Semiclassical theory of nuclear fission

Nuclear Physics A367 (198I ) 269-3 I2 0 North-Holland Publishmg Company

SEMICLASSICAL

THEORY

OF NUCLEAR

FISSION

H. REINHARDT

Received 14 January 1980 (Revised 13 October 1980) Abstract: The semtclasstcal approach recently developed for large-amplitude collective oscillations is extended to tunnehng phenomena in order to obtain a mtcroscoptc description of nuclear f&ton. For thts purpose the imaglna~-time partition function IS evaluated via its euclidean path tntegral representation. The stationary phase condrtton leads to the imagina~-time-de~ndent Hartree-Fock equation. Its classical solution gtves a dynamical picture of the fission process. The leading-order quantum fluctuations yield a finite lifetime of the ground state against &ton.

I. Intr~u~tion

Recently we proposed a semiclassical treatment of large-amplitude collective motion ’ - 3). The central idea of the semiclassical approach is based on the observation that for a collective motion with a large amplitude the action changes substantially during the motion, i.e. the changes of the action are large in comparison with h. We therefore expect a semiclassical description to be appropriate for such a motion. In ref. “) the semiclassical approach has been developed for quantized bound states, i.e. for a periodic collective motion. The semiclassical treatment of some schematic models has given very encouraging results ’ 3‘). Starting from an exact path integral representation for the time evolution operator we have shown in ref. 4, that the time-dependent Hartree-Fock (TDHF) equation can be considered to be the classical equation of motion of Mary-fe~ion systems in the sense that this equation follows from a stationary phase condition. The essential point, which is new compared to the standard formulation of the TDHF approximation, is that the path integral approach, which corresponds to the full quantum theory, allows one to go beyond the TDHF approximation. In the path integral approach corrections to the TDHF approximation appear as quantum ~uctuations around the classical solution of the TDHF equation. In the semiclassical approximation one retains only the leading-order fluctuations. From the analysis of these fluctuations we derived in ref. 3, a semiclassical quantization prescription for periodic solutions of the TDHF equation; which are capable of describing largeamplitude collective oscillations. A path integral formulation of the TDHF theory similar to the one given in refs. 4, 3, has been recently proposed in ref. r6), where the application of the TDHF theory to large-amplitude collective motion has been 269

270

H. Reinhardt 1 Semiclassical theory

also discussed. However, there the considerations have been restricted to the pure classical limit (TDHF) neglecting the quantum fluctuations. Furthermore the semiclassical quantization rule of periodic TDHF solution (with the omission of the quantum fluctuations) has also been derived in ref. I*) in a rather different approach. Analogously to the semiclassical treatment of quantized bound states we develop in the present paper the semiclassical approach for tunnelling phenomena in manyfermion systems like spontaneous fission of a nucleus. Nuclear fission is a typical largeamplitude collective motion and hence a semiclassical description should be appropriate. The tunnelling phenomena (barrier penetration) are quantum effects and could not be described by the TDHF theory unless we knew how to go beyond the pure classical approximation. In fact, as we shall see later, it is the quantum fluctuations which give rise to the finite lifetimes of fissioning states. Following the standard method in field theory we treat the tunnelling phenomena in many-body systems by studying the imaginary-time partition function ‘). Formally the evaluation of the corresponding euclidean functional integral (the integration variable is the density field) is done analogously to the real time case treated in ref. 3, [hereafter referred to as I ‘1. Especially, in the “classical” limit the euclidean fictional integral is dominated by the solutions of the imaginarytime-de~ndent Hartree-Fock (ITDHF) equation. However, for imaginary times some new features emerge which require special attention. For an understanding of the present paper it is therefore unavoidable to repeat some of the formal developments of I, here for imaginary times. To make the paper self-contained we give in sects. 2-4 the basic ingredients of the functional integral formulation of the THDF theory for imaginary times. For simplicity we shall restrict ourselves to the derivation of the imaginary-time-dependent Hartree equation. The exchange matrix elements can be easily included in the same way as in I. Because of habit we shall refer to the time-dependent Hartree approximation as the TDHF approximation, too. In sect. 2 we formulate a collective (euclidean) quantum field theory which is convenient for the description of tunnelling phenomena in many-fe~ion systems. In sect. 3 we evaluate the corresponding euclidean action. In sect. 4 we consider the classical limit to the collective quantum field theory. This leads to the (imaginary) TDHF approximation. In sect. 5 we consider quantum fluctuations around the classical ITDHF solution. The fluctuations of the density field are treated in second order in This approximation allows us to find a a kind of “quasiboson approximation”. closed expression for the integral over the fluctuations, which is derived in sect. 6. In sect. 7 the full semiclassical evaluation of the imaginary-time partition function is performed. We shall find there an expression for the lifetime of spontaneously fissioning nuclei. A short summary and some concluding remarks are given in sect. 8. Some mathematical proofs are presented in appendices.

t References to the equations and sections of this paper are prefixed by I.

H. Remhardt / Semiclassical theory

271

2. The collective quantum field theory in euclidean space Starting from the path integral representation of the imaginary-time grand partition function we formulate in the following a collective quantum theory in the non-local density field, which in the classical limit reduces to the TDHF approximation. Consider the (imaginary-time) grand partition function

(2.1) where H is the hamiltonian

of the many-fermion

H = xe,,a,+a,++

system

1 a,+a,V,,,gaaS+a,.

(2.2)

Here the cay are the matrix elements of a one-body operator (e.g. the kinetic energy of a nucleon) and V denotes the matrix elements of the two-body interaction Vay,pd = s dxdy&(x)@,dyMx,

Y)cP,(x)cP,(Y).

Furthermore, the E,,(N) are the exact energies of a system of N particles and the index n labels the different eigenstates corresponding to the same N. Thus 2: is the partition function of a system with a fixed particle number N. If T- ’ is identified with the temperature and l/h replaced by Boltzmann’s constant, .Yr becomes the ordinary thermodynamic partition function. Once ST has been evaluated all thermodynamic properties (as e.g. pressure, entropy etc.) can be calculated as function of the temperature T- ’ and of the volume of the system. For the evaluation of LP’~it is important to note that it can be represented by the following path integral 6, 2 ‘. = A“, s Da*Da exp (-Y&z,

a*)/h).

(2.4)

Here Y,(a, a*) is the euclidean action which is defined as ( - i) times the continuation of the ordinary (real time) action to imaginary times t = - i~(z - real) +. Hence the euclidean action is given by ++

s s T/2

Y,(a,

a*) =

-T/Z

d~[a,*)W,$4

+ e&a,(z) + iaZ(z)a,(3K/,,, paajX+ckGl

7’12

=

d~[4(W,a,(~) + W44, a*(@)],

(2.5)

-T/2

t Formally, the path integral representation (2.4) for TT can be obtained from the corresponding one of the real-time case, eq. (12.3), by a so-called “Wick rotation” 7 = it. ” In the following we agree that repeated indices are summed over. If confusion is excluded we shall not write down these indices.

272

H. Remhardt 1 Semiclassical

theor)

where a(z), u*(z) are the continuations of the fermion field operators defined in the Heisenberg picture at imaginary times t = - ir &Jt = -jr)

a(r), b+(t)

= eHrlh& eeHrlfi = a,(r), (2.6)

c;,‘(t = --it) = eH7!*Cia+ eeHrifi = a,*(r)+

The so-defined involution ‘) “*” differs from the usual one {denoted by ‘*-I-“) by the additionally included time reversal : a*(7) = [u(z)]”

= [a( - 7)]+.

(2.7)

Under the functional integral the a are no longer operators but classical, though anticommuting, (Grassmann) variables. Because of the trace operation in (2.1) they have to satisfy the antiperiodic boundary condition 5, a($T) = -a(-+T).

(2.8)

The no~ali~tion constant of the functional integral, N,, is determined by comparison with free field theory (see sect. 3). We proceed now as in I and linearize the two-body interaction by means of a collective Bose field + [cf. also ref. “)I exp [-- 4

dza*aVa*a] = (Ret V)*

d&VP-pVa*u)

s Inserting this into eq. (2.4) we find that at the stationary points the collective field takes the value P&I

(2.9)

= @N,(r).

Hence it satisfies periodic boundary conditions p(+T) = ~(-47’) the symmetry relation

and, furthermore, (2.10)

Here, for the commuting (Bose) field p, according to the definition (2.7), the involution * means time reversal, as well as the usually included complex conjugation (denoted by “ - “) : (2.11) The operation “*” defined here will be used below for any c-number i.e. we define

function

f{7),

f*(r)

= f(

-7)v

(2.12)

+ In this paper we use the same conventions as in I. The symbol “det” means a determinant of a matrix in configuration space only, while “Det” refers to a matrix in the configuration-time (functional) space. The trace symbols “tr” and “Tr” have analogous meanings.

173

H. ~einhardt 1 ~emtc~ass~caltheory

Let us now consider the set of the components of the collective lield p,,(r) as a quadratic (time-dependent) matrix a(~) with elements (&z))~.,. = p,,(r). From eqs. (2.10) and (2.11) it follows that this matrix is not hermitean in the usual sense (as in the real time case) but satisfies (2.13)

P,*(r) = &J-r).

In agreement with the above-defined involution for commuting (Bose) lields [see eqs. (2.1 l), (2.12)] we define the adjoint operation “+ ” for an imaginary-timedependent matrix A(z) with elements (&(7),, = Mxy(7), which are c-numbers, by (2.14)

(~+(~)~~ = M;@(t) = SQ( -z).

Then with respect to the so-defined adjoint operation ‘*+” the density field matrix P(T) is self-adjoint: (2.15)

p+(z) = b(T).

What we observe here is a genera1 feature of the imaginary-time approach: Most of the relations derived in I remain valid for imaginary times t = -ir if we complement the complex conjugation by time reversal (z -+ -z). After the linearization of the two-body interaction has been performed the exponent in the path integral is quadratic in the fermion variables. Integrating out the fermion variables we are left with a theory in the collective field p. Repeating the calculations of I step by step but for imaginary times we get

27’ = /Ir,(Det V)a JDp exp(-,SP,[p]/h).

(2.16)

Here cp,[p] defines the euclidean collective action

YJPI = The quantity

jd~ C:PYp+htr(log(-Y-‘[p]))(z,

T+o>].

(2.17)

9?[p] is the euclidean s.p. Green function defined by [cf. eq. (1.213)] 9 - ‘CPX? 0 = c- ha,- o(m4?

a

(2.18)

where

431 = e + 4dr>l,

%ycPl = vm,. &?&)~

(2.19)

is the mean field hamiltonian. Note, like p, the mean-tield hamiltonian h[p] is here not hermitean in the usual sense but self-adjoint with respect to the adjoint operation defined by eq. (2.14) :

CJml

= q%wl

= &Ed-211 = ~&WI.

