fulltext

Z

-graded point particle theory

W.-S. CHUNG

Theory Group, Department of Physics, College of Natural Sciences Gyeongsang National University - Jinju 660-701, Korea

(ricevuto il 14 Maggio 1996; approvato il 23 Luglio 1996)

Summary. — In this paper Zp-graded point particle theory is constructed by using

the Zp-extension of the Grassmann variable through the quantum group. It is shown

that the action for Zp-graded point particle is covariant under the Zp-transformation

extending the supersymmetry transformation.

PACS 02.90 – Other topics in mathematical methods in physics.

1. – Introduction

Since Jimbo [1], Drinfeld [2] and Faddeev [3] constructed the new mathematical object called quantum group, this topic has been being studied by many theoretical physicists and mathematicians. In spite of much work in this direction, it is suspected whether this new mathematical object will bring the new “phenomena” in the fothcoming physics or not. Since the symmetries play an important role in physics, it is worth extending them to the deformed concept of symmetries which might be used in physics as well. If quantum groups are applied to physics, they are supposed to create a kind of “new” physics which goes back to its classical versions when the deformation parameters take particular values. To this end, it is worthwhile to make out the fundamental concepts and the computational techniques of quantum groups.

Para-Grassmann algebra is a very interesting subject in that it is related to conformal field theory [4, 5] and unusual statistics [6, 7] such as parastatistics and fractional statistics. Kerner [8, 9] studied a Z3-graded algebra to obtain the cubic root of the supersymmetry and Dirac equation. Recently, Filippov and his coworkers [10] used the idea of quantum groups to construct the para-Grassmann algebras with one and many para-Grassmann variables.

The aim of this paper is to construct the Zp-covariant Lagrangian for the point

particle by using the Zp-graded variables and to show that this Lagrangian is covariant

under the Zp-transformation extending the supersymmetry transformation. This

means the extension of ordinary SUSY to Zp-graded version.

2. – Zp-graded space

In this section we consider the Zp-graded variable u satisfying the relation up

4 0 , (1)

where the p 42 case corresponds to the ordinary Grassmann variables. Let f(u) be an arbitrary function defined on the domain where variables are Zp-graded variables.

Then f(u) becomes a polynomial of degree p 21 in u,

f (u) 4

!

cnun,

(2)

where cn belongs to ( p 2n)-sector and u belongs to 1-sector. Now we define the q-commutator as

[A , B]q4 A B 2 qrsBA ,

(3)

where A and B belong to r- and s-sector, respectively and q 4exp [2piOp]. If A and B are q-commutative, [A , B]q4 0 , (4) then we have A B 4qrs_{BA .} (5)

Since we extend the case of fermions, we can promise that the product A B belongs to (r 1s)-sector (mod p) when A and B belong to r-sector and s-sector, respectively. Then we have the following properties for the q-commutator:

[A , B]q4 2qrs[B , A]q21,

(6)

[A , BC]q4 [A , B]qC 1qrsB[A , C]q,

(7)

where A, B and C belong to r-, s- and t-sector, respectively.

Since an ordinary number (real or complex) belongs to 0-sector and (ˇOˇu) u41, then the partial derivative with respect to u, ¯u, belongs to (21)-sector. Then the q-commutation relation between ¯u and u is given by

[¯_u, u]q4 ¯uu 2q21u¯u4 1 .

(8)

Using this we can obtain

[¯u, uk]q4 ¯uuk2 q2kuk¯u4 [k] uk 21, (9) where [x] 4 1 2q 2x 1 2q21 .

From the above q-commutation relation we can obtain the following results: ¯p_{uF(u) 40 ,}

which results from the fact that f(u) is a polynomial of degree p 21 in u. The Zp-graded

integral is easily defined like Grassmann integral, which is written as



4 dn , p 21[p 21]!

(11)

If we consider the n non-commutative Zp-variables, u1, u2, R , un, then the q-commutation relation among them is defined as

[ui, uj]q4 uiuj2 qujui4 0 (i Ej) ,

(12)

[ui, uj]q214 u_iu_j2 q21u_ju_i4 0 (i Dj) ,

(13)

These variables will be useful in the formation of the Zp-graded multi-Fock space.

