An Introduction to Supersymmetry in Quantum

Mechanical Systems

T. Wellman

April 23, 2003

Contents

1 Introduction 2

2 An Analogy of Oscillators 42.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 2-Dimensional Oscillator . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Supersymmetrical Oscillator . . . . . . . . . . . . . . . . . . 7

2.3.1 Q operators and the Supercharges . . . . . . . . . . . . . 82.3.2 The Energy of States . . . . . . . . . . . . . . . . . . . . . 9

2.4 The N-Particle Bosonic Oscillator . . . . . . . . . . . . . . . . . . 102.4.1 The Ground State and Construction of Higher States . . 112.4.2 The Partition Function . . . . . . . . . . . . . . . . . . . 11

3 A More General Look at Supersymmetry 123.1 Superpotentials and the Ground State Function . . . . . . . . . . 12

3.1.1 The Relationship of HB and HF . . . . . . . . . . . . . . 133.1.2 Properties of the Superalgebra . . . . . . . . . . . . . . . 143.1.3 SUSY Operators in terms of the Superpotential . . . . . . 143.1.4 Transmission and Reflection Coefficients . . . . . . . . . . 15

3.2 The N-Dimensional Supersymmetric Oscillator . . . . . . . . . . 163.2.1 The Construction of States . . . . . . . . . . . . . . . . . 183.2.2 The Partition Function . . . . . . . . . . . . . . . . . . . 19

4 Supersymmetry Breaking 204.1 What is SUSY Breaking? . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Why SUSY Breaking is Important . . . . . . . . . . . . . 214.2 Methods of Determining SUSY Breaking . . . . . . . . . . . . . . 22

4.2.1 The Witten Index . . . . . . . . . . . . . . . . . . . . . . 224.2.2 The Lattice Method . . . . . . . . . . . . . . . . . . . . . 24

4.3 SUSY Breaking in the N-Particle Oscillator . . . . . . . . . . . . 254.3.1 Center of Mass Separation . . . . . . . . . . . . . . . . . . 254.3.2 The Ground State Doublet . . . . . . . . . . . . . . . . . 274.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1

Chapter 1

Introduction

Supersymmetry, sometimes abbreviated SUSY, is a fairly new area of theoreticalphysics that promises to answer several outstanding questions relating to parti-cle physics, string theory and cosmology. It was discovered in 1971 by Gel’fandand Likhtman, Raymond, and Neveu and Schwartz. Later it was rediscoveredby several other groups. It was first introduced in the context of string theory,and soon worked into a theory called supergravity. In theories of supergravity,Einstein’s general relativity was found to be a necessary consequence of the localgauged supersymmetry. Because of this, physicists are hopeful that supersym-metry could provide the framework for the unification of gravity with the otherfundamental interactions of nature.

Supersymmetry is a model that relates bosons to fermions. One consequenceof supersymmetric particle theory is the existence of a SUSY partner particle forevery known particle. Every fermion would have a bosonic counterpart, and viceversa. So far, none of these predicted particles have been observed. The reasonfor this, according to current thinking, is that the supersymmetry is a brokenone. We will discuss supersymmetry breaking in chapter 4, but what it means isthat the energies of the bosonic and fermionic ”partner states” are different. Insome cases, radically different. The lightest postulated supersymmetric partnerto a known particle would be on the order of 100 GeV. The mass of a proton, bycontrast, is on the order of one GeV. Therefore, the proposed supersymmetricpartner particles are too heavy to be produced in any observable quantity byany particle lab in existence today.

The Large Hadron Collider (LHC), a particle accelerator under constructionat the European Laboratory for Particle Physics (CERN), is supposed to beable to produce particle collisions at upwards of 14 TeV. With that amount ofenergy available, physicists should be able to confirm or disprove the existence ofsupersymmetric partner particles. Either way, this will be a huge step forwardfor SUSY theory. The LHC is scheduled for completion in 2005.

In the meantime, there is more circumstantial evidence for the existence ofpartner particles. For instance, supersymmetric quantum field theory mightbe able to explain the observed breaking of the SU(2) × U(1) symmetry by

2

the Higgs mechanism at the 300 GeV scale, and it would require the SUSYpartner particles to be within the supposed mass range. Another argument isthat the unification of electroweak and strong nuclear forces at very high energyis inconsistent with experimental data, unless supersymmetric particles of mass100-1000 GeV are included in the theory. Finally, it is postulated that thelightest SUSY particle could be a form of dark matter, and account for a sizableamount of the mass in the universe. This also requires the particle to be on theorder or 100 GeV. While field theory and cosmology are not the topics of thispaper, they are presented here as examples of only a few of the applicationsthat supersymmetry may have on diverse areas of physics.

The purpose of this paper is to provide an introduction to the topic ofsupersymmetry. It is designed to be accessible and understandable to someonewho has a solid background in quantum mechanics, but no prior knowledgeof supersymmetry. While SUSY has many applications to field theory, thispaper does not require an in-depth understanding of it. We will mention fieldtheory briefly in places to give an idea of common applications of the conceptsdeveloped in this paper. This paper is intended to give the reader a generalbackground in the concepts and methods of supersymmetry, and to provide alaunching board for more specific study of SUSY as it relates to string theory,field theories, cosmology, particle physics, or any of the other fields that haveincorporated supersymmetry into current and developing theories.

3

Chapter 2

An Analogy of the2-Dimensional andSupersymmetricalHarmonic Oscillators

2.1 Harmonic Oscillator

Let us first develop a comparison of the supersymmetrical harmonic oscillator inits simplest form with a standard 2-dimensional quantum harmonic oscillator.The harmonic oscillator in one dimension has a Hamiltonian of the form:

H =12(p2 + ω2x2) (2.1)

We can define raising and lowering operators a and a† by:

a =1√2ω

(p− iωx) (2.2)

which gives:

[a, a†] = 1 (2.3)

This allows us to rewrite the Hamiltonian as:

H =ω

2(a†a+ aa†) (2.4)

which can also be written:

H = ω(a†a+12) (2.5)

4

If we have some wavestate ψ(x)n = ψn(x), Then naturally it is an eigenstate

of the hamiltonian withHψ(x)

n = Enψ(x)n

We can write the same thing using Dirac’s notation. Then ψ(x)n = |n〉, and

given that we know the energy of a harmonic oscillator wavestate, we can write:

ω(a†a+12)|n〉 = ω(n+

12)|n〉 (2.6)

Which leads us algebraically to the statement

a†a|n〉 = N̂ |n〉 = n|n〉 (2.7)

Where we have defined a new operator N̂ = a†a which we can call thenumber operator because it returns the number of the wavestate, or specificallythe number of times the raising operator a† has acted upon the ground state|0〉 to get to the current state |n〉.

