bose condensation in supercritical external fields

P H Y S I C A L R E Y I E L Y D \ O L U h I E 11, N U X I B E R 2 1 5 J A X U A R Y 1 9 7 5

Bose condensation in supercritical external fields*

Abraham Klein and Johann Rafelski Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 191 74

(Received 12 August 1974)

We study the relativistic field theory of a charged spin-zero boson field in the presence of an external field, such as the Coulomb field of a prescribed charge distribution. It is shown that for a sufficiently intense field the ground state is unstable against the formation of a Bose-Einstein condensate of charged boson pairs. A consistent quantum theory can be formulated when known nonlinear couplings such as the Coulomb interactions of the bosons are properly included in the Hamiltonian. Speculations are offered concerning the possible stability of nuclei with charge number Z 2 lo3.

I. INTRODUCTION

In th i s paper , we study the problem of the quantization of a charged spin-zero field in the p r e - sence of a localized positive-charge distribution of a r b i t r a r y s t rength ("Coulomb field of a nucleus").

When the nuclear charge number Z exceeds a cer tain cr i t ical value, Z,, solutions of the l inear Klein-Gordon (KG) equation in the external field no longer f o r m a complete s e t and s tandard quantization of the quadratic Hamiltonian operator fa i l s . With conventional nuclear charge densities, the fact that this phenomenon occurs fo r Z 2 lo3 may explain the paucity of attention which it has so f a r received in the It tu rns out that when one examines the usual quantum theory one notices that a t Z = Z , the vacuum becomes un- s table against the production of a n indefinite num- b e r of charged boson p a i r s . To guarantee a stable ground state fo r Z > Z , one must, a t the v e r y least , take account of the self -Coulomb interaction of the boson f ie ld. The basic physics of the resu l t - ing nonlinear field theory a s well a s some of the mathematics a r e given in Migdal 's paper,3 which i s , however, technically incomplete. The c o r e of our work, Secs . III and IV, thus has minimal overlap with the f o r m e r ' s work.

Thus, a t the very least , we exhibit in the mate r i - a l of Secs . 11-IV a n amusing example of a spat ial- ly inhomogeneous Bose-Einstein (BE) condensation phenomenon whose occurrence i s in a basic sense attributable to the special p roper t i es of a relat ivis t ic many -body theory.

In the l a s t section Sec. V, we speculate that ultimate connection with physical real i ty may not be beyond the possible. F o r nuclei with Z > Z , to be s table we argue that two conditions a r e neces - s a r y . The f i r s t is that ordinary >\i Z nuclei contain a neutral BE condensate of equal mix tures of neutral and charged pions. (Bounds on the density of such a condensate s e t by presen t experilnellts

will be the subject of separate investigations.) In the domain of l a r g e Z > Z,, the considerations of Sec. IV indicate that the Coulomb energy will be reduced in the p resence of the pion condensate. Physically this can only happen if the charge density s i t s well outside the mat te r density, the net charge residing on the condensate. This leads to our second condition, which is that this effect be sufficiently pronounced to prevent nuclear f iss ion of such superheavy nuclei. Though we a r e somewhat skeptical concerning the possible concurrence of these two conditions, such a possibility prob- ably m e r i t s fu r ther investigation.

The behavior of pions in the Coulomb field of a "heavy" nucleus i s in marked contrast with the behavior of e lectrons, a subject which has been thoroughly investigated.' These a r e so f a r the only c a s e s fo r which the theory has been worked out.

11. KLEIN-GORDON EQU.-ITTON IN A STRONG COULOMB FIELD. INSTABILITY AGAINST CHARGED PAIR

COh'DENSATION

.4. Forms of the KG equation

We study (with E = c = 1) the equation

(E - V)' y(F) : (pZ + P I ' ) +(F), (2.1)

where 4 7 ) i s a complex sca la r function and V ( v ) i s the potential energy of a negatively charged sca la r par t ic le of m a s s nz (henceforth called a pion) in the field of a fixed extended charged d i s - tribution. F o r example, we may take

i(Y) = 1 - e - K ~ ( 2 . 3 )

where K - I g R , the nuclear radius. The formal propert ies of the equation a r e more

easi ly studied if we introduce a fo rmal i sm of f i r s t o r d e r i n the energy (time d e r i ~ a t i v e ) . ~ In t e rn i s

11 - B C S E C O N D E N S A T I O N I N S U P E R C R I T I C A L E X T E R N A L F I E L D S 3 01

of the two-component vector

~ q . (2.1) becomes

E T ~ + = x $ , (2.5)

x = + ( 1 + ~ 3 ) ( f i 2 + 1 1 2 2 ) + $ ( 1 - T ~ ) + T ~ V . (2.6)

Here 7i ( i = 1 , 2 , 3 ) a r e the usual Paul i spin m a - t r i c e s . Since X is Hermitian, we recognize that the fundamental s c a l a r product is the integral of the density

deb(?) =$:(a T ~ & ( F ) , (2.7)

providing an orthogonality theorem f o r two solutions of (2.5) belonging to different energ ies .

