Zee A. Quantum Field Theory in a Nutshell

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VIII.3

Effective Field Theory Approach

to Understanding Nature

Low energy manifestation

The pioneers of quantum field theory, Dirac for example, tended to regard field theory

as a fundamental description of Nature, complete in itself. As I have mentioned several

times, in the 1950s, after the success of quantum electrodynamics many leading particle

physicists rejected quantum field theory as incapable of dealing with the strong and weak

interactions, not to mention gravity. Then came the great triumph of field theory in the early

1970s. But after particle physicists retrieved field theory from the dust bin of theoretical

physics, they realized that the field theories they were studying might be “merely” the low

energy manifestation of a deeper structure, a structure first identified as a grand unified

theory and later as a string theory. Thus was developed an outlook known as the effective

field theory approach, pace Dirac.

The general idea is that we can use field theory to say something about physics at low

energies or equivalently long distances even if we don’t know anything about the ultimate

theory, be it a theory built on strings or some as yet undreamed of structure. An important

consequence of this paradigm shift was that nonrenormalizable field theories became

acceptable. I will illuminate these remarks with specific examples.

The emergence of this effective field theory philosophy, championed especially by Wil-

son, marks another example of cross fertilization between condensed matter and particle

physics. Toward the late 1960s, Wilson and others developed a powerful effective field the-

ory approach to understanding critical phenomena, culminating in his Nobel Prize. The

situation in condensed matter physics is in many ways the opposite of that in particle phys-

ics at least as particle physics was understood in the 1960s. Condensed matter physicists

know the short distance physics, namely the quantum mechanics of electrons and ions.

But it certainly doesn’t help in most cases to write down the Schr

odinger equation for the

electrons and ions. Rather, what one would like to have is an effective description of how

a system would respond when probed at low frequency and small wave vector. A striking

example is the effective theory of the quantum Hall fluid as described in chapter VI.2: The

VIII.3. Effective Field Theory | 453

relevant degree of freedom is a gauge field, certainly a far cry from the underlying elec-

tron. As in the σ model description (chapter VI.4) of quantum chromodynamics, it is fair

to say that without experimental guidance theorists would have a terribly hard time de-

ciding what the relevant low energy long distance degrees of freedom might be. You have

seen numerous other examples in condensed matter physics, from the Landau-Ginzburg

theory of superconductivity to Peierls instability.

The threshold of ignorance

In our discussion of renormalization, I espouse the philosophy that a quantum field

theory provides an effective description of physics up to a certain energy scale ,a

threshold of ignorance beyond which physics not included in the theory comes into

play. In a nonrenormalizable theory, various physical quantities that we might wish to

calculate will come out dependent on , thus indicating that the physics at or beyond

the scale  is essential for understanding the low energy physics we are interested in.

Nonrenormalizable theories suffer from not being totally predictive, but nevertheless they

may be useful. After all, the Fermi theory of the weak interaction described experiments

and even foretold its own demise.

In a renormalizable theory, various physical quantities come out independent of ,

provided that the calculated results are expressed in terms of physical coupling constants

and masses, rather than in terms of some not particularly meaningful bare coupling

constants and masses. Low energy physics is not sensitive to what happens at high

energies, and we are able to parametrize our ignorance of high energy physics in terms of

a few physical constants.

From the late 1960s to the 1970s, one main thrust of fundamental physics was to

classify and study renormalizable theories. As we know, this program was “more than

spectacularly successful.” It allowed us to pin down the theory of the strong, the weak, and

the electromagnetic interactions.

Renormalization group flow and dimensional analysis

The effective field theory philosophy is intrinsically tied to renormalization group flow.

In a given field theory, as we flow toward low energies, some couplings may tend to zero

while others do not (and if they tend to infinity as in QCD, then we are unable to figure

out the effective theory without experimental input). Thus, the first step is to calculate the

renormalization group flow. A simple example is given in exercise VIII.3.1.

In many cases, we can simply use dimensional analysis. As I explained in our earlier

discussion on renormalization theory, couplings with negative dimensions of mass are

not important at low energies. To be specific, suppose we add a gϕ

term to a λϕ

theory.

The coupling g has the dimension of inverse mass squared. Let us define M

≡ 1/g.At

low energies, the effect of the gϕ

term is suppressed by (E/M)

454 | VIII. Gravity and Beyond

How do we understand Schwinger’s spectacular calculation of the anomalous magnetic

moment of the electron in the effective field theory philosophy?

