Separable bound state equations for three relativistic fermions

Nuclear Physics A.508 (1990) 305~31Oc North-Holland

305c

SEPARABLE BOUND STATE EQUATIONS FOR THREE RELATIVISTIC FERMIONS.

J. BIJTEBIER(*) Theoretische Natuurkunde, Vrije Universiteit Brussel,

Fakulteit Pleinlaan

der Wetenschappen, 2, 1050 Brussel (Belgium)

Abstr& We describe the bound states of two relativistic fermions plus an external potential and the bound states of three relativistic fermions by sysThe physical requirement of septems of compatible coupled Dirac equations. arability (into two or three independent clusters by “switching off” the corresponding interactions) is satisfied.

1. INTRODUCTION Somewhere quantum which

field

between

theory,

a system

nal field)

to

into two

or more The

body

literature.

N)-body results

last years,

the

The

two

(a more detailed

fermions

plus

quark

external

helium-like

present

fermions

atoms,

potential while

results our

mechanics,

also with

equations.

of these

Onderzoek

navorser

easiest

been

trying

systems

and

for

will be found in refs [l] could

three-fermion

be tested results

treat,

bij het Nationaal

0375-9474 / 90 / $3.50 0 Elsevier Science Publishers B.V. worth-Holland)

Fonds voor

interso that

can be found

to combine

to a short

potential

to

equations

and manageable

is devoted

(Belgium).

It

equations

these

two-

of three

(and

presentation the

three

and [2]).

of our fermion Our two-

in the

phenomenology

could

be

used

models.

(*)Bevoegdverklaard

in

an exter-

off” the corresponding the

pairs of coupled

external

presentation

relativistic

(we must be able to split the system

we have

talk

plus

compatibility

of course

separable

quantum

and possibly

by “switching

are

and

Dirac and Klein-Gordon

mutual

and compact

into compatible,

problems the

clusters

separable

equations. for

the

problems

These

equations

of coupled

of separability

independent

compatible,

in the

for a relativistic mutually

simultaneously

two-body

mechanics

is a place

requirement

actions).

quantum

(interacting

by a system

insure

and the physical

several

there

of particles

is described

is difficult

non-relativistic

Wetenschappelijk

to

of build

306~

J. Bijtebier I Separable bound state equations

2. TWO-FERMION SEPARABLE MODELS

We start with a pair of coupled K&Q

i Ki,* = Pf.Yt - mf - “1 *Vt 2

= 0,

and we denote particle

by xi, pi, mf, ytp the position,

i (= 1,2)

respectively.

x1 -x2. pi I p2, yl,

with

general)

the

[<,,

usual

x,,

They

0

to satisfy

built

(weakly

in

2

(2.3)

Pt2 =+ (P, - Pp)

(2.4)

x,2=x,

(2.5)

-x2 0 - H2)

VI2 =5l+v;,

(2.6)

- !32VY2)

(2.7)

(2.1) in the form

0 = (52

in an arbitrary 52(‘402

in order

of the

invariants

di = 70%

0 10 h,2 = TV,

0

1 v12=W,2+~2v,2

0 P12yi2

be chosen

Lorentz

(2.2)

yio 9

Pi =

+H2

we can write

must

mass and rtr matrix

V { 2 are

definitions

1 =5- (9 + 3)

0

momentum,

potentials

condition

+P2

H,2=H,

(2.1)

+ Cf2KT2 = 0.

H tc = di.Pi + 8tmf , F,,=P,

The

y2.

compatibility

KT2] = Ci2Ki2

With the

Dirac equations

+m12v12)*i2

Lorentz

frame.

We shall

(not unitary

= 9) = U&,

0 p12y12

’

= (hY2+

make

in general)

an inversible which

(2.8 a-b)

m12v12)y12

solves

transformation

?-‘t2 =

(2.8b): (2.9a)

Op12yi2

(2.9b)

=O

with

“z,2= v,2 +k(iy2

- Hy2).