F o r the norm of any given solution, we compute

Remembering the sign of V, Eq . (2.2), we s e e that (2.8) is certainly positive fo r E 2 0 . On the other hand, fo r negative-energy continuum solutions, the norm mus t be negative, by analytic continuation f r o m the l imit V=O. Thus if we choose the or iginal KG s c a l a r function to be normalized according to

where (v)=J;c* v$/J+*+, we s e e that, in general, the solution s e t f o r given V divides into two sub- s e t s #, and +, character ized a s follows:

The considerations above would obviously fai l should there occur a n eigenvalue f o r which the in- equalities (2.10) a r e replaced by an equality. The resolut ion of this difficulty, c a r r i e d out below, r e p r e s e n t s the bas ic goal of this par t of our work.

F o r the normally occur r ing situations fo r which (2.10) applies, the completeness relat ion for the solution of (2.5) takes the dyadic f o r m

and I i s the unit two-by-two matr ix. A third f o r m of the KG equation is sometimes

useful because of i t s analogy with the nonrelativ-

is t ic SchrEdinger equation. F o r this fo rm, we wr i te

where

B. Bound - state spectrum and approach to the critical point.

F o r full understanding of the situation when Z becomes very la rge , we consider together the s o - lutions fo r both negative and positive pions (n'). The behavior of the most deeply bound orb i t s is i l lustrated schematically in Fig. 1 . We shal l f i r s t descr ibe the r e s u l t s and then indicate how they follow f r o m the KG equation. F o r a potential en- e rgy of the f o r m (2.2), the curve m a r k e d E - r e p - r e s e n t s i n i t s solid par t the lowest bound s ta te of a n-. Two special values of Z a r e to be noted. F o r Z > 2, the re e m e r g e s f r o m the negative -energy continuum a new bound-state b ranch f o r n- which m e e t s the b ranch E- a t a point of ver t i ca l tangency, Z=Z,. F o r Z> Z, the re is simply no bound- s ta te solution corresponding to this branch. The curve marked E+ is a reflection of E- with r e - spect to the absc i ssa E = 0 and r e p r e s e n t s a solution branch f o r 8 +.

We shal l now indicate the derivation of these r e s u l t s f r o m the KG equation. To understand their physical implication we mus t r e s o r t to the quantum field theory described in the next s e c - tion.

It follows most easily f r o m (2.1) that

FIG. 1. Schematic representation of the lowest bound- state branch for a negative pion in the Coulomb field of a superheavy nucleus (neglecting strong interactions). The branch for the positive pion related by charge con- jugation i s also shown.

3 02 A B R A H A M K L E I N A N D J O H A N N R A F E L S K I 11 -

This shows that if branch E - occurs f o r n-, branch E + occurs fo r n t . Next consider the equation for n+ near ze ro binding energy. F r o m (2.13), remember ing to change the sign of the charge, we have

F o r Z l a rge enough, the second t e r m mus t dominate and give (2 > 2,) a .rr+ bound s ta te -even in the field of a positively charged nucleus-as shown in the solid p a r t of the branch E + . The combination of (2.14) and (2.15) establ ishes the qualitative charac te r of Fig. 1 .

One naturally inquires af ter the physical picture which explains how a n" can be bound in the field of a positively charged nucleus. Here, the fact that the charge density is not positive-definite plays a n essent ial r o l e . Though the total charge is one unit, this is obtained by adding individually l a rge contributions of negative and positive charge. If the negative charge i s on the average c loser to the nuclear charge, we obtain the r e - quired binding force .

It i s in fact easy to s e e f r o m the considerations of Sec. IIA that points on the solid p a r t s of curves E , correspond to solutions with positive norm whereas dashed portions represen t solutions with negative norm. The meeting point mus t therefore b e one a t which E = ( V ) . Now consider Eq. (2.1) f o r V - hV and f o r m the expectation value,

With the help of the KG equation, the f i r s t der iva- tive of (2.16) a t h = 1 becomes

At the point E = ( V ) , the right-hand side cannot vanish. Therefore we mus t have dE/dh-.o, a point of ver t ical tangency.

Let E - = - p. represen t this point occur r ing a t Z = 2,. Then we s e e that fo r the branch E + , we have E + = p . Therefore in a quantum field theory we could produce an indefinite number of n* p a i r s without energy cost (in the absence of other inter - act ions) . This apparent instability will be dealt with in the next section.

