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lished in 1975 for the genuine interaction between mechanics and
mathematics. New phenomena in mechanics require the develop-
ment of fundamentally new mathematical ideas leading to mutual
enrichment of the two disciplines. The society fosters the interests of
its members, elected from countries worldwide, by a series of bian-
nual international meetings (STAMM) and by specialist symposia
held frequently in collaboration with other bodies.




© 2001 by Chapman & Hall/CRC
CHAPTER 1

General Remarks on
Supersymmetry

1.1 Background
It is about three quarters of a century now since modern quan-
tum mechanics came into existence under the leadership of such
names as Born, de Broglie, Dirac, Heisenberg, Jordan, Pauli, and
Schroedinger. At its very roots the conceptual foundations of quan-
tum theory involve notions of discreteness and uncertainty.
Schroedinger and Heisenberg, respectively, gave two distinct but
equivalent formulations: the con¬guration space approach which deals
with wave functions and the phase space approach which focuses on
the role of observables. Dirac noticed a connection between commu-
tators and classical Poisson brackets and it was chie¬‚y he who gave
the commutator form of the Poisson bracket in quantum mechanics
on the basis of Bohr™s correspondence principle.
Quantum mechanics continues to attract the mathematicians
and physicists alike who are asked to come to terms with new ideas
and concepts which the tweory exposes from time to time [1-2]. Su-
persymmetric quantum mechanics (SUSYQM) is one such area which
has received much attention of late. This is evidenced by the fre-
quent appearances of research papers emphasizing di¬erent aspects
of SUSYQM [3-9]. Indeed the boson-fermion manifestation in soluble
models has considerably enriched our understanding of degeneracies


© 2001 by Chapman & Hall/CRC
and symmetry properties of physical systems.
The concept of supersymmetry (SUSY) ¬rst arose in 1971 when
Ramond [10] proposed a wave equation for free fermions based on
the structure of the dual model for bosons. Its formal properties
were found to preserve the structure of Virasoro algebra. Shortly
after, Neveu and Schwarz [11] constructed a dual theory employing
anticommutation rules of certain operators as well as the ones con-
forming to harmonic oscillator types of the conventicnal dual model
for bosons. An important observation made by them was that such
a scheme contained a gauge algebra larger than the Virasoro algebra
of the conventional model. It needs to be pointed out that the idea
of SUSY also owes its origin to the remarkable paper of Gol™fand and
Likhtam [12] who wrote down tne four-dimensional Poincare super-
algebra. Subsequent to these works various models embedding SUSY
were proposed within a ¬eld-theoretic framework [13-14]. The most
notable one was the work of Wess and Zumino [14] who de¬ned a
set of supergauge transformation in four space-time dimensions and
pointed out their relevance to the Lagrangian free-¬eld theory. It
has been found that SUSY ¬eld theories prove to be the least diver-
gent in comparison with the usual quantum ¬eld theories. From a
particle physics point of view, some of the major motivations for the
study of SUSY are: (i) it provides a convenient platform for unifying
matter and force, (ii) it reduces the divergence of quantum gravity,
and (iii) it gives an answer to the so-called “hierarchy problem” in
grand uni¬ed theories.
The basic composition rules of SUSY contain both commutators
and anticommutators which enable it to circumvent the powerful
“no-go” theorem of Coleman and Mandula [15]. The latter states
that given some basic features of S-matrix (namely that only a ¬-
nite number of di¬erent particles are associated with one-particle
states and that an energy gap exists between the vacuum and the
one-particle state), of all the ordinary group of symmetries for the
S-matrix based on a local, four-dimensional relativistic ¬eld theory,
the only allowed ones are locally isomorphic to the direct product
of an internal symmetry group and the Poincare group. In other
words, the most general Lie algebra structure of the S-matrix con-
tains the energy-momentum operator, the rotation operator, and a
¬nite number of Lorentz scalar operators.


