648
Progress of Theoretical Physics, Vol. 71, No.3, March 1984
Elimination of Redundant Variables
in a Partially Supersymmetric Bi-Local Model
Shigefumi N AKA and Hiroshi
KAKUHATA
Department of Physics, College of Science and Technology
Nihon University, Tokyo 101
(Received October 3, 1983)
The elimination of redundant variables is discussed with respect to the bi-Iocal model which have
fermionic variables as supersymmetric counterparts of relative time's degrees of freedom. It is shown that
in this model, the supplementary condition defining the physical space becomes a fermionic one which has
the meaning of the square root of the ghost number variable. The interaction of the bi·local system with
an external scalar field is also discussed.
§ 1_
Introduction
The relativistic treatment oftwo-body problems (the bi-Iocal model) as a constrained
dynamical system l ) is known to give a useful quasiphenomenological approach to the
bound states in the quark model of hadrons. Especially the model, to which the harmonic
oscillator potential is assumed, is interesting since the bound state in this case can be
solved exactly and allows linear rising Regge trajectories.
In such a model, however, there arises the problem of eliminating the degrees of
freedom of the relative time of the system, which provides the ghost states in q-number
theory. In classical treatment of such a bi-Iocal model, one frequently starts with the
classical action which is invariant under the reparametrization of time ordering parameters of respective particles. Then, as the result of this invariance, the system is supplemented by two kinds of primary constraints which correspond to the classical counterpart of the master-wave equation of the system Ho=O and the supplementary condition T
=0 characterizing the physical subspace in the phase space. For instance, for the system
of massless particles, these constraints become
(x=const)
T=P-P=O,
(1-1)*)
where f is the relative coordinate of the system and P and p represent the total
momentum and relative momentum of the system respectively. In Eq. (1-1), the supplementary condition T=O seems to play the role freezing the degrees of freedom of the
relative time of system. The Poisson bracket of T with H o, however, leads to the
following secondary constraints:
(1-2)
from which follows that the particles can move only to the opposite directions with light
*) In this paper, we use the metric: diag(gpv)=( + - - -).
We also use the units: h=c=1. As for the
action, from which Eq. (1·1) can be derived, see Appendix A.
Elimination of Redundant Variables in a Partially Supersymmetric Bi-Local Model
649
velocity; that is, the particle is free from the potential.
This situation can also be understood in relation to the no-interaction theorem2 ) which
asserts the impossibility of constructing a relativistic formalism of interacting particles in
terms of the canonical variables defined on a space-like surface. In order to avoid this
difficulty, we usually treat the supplementary condition T=O in the sense of expectation
value by physical states in the stage ofq-ntlffiber theory.3) It is, however, desirable to get
a formalism of the bi-Iocal model which is applicable to both c-number and q-number
theories. The purpose of the present paper is to solve this problem by introducing the
fermionic counterparts of the relative time's degrees of freedom into the model.
In the next section, we investigate the canonical formalism of a bi-Iocal model which
has a supersymmetry between the relative time's degrees of freedom and its fermionic
counterpart. There, it is shown that the supplementary condition T=O in Eq. (1·1) is
replaced by a fermionic constraint Q = 0 which is compatible with the constraint of the
master-wave equation in both cases of c-ntlffiber and q-number theories. In §3, we
investigate the physical space and physical variables defined by that fermionic constraint.
We also discuss the interaction of the bi-Iocal system with an external scalar field from the
viewpoint of constructing interaction vertex in terms of physical variables. Section 4 is
devoted to the summary and remaining discussion.
§ 2.
Bi-Iocal model with fermionic ghosts
The bi-Iocal model having a supersymmetry 4) between its bosonic and fermionic ghost
variables can be described in terms of position variables Xi, (i=1, 2) of respective
particles and scalar Grassmann variables ~i, (i=1, 2) associated with each particle. We
assume that the action of the system is the following (Appendix B):
(2·1)
where the dot represents the derivative with respect to the time ordering parameter r
which is assigned to be common to both particles. Further, e and % are einbein variables
of the usual canonical variable and the Grassmann variable respectively.
We note that the above action is invariant under the reparametrization of r and the
supersymmetric transformation defined by
O%=-c,
(c; Grassmann parameter)
(2·2)
In the canonical formalism, these invariances lead to the following constraints:
~;
= -
H=
-{! p2+ j}2+X2.f2+2iX~1~2}=O ,
(2·3)
s.
650
Naka and H Kakuhata
Varying the action with respect to ~/s, we also obtain the second class constraints which
are peculiar to the Grassmann variables:
(j=1,2)
where Il/s are momenta conjugate to
~/s.
(2·4)
Then the variables defined by
(2·5)
become oscillator variables with respect to the Dirac bracketS) { , }* which establishes Eq.
(2·4) as identities; that is, we get
{C, c+}*={c+, c}*=-i.
(2·6)
We also define the bosonic oscillator variables so that
1 (_
'ap +_
- ffx
xxp-z.p-)
p,
(2·7)
which satisfy
(2·8)
Then, in terms of these oscillator variables, the Hand Q can be expressed as :
H=2x(a'
P2+~{ap+, aP}-'-~[c+,
c]),
Q = p. ac+ + p. a+ c ,
(2·9)
where a' = 1/ 8x and { , } and [ , ] represent the anticommutator and commutator respectively.
