IL NUOVO CIMENTO

VOL. 106A, N. 10

Ottobre 1993

Simple Approach to Study the Collapse of Relativistic Particles. B. CARAZZA(1)(3) and P. QUARATI(2)(8) (1) Dipartimento di Fisica dell'Universitd - 43100 Parma (2) Dipartimento di Fisica del Politecnico di Torino - 10129 Torino (3) INFN, Sezione di Cagliari - 09127 Cagliari

(ricevuto 1'8 Marzo 1993; approvato il 15 Giugno 1993)

Summary. - - Using a semi-relativistic approach, we derive in a straightforward way the conditions for the collapse of one charged particle of angular momentum l bound to a pointlike and massive nucleus. We study also the collapse of a system of identical particles interacting through a Coulomb or a gravitational potential. PACS 12.20.Ds - Specific calculations and limits of quantum electrodynamics. PACS 25.70 - Heavy-ion-induced reactions and scattering. PACS 03.65 - Quantum theory; quantum mechanics.

1. - Introduction.

The early works by Chandrasekhar [1], Popov [2] and Lieb and Thirring [3] on collapse, in addition to the intrinsic value of the results obtained, have certainly stimulated, in the past and recently, the research in the physics of strong fields in QED, gravity and QCD (see for instance Greiner, Mtiller and Rafelski [4]). Martin [5] has recently re-examined the collapse of systems of relativistic particles (we refer the reader to Martin [5] for a list of relevant papers on this subject). This field of research is not only theoretical; many experiments have now been completed with heavy-ion beams at GSI Laboratory in Darmstadt in order to study QED of strong fields. We present here a simple way to find the condition for the collapse of one relativistic particle of a given angular momentum l subjected to a Coulomb or a gravitational potential, and of a system of self-gravitating identical particles (see also Jetzer [6]). The results we obtain are not a novelty expecially after the recent papers by Martin [5], by Basdevant, Martin and Richard[7] and by Martin and Roy[8]. However, we think it is worth describing our approach which consists in a handy and simple way to study that kind of problems, especially as we hope that it can be extended to systems composed of different species of particles having different interactions and we think it can be used to study the evolution in time toward collapse of the physical systems [6, 9]. Our approach is based on a variational method within the use of the ordinary SchrSdinger wave function, but considering a semi-relativistic Hamiltonian. It works also for a system of bosons, whereas for an assembly of 1353

1354

B. CARAZZAand P. QUARATI

identical fermions we shall use the semi-classical approximation, deriving the appropriate Thomas-Fermi-like functional, instead of a cumbersome-to-handle antisymmetrized probe function. 2. - C o l l a p s e

of an / = 0 particle.

If we have a particle of mass m bound by an attractive potential - g/r to a nucleus of infinite mass, where g is the coupling constant, we may introduce, as it is usual in these cases [5], the Hamiltonian (in the units h = c = 1 to be used in the following): (1)

H = ~V/~2 + m 2 - g/r.

Using the well-known inequality for the ground-state energy Eo we can set (2)

Eo < (~biHI ~),

where ~b is any normalized-state vector. Then, having in mind the following relation for the average value involving the operator fi2 + m2: (8)

+ m 2) -<

2 + m2),

we may conclude that for any normalized-state vector the ground-state energy satisfies also the inequality (4)

So ~ ~/(~]fi2 + m 2 [~) _ g(~l 1/r]r

The calculation of a bound to E0 reduces thus to finding the minimum of the r.h.s. of eq. (4) when the normalized wave function ~b is varied. The variational calculus, taking into account the normalization condition via a Lagrange multiplier, gives as a necessary condition to be satisfied by r a differential equation which, apart from the coefficients and their meaning, coincides with the usual stationary Schr5dinger equation of the hydrogen atom. We then simply assume the probe function (s-wave) (5)

T(r) = D(f) exp [ -fir],

where D(fl) is a normalization coefficient and fl a parameter to be varied. Using sueh a probe function it is straightforward to obtain from eq. (4) (6)

Eo <- ~/ff2 + m2 _ gfl.

We may look for the value fl + of fl for which the r.h.s, of the above equation has a minimum. We find 1 (7)

f + = mg ~/1

-

g2

'

which is meaningful when g < 1 . With the expression of fl+, the condition (6) becomes (8)

Eo ~< m ~

- g2

SIMPLE APPROACH TO STUDY T H E COLLAPSE OF RELATIVISTIC PARTICLES

1355

and, in the limit of g approaching one from below, (9)

E0 ~< 0.