2.2 2-Dimensional Oscillator

A harmonic oscillator in two dimensions with a single restoring force towards theorigin is a simple generalization if the one dimensional case. The hamiltonianin this case is:

H =12(p2

x + p2y) +

ω2

2(x2 + y2) (2.8)

Which is just the addition of two one-dimensional oscilators:

H = Hx +Hy (2.9)

Where Hx, Hy are given individually by (2.1).We can define ax, a†x as in (2.2), and ay, a†y in a similar fashion:

ax =1√2ω

(px − iωx) ay =1√2ω

(py − iωy) (2.10)

These follow the commutation relations (2.3), with the addition that the ax’sand ay’s commute with each other. This allows the hamiltonian to be writtenin a form similar to (2.4) or (2.5):

H =ω

2(a†xax + axa

†x + a†yay + aya

†y) (2.11)

or

H = ω(a†xax + a†yay + 1) (2.12)

Fortunately, we know that separation of variable can be used on the equa-tions for the wavestates of the two-dimensional oscillator. That is to say,

5

ψn,m(x, y) = ψn(x)ψm(y) = ψ(x)n ψ

(y)m . In Dirac’s notation we label the same

state as |n m〉.There is an inherent degeneracy in the eigenstates of H with total energy EN ,since we have:

Hψn,m(x, y) = ENψn,m(x, y) = H|n m〉 = (En + Em)|n m〉 (2.13)

This means that for every allowed energy EN there are N + 1 wavestates ofthe two-dimensional oscillator that have total energy equal to EN . Now we candefine a counting operator N̂ for the two-dimensional case:

N̂ = a†xax + a†yay , N̂ |n m〉 = (n+m)|n m〉 (2.14)

Now let us examine the symmetry of the two-dimensional oscillator. Thehamiltonian has rotational symmetry in the two-dimensional Hilbert space. Letus first write a given wavestate ψn,m(x, y) = |n m〉 in vector coordinates, andthen multiply by a standard 2-dimensional rotation matrix (a general elementof SO(2)).

[cos θ sin θ− sin θ cos θ

] [ψn

ψm

]= cos θ

[ψn

ψm

]+ sin θ

[ψm

−ψn

](2.15)

So we see that the wave function in rotated coordinates is a (normalized)linear combination of two wave functions with the same energy. Applying theHamiltonian to the rotated wavestate gives:

Hψ′n′,m′ = E′Nψ′n′,m′ = H(cos θψn,m+sin θψm,n) = (cos θENψn,m+sin θENψm,n)

for a total energy

E′N =√

cos2 θE2N + sin2 θE2

N = EN

Thuerefore we see that the enegry is conserved under rotation, so the two-dimensional oscillator has rotational symmetry.

There is another way to look at the rotational symmetry of the system,which is in terms of the total angular momentum, which must also be conserved.Let L̂x, L̂y, and L̂z be the angular momentum operators for the components ofangular momentum along each axis. Recall that L = r × p, that is, angularmomentum is the cross product of the momentum and radius. Our oscillatoris 2-dimensional by definition, so it is confined to the x, y plane. This meansthat L̂x, L̂y = 0 and the total angular momentum is just L̂z, so this must beconserved under rotations in the plane. Common sense will of course tell you itis, but let’s look at it in more detail. We have:

L̂ = L̂z = xpy − ypx (2.16)

6

We can derive the infinitesimal transformation from the properties of thealgebra:

δx = ε[L̂, x] = −εyδy = ε[L̂, y] = εx (2.17)δz = 0

Which shows what we would have expected. The motion is constrained toan ellipse in the x, y plane.

2.3 The Supersymmetrical Oscillator

A supersymmetrical harmonic oscilator can in many repsects be likened to thetwo-dimensional case. Instead of being a combination of x and y oscillators, itis a combination of bosonic and fermionic oscillators.The hamiltonian for the bosonic oscillator, HB , is given by equations (2.4) and(2.5), while the hamiltonian for the fermionic oscillator, HF , is composed ofnew raising and lowering operators, c and c†. The c’s follow anti-commutationrelations sililiar to (2.3):

{c, c†} = 1 (2.18)

With the minor inclusion that [a, c] = 0.We can now write the hamiltonian:

HF =ω

2(c†c− cc†) (2.19)

or:

HF = ω(c†c− 12) (2.20)

As in the two dimensional case (equation (2.9)), the supersymmetrical oscil-lator hamiltonian is just the addition of the two simpler hamilitonians:

H = HB +HF (2.21)

Which allows several equivalent ways of writing it:

H =ω

2((a†a+ aa†) + (c†c− cc†)

)(2.22)

H =ω

2({a†, a}+ [c†, c]

)(2.23)

H = ω(a†a+ c†c) (2.24)

7

Notice in the last equation, (2.24), the −12 from equation (2.20) cancells the

12 from equation (2.5).

Just as there is a formula for a, namely (2.2), there is a cresponding math-ematical representation for c:

c =[

0 01 0

](2.25)

Which finally allows us to write down the anaolgy to (2.1) and (2.8):

H =12(p2 + ω2x2) +

ω

2σ3 (2.26)

Where σ3 is the third Pauli matrix.

2.3.1 Q operators and the Supercharges

In a supersymetrical system, there are operators Qi that together with theHamiltonian form a super-algebra, that is, and algebra defined by both commu-tation and anti-commutation relations. The super-algebra for an N-dimensionalsystem is as follows:

[Qi,H] = 0 for i = 1 · · ·N{Qi, Qj} = 2δi,jH (2.27)

Also, the Q’s are hermitian; Q†i = Qi.In our oscillator system, there are only 2 dimensions. The operators are asfollows:

Q1 =√ω(a†c+ ac†

)Q2 = i

√ω(a†c− ac†

)(2.28)

Which can also be written:

Q1 =12(σ1p+ σ2ωx

)Q2 =

12(σ2p− σ1ωx

)(2.29)

It is a simple matter of algebra (perhaps easiest using equations (2.28)and (2.24)) to verify the conditions for the super algebra: {Q1, Q2} = 0,H = Q2

1 = Q22, and Q†1 = Q1; Q

†2 = Q2.