Eqs. (2.16) and (2.17) define a collective (euclidean) quantum field theory which is completely equivalent to the starting fermion theory defined by eqs. (2.4) and (2.5).

274

If. Re~nh~~dt1 ~erni~l~s~~l theory

However, the Bose field theory is more convenient for the study of collective tunnelling phenomena although it may seem that the collective field theory is even disadvantageous due to complicated structure of the action Y&1(2.14). In the next section we show how the collective action can easily be evaluated. 3. Evaluation of the euclidean collective action In what follows we shall derive a more explicit expression for the euclidean collective action 9&] (2.17), which is directly accessible to numerical calculations. Since the evaluation of the euclidean action proceeds along the same line as the evaluation of the ordinary (r~l-time) action we quote only the main results without detailed derivation. What preludes a direct numerical evaluation of the collective action (p&p] (2.17) is the term tr log (-$ -‘[PI), which is a non-local functional of the collective field. For its evaluation we need an explicit representation of the Green function SCp]. For this purpose let us consider the corresponding (imaginary) time-dependent Schriidinger equation, - Ad,I&(r)) =

h&a I%(~))-

(3.1)

Dhe last equation is the continuation of eq. (1.3.1) to imaginary times 1 = -ir.] For convenience we define the bracket notation in such a way that the bra-vectors are the time-reversed states of the ket-vectors. For instance, in the coordinate representation we have then MY,(r)> = &OYr)~ (g,(r)Ix)

= gXx,r)

(3.2)

= &(x1-r)*

The formal solution to eq. (3.1) is

I&W) = WM

-m h[p(r’)]dr’ r/2

where /g,( - $1”)) are some initial more, U(r) is the time-evolution whereas the arrow indicates that In appendix A we prove that the we can choose the normalization

1

conditions which will be specified later. Furtheroperator and F means imaginary-time ordering the time z’ is growing from the right to the left. quantity (g,(z)lg,(r)) is independent of r. Hence

(g,(r) Is,W

= 4,.

(3.4)

Since the collective field satisfies periodic boundary condition ~($7’) = p( --jr) the functions Ig,(r)) may be chosen to obey the following representation g,(z) = e-evr’“f(r),

(3.5)

H. Reinhardt

/ Semclassieal

where the E, are some (“quasi”) energies and thef,(r)

f,W

= L( -

215

theory

are periodic functions

+n.

(3.6)

Because of eq. (3.1) the f, satisfy the eigenvalue equation

(-fi~,-Mmlx(a

= --E,K(9).

(3.7)

The initial functions g,( - +T) can be consider as eigenfunctions of the time-evolution operator U(r = $T) since, by virtue of eqs. (3.3) and (3.6), they satisfy ls,(3T)> = U(+Ulg,(-+T))

= e-a~‘hlgy(-ST))r

ci, =

E,T.

The CC,are the counterparts of the Floquet indices defined in sect. 13. The Green function, 3 [p](r, 7’) can be expressed by the functions (cf. subsect. I. 3.2)

g[P](?T’) =

;~~gv(T))[-~(T-r.)(l-n~)+t(T'-T)n~](gv(T)~.

(3.8) Ig,(z)) as

(3.9)

Y

The generaliied occupation numbers 11: are determined by the boundary condition on C!Y[p](z,T'). Like the thermodynamic Green function 9), 3[p](z, T') satisfies antiperiodic boundary conditions + g[Cp](+, T') = -9l-p](-+T,

T').

(3.10)

With the properties of the functions g,(r), eqs. (3.8) (3.6), the antiperiodic boundary condition (3.10) implies that n:

=

1

(3.11)

exp (E,T/~)+ 1’

Thus the H,’ agree with the finite temperature (T- ‘) fermion occupation numbers except that the chemical potential p does not appear. [Since we are dealing here with systems having a fixed particle number N there is no need to include the constraining term -@, which ensures conservation of particle number on average, into the definition of the grand partition function. The reason why we have started from the grand partition function, ST, is that for 2’ the path integral representation (2.4) with the simple antiperiodic boundary condition (2.8) exists]. With the representation of the s.p. Green function Y defined by eqs. (3.9) and (3.11) at hand, and using the result of appendix A, one shows the following relation [cf. the proof of eq. (13.42)] : JV, exp [Tr log 3 - ’ [p]] = 1 exp [ - 1 n,o$r].

’ The antiperiodlc [see e.g. ref. 13)].

boundary

conditions

are a direct

consequence

of the trace operation

(3.12)

in eq. (2.1)

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H. Reinhardt / Semiclassical theory

Here the n, are fermion occupation numbers which take the values zero and one for empty (particle) and occupied (hole) states, respectively, and the summation on the r.h.s. of eq. (3.12) is performed over all different sets {n,>. Inserting the last relation into eq. (2.16) the grand partition function S!’’ reduces to LP=CZ;, n

(3.13)

ZE = (Det V)-k Dp c e-sEtP1’fi,

s where the summation relation

(3.14)

in,lh‘

in eq. (3.14) is restricted to such sets (n,>, which fulfil the

iv = &,,

(3.15)

with N the total number of particles. Hence 2: is the partition function of a system with a definite total number of particles. The euclidean collective action in eq. (3.14) is given by

Hereafter we shall always deal with systems of definite particle number N. We find it therefore convenient to introduce a new s.p. Green function G[p], which corresponds to a definite distribution of the N fermions, i.e. to a definite set of occupation numbers, (n,,\ N. Its inverse is defined as above, i.e. G-

‘blk

0 = (- f+--&W(~, G

(3.17)

but its boundary condition differs from the one of ‘22:

(3.18) ~C/g1~))[-e(~--~‘~l-n~)+~(r’-i)n,l( V

Gblfv’) =

For subsequent considerations it is convenient to express the second term in S,[p] (3.19 by the new Green function G[p]. Using the results of appendix A one easily proves the following relation [the proof proceeds analogously to the one of eq. (14.3)] : T/2 np, = (3.19) d&r[log (- G- ‘[p]) - log (h&6(z, z’))](r, r + 0). -1 s -T/Z The collective action

S,[P] = -

S&I]

(3.16) can then be written as

f J

dr[+Vp+

htr(log (-G-

‘[p]) - log (hQ(r, r’))].

(3.20)

Note that action S&Y] corresponding to a definite set of occupation numbers, (n,>,, has the same structure as the action .!Y[p] appearing in the grand partition

277

H. Reinhardi / Semiclassical theor?

function (2.17) except that g[p] is replaced by G[p]. [The term trlog (rZ?r(z, T’)), which arises from the normalization constant _N, [see eq. (3.12)], is an irrelevant (infinite) constant which is needed, however, to make the action finite.] The outcome of the present section can be summarized as follows: The grand partition function has been reduced to a sum of partition functions corresponding to definite particle numbers. This has brought the collective action to a form that can easily be evaluated qnce the eigenvalue problem (3.7) has been solved. While eq. (3.16) is convenient for the numerical evaluation of S&l, the form (3.20) is advantageous in the theoretical investigations of the collective properties of a system with a definite total number of particles. 4. The classical limit of the euclidean collective quantum field theory 4.1. THE IMAGINARY-TIME-DEPENDENT CAL SOLUTION

HARTREE-FOCK

EQUATION

AND ITS CLASSI-

The collective action S&I (3.20) is a complicated (non-local) functional in the collective field p and the path integral (3.14) can be evaluated only approximately. The fact that during a large-amplitude motion, like the fission process, the collective action S&l changes substantially suggests an evaluation of the path integral by the method of stationary phases. The stationary phase points (defined by SS,[p]/Sp = 0) occur for such Y that satisfy the relation P,,(T) =

fiG,,Cd(~,T+ 0).

(4.1)

Using the definition of the Green function G[p] (3.18) this self-consistent condition becomes P,,(T) = C(%(~)> Y

~V(&(~)lY>~

(4.2)

that is to say at the stationary points the collective field coincides with the imaginarytime density matrix. Therefore p is called the density field. This interpretation is also suggested by eq. (2.9). It is worth emphasizing that the density matrix, in particular, like the field p(r), in general, is here not hermitean in the usual sense but selfadjoint [see eqs. (2.14), (2.15)]. In the same way as in I one derives from the stationary phase condition (4.1) the ITDHF equation - %%r)

=

Ckwl~ P(m

(4.3)

which therefore may be interpreted as a classical equation of motion. This equation is also immediately obtained from the ordinary TDHF equation (14.8) by continuing the real time t to pure imaginary values t = - ir. Like the ordinary TDHF equation, eq. (4.3) has, of course, the trivial static Hartree-Fock-solution p = pO. In addition, the ITDHF equation may admit non-

278

H. Reinhardt

/ Semiclassical

theory

trivial time-dependent solutions ’ b(r). As is well known the solutions of the classical equation of motion continued to pure imaginary times describe a tunnelling motion [barrier penetration 5)]. For a system which shows spontaneous fission the static Hartree-Fock solution may be classically stable but becomes unstable if quantum tunnelling is switched on. The ITDHF equation must then have non-trivial solutions of period T, which for T + 00 describe the time-evolution of the system from the static Hartree-Fock ground state at r = - $T to the scission point at r = 0 where the system is bounced off. At time r = 4 X it reaches again the static Hartree-Fock solution. Such solutions are called “bounces” 5). As is well known bounces are well localized objects (see fig. la). In the present case this means that the bounce b(z) differs essentially from the static Hartree-Fock solution p” only in a small time interval around z = 0, the center of the bounce. T

T small

large

Fig. la. Schematrc representation of a solution to the ITDHF equation, P(T). For T + co this solution (bounce) becomes a well localized object.

(bl Fig. lb. Schematic representation

of a string of n = 4 widely separated r, (i = 1,. ., 4).

bounces with centers at

t Throughout the paper quantities refering to a stationary point, in general, and to a true TDHF trajectory, in particular, are indicated by a tilde.

H. Reinhardt / Semiclassical theor)

279

To illustrate the qualitative behaviour of the bounce p(z) let us consider it for large r where j?(r) approaches the static Hartree-Fock solution p”: G(z) + P”

for 171--+$T+

Co.

