Journal of Applied Mathematics and Physics, 2013, 1, 105-109
Published Online November 2013 (http://www.scirp.org/journal/jamp)
Open Access JAMP
Path Integral Quantization of Superparticle with 1/4
Nasser Ismail Farahat, Hanaa Abdulkareem Elegla
Physics Department, Islamic University of Gaza, Gaza, Palestine
Email: email@example.com, firstname.lastname@example.org
Received September 28, 2013; revised October 28, 2013; accepted November 5, 2013
Copyright © 2013 Nasser Ismail Farahat, Hanaa Abdulkareem Elegla. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
We present path integral quantization of a massive superparticle in 4d
which preserves 1
4 of the target space su-
persymmetry with eight supercharges, and so corresponds to the partial breaking 8NN2
. Its worldline action
contains a Wess-Zumino term, explicitly breaks 4d
Lorentz symmetry and exhibits one complex fermionic
-symmetry. We perform the Hamilton-Jacobi formalism of constrained systems, to obtain the equations of motion of
the model as total differential equations in many variables. These equations of motion are in exact agreement with those
obtained by Dirac’s m ethod.
Keywords: Hamilton-Jacobi Formalism; Singular Lagrangian; Supersymmetry
The theory of constrained systems is a basis of modern
physics: gauge field theories, quantum gravity, supergra-
vity, string and superstring models are examples of sys-
tems with constraints. For such theories, one should sp ec i-
fy not only an evolution equation but also additional re-
quirements (constraints) being imposed on initial condi-
A standard consistent way of dealing with singular
systems was first formulated by Dirac . In Dirac for-
mulism, when a singular Lagrangian in configuration
space is transformed into a singular Lagrangian in phase
space, the inherent constraints would be generated, which
are called Dirac primary constraints [3-5]. Through the
consistency conditions, step by step, using these primary
constraints may generate more new inherent constraints,
which are called Dirac secondary constraints. Such a way
to calculate different constraints in Dirac formalism is
named as Dirac-Bergmann algorithm, which was first
propos ed by Bergmann [6,7].
Canonical path integral method is a kind of quantiza-
tion method [8,9], which depends on Hamilton-Jacobi
formalism shown by Güler [10,11]. This method has
some very useful properties of obviating the need to dis-
tinguish primary and secondary constraints and the first
and the second types of constraints. The method is sim-
pler, and does not have such a hypothesis of Diracs con-
jecture, thus it has evoked much attention [12-20].
Partial breaking of global supersymmetr y (PBGS) [21-
24] is widely understood to be an inborn feature of su-
persymmetric extended objects. The concept of PBGS
provides a manifestly off-shell supersymmetric world-
volume description of various superbranes in terms of
Goldstone sup erfields
. The physical wo r l d -
volume multiplets of the given superbrane are interpreted
as Goldstone superfields realizing the spontaneous brea-
king of the full brane supersymmetry group down to its
unbroken worldvolume subgroup [25,26]. The technical
tools here are the method of nonlinear realizations
[27-29]. Recently, there has been a growing interest in
PBGS options other than the 1/2 breaking [30-34]. This
is essentially due to the discovery of the super-
gravity solutions preserving 1/4 or 1/8 of the
supersymmetry  and their subsequent interpretation
in terms of intersecting branes. Since branelike world-
volume effective actions which would be capable of de-
scribing those solutions are still unknown, it seems inter-
esting to study pointlike models that mimic the exotic
supersymmetry breaking options inherent in the inter-
secting branes. Such models could share some character-
N. I. FARAHAT, H. A. ELEGLA
istic features of the systems of intersecting branes, much
like the ordinary superparticle bears a similarity to the
Green-Schwarz superstring. Superparticle models exhib-
iting 3/4 or 1/4 PBGS have been constructed [35-38].
The 1/4 breaking of the original supersymmetry
down to manifests itself in the presence of only
one complex-symmetry in the corresponding worldline
action. This is achieved at the cost of the explicit break-
ing of the target space Lorentz symmetry down to SO(3)
symmetry (in the fermionic sector).