For the single particle supersymmetrical oscillator one may also use non-hermitianoperators Q and Q† defined as:

Q =i

2(Q1 + iQ2) = iac† (2.30)

8

In which case the algebraic relations become :

{Q,Q} = {Q†, Q†} = 0{Q,Q†} = HS (2.31)[HS , Q] = [HS , Q

†] = 0

It is these operators Q and Q† that are commonly referred to as the ”su-percharges” of the system. We will speak of them more generally in section3.1.

2.3.2 The Supersymmetric Number Operator and the En-ergy of States

This notation is easier to work with in some cases which we will look at shortly.Before we go farther with the Q’s, let us develop the number operator N̂ forthe supersymmetrical oscillator. The form is just as you would expect:

N̂ = a†a+ c†c (2.32)

So that HS = ωN̂ .This means that the energy of the supersymmetric oscillator is just the Bosonnumber plus the Fermion number, multiplied by constant.

However, it behaves somewhat differently from it’s counterpart in equation(2.14). As is clear from equation (2.25), c2 = (c†)2 = 0, which also follows fromthe anti-commutation relation (2.18).Let ψ(F )

0 be the ground state wave function for the fermionic oscillator so thatwe have:

cψ(F )0 = 0

c†ψ(F )0 = ψ

(F )1 (2.33)

c†ψ(F )1 = (c†)2ψ(F )

1 = 0 ∗ ψ(F )1 = 0

So that there are only two possible fermionic states.Now we write the supersymmetrical wavestate, which must be a two-component

spinor:

ψ(SUSY )(x) =[ψ(B)(x)ψ(F )(x)

]= |nB , nF 〉 (2.34)

So that we have:

a|nB , nF 〉 = |nB − 1, nF 〉, a†|nB , nF 〉 = |nB + 1, nF 〉c|nB , nF 〉 = |nB , nF − 1〉, c†|nB , nF 〉 = |nB , nF + 1〉 (2.35)

a|0, nF 〉 = 0, c|nB , 0〉 = 0

9

Now we can see that the operator Q = iac† acts to change a boson into afermion without changing the energy of the state, that is:

Q|nB , nF 〉 = |nB − 1, nF + 1〉 (2.36)

While the operator Q† has the opposite effect. Also, because Q and Q†

commute with HS (2.31), we see that H(Q|nF , nB〉) = Q(H|nF , nB〉) = (EB +EF )(Q|nF , nB〉) so that the energy remains unchanged by these operators.

Now we must take into account the property that the fermionic raisingand lowering operators (and by extension the supercharges) anti-commute withthemselves: {c†, c†} = 0 implies c†2 = 0, and the same holds for c, Q, and Q†.This means:

c†|nB , 1〉 = c†2|nB , 0〉 = 0 (2.37)

So that nF is always either 0 or 1. The energy of any state of the SUSYoscillator is therefore:

ES ={EB in the bosonic sector, nF = 0;E ′

B + 1 in the fermionic sector, nF = 1; (2.38)

This implies that the degeneracy for any given energy of the supersymmetricoscillator is two. The exception is the ground state, which is the only state ofzero energy.

2.4 The N-Particle Bosonic Oscillator

We now examine a system of N particles, with attractive harmonic forces be-tween them, and in addition a short-range repulsive force1. For the Bosoniccase, the Hamiltonian is relatively simple:

HB =12

N∑i=1

p2i +

ω2

2N

∑i<j

(xi − xj)2 + g2∑i<j

(xi − xj)−2 (2.39)

Where the first part of the potential, which is proportional to ω2, just rep-resents the attractive force between each particle and every other particle. Thesecond part of the potential, proportional to g2, represents a repulsive force thatwill dominate at short distances and push the particles away from each other.We can also generalize the raising and lowering operators, as per (2.10). Callthese operators An and A†n where n = 1 · · ·N . The exact construction of theseoperators increases in complexity as n increases, but in general terms:

An ∼N∑

i=1

ani + . . . (2.40)

1Freedman and Mende

10

Where ai ∼ pi + iωxi as in (2.10), and the . . . represent additional termswhich are more complicated.The operators obey the commutation relations:

[An, Am] = [A†n, A†m] = 0, [HB , A

†n] = nωA†n (2.41)

2.4.1 The Ground State and Construction of Higher States

We can write the ground state wave-function and energy of the system, withmotion relative to the center of mass:

|ψ0〉 =∏i<j

|xi − xj |αexp(− ω

2N

∑i<j

(xi − xj)2)

E0 =12(N − 1)(Nα+ 1)ω (2.42)

where α =12

+

√14

+ g2

So that |ψ0〉 is annihilated by all An and higher states can be constructedas:

|(n2n3 · · ·nN )〉 = A†n22 A†n3

3 · · ·A†nN

N |0〉

E = E0 +N∑

k=2

nkkω (2.43)

So the spectrum is basically that of N − 1 independent oscillators, withfrequencies 2ω, 3ω, . . . , Nω.

2.4.2 The Partition Function

We can find the degeneracy of states by using the statistical mechanical partitionfunction:

Z = Tr(e−αHB ) = e−αE0

N∏k=2

11− e−αkω

(2.44)

Where α = 1/kBT and kB is Boltzmann’s constant.We will return to the N-dimensional oscillator in the next chapter, once we

have developed a few more ideas that will aid us in examining the supersym-metrical case.