Le t u s es t imate the value of Z , predicted by our model, using the pion m a s s . This can be done with the help of a v i r ia l theorem. F r o m the s tatement

we derive f o r bound s ta tes

Combining this with Eqs . (2.16) (h = 1) and (2.2) and (2.3) we derive ( a = e 2 / E c )

The c o r r e c t o r d e r of magnitude for Z , should be obtained if we examine (2.20) f o r E = 0. Thus we have

using the essent ial fact that the pion wave function is largely confined to the nuclear inter ior . If we s e t R~ = (102/1112) we find Za - 10, which is roughly consistent but appears to render our problem somewhat academic, a t l eas t under present ly known conditions. (If we had a pion of electronic m a s s , Z , would be reduced by an order of magnitude .)

Returning now to the technical aspects of our problem, we notice that a t Z = Z , the re is only one solution remaining of the two we had for Z < 2,. To understand the approach to the limit, E e define in t e r m s of the two solutions of in te res t +, (+ re fe r r ing to norm) two new unclza~zged lin- e a r combinations:

* @ e = ( l / d z ) ( G + + $ - ) ,

+ (2.22)

4 0 = ( l / m ( $ + - 5 - ) .

With

etc., we have straightforwardly

but

It may be instructive to exhibit expressions fo r these densi t ies . In the following we make u s e of the orthogonality between $+ and 5-, which if we assume r e a l wave functions can be expressed in the f o r m

where

and ( a s will be needed below) V,, = V,. We a lso r e c o r d our expressions using unit normalization f o r q , . We then have

11 - B O S E C O N D E N S A T I O N IN S U P E R C R I T I C A L E X T E R N A L F I E L D S 3 03

From these expressions, we verify (2.24) and (2.25). From (2.28) we see that the limit 2-2, i s a delicate one. If we define

i t can be shown that de behaves like ~ ( a - I ) . By contrast, do, and do remain finite and vanish r e - spectively a s a - 0.

111. QUANTIZATION O F THE KLEIN -CORDON EQUATION IN A COULOMB FIELD. STABILIZATION O F THE

VACUUM BY NONLINEAR INTERACTIONS

A. Quantization below the critical point.

This i s achieved most elegantly in an external field by use of the f irst-order formalism. We adopt the Lagrangian

This yields the canonical momentum

Thus the required commutation relations a r e

[ @ ,,,(?, t ) , @I,, (F ' , L)I=T,F(F -:') . (3.3)

The Hamiltonian which follows from (3 . I ) and (3.2) i s

To this we adjoin our candidate for total charge operator, namely

It is understood that H and Q a r e to be taken in normal form with respect to the vacuum state to be defined below.

The completeness relation (2.11) will guarantee a satisfactory quantum theory when used in con- junction with the expansion

if we assume that

(and of course if the a ' s and b's commute, etc.) . Thus (3.6) and (3.7) satisfy (3.3). Furthermore, if the vacuum i s annihilated by a, and b,, we find for (3.4) and (3.5)

As long a s Z< Z,, this represents an unequivo- cally satisfactory theory of noninteracting bosons of either charge in the external field of a positively charged nucleus. As we approach the point Z = Z,, we ar r ive , now on a proper quantum basis , a t the situation already described: The vacuum state i s no longer unique, but approaches degen- e racy with an infinite se t of other s tates contain- ing various numbers of ni of energy i y each.

To remedy this situation, i t i s sufficient to in- clude in the quantum theory the mutual Coulomb interaction of any produced pions. This will be demonstrated with sufficient r igor in the next two sections, but the resul t s can be anticipated on physical grounds. Thus, a s we approach Z=Z, , including the Coulomb interaction of the pions, multipion states of small excitation energy can now mix with the previously defined vacuum. But the expectation value of the Coulomb interaction of any assembly of charges is positive, and since i t i s quartic in the amplitudes i t must ultimately dominate and prevent collapse. Moreover, if we consider the equation of motion for excitations of negative charge, which i s the appropriate general- ization of the unquantized KG equation, we must find that the charge distribution of the other pions screens the nuclear field on the average. The net result i s an effective nuclear field which remains subcritical. A vacuum state i s thus stabilized, but in t e rms of the eigenstates of the Hilbert space defined by the expansion (3.6) i t s description will involve many components.

B. Quantization beyond the critical point. The reduced Hamiltonian.

Formally the quantization can be car r ied out precisely a s above. We need only replace (3.4)

304 A B R A H A M K L E I N A N D J O H A N N R A F E L S K I 11 -

by the Hainiltonian

Since V in X i s now, by assumption, beyond c r i t i - cality, I V / > / V,; , we choose V , such that 1 Vo/ < / Vcl and 1 V , - V,/ / 1 V,1<< 1 and wr i te

We a l so define - -. E i 71 @ + =XO @ + ,

where these a l l r e f e r to the near-cr i t ical eigenvalues .

