Zee A. Quantum Field Theory in a Nutshell

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462 | VIII. Gravity and Beyond

Inventing supersymmetry

Suppose one day you woke up wanting to invent a field theory with a symmetry relating

bosons to fermions. The first thing you would need is the same number of fermionic and

bosonic degrees of freedom. The simplest fermion field is the two-component Weyl spinor

ψ. You would now have one complex degree of freedom,

so you would have to throw in

a complex scalar field ϕ. You could proceed by trial and error: Write down a Lagrangian

including all terms with dimension up to four and then adjust the various parameters in

the Lagrangian until the desired symmetry appears. For instance, you might adjust μ in

the mass terms μ

†

ϕ + m(ψψ +

ψ) until the theory becomes more symmetrical so

that the boson and the fermion have the same mass.

If you were to try to play the game by using a Dirac spinor  and a complex scalar

ϕ you would be doomed to failure from the very start since there would be twice as

many fermionic degrees of freedom as bosonic degrees of freedom. I believe that the

development of supersymmetry was very much retarded by the fact that until the early

1970s most field theorists, having grown up with Dirac spinors, had little knowledge of

Weyl spinors. That was a hint that now is the time for you to get thoroughly familiar with

the dotted and undotted notation of appendix E. To read this chapter, you need to be fluent

with that notation.

Supersymmetric algebra

It is perfectly feasible to construct this supersymmetric field theory, known as the Wess-

Zumino model, by trial and error, but instead I will show you an elegant but more abstract

approach known as the superspace and superfield formalism, invented by Salam and

Strathdee. We will have to develop a considerable amount of formal machinery. Everything

is very super here.

Write the supersymmetry generator taking us from ϕ to ψ

as Q

(known as the

supercharge). The statement that Q

transforms as a Weyl spinor means [J

μν

, Q

] =

−i(σ

μν

)

, where J

μν

denotes the generators of the Lorentz group. Of course, since

is independent of the spacetime coordinates [P

, Q

] =0. From appendix E we denote

the conjugate of Q

˙α

and [J

μν

˙α

] =−i(¯σ

μν

)

˙α

We have to write down the anticommutation relation between the Grassman objects Q

and

and now the work we did in appendix E really pays off. The supersymmetry algebra

is given by

}=2(σ

)

(1)

One complex degree of freedom on mass shell and two complex degrees of freedom off mass shell. See the

superfield formalism below.

VIII.4. Supersymmetry | 463

We argue by the “what else can it be?” method. The right-hand side must carry the indices

α and

β and we know that the only object that carries these indices is σ

. The Lorentz

index μ has to be contracted and the only vector around is P

. The factor of 2 fixes the

normalization of Q.

By the same kind of argument we must have {Q

, Q

}=c

(σ

μν

)

μν

+ c

. Com-

muting with P

we see that the constant c

must vanish. Recalling that Q

= ε

γβ

,we

have {Q

, Q

}=c

γα

; but since the left-hand side is symmetric in α and γ we have c

=0.

Thus, {Q

, Q

}=0 and {

˙α

}=0 (see exercise VIII.4.2).

A basic theorem

An important physical fact follows immediately from (1). Contracting with ( ¯σ

)

βα

obtain

= ( ¯σ

)

βα

} (2)

In particular the time component tells us about the Hamiltonian

4H =



˙α



, Q

†



†

+ Q

†

) (3)

We obtain the important theorem that in a supersymmetric field theory any physical state

|S must have nonnegative energy:

S|H |S=





|S



|S|

≥ 0 (4)

Superspace

Now that we have constructed the supersymmetric algebra let us keep in mind our goal of

constructing supersymmetric field theories. To do that, we need to figure out and classify

how fields transform under this supersymmetric algebra. We have to go through a lot of

formalism, the necessity for which will become clear in due course.

Imagine that you are trying to invent the superspace formalism. Let us motivate it by

staring at the basic relation (1) {Q

}=2(σ

)

. A supersymmetric transformation

Q followed by its conjugate

generates a translation P

. Hmm, let’s see, P

≡i(∂/∂x

)

generates translation in x

, so perhaps Q

, being Grassmannian, would generate transla-

tion in some abstract Grassmannian coordinate θ

? (Similarly,

would generate trans-

lation in

Salam and Strathdee invented the notion of a superspace with bosonic and fermionic

coordinates {x

, θ

} with the supersymmetry algebra represented by translations in

this space.