© 2001 by Chapman & Hall/CRC
Some of the interesting features of a supersymmetric theory may
be summarized as follows [16-28]:

1. Particles with di¬erent spins, namely bosons and fermions, may
be grouped together in a supermultiplet. Consequently, one
works in a framework based on the superspace formalism [16].
A superspace is an extension of ordinary space-time to the one
with spin degrees of freedom. As noted, in a supersymmetric
theory commutators as well as anticommutators appear in the
algebra of symmetry generators. Such an algebra involving
commutators and anticommutators is called a graded algebra.

2. Internal symmetries such as isospin or SU (3) may be incorpo-
rated in the supermultiplet. Thus a nontrivial mixing between
space-time and internal symmetry is allowed.

3. Composition rules possess the structure [28]

Xa Xb ’ (’)ab Xb Xa = fab Xc
c


where, a, b = 0 if X is an even generator, a, b = 1 if X is an odd
c
generator, and fab are the structure constants. We can express
X as (A, S) where the even part A generates the ordinary n-
dimensional Lie algebra and the odd part S corresponds to the
grading representation of A. The generalized Lie algebra with
generators X has the dimension which is the sum of n and the
dimension of the representation of A. The Lie algebra part of
the above composition rule is of the form T — G where T is
the space-time symmetry and G corresponds to some internal
structure. Note that S belongs to a spinorial representation of
a homogeneous Lorentz group which due to the spin-statistics
theorem is a subgroup of T .

4. Divergences in SUSY ¬eld theories are greatly reduced. In-
deed all the quadratic divergences disappear in the renormal-
ized supersymmetric Lagrangian and the number of indepen-
dent renormalization constants is kept to a minimum.

5. If SUSY is unbroken at the tree-level, it remains so to any order
of h in perturbation theory.
¯


© 2001 by Chapman & Hall/CRC
In an attempt to construct a theory of SUSY that is unbroken
at the tree-level but could be broken by small nonperturbative cor-
rections, Witten [29] proposed a class of grand uni¬ed models within
a ¬eld theoretic framework. Speci¬cally, he considered models (in
less than four dimensions) in which SUSY could be broken dynam-
ically. This led to the remarkable discovery of SUSY in quantum
mechanics dealing with systems less than or equal to three dimen-
sions. Historically, however, it was Nicolai [31] who sowed the seeds
of SUSY in nonrelativistic mechanics. Nicolai showed that SUSY
could be formulated unambiguously for nonrelativistic spin systems
by writing down a graded algebra in terms of the generators of the
supersymmetric transformations. He then applied this algebra to
the one-dimensional chain lattice problem. However, it must be said
that his scheme did not deal explicitly with any kind of superpoten-
tial and as such connections to solvable quantum mechanical systems
were not transparent.
Since spin is a well-de¬ned concept in at least three dimensions,
SUSY in one-dimensional nonrelativistic systems is concerned with
mechanics describable by ordinary canonical and Grassmann vari-
ables. One might even go back to the arena of classical mechanics
in the realm of which a suitable canonical method can be devel-
oped by formulating generalized Poisson brackets and then setting
up a correspondence principle to derive the quantization rule. Con-
versely, generalized Poisson brackets can also be arrived at by taking
the classical limit of the generalized Dirac bracket which is de¬ned
according to the “even” or “odd” nature of the operators.
The rest of the book is organized as follows.
In Chapter 2 we outline the basic principles of SUSYQM, start-
ing with the harmonic oscillator problem. We try to give a fairly
complete presentation of the mathematical tools associated with
SUSYQM and discuss potential applications of the theory. We also
include in this chapter a section on superspace formalism. In Chapter
3 we consider supersymmetric classical mechanics and study gener-
alized classical Poisson bracket and quantization rules. In Chapter
4 we introduce the concepts of SUSY breaking and Witten index.
Here we comment upon the relevance of ¬nite temperature SUSY
and analyze a regulated Witten index. We also deal with index con-
dition and the issue of q-deformation. In Chapter 5 we provide an


© 2001 by Chapman & Hall/CRC
elaborate treatment on factorization method, shape invariance con-
dition, and generation of solvable potentials. In Chapter 6 we deal
with the radial problem and spin-orbit coupling. Chapter 7 applies
SUSY to nonlinear systems and discusses a method of constructing
supersymmetric KdV equation. In Chapter 8 we address parasuper-
symmetry and present models on it, including the one obtained from
a truncated oscillator algebra. Finally, in the Appendix we broadly
outline a mathematical supplement on the derivation of the form of
D-dimensional Schroedinger equation.