To establ ish the stability of the Hamiltonian (3 . lo ) , we wri te

F o r the remainder of the p resen t section we drop the t e r m s a r i s ing froin x 0 , , since these make r e f - e rence to the nondangerous levels; a full discussion including these t e r m s will be c a r r i e d out in Sec. IV. We a lso rewr i te the f i r s t t e r m s of (3.15) by rneans of the opera tors

yielding

+ , , ( F ) = ~ ? , ( F ) + ~ P ' ? ~ ( F ) , (3.17)

where T, and $, a r e the combinations defined in (2.22). With the aid of definitions (3.11)-(3.14) and the fur ther definitions

we obtain by straightforward t ranscript ion (where Li,,,, -U,, e tc . )

H 1 ? , = ( 6 - ' 0 0 ) p t ~ + ( 6 - h e , ) q f q + ( - P + h , , ) L

+ + ~ , ( q ~ q ) ~ + i ~ r ~ , " { q ~ ~ , j ~ ~ ) + - : u , ( ~ ~ ~ l ) ~

The subscript o n H reminds u s that we a r e d e - scr ibing only two degrees of f reedom.

F o r Z - Z, " 1, the bas i s defined by (3.11) and (3.12) should be v e r y satisfactory for the diagonal- ization of the Hamiltonian (3.21). Tile unperturbed vacuum (we consider the subspace L = 0 ) will mix with s ta tes with at most a few pion p a i r s to yield a new stable vacuum. This is guaranteed by the positive-definite charac te r of the quartic t e r m s of a.

F o r Z - Z, = Z' >> 1, the mixing becomes la rge and the t reatment by means of the bas i s defined above becomes cumbersome. In this case the stability of the ground s ta te may be inferred f r o m a classical approxi~nat ion f o r which

e t c . With

we have ( L = 0)

A s will be seen below, we can, without l o s s of generality, choose pZ = 0. The variation of W with respec t to q2 then yields the minirnu~n a t

and

which i s proportional to 2''. Fur ther p r o g r e s s and substantiation of the above

simplification depends on recognizing that the variat ional expression (3.24) may be derived a s the expectation value of H r c , with respec t to a coherent t r i a l function

la', b ' , 6)=exp(a"a'+O ' ~ ' e - ~ " ) j v a c ) . (3.27)

L ~ h i s i s t rue insofar a s l a ' /2 >> 1, 1 b'j2 >> 1, and r e - qu i res remember ing that (3.27) is not normalized.] With the identifications

we have (immediately dropping the subscript ze ro)

Average charge neutrality i s assured by choosing a ' =Or. The choice O = 0 fo r the a r b i t r a r y phase fur thermore guarantees p = 0. Both here and b e -

11 - B O S E C O N D E N S A T I O N I N S U P E R C R I T I C A L E X T E R N A L F I E L D S 305

low, however, it i s useful to re ta in this phase until the final s tages of the formulation.

The observations just made suggest that the p r e - vious classical approximation can be improved by a l so varying the opera tors a t and b f , i.e., by varying the functions ?,(F) and $,(F) in (3.17). The expectation value of Hi,, with respec t to a general t r i a l s tate (3.27) again has the fo rm (3.24) with the more concise definitions of A and B,

It i s important in the following to notice that (3.24) i s a functional of the quantities q ;, and p 5, (and their Hermitian conjugates).

The variation i s to be car r ied out subject to the constraints that ?* given by -

@* = (2)-1'2($e*T0) (3.33)

have positive and negative norm, respectively, a s required by (2.10). F rom the variational expres - sion (3.24), we thus subtract

the coefficients of the norms represent ing a definition of Lagrange multipliers. This i s t rans - formed by means of (3.29), (3.30), and (3.33) into the expression

where we have also assunled charge neutrality and used the definitions (3.13) and (3.14). Var i - ation of the sum of (3.24) and (3.35) with respect to 1 qI2 $? and p / $1 yields the equations

xitt ?e=@71?e+671$o, (3.36a)

xett $ o = ~ ~ 1 T o + 6 7 1 ? e , (3.36b)

o r equivalently

xeii 3* = v 1 a * > (3.37)

where

An understanding of the s t ruc ture of the nonlin- e a r problem posed by (3.36)-(3.40) r equ i r e s f u r - ther discussion. Fo r instance, what happens when we s e t 1 p 12=0? Since this corresponds to setting the phase 8 = 0, we a r e led to study the equivalent transformation -. - - . -

@:=ei8 4-, +:= ++ , (3.41)

o r in t e r m s of 5, and 5, * $: = $(I+ ei" $e +$(I - e iB) T o , ( 3 . 4 2 ~ ~ ) -. 0: = $(I - e i 8 ) T e + h(1 + e i H ) Go. (3.42b)

With the help of (3.29) and (3.30), we can fur ther - more wri te

where

By means of t r ivial a lgebra we now find a s ex- pected that (3.36) o r (3.37) i s invariant in f o rm under the transformation (3.42), but because

P : = x P ~ + ( ~ - x ) P ~ (3.45)

the definition (3.40) i s simplified to

pet , (?) = n p ; ( F ) ~ p ( F ) . (3 .401)

Thus we have shown that the solution for 1 pj2+ 0 can be t ransformed into one with / P j 2 = 0 without affecting any physical resu l t s . But to obtain a general formulation of the problem this should be done af ter the variation has been car r ied out.

