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Quantum Field Theory*

Quantum Field Theory*



Quantum

y

Frank Wilczek

Institute for Advanced Study, School of Natural , Olden Lane, Princeton, NJ 08540

I discuss the general principles underlying eld theory, and attempt to identify its

most profound consequences. The deep est of these consequences result from the in nite number of

degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes,

b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the

of its history.

I. SURVEY

Quantum eld theory is the framework in which the regnant of the electroweak and strong ,

which together form the Standard Mo del, are formulated. Quantum electro (QED), b esides providing a com-

plete foundation for atomic and , has supp orted calculations of physical quantities with unparalleled

precision. The exp erimentally measured value of the magnetic dip ole moment of the ,

11

(g 2) = 233 184 600 (1680)  10 ; (1)

 exp:

for example, should b e compared with the theoretical

11

(g 2) = 233 183 478 (308)  10 : (2)

 theor:

In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet

QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum

eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena

characteristic of quantum eld theory. These include esp ecially running of the e ective with distance or

scale and the phenomenon of con nement. QCD has supp orted, and rewarded with exp erimental con rmation,

b oth heroic calculations of multi-lo op diagrams and massivenumerical simulations of (a discretized version of ) the

complete theory.

Quantum eld theory also provides p owerful to ols for condensed physics, esp ecially in connection with the

quantum many-b o dy problem as it arises in the theory of metals, sup erconductivity,thelow-temp erature b ehavior

3 4

of the quantum liquids He and He , and the quantum Hall e ect, among others. Although for reasons of and

fo cus I will not attempt to do justice to this asp ect here, the continuing interchange of ideas b etween condensed matter

and high energy theory, through the medium of quantum eld theory, is a remarkable phenomenon in itself. A partial

list of historically imp ortant examples includes global and lo cal sp ontaneous breaking, the

, e ective eld theory, , , and fractional and .

It is clear, from all these examples, that quantum eld theory o ccupies a central p osition in our description of .

It provides b oth our b est working description of fundamental physical laws, and a fruitful to ol for investigating the

b ehavior of complex systems. But the enumeration of examples, however triumphal, serves more to p ose than to

answer more basic questions: What are the essential features of quantum eld theory? What do es quantum eld

theory add to our understanding of the world, that was not already presentinquantum and classical eld

theory separately?

The rst question has no sharp answer. Theoretical are very exible in adapting their to ols, and no

axiomization can keep up with them. However I think it is fair to say that the characteristic, core ideas of quantum

eld theory are twofold. First, that the basic dynamical degrees of freedom are op erator functions of space and

{ quantum elds, ob eying appropriate commutation relations. , that the interactions of these elds are lo cal.

Thus the equations of and commutation relations governing the evolution of a given quantum eld at a given

p oint in space-time should dep end only on the b ehavior of elds and their derivatives at that p oint. One might nd it

convenienttouseothervariables, whose equations are not lo cal, but in the spirit of quantum eld theory there must

always b e some underlying fundamental, lo cal variables. These ideas, combined with p ostulates of symmetry (e.g., in



To app ear in the American Physical So cietyCentenary issue of Reviews of Mo dern Physics, March 1999.

y

[email protected] IASSNS-HEP 98/20 1

the context of the standard mo del, Lorentz and gauge invariance) turn out to b e amazingly p owerful, as will emerge

from our further discussion b elow.

The eld concept came to dominate physics starting with the of Faraday in the mid-nineteenth century.Its

conceptual advantage over the earlier Newtonian program of physics, to formulate the fundamental laws in terms of

among atomic , emerges when we takeinto account the circumstance, unknown to (or, for

that matter, Faraday) but fundamental in sp ecial relativity, that in uences travel no farther than a nite limiting

sp eed. For then the on a given at a given time cannot b e deduced from the p ositions of other particles

at that time, but must b e deduced in a complicated way from their previous p ositions. Faraday's intuition that the

fundamental laws of could b e expressed most simply in terms of elds lling space and time was of

course brilliantly vindicated byMaxwell's mathematical theory.

The concept of lo cality, in the crude form that one can predict the b ehavior of nearby ob jects without reference

to distant ones, is basic to scienti c practice. Practical exp erimenters { if not astrologers { con dently exp ect, on

the of much successful exp erience, that after reasonable (generally quite mo dest) precautions to isolate their

exp eriments they will obtain repro ducible results. Direct quantitative tests of lo cality, or rather of its close cousin

, are a orded by disp ersion relations.

The deep and ancient historic ro ots of the eld and lo cality concepts provide no guarantee that these concepts

remain relevantorvalid when extrap olated far b eyond their origins in exp erience, into the subatomic and quantum

domain. This extrap olation must b e judged by its fruits. That brings us, naturally, to our second question.

Undoubtedly the single most profound fact ab out Nature that quantum eld theory uniquely explains is the existence

of di erent, yet indistinguishable, copies of elementary particles.Two anywhere in the , whatever

their origin or history, are observed to have exactly the same prop erties. We understand this as a consequence of the

fact that b oth are excitations of the same underlying ur-stu , the eld. The electron eld is thus the primary

. The same , of course, applies to or , or even to comp osite ob jects such as atomic nuclei,

, or . The indistinguishability of particles is so familiar, and so fundamental to all of mo dern physical

science, that we could easily take it for granted. Yet it is by no means obvious. For example, it directly contradicts

one of the pillars of Leibniz' , his \principle of the identity of indiscernables," according to whichtwo

ob jects cannot di er solely in numb er. And Maxwell thought the similarity of di erent molecules so remarkable that

he devoted the last part of his Encyclopedia Brittanica entry on Atoms { well over a thousand words { to discussing

it. He concluded that \the formation of a is therefore an not b elonging to that order of nature in

whichwe live ... it must b e referred to the ep o ch, not of the formation of the or the solar system ... but of the

establishment of the existing order of nature ..."

