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An Introduction to the Concepts and
Methods of Analysis of Supersymmetry

Many people interested in physics as a science are perplexed at how much of it consists of speculations about entities that may or may not exist in the physical world. The prime example of this is the various manifestations of what is known as string theory. Various versions have been found to be flawed and discarded but the reformulations continue to be called string theory, although more recent versions are referred to as M theory where M is not defined but is suspected to stand sort-of for membrane. Edward Witten who proposed the term M theory has stated that supersymmetry is an implication of string theory.

The mathematics that has come out of string theory is esoteric but rigorous but also with a dubious connection to the real world. As an illustration of how dubious is the connection of string theory is to the world consider that string theories were intended to explain the system of subatomic particles and then one string theorist decided that his formulation really applied to astronomically massive black holes. The formulations of string theory became so abstruse that the Bogdanov brothers who were honestly trying to make some contribution to the field ended producing an article of pure nonsense that was never-the-less still published in a reputable journal of physics.

So the bold speculations of Supersymmetry Theory need to be questioned lest it turn into another field in which it is hard to distinguish between sense and nonsense.

The most significant element of the supersymmetric model is that particle types occur in pairs. For each boson type (a particle with integral spin) there is a fermion type (a particle with half integral spin). This is what is meant by supersymmetry.

The motivation for the model came from the success of P.A.M. Dirac in predicting the existence of anti-particles. Dirac proposed that particle physics should be invariant for a conjugation (switch) in particle charges. This required the existence for each particle a partner particle of the same mass but opposite charge. It was conjectured that there would be a similar invariance for a switch in particle spins and would be equivalent to switching fermions to bosons and bosons to fermions. Initially this was pure conjecture but later physicists realized that such invariance and the existence of the partners required to make it possible would explain certain things in particle physics. This gave physicists confidence that such partner particles do exist.

The physicists in this field have already adopted names for the supposed particles. The name for the symmetry partner of a fermion particle is the name of that particle with the letter s affixed at the front. Thus the boson partner of the proton fermion is a sproton. This generally works in the sense of giving a pronounceable name which has no alternate meaning. One exception is the case of the top quark. The partner would be a stop and this can easily make for confusion in a sentence. It would be better to represent this name as s/top.

The name of a boson is the name of the boson with ino at the end, either simply added on or replacing the on in the boson's name. Thus the partner of the photon is the photino. Those of the W and Z particles are the Wino and the Zino. Such is the way sphysicists talk.

The symbol for supersymmetr partner of a standard particle is the symbol of the particle with a tilde written over it. Thus symbol for a sneutron is ñ.

The above simple system of naming the partner particles is modified for pairs that are close enough in their characteristics that mixtures of the pair may be observed. One such pair is the partners of a W boson and an electrically charged Higgs boson;, a Wino and a Higgsino. A mixture of these is known as a chargino. A mixture of the partners of a Z particle, a photon and a Higgs neutral boson, (a Zino, a photino and two Higgsinos). is known as a neutralino.

An Illustration of the Mathematics
of the Supersymmetry Model


A state of a particle is a vector in an infinite dimensional Hilbert Space whose components are complex numbers. An operator is a linear transformation of all of the vectors of that Hilbert Space. For an operator H there are special vectors such that

HX = λX

where λ is a complex number (real numbers being special cases of complex numbers).

Such a vector X is called an eigenvector for the operator and λ is called its eigenvalue. An operator generally has more than one eigenvector. P.A.M. Dirac introduced a brilliant bit of notation called bras and kets. An infinite dimensional column vector in Hilbert Space is denoted by the ket |L> where L is a label. The label for an eigenvector may be its eigenvalue. Thus in the above X=|λ> so

H|λ> = λ|λ>

A row vector is denoted by a bra <L|.

This is all standard quantum theory. What follows is close to what is known as Second Quantization in quantum theory,

The Hamiltonian of Paired Oscillators

Define a pair of one dimensional oscillators in potential fields V+ and V. The Hamiltonian functions are then

H± = p²/(2m) + V±

where p and m are the momentum and mass of the particle, respectively.

Let W(x) be a real valued function that generates the pair of potential functions through the equation

V± = W² ± (h/(2m)½)(dW/dx)

Also from the function W(x) two operators, A+ and A can determined through the formulas

A± = ±(i/(2m)½)p + W(x)

where i is the imaginary unit.

The significance of the operators A± is that A+A=H+ and AA+=H.