(4.4)

For small amplitudes Q(r) = p(r) - p”, i.e. near to the static Hartree-Fock minimum, we can linearize the ITDHF equation (4.3) with respect to e(r). Using the fact that p” itself solves the ITDHF equation this yields -h%@(r) =

C&a @I+bm

PI = ~CPOl~,

(4.3

This equation has 2Np,, linear independent solutions (N,, is the number of particle and hole states included) : (pi(r) = eC”Z’XE, where xi, xi’ are the ordinary equation

q:+(r)

RPA eigenvectors,

= e”z*X,“+,

(4.6)

which satisfy the (static) RPA

(4.7) and the ok are the corresponding RPA frequencies. For large 1rJ the amplitude $5(z) can be represented as a linear combination of the functions (p;(r), q:‘(r). Furthermore, because of the boundary condition (4.4) the amplitude G(r) = p(r)-@ has to vanish for 1~1-+ +T + co.From this, it follows; by using the symmetry relation (2.1 S), that @5(r)obeys the following asymptotic representation

The expansion coefficients qk cannot be obtained by investigating only the asymptotic limit. This is because the RPA equation (4.7) is linear, and consequently does not determine the amplitudes qk themselves but merely assumes them to be small. They can only be determined from the full solution of the ITDHF equation, which is non-linear and, hence, the absolute magnitude of its solution is uniquely determined. As we can see from eq. (4.8) for 171 + +T + coonly the lowest lying (“collective”) RPA mode (k = 0) contributes to G(7). Furthermore we can then neglect in eq. (4.8) the terms involving the factor exp (-o:T). Then G(z) becomes

(4.9) This asymptotic form of 4,(z) = p(7) - p” shows : (i) The fission mode described by the bounce solution of the ITDHF equation 3(z), starts from the static Hartree-Fock minimum p” at r + - co in the direction of the lowest lying RPA mode 1:: o as exp ( - ozlrl).

280

H. Remhardt

/ Semiclassical

theor)

(ii) Due to this exponential fall off [4(z) - e -“:“‘I the bounce solution is a well-localized object having a size of the order l/o:. The importance of the bounce solution G(r) is given by the fact that for a system which decays via spontaneous fission it gives the dominant contribution to the functional integral (3.14) in the limit h -+ 0. Note furthermore with a suitable phase convention, in which the s.p. wave functions ICC),I/?), . . . are real, the solutions of the ITDHF equation, p(r), are also real. 4.2. THE EUCLIDEAN

The expression general and holds interesting case p the self-consistent SE@] can be cast

ACTION

ALONG

THE CLASSICAL

TRAJECTORY

(3.16) derived for the euclidean collective action S&l is quite for an arbitrary periodic p(r). Let us now consider the potentially = p, where 6 denotes a solution to the ITDHF equation. Using condition (4.2) and the eigenvalue equation (3.7) the action into the form

MPI =

r/2 s -T/2

dz 1 n,( %)lW %D + ~%diY~)l9 > ( Y

(4.10)

where (4.11) is the Hartree-Fock energy which is conserved for p = 6 being a solution of the ITDHF equation. The expression (4.10) obtained above for the euclidean collective action suggests the following interpretation: If Ifly( and @,(r)l are considered as conjugated coordinates + the first term in eq. (4.10) corresponds to the term SpQ dr in the hamilton form of the classical action of a mechanical system. Furthermore, the Hartree-Fock energy E&(r)] may be considered as the “classical” Hamilton function of the many-fermion system. Note the actionS&] (4.10) obtained above in the path integral approach is different from the one which follows from the customary time-dependent variation principle l O) continued to pure imaginary times t = -ir. While the former is invariant with respect to space-independent but time-dependent gauge transformations the latter is gauge dependent (for details we refer the reader to sect. 14). Let us now consider the action for the bounce p(t)( T + co), which for large (r) approaches the static Hartree-Fock solution p” [see eq. (4.9)], i.e. p(c)-p”

Therefore,

a

ewuglrl,

because of the conservation

Iz( -+

$T+

of EuJP(r)],

4-&(~)1 = 4.&“l. t A similar point of view has been adopted

in ref. lo).

co.

(4.12)

we have (4.13)

H. Reinhard

/ Semiclassrcal theory

Let us now apply the same considerations

ww7x~D

( - h?, -

281

to eq. (3.7):

= - qvw).

For large 1~1values &j(z)] becomes independent of r and equal to the static HartreeFock hamiltonian h[p”]. From this and from the periodic boundary conditions for c(z) and&t), it follows that M&r))

+ 0

(4.14)

for (r( + cc.

Hence the functions f;,(z = + iv?, T + cc have to diagonalize the hamiltonian h[p”]. But this identifies the functions f,( + 4 7’) uniquely to be the static HF s.p. functionsx : _Q+m

-.c

for T+

0~.

(4.15)

Consequently, for T + cc the Z, are the ordinary Hartree-Fock s.p. energies. Eq. (4.15) is also in agreement with eq. (4.12) since the s.p. Green function obeys the representation (3.18) and by virtue of eqs. (4.2) and (3.5) we have (4.16) With the%symptotic

limit (4.12) of p(z) from the last relation it follows that TV(r)-jt

cc e-@l’l

for (71--f +T+

co.

(4.17)

Having evaluated the classical action .S&], in the next section we consider the quantum fluctuations around the classical solution b(t).

5. Quantum fluctuations around the bounce As mentioned above, during the fission process the collective action changes substantially and we expect that the variations in S&l are large in comparison with the magnitude of h. Hence it is sufficient to consider small fluctuations around the classical solution i and to evaluate the path integral in the semiclassical approximation. Let us suppose we have found the classical solution to the ITDHF equation, p(r), We can then expand the collective action SJp] around p = 6. To second order in the fluctuations cp = p -8 [i.e. in the stationary phase approximation (SPA)] we find for the partition function 2;

=

1 GCPlz,‘Cdl, ‘(%)N

(5.1)

282 where

H. Re~hardt / ~emici~s~al

theory

t

GGI = ev ( - ~,CPl/~)

(5.2)

is the partition function in the classical (TDHF) approximation .Z$[fi] = (Det V)l- j

Dqexp

[Aj

and

dzdz’cp(z)A- ’ [fi](z, z’)cp(z’)]

(5.3)

is the integral over the fluctuations. Restricting to second order implies that the fluctuations are small. The second variation A-‘[7J(z, z’) follows from eq. (3.20) to be

= Vd(z, 6) - vT[jq(z, r’)V,

(5.4)

where L,,,rsCPl(r> r’) = ~G,,CPl(r,

fFaolCPl(~‘, 9.

(5.5)

The propagator function r@J represents the density correlation function of a system of non-interacting particles in an “external” field !&?I. Furthermore, the quantity A@] appears here as the propagator of the ~uct~tions. However, in subsect. 5.2 we shall find that A@] is not the true propagator. This is because A[fi] contains a contact term (see sect. IS).

5.1. PECULIARITIES FLUCTUATIONS

IN THE EVALUATION

OF THE FUNCTIONAL

INTEGRAL

OVER THE

The standard way to evaluate the functional integral (5.3) is to expand the field variable V(Z) in terms of the eigenfunctions of (-A- ‘@J) W - A - ’ [P]h

O#@)

= ~~~~~~),

66)

s

which have to satisfy periodic boundary &Or)

conditions = #k(--$9

and rewrite then the functional integral as a multiple integral over the expansion coeffkients. Since the operator A-‘[p] is symmetric, A,,‘,,[Cl(r,

7’) = A&Jijl(r’r

r),

(5.7)

* Note: the normalization constant .X, has already been absorbed into the collective action S&p] [see eq. (3.12)). In the following we drop the summation over the sets in), since in the Limit T -+ a3 only the configuration corresponding to the Hartree-Fock ground state contributes.

383

H. Reinhardt / Semiclassical theor)

and real the eigenfunctions corresponding to different eigenvalues are orthogonal to each other. Hence with a suitable normalization we have (5.8) Furthermore, using the definition symmetry relation

of A-‘[fi]

eqs. (5.4) and (5.5) one proves the

from which it follows that the eigenfunctions of A- ‘[fi] are either self-adjoint (G(r) = &( r )) or antiself-adjoint (c#$(z) = -C&(Z)) or they come in pairs (&(r), c$$(T)) implying that the eigenvalues are doubly degenerate. Let us here suppose, for simplicity, that all eigenfunctions are self-adjoint. (The subsequent considerations can be analogously performed for the other cases.) We can then expand the field variable 9 as (5.10) cp(r) = CG”(T)Y ” where the expansion coefficients must be real since p(r) is self-adjoint ((p +(T) = q(z)), too. Inserting the expansion (5.10) into eq. (5.3) and using eqs. (5.6) and (5.8) the functional integral becomes an ordinary multiple integral Zi[p]

= (Det V)* nk J,+exp[-$&I.

If all eigenvalues pLkwere positive definite the integrations carried out and we would get ~‘,[fi] = (Det V)* n L k

(5.11) could be immediately

= (Det V)*(Det (- A- l[P]))-*.

(5.12)

hk

However, not all of the eigenvalues c(~are porisive and, consequently, the integral over the corresponding expansion coefficients diverges. But this means that the functional integral over the fluctuations is ill defined in the SPA. In the following two subsections we explain why such extraordinary eigenvalues must emerge, and remedy the functional integral Z&7]. 5.1.1. The zero mode problem. There is at least one zero eigenvalue of A - ‘[PI. This can be easily seen in the following way: Time translation invariance implies that if P(r) is a solution to the ITDHF equation then any P(r - T,,)with z. E [ -37; $T] is a solution, too. When expanding the collective action around a definite classical solution P(T - TV) we fix the position of the center of the bounce, zo, and, hence, violate time-translation invariance. This violation, like any symmetry breaking by the classical solution, gives rise to a zero eigenvalue of c?~S,/~~C~~.In the present case the corresponding eigenfunction is given by i (see sect. 15). Due to the zero eigenvalue of A-t the functional integral

H. Reinhardt / Semiclassrcal theory

284

over the fluctuations in the ordinary SPA, eq. (5.3), is ill-defined since Det( - A- ‘[jj]) = 0. To make it well defined we have to restore the broken symmetry. This makes the zero eigenvalue disappear from the functional integral Zi[fi]. The time-translation invariance can be easily restorted by summing over all stationary points p(r--rO) differing in the position of the center of the bounce, rO, (i.e. by integr a t ing over rO) so that no particular r0 is favoured. Following the collective coordinate method 8, we convert the integral over the expansion coefficient corresponding to the spurious mode, cO, into an integral over the position of the center of the bounce +. For an infinitesimally small change in the location of the center of the bounce, r0 the change in the fluctuation field cp(z) is dq=

-do=

-

dd(r - r,,) dz dr, = j(r-r,)dz,

(5.13)

0

On the other hand the change in cp induced by a small change in the expansion coefficient co of the (properly normalized) spurious mode 40(r) = JL

;

(5.14)

is dq = &,(r)dc,. The normalization factor ./lro will be determined (5.13) and (5.12) we get +,

=

\i

$dr,.

(5.15) in subsect. 5.2. Equating eqs.