The existence of classes of indistinguishable particles is the necessary logical prerequisite to a second profound insight

from quantum eld theory: the assignment of unique quantum statistics to each class. Given the indistinguishabilityof

a class of elementary particles, and complete invariance of their interactions under interchange, the general principles of

teach us that solutions forming any representation of the p ermutation retain that

prop erty in time, but do not constrain which representations are realized. Quantum eld theory not only explains the

existence of indistinguishable particles and the invariance of their interactions under interchange, but also constrains

the symmetry of the solutions. For b osons only the identity representation is physical (symmetric functions),

for only the one-dimensional o dd representation is physical (antisymmetric wave functions). One also has

the -statistics , according to which ob jects with spin are b osons, whereas ob jects with half o dd

integer spin are fermions. Of course, these general have b een veri ed in many exp eriments. The

character of electrons, in particular, underlies the stability of matter and the structure of the p erio dic table.

A third profound general insight from quantum eld theory is the existenceofantiparticles. This was rst inferred

by Dirac on the basis of a brilliant but obsolete interpretation of his equation for the electron eld, whose elucidation

was a crucial step in the formulation of quantum eld theory. In quantum eld theory,were-interpret the Dirac

wave as a p osition (and time) dep endent op erator. It can b e expanded in terms of the solutions of the Dirac

equation, with op erator co ecients. The co ecients of p ositive-energy solutions are op erators that destroy electrons,

and the co ecients of the negative-energy solutions are op erators that create p ositrons (with p ositive energy). With

this interpretation, an improved version of Dirac's hole theory emerges in a straightforward way. (Unlike the original

hole theory, it has a sensible generalization to b osons, and to pro cesses where the numb er of electrons minus p ositrons

changes.) Avery general consequence of quantum eld theory,valid in the presence of arbitrarily complicated

interactions, is the CPT theorem. It states that the pro duct of charge conjugation, , and time reversal is always

a symmetry of the world, although eachmay b e { and is! { violated separately. are strictly de ned as

the CPT conjugates of their corresp onding particles.

The three outstanding facts wehave discussed so far: the existence of indistinguishable particles, the phenomenon

of quantum statistics, and the existence of antiparticles, are all essentially consequences of free quantum eld theory.

When one incorp orates interactions into quantum eld theory,two additional general features of the world immediately

b ecome brightly illuminated. 2

The rst of these is the ubiquity of particle creation and destruction processes.Localinteractions involve pro ducts of

eld op erators at a p oint. When the elds are expanded into creation and op erators multiplying mo des,

we see that these interactions corresp ond to pro cesses wherein particles can b e created, annihilated, or changed

into di erent kinds of particles. This p ossibility arises, of course, in the primeval quantum eld theory, quantum

electro dynamics, where the primary arises from a pro duct of the electron eld, its Hermitean conjugate,

and the eld. Pro cesses of and absorption of photons by electrons (or p ositrons), as well as electron-

p ositron pair creation, are enco ded in this pro duct. Just b ecause the emission and absorption of lightissucha

common exp erience, and electro dynamics such a sp ecial and familiar classical eld theory, this corresp ondence b etween

formalism and reality did not initially make a big impression. The rst conscious exploitation of the p otential for

quantum eld theory to describ e pro cesses of transformation was Fermi's theory of b decay. He turned the pro cedure

around, inferring from the observed pro cesses of particle transformation the nature of the underlying lo cal interaction

of elds. Fermi's theory involved creation and annihilation not of photons, but of atomic nuclei and electrons (as well

as ) { the ingredients of `matter'. It b egan the pro cess whereby classic , involving stable individual

ob jects, was replaced by a more sophisticated and accurate picture. In this picture it is only the elds, and not the

individual ob jects they create and destroy, that are p ermanent.

The second is the association of forces and interactions with particle exchange. When Maxwell completed the

equations of electro dynamics, he found that they supp orted source-free electromagnetic . The classical electric

and magnetic elds thus to ok on a life of their own. Electric and magnetic forces b etween charged particles are

explained as due to one particle acting as a source for electric and magnetic elds, which then in uence others. With

the corresp ondence of elds and particles, as it arises in quantum eld theory, Maxwell's discovery corresp onds to

the existence of photons, and the generation of forces byintermediary elds corresp onds to the exchange of virtual

photons. The asso ciation of forces (or, more generally,interactions) with exchange of particles is a general feature

of quantum eld theory.Itwas used byYukawa to infer the existence and of from the range of nuclear

forces, and more recently in electroweak theory to infer the existence, mass, and prop erties of W and Z b osons prior

to their observation, and in QCD to infer the existence and prop erties of jets prior to their observation.

The two additional outstanding facts we just discussed: the p ossibility of particle creation and destruction, and the

asso ciation of particles with forces, are essentially consequences of classical eld theory supplemented by the connection

between particles and elds we learn from free eld theory. Indeed, classical waves with nonlinear interactions will

change form, scatter, and radiate, and these pro cesses exactly mirror the transformation, interaction, and creation of

particles. In quantum eld theory, they are prop erties one sees already in treegraphs.

The foregoing ma jor consequences of free quantum eld theory, and of its formal extension to include nonlinear

interactions, were all well appreciated by the late 1930s. The deep er prop erties of quantum eld theory, which will form

the sub ject of the remainder of this pap er, arise from the need to intro duce in nitely many degrees of freedom, and the

p ossibility that all these degrees of freedom are excited as quantum-mechanical uctuations. From a mathematical

p oint of view, these deep er prop erties arise when we consider loop graphs.

>From a physical p oint of view, the p otential pitfalls asso ciated with the existence of an in nite numb er of degrees of

freedom rst showed up in connection with the problem which led to the birth of , that is the ultraviolet

catastrophe of blackb o dy radiation theory. Somewhat ironically, in view of later history, the crucial role of the quantum

theory here was to remove the disastrous consequences of the in nite numb er of degrees of freedom p ossessed by

classical electro dynamics. The classical electro dynamic eld can b e decomp osed into indep endent oscillators with

arbitrarily high values of the wavevector. According to the of classical , in

thermal equilibrium at temp erature T each of these oscillators should haveaverage energy kT . Quantum mechanics

alters this situation by insisting that the oscillators of frequency ! have energy quantized in units of h! . Then the

h!

kT

h! e

.The high-frequency mo des are exp onentially suppressed by the Boltzmann factor, and instead of kT receive

h!

kT

1e

role of the quantum, then, is to prevent accumulation of energy in the form of very small amplitude excitations of

arbitrarily high frequency mo des. It is very e ective in suppressing the thermal excitation of high-frequency mo des.