This result means that the binding energy of the particle is at least equal to its mass. Of course, if Eo ~ - - m , the bound level dives into the negative Dirac sea and particle-antiparticle pairs can be emitted [2,4]. The pair production, being due to a strong binding energy of value at least 2m, can be interpreted as an indication that the system approaches the collapse. This event is reached when the binding energy is infinite or when the wave function is strictly localized around the centre of attraction. Of course, if g > 1, the r.h.s, of eq. (8) becomes imaginary. This fact can be simulated by the presence of an imaginary potential in the Hamiltonian, as it happens in an atomic system composed by a particle and its antiparticle, attracting each other through a Coulomb potential in the presence of an imaginary annihilation potential. We may again interpret this as an indication of collapse. At last we can directly see from eq. (6) that, when g > 1, in the limit in which ~ -+ + ~ the r.h.s, of eq. (6) goes to - oo and therefore the same happens to E0: this is the condition usually accepted for the collapse [2]. As fl ~ + oo our probe wave function becomes like a delta function, showing once more very clearly and in a pictorial way the collapse, since in that case the particle is essentially localized on the centre. In the particular case when the interacting potential is the Coulomb potential the condition is g = Z e 2 >t 1. As is well known, the Dirac equation in the Coulomb field of a point charge can be solved exactly. In particular the energy of the lowest level 1S1/2 of the discrete spectrum reaches zero when g = 1, and the ,,collapse to the centre), occurs [2] for g > 1 which coincides with our result although we considered the SchrSdinger wave function (but with relativistic expression for the kinetic energy in the Hamiltonian) for a spinless particle. This is somewhat surprising since the exact calculation developed by Herbst[10] considering the SchrSdinger equation for the same Hamiltonian as in eq. (1) gives the condition g > 2/7:. 3. - Collapse of an l ~ 0 particle. We consider now a particle with angular momentum l different from zero. Having in mind the result of the variational calculus we discussed previously, we assume as a probe function the following wave function, depending on flz, which behaves near the origin like the usual solution for the hydrogenlike system with the same angular momentum: (13)

Fz = Dz(~z) rt e x p [ - f l l r ] Yz, m(~, 9),

where, as before, D~(~l) is a normalization coefficient. We consider again the relation

(14)

Eol

<

2 + m 2

-

11/rl

After calculating the average values, we obtain (15)

89 - II Nuovo Cimento A

Eol ~ ~

+ fl~

gill l+1

).

1356

B. CARAZZAand P. QUARATI

Since if g < 1 + 1 the minimum of r.h.s, of eq. (15) is reached, when fit has the value fll+ given by fl~ = g m / ~ / ( l + 1)2 _ g2,

(16) we have

Eol <<-m ~ / ( 1 + 1)2 - g~.

(17)

Of course the r.h.s, of eq. (17) becomes imaginary if the result is extended to the case g > 1 + 1. In the limit of g approaching 1 + 1 from below (18)

E0t ~< 0.

For g > l + 1 we also directly see that if fit ~ + ~ the value of r.h.s, of eq. (15) goes to minus infinity. In conclusion, the particle collapses when g > l + 1. We recall that we considered spinless particles and the SchrSdinger equation. By using the KleinGordon equation one obtains the bound g > l + 1/2 [2].

4. - Collapse of N interacting bosons. Let us now consider the case of a system of N self-gravitating spinless bosons each having the same mass m. The Hamiltonian is N

(19)

HB: •

N

~+m

2+ •

i
i=1

rjl

'

where K = G m 2, G being Newton's constant, is measured in our units. Again, we may set the condition Eo < (HB } from which it derives that N

(20)

N

1

Eo ~< E V@B Ifi~ + m2 I~bB) + K(~bB ] i~<~ i=1

"

--

" lri

rj l

]~bB)

for any normalized state vector ~bu of the many-particle system. We can evaluate Eo using the following N-body normalized wave function: N

(21)

F S = D(fl) N l-I exp [ -flri ], i=l

where D(/~) is the same as in eq. (5). It is straightforward to obtain (22)

Eo <<-N ~ / f l 2 + m s - ~ N ( N - 1)58/16.

The r.h.s, of eq. (25) goes asymptotically, for large and positive 8, to the value N/~[1- ( N - 1)K5/16], therefore the bosons collapse if (23)

~c > 3.2/(N - 1).

We note that Basdevant, Martin and Richard [7] obtained for a system of bosons an expression which differs from eq. (23) only for the numerical coefficient which the authors above find to be 3.037.