Because operators are not communicative extreme care be exercised in evaluating products.

A+A = p²/(2m) + i/(2m)½p(W) − i/(2m)½W(p) + W²

But for any wave function φ(x)

p(Wφ) − Wp(φ) = φp(W) + Wp(φ) − Wp(φ) = p(W)φ

The momentum operator for p is −ih(d/dx), thus

p(W)φ = −ih(dW/dx)φ

where h is Planck's constant divided by 2π


A+A = p²/(2m) + [ i/(2m)½][−ih(dW/dx)] + W²
which reduces to
A+A = p²/(2m) + h/(2m)½(dW/dx) + W²
further to
A+A = p²/(2m) + V+ = H+


AA+ = p²/(2m) + V = H

It is notable that the complex conjugates of A+ and A are such that

(A+)* = ((i/(2m)½)p + W(x))*
= −i/(2m)½)p + W(x) = A
and similarly
( A)* = A+

The Wavefunctions

Let n be the principal quantum number for the systems and let ψn and φn be eigenvectors of H+ and H, respectively, for principal quantum number n and μn and νn the corresponding eigenvalues. A wavefunction is conventionally interpreted as constituting the state of a field which may contain multiple particles. For eigenvectors the eigenvalues are interpreted as representing the number of particles in the field, but more generally the eigenvalues represent the number of energy quanta, which may or may not correspond to the number of particles. In the case of the quantum version of a harmonic oscillator the number of energy quanta correspond to paired peaks in the probability density function and these do not represent particles in any conventional notion of a particle. However, in the following the conventional view of eigenvalues corresponding to the number of particles will be adhered to.

A Second Quantization analysis establishes that

A+n> = ν½n-1>
An> = μ½n+1>

It is an awkward aspect of the notation that A+ is the annihilator operator and A is the creation operator.

The other important relation is

μn = νn+1

Further analysis establishes that the eigenvalues are real and integral; i.e., μn=n .

Ground States

The ground state is φ0. If the annihilator operator is applied to this ground state it cannot produce a state with eigenvalue −1. This means it has to annihilate the ground state; i.e.,

(A+0 = 0

where 0 stands for the zero vector.

The equation (A+0 = 0 is equivalent to the differential equation

(h/(2m)½)(dφ0/dx) + W(x)φ0 = 0

This equation has the solution

φ0(x) = c·exp(−((2m)½/h)∫0xW(z)dz)

where c is a constant.

The Hamiltonian for a Supersymmetric Model

The model is formulated in terms of 2×2 matrices whose components are which were introduced previously in the above material. The Supersymmetric Hamiltonian is defined in terms of the supersymmetric partners, H+ and H−, as

| H0    |
HS=|           |
|  0  H+ |

The eigenvectors of HS for a principal quantum number of n are the column vector versions of the row vectors

n, 0) and (0, ψn)

Let these eigenvectors be denoted as Φn and Ψ, respectively. Their eigenvalues are n and (n+1).

The properties of the model can be expressed in terms of a supersymmetry particle operator PS, defined as

| 00 |
PS=|        |
|  A+0 |

From the fact stated previously that the complex conjugate of A+ is A it follows that the conjugate transpose of PS is given by

| 0A |
P*S=|           |
|  0  0  |

From this property several other important properties follow; i.e.;

PSΦn = n½Ψn-1
    P*SΨn = (n+1)½Φn+1
    (PS)² = (P*S)² = 0^
    HS = PSP*S + P*SPS

In the above 0^ stands for the zero operator.

The RHS of the last equation is called the anti-commutator of PS and P*S. Thus HS is the anti-commutator of PS and P*S.

The commutator of two operations B and C, [B, C], is defined as BC−CB. The commutator of PS with HS is given by


The above evaluates to

| 00 |
| H+A+−A+H0 |

Because H=AA+,

A+H = A+(AA+)
is the same as
(A+A)A+ = H+A+

Thus (H+A+−A+H) is equal to the zero operator so [PS, HS]=0^ and hence PS and HS commute. Similarly P*S and HS commute.

Since HS, PS and P*S close under commutation and anti-commutation they form a graded Lie algebra.

(To be continued.)

For a timeline history of the development of the theory of supersymmetry see History.


O.L. De Lange & R.E. Raab, Operator Methods in Quantum Mechanics, Clarendon Press, Oxford, 1991.

Gordon Kane, Supersymmetry, Perseus Publishing, Cambridge, Mass., 2000.

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