So let us try Q

and

being something like ∂/∂θ

and ∂/∂

, respectively. But then

}=0 and we don’t get (1). We have to keep playing around modifying Q

and

You may already see what we need. If we add a term such as θσ

∂

, then the ∂/∂θ

464 | VIII. Gravity and Beyond

in Q

acting on θσ

∂

will produce something like the right-hand side of (1). Similarly,

we will want to add a term such as

θσ

∂

to Q

. (Once again, the dotted and undotted

notation we worked hard to develop fixes what we must write, namely (σ

)

α ˙α

˙α

∂

so that

the indices match and obey the “southwest to northeast” rule.) Thus, we represent the

supercharges as

∂

∂θ

− i(σ

)

α ˙α

˙α

∂

(5)

and

=−

∂

+ iθ

(σ

)

∂

(6)

You see that (1) is now satisfied. Interestingly, when we translate in the fermionic direction

we have to translate a bit in the bosonic direction as well.

Superfield

A superfield (x

, θ

), as the name suggests, is just a field living in superspace. An

infinitesimal supersymmetry transformation takes

 → 



= (1 + iξ

+ i

˙α

) (7)

with ξ and

ξ two Grassmannian parameters.

It turns out that we can impose some condition on  and restrict this rather broad

definition a bit. After staring at (5) and (6) for a while, you may realize that there are two

other objects,

∂

∂θ

+ i(σ

)

α ˙α

˙α

∂

and

=−



∂

+ iθ

(σ

)

∂



that we can define, sort of the combinations orthogonal to Q

and

. Clearly, D

and

anticommute with Q

and

. The significance of this fact is that if we impose the

condition

 =0 on the superfield , then according to (7) its transform 



also satisfies

the condition.

A superfield  satisfying the condition

 =0 is known as a chiral superfield. The con-

dition is actually easy to implement:

Observe that if we define y

≡ (x

+ iθ

(σ

)

α ˙α

˙α

)

(note we are adding two bosonic quantities here), then

=−



∂

+ iθ

(σ

)

∂



=−



−iθ

(σ

)

+ iθ

(σ

)



= 0

Thus, a superfield (y, θ) that depends on y and θ only is a chiral superfield.

This is analogous to the problem of constructing a function f(x, y) satisfying the condition Lf = 0 with

L ≡ [x(∂/∂y) − y(∂/∂x)]. We define r ≡ (x

+ y

)

and observe that Lr =0. Then any f that only depends on

r satisfies the desired condition.

VIII.4. Supersymmetry | 465

Let us expand  in powers of θ holding y fixed. Remember that θ contains two compo-

nents (θ

, θ

). Thus, we can form an object with at most two powers of θ , namely θθ, which

you worked out in exercise E.3. Thus, as usual, power series in Grassmannian variables

terminate, and we have

(y, θ) =ϕ(y) +

√

2θψ(y) + θθF(y)

with ϕ(y), ψ(y), and F(y) merely coefficients in the series at this stage. We can Taylor

expand once more around x:

(y, θ) =ϕ(x) +

√

2θψ(x) + θθF(x)

+ iθσ

θ∂

ϕ(x) −

θσ

θθσ

θ∂

∂

ϕ(x) +

√

2θiθσ

θ∂

ψ(x)

(8)

We see that a chiral superfield  contains a Weyl fermion field ψ , and two complex scalar

fields ϕ and F .

Finding a total divergence

Let’s do a bit of dimensional analysis for fun and profit. Given that P

has the dimension

of mass, which we write as [P

] = 1 using the same notation as in chapter III.2, then (1),

(5), and (6) tell us that [Q] = [

Q] =

and [θ ] = [

θ] =−

. Given [ϕ] = 1, then (8) tells us

[ψ] =

, which we know already, and [F ] = 2, which we didn’t know. In fact, we have never

met a Lorentz scalar field with mass dimension 2. How can we have a kinetic energy term

for F in L with dimension 4? We can’t. The term F

†

F already has dimension 4, and any

derivative is going to make the dimension even higher. Also, didn’t we say that with ϕ and

ψ we balance the same number of bosonic and fermionic degrees of freedom?

The field F(x) definitely has something strange about him. What is he doing in our

theory?