(5.16)

Thus if we restore the time-translation invarianee by integrating over zo, the location of the center of the bounce, we must no longer integrate over the expansion coefficient of the spurious mode but we have to supply a factor ,/(.~,/27rti). In this way the zero eigenvalue of A-’ disappears from the evaluation of the functional integral. Hence assuming that all of the remaining eigenvalues are positive we obtain (5.17) where the reduced determinant De&J-A-‘) is defined with the zero eigenvalue omitted (see subsect. 5.1.3.). Note that the quantity De&(-A‘[p]) is independent of z. (see sect. 15). 5.1.2. The negative eigenvalue. A second difficulty in evaluating the functional + Here we use the collective coordmate method m a somewhat different way than in I. We hope that this different point of view will make clearer the content of this method. The result obtained is of course the same as in I.

H. Reinhard!

/ Semwlasslcal

theor)

285

integral over the fluctuations in the SPA arises from the fact that a classical solution of the bounce type gives rise to a negative eigenvalue of the second variation of the euclidean action [see ref. “)I. Consequently the integral over the corresponding expansion coefficient is hopelessly divergent. To make the functional integral convergent one has to deform the corresponding contour of integration following the method of steepest descent. This treatment of the functional integral, which is rather lengthy, is presented in appendix B. Here we only quote the result, which can be summarized as follows: The functional integral over the fluctuations becomes pure imaginary. In addition a factor 4 arises. Taking into account these modifications we find from eq. (5.17) for the fluctuation part of the partition function (5.18) What remains to be done is the evaluation of Detred( - A-‘). This task requires finding the eigenvalues of (- A-‘) [see eqs. (5.6), (5.12)]. In principle, this could be done numerically. The result would not, hoever, be very instructive. Therefore we prefer to present in the following subsection an approximate evaluation of Det( - A- ‘), which exhibits the underlying physics of the semiclassical approximation for many-fermion systems better than the direct evaluation of Det( - A- ‘[p]). In addition this approximation leads to computational simplifications. 5.1.3. The “quasiboson approximation”. Having tackled the problems caused by the presence of a zero and a negative eigenvalue the functional integral Z,‘[p] (5.3) is now well defined and the integration over the fluctuation field cp could be done, in principle straightforwardly, yielding the result given by eq. (5.18). However, the straightforward evaluation of De&J-A - ‘[PI) would not reproduce in the static limit (6 = p”) the RPA, which is supposed to give a sophisticated description of small amplitude oscillations. In I it has been shown how the functional integral over the fluctuations is approximated in such a way that in the static limit the RPA result is recovered. The proposed approximation is obtained in the following way : We rewrite A-’ as A-‘[PI

= VT[/F]A,‘[fi],

(5.19)

= r-‘CD]-I’

(5.20)

where the matrix A,‘[/71

has also a zero eigenvalue with the same eigenfunction j as A- ‘[PI (for the proof see I). The point is that the quantity AJP], which is the collective part of A-‘[PI (see sect. 15), rather than A- ‘[p] itself, represents the true propagator of the fluctuation tield cp (see subsect. 5.2). Therefore instead of eliminating the zero eigenvalue of A- ‘[PI we exclude it from A, ‘CD] and define the reduced, non-zero determinant

N. Reinhardl / Sem~riassiral theory

2%

of(-A-‘@])

appearing in eq. (5.18) as IDet,,,(-AA-l[~I)I

= Det(V)IDet(-r(fi])l

lDet’(--A;‘@])l.

(5.21)

where the prime indicates that the zero eigenvalue of A; ‘[$I has to be omitted. With the factorization (5.12) the fluctuation part of the partition function Zi[$J (5.181 becomes (5.22)

Besides computational simpli~~atio~s (see below) the approximation presented above is suggested on physical grounds: (i) As we shall see later this approximation defines bosonic normal modes of the small fluctuations, which describe (intrinsic) excitations orthogonal to the classical path and which in the static limit become the usual RPA phonons. Therefore, we call this treatment of the fluctuations the “quasiboson approximation”. {ii) Linearization of the (I)TDHF equation (4.3) with respect to the flctuations q = p-p yields a so-called linearized stability equation [eqs. (I6 15), (5.30)] which is given by A; ‘[Flu, = 0 instead of A - ‘[p]tp = 0. For a more detailed justification of this approximation, eq. (5.21), we refer the reader to I. In I the ~u~tuation part of the real-time partition function, Zi[fi]$ has been evaluated. In the same way we could evaluate the functional determinants appearing in Z,T[p] for imaginary times. Indeed as we shall see later the value of Z,‘[c] for imaginary times, except for the modifications which are originated by the negative eigenvalue (i.e. the factor ii, see subsect. 5.1.2),can be obtained directly from I by a Wick rotation (it + z), In subsect. 5.3 we prefer, however. to give adifferent evaluation of the functional determinants than in I because this will provide us with some useful by-products which we shall need later. But before doing this it is necessary to study the quantum ~~ctuations around the bounce solution in more detail. This is done in the next subsection. Before proceeding with the study of the ~u~tuations it is convenient to switch to the periodic s.p. basis v&r)] defined by eq. (3.7) with p = Z;,and to adopt the usual particle-hole picture for the time-dependent states&,(r), i.e. we denote empty (n, = 0) and occupied (n, = 1) statesf,(r) as particle (v = p) and hole (v = h) states. In the basis IT,(z)} the inverse propagator r -‘[PI (5.5) becomes r - ‘[fi](2, r’) = - [b- ‘h&-t E]6(r, z’),

(5.23)

where the matrices b and E are defined by (5.24) (5.25)

187

H. Remhardt / Semiclassical theor)

Hence the quantity A, ’ [TJ (5.20) is given by A,‘[Yj(z,

7’) =

-h-.‘(hr:,+LLld](~))s(z,

(5.26)

7’).

where the matrix f.[~] is the same as in eq. (4.5) and L[I, = jj] is given in the periodic s.p. basis r,,(r)) by L[P](r) Note in the basis f,(z); are defined by

(5.27)

= hE+bV[p](z).

the matrix elements of the two-body interaction,

w$,~..,(~) = ~p(eo cT,WlP>Yw?,(YET,(~)> @IL:,(~D~

which (5.28)

depend on the classical solution j?(t) via y”(z) and, hence, are periodic, too. They satisfy the symmetry relation ‘ZV.Kj.CPlG) = ‘NV, h_j.Ci%‘1 =

‘vp,j~~Ci2(~).

(5.29)

In concluding this subsection we would like to mention the following: While the matrix A ,,,Jfi](r) (5.4) is deIined in the full space of the two-particle configurations {(TV) = (pp), (ph), (hp), (hh)], the matrix (AJYj),,.,, has non-vanishing matrix elements only in the particle-hole subspace [(pv) = (ph), (hp)] and, hence, is detined only in this subspace (see subsect. 15.3). Thus the quasiboson approximation implies restriction to the particle-hole part of the fluctuations ((Pi,,, (~~~(7)). Notice further that, unlike A-‘[P](z, z’), the quantity A; ‘[;](7, 7’) (5.26) is local in time in the sense that it contains the d-function 47, z’). But this simplifies essentially the evaluation of the functional integral Z&S] provided the quasiboson approximation is used (see sect. I6). 5.2. THE EIGENMODES

OF THE FLUCTUATIONS

For the evaluation of the fluctuation part of the partition function Z,T[?J in the quasiboson approximation it is conv~ient to study at the beginning small fluctuations around the large-amplitude fission motion, which is described by the ITDHF solution p(7). To this aim we put p = fi-tcp into the classical equation of motion (4.3) and linearize it with respect to the fluctuations cp assuming that they are small [cf. also eq. (4.5) and subsect. 1.6.2.].‘,Transforming the result into the periodic s.p. basis (irYfwe obtain the equation of motion of the fluctuations rp to be

-(~,~+mm

= 0,

(5.30)

where the matrix L[i;] has been defined by eq. (5.27). A glance at eq. (5.26) shows that the equation of motion of the fluctuations corresponds to T/2 A~‘[~](7,7’)~(7’)d7’

s -T/Z

= -b-‘(ha,+L[Zil(z))cp(z)

=

0.

(5.31)

H. Rernhardt f Semiclassical theory

288

Hence A,@], which is a natural generalization of the density correlation function in the RPA [see eq. (5.20)], represents the true propagator of the fluctuations. This fact motivates the factorization (5.21) (quasiboson approximation). Since L[fi](r) is periodic the solutions to eq. (5.30) obey the representation (am

(5.32)

= e-“%(r),

where the olr are some frequencies and the xk(r) are periodic functions, which by virtue of eq. (5.30) satisfy the following eigenvalue equation

-@&+~C3Ikd~) = --twain), x&T) = Xkf-:T).

(5.33)

Because of eq. (5.32) the functions Y)~(z)satisfy the relation (P&T) = e-“%p,( -iv,

(5.34)

vk = w,T.

The V~are the analogues of the stability angles (see sect. 16). Using the symmetry properties of V and b one shows that if ~~(7) is a solution to eq. (5.30) then p:(z), with (5.35) is also a solution, and satisfies the relation (5.36) In the same way as in I one proves that the vk are real. (Notice, that with a suitable phase convention of the starting s.p. states la), I@, . . ., the matrix L[p] and, hence, the functions (Pi themselves are real.) Therefore the vii come in pairs (+ v,J and we can choose the vectors cpk(z), q:(z) in such a way that vk > 0. The functions C&(Z),cp:(r) have similar properties as in the real-time case (see sect. 16). In fact they obey the same orthogonality relations provided that the complex conjugation is here again supplied by time reversal. It is therefore convenient to adopt the bracket notation introduced in (I) with the modi~cation that the bravector is time reversed with respect to the ket-vector [cf. eq. (16.21)]. W%(r)~

= (4$(r)),fi

(5.37) Using eqs. (5.30), (5.34) and (5.36) and the symmetry properties of the matrices b, E and V one shows that 0

forkfk

<(PkW I%A@)= const for k = k’, vlr #

0.

(5.38)

289

H. Rei~~a~d~ i Semicfass~ca~ theor?

Hence we can choose the normalization

for all (p,, for which vk # 0. We observe that the vectors qk(r), (ok+ have similar properties as the ordinary RPA eigenvectors. In analogy to the real time case we call the q&z) the eigenmodes of the small fluctuations in the density field p. Consider now a small fluctuation (pfk)around p(7) in the direction of the kth eigenmode. Since the fluctuation field cp(t) must satisfy periodic boundary condition C4of3T) = r;o(-ST)1 and further the symmetry relation p’(z) = q(z) the fluctuation #k’(z) is given by (5.40)

q’k’(7) = Ck&(7) + C,&?r). It leads to a change in the euclidean action S&I

(3.20) which is given by [see eq.

(5.4)1 S’S, = S&?-t @‘] -Se@]

= 3 l dzdz’cp’k’(z’)(-A;

‘[fi](z,

~‘))cp’~‘(z’).

(5.41)

According to the quasiboson approximation we have here replaced A-‘@] by A;‘[61 which is the dominating (collective) part of A-‘@]. Using eqs. (5.32) and (5.39) we get CS,

= 2Ek(Ckl%Ok.