SIMPLE APPROACH TO STUDY THE COLLAPSE OF RELATMSTIC PARTICLES

1357

5. - Collapse of N interacting fermions. In this case we should use antisymmetrized wave functions as probe functions. However, with these functions the calculations are quite involved; we rather prefer to use the semi-classical approximation. In that approximation one refers to the spatial particles distribution given by (24)

~(r) =

q I dp, (27:)31pl -
where q is the occupation number (q = 2 for fermions). The non-relativistic density of kinetic energy is PM (r)

(25)

TNR(r) -- 47:q

I

p2 p2 dp

(27:) 3

'

o

which can be expressed as a function of ~(r) by means of eq. (24) and the non-relativistic expression of the Thomas-Fermi functional can be found. Let us consider the relativistic kinetic-energy density rR(r). We can derive the following inequality: PM (r)

(26)

~a(r) _

PM (r)

m~p2

47:q

47:q

o

+ 2pm + m2p 2 dp = o

PM (r)

(p + m)p 2 dp

(27:)3

(27:)347:q p4 (r)4

m p~(r)3

o

From eq. (24) we have 67:2 \ 1/3

(27)

pM(r) =

q )

tzl/3(r)'

therefore

4~ q

(28)

pt/3(r) + rap(r).

In the semi-classical approximation, the ground-state energy is given by the minimum value of the exact Thomas-Fermi functional. If we consider in place of the exact expression T R (r), as derived from the Hamiltonian of eq. (1), the value given by the r.h.s, of eq. (26), we may conclude that the ground-state energy must satisfy the following condition: (29)

Eo< J[4 \ q

P4/3(r) + mp(r) dr

-~

- ~ - - f i drdr',

where K is the coupling constant and ~(r) is any positive function which satisfies the

1358

B. CARAZZA

and P. QUARATI

normalization condition ~ p(r) d r =N,

(30)

N being the number of particles considered. Once more, instead of solving for a density p(r) which minimizes eq. (29) we shall use a probe function. Let us assume that p(r) normalized as in eq. (30) is a radial function which also in this case has an exponential behaviour: (31)

p(r) = NT 8 exp [ - Tr]/(8r:),

where ~, is a parameter to be varied. We obtain from eq. (32)

Eo <~

(32)

\ ~ ] \--~qq

$ - tcN257/32 + mN.

Usually N is a number sufficiently large; we can disregard the term mN in eq. (32). In the limit in which y--* ~, E0 < - ~, that is the system collapses, if

In the case the particles are the bosons of sect. 4 (q = N), from eq. (33) we derive the condition (34)

~N> y~-]

\~1

=2.6946,

which is close to the result obtained before (see eq. (23)). If the particles are fermions (q = 2) we obtain as condition for the collapse (35)

KN=/3> ~ ~-41 \-~1

=2.1387,

or

•3/2N > 3.1277.

(36)

This result is to be compared with the one given by Martin [5]:

K3/2N > 3.471

(37)

and to the rigorous lower limit obtained by Lieb and Yau [11]:

K3/2N > 3.09.

(38)

6. -

Conclusions.

We have derived upper bounds for the coupling constants of collapsing systems composed of relativistic particles. These results can be obtained by means of a very simple derivation using a variational approach and probe exponential wave functions and are very close to the ones obtained by other authors. The above method can be

SIMPLE APPROACH TO STUDY THE COLLAPSE OF RELATIVISTIC PARTICLES

1359

extended in order to study the collapse of a system of fermions like neutron stars, real neutral atoms or ions, with very high Z. The evolution in time towards collapse of these systems of particles is the subject of a further inquiry.

REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11]

S. CHANDRASEKHAR:Philos. Mag., 11, 592 (1931). V. S. PoPov: Soy. J. Nucl. Phys., 12, 235 (1971). E. n. LIEB and W. THIRRING:Ann. Phys., 155, 494 (1984). W. GREINER, B. M/JLLERand J. RAFELSKI:in Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985). A. MARTIN: Phys. Lett. B, 214, 561 (1988). P. JETZER: Phys. Rep., 220, 163 (1992). J. L. BASDEVANT,A. MARTIN and J. M. RICHARD:Nucl. Phys. B, 343, 60 (1990). A. MARTIN and S. M. RoY: Phys. Lett. B, 233, 407 (1989). C. J. PETHICK: Rev. Mod. Phys., 64, 1133 (1992). I. HERBST: Commun. Math. Phys., 53, 285 (1977); 55, 316 (1978). E. H. LIEB and H. T. YAU: Princeton University preprint Print-87-0372 (1987).