Under an infinitesimal supersymmetric transformation the superfield changes by δ =

i(ξQ +

Q). Referring to (8), (5), and (6), you can work out how the component fields ϕ ,

ψ , and F transform (see exercise VIII.4.5). But we can go a long way invoking symmetry

and dimensional analysis. For example, δF is linear in ξ or

ξ, which by dimensional

analysis must multiply something with dimension [

] since [F ] = 2 and [ξ ] = [

ξ] =−

The only thing around with dimension [

]is∂

ψ, which carries an undotted index. Note it

can’t be ∂

ψ since  does not contain

ψ . By Lorentz invariance we have to find something

carrying the index μ, and that can only be (σ

)

α ˙α

. The dotted index on (σ

)

α ˙α

can only be

contracted with

ξ . So everything is fixed except for an overall constant:

δF ∼ ∂

(σ

)

α ˙α

˙α

(9)

Arguing along the same lines you can easily show that δϕ ∼ ξψ and δψ ∼ ξF + ∂

ϕσ

ξ .

The important point here is not the overall constant in (9) but that δF is a total diver-

gence.

Given any superfield  let us denote by []

the coefficient of θθ in an expansion of 

[as in (8)]. What we have learned is that under a supersymmetric transformation δ([]

)

is a total divergence and thus



x[]

is invariant under supersymmetry.

466 | VIII. Gravity and Beyond

Our next observation is that if

 = 0, then



= 0 also. In other words, if  is a

chiral superfield, then so is 

(and by extension, 

, 

, and so forth).

Supersymmetric action

What do we want to achieve anyway? We want to construct an action invariant under

supersymmetry.

Finally, after all this formalism we are ready. In fact, it is almost staring us in the face:



m

g

...

]

is invariant under supersymmetry by virtue of the last two

paragraphs. Squaring (8) and extracting the coefficient of θθ we see by inspection that

[

]

= (2Fϕ − ψψ). Similarly, [ 

]

= 3(F ϕ

− ϕψψ). Now do exercise VIII.4.6.

Looks like we have generated a mass term for the Weyl fermion ψ and its coupling to

the scalar field ϕ, but where are the kinetic energy terms, such as

˙α

( ¯σ

)

˙αα

∂

Vector superfield

The kinetic energy terms contain

˙α

, which does not appear in . To get the conjugate

field

˙α

, we obviously have to use 

†

, and so we are led to consider 

†

. More formalism

here! We call a superfield V(x, θ ,

θ) a vector superfield if V =V

†

. For example, 

†

 is a

vector superfield.

Imagine expanding 

†

 = ϕ

†

ϕ +

...

or any vector superfield V in powers of θ and

θ .

The highest power is uniquely

θθθ since by the properties of Grassmannian variables the

only object we can form is

. Any object quadratic in θ and quadratic in

θ, such as

(θσ

θ)(θ σ

θ), can be beaten down to

θθθ by using the kind of identities you discovered

in the exercises in appendix E. Let [V ]

denote the coefficient of

θθθ in the expansion

of V .

Again, dimensional analysis can carry us a long way. If V has mass dimension n,

then [V ]

has mass dimension n + 2 since θ and

θ each has mass dimension −

. Let

us study how [V ]

changes under an infinitesimal supersymmetry transformation δV =

i(ξQ +

Q)V . We use the same kind of argument as before: δ([V ]

) is linear in ξ or

ξ,

which by dimensional analysis must multiply something with dimension n +

since [ξ ] =

[

ξ] =−

. This can only be the derivative ∂ of something with dimension n +

, namely

the coefficients of

θθ and

θθθ in the expansion of V . We conclude that δ([V ]

) has to

have the form ∂

(...), namely that δ([V ]

) is a total divergence. This is the same type of

argument that allows us to conclude that δ([]

) is a total divergence.

Thus, the action



x[

†

]

is invariant under supersymmetry.

Staring at (8), which I repeat for your convenience,

(y, θ) =ϕ(x) +

√

2θψ(x) + θθF(x) (10)

+ iθσ

θ∂

ϕ(x) −

θσ

θθσ

θ∂

∂

ϕ(x) +

√

2θiθσ

θ∂

ψ(x)

VIII.4. Supersymmetry | 467

we see that



x[

†

]

contains



xϕ

†

∂

ϕ (from multiplying the first term in 

†

with

the fifth term in ),



x∂ϕ

†

∂ϕ (from multiplying the fourth term in 

†

with the fourth

term in ) ,



ψ ¯σ

∂

ψ (from multiplying the second term in 

†

with the sixth term

in ), and finally



†

F (from multiplying the third term in 

†

with the third term

in ). It is quite amusing how derivatives of fields arise in supersymmetric field theories:

Note that the action



x[

†

]

does not contain derivatives explicitly.