If we apply (3.42) to the energy (3.24) (or equivalently s e t 1 p / = O), we find

which i s a minimum for [cf. (3.25)]

where we henceforth drop the pr imes . Remember- ing the definitions (3.19) and (3.23), we see that p,,, ( r ) , (3.400, i s independent of the sca le of 4,.

The las t observation i s the essent ial one to understand: the l imit 6 - 0 . In this l imit pet, will remain finite, but of the separate fac tors i z = q2 - 0 and p, -a. It i s str2ng;y suggested that we r e - scale the functions @,, @,. We therefore define -.

b = q T e (3.48a)

thus retaining the sca la r product between them [ ~ q . (3.53) below]. With

306 A B R A H A M K L E I N A N D J O H A N N R A F E L S K I

X= (6/q2),

E q s . (3.36) become .+

(Xett - 11 71) ; =q4h71 V, ,

(Kc,, - 1 - ~ 7 1 ) ? = ~ 7 ~ $ ,

where

X , , , = X + v ,

-. - * ijr = + t ~ l , e tc . , (3.54)

In this new form the s t ruc ture of the self-consistency problem is completely c l e a r . The s e t (3.50)-(3.55) will have a self-consistent solution f o r any (reasonable) g 2 0 . [ ~ t self -consistency q2 will, in fact , be given by (3.47).] The l imit q = 0 may be taken with inpunity. In this limit, the self-consistency problem becomes independent of * +! and of i t s Eq . (3.50b). This means, a s we know, that we have lost one solution of our self-consis t- ent KG equation. Eqz~ation (3. 50b) nozv defines a ~zonuanislzi~zg huzctioiz s~titably ovthogonal to all solzttions of the KG equation, tlzzts pyoviding a missi~zgfit~zction to make ctp a complete set.

Let $a be a l l solutions of

other than $ i tself. A s described in Sec. 11, these will divide neatly into two s e t s qp and qn, with positive and negative n o r m . F o r an a r b i t r a r y vector

we have (including the l imit 6 = 0, but not confined to i t ) an expansion theorem

where

The discussion just completed provides u s with the basic machinery needed to c a r r y out a complete quantization in the supercr i t ical domain Z > 2,. This i s done below.

IV. CONSISTENT QUANTIZATION FOR SUPERCRITICAL

FIELDS.

A s we have seen in the preceding section, f o r Z > 2, we a r e dealing with a Bose condensation phenomenon. In our discussion of the stability of the vacuum based on the coherent s ta te approximation, we have already mixed eigenstates of dif - fe ren t charge. F o r a general quantization we gen- e ra l ize this procedure by replacing H, Eq. (3.10), by

H 1 = H - p L , (4.1)

where

is proportional [cf. (3.5)] to the total charge op- e r a t o r . We now wr i te

=?(a ~ x " ~ ( ; ) , (4.3)

where -. @ ( ~ ) = c r ? , , (4.4) -.

and $,, q a r e solutions of the self-consistency problem (3.36) with /I = 0 . We shall ultimately con- s ider the l imit 6 - 0. We have argued that the function $(Y) remains well defined in this limit. A s long as 6 + 0, X,,, defines a complete s e t of single-particle functions in the sense of Eq . (2.11). F o r 6 = 0, we have learned how to adjoin a function to the solutions of the KG equation so a s to achieve completeness.

If we substitute (4.3) into (4.1), using (3.36), we find straightforwardly, omitting also the sub- sc r ip t f r o m x , , ~ , that

H I = - ic2[ P,(!')c!~) r - r 1 + q 2 6 + 4 6 (fXfrl;h

+ [ X ~ ( K ~ , ~ - P T ~ ) X + ~ ~ ' J CII(F)+v'(F)1[v(F1)+~f(F')J I F -F ' I

J u ( 3 ~ ~ x(T)I[x t ( F 1 ) ~ l x ( F 1 ) ] IF -FII IT. -FII 9

11 - B O S E C O N D E N S A T I O N IN S U P E R C R I T I C A L E X T E R N A L F I E L D S 3 07

where [ c f . (3.40'))

~ ( 3 = G T ( r ) r l G ( 4 , . r l (F) =Gt(r)r1 X ( Y ) .