11

Chapter 3

A More General Look atSupersymmetry

3.1 Superpotentials and the Ground State Func-tion

In this section we will elaborate on the concepts we introduced in section 2.3.For a system to exhibit supersymmetry, there must be a hamiltonian and setof operators Qi that obey the algebra of equation (2.27) or (2.31). We will usethe latter in this section, and it is reprinted here for convenience:

{Q,Q} = {Q†, Q†} = 0{Q,Q†} = HS (3.1)[HS , Q] = [HS , Q

†] = 0

Let HB be the bosonic hamiltonian, and assume there is a ground state ofzero energy, ψ0(x) so that from Schrodinger’s equation we have:

HBψ0(x) = − ~2

2md2ψ0

dx2+ VB(x)ψ0(x) = 0 (3.2)

Then solving for the potential energy:

VB(x) =~2

2mψ′′0 (x)ψ0(x)

(3.3)

We can thus globally reconstruct the potential VB from the ground statewave function, which has no nodes. From here we see that we can factor theHamiltonian into two operators:

HB = A†A (3.4)

12

where

A = ~√2m

ddx +W (x)

A† = −~√2m

ddx +W (x) (3.5)

The quantity W (x) is know as the superpotential. The bosonic potential, interms of the superpotential, is given by the Riccati equation:

VB(x) = W 2(x)− ~√2m

W ′(x) (3.6)

where W ′(x) just means ddxW (x).

The superpotential is given in terms of the ground state wave function by:

W (x) =−~√2m

ψ′0(x)ψ0(x)

(3.7)

We can define a new hamiltonian by reversing the order of the operators Aand A†. This will of course correspond to the fermionic hamiltonian, althoughthat will connection will not be made until a little later. Still, we will referto it as HF to avoid confusion in the long run. Now we have HF = AA†.Simplification shows that:

HF =−~2

2md2

dx2+ VF (x) (3.8)

where

VF (x) = W 2(x) +~√2m

W ′(x) (3.9)

3.1.1 The Relationship Between Eigenstates of HB and HF

The potentials VB(x) and VF (x) are know as supersymmetric partner potentials.It is easy to show that the eigenstates of the the two hamiltonians, ψ(B)

n andψ

(F )n , are related:

HF (Aψ(B)n ) = AA†Aψ(B)

n = AHBψ(B)n = E

(B)n (Aψ(B)

n )

HB(A†ψ(F )n ) = A†AA†ψ(F )

n = A†HFψ(F )n = E

(F )n (A†ψ(F )

n ) (3.10)

So that

ψ(F )n = (E(B)

n+1)−1/2Aψ

(B)n+1

ψ(B)n+1 = (E(F )

n )−1/2A†ψ(F )n (3.11)

13

and,

E(F )n = E

(B)n+1 (3.12)

The spectra of the two hamiltonians is the same, except for one fact worthnoting. In particular, ψ(F )

n ∝ ψ(B)n+1, and equation (3.12) above, imply that the

only state with zero energy is ψ(B)0 ; there is no eigenstate of HF with zero

energy. What this means in practical terms is that the zero-energy ground stateis bosonic, and has no counterpart in the fermionic sector. This is the conditionfor supersymmetry to remain unbroken, and will be discusses in greater detailin Chapter 4.

3.1.2 Properties of the Superalgebra

Now we will look at the properties of a supersymmetry algebra, commonly calleda superalgebra because it is defined by anti-commutation as well as commutationrelations. We have already seen an example in equation (3.1), but let us see howit can be re-constructed following the method above. First, we can consider asupersymmetrical hamiltonian in matrix form:

HS =[HB 00 HF

](3.13)

Now we can define the operators:

Q =[

0 0A 0

]Q† =

[0 A†

0 0

](3.14)

It is simple enough to check that the condition of (3.1) are satisfied. Theserelations describe the closed superalgebra sl(1/1). As has been discussed insection 2.3, the operators Q and Q† can be thought of as operating of statesof the super-hamiltonian to change bosonic degrees of freedom into fermionicones, and vice-versa, without changing the total energy. The fact that theQ’s commute with HS is responsible for the degeneracy of supersymmetricalwavestates. For a supersymmetrical system in which the bosonic and fermionichamiltonians do not themselves exhibit degeneracy, the degeneracy for a givenEn is 2, because the first constraint of (3.1) requires nF = 0 or 1.

3.1.3 SUSY Operators in terms of the Superpotential

It will be useful at times to define the Hamiltonian and supersymmetric opera-tors in terms of the superpotential. For a general non-field theory case:

HS =12(p2 +W 2(x) + ~σ3W

′(x))

(3.15)

14

Although usually in this paper we let ~ = 1. The hermitian operators Qi

are given by:

Q1 =12(σ1p+ σ2W (x))

Q2 =12(σ2p− σ1W (x)) (3.16)

So the non-hermitian Q’s are given by equation (2.30) as:

Q = ic†(p− iW (x))Q† = −ic(p+ iW (x)) (3.17)

where c is the matrix of equation (2.25). These are the supercharges of thesystem mentioned briefly in section 2.3.1.

3.1.4 Transmission and Reflection Coefficients

The last thing we will look at in this section is how supersymmetry relates thetransmission and reflection coefficients of the two hamiltonians in systems wherethe potential allows for quantum scattering. For scattering to occur at all, thepotentials VB , VF must be finite as x→ +∞ or x→ −∞ or both. Define:

limx→±∞

W (x) = W± (3.18)

Then VB,F →W 2± as x→ ±∞.

Now consider a plane wave of energy E coming from the −∞ direction, ψ ∝eikx. After scattering from one or the other of the potentials, one would obtainthe the reflected waves RB,F (k)e−ikx and the transmitted waves TB,F (k)eik′x

where R and T are the reflection and transmission coefficients. This gives:

ψ(B,F )(k, x→ −∞) → eikx+ RB,F e−ikx

ψ(B,F )(k′, x→ +∞) → TB,F eik′x (3.19)

Using the relations of (3.11) and (3.12) between wave functions with thesame energy, we can derive the relationships:

eikx +RBe−ikx = N

((−ik +W−)eikx + (ik +W−)e−ikxRF

)TBe

ik′x = N((−ik′ +W+)eik′xTF

)(3.20)

where N is an overall normalization constant. We can equate terms with thesame exponent, eliminate N, and solve for R and T :

15

RB(k) =(W− + ikW− − ik

)RF (k)

TB(k) =(W+ − ik′

W− − ik

)TF (k) (3.21)

where k and k′ are given by

k = (E −W 2−)1/2

k′ = (E −W 2+)1/2 (3.22)

From this we can see several things. First of all, |RB |2 = |RF |2 and|TB |2 = |TF |2, which is to say the partner potentials have identical probabilitiesof reflection or transmission.In the case that the potential is an even function in the x → ±∞ regime, thatis, if W+ = W−, then TB(k) = TF (k).If W− = 0, then RB(k) = −RF (k).RB(TB) and RF (TF ) have the same poles in the complex plane, except thatRBTB has one extra pole at k = −iW−. If W− < 0 this pole is on the positiveimaginary axis and corresponds to a zero energy bound state.As a last note, it is clear that if one of either VB or VF is a constant potential(for example a free particle) then the other one must also be reflection-less.