To summarize, given a chiral superfield  we have constructed the supersymmetric

action

S =



x{[

†

]

− ([W ()]

+ h.c.)} (11)

Explicitly, with the choice W () =

m

g

, we have

S =



x{∂ϕ

†

∂ϕ + i

ψ ¯σ

∂

ψ +F

†

F − (mF ϕ −

mψψ +gFϕ

− gϕψψ + h.c.)} (12)

An auxiliary field

From the very beginning the field F seemed strange. Since [F ] = 2 we anticipated that it

cannot have a kinetic energy term with mass dimension 4 and indeed it doesn’t. We see that

it is not a dynamical field that propagates—it is an auxiliary field (just like σ in chapter III.5

and ξ

in chapter VI.3) and can be integrated out in the path integral



†

DFe

. Indeed,

collect the terms that depend on F in S, namely

†

F − F (mϕ + gϕ

) − F

†

(mϕ

†

+ gϕ

†2

) =|F −(mϕ + gϕ

)

†

−|mϕ + gϕ

So, integrate over F and F

†

and get

S =



x{∂ϕ

†

∂ϕ + i

ψ ¯σ

∂

ψ −|mϕ + gϕ

+ (

mψψ −gϕψψ + h.c.)} (13)

Note that the scalar potential V(ϕ

†

, ϕ) =|mϕ + gϕ

≥ 0 in accordance with (4) and

vanishes at its minimum, giving a zero cosmological constant. Note that we are no longer

free to add an arbitrary constant to V(ϕ

†

, ϕ) as we could in a nonsupersymmetric field

theory.

As expected, supersymmetric field theories are much more restrictive than ordinary

field theories, and, duh, also much more symmetric. The formalism described here can

be extended to construct supersymmetric Yang-Mills theory.

Another important generalization is to introduce, instead of one supercharge Q

, N su-

percharges Q

, with I = 1,

...

, N (exercise VIII.4.2). Since each charge Q

transforms

like the S

component of a spin

operator, it takes a state with S

= m in a super-

multiplet to a state with S

= m +

. Thus the integer N is bounded from above. For

supersymmetric Yang-Mills theory, the maximum number of supersymmetry generators

is N = 4 if we do not want to introduce fields with spin ≥ 1. Similarly, the most supersym-

metric supergravity theory we could construct (exercise VIII.4.3) has N = 8.

468 | VIII. Gravity and Beyond

As mentioned in chapter VII.3, if any nontrivial 4-dimensional quantum field theory

turned out to be exactly soluble, the supersymmetric N = 4 Yang-Mills theory is probably

our best bet. In all likelihood, the first relativistic quantum field theory to be solved exactly

would be N = 4 Yang-Mills in the planar large N limit of chapter VII.4.

I hope that this brief introduction gave you a flavor of supersymmetry and will enable

you to go on to specialized treatises.

Exercises

VIII.4.1 Construct the Wess-Zumino Lagrangian by the trial and error approach.

VIII.4.2 In general there may be N supercharges Q

, with I = 1,...,N . Show that we can have {Q

, Q

αβ

, where Z

denotes c-numbers known as central charges.

VIII.4.3 From the fact that we do not know how to write consistent quantum field theories with fields having

spin greater than 2 show that the N in the previous exercise cannot exceed 8. Theories with N = 8

supersymmetry are said to be maximally supersymmetric. Show that if we do not want to include gravity,

N cannot be greater than 4. Supersymmetric N =4 Yang-Mills theory has many remarkable properties.

VIII.4.4 Show that ∂θ

/∂θ

= ε

αβ

VIII.4.5 Work out δϕ, δψ, and δF precisely by computing δ = i(ξ

˙α

).

VIII.4.6 For any polynomial W () show that [W ()]

= F [dW (ϕ)/dϕ] + terms not involving F . Show that for

the theory (11) the potential energy is given by V(ϕ

†

, ϕ) =|∂W (ϕ)/∂ϕ|

VIII.4.7 Construct a field theory in which supersymmetry is spontaneously broken. [Hint: You need at least three

chiral superfields.]

VIII.4.8 If we can construct supersymmetric quantum field theory, surely we can construct supersymmetric

quantum mechanics. Indeed, consider Q

≡

[σ

P + σ

W(x)] and Q

≡

[σ

P − σ

W(x)], where the

momentum operator P =−i(d/dx) as usual. Define Q ≡Q

+iQ

. Study the properties of the Hamil-

tonian H defined by {Q, Q

†

}=2H.