If we ignore t e rms linear in 6 which will ultimate- l y - 0 , we see that (4.5) has the structure, in an obvious notation,

where

i s the condensation energy and

i s the part quadratic in the field operator X . W e shall seek a realization o f the operator X ( ? )

which brings (4.10) to diagonal f o rm and satisfies the commutation relations

[ X ( F ) , X t ( F f ) ] =r16(F - F f ) , (4.11a)

[ x ( ? ) , x ( Z f ) i = O . (4.11b)

This can be done most expeditiously as follows: Let u s define the amplitudes

(vac I x ( F ) l v) ~ F , ( F ) , ( 4 . 1 2 ~ ~ )

(vac l x r ( ? ) I v) =;:(?), (4.12b)

where the states I v) are quasiparticle states, which we study f i r st in an approximation which neglects HA and H:, E q . (4.8). In this approximation we obtain by taking matrix elements of the Heisenberg equations of motion for the operators x , x r ,

with tive to that of the vacuum. For an acceptable quantization, this must be positive for all v.

if(:) = J ( F ) ;(a, (4.14) W e demonstrate how (4.13) meets these require- ments. By forming the obvious scalar products,

and E , measures the energy o f the state 1 v ) rela- we derive f rom (4.13a) and (4.13b)

The third t e rm of (4.15) i s clearly positive, and by u_sing the expansion theorems (3.58) and (3.59) for ) f ; andg,, respectively, i t follows that these t e rms are positive as well. Since E,> 0 , we see th~t_two_cla,ses o f solu_tio_ns eme_rge. In the f i rs t S f , f , = ( L , f , ) > ~ and l(T,,f,)l>l(E,,E,)l. These are the appropriate continuations into the supercritical region of the positively normed a- solutions. For the gth_er class o f solutions (z,, E,) < 0 and / (g,,E,)/ > / ( f , , f , ) l , so that the left-hand side o f (4.15) remains positive. In this case - E , corresponds to the solution o f the Klein-Gordon equation and we are dealing with a continuation o f the n+ solutions. These assertions become intuitively clear if we allow the explicit e2 which occurs in (4.13) to approach zero, in which case the two equations would decouple to the l imits indicated.

W e next notice a symmetry property of the sys- t em (4.13). W e interchange f , with g,, take the Hermitian conjugate, and replace E,- -E,. This changes (4.13a) into (4.13b) and conversely. The result may be usefully reformulated as follows. W e f i rs t replace (4.13b) by i ts transpose so that g:r1 - Yl E$ = r lg$ , etc. A solution to (4.13) i s then represented by the double vector

F,(r) = ( ? u ( ~ ) ) . Hv* (r )

In this new two-component space, the Pauli ma- trices will be designated as u i . Then the symme- t ry property noted above i s that to every solution F, with energy E,> 0 there corresponds another solution G , with equal but opposite E,, where


G,,=o, F$ . (4 .17) standard methods.

With this theorem we have accounted f o r a l l solutions of the doubled sys tem (4 .13) . The physical solutions a r e those with positive energy E,>O. The unphysical solutions a r e , however, needed f o r completeness in the doubled space. Accord- ing to (4 .13) , the orthonormalization theorem f o r the F, should be

The orthogonality follows f r o m (4 .13) i tself; the normalization follows f r o m the orthogonality plus completeness relat ions which will be given below. In pract ice the G , a r e distinguished by the sign of the energy and by the sign of the norm, s ince f r o m (4 .17) and (4 .18) follows

The quantization procedure can now be s u m a r - ized by the expansion theorems

Insertion of (4 .20) into the commutators (4 .11) yields the completeness relat ions

These, together with their Hermit ian conjugates, can be rewri t ten succinctly in t e r m s of the vec- t o r s (4 .16) and (4 .17) a s

Together with the orthogonality implied by (4 .13) , the relat ions lead back to (4 .18) and (4 .19) and to the fur ther orthogonal.ity relation

It is finally straightforward to verify that the expansions (4 .20) fulfill their a im of diagonalizing H i . We obtain

the l a s t t e r m represent ing a (positive) zero-point energy.

The remaining t e r m s H,' and Hl, respectively cubic and quart ic in the new quasiparticle oper - a to rs , may be t reated by perturbative o r other

We cannot re f ra in f rom car ry ing the fo rmal de- velopment just a lit t le fu r ther . It i s mainly to ob- s e r v e that the completeness relat ions (4 .23) can be reduced to the f o r m (2 .11) appropriate to the KG equation by considering, a s af ter Eq. (4 .15) , two s e t s of solutions. F o r the f i r s t we wr i te

and f o r the second

F o r bound s ta tes we choose cV r e a l . F o r this case , we s ta te the equations determining this function c, and the angle B,, obtained by insertion of (4 .27) into (4 .13) and straightforward algebraic combination, utilizing also (4 .28) :

where

Equations (4 .31) and (4.32) have to be solved by self-consistent methods.