3.2 The N-Dimensional Supersymmetric Oscil-lator

Now we return to the system of N coupled particles which we introduced insection 2.4. In this section we will explore the supersymmetric counterpart tothe bosonic system defined by (2.39). We have 3N degrees of freedom, namelythe position vectors xi’s and their fermionic counterparts ci and c†i . These latterof course obey the anti-commutation rules of (2.18), that is:

{ci, cj} = {c†i , c†j} = 0

{ci, c†j} = δij (3.23)

The dynamics of any supersymmetrical quantum system are defined by thesuperpotential, since both the hamiltonian and supersymmetry operators Qand Q† are given in terms of it. The generalization of equation (3.17) gives thesupercharges in this case:

16

Q =∑

i

c†i(pi − iW

)Q† =

∑i

ci(pi + iW

)(3.24)

While the hamiltonian is the generalization of (3.15):

HS =12

∑i

(p2

i + (W )2)

+12

∑i,j

[c†i , cj ]∂W

∂xi(3.25)

These operators follow the rules of the superalgebra sl(1/1), that is: HS =12{Q,Q

†}, Q2 = Q†2 = 0, [HS , Q] = 0. We want to generalize the system of(2.39) to a solvable supersymmetric theory, so we define the superpotential tobe:

W (x1, . . . , xN ) =ω

N

∑i<j

(xi − xj) + g∑i<j

(xi − xj)−1 (3.26)

From this, we see that the supercharges are:

Q =∑

i

c†i

pi − iω

N

∑j

(xi − xj)− ig∑j 6=i

(xi − xj)−1

Q† =

∑i

ci

pi + iω

N

∑j

(xi − xj) + ig∑j 6=i

(xi − xj)−1

(3.27)

and the hamiltonian is:

HS =12

N∑i=1

p2i +

ω2

2N

∑i<j

(xi − xj)2 + g2∑i<j

(xi − xj)−2 +ω

2gN(N − 1)

+ω

2N

∑i<j

[c†i − c†j , ci − cj ] +12g∑i,j

[c†i , cj ]∂iUj (3.28)

WhereUj ≡

∑k 6=j

1xj − xk

.When you plug (3.26) into (3.25), it takes some work to produce the form of

the equation given above. In particular, the terms of the form (xi−xj)(xi−xk)−1

sum to the constant ω2N(N−1), and the sum of the cross terms (xi−xj)−1(xi−

xk)−1 with j 6= k cancel out.One thing to notice is that the first three terms of HS are just HB , the bosonichamiltonian of section 2.4. If you remove the terms that involve fermionicdegrees of freedom, you are left with the bosonic hamiltonian plus a constant.

17

3.2.1 The Construction of States

This model has two phases, depending on the value of the parameter g. Wheng < 0, supersymmetry is conserved, so there exists a unique ground state energyannihilated by Q and Q†. The wave function takes the form:

|φ0〉 = e−R

W (x)dx|0〉 =∏i<j

|xi − xj |−gexp

− ω

2N

∑i<j

(xi − xj)2

|0〉 (3.29)

where |0〉 is the state in the 2N dimensional fermionic vector field (Fockspace) that is annihilated by all operators ci for i = 1 · · ·N . This state isthe supersymmetric generalization of the system in section 2.4. The operatorsthat take the place of a and c in this model are the bosonic operators An fromequation (2.40) and the fermionic counterparts which we will call Cn and C†nwhich likewise are formed out of the operators ci. This of course means thatCn|0〉 = 0|0〉 for any value of n. The operators follow algebraic relations:

{Cm, Cn} = 0[Am, Cn] = 0 (3.30)[Am, An] = 0

{Q,C†n} = {Q†, Cn} = 0{Q†, C†n} = A†n

[Q,An] = [Q†, A†n] = 0 (3.31)[Q,A†n] = 2nωC†n

[HS , Cn] = nωCn

[HS , An] = nωAn (3.32)

Notice that the first condition of (3.30) implies that C2n = C†2n = 0. As in

the bosonic model, new wavestates can be created from the ground state byapplying combinations of the raising operators. If we let |φ0〉 = |φ(B)

0 , φ(F )0 〉

then:

|φ(B)(n2,...,nN ), φ

(F )(ν2,...,νN )〉 = A†n2

2 · · ·A†nN

N C†ν22 · · ·C†νN

N |φ0〉 (3.33)

Where each νn is either 0 or 1. The energy of the state is:

E = ωN∑

k=2

k(nk + νk) (3.34)

18

When we compare this with the energy of the eigenstates of HB , from equa-tion (2.43), it should become apparent that the spectrum of our N-particle oscil-lator is identical to that of N −1 supersymmetric oscillators, having frequencies2ω, 3ω, . . . , Nω.

3.2.2 The Partition Function

The center of mass of the N oscillator system is of course:

X = N−1N∑

i=1

xi,

while the fermionic counterpart is:

Ψ = N−1N∑

i=1

ci.

After doing supersymmetric center-of-mass separation, we can write the parti-tion function of the system:

Z = Tr(e−HS/kT eiθΛ) =N∏

j=2

(1 + eiθ(e

−ωkT )j

1− (e−ωkT )j

)(3.35)

where once again, k is Boltzmann’s constant. Λ =∑N

i=1 c†i ci − Ψ†Ψ is the

internal fermion number operator, and eiθ is a phase factor to count fermionnumber. Notice that the partition function for a singe-particle oscillator is:

Tr(e−HS/kT eiθc†c) =1 + eiθe−

ωkT

1− e−ω

kT(3.36)

so that the partition function for the whole system, (3.35), is just a prod-uct of the partition functions of individual oscillators, again with frequencies2ω, . . . , Nω.