(5.42)

If the bounce solution of the ITDHF equation is stable with respect to small fluctuations p (k) these fluctuations must lead to an increase in the euclidean action (d2SE > 0). Hence we get from eq. (5.42) the following stability criterion: If the bounce solution /s(7) is stable with respect to small fluctuations in the direction of the kth eigenmode ((Pk(r), q:(z)) the norm [see eq. (5.39)] of the vector q,(7) [or equivalently of x~(z)], which corresponds to the positive eigenvalue of (RS,-i-L[Yj), ok > 0 [see eq. (5.33)], must be positive, i.e. sk = 1. This stability criterion is analogous to the criterion for the stability of the static Hartree-Fock state, p* derived by Thouless 14) and in fact it is obtained also frotn our considerations if we take the static limit j?(z) 3 p’. As mentioned in subsect. 5.1 there is a particular (spurious) solution to eq. (5.30) (for the proof see sect. 16) cpo(7)

=

A%

(5.43)

which gives rise to the zero eigenvalue of A- ‘[P]. In fact, since j(z) is periodic, the corresponding stability angle v0 vanishes. Furthermore, since j(7) is tangential to p(7) the spurious solution describes fluctuations along the classical path, which proceed with zero energy and, hence, does not represent physical excitations.

290

H. Reinhard~

/ Semiclassical

theq

Because of eq. (2.15) 4p&) satisfies the relation (5.44)

cpo+(r)= --&j(r). Furthermore

with the help of eqs. (5.30), (5.34) and (5.36) one shows that ((P~(T)~(P,,(z)) = 0 for all k.

(5.45)

Since the (pk+ e(z) are orthogonal to the spurious mode p,, = 6, which is tangential to the bounce solution fir), they describe fluctuations in the density field p transverse to the fission mode &z). Hence the qn + 0 can be interpreted as (bosonic) “eigenmodes” of the intrinsic excitations. Because of the symmetry property of (p*(r), eq. (5.43), the solutions to eq. (5.30), (40~+ 0, rp:, e, cpe> do not form a complete set and the zero eigenvalue o0 = 0 is twofold degenerate. We face here the same situation as in the ordinary RPA theory 14). If the static Hartree-Fock solution breaks a continuous symmetry there exists a spurious solution with zero frequency and the set of the RPA eigenvectors is not complete. To get a complete basis we have to supply a complementary vector &-,which is conveniently chosen to satisfy the following equation [this method of completing the vector basis is similar to the ordinary RPA theory ‘“)I

= &>.

( - f4 - wl(mo(~)

(546)

Then it obeys the relation (5.47)

t,‘(r) = &j(r). In a similar way as in the proof of eq. (5.38) one shows that 0, =

(&,tz)tpk(Z)=

0

I

fork

# 0

const for k = 0.

(5.48)

