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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