Multiplet containing components with different masses

D.V. Soroka[1] and V.A. Soroka[2]

[1]E-mail: dsoroka@kipt.kharkov.ua[2]E-mail: vsoroka@kipt.kharkov.ua

Kharkov Institute of Physics and Technology 1 Akademicheskaya St., 61108 Kharkov,

Ukraine

Abstract

A principle possibility for the existence of a multiplet including the components with the different masses is indicated.

• 1. Introduction• 2. Tensor extension of the Poincaré algebra• 3. Two-dimensional case• 4. New coordinates• 5. Multiplet• 6. Conclusion• Acknowledgments• References

1. Introduction

• We start from the citation a very surprising (for us) appraisal of supersymmetry [1,2,3,4] given by Yury Abramovich Golfand during the Conference "Supersymmetry-85" at Kharkov State University in 1985. He said [5] that supersymmetry did not justify his hopes to find a generalization of the Poincaré group such that every its representation include the particles of different masses. Golfand and Likhtman had missed their aim, but had instead found supersymmetry, every representation of which contains the fields of different spins.

• So, the problem was raised and requires its solution. In the present paper we give a possible solution of the problem of the multiplet which components have the different masses[1]. We illustrate the solution on the example of the centrally extended -dimensional Poincaré algebra [7,8,9,10,11].

[1]Concerning another approach to this problem see the paper [6]

(1 1)

2. Tensor extension of the Poincaré algebra• In the paper [10] the tensor extension of the Poincaré algebra

in dimensions

was introduced and its Casimir operators

[ ] ( ) ( )M M g M g M c dab cd ad bc bc ad

[ ]M P g P g Pc a acab bc b

[ ]P P Za b ab

[ ] ( ) ( )M Z g Z g Z c dab cd ad bc bc ad

[ ] 0P Za bc

[ ] 0Z Zab cd

(2.1)

(2.4)

12 3 2 ( 1 2 )1 2 2 1 2

a aa akZ Z Z Z k …a a a a

k k

2 3 2 2 111 2 2 1 2 2 1

a aa aak kP Z Z Z Z Pa a a a a

k k k

2 2 111 2 2 1 2 2 1

a aaak kZ Z Z Z Ma a a a a a

k k k

( 01 2 )k …

1 2 2 1 2 21 2 2 1 2

a a …a ak kZ Z k Da a a a

k k

(2.3)

(2.2)

were constructed. Here are generators of rotations, are generators of translations, is a tensor generator

and , is the totally antisymmetric Levi-Civita

tensor in the even dimensions . Generators of the left shifts with a group element , acting

on the function

have the form

Mab

Pa Zab

1[ ( ) ]( ) ( ) ( )a abT G f u f G u u x z

12bP xa ax abz

Zab abz

cc zzM x x Saa a acbx zab b b bc abx z

1 2a…a

k 01 2 1 1… k

(2.5)

2D kG

( )f u

where coordinates correspond to the translation generators , coordinates correspond to the generators and is a spin operator. In the expressions (2.5) .

In the case of the extended two-dimensional Poincaré algebra the Casimir operators (2.2), (2.3) and (2.4) can be expressed as degrees of the following generating Casimir operators:

where , is the completely antisymmetric two-dimensional Levi-Civita tensor.

(3.1)

axPa

abz Zab

Sab u u

3. Two-dimensional case

12abZ Z

ab

a abC P P Z Ma ba (3.2)

ab ba 01 1

The relations (2.5) can be represented as

where is a time, is a space coordinate, is a coordinate corresponding to the central element and the space-time metric tensor has the following nonzero components:

(3.3) 0 2

xP P yt t

1 2tP Px x y

12abJ M t x Sx t 01ab

(3.4)

(3.5)

Z y (3.6)

0t x 1x x 01y zZ

111 00g g

The extended Poincaré algebra (2.1) in this case can be rewritten in the following form (see also [9]):

and for the Casimir operator (3.2) we have the expression

For simplicity let us consider the spin-less case . Then with the help of the relations (3.3) – (3.6) we obtain a mass square operator

where the notations and are used.