We can therefore choose the normalization

~~~~*~I~~(~))

(5.49)

and the properly normalized (i.e. to one) eigenvector of the spurious mode reads rpo(r) = M/VJ%(r).

(5.50)

Hence the factor .N, in eq. (5.22) arising in eliminating the spurious mode is given by [cf. eq. (5.14)] .N, = 10,11:

(5.51)

With this value for X0 the elimination of the spurious mode performed in subsect. 5.1.2 yields the same resuit obtained in I by using the collective coordinate method in its standard form.

H. Remhardt / Semiclassicui theoq

291

Having found the eigenmodes of the fluctuation field cp(r) we are now in a position to evaluate the functional determinants appearing in the fluctuation part of the partition function, Z,‘[fi] [see eq. (5.22)]. This will be done in the next section.

6. Evaluation

of the fluctuation

part of the partition function

In this section we evaluate the functional determinants appearing in the fluctuation part of the partition function, Z,‘[b] [see eq. (5.22)]. We shall do this in a way different from the one followed in I. The reason for doing so, is that in the course of the evaluation we shall find as a by-product some relations which are needed later. Besides the evaluation of functional determinants to be given below is rather instructive from the theoretical point of view. In the later considerations we shall need not only Z,‘[fi] but also the partition function of the fluctuations around the static Hartree-Fock solution, Zi[pO]. The equation of motion of these fluctuations [see eq. (5.30)] reduces then to the ordinary RPA equation (4.7). If the static Hartree-Fock solution p” is classically stable all eigenvalues of ( - A - ‘[p”]) are positive definite and the path integral over the fluctuations around p” can be straightforwardly evaluated. In the quasiboson approximation we find

GcP”1=

Det (-A;

‘[PO]) -*

Det(-T-‘[PO])

’

(6.1)

We prefer to evaluate the ratio Z,‘[p”[/Z,‘[p’] at the beginning. This will provide us with some useful relations needed later. But in+the same way as the ratioZ&]/ Z,[p”] we can also evaluate the quantities Znb] and Zi[p”], separately (see the end of this section). Since for T + co the quasienergies E, coincide with the ‘ordinary Hartree-Fock s.p. energies (see subsect. 4.2) we have by virtue of eq. (5.23)

rm

= rhoI

(6.2)

and from eqs. (5.22) and (6.1) it then follows that

=

s

dz,K,.

(6.3)

292

H. Reinhardr / Semiclassical theor)

To evaluate the ratio of the functional determinants, KB, we proceed as in sect. 3 in the evaluation of the fermion determinant Det( - 9). For this we need the bosonic Green function of the fluctuations, which will be constructed in the next subsection. 6.1. THE BOSONIC GREEN FUNCTION

OF THE FLUCTUATIONS

For the direct evaluation of the ratio Z~(p]/Z,‘[p”] matrix A,@;

z, 7’) = b-I(-h?,-R(p,

it is convenient to define a Z))6(T, r’),

WL, r) = UP01 +PL(m3(~)-m”I)~

(6.4)

(6.5)

which interpolates between A - ‘[;O”] and A- ‘[PI. Obviously we have A,‘(p

= 0) = A, ‘[p”],

(6.6)

A, i(p = 1) = A, ‘[PI.

(6.7)

Let us now define the Green function of the fluctuations A,Q; 9#,

z’)

7, 7’) =

=

b-‘9;‘(z,

-(ha,

g&z, 7’) by

e’),

+ qp,

W(?

(6.8) 0,

(6.9)

where the matrix b-l serves again as the metric tensor for the eigenfunctions of 9; ‘(z, t’) (Remember, that b-l just defines the RPA metric.) The Green function g&z, z’) itself is everywhere regular except for p = 1 where it has a pole (see below) since I 9;: 1 has a zero eigenvalue. To get an explicit representation for ga,(r, r’) we consider the related imaginary-time-dependent “Schrodinger” equation -WV;(r))

= R(C1,r)l@Xr)>Y

(6.10)

which is a generalization of the linearized stability equation (equation of motion of the fluctuations), eq. (5.30). Its formal solution reads I@(r)) = Uk

r)l4$(-3T))>

(6.11)

where r

U&

z) = T, exp

- i i

s -T/2

dr’R(p, 7’) 1

(6.12)

is the time-evolution

operator. The functions q;(z) have the same properties as the (Pi+ o(z) = cp:; h(r) defined by eq. (5.30). In particular, they satisfy the same orthogonality relations [see eq. (5.39) and appendix C], and since R@, z) is periodic they, too, can be represented

as C&(Z) = e-dk!P”Xf(z),

(6.13)

293

H. Remhardt / Semiclassical theor?

where the ;(j!(r) are periodic functions, which satisfy the eigenvalue equation: (6.14)

- (@,-I- R(@, ~))~~(~) = - ~~~~~(~), X%(:73 = x!(-30. Due to eq. (6.10) the v); satisfy the realtion lso#i3>

= e +‘“‘“‘l(~k-fT)>,

= tk0.k UYcpC(-+T))

rI,(cO = o&)C

and, hence, they are the eigenfunctions of the time-evolution operator From the de~nition of the matrix Rfp,z)(6.5) it is clear that

(6.15)

U&E, +r).

Vk(P= 1) = Vk,

(6.16)

V&J = 0) = $ = o,oT,

(6.17)

where the c&’ are the ordinary RPA frequencies [cf. eq. (4.7)]. By means of the 937) the Green function LBJz, r’) can be represented as

+ Ig+(7))[@7‘-7)(1

+~kT(~)U))+~(fZ---7’f~kT(~)]
(6.18)

Here we have used the fact that the functions q:+(z) are time reversed to the ~~(~~. The occupation numbers n:(p) are determined by the boundary condition on g@(r, r’). Since 2,(r, 7’) represents a propagator of the bosonic fluctuation field it has to satisfy periodic boundary condition g’,($r,

7’)

=

sBp( -+r

With this boundary condition the occupation to be

(6.19)

7).

numbers n,‘(p) follow from eq. (6.15)

n&u) = [e”@@‘-13-l.

(6.20)

They are formaIly equivalent to the unite-temperature In fact gfi = Jr, z’) represents the unite-temperature phonons.

boson occupation numbers 9). Green function of the RPA

6.2. EVALUATION

OF THE FUNCTIONAL

For the evaluation quantity

of the functional

J(p)

=

DETERMINANT

determinants

in eq. (6.3) let us define a

~bc’(P~u)) -+ Det (-A;'(O)) '

(6.21)

J(p) is an analytic function of p everywhere in the comples g-plane except for p = 1 where it has a pole. This is because there is a doubly degenerate eigenvalue

H. Remhardt / Semiclassical theor?

194

of -A, i(p), O&L), which for p -+ 1 vanishes, since cpg= ‘(7) = j(r) is the spurious mode +. Hence the ratio of the functional determinants in eq. (6.3) J, =

Det’( -A;

‘[jj])

Det( -A;

‘[PO])

-+

(6.22)

’

[remember the prime indicates that the zero eigenvalue of -A, excluded from Det( -A; ‘[p])] can be expressed as

[ResWI,=

” = [Res l/o,(&],,

‘[b]

has to be

dqb4

1

i = rRes J(‘)la=l

[

(6.23)

dp 1 p= i’

Therefore let us now evaluate the function J(p). Proceeding as in the evaluation of the fermion determinant Det( -9 - ‘) (see sect. 13) we rewrite J(p) (6.21) as

(6.24) Using the definition of the Green function g&z, r’) [eq. (6.9)], J(p) becomes

Inserting the explicit representation

(6.18) here for g,,(r, z’), we find T’2 dr
r)ldAcp:(r))

1 .

(6.26)

Here we have used the fact that (cp:!r)JdR(ti, WA@(r))

= (cpf:+)ldR(p, r)ld/+:+(r)).

(6.27)

In appendix C we prove that

(6.28) Inserting the last relation into eq. (6.26) we get

+~~v&4 . 5’

(6.29)

+ Strictly speaking, by an eigenvalue of -A;‘(p) we mean an eigenvalue of C3,,whereas the matrix serves as the metric tensor of the eigenfunctions of 23, [see eq. (6.8)].

295

H. Remhardt / Semiclassrcal theor)

The remaining JL’integration

is elementary

s

v;(P)

P

0

4&%4U~‘~ = dp’ m

=

log

Il-e-vk(p) I 1

_e-“‘O’

.

6.30)

Hence we obtain

2sh$,( 0)

=r*Imj

(6.31)

2 ‘k

As we have seen in subsect. 5.2. vk= o = vk= o(fl = 1) = 0. Consequently, J(p) (6.31) has a pole at ~1= 1. But because of eq. (6.21) this means that Det (-A, ‘[j]) must have a zero. [This proves our statement of sect. 5 that okzo = 0 is a doubly degenerate eigenvaiue of Ftii,+ L[fi] .] The residuum of J(p) is given by fl2 sh &(O) [ResJ(p)],=,

= k fl2sh+v,(l)

V&L’= 1)’

(6.32)

k

Furthermore

we have

7-y.

(6.33)

fl2 sh iv; k T n 2 shiv, ’

(6.34)

v;(p) =

Hence the quantity (6.23) becomes J/=1

k+O

Inserting this value into eq. (6.3) we get (6.35) k#O

For T + ccthe quantities vk = wk T tend to infinity while the frequencies ok remain finite, since they are eigenvalues of the confined operator M,+L[b] ‘. Hence for large T the quantity Ka (6.25) reduces to K,

The asymptotic

=+i

(6.36)

behaviour of KB for T + 00will be investigated in sect. 7.

t The operator (ha,+L[b]) condition.

is confined in the space of the real functions satisfying periodic boundary

296

H. Reinhardt / Semiclassical theory

6.3. THE PARTITION FUNCTION OFTHE BOSONIC EIGENMODES OFTHE FLUCTUATIONS

For later applications we quote also the results for the functional integrals over the fluctuations around the bounce and the static Hartree-Fock solution, Z,‘[p] (5.22) and Z&O] (6. l), respectively. These quantities can be evaluated in the same way as the ratio Z,‘[fi]/Z&“]. For this we have merely to redefine the matrix R(p, z) (6.5). Defining QL, r) = and repeating the calculations obtain

bE+tiwq4

(6.37)

of subsects. 6.1 and 6.2 with this new R(p, z) we

(6.38) where v;h = (c$ - a,)/h = (Er- Eh)T/h.

(6.39)

Here we have introduced a common index k to label a particle-hole pair (ph). Except for the modifications which are originated by the presence of the negative eigenvalue of -A - ‘[p] (i.e. the factor 42) eq. (6.38) just gives the result which one obtains by continuing eq. (16.63) to pure imaginary times t = - iz. Note, that in I the functional determinants have been evaluated in a different way to here. The quantity Zi[p] (6.38) can be brought to the standard form of a bosonic partition function by decomposing the denominator into a geometric series (“it F 2sh*vFh c exp [ - kfO 1 (n,+

+)v,],

(6.40)

nk = 0, 1,2, . . . Here the nk appear as the occupation numbers of the bosonic eigenmodes of the fluctuations which describe the (intrinsic) excitations orthogonal to the (classical) fission motion. We are interested in the partition function in the limit T + 00, where the classical solution of the ITDHF equation describes the time evolution of the system from the static Hartree-Fock ground state to the scission point and its return. In this limit only the “ground state” with all nk = 0 contributes to the partition function and the fluctuation part Zi[fi] reduces to dro

exP [ -3

1 k#O

vk -

1

%“,I.

(6.41)

k

We observe that the effect of the fluctuations is twofold. Firstly the partition function becomes pure imaginary. Secondly, even for the lowest lying state with all nk = 0

297

H. Reinhardt 1 Semrclassical theor)

there is a contribution to the energy from the zero-point motion, +(Ck _+Ovk- CkrEh), which is neglected in the classical approximation. The functional determinants in Z&O] (6.1) can be evaluated as above by redefining the matrix R@, r) in eq. (6.5) as R(& r) = bE+tiWOl

(6.42)

(Here R does not depend on r). The result is 2 sh lvPh z,T[Pol = ~2$$, 2k

v”

k

=

w”T k ’

(6.43)

with CO,”being the RPA frequencies. If we use eqs. (6.38), (6.43) and form the ratio Z&?]/Z,‘[pO] we obtain the same value as in eq. (6.35), where we directly evaluated the ratio. For a large period T > l/ok0 the partition function of the fluctuations around the static Hartree-Fock solution p” reduces to Z&O]

= ,-MOT/h,

(6.44)

where LlEO = *c(W; - oph), wp” = Ep- Eh,

(6.45)

k

is the zero-point energy in the RPA. Having evaluated the contributions of the (small) fluctuations to the imaginarytime partition function we are now in a position to derive the expression for the finite lifetime of the fissioning nuclei. This will be done in the next section.

7. The partition function in the semiclassical

approximation:

the finite lifetime

In the previous section we evaluated the integral over the fluctuations around the classical bounce solution in the gaussian approximation. As shown in sect. 4 in the limit we are interested, of large T the bounce is a well-localized object and, consequently, there are also approximate solutions to the ITDHF equation consisting of strings of widely separated bounces (see fig. lb). (These strings become exact solutions only if the distances between the centers of all bounces go to infinity.) Except for the small intervals around the centers of the bounces, zi, a string of bounces would give the same contribution to the partition function as the sta’tic HartreeFock solution p”. This suggests that one writes the partition function in the presence of one bounce as