F o r the s e t of solutions (4 .29) and (4 .30) , a se t analogous to (4.31) and (4 .32) can be obtained dif - fe r ing f rom the la t t e r in f o r m by the replacement E,- -E,.

With the aid of the considerations flowing f rom ( 4 . 2 7 ) forward, we can rewr i te the expansion (4 .20) so a s to suggest continuity with the subcri t i - cal expansion given in (3 .6 ) . F o r this purpose a change in notation i s de rzgezcv:

+ +

X v t ' X n , + .+

X u - ' X n ,

11 - B O S E C O N D E N S A T I O N IN S U P E R ( Z R I T I C A L E X T E R N A L F I E L D S

In place of (4.20) we therefore write

As (2 - Z,) - 0 the condensate wave function 6 and all the phase angles 0 - 0, so that quite for - mally the two expansions (3.6) and (4.36) look similar in this limit.

In point of fact, however, neither exists mathe- matically in this limit. The physical counterpart of this i s that in the neighborhood of Z - Z, = 0 the excitations of unit charge, o r more accurately those that can be reached by applying the field operator once to the vacuum, cannot be described simply either a s part icles o r quasiparticles. The description of the eigenstates in t e rms of either limiting description i s more complicated. In the language of thermodynamics this corresponds to an increase in the size of the fluctuations in the neighborhood of a phase transition.

The same point can be emphasized by considera- tion of Fig. 2 . In Fig. 1, the par t s of the solid curves marked E - and E+ not too neav Z = 2, may be considered eigenvalues of the field theory. In Fig. 2 we indicate schematically the continuation of these eigenvalues for increasing Z according to the nonlinear field theory of this paper. The pa r t s of these curves corresponding to (2 - 2,) >> 1 would emerge from the solution of (4.31) and (4.32) [or (4.13)]. However, for Z - 2, we would have to mix either multiparticle states with the single- particle states valid for Z < Z, o r multi-quasipar - ticle s tates with the single-quasiparticle states valid for Z > Z, in order to reproduce the curves of Fig. 2 in the transition region.

V. POSSIBLE APPLICATION AND EXTENSION OF RESULTS.

To provide some balance to the very formal considerations of the preceding portions of this paper, we now attempt to relate our discussion to the structure of nuclei, known and unknown.

The f i r s t point which must be emphasized i s that the results developed so f a r a r e applicable to a situation where the nucleus of charge Z - lo3 i s held together by a dezis e x machina. C'ndev no c ircu~~zs tances i s the condensation energy sufficient to stabilize the nucleus. This i s easily seen a s follows: The sum of the condensation energy and of the Coulomb energy of the nucleus can, in the semiclassical approximation in which we a r e working, be written in the form

FIG. 2 . Schematic representation of those eigenvalues of the field theory, continued into the range 2 >Z,, which correspond for Z iZ, to single T- and .iit mesons bound to the nucleus.

where $ i s the K+G amplitude corresponding to the two-component @, and p, i s the sum of the charge densities of the protons and of the pion condensate. As previously noted, the lat ter a r ranges itself so a s to cancel a s much a s possible the proton charge inside the nucleus. This must be balanced for overall charge neutrality of the condensate by a positive charge density outside the nucleus. Depending on the actual range of this charge, it i s concezuable (though we have no reliable numer- ical results) that the overall Coulomb repulsion, the last t e rm in (5.1), i s sufficiently reduced so a s to res tore stability against fission if the nucle- u s i s otherwise stable.

In this connection, of course, one immediately thinks of the obvious advantages of a negatively charged condensate. We have, however, proved in the last section that small charge fluctuations about a neutral condensate increase the total energy of the system. This does not rule against the interest of studying condensates carrying charges of order Z, especially if one simultane- ously considers pion-nucleon interactions (see below).

In any event, and this i s our elementary point, the best we can do without strong interactions i s to reduce the last t e rm of (5.1) to impotence. We ave left wzth the r e s t and kznetzc enevgzes of the condensate. These can be compensated, if at all, by the pion-nucleon interaction. It has been claimed7 that ordinary nuclei contain a neutral pion condensate. This i s an extremely interest- ing suggestion, even if i t i s moot.8 If this turns out to be the case, a number of questions come to the fore: (i) Why has this condensate not been observed a s yet? (ii) How would we look for i t? (iii) What a r e the implications for the ideas dis- cussed in the body of this paper?