Let me first tell the traditional (i.e., pre-Wilsonian) version of the story. A student could

have asked, “Professor Schwinger, why didn’t you include the term (1/M)

ψσ

μν

ψF

μν

the Lagrangian?”

The answer is that we better not. Otherwise, we would lose our prediction for the

anomalous magnetic moment; it would depend on M. Recall that [ψ] =

and [A

] = 1,

and hence

ψσ

μν

ψF

μν

has mass dimension

+ 1 + 1 = 5 > 4. The requirement of

renormalizability, that the Lagrangian be restricted to contain operators of dimension 4 or

less, provides the rationale for excluding this term.

Actually, the “real” punchline of my story is that Schwinger probably would not have

answered the question. When I took Schwinger’s field theory class, it was well known

among the students that it was forbidden to ask questions. Schwinger would simply

ignore any raised hands. There was no opportunity to ask questions after class either:

As he uttered his last sentence of an invariably beautifully prepared lecture, he would sail

majestically out of the room. Dirac dealt with questions differently. I was too young to have

witnessed it, but the story goes that when a student asked, “Professor Dirac, I did not under-

stand...,”Dirac replied, “That is an assertion, not a question.”

The modern retelling of the magnetic moment story turns it around. We now regard the

Lagrangian of quantum electrodynamics as an effective Lagrangian which should include

an infinite sequence of terms of ever higher dimensions, with coefficients parametrizing

our threshold of ignorance. The physics of electrons and photons is now described by

L =

ψ(iγ

(∂

− ieA

) − m)ψ −

μν

ψσ

μν

ψF

μν

...

Yes, the term (1/M)

ψσ

μν

ψF

μν

is there, with some unknown M having the dimension of

a mass. Schwinger’s result, that quantum fluctuations generate a term

(α/2π)(1/2m

)

ψσ

μν

ψF

μν

, should then be interpreted as saying that the anomalous

magnetic moment of the electron is predicted to be [(α/2π)(1/2m

) + 1/M]. The close

agreement of (α/2π)(1/2m

) with the experimental value of the anomalous magnetic

moment can then be turned around to set a lower bound on M  (4π/α)m

Equivalently, Schwinger’s result predicts the anomalous magnetic moment of the elec-

tron if we have independent evidence that M is much larger than [(α/2π)(1/2m

)]

−1

want to emphasize that all of this makes total physical sense. For example, if you speculate

that the electron has some finite size a, then you would expect M ∼1/a. The anomalous

magnetic moment calculation gives an upper bound for a, telling us that the electron must

be pointlike down to some small scale. Alternatively, we could have had independent evi-

dence, from electron scattering for example, that a has to be smaller than a certain length,

thus giving us a lower bound on M.

To underscore this point, imagine that in 1948 we followed Schwinger and quickly

calculated the anomalous magnetic moment of the proton. We could literally have done

it in 3 seconds, since all we have to do is replace m

by m

in the Lagrangian, thus

obtaining (α/2π)(1/2m

)

ψσ

μν

ψF

μν

, which would of course disagree resoundingly with

VIII.3. Effective Field Theory | 455

experiment. The disagreement tells us that we had not included all the relevant physics,

namely that the proton interacts strongly and is not pointlike. Indeed, we now know that

the anomalous magnetic moment of the proton gets contributions from the anomalous

magnetic moments and the orbital motion of the quarks inside the proton.

Effective theory of proton decay

It may seem that with the effective field theory approach we lose some predictive power. But

effective field theories can also be surprisingly predictive. Let me give a specific example.

Suppose we had never heard of grand unified theory. All we know is the SU(3) ⊗SU(2) ⊗

U(1) theory. An experimentalist tells us that he is planning to see if the proton would decay.

Without the foggiest notion about what would cause the proton to decay we can still

write down a field theory to describe proton decay. The Lagrangian L is to be constructed

out of quark q and lepton l fields and must satisfy the symmetries that we know. Three

quarks disappear, so we write down schematically qqq, but three spinors do not a Lorentz

scalar make. We have to include a lepton field and write qqql.

Since four fermion fields are involved, the terms qqql have mass dimension 6 and so in

L they have to appear as (1/M

)qqql with some mass M , corresponding to the mass scale

of the physics responsible for proton decay. The experimental lower bound on the lifetime

of the proton sets a lower bound on M.