= ~“cP”l exp C- WPI - &CP~IYW$f$j. B

(7.1)

‘98

H. Reinhardt

The ratio Zi[p]/Z,‘[p”] eq. (6.3) we have

/ Semiclassical

theor)

has been evaluated in the previous section. By means of

zr[j]

= z’[pO]K

dr,,

(74

s where the quantity K is defined by K = K, exp C- bV31 Using the fact that the Hartree-Fock we obtain from eq. (4.10) S,[?il-

- ~,C~~l)/~l.

energy E&(Z)]

(7.3)

is conserved [see eq. (4.13)],

S,[IP”I = c n, 1”’ dr(.&)lW.%)). Y -T/Z

(7.4)

Inserting now the last relation and the explicit value for KB (6.36) into eq. (7.3) the quantity K becomes K = ii

oOeWOr _ W,h J- %Ee

(7.5)

’

with T/2 _ II = En, dr Y s -T/2

+ ; C WV,- I$).

(7.6)

kfO

K describes the effect of the presence of the bounce on the partition function. Clearly for a string of n separated bounces we get the factor K n times. Furthermore within

a string the locations of the centers of the bounces come in a definite order, e.g. -+T

< z, < z2 < . . . < T,, < $T.

Hence the integrals over the positions of the centers of the bounces read SI’;d%[;T,dL1...~;T,d5

=;.

Thus a string of n bounces, p,,, yields, in the semiclassical approximation, function ZT[&]

= ZT[po] ;

K”.

(7.7) a partition

(7.8)

To get the full semiclassical partition function we have to sum over all strings with an arbitrary number of bounces. (The “string” with no bounce is the static HartreeFock solution p”.) This yields ZT = ZTIIpo]

nto(T = gT[po]

eKT.

(7.9)

H. Remhardr

/ Semcfassical theoq

‘99

Note, in deriving eq. (7.9) we have assumed that within a string the bounces are widely separated, that is to say their density is small, so that they do not interfere with each other, and every bounce in a string contributes to the partition function as a single bounce. This treatment corresponds to the dilute-gas approximation in statistical mechanics. In fact the most important terms in the exponential series (7.9) are those for which ill?- 5 K h exp (- ~1~).

(7.10)

Thus in the limit fr + 0 the partition function is dominated by strings that have a small bounce density. Using the fact that for the static Hartree-Fock solution p* we have [see eq. (4. lo)]

~&*I = q&“lT

(7.11)

from eqs. (5.1) (5.2) and (6.44) the partition function in the static (HFf RPA) approximation follows:

Z’tp’l

= exp [ -(En,[p”]

+dEO)T/h].

(7.12)

The second term in the exponent is the energy of the zero-point motion within the RPA, eq. (6.45). It lowers the ground-state energy relative to the static HartreeFock energy EnJp”] since for an attractive interaction the correlation energy involved in a RPA phonon (i.e. the particle-hole binding energy), defined by

is positive. With eqs. (7.9) and (7.12) the total partition approximation becomes, in the limit T -+ co,

function

.ZT = exp I-(E~FIC~o]+dEo+:iT)T/h],

in the semiclassical (7.13)

where

Here we have used eq. (3.7) to get the last equation. Besides the fact that the energy of the ground state has been corrected by the energy of the zero-point motion it also has an imaginary part. This means the system is unstable, having in its initial

300

H. Reinhardr

/ Semiclassical

theory

ground state a finite lifetime of l/r. This result was expected since, as assumed, the system shows spontaneous fission, and, consequently, it cannot have a stable ground state. For T + co the quantity W goes to a finite limit. One easily sees that the first term in W(7.15) remains finite for T -+ cc. Indeed, for (~1+ $T -+ co, the functions f,(z) approach the static Hartree-Fock s.p. functions f;9 exponentially; and by virtue of eq. (4.12) we have 13f”(r) cc ~:~-@8’~~,1~1+ +T+

co.

(7.16)

Hence the integral over the first term in eq. (7.15) remains finite. Now let us consider the difference (vk- v:). For T -+ 00, v, and v: go separately to infinity like T. Nevertheless, the difference vk- vz remains finite for T --) co. This can be shown in the following way. Due to eq. (6.28) we have 1

d/Lv;(p) = ; l’dp{r’2 d~(cp~(~)l~[P](~)-L[p”]lcp~(~)). (7.17) 0 -T/2 s0 As we have seen in sect. 5, the scalar product ((p~(z)I(p~(z)) is independent of z and, due to our normalization, equal to one. Thus the product of the functions l@(z)), (C&(Z)/ is of order one. Furthermore, as we have shown in subsect. 4.2, for /z/ -+ )T -+ co, the bounce 6(r) approaches exponentially the static Hartree-Fock solution p”. By virtue of eq. (4.12) the difference (~[~](~)-~[~O]) vanishes exponentially vk

-p

k

=

(7.18) Thus the z-integral in eq. (7.17) is, for T --f cc, absolutely convergent and, hence, gives a finite contribution. Thus the difference vk-vg remains finite for T --f co, although vk and vz separately go to infinity. In order that the, lifetime r(7.14) remain finite for T -+ cc the quantity as 00 = O,(T) = 5,b-‘j? [see eq. (5.48)] must vanish ~~ptotically 6,(T)

cc Te-“OT, for T+

00

(7.19)

Although we cannot strictly prove this asymptotic behaviour since for a finite T the complementary vector to(z) cannot be constructed analytically, we can at least show that O,(q vanishes asymptotically for T + 00: From sect. 4 we know the asymptotic behaviour of the bounce for T + 00. Differentiation of eq. (4.8) yields [cf. also eq. (4.9)] 1 oiqoxi

&* = _e-wd i.

- e&x:

, for z + cc + , forz + --co

(7.20)

Since for 1~1+ ST + 00 the periodic functions T,,(Z) can be replaced by the static Hartree-Fock s.p. f~ction~ the asymptotic behaviour of j?(z), eq. (7.20), is not changed when switching to the periodic s.p. basis f7t(z)). Thus (~)~~(~) vanishes exponentially as e+@lrl for 1213 )T -+ M, and for a finite T, ($)( +_fZJ should

H. Reinhardr / Semiclassical theory

301

be of the order e-“‘OTJ2. From the definition of the complementary vector 4,,, eq. (5.46) it is clear that
8. Summary

and conclusion

In the present paper the semiclassical approach to large-amplitude collective motion in many-fermion systems previously developed for quantized bound states 3), has been extended to the description of quantum tunnelling phenomena like spontaneous fission of a nucleus. In the classical limit the approach developed reduces to the TDHF approximation. The classical solutions of the ITDHF equation yield a dynamical picture of the fission process. They describe the time-evolution of the fissioning nucleus between the static Hartree-Fock “ground state” and the scission point. From the asymptotic behaviour of J(z) we have seen that the fission mode leaves the static Hartree-Fock ground state in the direction of the lowest RPA mode. Performing realistic calculations one should be able to understand the dynamics of the fission process as e.g. the competition between symmetric and asymmetric fission. Evaluation of the functional integral over the quantum fluctuations in the semiclassical approximation yields the finite lifetime of the fissioning ground state. In this way the lifetime of spontaneously fissioning nuclei can be microscopically evaluated in the mean-field approximation. Realistic calculations within the developed approach here should be feasible since the TDHF equation can be solved numerically on the computers available nowadays. Similar to the application of the TDHF theory to heavy-ion reactions l 5, the present approach requires, on the classical level, that one solves the ITDHF equation. In the numerical calculations one will, however, prefer to solve the eigenvalue problem (3.7) for a large but finite T,just as in the semiclassical treatment of large-amplitude oscillations considered in I. But compared to the application of the semiclassical theory to the bound-state problem, practical simplifications arise in the treatment of the fission problem since it is sufficient to consider a real density matrix.

302

H. Rein~ardf / Se~ic~ass~al

Note

theory

added in proof:

After the paper was submitted for publication the author has been able to show that the way of evaluating the functional integral over the fluctuations given in ref. 3, (called above “quasiboson approximation”, see subsect. X1.3) does not involve any approximation but yields the exact result 19).

Appendix A SOME PROPERTIES OF THE TIME-DEPENDENT S.P. WAVE FUNCTIONS

At the beginning we prove the orthogona~ity of the time-de~ndent defined in sect. 3. Consider the quantity

states \g,(r))

CA.1)

J&l(r) = (g,(r) l&W. Taking the time derivative and using eq. (3.1) one shows that

64.2) Hence we can choose the normalization G?,W~,W

(A.3)

= %,.

For evaluating the term tr log (-@ - ‘) it is convenient to represent the time-dependent states jg,(z)) and (gy(r)I by means of the time-evolution operator U(r) [see eq, (3.3)]. For this we note that the (g,(r)) obey an alternative representation

fA.4) since O(z)U(+T) = U(z).

(A.9

The last relation is easily proved using the definition of the time-ordered product on the time lattice. To get the corresponding representation for the bra-vector (g,(r)1 we write eq. (A.4) explicitly in the time-independent basis (f&~,(r)) = (a\?,

exp jl!1

~2d7’h[p(i)ljiu)<~lg~~~~)~- (A-6)

Taking the complex conjugate of this equation we get t =
exp (’i I”lh’(r’)dr.)

la>.

(A.7)

+ Note: according to our defmition (3.2) the bra-vectors are time-reversed compared to the ket-vectors.

303

H. Reinhardt / Semiclassical theory

With the help of the relation h’(z) = h(z) we obtain after a change in the integration variable s’ = -7’
=
exp [ - :, j:rT’2h(s.)ds’] ,a>

=
exp [hI ~;,Ks’)ds’]

,a>.

(A.8)

Here we have used the fact that ‘I‘,,= f,,. Eq. (A.8) implies that (gy(r)Ia) = (9,(-*TNU-

‘(4,

l(z) = T,. exp [a ~;T,&?dsj.

(A.9)

With the relation (A.9) at hand it is now easy to prove eq. (3.12). The proof proceeds in excatly the same way as given in sect. 3 and Appendix A of I and for the proof we refer the reader to that paper.

Appendix B HANDLING

OF THE

NEGATIVE

EIGENVALUE

As we have seen in sect. 5, there is a negative eigenvalue of (-A- ‘[Yj) which gives rise to a diverging integral. In this appendix we show how the functional integral is treated by using the method of steepest descent making the integral converging. This will provide also the factor $i mentioned in sect. 5. For pedagogical clarity as well as for economy of notation let us ignore in the following considerations the zero-mode problem. (It can be treated as well simultaneously with the negative eigenvalue. However, the equations would then become very cluttered.) Following the idea of ref. l’) we consider a path in the function space of the field variable p,,(r), CJz, r), which is parametrized by a real variable z. The total functional integral (i.e. the integral over the total function space) can be written as an integral over the variable z times the functional integral over the hyperplane perpendicular to the path i(z). The equation of this hyperplane in the function space of the density field P(p-space) is given by

s T/2

dT

Qbl(4

=

dC’(z, 4

(B.1)

304

H. ~elnhardt / Se~i~~~sical theory

(The quantity d?Jdz represents the “normal vector” of this hyperplane.) Therefore let us introduce into the functional integral (3.14) the following identity l=

m dz~(Q[p](z))lQ'[pl(z)I, s -cc

where the prime denotes differentiation

Q’[p](z) =

VW

with respect to z:

s” di[dd(j$r)-&z,r))--CT2

(~>‘I.

(B.3)

Then the functional integral for the partition function (3.14) becomes t 2: = (Det V)*

Dp s

e-Sa[P1in.(B.4) a: ~~(Q[p](z))lQ'[pl(z)l s --a

The constraint S(Q[p]( z )) now confines the functional integral over the field variable p(z) to the hyperplane perpendicular to the path i(z). We now choose the following properties for the path i(z): (i) At .z = 0 th e path goes through the static Hartree-Fock solution, i.e. [(z = 0, z) = po.

(B.5)

(ii) At z = 1 the path goes through the bounce, i.e. [(z = 1, Z) = r?(r).

(B-6)

(iii) The tangent vector to the path at z = 1 is given by the eigenvector of - A- ‘[j?], #k= _(z), which corresponds to the negative eigenvalue P- < 0, i.e.

I I dik 7) dz

= gb_(z).

(B.7)

z= 1

As in sect. 5 we suppose here that the eigenvalues of -A -‘[PI normalized

are properly

T/2 d~~-(~)#_(~) = 1. s -Tj2

03.8)

For the &function in eq. (B.4) we use the representation S(Q)= iii

&e-y’i’.E.