Now, in consideration of Eqs. (2·6) and (2·8), one can verify the following algebra
easily:
{Q, Q}*=2iP 2N ,
{Q, H}*= {Q, N}*= {H, N}*=O,
(2·10)
where
N=-
~ {a ll +,
all}+
~ [c+, c]
(2·11)
with the notation a ll =a·P/fP2. Equations (2·10) and (2·11) mean that Q is proportional to the square root of the mass term which arises from the time-like and fermionic
oscillations of the system. Thus, with the secondary constraint N = 0, Q = 0 can suppress
the degrees of freedom of the ghosts. The above formulation can be easily translated to
the q-number theory according to the substitution: [A, B] or {A, B}-+ i {A, B}*.
§ 3.
Physical variables and the interaction with an external scalar field
In q-number theory, the set of constraints H=O and Q=O, (N=O) are understood
Elimination of Redundant Variables in 'a Partially Supersymmetric Bi-Local Model
651
respectively as the master wave equation of the system and the subsidiary conditions
which characterize. physical states; that is,
H/Phys) =0
(3·1)
Q/Phys)=N/Phys) =0 .
(3·2)
and
The explicit form of the physical states is determined depending on the choice of the
ground state for the oscillator variables (ap +, ap) and (c+, c). In order to guarantee the
manifest Lorentz covariance, we hereafter define the ground state by
(3·3)
ap/O)=c/O)=O.
In this case; since the mass square operator defined from H becomes
M2=~ -ap+aP+c+c+ ~»O,
(3·4)
the subsidiary condition (3·2) really chooses the positive norm states in consideration of
P2>O.*> "
The projection operator for the physical states defined by Eq. (3·2) can easily be
'
obtained to give
1 dBe
l'
1=27l'
2
"
iBN
(3·5)
•
0
Then we can see that the physical part [A] of an operator A defined by IAI = [A]I
becomes
[A] =_1_12" dfJe iBNAe- iBN
27l'
(3·6)
0
apart from the additional terms which vanish on the physical states.
we can get, for example, the following:
By using Eq. (3·6),
[Xl] = [X] + ~ [x],
(3·7)
where Xl is the position variable of particle 1,
pv
[Xp]=X p+ p 2 Svp,
and
-]_(
PpPv)-_[XI'
- gpv-~
Xv=X.L,P (P) .
(3·8)
Now, in terms of the above physical variable (3·7), one
Fig. 1.
*> In this case, N/Phys)=O becomes, in the reference frame (P)=(P O, 0, 0, 0), (-ao+ao+c+c)/Phys)=O.
Thus, the physical state does not allow the excitation of ao + and c+: that is, the degrees of freedom of the ghosts.
One can verify that with the definition of a,/O>= ao+/O> = c/O>=O instead of (3'3), the subsidiary condition (3'2) also
choose positive norm states provided p2>0.
s.
652
Naka and H. Kakuhata
can naturally define the interaction vertex of the bi·local system with an external scalar
field (Fig. 1) by
(3' 9 )*)
(g =const)
In this case, we can obtain the form factor of the ground state in the following form
(Appendix C):
<p",ol V(k)IO, p')=g2a(4)(p" - p' -k)exv [
t~:;2 ( sin-
! _t~!m2 YJ.
1
(3-10)
where t = k 2 and
10, P') = 10)®IP'), (PiP') = P'IP'),· <P"IP') = a(4)(p" - P'».
It can be seen that the above form factor behaves asymptotically as different Gaussian
functions of k in the respective regions of - t~4m2 and ~ t~4m2. One can also calculate
the scattering amplitude of the bi-Iocal field by using the vertex operator (3-9). Then, it
can be verified that the second order amplitude exchariging the bi-Iocal field shows Regge
behaviour in an asymptotic region.
We note that the physical interaction vertex is not determined uniquely and thus, the
same is true for the form factor. For example, if we adopt
(3-11)
as the interaction vertex instead of (3-9), then the form factor ofthe ground state becomes
(3-12)
In this case, however, the meaning of the physical position variable is not apparent, though
V'(k) is a physical quantity.
Another candidate for the physical interaction vertex is the following:
V"(k)=gexv[ -ik-([X]+~
.n],
(3-13)
where
(3-14)
Indeed, since [Q, X]=4X-l~1~2PQ, Eq. (3-14) really defines a physical relative coordinate
without depending on the total momentum of the system. Unfortunately, however, x is
not a hermitian operator due to the reason xlPhys)= -/2{x a.LIPhys), from which also
follows that the form factor of the ground state defined from V"(k) becomes simply a
constant.