It is instructive to contrast this analysis with an (imagined) effective field theory analysis

of proton decay long before the concept of quarks was invented, say around 1950. We would

construct an effective Lagrangian out of the available fields, namely the proton field p,

the electron field e, and the pion field π , and thus write down the dimension 4 operator

f ¯pe

with some dimensionless constant f . To estimate f , we would naively compare

this operator with the one describing pion-nucleon coupling (chapter IV.2) g ¯pnπ

in the

effective Lagrangian. Since f ¯pe

violates isospin invariance, we might expect f ∼αg,

namely the same order as g multiplied by some measure of isospin breaking, say the fine

structure constant. But this would give an unacceptably short lifetime to the proton. We

are forced to set f to a ridiculously small number, which seems highly unnatural. Thus, at

least in hindsight, we can say that the extremely long lifetime of the proton almost points

to the existence of quarks. The key, as we saw above, is to promote of the mass dimension

of the term in the effective Lagrangian responsible for proton decay from 4 to 6. (Can the

cosmological constant puzzle be solved in the same way?)

Another way of saying this is that SU(3) ⊗SU(2) ⊗ U(1) plus renormalizability predicts

one of the most striking facts of the universe, the stability of the proton. In contrast, the

old pion-nucleon theory glaringly failed to explain this experimental fact.

In accordance with our philosophy, L must be invariant under SU(3) ⊗SU(2) ⊗ U(1),

under which quark and lepton fields transform rather idiosyncratically, as we saw in

chapter VII.5. To construct L we have to sit down and list all Lorentz invariant SU(3) ⊗

SU(2) ⊗ U(1) terms of the form qqql.

456 | VIII. Gravity and Beyond

Sitting down, we would find that, assuming only one family of quarks and leptons for

simplicity, there are only four terms we can write down for proton decay, which I list here

for the sake of completeness: (



)(u

), (e

)(q

), (



)(q

), and

)(u

). Here l





and q





denote the lepton and quark doublet

of SU(2) ⊗ U(1), the twiddle is defined by



= l

with SU(2) indices i , j = 1, 2 (see

appendix B), and C denotes the charge conjugation matrix. Color indices on the quark

fields are contracted in the only possible way. The effective Lagrangian is then given by the

sum of these four terms, with four unknown coefficients.

The effective field theory tells us that all possible baryon number violating decay pro-

cesses can be determined in terms of four unknowns. We expect that these predictions will

hold to an accuracy of order (M

/M)

. (If M

were zero, SU(3) ⊗ SU(2) ⊗ U(1) would

be exact.)

Of course, we can increase our predictive power by making further assumptions. For

example, if we think that proton decay is mediated by a vector particle, as in a generic grand

unified theory, then only the first two terms in the above list are allowed. In a specific grand

unified theory, such as the SU(5) theory, the two unknown coefficients are determined in

terms of the grand unified coupling and the mass of the X boson.

To appreciate the predictive power of the effective field theory approach, inspect the

list of the four possible operators. We can immediately predict that while proton decay

violates both baryon number B and lepton number L, it conserves the combination B − L.

I emphasize that this is not at all obvious before doing the analysis. Could you have told

the experimentalist which of the two possible modes n → e

−

or n → e

−

he should

expect? A priori, it could well be that B + L is conserved.

Note that Fermi’s theory of the weak interaction would be called an effective field theory

these days. Of course, in contrast to proton decay, beta decay was actually seen, and the

prediction from this sort of symmetry analysis, namely the existence of the neutrino, was

triumphantly confirmed.

Along the same line, we could construct an effective field theory of neutrino masses.

Surely one of the most exciting experimental discoveries in particle physics of recent

years was that neutrinos are not massless. Let us construct an SU(2) ⊗ U(1) invariant

effective theory. Since ν

resides inside l

, without doing any detailed analysis we can see

that a dimension-5 operator is required: schematically l

contains the desired neutrino

bilinear but it carries hypercharge Y/2 =−1; on the other hand, the Higgs doublet ϕ

carries hypercharge +

, and so the lowest dimensional operator we can form is of the

form llϕϕ with dimension

+ 1 + 1 = 5. Thus, the effective L must contain a term

(1/M)llϕϕ, with M the mass scale of the new physics responsible for the neutrino mass.