More could be said about the group structure of the theory at this point.The equations (3.30) through (3.32) mention nothing of the relations [Am, A

†n],

{Cm, C†n}, and especially [Am, C

†n]. These commutators do not vanish in general;

the latter is interesting because for an individual oscillator [a, c†] does vanish.This means that the algebra of the system is considerably more complicatedthan that of N − 1 individual oscillators taken together. The operators Q, A2,and C2, together with their adjoints and the operators HS and Λ, form thesuper-algebra OSp(2|2), which contains the bosonic sub-algebra O(2, 1)×U(1).To restate, a bosonic algebra is one with no anti-commutator relations in thedefinition.

19

Chapter 4

Supersymmetry Breaking

4.1 What is SUSY Breaking?

Supersymmetry breaking is the term for a system in which the hamiltonian ingeneral obeys the laws of supersymmetry, given by:

HS = {Q,Q†}{Q,Q} = {Q†, Q†} = 0 (4.1)[HS , Q] = [HS , Q

†] = 0

But the ground state does not. Let |0〉 be the ground state of a supersym-metrical hamiltonian. When there is a ground state of energy E0 = 0, thenHS |0〉 = 0 implies (QQ† +Q†Q)|0〉 = 0 which implies Q|0〉 = 0 and Q†|0〉 = 0.So when there exists a ground state function with zero energy, it is clear that allthe conditions of (4.1) are met when the supersymmetry operators are appliedto the ground state, and therefore to every state of the hamiltonian. This is thecondition for supersymmetry to be unbroken.

Remember that the operator Q† has the effect of transforming one bosoninto one fermion while conserving the energy of the system as a whole. How-ever, when the ground state energy is zero, Q† annihilates the ground state. Itis therefore a result of unbroken supersymmetry that the ground state functionhas no fermionic counterpart. In other words, there is one more state in thebosonic sector than in the fermionic sector.

Now let us look at the converse case, when the ground state has energyE0 6= 0. Here we will refer to this non-vacuum ground state as |GS〉. ThenHS |GS〉 = E0|GS〉 implies (QQ† +Q†Q)|GS〉 = E0|GS〉.So we see that the operators Q and Q† do not annihilate the lowest-energy statein this case. However, it is almost the definition of a ground state that it isannihilated by the supercharges. Remember that Q ∼ AC† and Q† ∼ A†C

20

where A and C are general symbols representing the lowering operators in thebosonic and fermionic sectors respectively. We would expect that when a low-ering operator of either sector was applied to the ground state it would returna zero value, that is A|GS〉 = C|GS〉 = 0. If this is not the case, one mightexpect to find a state of lower energy than our ground state candidate.

In essence, the condition of broken supersymmetry implies the absence of anormalizable ground state function for the system. Without such a state, thesystem is incomplete and cannot be said to truly exhibit supersymmetry. Noticealso that Q†|GS〉 6= 0 implies their is a fermionic counterpart to the (bosonic)ground state. This means that in systems with broken supersymmetry (butthat still have a hamiltonian obeying (4.1) above), their is a precise one-to-onecorrespondence between states in the bosonic and fermionic sectors, and suchstates have equal energy.

4.1.1 Why SUSY Breaking is Important

At this point it is appropriate to say something about the physical importanceof supersymmetry breaking. One of the most notable aspects of current su-persymmetry theory is that so far there has been no supersymmetry actuallyobserved in nature. There is no physical representation of a supersymmetricharmonic oscillator, for example. On the other hand, supersymmetry has manyvery nice properties that we would expect to be able to use. It has so far beenthe case that every mathematical symmetry has been reflected in nature. Thereare of course the rotational and translation symmetries we are quite familiarwith. Special relativity shows that the Lorentz transformation is a symmetryobserved by nature. The unification of the electromagnetic and weak forces isalso a symmetrical relationship. This implies that nature is very balanced, atleast from a physicist’s point of view. So why do we not see direct evidence ofsupersymmetry, when it is one of the most balanced symmetries of all?

One theory is that the supersymmetry of the universe was spontaneouslybroken at the very beginning. According to theory, roughly 10−35 seconds af-ter the big bang nature was in a perfectly balance and supersymmetrical state.After that, the symmetry was broken, and has been ever since. No one reallyknows why. What this means is that supersymmetrical relationships might stillexist in nature, but they would not be balanced in the nice way that unbrokenSUSY would require.

Supersymmetry requires the existence of a fermion for every boson, and vice-versa. In an unbroken supersymmetrical system, we would expect the partnerpairs to have equal energy. However, in a broken SUSY system, this mightnot be the case, even if the partner particles did still exists under the SUSYrelations. This is the basis for the current theory of supersymmetric partnerparticles. None of the known bosonic or fermionic particles are SUSY partnersof each other. This means that there should be an undiscovered particle for

21

every known one. The somewhat whimsical naming scheme has been to add an’s’ the front of the names of fermionic particles and an ’ino’ to the end of thenames of bosonic ones. So the proposed supersymmetric particles have nameslike squark, selectron, and sneutrino for the bosons and photino, gluino, andgravitino for the fermions.

None of these particle have yet been observed, but again that goes back tothe breaking of supersymmetry. Because the energies of the partner particles areuneven, it happens that the proposed particles are 100 to 1000 times as massiveas a proton. This is heavy enough that current experimental techniques wouldbe unable to detect such particles, although hopefully that will change in thenear future.

In general however, the fact that supersymmetry has not been observed innature so far does not imply that there are no practical uses for supersymmetrictheory. It could be that every naturally occurring supersymmetry is a brokenone. The existence of a broken supersymmetry would still imply the existenceof a hamiltonian and supercharges that obey the supersymmetric relations. Theexistence of such a system would in turn imply a fundamental relationship be-tween fermions and bosons in the system, even if the relationship is not as”balanced” as we might like. The fact that supersymmetry has not yet beenobserved does indicate the importance of understanding the causes and effectsof supersymmetry breaking.

4.2 Methods of Determining SUSY Breaking

In this section we will look at ways to tell whether supersymmetry is brokenin any given case. For the first part we will use the example of the super-symmetric oscillator from section 2.3. In the second part we will examine anon-perturbative method for checking for broken supersymmetry in quantumfield theory.

