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Covariance, CPT and the Foldy-Wouthuysen transformation for the Dirac oscillator

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1989 J. Phys. A: Math. Gen. 22 L821

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J . Phys. A: Math. Gen. 22 (1989) L821-LS25. Printed in the UK

LETTER TO THE EDITOR

Covariance,

CPT

and the Foldy-Wouthuysen transformation for

the Dirac oscillator

Matias Moreno and Arturo Zentella

Instituto de Fisica, Universidad N acio nal Aut6n om a de MCxico, Ap. Postal 20-364, Del.

Alvaro Obregbn, D.F. 01000 MCxico, Mexico

Received 9 Jun e 1989

Abstract. The covariance propert ies and th e Foldy-Wouthuysen and Cini-Touschek trans-

formations

for

the recently p roposed Dirac oscilla tor are found. The charge conjugation,

parity and time reversal properties ar e established. A physical picture as an interaction

of

an anomalous (chromo) magnetic dipole with a specif ic (chromo) electr ic f ie ld emerges

for this system. This interaction suggests an alternative confinement potential for heavy

quarks in QCD

In a recent work, Moshinsky and Szczepaniak l ] introduce d an interesting force in

the Dirac equation. In an analogy with the original Dirac procedure, they require an

interaction linear in the coordinates. Subsequently, they show that the non-relativistic

form of the interaction reduc es to one of the harm onic oscillator type. Following [

11,

we will refer to this system as the D irac oscillator. Its explicit form is

id,V = HOD”

=[a ( p

i r p )mp ]V (1 )

where the usual conventions have been taken [2]. This equation has the remarkable

property of having exact solutions [llt.

In this let ter we study equation (1) for the single-particle case; we focus on two

points. First, we presen t a manifestly covarient form of equa tion ( 1) .

Second, we

deduce for it an exact Foldy-Wouthuysen transformation [4] (FWT). With these two

elements a covariant field theory can be defined for the Dirac oscillator. Moreover,

from the

FWT

the energy spectrum and the eigenfunctions are immediately obtained

using the non-relativistic harm onic oscillator solutions. In or der to build the covariant

form of equ ation ( l ) , we notice that the interaction term in it,

i cu - r (2 )

can be p ut in to the form

i = l

Introducing the frame dependent velocity vector

U”

=

(1,O)

f Exact solutions

for

a minimal electromagnatic interaction have been known for som e time; see [3].

0305-4470/89/ 170821 +05 02.50 @ 1989 IOP Publishing Ltd L821


3/6

L822 Letter t o the Editor

the interaction can be p ut into the form

ia r = u p u x u .

Therefore, equation

(1)

can be rewritten as

with

7)

PY= p c x ”

and for the Dirac oscil lator

K =

2m /e. This form shows explicitly the covariance of

the D irac oscillator. We remark that the external ‘electromagn etic field’ of equa tion

7) is, as usual, frame de pen den t [5]. Additionally, this form shows

a

possible physical

interpretation of the Dirac oscillator interaction as stemming from the anomalous

magne tic mom ent. It also shows, without furthe r ad o, its charge conjugation, parity

and t ime reversal invariance [6], [2, ~ 6 1 1 .

We can now address to the problem of the FWT. Let us first recall that this

transformation defines an operator HFWhat is related to the Ham iltonian

H

by means

of a unitary transformation

id,

HFw

= eis( id , H)e-iS

8)

where

S

is to be chosen such that H,, is purely even. More explicitly, one splits

H

into two Hermitian operators: an odd,

OH,

nd an even,

EH

part

H

= OH EH 9)

is = B O H 8 10)

The usual c hoice [2] for i S is

with B and 19 Hermitian and even, 8 commutes with both B and OH; he additional

requirement that

{ B ,

OH}

0

must be set for consistency

so

that S is Hermitian. In

order to define the FWT one must specify the ope rato r an d the ‘angle’

8.

We also remark that, from the well known identity

AI=

e i S ~

-iS

= A + [ i S , A ] + & [ i S , A ] ] + .

. . 11)

{ A , s } =

0

(12)

1 3 )

and since for the Dirac oscillator HOD oes not depend explicitly on time there are

no contributions to the Foldy-Wouthuysen Hamiltonian coming from the transforma-

tion of 8 . Defining

one can show that if

S

satisfies

then

AI

= A e-2iS

=p p r

14)

HOD a w + p m . 15)

one can write


4/6

Letter to the Edi tor

L823

If the od d an d even parts are defined according to

O H = a - m a nd EH

= p m

one arrives at the following proposal for the FWT

iS=pa IzO

=

7 -

me.