In order to speculate on lhese questions, let u s


consider briefly a highly schematic modelQ which tegral J $'(r), the re a r e formally a n infinite num- pred ic t s such a condensation phenomenon. We b e r of solutions f o r +(Y). These can be classified descr ibe the condensate by a r e a l (KG) wave func- by wave-numbers k,, where it can be shown by tion y ( 7 r ) and postulate that the energy functional s tandard methods that the values of k, which de te rmines q ( r ) has the f o r m

k , = ~, / [C , f (o) - 11 (5.7)

which contains two new constants C and D, both positive, and a function f ( r ) , which we shal l ten- tatively identify with the nucleon mat te r density. Actually the t e r m proportional to D is not sufficient to stabilize the sys tem, because there is no cutoff on the p-wave interaction. This can be done if we make the replacement

dc V+(Y) - [ c ( v z ) j1'2v#(~) , (5.3)

where C(V2) will cut off short-wavelength osci l la- tions of $(Y). F o r the t ime being we neglect the operator charac te r of C .

If we then vary (5.2), we obtain the equation

-[1 - C/'(r)] V2$(Y) + K'?/J(Y) = 0, (5.4)

where f o r purposes of qualitative discussion we have a l so dropped a t e r m proportional to the g r a - dient of the function f (9,). If we choose f (Y) =con- stant inside the nucleus and zero outside and a s - sume that

then a solution of (5.4) corresponds to finding an s-wave bound s ta te of ze ro energy f o r a par t ic le subject to an at t ract ive square-well potential in- s ide the nucleus and a b a r r i e r outside. Since both of these can be adjeisted by adjusting the in-

f o r m a monotonically increasing sequence. We shall understand that the main effect of the p roper incorporation of the requirement (5.3) is to select the lowest value of k,. This can s t i l l correspond to a solution in which $ ( r ) osci l la tes a number of t imes inside the nucleus.

Such a condensate, if i t s amplitude were a l so small , might be difficult to detect. Such an a m - plitude must , however, induce corresponding o s - cillations in the nuclear density which could be detected a t sufficiently l a rge momentum-transfer e lectron and nuclear e last ic scat ter ing. It has been a s s e r t e d that such effects have been seen in electron ~ c a t t e r i n g . ~ Effects on nuclear spec t ra should a l so be examined.

F o r heavier nuclei, we can no longer ignore the Coulomb interaction of the pion cloud with the proton charge distribution. This will r esu l t in charge separat ion in the condensate proper , ultimately to the ex t reme separation described in the body of of the paper . One may then anticipate additional observable effects in electron scat ter ing and muonic x r a y s . F i r m conclusions (from the mod- e l ) would, however, requ i re a self-consistent t reatment of the nuclear and electromagnetic ef - fec t s .

It thus i s c lear ly worthwhile to t r y to understand f r o m f i r s t pr inciples if nuclei a r e condensed in the sense of the present discussion. It may be worthwhile quite independently to investigate phenomenologically the experimental consequences of such a condensate. Finally, one may t r y to understand if super-sl.lperheavy nuclei of the type envisaged in this paper could indeed exis t .

*Work supported in part by the U . S. Atomic Energy Commission.

III . Snyder and J. Weinberg, Phys. Rev. 57, 307 (1940). 2 ~ . I. Schiff, H. Snyder, and J. Vbreinberg, Phys. Rev. 57,

315 (1940). 3 ~ . B. Migdal, Zh. Eksp. Teor. Fiz. 61, 2209 (1971) [Sov.

Phys.-JETP g, 1184 (197211. ' ~ f t e r the present work was essentially completed, one

of the authors (A.K.) was reminded by K. Johnson that his Barvard Ph.D. thesis [I954 (unpublished)] dealt with the problem of the quantization of the Iclein-Gordon equation in an external field.

5 ~ o r two recent discussions and further references, see A. Klein and J. Rafelski, in Fu~zdame?ztal Theories

i?z Phys ics , edited by S . L. Mintz, L. Mittag, and S. M. Widmayer (Plenum, New Yorli, 1974); and J . Rafelski and A. Iclein, in proceedings of the Interna- tional Conference on Reactions between Complex Nuclei, Nashville, Tennessee, 1974 (to be published). The first art icle also contains a preliminary account of the present work.

'H. 17eshbach and I:. Villars, Rev. Mod. Phys. 30, 24 (1958). Our representation differs by a rotation in 7

space from the one given in this papzr. Notice that the "Hamiltonian" 3C and the vector d a r e dimensional hybrids.

7 ~ . B. Migdal, Phys. Rev. Lett. 31, 257 (1973); Phys. Lett. 455 448 (1973); Nucl. Phys. m, 421 (1973).

11 - B O S E C O N D E N S A T I O N I N S U P E R C R I T I C A L E X T E R N A L F I E L D S 311

Barshay and G. E. Brown, Phys. Lett. 9, 107 account of a possibly more realist ic attempt to des- (1973). cribe a neutral condensate came to our attention just a s

his model has so far no known basis in fact, but i s this paper was finished: A. B. Migdal, ii. A. Kirichenko, introduced only to i l lustrate the kind of effects to be and G . A. Sorokin, Phys. Lett. g, 411 (1974). looked for when a condensate appears. A preliminary

bose condensation in supercritical external fields

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