IL NUOVO CIMENT0 VoL. 87B, N. 2 11 Giugno 1985

Path Integrals with Coalescent Dominant Saddle Points.

F. AXEL

Laboratoire de Physique des Solides, (LA 2), Bat. 510, Universitd Paris-Sud Centre d'Orsay - 91405 Orsay Cedex, .France

M. ~ASETTI

Dipartimento di ~isiea del Politeenico - 10129 Torino, Italia Istituto 1Vazionale di ~'isica •ucleare - Gru/2po IV, Torino, Italia

(ricevuto il 9 0 t t o b r e 1984)

S u m m a r y . - The method of stationary phase, usually adopted to evaluate oscillatory integrals, such as Feynman's path integrals, leads to difficulties when the integrand phase--which is controlled by the position of a point in a space of parameters--exhibits a configuration characterized by the coalescing of several dominant saddle points into one. A method is proposed to deal with such a pathological occurrence and the case of two merging saddle points is thoroughly discussed.

PACS. 02.30. - Function theory; analysis. PACS. 03.65. - Quantum theory; quantum mechanics.

1 . - I n t r o d u c t i o n .

F e y n m a n p a t h in tegra ls can be t h o u g h t of as osc i l la tory in tegra ls over

Hi lbe r t space for which, in general , the phase func t ion can be reduced to be

wr i t t en as t he sum of a pos i t ive quadra t i c form and a cont inuous bounded

func t ion (1).

The same occurs for the gene ra t ing funct ions of s t a t i s t i ca l mechanics ,

(1) S. ALB]EV'EI~IO and R. HO~G-K~oH~: In/ventiones .Math., 40, 59 (1977).

157

158 P. ~X~L and ~ . RAS~TT:

provided one writes t h e m as n-point funct ions of a q u a n t u m theo ry with periodic pure imaginary t ime of period fl~ (2.4).

I n b o t h cases the osci l la tory integrals are control led by the posi t ion of a point in a space of pa r ame te r s ~I of g iven - -gene ra l ly f in i te - -d imension.

The t r e a t m e n t of such integrals is s t andard (5.7), a t least in the f ini te-dimen- sional ease, and essential ly resor ts to the use of the me thod of s t a t ionary phase (saddle point) , even when the n u m b e r of s t a t ionary points is higher t h a n uni ty .

Very l i t t le is known, however , in the c~se in which, upon vary ing the con- t ro l p a r a m e t e r in gt, a finite subdomain ~t ~ of ~t is h i t so t h a t several of such saddle points coalesce. Only ad hoc discussions for par t icu lar cases have been produced thus far, in pa r t i cu la r in view of the appl ica t ion to renormal iza t ion group calculations (s).

The p rog ram we discuss here is a iming to devise a general m a t h e m a t i c a l t echn ique to evaluate func t iona l integrals in th is case.

Phys ica l applicat ions could range f rom the theo ry of phase t rans i t ions to gauge q u a n t u m field theory .

I n sect. 2 we briefly review the role of F e y n m a n ' s pa th integrals in con- nect ing classical s tat is t ical mechanics wi th q u a n t u m mechanics.

I n sect. 3 it is described how the a sympto t i c expans ion of such integrals , which are rap id ly oscillating, is dominated b y s t a t iona ry points of the phase funct ion (which, in tu rn , is connected with the phys ica l action).

I n sect. 4 the case in which such s ta t ionary points coalesce on some finite subset of the space of p a r a m e t e r s is handled b y a general series expansion. The ra t e of convergence of the l a t t e r is thoroughly discussed, by finding uppe r bounds to the remainders which are exponent ia l ly small .

The funct ions enter ing the above expansion are solutions of a compl ica ted set of recursion relations, which are solved in detai l in sect. 5 in the special ease of two coalescing s t a t i ona ry points.

Sect ion 6 contains a few conclusive remarks .

2. - F e y n m a n p a t h i n t e g r a l s a s o s c i l l a t o r y i n t e g r a l s .

F e y n m a n ' s p a t h in tegra l formula t ion of q u a n t u m mechanics reveals the deep connect ion between classical s tat is t ical mechanics and q u a n t u m theory .

