The N53 law for bosons

Volume 70A, number 2

PHYSICS LETTERS

19 February 1979

THE N5!3 LAW FOR BOSONS Elliott H. LIEB’ Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08540, USA Received 13 December 1978

Non-relativistic negative bosons interacting with infinite mass positive particles via Coulomb forces are shown to be unstable in the sense that E 5~’3.This agrees with the previously known lower bound E 513.

0 ~ —CN

0 ~ —AN

In a celebrated series of papers [1—3],Dyson and Lenard proved that matter is quantum-mechanically stable under the action of Coulomb forces provided all the particles of at least one sign of charge (say negative) are fermions. In other words, the ground state energy E 0 satisfies E 0 ~ —AfN (negative fermions) (1) ,

where N istothe number of negative particles. It is not necessary assume neutrality or that the positive particles have finite mass. It is necessary to assume that all the charges are bounded however. (1) was subsequently rederived by Federbush [4] and by Lieb and Thirring [5]. The best current value [6] is Af ~ —22.24 Ry, (2) for electrons and protons, and with the assumption of neutrality. If all the particles are bosons or, what is the same thing, are not subject to any statistics, the best available lower bound [1—7] is 5I3

E

(all bosons),

(3)

0 ~ —AbN

(4)

when the positive and negative particles have charges ±e and the negative particle mass is me. The positive particles can have infinite mass, partially supported by U.S. Nationai Science Foundation grants MCS 75-21684 A02 and INT 78-01160.

1 Work

0 ~

if all particles have finite mass. It was conjectured 7/5 is the correct law for bosons and not [1,8] that N5!3. WhileN this question might have only moderate practi~calimportance (it for would be relevant ir the4He nuclei, example), it hasforgreat mesons and oretical importance and it is to be hoped that its solution will soon be forthcoming. It is interesting because at this point there is no simple, compelling physical argument, as distinguished from a computational argument [9], why the N7!5 law is correct. Subtle correlation effects are yet to be understood fully and rigorously. The purpose of this paper is to add a minor cornmentary on the problem. By means of a simple variational calculation, it will be shown that the N5!3 law is indeed correct if the positive particles have infinite mass and charge zIel >0, i.e. 513 (6a) —CN If the negative particles have mass me and charge —let,

E

.

0

with [6] Ab ~ 14.01 Ry

Dyson [8] then proved, b~’a complicated variational calculation, that in the boson case E
~

and if the system is neutral, then C~(l/108)z4/3Ry.

(6b)

Thus, the limits N °° and the mass of the positive particles are not interchangeable if the N7/5 conjecture is correct. We note in passing that (1) alone does not imply that e 0 = limN~,,Eo/Nexists. However, the method -~

—~

71

Volume 70A, number 2

PHYSICS LE’FTERS

developed to prove the existence of the thermodynamic limit [10] also proves that this limit exists. Likewise (3) and (6a) do not imply that limN~OEO/NS!3 exists. The aforementioned method [10] is not suitable for this task and we do not know an adequate substitute. For simplicity of exposition we assume there is one kind each of positive and negative particle. The N negative particles have a mass =1/2 and charge —1. If 112 = 1, then one Rydberg = 1/4. The K infinite mass positive particles have charge z >0 and we assume N=Kz (neutrality) with K = 8 n3, n an integer. Other cases can easily be handled by this method but we omit details for simplicity. The hamiltonian for the negative particles is =

—

~

E

1 ~i
where R

=

+

Vj~(P’~)}

1r —

r1l~+ U(R)

(7)

lxl ~ 1

,

=0,

Ixl~l.

(12)

The explicit choice in (12) is neither important nor optimal. With T

=fi Vg(r)t2 d3r

=

9

(13)

,

HN,1~~X2NT + XW(N, R), W(N,R) = ~N2 g2(r)g2(r’) I r r’ I_i d3r d3r’

ff

(14)

—

Nfg2(r) VR(r) d3r

+

U(R)

(15)

.

1/2z2!3N4/3 (16) If so, (6) is _(12) proved by minimizing (14) with respect to W(N,R) ~

{R . . ,

K lr—R VR(r)zZ~

1I~

1=1

2

—

.

1

1, R~}is the collection of fixed coordinates of the positive particles and

U(R)z

\/~7~ [1 xl]

—

1=1 +

=

There is ~ in (14) because we should haveN(N 1) instead of N2 in (15). We claimR can be chosen so that

N

HNR

f(x)

19 February 1979

X.