But while removing arbitrarily small amplitude excitations, quantum theory intro duces the idea that the mo des are

always intrinsical ly excited to a small extent, prop ortional toh  . This so-called zero p oint motion is a consequence

of the . For a of frequency ! , the energy is not zero, but

1

h! . In the case of the electromagnetic eld this , up on summing over its high-frequency mo des, to a highly

2

divergent total ground state energy.For most physical purp oses the absolute normalization of energy is unimp ortant,

1

and so this particular divergence do es not necessarily render the theory useless. It do es, however, illustrate the

dangerous character of the high-frequency mo des, and its treatmentgives a rst indication of the leading theme of

1

One would think that should care ab out the absolute normalization of energy. The zero-p oint energy of the elec- 3

renormalization theory: we can only require { and generally will only obtain { sensible, nite answers when weask

questions that have direct, op erational physical meaning.

The existence of an in nite numb er of degrees of freedom was rst encountered in the theory of the electromagnetic

eld, but it is a general phenomenon, deeply connected with the requirementoflocalityintheinteractions of elds.

For in order to construct the lo cal eld () at a space-time p oint x, one must take a sup erp osition

Z

4

d k

ik x

~

(x)= e (k ) (3)

4

(2 )

~

that includes eld comp onents (k ) extending to arbitrarily large momenta. Moreover in a generic interaction

Z Z Z

4 4 4

d k d k d k

2 3 1

3 4 4

~ ~ ~

L = (x) = (k ) (k ) (k )(2 )  (k + k + k ) (4)

1 2 3 1 2 3

4 4 4

(2 ) (2 ) (2 )

we see that a lowmomentum mo de k  0 will couple without any suppression factor to high- mo des k

1 2

and k k . Lo cal couplings are \hard", in this sense. Because lo cality requires the existence of in nitely many

3 2

degrees of freedom at large momenta, with hard interactions, ultraviolet divergences similar to the ones cured by

Planck, but driven by quantum rather than thermal uctuations, are never far o stage. As mentioned previously,

the deep er physical consequences of quantum eld theory arise from this circumstance.

First of all, it is much more dicult to construct non-trivial examples of interacting relativistic quantum eld

theories than purely formal considerations would suggest. One nds that the consistent quantum eld theories form

a quite limited class, whose extent depends sensitively on the of space-time and the spins of the particles

involved. Their construction is quite delicate, requiring limiting pro cedures whose logical implementation leads directly

to renormalization theory, the running of couplings, and .

Secondly, even those quantum theories that can beconstructed display less symmetry than their formal properties

would suggest. Violations of naive scaling relations { that is, ordinary { in QCD, and of

numb er conservation in the standard electroweak mo del are examples of this general phenomenon. The original

o

example, unfortunately to o complicated to explain fully here, involved the decay pro cess  ! , for whichchiral

symmetry (treated classically) predicts much to o small a rate. When the correction intro duced by quantum eld

theory (the so-called `') is retained, excellent agreement with exp eriment results.

These deep er consequences of quantum eld theory, whichmight sup er cially app ear rather technical, largely dictate

the structure and b ehavior of the Standard Mo del { and, therefore, of the physical world. My goal in this preliminary

survey has b een to emphasize their profound origin. In the rest of the article I hop e to convey their main implications,

in as simple and direct a fashion as p ossible.

II. FORMULATION

The physical constants h and c are so deeply emb edded in the formulation of relativistic quantum eld theory that

it is standard practice to declare them to b e the units of and velo city, resp ectively. In these units, of course,

h = c = 1. With this convention, all physical quantities of interest have units whicharepowers of mass. Thus

1

the dimension of momentum is (mass) or simply 1, since massc isamomentum, and the dimension of length is

1

(mass) or simply -1, since hc/mass is a length. The usual way to construct quantum eld theories is by applying

the rules of to a continuum eld theory, following the canonical pro cedure of replacing Poisson brackets

1

by (or, for fermionic elds, anticommutators). The eld theories that describ e free spin 0 or free spin

2

elds of mass m;  resp ectively are based on the Lagrangian densities

2

1 m

2

L (x)= @ (x)@ (x) (x) (5)

0

2 2



1

L (x)= (x)(i @ ) (x): (6)

2

tromagnetic eld, in that context, generates an in nite . This might b e cancelled by similar negative

contributions from fermion elds, as o ccurs in sup ersymmetric theories, or it might indicate the need for some other profound

mo di cation of physical theory. 4

R

4

Since the action d xL has mass dimension 0, the mass dimension of a eld like  is 1 and of a eld

3

.For free spin 1 elds the Lagrangian densityisthatofMaxwell, like is

2

1

L (x)= (@ A (x) @ A (x))(@ A (x) @ A (x)); (7)

1

4

so that the mass dimension of the vector eld A is 1. The same result is true for non-ab elian vector elds (Yang-Mills

elds).

Thus far all our Lagrangian densities have b een quadratic in the elds. Lo cal interaction terms are obtained from

Lagrangian densities involving pro ducts of elds and their derivatives at a p oint. The co ecient of such a term is a

, and must have the appropriate mass dimension so that the Lagrangian density has mass dimension

4. Thus the mass dimension of a Yukawa coupling y , whichmultiplies the pro duct of two spinor elds and a scalar

eld, is zero. Gauge couplings g arising in the pro cedure @ ! @ + ig A are also evidently of mass

dimension zero.