(~) A.S. WIGHT~A~: Phys. Rev., 101, 860 (1956). (3) S . W . HAWKING: Commun. Math. Phys., 55, 133 (1977). (4) R. BAL:A~ and C. 17~ DoMI~ICIS: Ann. Phys. (N. Y.), 62, 229 (1971). (6) J . J . DVIST~MAAT: Commun. Pure Appl. Math., 27, 207 (1974). (e) V . I . A ~ O L ' D : Russ. Math. Surveys, 28, 19 (1976). (v) V.P . MASLOV: Theor. Math. Phys. (USSIr 2-3, 21 (1970). (s) E. BR]~ZIN, J. C. LE GUILLOU and J. Z:~N-JUST:~ : in Phase Transitions and Critical Phenomena, Vol. 6, edited by C. DOMB and M.S. Gnx~- (Academic Press, London, 1976), Chapt. 3.

P A T H I N T E G R A L S W I T H C O A L E S C E N T D O M I N A N T S A D D L E P O I N T S 159

Indeed, in an imaginary t ime formalism the F e y n m a n integral is math- ematical ly equivalent to a par t i t ion funct ion (~).

In short , the original version of l~eynman's pa th integral defines the Green's funct ion in the form

(2.1) Z ( x , , x , ) : ~. e x p [ i s M ] , Paths u~ j

where the formal sum is over all world-lines tha t connect the initial (xi) and final (x,) points in space-time and S~ is the action in configuration space M fo~ a par t icu lar path.

A precise definit ion of the sum over all paths in the measure theoret ical sense of Wiener (lo) leads in a s t ra ightforward way to the cus tomary funct ional integral formula t ion of (2.1).

The in teres t ing fact is tha t (2.1), which is a sum of rapid ly oscillating phases, by analyt ic cont inuat ion to imaginary t ime of each pa th tu rns into a sum over paths of ordinary damped exponent ia l factors. This of course makes it easier to dist inguish impor tan t pa ths f rom un impor tan t ones.

In the funct ional integral representa t ion we have the Eucl idean version of (2.1) in the form

i

where d~{x} is a suitable measure. By denot ing by ]xi> , Ix,> posit ion eigenstates for the Hami l ton ian H as-

sociated with the action S, (2.2) can be wri t ten by i tera t ion of the cus tomary method of wri t ing the exponent ia l as a l imit (11):

IT] (2.3) Z,,---- (x, l exp -~ H Ix,>,

where T ~- fl~/ is a positive t ime. By introducing a complete set of energy eigenstates, {In>}, (2.3) reads

finally

(2.4) z~, = ~ cxp [ - ~B.] <x,l~> <nlx~>, la

which indeed identifies it as a pa r t i t ion funct ion. On the other hand, assuming a general expression for H as the sum of a

quadrat ic form and a continuous bounded funct ion, the p a th integral (2.2)

(~) M. CREVTZ and B. FREEDMAN: Ann. Phys. (N.Y.), 132, 427 (1981). (lo) 1N. WXENER: J. Math. Phys. (N. Y.), 2, 131 (1923); Acta Math., 55, 117 (1930). (11) E. NELSON: J. Math. Phys. (N. :Y.), 5, 332 (1964).

160 ~'. AXEI., and ~t. :RAS:ETTI

is, in fact, an oscillatory integral over a Hilbert space (we shall briefly review this point in next section).

In it both the phase S({x}) and the amplitude a ({x)), where--if we de- note by d{x} the customary Lebesgue measure--d#{x}----a({x}) d{x}, might depend, in general smoothly, also on various parameters.

The detailed study of integrals of the form (2.2) leads then directly to a set of problems connected with the method of stationary phase.

)/fore specifically, if S({x}) is real valued, one has to resort to a proper sta- tionary-phase method. On the other hand, if S({x)) is pure imaginary (which is the case in which the oscillations are damped, as mentioned before) (2.2) is indeed a Laplace integral. In general, S({x)) is complex and a steepest-descent method is what has to be adopted.

The asymptotics of (2.2) is what one is usually interested in, in that the limit of vanishing ~ is what connects quantum and classical behaviours.

This has been studied in the case of real phase by decomposing the integral into various dominant contributions coming from the submanifolds of M where the phase itself is stationary (i.e. it has vanishing gradient).

The structure of stationary points can be classified from the homotopy equivalence point of view (Morse theory) (e) as well as from the point of view of catastrophe theory (5).

