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Supersymmetric Quantum Mechanics
Usman Naseer , Syed Moeez Hassan , Saad Pervaiz , Rabia Aslam Chaudary , Syed Muntazir Mehdi Abidi
School of Science and Engineering , Lahore University of Management Sciences
Where S[x] is the action associated with the path x(t) and Dx
means we have to sum over all possible paths.
Abstract
In in our work, we studied a simple one dimensional
supersymmetric quantum mechanical system : the supersymmetric
harmonic oscillator. It was studied using both operator method
and path integral formalism and the results were found to be
consistent.
It is not always easy to calculate Tr(U) from Equation (A), path
integrals present a much easier way of doing it.
This figure shows different
paths that are possible
between points A and B.
If this was a free particle,
It would take the straight
line path between the two
points. However, in
Quantum Mechanics, we
have to consider all paths.
p
Introduction
Symmetries play a great role in physics and due to these
symmetries, many important quantities like momentum, energy
and charge are conserved. Simple symmetries include translational
symmetry, rotational symmetry , time invariance etc.
Supersymmetry is a greater symmetry of nature. Through
supersymmetry, bosons and fermions are paired to each other i.e.
for every boson there is a corresponding fermion and vice versa.
Richard Feynman gave another approach for Quantum Mechanics
than Schrodinger’s Equation. Although his method was
theoretically totally different than Schrodinger’s equation , It gave
the same results. Here is the great idea:
In classical mechanics, if a particle has to go from a point A to
another point B, it can only follow one path. In quantum
mechanics, it follows all possible paths between the points A and
B. Each path is weighted by a phase proportional to the action of
that particular path. Then we sum over all these paths. This is
known as path integral or sum over paths.
Path Integral
Formulation
Supersymmetric harmonic
oscillator
In the lagrangian mentioned before, if we set
obtain the supersymmetric harmonic oscillator.
--------(3)
Now, finally, we combine both path integrals. If we use (1) and
(2), we get
Then the path integral becomes :
For the bosonic part, we divide the path into classical path and the
remaining paths as:
(where
)
and we note that since end points are same (
) for
both classical path and x(t) , q(0) = 0 = q(T) (Periodic boundary
conditions). This allows us to write q(t) as :
However, if we use (1) and (3) , we get the trace of a very
special operator
where
This quantity :
is known as the Witten Index. It
gives the number of bosonic ground states minus the
fermionic ground states.
Then, the variables (DA’s) are separated and integration is carried
out (this is a Gaussian integral) and after further simplification:
Where ,
is some generic potential and
and
are
fermionic variables (Grassmann numbers) that anti commute
with each other and square to zero i.e
Now, we note that,
Then under the following supersymmetric transformations, the
action remains invariant:
;
;
and
are Grassmann numbers.
is the Hamiltonian.
They satisfy the algebra :
and
The constant is fixed by
comparing with the free particle case whose constant is fixed
using zeta function regularization. We obtain:
Conclusions

Schrodinger’s equation and path integral formalism are
equivalent.

The ground state energy of a supersymmetric harmonic
oscillator is zero !!!
Action and integrating over dx to obtain the trace, we get the
simple expression:
-------(1)
Now, we consider the fermionic action.
In this case, instead of breaking the path, we simply impose the
boundary conditions. Here there is a choice, for these
(Grassmann) variables, both periodic and anti-periodic boundary
conditions can be considered. We will see both of them.
The path integral is very similar to that done before except that
the following rule is used for integrating Grassmann variables:
The following identities were also used:
----- Eqn (A)
While, by path integral method, we have :
)
This gives the result:
,
The Lagrangian is given by:
and
Then by Schrodinger’s equation and using bra-ket notation , we
find that :
2) Periodic : (
Then, using the decomposition of x(t) in
bosonic action and simplifying, we obtain:
The generators of this supersymmetry are called supercharges
and were found to be :
Where
---- -------(2)
The action can be divided into two parts , a bosonic part and a
fermionic part.
Supersymmetric
Quantum Mechanics
)
This gives the result:
, we
Now solving for classical
The time evolution operator (the one which evolves the wave
function in time) is defined as:
1) Anti-periodic : (
Furthermore, these supercharges commute with the
Hamiltonian :
;
Bibliography
1. Mirror Symmetry (Cumrun Vafa et al) :
http://www.claymath.org/library/monographs/cmim01c.
pdf
2. Supersymmetric hadronic mechanical harmonic
oscillator (A.K.Aringazin) :
http://www.i-b-r.org/docs/susy.pdf