The p ossibilities for couplings with non- dimension are very restricted. This fact is quite imp ortant,

for the following reason. Consider the e ect of treating a given interaction term as a p erturbation. If the coupling 

asso ciated to this interaction has negative mass dimension -p, then successivepowers of it will o ccur in the form of

p

powers of  , where  is some parameter with of mass. Because, as wehave seen, the interactions in a

lo cal eld theory are hard, wecananticipate that  will characterize the largest mass scale we allow to o ccur (the

cuto ), and will diverge to in nity as the limit on this mass scale is removed. So we exp ect that it will b e dicult

to make sense of fundamental interactions having negative mass dimensions, at least in p erturbation theory.Such

interactions are said to b e nonrenormalizable.

The standard mo del is formulated entirely using renormalizable interactions. It has b een said that this is not in

itself a fundamental fact ab out nature. For if non-renormalizable interactions o ccurred in the e ective description

of physical b ehavior b elow a certain mass scale, it would simply mean that the theory must change its nature {

presumably by displaying new degrees of freedom { at some larger mass scale. If we adopt this p oint of view, the

signi cance of the fact that the standard mo del contains only renormalizable op erators is that it do es not require

mo di cation up to arbitrarily high scales (at least on the grounds of divergences in p erturbation theory). Whether

or not we call this a fundamental fact, it is certainly a profound one.

Moreover, all the renormalizable interactions consistent with the gauge symmetry and structure of the

standard mo del do seem to o ccur { \what is not forbidden, is mandatory". There is a b eautiful agreementbetween the

of the standard mo del, allowing arbitrary renormalizable interactions, and the symmetries of the world.

One understands why is violated, but baryon numb er is not. (The only discordant element is the so-called

 term of QCD, which is allowed by the symmetries of the standard mo del but is measured to b e quite accurately

zero. A plausible solution to this problem exists. It involves a characteristic very light eld.)

The p ower counting rules for estimating divergences assume that there are no sp ecial symmetries cancelling o the

contribution of high energy mo des. They do not apply, without further consideration, to sup ersymmetric theories, in

which the contributions of b oson and fermionic mo des cancels, nor to theories derived from sup ersymmetric theories

by soft sup ersymmetry breaking. In the latter case the scale of sup ersymmetry breaking plays the role of the cuto

.

2

The p ower counting rules, as discussed so far, are to o crude to detect divergences of the form ln  .Yet divergences

of this form are p ervasive and extremely signi cant, as we shall now discuss.

III. RUNNING COUPLINGS

The problem of calculating the energy asso ciated with a constant magnetic eld, in the more general context of

1

an arbitrary nonab elian coupled to spin 0 and spin charged particles, provides an excellent concrete

2

illustration of how the in nities of quantum eld theory arise, and of how they are dealt with. It intro duces the

concept of running couplings in a natural way, and leads directly to qualitative and quantitative results of great

signi cance for physics. The interactions of concern to us app ear in the Lagrangian density

1

I  y  2 I



G + (i D ) +  (D D m ) (8) G L =

 

2

4g

I I I IJ K J K I I

where G  @ A @ A f A A and D  @ + iA T are the standard eld strengths and covariant deriva-

 



IJ K I

tive, resp ectively. Here the f are the of the gauge group, and the T are the representation

matrices appropriate to the eld on which the acts. This Lagrangian di ers from the usual one 5

by a rescaling gA ! A, whichserves to emphasize that the gauge coupling g o ccurs only as a prefactor in the rst

term. It parametrizes the energetic cost of non-trivial gauge curvature, or in other words the sti ness of the gauge

elds. Small g corresp onds to gauge elds that are dicult to excite.

I

From this Lagrangian itself, of course, it would app ear that the energy required to set up a magnetic eld B is

1

I 2

(B ) . This is the classical energy, but in the quantum theory it is not the whole story. A more accurate just

2

2g

calculation must takeinto account the e ect of the imp osed magnetic eld on the zero-p oint energy of the charged

elds. Earlier, we met and brie y discussed a formally in nite contribution to the energy of the ground state of a

quantum eld theory (sp eci cally, the electromagnetic eld) due to the irreducible quantum uctuations of its mo des,

which mapp ed to an in nite numb er of indep endent harmonic oscillators. Insofar as only di erences in energy are

physically signi cant, we could ignore this in nity.Butthechange in the zero-p oint energy as one imp oses a magnetic

eld cannot b e ignored. It represents a genuine contribution to the physical energy of the induced by

the imp osed magnetic eld. As we will so on see, the eld-dep endent part of the energy also diverges.

Postp oning momentarily the derivation, let me anticipate the form of the answer, and discuss its interpretation.

Without loss of generality, I will supp ose that the magnetic eld is aligned along a normalized, diagonal of

the gauge group. This allows us to drop the index, and to use terminology and intuition from electro dynamics freely.

If we restrict the sum to mo des whose energy is less than a cuto , we ndfortheenergy

1 1

2 2 2

E (B )=E +  E = B B (ln( =B ) + nite) (9)

2 2

2g ( ) 2

where

1 1

1 1

)+2T ( ))] + )+8T (R ))]; (10)  = [(T (R ) 2T (R [3(2T (R

1 1 o

2 2

2 2

96 96

2 2

and the terms not displayed are nite as  !1. The notation g ( ) has b een intro duced for later convenience.

The factor T (R ) is the trace of the representation for spin s, and basically represents the sum of the squares of the

s

charges for the particles of that spin. The denominator in the logarithm is xed by dimensional analysis, assuming

2 2

B>> ;m .

The most striking, and at rst sight disturbing, asp ect of this calculation is that a cuto is necessary in order

to obtain a nite result. If we are not to intro duce a new fundamental scale, and thereby (in view of our previous

discussion) endanger lo cality,wemust remove reference to the arbitrary cuto  in our description of physically

meaningful quantities. This is the sort of problem addressed by the renormalization program. Its guiding idea is

the thoughtthatifweareworking with exp erimental prob es characterized by energy and momentum scales well

b elow,we should exp ect that our capacity to a ect, or b e sensitive to, the mo des of much higher energy will b e

quite restricted. Thus one exp ects that the cuto , whichwas intro duced as a calculational device to removesuch

mo des, can b e removed (taken to in nity). In our magnetic energy example, for instance, we see immediately that

the di erence in susceptibilities

2 2

E (B )=B E(B )=B = nite (11)

1 0

1 0

is well-b ehaved { that is, indep endent ofas!1.Thus once we the susceptibility, or equivalently the

coupling constant, at one reference value of B , the calculation gives sensible, unambiguous predictions for all other

values of B .