From now on, we shall independent

0

on P,~.

restrict

(2.10) ourselves

In this case, U,,

to the models

in which

V,2

and v, 2 are

is given by

U 12 = exp(-ihy2xy2) L(xy2, 0 ;q2q2)

(2.11)

with Z

12

-- eWh~2x~2)

VI 2ew(-ihl

0

2x1

0

2)

(2.12)

J. Bijtebier I Separable

where

we define

(more

generally)

307c

bound state equations

an operator

L(t’,t;z)

in terms

of an operator

z(t)

by + (-i)2;

L(t’,t;z) = 1 + (-i) ; dt, z(t,) The compatibility

condition

[pj12. Fy2 - f-fy2 - ml2 expliciting

the

;’ dt,z(t,)

t

(2.2) written

in terms

of the operator

~,2(‘402)1 = q2C,242W;2

dependence

on

0

assumed

indedpendence

equation

in x,~, we get

0

0

p12

- %2

right-hand

side

- f-f2

(2.14)

could

a priori

as a consequence

p ’,2. Treating

on

= D,2(x;2V;2

reads

(2.14)

as

an

92&2P)1

-

is

in fact

equal

)402 = 0 and

that

have of the

evolution

(2.15) (2.16)

O 0 ; w,$,,). = L(x,~’

for

assume

the potentials

- a1 2z, 2(x;2)

0 D,2(x,2) As ?!,2(O)

of

z,,,

- t-f2 - q24242)1

The

x ,2.

(2.13)

+ . ..

’ but this term does not appear, also a term in P,~,

contained

written

dt2z(t2)

t

I

to V, 2(O), we multiplied

VI 2(O) is hermitian,

see

that

at left by the we

solving functions at xy2 = 0 by 0 x,~ + 0 are then obtained by applying

obtain

this

the

inversible energy

principal

the operator

eq. (2.9a) is simply operator

spectrum

equation.

eq.(2.8a),

D, 2. and

If we

the

wave

The wave functions

at

U, 2.

3. TWO FERMIONS AND AN EXTERNAL POTENTIAL The equations

of two independent

fermions

in an external

potential

are

[pi.Yi - mi - eJi] ‘Z’t2 = 0 where

Ji depends

refers to a given get

compatible

interactions

Xi, pi, c

Lorentz and

by

(we

frame

separable

take a static and energy (which

independent

we can call the laboratory

equations

combining

the

potential)

frame).

mutual

and

-

BJ,~I+,~

= 0

external

(3.2a) (3.2b)

= 0

with Y ,2 = U,2?,2,

J,2 = P,J,

+ P2J2v

U,,

= exp ( -i WY2 + ej,2)x~2)Ux~2p

j,, =i(P,J,

- P2J2).

0 ; q2q2)

and

We can

writing

0 0 IPi 2 - f% 2 - ~,2V,2~O)

0 D12(x12)

p;2+,2

on

(3.1)

(3.3)

(3.4)

308c

J. Bijtebier I Separable

The system

(3.2)

(separability). obtained

is equivalent

An

infinity

by introducing

to (2.1)

of other

arbitrary

bound state equations

when

e = 0 and to (3.1)

compatible

and

x:2-independent

separable

when

systems

and eo, 2-proportional

o12

= 0

could

be

operators

into (3.2a) and (3.3). 4. THREE FERMIONS We can equations,

obtain built

two-fermion

a system

with

the

models,

of three

mutual

and

given

separable

by three

coupled

(different

GJ +

(4.la)

Op,2Y=p;3;Y=p;,+=o for G = Y(xi quantities The

refer

Pp2 replaced interact

(4.1 b)

= 0), with P = p, + p2 + p3, n,ow to the total

potential

center

Vt 2 is the potential

by PO - Hi

with the two

that eq. (4.la) evident.

The

splitting

(to which

others).

contains

example),

Dirac

or identical)

by writing

I\ I\ 0 +o,~V,~+~~V~~+W~,

OP y=[H

compatible

interactions

can

the system

non-covariant

of mass frame.

it is equiva;ent

(1,2)

whet

demonstrated

into the two clusters

model,

two-fermion the third

V23 and V3,

time or energy. be

The

fermion

are defined

with

does

not

similarly,

so

The compatibility

of eqs. (4.1) is

by

= 03,

making

(1,2) + (3).

~2~

= 0 (for

Eqs. (4.1) become (4.2a)

(P;2+

pi)+

OP,zY

= 0,

(4.2b)

0 (P,2

- 2p,o,G = 0.

(4.2~)

We should

= (H;2 + o,$,~

+ Hz + Hi.

V, 2(O) of the

The potentials

no relative

separability

Ho = H;

have a non-zero

+ Hi)+,

term in the right-hand

eqs. (a) and (c) into an equation The

fermion.

brings

the

equations

desired*

i 0 U31=exp[~(H,2+~,2

the

to

commut!s use of Vii

(4.3) and combining

P;2k,2

= (H;2 + a, $,&

p;2;i’,,

= 0

2

C ,2-2Hi)(xy+$

without (4.2c), 0 0 0 with x, + x2 - 2x3.

modification

(note that V,2

for V, 2(O), whence transformation

(1,2) and an equation

for the third

transformation

,

$=U3’%,2

for the system

side of (4.2~) in order to combine

instead

of Vij(O)

eqs. (a) and

in

-2x$]

(4.3)

modifying

the

This would

not be the case

(4.la)).

two

other

Performing

(c) we obtain (4.4a) (4.4b)

the

309c

J. Bijtebier I Separable bound staie equations

p&2 =H ;%,2.

