1. 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
Supersymmetric harmonic oscillator
In the lagrangian mentioned before, if we set , we obtain the supersymmetric harmonic oscillator.
The action can be divided into two parts , a bosonic part and a fermionic part. ,
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 :
Then, using the decomposition of x(t) in bosonic action and simplifying, we obtain:
Then, the variables (DA’s) are separated and integration is carried out (this is a Gaussian integral) and after further simplification:
Now, we note that,
The constant is fixed by comparing with the free particle case whose constant is fixed using zeta function regularization. We obtain:
Now solving for classical
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: ;
Anti-periodic : ( )
This gives the result:
(2)
2) Periodic : ( )
(3)
Now, finally, we combine both path integrals. If we use (1) and (2), we get
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.
Abstract
Where S[x] is the action associated with the path x(t) and Dx means we have to sum over all possible paths.
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 p 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.
Supersymmetric Quantum Mechanics
The Lagrangian is given by:
Where , is some generic potential and and are fermionic variables (Grassmann numbers) that anti commute with each other and square to zero i.e
Then under the following supersymmetric transformations, the action remains invariant:
; ;
and are Grassmann numbers.
The generators of this supersymmetry are called supercharges and were found to be :
and
They satisfy the algebra :
Furthermore, these supercharges commute with the Hamiltonian :
Conclusions
Schrodinger’s equation and path integral formalism are equivalent.
The ground state energy of a supersymmetric harmonic oscillator is zero !!!
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.
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
The time evolution operator (the one which evolves the wave function in time) is defined as:
Where is the Hamiltonian.
Then by Schrodinger’s equation and using bra-ket notation , we find that :
----- Eqn (A)
While, by path integral method, we have :
Bibliography
Mirror Symmetry (Cumrun Vafa et al) :
Supersymmetric hadronic mechanical harmonic oscillator (A.K.Aringazin) :