This simple example illustrates a much more general result, the central result of the classic renormalization program.

It go es as follows. A small number of quantities, corresp onding to the couplings and in the original Lagrangian,

that if calculated formally would diverge or dep end on the cuto , are chosen to t exp eriment. They de ne the physical,

1

as opp osed to the original, or bare, couplings. Thus, in our example, we can de ne the susceptibilitytobe at

2

2g (B )

0

some reference eld B .Thenwehave the physical or renormalized coupling

0

1 1

2

=  ln( =B ): (12)

0

2 2 2

g (B ) g ( )

0

(In this equation I have ignored, for simplicity in exp osition, the nite terms. These are relatively negligible for large

2

B . Also, there are corrections of higher order in g .) This of course determines the `bare' coupling to b e

0

1 1

2

= +  ln( =B ): (13)

0

2 2 2

g ( ) g (B )

0

In these terms, the central result of diagrammatic renormalization theory is that after bare couplings and masses

are re-expressed in terms of their physical, renormalized counterparts, the co ecients in the p erturbation expansion 6

of anyphysical quantity approach nite limits, indep endent of the cuto , as the cuto is taken to in nity.(Tobe

p erfectly accurate, one must also p erform wave-function renormalization. This is no di erent in principle; it amounts

to expressing the bare co ecients of the kinetic terms in the Lagrangian in terms of renormalized values.)

The question whether this p erturbation theory converges, or is some sort of asymptotic expansion of a soundly

de ned theory, is left op en by the diagrammatic analysis. This lo ophole is no mere technicality,aswe will so on see.

Picking a scale B at which the coupling is de ned is analogous to cho osing the origin of a co ordinate system in

0

. One can describ e the same physics using di erentchoices of normalization scale, so long as one adjusts the

coupling appropriately.We capture this idea byintro ducing the concept of a running coupling de ned, in accordance

with equation (12), to satisfy

1 d

= : (14)

2

d ln B g (B )

With this de nition, the choice of a particular scale at which to de ne the coupling will not a ect the nal result.

It is profoundly imp ortant, however, that the running coupling do es make a real distinction b etween the b ehavior

at di erent mass scales, even if the original underlying theory was formally scale (as is QCD with massless

quarks), and even at mass scales much larger than the mass of any particle in the theory.Quantum zero-p oint motion

of the high energy mo des intro duces a hard source of scale symmetry violation.

The distinction among scales, in a formally scale-invarianttheory,emb o dies the phenomenon of dimensional trans-

mutation. Rather than a range of theories, parametrized by a dimensionless coupling, wehave a range of theories

2

di ering only in the value of a dimensional parameter, say (for example) the value of B at which1=g (B )=1.

2

Clearly, the qualitative b ehavior of solutions of eq. (14) dep ends on the sign of  .If>0, the coupling g (B ) will

get smaller as B grows, or in other words as we treat more and more mo des as dynamical, and approach closer to the

`bare' charge. These mo des were enhancing, or antiscreening the bare charge. This is the case of asymptotic freedom.

2

In the opp osite case of <0 the coupling formally grows, and even diverges as B increases. 1=g (B ) go es through

zero and changes sign. On the face of it, this would seem to indicate an instability of the theory,toward formation

of a ferromagnetic at large eld strength. This conclusion must b e taken with a big grain of salt, b ecause

2

when g is large the higher-order corrections to eq. (13) and eq. (14), on which the analysis was based, cannot b e

neglected.

In asymptotically free theories, we can complete the renormalization program in a convincing fashion. There is

no barrier to including the e ect of very large energy mo des, and removing the cuto . We can con dently exp ect,

then, that the theory is well-de ned, indep endent of p erturbation theory. In particular, supp ose the theory has b een

discretized on a space-time lattice. This amounts to excluding the mo des of high energy and momentum.Inan

asymptotically free theory one can comp ensate for these mo des by adjusting the coupling in a well-de ned, controlled

way as one shrinks the discretization scale. Very impressive nonp erturbative calculations in QCD, involving massive

computer simulations, have exploited this strategy. They demonstrate the complete consistency of the theory and its

ability to accountquantitatively for the masses of .

In a non-asymptotically free theory the coupling do es not b ecome small, there is no simple fo olpro of wayto

comp ensate for the missing mo des, and the existence of an underlying limiting theory b ecomes doubtful.

Now let us discuss how  can b e calculated. The two terms in eq. (10) corresp ond to twodistinctphysical e ects.

The rst is the convective, diamagnetic (screening) term. The overall constant is a little tricky to calculate, and I do

not have space to do it here. Its general form, however, is transparent. The e ect is indep endent of spin, and so it

simply counts the numb er of comp onents (one for scalar particles, two for spin-1/2 or massless spin-1, b oth with two

helicities). It is screening for b osons, while for fermions there is a sign ip, b ecause the zero-p oint energy is negative

for fermionic oscillators.

The second is the paramagetic spin susceptibility.For a with spin s and g

m

the shift, giving rise to the altered zero-p oint energy

Z

E =

3

p p p

d k 1

2 2 2

E = k + g sB + k g sB 2 k ): (15) (

m m

3

(2 ) 2

0

This is readily calculated as

2

1 

2 2

E = B (g s) ln( ): (16)

m

2

32 B

1

With g =2, s = 1 (and T = 1) this is the spin-1 contribution, and with g =2, s = , after a sign ip, it is the

m m

2

1

spin- contribution. The preferred moment g = 2 is a direct consequence of the Yang-Mills and Dirac equations,

m

2

resp ectively. 7

This elementary calculation gives us a nice heuristic understanding of the unusual antiscreening b ehavior of non-

ab elian gauge theories. It is due to the large paramagnetic resp onse of charged vector elds. Because weareinterested

in very high energy mo des, the usual intuition that charge will b e screened, which is based on the electric resp onse of

heavy particles, do es not apply. Magnetic interactions, which can b e attractive for likecharges () are,

for highly relativistic particles, in no way suppressed. Indeed, they are numerically dominant.