(4.4c)

Eq. (4.4~) is the equation Vt 2(O), using system

(4.4c).

studied

of a free fermion. The

in section

system

In eq. (4.4a),

(4.4a,b)

The

2.

U, 2 g iven by (2.11).

transformation

h VI2

is then

equations

equivalent

for T1 2 are

can be replaced to the

obtained

by

using

(4.5)

we have

to consider

like to find an operator coupling

constants

also the two

which

U-’

extrapolates

differ from zero.

other

two-cluster

the three

limits

U k ‘Ui”j’

and we would when

all three

We can choose

$ = u-‘y u-l

the

At the ~23 = w3,= 0 limit, we have thus

Y = u &-;Y,2. However,

by

two-fermion

(4.6)

=u;‘u;~+u;‘u;,1+u;‘u;~

as each Uk’U

-2lJ;’

i-i’ depends

(4.7)

only on Oii while U o’

IS their

common

no-interaction

limit: -I 2i 0 0 UO = exp 1T(ht2xt2 Once

again,

the

’ * + h~,x~,)]. + h22x22

solution

given

in this

(4.8)

section

to

the

compatibility-separability

puzzle is by no means unique. 5. CHOICE OF THE TWO-FERMION MODELS. In section

2, the

to technical and V,2(0) others Some

choice

restrictions: must

potentials

be hermitian.

must be transformed models

of the starting

the

This

before

can be used directly,

has been

submitted

V, 2 and v, 2 must be independent

two-fermion

on p y2

excludes

some

models possible

use (as the very general as the harmonic

oscillator

models, model model,

while

some

of Sazdjian given

with

1 Vf2 = iy,.(xt2)~,2y,5y25,

“12 = -iy2.(x1 2)P t2Yi 5y25’ (x12)p,, The potential

= x12 - [P:2ii(x12.P12) of the principal

Pt*.

equation

[3]).

by (2.1)

(5.1) (5.2)

is then

(5.3)

31oc

J. Bijtebier

In the two-fermion (1,2) center

P,*

of mass frame where

&2)P,,

bound state equations

can be diagonalized,

the

little

function

(in this case

components

harmonic

of

identical

the

oscillator

two-fermions

work in a “laboratory

plus

spinors

equation

potential

frame”.

are

for

= x,2

In the three

2

+ UP,21

fermion

model,

We use 0

052)p,*(O)

the

model,

2

-p,21

only

V,2(0) -1

can be computed

eliminated,

we

get

“internal”

part

of

exactly: a

three-

the

wave

@12)P,,

no component

If we take x12,

the

and

separable

interactions) are

side

This the

(we

- P8m3)2

two-body free

but not separable

into

non-local

of (5.5)

meets

get

not a specific be solved

right-hand

as in (5.4).

hamiltonians

globally

d3.P3

+ 1 (PO-

= xl2

relativistic

complication,

of P12 can bf P = 0. 2

and we

with

-1

p,21 &J,2P,2'

-

or (5.6)

at order

practice

interaction

terms.

Dirac

diagonalized,

We useV, 2 built

equations

arbitrary

zero

of simply Such

if

clusters.

we

a model

switch

x12-only

we find the

equations

[I]

Preprint

VUB/TENA/89/01.

Submitted

to II Nuovo

Cimento

A.

[2]

Preprint

VUB/TENA/89/02.

Submitted

to II Nuovo

Cimento

A.

Dirac

would

off

all

must

be

three

of (5.5)

as perturbations.

REFERENCES.

Phys. Rev. D 33 (1986) 3401.

in P,, adding

The last terms

to be treated

the simpler

(5.6)

by perturbations.

[3] H. SAZDJIAN:

and we

(5.5)

the usual

corrections, as even

P!2 can be diagonalized

built with

(x,p,2P,,.

work in the total center of mass frame where

(5.6)

to (2.15))

[l].

In the

free

in the

(5.4)

of (2.8a)

dimensional

simply

so that we can work

P,2 = 0 and

= X12.

The solutions when

model,

I Separable

and

This

is

already