and

again let J denote the induced complex structure

on TX. From a function  : X ! R, we can define a

2-form ! = d(d  J) and a symmetric form

g(v, w) = !(v, Jw). If this symmetric form is positive

definite, the function  is called ‘‘strictly plurisub-

harmonic.’’ The manifold X is a Stein manifold if X

634 Contact Manifolds