Though I have presented it in the very sp eci c context of vacuum magnetic susceptibility, the concept of running

coupling is much more widely applicable. The basic heuristic idea is that in analyzing pro cesses whose characteristic

2 2

energy-momentum scale (squared) is Q , it is appropriate to use the running coupling at Q , i.e. in our earlier

2 2

notation g (B = Q ). For in this waywe capture the dynamical e ect of the virtual oscillators which can b e

appreciably excited, while avoiding the formal divergence encountered if we tried to include all of them (up to in nite

mass scale). At a more formal level, use of the appropriate e ective coupling allows us to avoid large logarithms in

the calculation of Feynman graphs, by normalizing the vertices close to where they need to b e evaluated. There is

a highly develop ed, elab orate chapter of quantum eld theory which justi es and re nes this rough idea into a form

where it makes detailed, quantitative predictions for concrete exp eriments. I will not b e able to do prop er justice

to the dicult, often heroic lab or that has b een invested, on b oth the theoretical and the exp erimental sides, to

yield Figure 1; but it is appropriate to remark that quantum eld theory gets a real workout, as calculations of two-

and even three-lo op graphs with complicated interactions among the virtual particles are needed to do justice to the

attainable exp erimental accuracy.

FIG. 1. Comparison of theory and exp eriment in QCD, illustrating the running of couplings. Several of the p oints on this

curve representhundreds of indep endent measurements, any one of which mighthave falsi ed the theory. Figure from M.

Schmelling, hep-ex/9701002.

2

An interesting feature visible in Figure 1 is that the theoretical prediction for the coupling fo cuses at large Q ,

2 2

in the sense that a wide range of values at small Q converge to a muchnarrower range at larger Q .Thus even

2 2

crude estimates of what are the appropriate scales (e.g., one exp ects g (Q )=4  1 where the

p

2 2

2

< <

Q is strong, say for 100 Mev 1 Gev) allow one to predict the value of g (M )with10% accuracy.The

  Z

original idea of Pauli and others that calculating the ne structure constantwas the next great item on the agenda of

now seems misguided. We see this constant as just another running coupling, neither more nor less

fundamental than many other parameters, and not likely to b e the most accessible theoretically. But our essentially

parameter-free approximate determination of the strong interaction analogue of the ne structure constant

realizes a form of their dream.

The electroweak interactions start with much smaller couplings at low mass scales, so the e ects of their running

are less dramatic (though they have b een observed). Far more sp ectacular than the mo dest quantitative e ects we can

test directly,however, is the conceptual breakthrough that results from application of these ideas to uni ed mo dels

of the strong, electromagnetic, and weak interactions. 8

The di erentcomponents of the standard mo del have a similar mathematical structure, all b eing gauge theories.

Their common structure encourages the sp eculation that they are di erent facets of a more encompassing gauge

symmetry, in which the di erent strong and weak color charges, as well as electromagnetic charge, would all app ear

on the same fo oting. The multiplet structure of the quarks and in the standard mo del ts b eautifully into small

representations of uni cation groups suchasSU (5) or SO(10). There is the apparent diculty,however, that the

coupling strengths of the di erent standard mo del interactions are widely di erent, whereas the symmetry required for

uni cation requires that they share a common value.The running of couplings suggests an escap e from this impasse.

Since the strong, weak, and electromagnetic couplings run at di erent rates, their inequality at currently accessible

scales need not re ect the ultimate state of a airs. We can imagine that sp ontaneous symmetry breaking { a soft

e ect { has hidden the full symmetry of the uni ed interaction. What is really required is that the fundamental, bare

couplings b e equal, or in more prosaic terms, that the running couplings of the di erentinteractions should b ecome

equal b eyond some large scale.

Using simple generalizations of the formulas derived and tested in QCD, we can calculate the running of couplings,

to see whether this requirement is satis ed in reality. In doing so one must makesomehyp othesis ab out the sp ectrum

of virtual particles. If there are additional massive particles (or, b etter, elds) that havenotyet b een observed, they

will contribute signi cantly to the running of couplings once the scale exceeds their mass. Let us rst consider the

default assumption, that there are no new elds b eyond those that o ccur in the standard mo del. The results of this

calculation are displayed in Figure 2.

FIG. 2. Running of the couplings extrap olated toward very high scales, using just the elds of the standard mo del. The

couplings do not quite meet. Exp erimental uncertainties in the extrap olation are indicated by the width of the lines. Figure

courtesy of K. Dienes.

Considering the enormity of the extrap olation this calculation works remarkably well, but the accurate exp erimental

data indicates unequivo cally that something is wrong. There is one particularly attractiveway to extend the standard

mo del, by including sup ersymmetry. Sup ersymmetry cannot b e exact, but if it is only mildly broken (so that the

<

sup erpartners have masses 1Tev) it can help explain why radiative corrections to the Higgs mass parameter, and



thus to the scale of weak symmetry breaking, are not enormously large. In the absence of sup ersymmetry p ower

counting would indicate a hard, quadratic dep endence of this parameter on the cuto . Sup ersymmetry removes the

most divergent contribution, by cancelling b oson against fermion lo ops. If the masses of the sup erpartners are not

to o heavy, the residual nite contributions due to sup ersymmetry breaking will not b e to o large.

The minimal sup ersymmetric extension of the standard mo del, then, makes semi-quantitative predictions for the

sp ectrum of virtual particles starting at 1 Tev or so. Since the running of couplings is logarithmic, it is not extremely

sensitive to the unknown details of the sup ersymmetric mass sp ectrum, and we can assess the impact of sup ersymmetry

on the uni cation hyp othesis quantitatively. The results, as shown in Figure 3, are quite encouraging. 9

FIG. 3. Running of the couplings extrap olated to high scales, includin g the e ects of sup ersymmetric particles starting at 1

Tev. Within exp erimental and theoretical uncertainties, the couplings do meet. Figure courtesy of K. Dienes.