In order for supersymmetry to exist and be unbroken, we require a groundstate such that HB |0〉 = HF |0〉 = 0|0〉 so of course, HS |0〉 = 0|0〉. This requiresthat the ground state, |0〉, be annihilated by both Q and Q† as discussed insection 4.1.

4.2.1 The Witten Index

E. Witten defined an index to determine whether supersymmetry was brokenin complex situations, such as supersymmetric field theory. The Witten indexis written:

∆ = Tr(−1)F (4.2)

Where F is the fermion number, and the trace is over all bound and contin-uum states of the super-Hamiltonian, HS .

22

We can write the eigenstates of HS in vector form, as:

ψn =

[ψ

(+)n

ψ(−)n

](4.3)

so that ψ(+)n = |nB , 0〉 and ψ

(−)n = |0, nF 〉 in the example in section 2.3,

and in general ψ(+)n corresponds to an eigenstate of HB , and represents the

bosonic sector while ψ(−)n corresponds to HF and the fermionic sector. In this

representation, the fermion number nF = F is represented by the operator12 (1− σ3) and we can represent (−1)F by σ3.Now the bosonic and fermionic bound states, that is, the eigenstates of HB

and HF are paired, except in the case of unbroken supersymmetry. Whensupersymmetry is unbroken, there is a ground state of zero energy in the bosonicsector, which has no counterpart in the fermionic sector. This extra eigenstatewith eigenvalue +1 is not cancelled by a corresponding eigenstate of value −1so the Witten index yields a result of +1. In following our example from section2.3, we see that indeed |0, 0〉 has no fermionic counterpart. Of course, theexistence of a zero-energy ground state that we can see is annihilated by Q andQ† is itself enough to tell us that supersymmetry is unbroken for this case.In general however, we expect:

∆ = 0 for broken supersymmetry∆ = 1 for unbroken supersymmetry (4.4)

For field theories, the Witten index needs to be regulated to be well defined.We can consider:

∆(α) = Tr(−1)F e−αH (4.5)

Which for supersymmetry quantum mechanics is:

∆(α) = Tr(e−αHB − e−αHF ) (4.6)

The question of the breakdown of supersymmetry is related to the questionof whether there exists a normalizable ground state wave function that satisfiesQ|0〉 = 0|0〉, which implies:

|0〉 = ψ0 = Ne−R

W (x)dx (4.7)

So that if ψ0 does not fall off fast enough at ±∞ then Q does not annihilatethe ground state and supersymmetry is spontaneously broken. For the exampleof the oscillator we’ve been using, we have:

W (x) = ωx (4.8)

for the superpotential, which can be seen easily by comparing equations(2.26) and (3.15). As we already know what the hamiltonian and Q operatorsare, equations (4.7) and (4.8) tell us the ground state wave function should be:

23

ψ0 = e−ω2 x2

(4.9)

We can see immediately that ψ0 falls off quite rapidly as (x → ±∞). Ap-plying Q explicitly we obtain:

Qψ0 = c†(−ωxe−ω2 x2

+ ωxe−ω2 x2

) = 0 (4.10)

This is a very simple case to check, and we did not have to evaluate theWitten index because we already knew the ground state function.For a general supersymmetry hamiltonian, given by equation (3.15), the Wittenindex (4.5) can be evaluated:

∆(α) = Trσ3

∫dpdx

2πe−

α2 (p2+W 2)

(e−σ3W ′(x)/2

)(4.11)

We can expend the term proportional to σ3 in the exponent, which in theabove equation has been separated inside parentheses. Then, taking the tracewe obtain:

∆(α) =∫dpdx

2πe−

α2 (p2+W 2) sinh(αW ′(x)/2) (4.12)

We are interested in the index as α → 0 so we only need to evaluated theequation when sinh(αW ′(x)/2) ≈ (αW ′(x)/2).

∆(α) =∫dpdx

2πe−

α2 (p2+W 2)(αW ′(x)/2) (4.13)

If we now choose a superpotential of the form W (x) = gx2n+1, (such as ωx)the integrals become gamma functions and we obtain ∆(α) = 1. Also, since|∫gx2n+1dx| ≥ 0, a superpotential of this form yields a normalizable ground

state function, (4.7). If instead we choose W (x) = gx2n, then the ground statefunction is not normalizable and the integrand becomes an odd function so that∆(α) = 0. Therefore the Witten index does indeed give us the answer we wouldexpect for examples we can check directly.

4.2.2 The Lattice Method

For the last part of this section, we will look at another non-perturbative methodfor studying supersymmetry breaking. This method is used in field theories, andthis treatment of it will merely be a summary.The general idea is to explicitly break the symmetry by placing the field on adiscrete lattice, and evaluating the path integral numerically or via some othernon-perturbative method. This introduces a new parameter, namely the latticespacing, a. This parameter breaks supersymmetry so that the ground stateenergy is no longer 0 in any case. In the case of unbroken supersymmetry, oneexpects the ground state energy to go to 0 as a→ 0. That is, we expect:

E0(a) = caγ (4.14)

24

Where γ is a critical index, which if supersymmetry is unbroken should begreater than 0. Comparing the ground state energies at two different latticespacings a and a′, we find:

γ = lima′→0a→0

lnE0(a′)E0(a)

/ lna′

a(4.15)

In the case of broken SUSY, we expect γ = 0 which is a more difficultmeasurement to make numerically. In this case, an easier method may be tomeasure the ground state energy itself, and show that it remains non-zero asa→ 0.

4.3 SUSY Breaking in the N-Particle Oscillator

In section 3.2 we discussed the N-particle oscillator in the conserved supersym-metry case; that is when g < 0. In this section we will look at the case wheng > 0 and supersymmetry is broken. Recall from section 3.2 the hamiltonian ofthe system is:

HS =12

N∑i=1

p2i +

ω2

2N

∑i<j

(xi − xj)2 + g2∑i<j

(xi − xj)−2 +ω

2gN(N − 1)

+ω

2N

∑i<j

[c†i − c†j , ci − cj ] +12g∑i,j

[c†i , cj ]∂iUj (4.16)

Recall that Uj ≡∑

k 6=j1

xj−xk.