To prove (16), let 0 = a(0)
—

E

lR—Rl~

1~i~j
‘

(8)

.

/

f f(x)2 dx

=

(2n)~,

for allj.

(17)

L (j)

We want to find a normalized ~1i (r 1,. r~)and5!3. R (depending on N) such that (~ I HNR I ~ ~ cN ,li is chosen to be a simple product of identical functions: N Ø~(r~). (9) ..

,

I~I i=1

~ depends on the parameter (which will turn out 3) asX follows: to be proportional to Ni! = X3!2g(Xr), (10) where g(r) is the fixed, normalized function g(r)f(x)f(y)f(z),

and

r(x,y,z),

The rectangles r(i, J, k) = L(i) X L(j) X L(k), —n ~ i, /, k
f

g(r)~d3r~’1/K.

(18)

F(i,/,k)

Returning to (15), place oneR 1 in each oftheK rectangles and then average W(N, R) over the positions, R1, of theweight positive particles within the a relative g(R 2.Trectangles his averagewith is

1 )2

given by W(JV) =

—

~N2 ~

ff

m F(m)

. . .

g(R~)

g(r)2g(r’)2

(11)

X Ir

—

r’I’ d3rd3r’

,

(19)

with m = (i, f~ k). Since the weight is nonnegative, there is at least one choice of R (with one particle per 72

Volume bA, number 2

PHYSICS LETVFERS

rectangle) such that W(N, R) ~ W(N). Thus, we are done if W(N) ~ right side of (16). Each integral in (19) is the self-energy of a charge 1/K confined to a rectangle. This, in turn, is greater than the minimum self-energy of a charge 1/K confined to lie only in a circumscribed sphere. If F(m) has sides of length (s, t. u), this sphere has radius p(m) 2 + u2] 1/2, It is well known that the mini=

~ [~2 + t

mum self-energy occurs when the charge is distributed uniformly on the surface of the sphere and is p(m)~ X(1/K)2.Thus W(N)

~—~z2Em p(m)~

~ —z2K [a2

+

+ ,~2] 1/2

(20) (by convexity),

where a, r and p are the mean lengths of the sides of the rectangles. However, a = r = p = 1/n, and (16) is proved. The author would like to thank the Research Institute for Mathematical Sciences, Kyoto University, in particular Professors H. Araki and K. ItO, for their generous hospitality.

19 February 1979

[2] F.J. Dyson, in: Brandeis University Summer Institute in Theoretical physics (1966) eds. M. Chretien, E.P. Gross and S. Deser (Gordon and Breach, New York 1968), Vol. 1, p. 179. [3] A. Lenard, in: Statistical mechanics and mathematical problems, ed. A. Lenard, Lecture Notes in Physics (Springer, New York, 1973), Vol. 20, p. 114. [4] P. Federbush, J. Math. Phys. 16 (1975) 347, 706. [5] E.H. Lieb and W.E. Thirring, Phys. Rev. Left. 35 (1975) 687. [6] E.H. Lieb, Rev. Mod. Phys. 48 (1976) 553. [7] D. Brydges and P. Federbush, J. Math. Phys. 17 (1976) 2133. [8] F.J. Dyson, J. Math. Phys:8 (1967) 1538. [9] L.L. Foldy, Phys. Rev. 124 (1961) 649; M. Girardeau and R. Arnowitt, Phys. Rev. 113 (1959) 755; M. Girardeau, Phys. Rev. 127 (1962) 1809; W.H. Bassichis and L.L. Foldy, Phys. Rev. 133 (1964) A935; W.H. Bassichis, Phys. Rev. 134 (1964) A543; J.M. Stephen, Proc. Phys. Soc. (London) 79 (1962) 994; D.K. Lee and E. Feenberg, Phys. Rev. 137 (1965) A731; D.Wright,Phys. Rev. 143 (1966) 91; E.H. Lieb, Phys. Rev. 130 (1963) 2518; E.H. Lieb and A.Y. Sakakura, Phys. Rev. 133 (1964) A899. [101 E.H. Lieb and J.L. Lebowita, Adv. Math. 9 (1972) 316.

References [1] F.J. Dyson and A. Lenard, J. Math. Phys. 8 (1967) 423; A. Lenard and FJ. Dyson, J. Math. Phys. 9 (1968) 698.

73