Thus, by dimensional analysis we can estimate m

∼m

/M, with m

some typical charged

lepton mass. If we take m

to be the muon mass ∼ 10

Mev and m

∼ 10

−1

ev, we find

M ∼ (10

Mev)

/10

−1

(10

−6

Mev) = 10

Gev.

The philosophy of effective field theories valid up to a certain energy scale  seems

so obvious by now that it is almost difficult to imagine that at one time many eminent

physicists demanded much more of quantum field theory: that it be fundamental up to

arbitrarily high energy scales.

VIII.3. Effective Field Theory | 457

Indeed, we now regard all quantum field theories as effective field theory. For all we

know, spacetime on some short distance does consist of a lattice, and so the Yang-Mills

action is but the leading term in an expansion of the Wilson lattice action. The Einstein-

Hilbert Lagrangian, being nonrenormalizable, is a fortiori “merely" the leading term in

an effective field theory

L =

√

−g(M



+ M

R + c

+ c

μν

+ c

μνσρ

...

) +

...

)

Here c

1,2 , 3

and d

are dimensionless numbers presumably of order 1. The three terms

quadratic in the curvature involve four powers of derivatives versus the two powers in

the Einstein-Hilbert term, and hence their effects relative to the leading terms are sup-

pressed by (E/M

)

with E an energy scale characteristic of the process we are studying.

Thus, these so-called Weyl-Eddington terms could be safely ignored in any conceivable

experiment. [A technical aside: The Gauss-Bonnet theorem implies that the combination

−4R

μν

μνσρ

) is a total derivative, so c

can be effectively set to 0, but that

is besides the point here.] We have indicated only one representative dimension 6 term

(out of many). Its coefficient, in accordance with high school dimensional analysis, is

suppressed by two powers of some mass M.

What do we expect the mass scale M to be? Suppose we live in a universe with only gravity

(and of course we don’t, actually) then once again, we could risk being presumptuous and

take M to be the intrinsic mass scale of gravity, namely the Planck mass M

, but we have

not yet recovered from our third-degree burn from supposing that M



∼ M

. If we could

ignore the cosmological constant problem for a moment, then the standard (but quite

possibly wrong!) consensus is that in a universe of pure gravity our theory of gravity is an

effective expansion in powers of (E/M

)

Alternatively, we could treat L as the effective theory of gravity after we integrate out all

the matter degree of freedom. In that case, M would be of order m

(imagine gravitons

coupled to an electron loop; see exercise VIII.3.5), or perhaps even m

(generated by a

neutrino loop).

Effective field theory of the blue sky

As another application of the effective field theory philosophy, consider the scattering of

electromagnetic waves on an electrically neutral spinless particle described by a scalar field

. Since  is neutral, the lowest dimension gauge invariant term that can be added to

L = ∂

†

∂ + m



†

 +

...

is (1/M

)

†

F

μν

. A factor of 1/M

, with M some mass

scale, has to be included with the dimension 1 + 1 + 2 + 2 = 6 operator to bring the high

school dimension down to 4. The two powers of derivative in F

μν

tell us immediately

that the amplitude for photon scattering on this neutral particle goes like M ∝ ω

, with

ω the frequency of the electromagnetic wave. Thus we conclude that the scattering cross

section varies like σ(ω)∝ω

458 | VIII. Gravity and Beyond

We have arrived at Rayleigh’s celebrated explanation of the color of the sky. In passing

through the atmosphere red light scatters less than blue light on air molecules and hence

the sky is blue.

For application to spinless atoms or molecules, we can pass to the nonrelativistic limit

as described in chapter III.5, setting  = (1/

√

2m)e

−imt

ϕ, so that the effective Lagrangian

now reads

L = ϕ

†

i∂

ϕ −

∂

†

∂

ϕ +

†

ϕ(c



− c



) +

...

In this case, since we understand the microscopic physics governing atoms and

molecules, we know perfectly well what the mass scale M represents. The coupling of

a photon to an electrically neutral system such as an atom or a molecule must vanish like

the characteristic size d of the system, since as d → 0 the positive and negative charges are

on top of each other, giving a vanishing net coupling to the photon. Rotational invariance

implies that the coupling ∼



d. The scattering amplitude then goes like M ∝ (ωd)

since the coupling has to act twice, once for the incoming photon and once for the out-

going photon. (Note that by rotational invariance the expectation value of the operator



vanishes, but we are doing second order perturbation theory so that we have to evalu-

ate the expectation value of a quantity quadratic in



d.)