We intend to exhibit in the sequel a general algorithm which allows us to extend the customary saddle point method to those subvarieties of the space of parameters in correspondence to which the stationary-phase points degenerate into singular configurations (coalescent saddle points).

3. - A s y m p t o t i c expans ion o f the osc i l la tory integral .

Following DUISTER~AAT (s), let us consider real-valued smooth functions 9(x), a(x, v); x e X c M, ~ e R; the latter having an asymptotic expansion of the form

co

(3.1) a(x, "c) ~.~ ~ a,(x) "c ~ r~O

where @ is a real number. Assume 9(x) as phase function and a(x, 3) as the amplitude for a highly

oscillatory wave v:

(3.2) v(x, 3) = e x p [i~9(x)] a(x, 3).

Locally finite (in x) continuous superpositions of functions of type (3.2)

(3.3)

P A T H I N T E G R A L S W I T H C O A L E S C E N T D O M I N A N T S A D D L E P O I N T S 161

where r (a,, ..., a , ) e ~ l c R *, are global extensions of v (x , r ) to a n y con- nected subspace of M.

B y repea ted par t ia l in tegrat ion, one can check t h a t the only contr ibut ions to (3.3) which are not rap id ly decreasing come f rom the set

(3.4)

of points , where ~0 is s tu t ionary as a func t ion of a, if d*_q~(x, a) is a nondegen- erate bi l inear form.

At points in which d~o(x, a) is degenerate , the c u s t o m a r y wisdom is t ha t one has to compu te integrals of t he fo rm

(3.5) ~2~/ j j

The l a t t e r are the most generic fo rm of F e y n m a n ' s t y p e oscil latory integrals. Indeed, if one considers the a s y m p t o t i c (~--> 0) solution of the t ime-de-

pendent SchrSdinger 's equation,

(3.6) i~ = H , H - 2Me A + $/"

for a given ini t ial condit ion v(x, 0) ~ exp [i](x)/]~] w(x), in t e rms of the so]u~ions of the classical equat ions of mo t ion for a par t ic le of mass Me moving under the field of f o r c e - - g r a d ~ , wi th v e l o c i t y - - g r a d ] at t i m e r --~ 0, one can write v as the superposi t ion of t e rms of the fo rm (~)

i [~m(p~(.r (3.7) (2~)-'l~(p~(~))]-�89 i

where m is the 1VIaslov index (~) associated wi th the classical orbi t (path) p~(r) ending in x at t ime r - - a n d ~(p~(r)) is the de t e rminan t coming f rom the Gaussian in tegra l connected wi th the classical act ion func t iona l

(3.8) 7 1 So(p~) | - ~(p~(r)) d r = ~mp~

e l

o

(1~) S. ALBEVE~IO, PH. BLANCHARD and R. tt0EG-KRoHN: Commun. Math. Phys., 83, 49 (1982); M.C. GUTZWlLLEn: in Path Integrals, edited by G.J . PAPADOPOVLOS and J. T. DEV~E]~SE (Plenum Press, New York, N. Y., 1978), p. 163.

162 ~'. AX~L and ~. ~AS~TTI

The formal solution in te rms of F e y n m a n pa th integral can then be wr i t t en as

(3.9) v(x,v) =fa(p~} exp [~s((p.})]v(p~(o),o) ,

the integral being over all pa ths ending in x at t ime ~. The expansion of S({p,}) a round So(P,(V)) leads to the general form (3.5),

where O = O(a, x; ~) ~ Z(a) exp [-- i~v(x)] can be thought of as a complex- valued tes t funct ion with Z e C~(a) and V a smooth real-valued phase funct ion.

The hope is tha t W can be chosen so tha t the new phase funct ion (~o(x, a) -- -- ~(x)) has only nondegenerate s ta t ionary points as a funct ion of (x, o O.

(Xo, no) is a s ta t ionary point if and only if

(3.1o) d,q (x o, = @(Xo),

"d 8 d ~

is nondegenerate at (x0, no). This implies tha t

(3.11) r ank (dr162 Ore) ) z k .

d~,qXXo, Ore) = 0 ,

d~d~q~

l

Applying the method of s ta t ionary phase, t hen J~ tu rns out to have the asymp- to t ic expansion

, -12~r\ ~'/8 [.7~4 ] (3.12) exp ['/,~:~(xo, ao)] ( 7 / [det.Z~,]-�89 exp , sgn .,~, Z(Xo).

