(4) moreno1989the dirac oscillator is an exactly soluble model recently introduced in the context of...
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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
(http://iopscience.iop.org/0305-4470/22/17/003)
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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
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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
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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 ) .
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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)
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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