In the present paper we study the canonical path in-
tegral quantization for model as a ty-
pical example of massive superparticles with 1/4 PBGS
in an ordinary four dimensional Minkowski spacetime
(with as the target superspace and explicitly bro-
ken Lorentz symmetry) . Prior to quantization, Ham-
iltonian analysis is accomplished in full detail, by ob-
taining the set of inherent constraints, and the equations
of motion as total differential equations. Our paper is
organized as follows. Hamilton-Jacobi formulation is
presented in Section 2. In Section 3, the canonical path
integral qu antizatio n of our model is investigated. In Sec-
tion 4, the conclusion is given.
2. Hamilton-Jacobi Formalism of
The system that is described by the Lagrangian
, , is constrained system if the
Hessian matrix 1, ,i
has a rank
, . In this case we have
momenta are dependent of each other. The generalized
momenta i corresponding to the generalized coor-
dinates are defined as,
Since the rank of the Hessian matrix is
may solve (2) for as
Substituting (4) into (3), we obtain relations in
and t in the form
The canonical Hamiltonian 0
is defined as
The set of Hamilton-Jacobi partial differential equa-
tions (HJPDE) is ex p re ss ed as
and being the action. The equations of
motion are obtained as total differential equations in
many variables such as,
.These equation are integrable if
and only if
If the conditions (13) and (14) are not satisfied iden-
tically, we consider them as new constraints and we ex-
amine their variations. Thus repeating this procedure, one
may obtain a set of constraints such that all the variations
vanish, then we may solve the equations of motion (10)
and (11) to get the canonical phase-space coordinates as
qq ttpp ttr
In this case the path integral representation may be
1, ,,0,1, ,.anr nrn
We should notice that the integral (16) is an integra-
tion over the canonical phase space coordinates
3. Hamilton-Jacobi Formulation of
Superparticle with 1/4 Supersymmetry
(6) The action functional of a massive superparticle model
exhibiting 1/4 PBGS, is written as 
Open Access JAMP
N. I. FARAHAT, H. A. ELEGLA 107
are four complex fermions parameterizing the
odd sector of the model. The Lagrangian is
The singularity of the the Lagrangian follows from the
fact that the rank of the Hessian matrix ij
The canonical momenta defined in (2) and (3) read as
Now the velocities 0
P can be expressed in
terms of the momenta and respectively as
The canonical Hamiltonian
is obtained as
The set of HJPDE’s are
Therefore, the total differential equations for the cha-
racteristics read as
xeP ii ,
To check whether the set of Equations (35)-(43) are
integrable or not, let us consider the total variations of
the set of (HJPDE)’s. The variati on of
are identically zero, whereas
Open Access JAMP
N. I. FARAHAT, H. A. ELEGLA
is a new constraint. Thus the equations of motion (35)-
(43) and the new constraint (50) represent an integrable
Now to obtain the path integral quantization of this
system, we can use (12) to obtain the canonical action
By using (51) and (16) the path integral for the system
is expressed as
The path integral qantization of constrained systems is
obtained for using the canonical path integral method,
which based on the constrained Hamilton theory. The
equations of motion are obtained as total differential
equations in many variables, and the integrability con-
ditions were shown to be equivalent to the vanishing of
the variation of each
, i.e. ’s, then the sys-
tem is integrable. In this paper, we examined Hamilto-
nian treatment of a massive superparticle model with 1/4
partial breaking of global supersymmetry which propa-
gates in four dimensional flat spacetime. We obtain con-
straints in phase space, which contains all kinds of con-
straints (primary and secondary, first and second class
ones). This example is very illustrative, since it allows a
comparison between all features of Diracs and Hamil-
ton-Jacobi formalisms. In Dirac’s formalism, we must
reduce any constrained singular system to one with first-
class constraints only, and we must call attention to the
presence of arbitrary variables in some of the Hamil-
tonian equations of motion due to the fact that we have
gauge dependent variables, therefore we have made a
gauge fixing. This does not occur in Hamilton-Jacobi
formalism since it provides a gauge-independent descrip-
tion of the systems evolution due to the fact that the
Hamilton-Jacobi functio n S contains all the solu tions that
are related by gauge transformations. Our results are in
agreement with those given in Dirac’s method .
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