VIII.5

A Glimpse of String Theory

as a 2-Dimensional Field Theory

Geometrical action for the bosonic string

In this chapter, I will try to give you a tiny glimpse into string theory. Needless to say, you

can get only the merest whiff of the subject here, but fortunately excellent texts do exist and

I believe that this book has prepared you for them. My main purpose is to show you that

perhaps surprisingly the basic formulation of string theory is naturally phrased in terms

of a 2-dimensional field theory.

In chapter I.11 I described a point particle tracing out a world line given by X

(τ ) in

D− dimensional spacetime. Recall that the action is given geometrically by the length of

the world line

S =−m



dτ



dτ

(1)

and remains unchanged under reparametrization τ → τ



(τ ). Recall also that classically, S

is equivalent to

imp

=−



dτ



dτ

+ γm



(2)

Now consider a string sweeping out a world sheet given by X

(τ , σ) in D-dimensional

spacetime, which we have already encountered in chapter IV.4 in connection with differ-

ential forms. In analogy with (1), Nambu and Goto proposed an action given geometrically

by the area of the world sheet

= T



dτdσ



det(∂

∂

) (3)

where ∂

≡∂X

/∂τ , ∂

≡∂X

/∂σ , and (∂

∂

) denotes the ab element of a 2 by

2 matrix. Here, as in (1), μ ranges over D values: 0, 1, . . . , D −1. The constant T(≡1/2πα



with α



the slope of the Regge trajectory in particle phenomenology) corresponds to the

string tension since stretching the string to enlarge the world sheet costs an extra amount

of action proportional to T .

470 | VIII. Gravity and Beyond

In a precise parallel with the discussion for the point particle, it is preferable to avoid

the square root and instead use the action

S =



dτdσγ

(∂

∂

) (4)

with γ = det γ

in the path integral to quantize the string. We will now show that S is

equivalent classically to S

As in (2), we vary S with respect to the auxiliary variable γ

, which we then eliminate.

For a matrix M , δM

−1

=−M

−1

(δM)M

−1

and δ det M = δe

tr log M

= e

tr log M

trM

−1

δM =

(det M)trM

−1

δM. Thus, δγ

=−γ

δγ

and δγ = γγ

δγ

. For ease of writing,

define h

≡ ∂

∂

. The variation of the integrand in (4) thus gives

δ[γ

] =γ

[

δγ

(γ

) − γ

δγ

]

Setting the coefficient of δγ

equal to 0 we obtain

(γ

) (5)

where the indices on h are raised and lowered by the metric γ . Multiplying (5) by h

(and

summing over repeated indices) we find γ

=2 and thus γ

. Plugging this into

(4) we find that S = T



dτdσ(det h)

. Thus, S and S

are indeed equivalent classically.

The action (4), first discovered by Brink, Di Vecchia, and Howe and by Deser and Zumino,

is known as the Polyakov action.

Note that (5) determines γ

only up to an arbitrary local rescaling known as a Weyl

transformation:

(τ , σ)→ e

2ω(τ , σ)

(τ , σ) (6)

Thus, the action (4) must be invariant under the Weyl transformation.

Staring at the string action (4), you will recognize that it is just the action for a quantum

field theory of D massless scalar fields X

(τ , σ) in 2-dimensional spacetime with coor-

dinates (τ , σ), albeit with some unusual signs. The index μ plays the role of an internal

index, and Poincar

e invariance in our original D-dimensional spacetime now appears as

an internal symmetry. Indeed, a good deal of string theory is devoted to the study of quan-

tum field theories in 2-dimensional spacetime! It is amusing how quantum field theory

manages to stay on the stage.

To this bosonic string theory we can add fermionic variables in such a way as to make

the action supersymmetric. The result, as you surely have heard, is superstring theory,

thought by some to be the theory of everything.

“To understand macroscopic properties of matter based on understanding these microscopic laws is just

unrealistic. Even though the microscopic laws are, in a strict sense, controlling what happens at the larger scale,

they are not the right way to understand that. And that is why this phrase, “theory of everything,” sounds sleazy.”—

J. Schwarz, one of the founders of string theory.

VIII.5. A Glimpse of String Theory | 471

This infinitesimal introduction to string theory is all I can give you here, but I hope that

this book has prepared you adequately to begin studying various specialized texts on string

theory.

For a brief but authoritative introduction, see E. Witten, “Reflections on the Fate of Spacetime,” Physics Today,

April 1996, p. 24.