�9 ao(Xo, .o).O{1+ e(!)}, where /z = dim d~.

Not ice tha t the leading t e r m in the expansion of ~ depends on the choice of the tes t ing phase funct ion only through d:V(Xo).

F r o m the computat ional point of view, s ta t ionary point degeneracy is, however, difficult to remove in this way.

Often the occurrence of degenerate critical points is t rea ted by considering families of oscillatory integrals depending on an enlarged space of parameters .

I t is expected tha t the asympto t ic behaviour of the in tegra l s - -d ic ta ted by the type of degeneracy- -migh t be recovered in this case by using the ad- dit ional parameters at disposal and changing variables in such a way tha t the phase funct ion is brought into s tandard form.

The idea is t ha t by ~r theorem (18,1,) the funct ionals S({x}) which have

(18) j . ~V[ILNOR: Morse Theory (Princeton University Press, Princeton, N.J . , 1973). (14) B. MALGRA~G]~: Springer ;beet. Notes Phys., 126, 170 (1980); Ann. Sci. Ec. .Norm. Sup., 4$me Sdrie, 7, 405 (1974).

PATH INTEGRALS W I T H COAL:ESCENT DOMINANT SADDLE POINTS 1 6 3

only nondegenera te cri t ical points fo rm an open and dense set in C ~ the Whi tney t o p o l o g y - - w i t h a complemen t of codimension 1. Hence degenerate critical points should be uns table in the sense t ha t t h e y are expec ted to disap- pear under smal l C ~ per turba t ions .

On the o ther hand, b y Thom' s t r ansve r sa l i t y theorem~ the re is a countable intersect ion of dense open sets of phase functions, depending paramet r ica l ly on k variables , such t ha t the funct ions induced in the je t bundle over the Hi lbe r t space in tersect the critical manifo ld in this jet bundle t ransversa l ly .

Hence s ingular manifolds of codimension at mos t k are in te rsec ted and the in tersect ions are stable.

Thus the s t u d y of oscil latory in tegrals depending on k p a r a m e t e r s in the case of degenerate critical points wi th degeneracy of codimension k requires to be r e fo rmula t ed on its own ground.

When the correspondence be tween F e y n m a n p a t h in tegra l and stat is t ical me- chanics is carr ied on the way discussed above, i .e. b y ident i fy ing the correlat ion funct ions of s ta t is t ical mechanics wi th n-point funct ions of a q u a n t u m theory with periodic pure imaginary t i m e of per iod fib, a different set of problems

arises as well. I n p r o x i m i t y of the phase t rans i t ion submanifo ld in the space of pa ram-

eters, the eodimension of the cri t ical manifold decreases at least one uni t and the a sympto t i c expansion ceases to converge even in the sense of dis t r ibut ions

of eqs. (3.5) and (3.12). One way out in this case would be to set up a recurs ive scheme, as is

done in renormal iza t ion group theory , whereby the ent i re in tegra l is redefined

at each step b y a suitable r epa rame t r i za t ion of the space of paramete rs . I n the nex t section we will discuss an a l te rna t ive m e t h o d which f rom the

ve ry s t a r t moves on the cri t ical submani io ld .

4. - The saddle point method in the critical region.

There exists a sui table diffeomorphic t r ans fo rma t ion of X c M, whereby the p rob lem s ta t ed a t the end of the previous section can be re fo rmula ted in

the following way. Consider contour integrals of the fo rm (iV oc 3)

(4.1) I(N, o,) =fg(z, exp =)] d z , P

where / and g are regular funct ions of z in a domain W_c C conta ining the contour F and of the complex p a r a m e t e r s a = (~1, . . . , o~) e D , D c_ C t D is

the complexif icat ion of ~t.

164 ~. AXET. and M. ~AS~TT~

:For fixed r an a s y m p t o t i c expansion (~/--~ oo) can be usual ly found b y the m e t h o d of s teepest descent: saddle points (zeros z : z(a) of ~,]) are identif ied and the expansion arises f r o m a local analysis in the neighbourhood of these points .