(B.9)

Then the constraint yields an addition to the collective action S,[p]. The functional integral becomes Z;f, = (Det V)* lim ~ f

exp1--&~CPI(~/~~~ Dp d.zlQ'[p](z)l (B.10) s

’ For economy of notation we drop, in this appendix, the summation over the

InJ,, too.

occupationnumbers

H. Reinhard

/ Sernielasszcal rheorl

305

where the effective action is given by

&LPlr4 = S&l +

&fQCPlm2.

(B.1 I)

The functional integral over p is now evaluated in the SPA. The phase S,&](z) becomes stationary for (B.12) Using the definition of the eulcidean action S&l, takes the form [cf. also eq. (4. l)] :

4&Jl(4 =

-

v,,,

,&%&7)

-

G&,

z +O,]

eq. (3.20), the equation of motion

+

t

Q[p](z) y

= 0.

(B.13)

&3,(7)

Let us denote its solution by cE(z, r). Note that this solution depends on the path i(z) and, in general, also on the value of the small quantity E. According to our choice of the path &z, r) this solution coincides for z = 0 and z = 1 with the static Hartree-Fock solution and the bounce, respectively, i.e. &(z = 0, r) = po,

(B.14)

P,(z = 1, r) = j?(r).

(B.15)

Indeed by virtue of eqs. (B.SHB.7)

we have then

QM,]WL = 0 = QCP,l(4I,= 1 = 0

(B.16)

and the equation of motion (B.13) reduces then to the Hartree-Fock self-consistent condition (4.1). As in sect. 5 we now shift the field variable p(r) by the classical solution fi,(z, r) P(Z) = P,(r, Cl+&).

sr,

dzew C-

&fCB,l(d/~l

(B-17)

Pet Vt

where [cf. also eq. (5.4)] (B-19)

306

H. Reinhardt / Semictassicai theory

For economy of notation let us write the integral (B.18) as m

(B.20)

where

What remains to be done is the z-integration, again in the SPA. But it is this integration which makes all the trouble since it diverges. To show this let us consider the behaviour of the effective action S,,,[~,](z) (B. 11) as function of z. As one easily proves this function has extrema at z = 0 and z = 1. The extrema occur for

where dQ/dz denotes the total derivative, which is given by

(3.23) and the quantity Q’@](z) has been defined by eq. fB.3). It is now obvious that eq. (B,22) is fulfilled only for z = 0, 1. This is because the first term [~~~~~]~~~I~= Fz vanishes for all z since p = &(z, r} makes the effective action S,,,[p](Z) stationary [see eq. (B.13)], while the second term vanishes only for z = 0, I due to eq. (B.16). Now let us suppose that the static Hartree-Fock solution is classically stable (as is usually the case). Then S,[p = p”] is a local minimum with respect to all directions in function space. Hence S,&?J(z) has, at z = 0, j5,(z = 0, z) = p”, a local minimum (S,,,[p,](z = 0) = SEtpa]). On the other hand, the bounce solution is stable with respect to small ~uctuat~ons in all directions except for the direction given by #_(z>, with which the negative eigenvalue of -A - “[p] is connected. To see the type of

307

H. Reinhardt / Semiclassical theory

the extremum of S,&,](z)

at z = 1 we take the second derivative

d2&,,[i%](z) = drdr, dz2 s

(B.24) To find the quantity dp,(z, r)/dz we take the derivative of the equation of motion, eq. (B. 13), with respect to z. This yields

s T/2

[-L‘-‘[P,](r,r’)]~&’

-T/2

+

1 dQCPJ4 dk 7) E

7

dz

+QCiil(d

d2&, d

1 o.

(B.25)

=

dz2

We are interested in (dp,(z, r)/dz), = 1. Let us therefore consider eq. (B.25) for z = 1. Using eqs. (B.6), (B.7) and (B.15) this reduces to

s T/2

C-A- ‘[iq? T’)][VI_

Idr’+ ; [dQy;)z)]Z;

r&(r)

= 0.

(B.26)

-T/2

Furthermore

from eqs. (B.23) and (B.3) we get

Using furthermore

the fact that 4_(r) is an eigenfunction ‘(-A-‘[P](~,~‘))~_(z’)dz’

of -A-

= p-+-(r),

‘[PI, (B.28)

s we find that the solution to eq. (B.26) is of the form

c1 -ditk 7) dz

=u +a#-(4

(B.29)

r=l

1=

where the quantity al depends on E. With this ansatz we obtain from eq. (B.27) dQCiW [

dz

a 8,

z=l

(B.30)

H. Reinhardt / Semiclassical theory

308

where we have used the normalization

(B.8). Eq. (B.26) reduces then to

[(l +c$)P- +CI,/s]&(Z)

= 0.

(B.31)

Since 4_(r) does not identically vanish this equation requires EP@&= - l+sj?_’

(B.32)

Inserting, now, eq. (B.30) into eq. (B.24) and using eq. (B.28) we obtain (B.33) In the limit E + 0 the quantity cle(B.32) goes like E to zero and we get X‘,(l),

= lii~2~~r$-J(z)]Z~1=

P_ < 0.

(B.34)

Hence X&L ,I(4 has a maximum at z = 1. Furthermore, since there are no more stationary points than the ones at z = 0 and z = 1 [see the discussion following eq. (B.23)] this function must tend to minus (plus) infinity if z + a(- co). Therefore S&,]( z ) as function of z has the qualitative behaviour illustrated in fig. 2a, and, consequently, the z-integral in eq. (B.21) is hopelessly divergent. To make it convergent we have to deform the contour of integration into the complex z-plane. Let us suppose that S&,]( z) is an analytic function of z. [dS&,](z)/ dz], = i = 0 implies, then, that z = 1 is a saddle point of Re(S,,,[P,](z))]. We can then distort the right-hand part of the integration contour into the complex z-plane to make the integral convergent. As usual in the evaluation of Laplace’s integrals “) the contour of integration is led along a path of steepest descent, which gives the dominant contribution to the integral in the limit /i + 0. The path of steepest descent is given by Im(S,,,[~,](z)) = const and since d2S,,,(z)/dz2 is real this path leaves the saddle parallel to the imaginary axis. This leads to the contour of integration shown in fig. 2b and the integral (B.20) becomes Z,T = zy+z;,

(B.35)

where 1

ZT = &l&m0 dzF,( z) e - h[p”clW,

(B.36)

s -al e-serrt~cl(z),

(B.37)

and C, is the contour shown in fig. 2b. The integral along the real axis, Zy, is standard. For k + 0 it is dominated by the minimum of S&J(z) at z = 0. If it were retained it would yield a real contribution to the ground-state energy which is

H. Reinhardt / Semiclassical theor]

309

(a) Fig.2a. Qualitative behaviour of the effective action S[fi,](z) (B. 11) as function of z. The minimum at z = 0 and the maximum at : = 1 correspond to the static Hartree-Fock solution p” and the bounce P(t)+ respectively.

b) Fig. 2b. Contour of integration over the variable z. In order to make the integral convergent, the contour C, on the r.h.s. of the saddle z = 1 is deformed into the complex z-plane. The contour of steepest descent C, leaves the saddle point parallel to the imaginary axis. In the limit h + 0 this contour can be replaced by the contour C; (see text).

exponentially small compared to the leading order term, E&p] and, hence, it can be neglected. The second integral Zt being also exponentially small, gives the leading-order contribution to the imaginary part of the energy (see sect. 7) and, hence, is retained. Therefore let us evaluate now Zr: For h --, 0 the main contribution to the integral comes from the contour near the saddle point, which is parallel to the imaginary axis while the endpoints of the contour of integration are irrelevant. We can therefore deform the contour from C, to C; see fig. 2b.. Furthermore near to z = 1, S&,](z) can be approximated as

w-Pcl(4 = &CA+ J&,(1>,(2 - w x;,(l), < 0,

(B.38)

where we have used eqs. (B.15) and (B.16). Along the contour C; we have dz = idy (z = x+y)

(B.39)

and the integral 2, becomes now an ordinary gaussian integral

(B.40) Here the factor 3 arises since the integration is only over half of the gaussian curve. We can now take the limit E -+ 0. Using eqs. (B.6), (B.7), (B.8) and (B. 15) thequantity Q’[p,](z = 1) (B.3) becomes Q’[&](z = 1) = - 1.

(B.41)

Hence we find for the function F,(z = 1) (B.21) 12

F, = lim F,(l) = (Det I’)*

E+O

Dq,S s

$-(r)co(r)dT K T/2 >

drdr’cp(r)[ - A- ‘(I?-&,

~‘)]447’)

1 .

(B.42)

Here we have used the definition (B.9) of the S-function. The functional integration over rp(z) is now restricted to the hyperplane perpendicular to the direction $_(7) connected with the negative eigenvalue, and involves now only positive (or zero) eigenvalues. In fact, proceeding as in subsect. 5.1 and expanding the field variable q(z) in terms of the eigenfunctions of -A-‘[P] [see eqs. (5.10), (5.1 l)] one finds that the constraint in eq. (B.42) eliminates the integration over the dangerous direction #_(7). The remaining integrals contain only positive (or zero) eigenvalues and we obtain (B.43) (kt

-1

Inserting this result into eq. (B.40) and using eq. (B.34) we get 2: = (Det I,‘)*e-s&1$ ekes

=

fi

(Det V)*lDet (- A-l[fi]}l-fe-SEt~.

This is just the result stated in subsect. 5.2.

WW

311

H. Reinhardt / Semiclassical theory

Appendix C PROOF

OF THE

EQUATION

(6.28)

In what follows we briefly sketch the proof of eq. (6.28) dv#)

--=-

dp

1

T’*

--

d~(cp~(~)l~[Pl(~)-~[~“ll~~(~)).

Cl)

h s -T/Z

The time-dependent states qpP(r) have been defined by eq. (6.10). By means of the time-evolution operator U&A, T) (6.12) they can be represented as k&(7))

=

f-J&4

eP3-3m

(C.2)

Using the symmetry properties of the matrices 6, E, V [see eqs. (5.24), (5.25), (5.29)] and the definition of the time-evolution operator U& r) one easily shows that the bra-vector can be represented as (the proof proceeds analogously to the proof of eq. (11.12): (cpix7)l

=

(cpix

-ilr)l

4i1(PL, q.

where LJ; ‘(p, z) = T,. exp

1 r H’

(C.3)

1.

dr’R(p, r’)

-T/2

(C.4)

Since u - l(P> r)V&,

r) = 1

(C.5)

from eqs. (C.2) and (C.4) it immediatelyifollows that the scalarproduct(@!(z)l q%(r)) is independent of r and we can choose the normalization (cf. also subsect. 5.2) (cp;(r)lcp&(r)) = &,,. With the representations (6.15)] that:

Furthermore,

(C.6)

(C.2) and (C.3), it follows from eq. (6.13) [cf. also eq.

using eq. (C.5) one easily shows the analogous relations U;‘(P,

-ST)Iv:(-+W


fQ

= e’~‘%$(-tr)>,

(C.9)

= e-‘“‘p~3ZF3ji.

(C.10)

Because of eqs. (C.6) and (C.7) we have %(P) = -log (cp:(--3n~&c,

f-m#--3w

((2.11)

312

H. R&hard1

Differentiating

/ Semiclassicaf

theory

the last equation with respect to p we obtain

t&(p) = - (cp$ +)I&

Y/-4$7

[” -&

U&2

iv 'I

lsD#-+-D.

(C.12)

Here we have used eq. (C.8) and further the fact that

1

0. (C-13)

This relation is easily proved by noticing that because of eqs. (C.7) and (C.10) it reduces to epvkca)

_._.~ 6 [(c&!( -+T)l&(

-fT))]

= 0.

(C. 14)

But the last equation is obviously true since the norm is independent of p. Representing U&L, +T), Vi’@, 47’) defined by eqs. (6.12) and (C.4) on a time lattice (c.f. appendix A of I) one easily verifies that

Inserting the last relation into eq. (C. 12) and using eqs. (C.2) and (C.3) one obtains the desired result: eq. (C. 1). References 1) 2) 3) 4) 5) 6) 7) 8)

H. Reinhardt, Nucl. Phys. A331 (1979) 3.53 H. Kleinert and H. Reinhardt, Nucl. Phys. A332 (1979) 331 H. Reinhardt, Nucl. Phys. A346 (1980) 1 H. Remhardt, J. of Phys. 65 (1979) L91 S. Coleman, Lectures delivered at the 1977 Int. School of Subnuclear Physics, &tore Majorana Y. Ohm&i and T. Kashiwa, Prog. Theor. Phys. 60 (1978) 548 F. A. Berezin, The method of second qu~t~ation (Academic Press, NY, 1966) J.-L. Gervais and B. Sakita, Phys. Rev. Dll (1975) 2943; J.-L. Gervais er al., Phys. Rev. D12 (1975) 1038; J.-L. Gervais, Lectures given at the XVI Universimtswoche fur Kernphysik, Schladmig (Austria), Feb., March, 1977, Acta Phys. Austr., Suppl. XVIII, (1977) 385 9) A. Fetter and J. D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, NY, 1971) Ch. 7 10) A. K. Kerman and S. E. Koonin, Ann. of Phys. 100 (1976) 332 11) H. Reinhardt, Nucl. Phys. A298 (1978) 77 12) C. Callan and S. Coleman, Phys. Rev. D16 (1977) 1762 13) L. P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, NY, 1962) 14) D. J. Thouless, Nucl. Phys. 21 (1960) 225; 22 (1961) 78 1.5) See e.g. I. Boncho et al., Phys. Rev. Cl3 (1976) 1226; S. E. Koonin et af., Phys. Rev. Cl5 (1977) 1359 16) J. W. Negele et al., Time-de~ndent Hartree-Fock method, eds. P. Bonche eta/. (Editions de Physique, Paris, 1979) p. 150; J. W. Negele, Proc. Int. Conf. on extreme states in nuclear systems, Dresden, DDR, February 1980 17) L. Sirovich, Techniques of asymptotic analysis (Applied Mathematical Science) (Springer, NY, 1971) 18) K. K. Kan, J. J. Griffin, P. C. Lichtner and M. Dworzecka, Nucl. Phys. A332 (1979) 109 19) H. Reinhardt, Nucl. Phys. A, in press