The parameter g is a measure of the short-range force between two particles,that is in addition to the harmonic force between them. If g is negative, it isa repelling force that keeps the particles from getting too close to each other.If g is positive, it is an attracting force, in which case the particles might endup arbitrarily close to each other. Actually, this is a very hand waving analogy,since the sign of g only matters for the constant term and one of the fermionicterms. If you refers back to section 3.2, you can see that the sign of g hasvirtually no bearing on the bosonic case at all. This is not a surprise, since wecannot have a supersymmetry breaking phase in a non-supersymmetric system.

4.3.1 Center of Mass Separation

To make the Superpotential and eigenstates of the system easier to deal with,we can separate out the center of mass, X and the fermionic center of massequivalents, Ψ and Ψ†. These are defined in section 3.2.2, but to restateX = N−1

∑i xi; Ψ and Ψ† are defined by taking the sum over all ci and c†i

respectively. We can also isolate the momentum of the center of mass,

P =1N

N∑i=1

pi

25

Now we can define new variables. For each i, let:

yi = xi −X p̂i = pi − P

λi = ci −Ψ λ†i = c†i −Ψ† (4.17)

These represent the motion of each particle relative to the center of mass.Since the motion of the system as a whole is defined by the motion of the centerof mass, once we have removed it we expect the net motion of the system to bezero. This can be seen easily from the fact that:∑

i

yi =∑

i

p̂i =∑

i

λi =∑

i

λ† = 0 (4.18)

These new variables allow us to re-write the internal fermion number oper-ator from equation (3.35) as:

Λ =∑

i

λ†iλi (4.19)

The new algebraic relations that do not vanish are:

[yi, p̂j ] = i(δij −1N

) {λi, λ†j} = δij −

1N

(4.20)

Of course, their sums also vanish. To do supersymmetrical center of massseparation for HS and Q we will use the identities:

∑j 6=i

(xi − xj) = Nyi

∑i<j

(xi − xj)2 = N∑

i

y2i∑

j

Uj = 0∑i<j

[c†i − c†j , ci − cj ] = N∑

i

[λ†i , λi] (4.21)

When we use these in equations (3.26), (3.27), and (3.28) we end up with:

W (y1 . . . yN ) = ω∑

i

yi + g∑i 6=j

(yi − yj)−1

(4.22)

for the superpotential. The supercharges for the internal degrees of freedomare:

Q =N∑

i=1

c†i

p̂i − iωyi − ig∑j 6=i

(yi − yj)−1

Q† =

N∑i=1

ci

p̂i + iωyi + ig∑j 6=i

(yi − yj)−1

(4.23)

26

and the internal hamiltonian is:

H =12

∑i

p̂2i + ω2y2

i + g2∑j 6=i

(yi − yj)−2

− ω

2(1−Ng)(N − 1)

+ωΛ +12g∑i 6=j

[λi, λ†i − λ†j ](yi − yj)−2 (4.24)

Supersymmetric center of mass separation does not affect the supersymmet-ric relationships of the operators. The Q’s and H of equations (4.23) and (4.24)still obey all of the conditions from (3.1).

4.3.2 The Ground State Doublet

Using the internal variables from the previous section, we can write the groundstate function (3.29) as:

|φ0〉 = e−R

W (y)dy|0〉 =∏i<j

|yi − yj |−gexp

(−ω

2

∑i

y2i

)|0〉 (4.25)

The normalization integral converges as yi → inf, which is what made it acandidate for the ground state function to begin with. For g < 0 we also haveconvergence for coincident points: yi → yj .

When g > 12 , the normalization integral diverges and supersymmetry is spon-

taneously broken. However, when 0 < g < 12 , the normalization integral still

converges. We require that for wave functions of the form∏

i<j |yi− yj |−gϕ(y),each pi be self-adjoint and ϕ(y) be regular at coincident points. This requiresg < 0, which is the necessary and sufficient condition for supersymmetry to beconserved in this system.

Another way to see that g > 0 leads to supersymmetry breaking is to lookfor a ground state based on a minimum of the bosonic potential function. Welook for:

V =12

∑i

(Wi)2 = 0 (4.26)

Where Wi is the ith component of the superpotential. That is, W =∑

iWi.We want solutions to the equations:

Wi = ωyi + g∑j 6=i

(yi − yj)−1 = 0 (4.27)

A set of yi that satisfy the above condition we would associate with a min-imum in the potential function V . We can multiply equation (4.27) by yi and

27

sum over i, since each Wi must vanish. We also use the trivial condition that0 ≤

∑i y

2i to write:

0 ≤∑

i

y2i = − 1

ωg∑j 6=i

yi

yi − yj= − g

2ωN(N − 1) (4.28)

The last term uses the fact that the sum over yi(yi−yj)−1 gives 12N(N −1),

which we first mentioned when introducing the supersymmetric hamiltonian forthis system (see equation (3.28)).

Clearly, the inequality in (4.28) has no solutions if g > 0, which should sug-gest that this is the region of supersymmetry breaking.

As we expect, the ground state is a doublet for the broken SUSY phase. Thetwo states have fermion numbers N −1 and N −2. Recall that in the N-particlecase, the fermion number is:

nF =N∑

i=1

νi (4.29)

where each νi is the fermion number of an individual oscillator part of thesystem, and can be either 0 or 1. Therefore nF can be between 0 and N − 1.

The energy of each state in the doublet is:

E0 = (1 +Ng)(N − 1)ω (4.30)

While the partition function is:

Z = Tr(e−HS/kT eiθΛ) = e−E0/kTN∏

j=2

(eiθ + (e−ω/kT )j−2

1− (e−ω/kT )j

)(4.31)

4.3.3 Conclusion

Even the N-particle supersymmetric oscillator is a fairly simple system as far assupersymmetric theory is concerned. It is nice in that it is exactly solvable andwell understood, despite the complications to the algebra as N increases. Mostsystems in theories involving supersymmetry are not nearly so well-mannered.In addition, as we have already mentioned, they must deal with supersymmetrybreaking - a mechanism that is not yet fully understood. The applicationsof supersymmetry to field theory, string theory, and other areas might be morecomplex than the systems we have examined in this paper, but the principles andmethods are largely the same. The author hopes that you will take away fromthis paper an understanding of the fundamental principles of supersymmetrictheory, and an idea of the uses and implications it brings to the developingworld of science.

28