Of course, as ~ moves a round in the space of p a r a m e t e r domain D, the saddle points move a round themse lves and the fac tors of the s teepest descent expans ion are funct ions of a.

Such an expansion tends to be locally un i fo rm wi th respect to a, bu t it cannot be un i form for a n y c{ in a ne ighbourhood of a poin t a0, where

i) as r --> ao two or more cr i t ical points t end to coincide,

ii) a t r162 = ~o the resu l t ing mul t ip le cri t ical po in t provides a dominan t cont r ibu t ion in the s teepes t -descent expansion of I (~ , a~).

The p rob lem is thus to find out in this case a un i fo rmly val id a s y m p t o t i c expans ion for I.

I n o ther words, supposing the re is a pa i r (no, zo)e D • W such t h a t

] ' ( zo , ~o) = i"(zo , ore) . . . . . i ( , ) ( zo , Ore) = o

(superscr ipts denote z der ivat ives) , whereas

lu'+~)(zo, ~o) r 0

and such t h a t the saddle po in t Zo is dominan t in (4.1) when ~ ---- co, we wan t to wri te a fo rmula equiva len t to (3.7).

Fo r ~ sufficiently close to ~0, there are k zeros of ]'(z, r mul- t ip l i c i t i es - - ly ing in a smal l ne ighbourhood of zo, a m o n g which are to be found the dominan t cont r ibu tors to (4.1).

Such zeros converge to zo as ~ --> ~o and t h e y will be t e rmed the p r inc ipa l saddle points . The other zeros of ]'(z, ~,) will be re fe r red to as subsidiary saddle points .

The first step is to find out a m a p p i n g f rom the var iab le z to ano the r va- r iable, say u, which t r ans fo rms ](z, ~) to a s t anda rd form.

The existence of the l a t t e r is guaran teed b y a t heo rem of Levinson (is)

s ta t ing tha t , under the assumpt ions discussed above concerning ](z, ct), there is a t r a n s f o r m a t i o n

(4.2) u ---- u(z, ~)

un i fo rmly regular and one to one (so tha t the inverse t r ans fo rma t ion z ~-- z(u, or) exists and is regular) for all z in a region conta in ing a disc 5o - - {Iz - - zol ~ ~o}

(15) ~T. L~.vINso~: D~ke Math. J. , 28, 345 (1961); J. DIEI~I)O~T}T~: Calcul Infinitesimal (I-Iermann, Paris, 1968).

PATH INTEGRALS WITH COALESCENT DOMINANT SADDL:E POINTS 16~

and all ~ in a ne ighbuorhood / ) I of 0~o~ such tha t

1 ~'--1 _ _ _ _ ~ k + l _ _ (4.3) ](z,~) k § ~ .~(~)d§ F(u,~,A),

~ 1

i.e. ] is mapped into something which is an exact (k -~ 1)-th degree polynomial minus the simplest funct ion capable of having k near ly coincident critical points.

The funct ions ~(a), j ~ 1,...~ k - - 1 and A(~) are regular in 1)1 and ~j(ao) --~ 0 for all j .

The image of 5o in the u-plane contains a disc ~1--{[u[ < s} for a ED~ such tha t the inverse image of ~ contains a disc W1-- {]z -- z0[ < Q1}.

The key poin t is t ha t D1 m a y be chosen so small t h a t for a e D1 there ure k critical points of ] in W1 (counting multiplicities) and these are precisely those corresponding to the dominant saddle points.

Le t us define then

= ~k+i _ Z ~, U ' . (dA) �9 ~(u, ~) - F ( u , ~, A) - -A -- k § i ,-1

I f a is sufficiently close to a0, one m a y - - w i t h exponent ia l ly small e r ro r - - replace the contour F in (4.1) by a contour C all lying in W~. Then ! can be approx imated by the s tandard form

(4.5)

where

(4.6)

exp [lVA(~)]fe(u, ~) exp [Nr ~(~))] du, 7

G(u, r ~- g(z(u, a), a ) J (u , a ) ,

J(u, ~ ) = 8zlSu(u , a) being the Jacobian of t ransformat ion (4.2). G(u, a) is holomorphie for u ~ 61 and a E Ds, and ? is the image of C, which- -aga in with exponent ia l ly small d i sc repancy- -can be deformed to be ent i re ly lying in ~1.