(3.7)

(3.8)

[ ] bP J Paa b

[ ]P P Za b ab

[ ] 0 [ ] 0P Z J Za

2aC P P ZJa 001S

2 2222 24

t xPM JZ ZPxx x ttt (3.9)

22xx x

22tt t

By a transition from and to the new coordinates

we obtain the following expressions for the generators:

(4.1)

(4.2)

(4.4)

4. New coordinates

t x y

2t xx 2 24

t xy y

P x 2P x y x

(4.3)

J x xx x Z y

(4.5)

We see that are step-type operators.

The Casimir operator (3.8) in the new coordinates takes the form

and the mass square operator is

(4.7)

(4.8)

P

2C P P Z ZJ

2 2M P P Z ZJ x x Z

As a complete set of the commuting operators we choose the Casimir operators , and rotation operator . Let us assume that there exist such a state that

The equations (4.2) and (5.1) mean that

independent on the coordinate . Then, as a consequence

(5.1)

(5.2)

5. Multiplet

(5.3)

Z C J( )x x yz j

( ) 0P x x yz j

( ) ( )Z x x y z x x yz j z j

( ) ( )J x x y j x x yz j z j

( )x x yz j x

of the relations (4.4), (4.5), (5.2) and (5.3), we come to the following expression for the state :

where is some constant.

For the states

we obtain

(5.4)

(5.5)

(5.6)

( )x yz j

( ) zyjxx y a ez j

a

( ) ( 01 2 )kP x y k …z j

( ) ( )k kP PJ x y x yz j z j

( ) ( )kPj k x yz j

from which we have

The states (5.5) are the components of the multiplet.

(5.7)

(5.8)

(5.9)

2 2( ) ( )k kP PM x y x yz j z j M

2[( 1 ) ] ( )kPk j z x x z x yz j

( ) (1 2 ) ( )k kP PC x y z j x yz j z j

j k

(5.10) 2 2( 1 )k j z x x z M

By excluding from the relations (5.9) and (5.10), we come to the Regge type trajectory

with parameters

(5.11)

(5.12)

(5.13)

2(0) M

k

(0) 2 1j x x z

1z

Thus, on the example of the centrally extended -dimensional Poincaré algebra we solved the problem of the multiplet which contains the components with the different masses.

It would be interesting to construct the models based on such a multiplet.

Note that, as can be easily seen from the commutation relation

where is an electromagnetic field and is its strength tensor, the above mentioned extended Poincaré algebra (3.7) is arisen in fact when an “electron” in the two-dimensional space-time is moving in the constant homogeneous electric field.

Authors would like to thank V.D. Gershun, B.K. Harrison, A. Mikhailov, D.P. Sorokin, I. Todorov and A.A. Zheltukhin for the useful discussions. The authors are especially grateful to E.A. Ivanov for the interest in the work and for a set of valuable remarks.

6. Conclusion(1 1)

[ ]a ba b abx xA A F

aA abF2D

Acknowledgments

[1] Yu.A. Golfand, E.P. Likhtman, JETP Lett. 13 (1971) 323.

[2] D.V. Volkov, V.P. Akulov, JETP Lett. 16 (1972) 438.

[3] D.V. Volkov, V.A. Soroka, JETP Lett. 18 (1973) 312.

[4] J. Wess, B. Zumino, Nucl. Phys. B 70 (1974) 39.

[5] Yu.A. Golfand, Private communication.

[6] H. van Dam, L.C. Biedenharn, Phys. Lett. 81B (1979) 313.

[7] A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Ann. Phys. 185 (1988) 1.

[8] A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Ann. Phys. 185 (1988) 22.

[9] D. Cangemi, R. Jackiw, Phys. Rev. Lett. 69 (1992) 233.

[10] D.V. Soroka, V.A. Soroka, Phys. Lett. B607 (2005) 302; hep-th/0410012.

[11] S.A. Duplij, D.V. Soroka, V.A. Soroka, J. Zhejiang Univ. SCIENCE A 7 (2006) 629

References