§ 4_
Summary and discussion
We have seen that the bi-Iocal model having a partial supersymmetry between its
bosonic and fermionic ghost variables allows constraints which are compatible with the
master-wave-equation in both cases of c-number and q-number theories. In q-number
*)
point. S)
This type of the vertex operator was investigated firstly by Got6 and Kamimura from a different view-
Elimination of Redundant Variables in a Partially Supersymmetric Bi-Local Model
653
theory, the constraints became subsidiary conditions characterizing the physical states
and really, could suppress the degrees of freedom of the ghost to the ground state. The
physical feature of the bi-Iocal model, however, have not been changed drastically by
introducing fermionic ghost variables to the theory. For example, the physical position
variable [Xl] became the same as the one which is defined by the usual subsidiary condition
p. alPhys) = O.
The above method eliminating redundant variables in the form of a supersymmetric
ghost pair will be applicable more naturally of a fully supersymmetric model such as the
counterparf) of the Neveu-Schwartz-Ramond stringS) in the bi-Iocal model. In such a
case, the system is characterized by two kinds of oscillator variables (aI', ap +: bose) and
(b p , bp +: fermi) which transform as four vectors. Then, the fermionic ghosts (c, c+)
appearing in this paper may be regarded as c = p. bl !P.
In the present paper, we have discussed the introduction of fermionic ghosts in the
context of the supersymmetry. It should be noticed, however, that we can treat the
constraints Q = N = 0 in relation with the BRS-symmetry 9) in an extended phase space;
that is, we can represent these constraints in a single form such aslO)
Q= Qy+ P 2N7}- iIIf/y 2=O,
where (y, py) and (7}, IIf/) are auxiliary bosonic and fermionic canonical pairs respectively. Then, in consideration of {llf/, 7} }*= {7}, IIf/}*= -1, one can verify the following easily:
{Q, Q}*=O;
that is, Q behaves as the BRS-charge in the extended phase space.
We, finally, comment on the other possibilities handling the additional ghosts (c, c+)
as the quantities quantized by {C, c+}=O or [C, c+]=1. The case {C, c+}=O is related
more directly to a BRS-symmetry instead of the supersymmetry in handling the fermionic
ghosts. In the case of [c; c+]=I, physical states allow the excitation of (all, Cl II +) and
(c, c+) in such a form as (all ++c+)n(a l +- ic+)nlo), (n=O, 1, ... ). Since these are positive
norm states, it may be possible to treat the additional ghosts as bosonic ones. These
modified approaches in handling the additional ghosts will be discussed in another place.
Appendix A
The action of the bi-Iocal system characterized by constraints (1·1) is the follow·
ing:
2
L = l! mCr)j :i ;2,
i=l
(A·I)
where the notations are the same as the ones used in §§1 and 2. Introducing, here, the
auxiliary (einbein) variables e;, U=I, 2), the above action can be rewritten in the
following form:
(A·2)
Then the variation of the action L2 with respect to e;'s leads to the set of constraints
654
S. Naka and H Kakuhata
(A'3)
(i = 1, 2). Equation (1'1) is obtained by taking linear combination of these constraints.
Appendix·B
The form of the action (2'1) is not unique. For example,
1 .
.
l · i 2
•
L=-2 [X+X{/Xf;1-(2eIX)-lf;2})2+-4 f 2-ex(xf 2+2i;16)--2 ~;j;j
e
J~l
(B'1)
and
(B'2)
( {I = ;2,
{2 = -;1)
are other expressions to action (QH). One can see that action (B'2) is obtained from
action (A· 2) through a slight moOification. In (A, 2), the system has the symmetry
characterized by two kinds of bosonic einbein variables ei, (i=1, 2). On the other side
in (B· 2), the symmetry of the system is specified by a common einbein e and an additional
fermionic einbein X.
Appendix C
We here show the method calculating the form factor given in Eq. (3'10). This can
be done by using the following equation l l ) repeatedly:
Texp[l,su ds {F( s )+ G( s )} ] = Texp[l,su dsF( s )] Texp[l,su ds G( s )].
(C'l)
where
(C'2)
and T stands for the chronological ordering with respect to s. The above equation can
be verified easily by solving the differential equation for G(s) which is obtained by
differentiating the both sides of Eq. (C'1) with respect to s".
In virtue of Eq. (C .1), we firstly get
e-iko[X(l)] =
Texp[ - i 11 dsk·
[x (1)] ]
(C'3)
where P s = P + sk and
ffl(P)=
P~lIfl + ~ f.L,fl(P),
(C'4)
The chronological ordering on the right-hand side of Eq. (C·3) can be calculated by using
Elimination of Redundant Variables in a Partially Supersymmetric Bi-Local Model
655
Eq. (C'l) again and in consideration of that the Spv's are the generator of the Lorentz
group, we have the following secondary:
(C'5)
where (P /\k) is the 4X4 matrix defined by (P /\KV=gIlPpIPk P] and
Q(s)=
i
s
o
ds'
P
s'
(C'6)
2 •
Now, it is not difficult to verify that
(C'7)
Thus, since the ground state is invariant under the Lorentz transformation, we finally
obtain the following:
2
-1/ -
_ S-(P" - P' - k) exp [ f-4m
f
(.
16xSill-t+4m2
-u
)2] .
(C'8)
In the last equality, we have used the mass-shell conditions p'2 = p"2 = m 2 and p'. k
= - (t/ 2), (t = k 2 < 0 ). Equation (C, 8) is the result which we want to prove.
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