Squaring M and invoking some

elementary quantum mechanics and dimensional analysis, we obtain the cross section

σ(ω)∼d

Appendix: Reshuffling terms in effective field theory

The Lagrangian of an effective field theory consists of an infinite sequence of terms arranged in an orderly

progression of higher and higher mass dimension, constrained only by the assumed symmetries of the theory.

In fact, some terms could be effectively eliminated. To explain this, we focus on a toy example:

L =

(∂ϕ)

− λϕ

(aϕ

+ bϕ

∂

ϕ + c(∂

ϕ)

) + O





(1)

We are secretly dealing with the action and thus we freely integrate by parts. For arithmetical simplicity, we did

not include a mass term, so that to leading order in 1/M the equation of motion reads simply ∂

ϕ = 0. The three

possible dimension 6 terms are shown explicitly [we integrate by parts to get rid of the term ϕ

(∂ϕ)

Are we allowed to use the equation of motion to eliminate the two dimension 6 terms that are proportional

to ∂

ϕ?

We know that we could make a field redefinition without changing the on shell amplitudes, so let us rede-

fine ϕ → ϕ + (1/M

)F . Then

(∂ϕ)

→

(∂ϕ)

− (1/M

)F ∂

ϕ + O(1/M

) and λϕ

→ λ(ϕ

+ (1/M

)ϕ

F +

O(1/M

)).SetF = pϕ

+ q∂

ϕ. We see that with an appropriate choice of p and q we can cancel off b and c.

Notice that in the process we also change a to some other value.

The answer to the question is yes, but the naive statement that the equation of motion ∂

ϕ = 0 empowers us

to simply set ∂

ϕ to zero in the nonleading terms in the effective field theory is, legalistically speaking, incorrect,

or at least misleading. We see that we actually generated O(1/M

) terms and changed the ϕ

term. Thus, more

correctly, a field redefinition allows us to shuffle terms around and to higher order. The net effect, however, is

the same as if we trusted the naive statement and set ∂

ϕ to zero in the nonleading terms.

For details, see, for example, J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, New York, 1967,

p. 47.

VIII.3. Effective Field Theory | 459

This procedure works for fermions also. As an example, consider the effective Lagrangian L =

ψ(iγ

∂

−

m)ψ + (1/M

)

ψ(iγ

∂

−m)ψ(

ψψ) +

...

. Then the field redefinition ψ →ψ −(1/2M

)ψ(

ψψ) gets rid of the

dimension 7 term shown.

We could also apply what we just learned to the effective theory of gravity if without any understanding we

set the cosmological constant to zero. Also, use the Gauss-Bonnet theorem to get rid of the R

μνσρ

term, so

that we have

L =

√

−g



R + c

+ c

μν

...

) +

...



(2)

Make a field redefinition g

μν

→ g

μν

+ δg

μν

and use



√

−gR =−



√

−g(R

μν

−

μν

R)δg

μν

Set δg

μν

=pR

μν

+qg

μν

R. Then we can cancel off c

and c

with a judicious choice of p and q. I emphasize that

this works only if we set the cosmological constant to zero without any ado.

Exercises

VIII.3.1 Consider

L =



(∂ϕ

)

+ (∂ϕ

)



− λ(ϕ

+ ϕ

) − gϕ

(3)

We have taken the O(2) theory from chapter I.10 and broken the symmetry explicitly. Work out the

renormalization group flow in the (λ − g) plane and draw your own conclusions.

VIII.3.2 Assuming the nonexistence of the right handed neutrino field ν

(i.e., assuming the minimal particle

content of the standard model) write down all SU(2) ⊗U(1) invariant terms that violate lepton number

L by 2 and hence construct an effective field theory of the neutrino mass. Of course, by constructing a

specific theory one can be much more predictive. Out of the product l

we can form a Lorentz scalar

transforming as either a singlet or triplet under SU(2). Take the singlet case and construct a theory. [Hint:

For help, see A. Zee, Phys. Lett. 93B: p. 389, 1980.]