1.2 References
[1] L.M. Ballentine, Quantum Mechanics - A Modern Develop-
ment, World Scienti¬c, Singapore, 1998.

[2] M. Chester, Primer of Quantum Mechanics, John Wiley &
Sons, New York, 1987.

[3] L.E. Gendenshtein and I.V. Krive, Sov. Phys. Usp., 28, 645,
1985.

[4] A. Lahiri, P.K. Roy, and B. Bagchi, Int. J. Mod. Phys., A5,
1383, 1990.

[5] B. Roy, P. Roy, and R. Roychoudhury, Fortsch. Phys., 39, 211,
1991.

[6] G. Levai, Lecture Notes in Physics, 427, 127, Springer, Berlin,
1993.

[7] F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep., 251, 267,
1995.

[8] G. Junker, Supersymmetric Methods in Quantum and Statisti-
cal Physics, Springer, Berlin, 1996.

[9] M.A. Shifman, ITEP Lectures on Particle Physics and Field
Theory, 62, 301, World Scienti¬c, Singapore, 1999.

[10] P. Ramond, Phys. Rev., D3, 2415, 1971.

[11] A. Neveu and J.H. Schwarz, Nucl. Phys., B31, 86, 1971.


© 2001 by Chapman & Hall/CRC
[12] Y.A. Gol™fand and E.P. Likhtam, JETP Lett., 13, 323, 1971.

[13] D.V. Volkov and V.P. Akulov, Phys. Lett., B46, 109, 1973.

[14] J. Wess and B. Zumino, Nucl. Phys., B70, 39, 1974.

[15] S. Coleman and J. Mandula, Phys. Rev., 159, 1251, 1967.

[16] A. Salam and J. Strathdee, Fortsch. Phys., 26, 57, 1976.

[17] A. Salam and J. Strathdee, Nucl. Phys., B76, 477, 1974.

[18] V.I. Ogievetskii and L. Mezinchesku, Sov. Phys. Usp., 18, 960,
1975.

[19] P. Fayet and S. Ferrara, Phys. Rep., 32C, 250, 1977.

[20] M.S. Marinov, Phys. Rep., 60C, 1, (1980).

[21] P. Nieuwenhuizen, Phys. Rep., 68C, 189, 1981.

[22] H.P. Nilles, Phys. Rep., 110C, 1, 1984.

[23] M.F. Sohnius, Phys. Rep., 128C, 39, 1985.

[24] R. Haag, J.F. Lopuszanski, and M. Sohnius, Nucl. Phys., B88,
257, 1975.

[25] J. Wess and J. Baggar, Supersymmetry and Supergravity, Prince-
ton University Press, Princeton, NJ, 1983.

[26] P.G.O. Freund, Introduction to Supersymmetry, Cambridge Mono-
graphs on Mathematical Physics, Cambridge University Press,
Cambridge, 1986.

[27] L. O™Raifeartaigh, Lecture Notes on Supersymmetry, Comm.
Dublin Inst. Adv. Studies, Series A, No. 22, 1975.

[28] S. Ferrara, An introduction to supersymmetry in parti-
cle physics, Proc. Spring School in Beyond Standard Model
Lyceum Alpinum, Zuoz, Switzerland, 135, 1982.

[29] E. Witten, Nucl. Phys., B188, 513, 1981.

[30] E. Witten, Nucl. Phys., B202, 253, 1982.


© 2001 by Chapman & Hall/CRC
[31] H. Nicolai, J. Phys. A. Math. Gen., 9, 1497, 1976.

[32] H. Nicolai, Phys. Bl¨tter, 47, 387, 1991.
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