18)

Here i s ant icommutes with

OH

nd

EH

provided that

8

commutes with them. Using

the relations

2

y P = -m t n i a ( m t x P

n t . ~ = p 2 + m 2 - 2 p

a nd

t

x

m

=

i 2 p ~

and defining

h 2: = a

P *

one gets

h 2 = - ( y .

P ~

= p 2 + r 2

3 + 4 ~

S )

( 2 2 )

which is a positive definite Hermitian operator. The ope rato r

h 2

has even and odd

roots, two odd roots of

h 2

are * a.

) ,

taking

h

to be an even root (which we will

show explicitly later) we obtain

23)

is

=

cos

hO + y ?r)h-

sin

he.

From

this set-up it follows that

HFW

e iSHoD - iS

h

a - cos 2 hO -- si n 2 h0 + p ( m cos 2h O + h sin 2hO) . (24)

Because the operator h and the mass m are Hermitian it makes sense to choose such

that

m

cos

2h0

sin

2h0 =

0

h

or, equivalently,

h

m

The last form implies the consistency condition

tan

2h0

=-.

= e h 2 ) .


5/6

L824 Letter to the Editor

We finally arrive at the desired solution

HFW p J m + h 2

=

@ J m 2 p 2

+ r2 3+ 4~

S )

28)

Eigensolutions for this Hamiltonian can easily be obtained because the non-

relativistic harmonic oscillator Hamiltonian, @, J 2 ,Jz 2 ,S and the coordinate reflec-

tion operator commute among themselves and with H F W . Furthermore,

HFW

s a

functional of these operators. Hence, the spectra of the Dirac oscillator can be directly

obtained from the non-relativistic oscillator solution and the eigenfunctions of these

operators [2,3]. These eigensolutions coincide with those first found in [ by squaring

the

HOD

Hamiltonian for the positive energy part of the spectrum.

Equation

27)

also gives an explicit representation of the operator h, which is

simply H F W with the mass set equal to zero. Once the negative energy states are

defined, one can proceed to construct the appropriate vacuum in order to define a

consistent field theory for this system.

One can also use equation

26)

to transform the Dirac oscillator to a form adequate

for the ultrarelativistic limit. This is done by selecting 6 such that the term multiplying

@ vanishes; the resulting Cini-Touschek [7] Hamiltonian is

We would like to notice that the remarkably simple properties of the Dirac oscillator

interaction and the transparent physical interpretation that follows from its covariant

form suggest that it might be a good candidate to be used as the confinement potential

in heavy quark systems.

If one tries to find an electric charge distribution that produces the linearly growing

electric field that interacts with the anomalous magnetic moment of a Dirac particle

to produce a Dirac oscillator, one reaches the peculiar conclusion that it is a uniform

(infinite) sphere of charge.

A different conclusion follows in the framework of quantum chromodynamics

QCD) if a slight modification of the usual ideas about confinement is adopted. In

QCD, ignoring at this stage the subtleties of Gauss law for the non-Abelian gauge

theories [ S I the lines of field are conceivably confined to strings which might simulate

the adequate potential without unphysical charge distributions. For example, an

effective chromoelectric field which grows with distance stems from a physical picture

in which the string is assumed to be of constant volume, as opposed to the more

conventional hypothesis of a string of constant cross section.

The authors would like to ,:knowledge many fruitful discussions about this work with

Jean Pestieau, Marcos Moshinsky and Rodolfo Martinez. We also are grateful for a

critical reading of the manuscript by Manuel Torres.

This work was supported by Conacyt and Fondo Zevada, Mexico.

References

[ l ] Moshinsky M and Szczepaniak A 1989 J. Phys.

A

Math. Gen.

22

L817

[2] Bjorken J and Drell 1964 Relarivistic Quanrum Mechanics Ne w York: McGraw-Hill)


6/6

Letter to the Editor

[3] Schweber S S 1961 An Introduction

to

Relativistic Quantu m Field Theory Ne w York: Harper and Row )

[4] Foldy L L and Wouthuysen S A 1950 Phys. Rev. 78 29

[5]

Hagedorn

R

1963

Relativistic Kinematics

New

York:

Benjamin)

[6] Foldy L L 1958 Rev. Mod. Phys. 30 471

[7] C h i M and Touschek T 1958 Nuouo Cimento

7

422

[8] Bjorken J 1980

1979

SLAC Summer Institute (SLAC Report 224) ed A Mosher Stanford,

CA:

SLAC)

p 103

p 219

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