With all its attractions, there is one general feature of sup ersymmetry that is esp ecially challenging, and deserves

mention here. We remarked earlier how the standard mo del, without sup ersymmetry, features a near-p erfect match

between the generic symmetries of its renormalizable interactions and the observed symmetries of the world. With

sup ersymmetry, this feature is sp oiled. The scalar sup erpartners of fermions are represented by elds of mass dimension

one. This means that there are many more p ossibilities for low dimension (including renormalizable) interactions that

violate avor symmetries including and baryon numb er. It seems that some additional principles, or sp ecial

discrete symmetries, are required in order to suppress these interactions suciently.

A notable result of the uni cation of couplings calculation, esp ecially in its sup ersymmetric form, is that the

uni cation o ccurs at an energy scale which is enormously large by the standards of traditional ,

1617

p erhaps approaching 10 Gev. From a phenomenological viewp oint, this is fortunate. The most comp elling

uni cation schemes merge quarks, antiquarks, leptons, and antileptons into common , and have gauge

b osons mediating transitions among all these particle typ es. Baryon numb er violating pro cesses almost inevitably

result, whose rate is inversely prop ortional to the fourth p ower of the gauge b oson masses, and thus to the fourth

power of the uni cation scale. Only for such large values of the scale is one safe from exp erimental limits on

instability.From a theoretical p oint of view the large scale is fascinating b ecause it brings us from the internal logic of

the exp erimentally grounded domain of particle physics to the threshold of ,aswe shall now discuss.

IV. LIMITATIONS?

So much for the successes, achieved and anticipated, of quantum eld theory. The fundamental limitations of

quantum eld theory,ifany, are less clear. Its application to gravity has certainly, to date, b een much less fruitful

than its triumphant application to describ e the other fundamental interactions.

All existing exp erimental results on gravitation are adequately describ ed byavery b eautiful, conceptually simple

classical eld theory { Einstein's . It is easy to incorp orate this theory into our description of the

world based on quantum eld theory,byallowing a minimal coupling to the elds of the standard mo del { that is,

p

g , and adding an Einstein- bychanging ordinary into covariant derivatives, multiplying with appropriate factors of

Hilb ert curvature term. The resulting theory { with the convention that we simply ignore quantum corrections

involving virtual { is the foundation of our working description of the physical world. As a practical matter,

it works very well indeed.

Philosophically,however, it might b e disapp ointing if it were to o straightforward to construct a quantum theory

of gravity. One of the great visions of , going back to , is that the prop erties of the 10

world are determined uniquely by mathematical principles. A mo dern version of this vision was formulated by Planck,

shortly after he intro duced his quantum of action. By appropriately combining the physical constants c,h as units

q

hc

of velo city and action, resp ectively, and the PlanckmassM = as the unit of mass, one can construct any

Planck

G

p

2

hc, and unit of measurement used in physics. Thus the unit of energy is M c , the unit of is

Planck

so forth. On the other hand, one cannot form a pure numb er from these three physical constants. Thus one might

hop e that in a physical theory where h, c, and G were all profoundly incorp orated, all physical quantities could b e

expressed in as pure numbers.

Within its domain, QCD achieves something very close to this vision { actually,ina more ambitious form! Indeed,

let us idealize the world of the strong interaction slightly,by imagining that there werejusttwo sp ecies with

vanishing masses. Then from the twointegers 3 (colors) and 2 ( avors), h, and c { with no explicit mass parameter { a

sp ectrum of hadrons, with mass ratios and other prop erties close to those observed in reality, emerges by calculation.

The overall unit of mass is indeterminate, but this ambiguity has no signi cance within the theory itself.

The ideal Pythagorean/Planckian theory would not contain anypurenumb ers as parameters. (Pythagoras might

22

have excused a few small ). Thus, for example, the value m =M  10 of the electron mass in Planck

e Planck

units would emerge from a dynamical calculation. This ideal mightbeoverly ambitious, yet it seems reasonable to

hop e that signi cant constraints among physical will emerge from the inner requirements of a quantum

theory which consistently incorp orates gravity. Indeed, as wehave already seen, one do es nd signi cant constraints

among the parameters of the standard mo del by requiring that the strong, weak, and electromagnetic interactions

emerge from a uni ed gauge symmetry; so there is precedent for results of this kind.

The uni cation of couplings calculation provides not only an inspiring mo del, but also direct encouragementfor

the Planck program, in two imp ortant resp ects. First, it p oints to a symmetry breaking scale remarkably close to the

2 3

Planck scale (though apparently smaller by10 10 ), so there are pure numb ers with much more `reasonable'

22

values than 10 to sho ot for. Second, it shows quite concretely howvery large scale factors can b e controlled by

22

mo dest ratios of coupling strength, due to the logarithmic nature of the running of couplings { so that 10 maynot

b e so `unreasonable' after all.

Perhaps it is fortunate, then, that the straightforward, minimal implementation of general relativityasaquantum

eld theory { whichlacks the desired constraints { runs into problems. The problems are of two quite distinct kinds.

First, the renormalization program fails, at the level of p ower-counting. The Einstein-Hilb ert term in the action comes

with a large prefactor 1=G, re ecting the diculty of curving space-time. If we expand the Einstein-Hilb ert action

around at space in the form

p

g =  + Gh (17)

we nd that the quadratic terms give a prop erly normalized spin-2 eld h of mass dimension 1, as the

powers of G cancel. But the higher-order terms, which representinteractions, will b e accompanied by p ositivepowers

of G. Since G itself has mass dimension -2, these are non-renormalizable interactions. Similarly for the couplings of

gravitons to matter. Thus we can exp ect that ever-increasing p owers of =M will app ear in multiple virtual

Planck

graviton exchange, and it will b e imp ossible to remove the cuto .