VIII.3.3 Let A, B , C , D denote four spin

fields and label their handedness by a subscript: γ

= hA

with

h =±1. Thus, A

is right handed, A

−

left handed, and so on. Show that

)(C

−h

) =−

−h

)(C

−h

) (4)

This is an example of a broad class of identities known as Fierz identities (some of which we will need in

discussing supersymmetry.) Argue that if proton decay proceeds in lowest order from the exchange of a

vector particle then only the terms (



)(u

) and (e

)(q

) are allowed in the Lagrangian.

VIII.3.4 Given the conclusion of the previous exercise show that the decay rate for the processes p → π

+¯ν ,

p → π

+ e

, n → π

+¯ν , and n → π

−

+ e

are proportional to each other, with the proportionality

factors determined by a single unknown constant [the ratio of the coefficients of (



)(u

) and

)(q

)].

For help on these last three exercises see S. Weinberg, Phys. Rev. Lett. 43: 1566, 1979; F. Wilczek and

A. Zee, ibid. p. 1571; H. A. Weldon and A. Zee, Nucl. Phys. B173: 269, 1980.

VIII.3.5 Imagine a mythical (and presumably impossible) race of physicists who only understand physics at

energies less than the electron mass m

. They manage to write down the effective field theory for the one

particle they know, the photon,

L =−

μν

{a(F

μν

)

+ b(F

μν

)

...

(5)

460 | VIII. Gravity and Beyond

with

μν

μνρσ

ρσ

the dual field strength as usual and a and b two dimensionless constants

presumably of order unity.

(a) Show that L respects charge conjugation (A →−A in this context), parity, and time reversal, (and

of course gauge invariance.)

(b) Draw the Feynman diagrams that give rise to the two dimension 8 terms shown. The coefficients

a and b were calculated by Euler and Kockel in 1935 and by Heisenberg and Euler in 1936, quite a

feat since they did not know about Feynman diagrams and any of the modern quantum field theory

set up.

μν

∂

μν

(d) Our mythical physicists do not know about the electron, but they are getting excited. They are going

to start doing photon-photon scattering experiments with a machine called LPC that could produce

photons with energy greater than m

. Discuss what they will see. Apply unitarity and the Cutkosky

rules.

VIII.3.6 Use the effective field theory approach to show that the scattering cross section of light on an electrically

neutral spin

particle (such as the neutron) goes like σ ∝ ω

to leading order, not ω

. Argue further that

the constant of proportionality can be fixed in terms of the magnetic moment μ of the particle. [Historical

note: This result was first obtained in 1954 by F. Low (Phys. Rev. 96: 1428) and by M. Gell-Mann and

Murph L. Goldberger (Phys. Rev. 96: 1433) using much more elaborate arguments.]

VIII.4 Supersymmetry: A Very Brief Introduction

Unifying bosons and fermions

Let me start with a few of the motivations for supersymmetry. (1) All experimentally known

symmetries relate bosons to bosons and fermions to fermions. We would like to have a

symmetry, supersymmetry, relating bosons and fermions. (2) It is natural for fermions to

be massless (recall chapter VII.6), but not for bosons. Perhaps by pairing the Higgs field

with a fermion field we can resolve the hierarchy problem mentioned in chapter VII.6.

(3) Recalling from chapter II.5 that fermions contribute negatively to the vacuum energy,

you might be tempted to speculate that the cosmological constant problem could be solved

if we could get the fermion contribution to cancel the boson contribution.

Disappointingly, it has been more than 30 years

since the conception of supersymmetry

(Golfand and Likhtman constructed the first supersymmetric field theory in 1971) and

direct experimental evidence is still lacking. All existing supersymmetric theories pair

known bosons with unknown fermions and known fermions with unknown bosons.

Supersymmetry has to be broken at some mass scale M beyond the regime already explored

experimentally, but then (as explained in chapter VIII.2) we might expect a cosmological

constant of order M

Be that as it may, supersymmetric field theories have many nice properties (hardly

surprising since the relevant symmetry is much larger). Supersymmetry has thus attracted

a multitude of devotees. I give you here as brief an introduction to supersymmetry as I can

write. In the spirit of a first exposure, I will avoid mentioning any subtleties and caveats,

hoping that this brief introduction will be helpful to students before they tackle the tomes

out there.

For a fascinating account of the early history of supersymmetry, see G. Kane and M. Shifman, eds., The

Supersymmetric World: The Beginning of the Theory.