It is convenient to define the Zp-covariant derivative [10] as follows: D 4¯u1 1 [p 21]!u p 21_¯ t. (14)

This derivative may be considered as a p-th root of ¯t since it satisfies Dp

4 ¯t.

(15)

This derivative will be widely used in defining the Zp-covariant action for point particle

in the following sections.

3. – Z3-covariant action

In this section we construct the action which extends the supersymmetric point particle into the Z3-graded point particle. To do so we must introduce the generalized

Z3-field described by the coordinates

(

)

X(t , u), we can relate it to its component field by using the Z3-graded variables

X(t , u) 4x(t)1uc(t)1u2

h(t) ,

(16)

where x(t), h(t) and c(t) are assumed to belong to 0-, 1- and 2-sectors, respectively. Then the Z3-field X(t , u) becomes 0-sector field. In this case we can introduce the

Z3-covariant action as follows:

S 4 1 2



Performing the Z3-integration we get

S 4 1 2



where we choose the mass equal to unity. This is very similar to the ordinary supersymmetric point particle case.

the following Z3 transformation:

dx 4ec , dh 42q21

e x., _{dc 4eh ,} (20)

where e belongs to 1-sector. Then the action submitted to this transformation behaves as

dS 4



.

c) 40 ,

(21)

where we assumed that

[u , h]q4 [u , c]q4 [e , h]q4 [e , c]q4 0 .

(22)

4. – Zp-covariant action

In this section we construct the action which extends the supersymmetric point particle into the Zp-graded point particle. To do so we must introduce the generalized Zp-field described by the coordinates

(

)

variables

(23) _{X(t , u) 4x(t)1uc}(p 21)_{(t) 1u}2c(p 22)(t) 1R1up 21c( 1 )4 x(t) 1

!

k 41 p 21

ukc.(p 2l), where x(t) and c(i)_{(t) are assumed to belong to 0- and i-sectors, respectively. Then the}

Zp-field X(t , u) becomes 0-sector field. In this case we can introduce the Zp-covariant

action as follows: S 4 1 2



Performing the Zp-integration we get S 4 1 2



g

!

h

Consider the following Zp transformation:

.

/

´

where e belongs to 1-sector. Demanding that dS 40, we have

.

/

´

Then the action submitted to this transformation behaves as

dS 4 1 2



where we assumed that

[e , c(i)_]

q4 0 .

(31)

5. – Conclusion

In this paper we used the Zp-generalization of the Grassmann variable provided by

quantum groups to construct the Zp-covariant action for point particle. Thus we came to

know that the supersymmetric theory can be extended to Zp-graded theory.

Supersymmetry is a symmetry between bosons and fermions. So this extended symmetry (so-called Zp-symmetry) becomes a symmetry between bosons and Zp-ons.

We think that this result will be applied to the new kind of field theory composed of bosonic field and Zp-fields and that it will be related to new statistical field theory

extending the Fermi statistics.

* * *

This paper was supported in part by NON DIRECTED RESEARCH FUND, Korea Research Foundation (1995). The Present Studies were supported in part by Basic Science Research Program, Ministry of Education, 1996 (BSRI-96-2413).

R E F E R E N C E S

[1] JIMBO M., Lett. Math. Phys., 10 (1985) 63; 11 (1986) 247.

[2] DRINFELDV., Proceedings of the International Congress of Mathematicians, Berkeley, 1986, Vol. 1 (Academic Press, New York, N.Y.) 1986, p. 798.

[3] FADDEEVL., Les Houches XXXIX, edited by J. ZUBERand R. STORA(Elsevier, Amsterdam) 1984.

[4] ZAMOLODCHIKOV A. B. and FATEEV V., Sov. Phys. JETP, 62 (1985) 215. [5] PASQUER V. and SALEURH., Nucl. Phys. B, 330 (1990) 523.

[6] MACKENZIER. and WILCZEK F., Int. J. Mod. Phys. A, 3 (1988) 2827. [7] GREENBERG O., Phys. Rev. D, 43 (1991) 4111.

[8] KERNERR., Class. Quantum Grav., 9 (1992) S137. [9] KERNERR., J. Math. Phys., 33 (1992) 403.