Second, one of the main qualitative features of gravity{ theweightlessness of empty space, or the vanishing of the

cosmological constant { is left unexplained. Earlier wementioned the divergent zero-p oint energy characteristic of

generic quantum eld theories. For purp oses of non-gravitational physics only energy di erences are meaningful, and

we can sweep this problem under the rug. But gravity ought to see this energy. Our p erplexityintensi es when we

recall that according to the standard mo del, and even more so in its uni ed extensions, what we commonly regard

as empty space is full of condensates, which again one would exp ect to weigh far more than observation allows. The

failure, so far, of quantum eld theory to meet these challenges might re ect a basic failure of principle, or merely

that the appropriate symmetry principles and degrees of freedom, in terms of which the theory should b e formulated,

have not yet b een identi ed.

Promising insights toward construction of a quantum theory including gravity are coming from investigations in

/M theory, as discussed elsewhere in this volume. Whether these investigations will converge toward an accurate

description of nature, and if so whether this description will take the form of a lo cal eld theory (p erhaps formulated

in many dimensions, and including many elds b eyond those of the standard mo del) , are questions not yet decided.

It is interesting, in this regard, brie y to consider the ro cky intellectual history of quantum eld theory.

After the initial successes of the 1930s, already mentioned ab ove, came a long p erio d of disillusionment. Initial

attempts to deal with the in nities that arose in calculations of lo op graphs in electro dynamics, or in radiative

corrections to b eta decay, led only to confusion and failure. Similar in nities plagued Yukawa's theory, and

it had the additional diculty that the coupling required to t exp eriment is large, so that tree graphs provide a

manifestly p o or approximation. Many of the founders of quantum theory, including Bohr, Heisenb erg, Pauli, and 11

(for di erent reasons) Einstein and Schro dinger, felt that further progress required a radically new innovation. This

innovation would b e a revolution of the order of quantum mechanics itself, and would intro duce a new fundamental

length.

Quantum electro dynamics was resurrected in the late 1940s, largely stimulated bydevelopments in exp erimental

technique. These exp erimental developments made it p ossible to study atomic pro cesses with such great precision,

that the approximation a orded bykeeping tree graphs alone could not do them justice. Metho ds to extract sensible

nite answers to physical questions from the jumbled divergences were develop ed, and sp ectacular agreement with

exp erimentwas found { all without changing electro dynamics itself, or departing from the principles of relativistic

quantum eld theory.

After this wave of success came another long p erio d of disillusionment. The renormalization metho ds develop ed for

electro dynamics did not seem to work for theory. They did suce to de ne a p erturbative expansion

of Yukawa's pion theory, but the strong coupling made that limited success academic (and it came to seem utterly

implausible that Yukawa's schematic theory could do justice to the wealth of newly discovered phenomena). In any

case, as a practical matter, throughout the 1950s and 1960s a o o d of exp erimental discoveries, including new classes

of weak pro cesses and a rich sp ectrum of hadronic with complicated interactions, had to b e absorb ed

and correlated. During this pro cess of pattern recognition the elementary parts of quantum eld theory were used

extensively, as a framework, but deep er questions were put o . Many theorists came to feel that quantum eld theory,

in its deep er asp ects, was simply wrong, and would need to b e replaced by some S- or b o otstrap theory; p erhaps

most thoughtitwas irrelevant, or that its use was premature, esp ecially for the strong interaction.

As it b ecame clear, through phenomenological work, that the weak interaction is governed by currentcurrent

interactions with universal strength, the p ossibility to ascrib e it to exchange of vector gauge b osons b ecame quite

attractive. Mo dels incorp orating the idea of sp ontaneous symmetry breaking to give mass to the weak gauge b osons

were constructed. It was conjectured, and later proved, that the high degree of symmetry in these theories allows one

to isolate and control the in nities of p erturbation theory. One can carry out a renormalization program similar in

spirit, though considerably more complex in detail, to that of QED. It is crucial, here, that sp ontaneous symmetry

breaking is a very soft op eration. It do es not signi cantly a ect the symmetry of the theory at large momenta, where

the p otential divergences must b e cancelled.

Phenomenological work on the strong interaction made it increasingly plausible that the observed strongly inter-

acting particles { and { are comp osites of more basic ob jects. The evidence was of two disparate

kinds: on the one hand, it was p ossible in this waytomake crude but e ective mo dels for the observed sp ectrum with

mesons as quark-antiquark, and baryons as quark-quark-quark, b ound states; and on the other hand, exp eriments

provided evidence for hard interactions of photons with hadrons, as would b e exp ected if the comp onents of hadrons

were describ ed by lo cal elds. The search for a quantum eld theory with appropriate prop erties led to a unique

candidate, whichcontained b oth ob jects that could b e identi ed with quarks and an essentially new ingredient, color

.

These quantum eld theories of the weak and strong interactions were dramatically con rmed by subsequent

exp eriments, and have survived exceedingly rigorous testing over the past two decades. They make up the Standard

Mo del. During this p erio d the limitations, as well as the very considerable virtues, of the Standard Mo del have

b ecome evident. Whether the next big step will require a sharp break from the principles of quantum eld theory or,

like the previous ones, a b etter appreciation of its p otentialities, remains to b e seen.

ACKNOWLEDGMENTS

I wish to thank S. Treiman for extremely helpful guidance, and M. Alford, K. Babu, C. Kolda, and J. March-Russell

for reviewing the manuscript. F.W. is supp orted in part by DOE grant DE-FG02-90ER40542

REFERENCES

For further ab out quantum eld theory, the reader may wish to consult:

1. T.P. Cheng and L.F. Li, Gauge Theory of Physics, (Oxford, 1984).

2. M. Peskin and D. Schro eder, Intro duction to , (Addison-Wesley, 1995).

3. S. Weinb erg, The Quantum Theory of Fields, I, (Cambridge, 1995) and The Quantum Theory of Fields, I I,

(Cambridge, 1996). 12