The nex t step to perform is t hen the cont inuat ion of the functions ~(a), A(a), fo rmer ly defined for a r D1, as well as of u(z, ~), defined for (z, a) E W~xD1 into a larger subregion of A - W x D.

Assuming tha t , for no value of a =/: no, ]'(z, a) is ident ica l ly zero as a func- t ion of z, and t ha t for at least one value of ~ e D1 the k pr incipal saddle points are distinct, i t is a simple applicat ion of Riemunn's extension theorem to show the existence of a connected r e g i o n / 9 , o D1 with the p rope r ty tha t there exists an analyt ic set H , _c D , such tha t the principal saddle points m ay be continued f rom D1 along any pa th in D , \ H , always remaining in W.

Along a pa th having a l imit point in H , , two or more principal saddle points t end to coincide.

11 - l l Nuovo ~imenfo B .

1 6 6 F. A ~ L a n d M. I~ASETTI

I f the principal saddle points are cont inued in the above way, at no point a pr incipal saddle point merges into a subsiduary saddle point and, if the principal saddle points are cont inued along a closed loop in D . \ H . beginning and ending in D~, they are e i ther unchanged or pe rmuted among themselves.

Thus in D . the principal saddle points zr j = 1, ..., k, are identified up to an a rb i t r a ry permuta t ion .

The funct ions zj(a) are locally regular, except on the set H . , defined by the vanishing of the de te rminan t of the a l t e rnan t ma t r ix P = {_P~,.--~ z~-~; 1 ~<j, n ~< k} (indeed the de te rminan t vanishes if and only if z~ = zj for some i # j ) . ~(a), A(a) are, in fact , defined and regular on a cut version of D to be denoted by D#.

For (z, or) c A # - - W• eq. (4.3) can be wr i t t en as

1 /r - u~+~ - - ]~ r + A ( ~ ) - - f ( z , a ) = o ( 4 . 7 ) F(u, ~, A.) - - / (z , el k + 1

Since now ~ and A are known regular funct ions of a, (4.7) is a (k + 1) - th- order polynomial equat ion in u with coefficients which are regular funct ions of (z, a) ; therefore, there are k -k 1 solutions which are locally regular funct ions of ~, A and / (and hence of (z, a)) except at points at which the discriminant of (4.7) vanishes.

The la t te r occurs if and only if the funct ion

(4.s) A(z, 0,) = I I {/(z, ~) - /~(~)}

vanishes. In (4.8)/~(~) = F(u,(~), ~, A) ; {u,; i = 1, ..., k} denot ing the set of roots of

(4.9) /~(u, ~, A) ~ ~ u (u, ~(r = O.

I f we set

(4.10) A# ---- {(z, a) eA#]A(z, a) -= 0},

A# tu rns out to be an ana ly t ic set in A#. Then all solutions u of eq. (4.7) are locally regular funct ions of z and r for

(z, r ~ A# \A# . u(z, or) is indeed a regular funct ion in a cut version A, of A#. W ha t is finally required is t ha t u(z, a) should be defined for the z's in a

region containing all the principal saddle points , and such tha t u(z, a) m a y be cont inued from each saddle point (z~(a), a) into a neighbourhood of a line consisting of points (z, a), where z lies on a pa th of s teepest descent f rom z~(a), for at least a finite descent f rom the saddle point i tself ((~ pa th of steepest descent at cons tant a ~>).

PATH INT:EGI~AX,S W I T H COALESC:ENT DO~IINANT SADDLE POINTS 1 6 7

Let us denote by D** such a region. Tha t D** is maximal depends, of course, on which branches of A can be removed by the Riemann 's extension theorem. The general procedure is the following.

Iden t i fy first a set of branches B e A# radiat ing f rom (zo, no) and including - - a m o n g the o the r s - - t he branches (z~(a), a)~ i : 1, ..., k~ for sui tably restr ic ted r

There is a compact region V, such tha t V ~ A # - - - - B . The parts /~ of B not lying in A will be removed. Le t I y --~ V~/~.

u(z, ~) can be uniquely defined on 17 by pathwise cont inuat ion f rom A . . Consider t he n cont inuat ion f rom points (zi(a), a) E V down paths of steepest descent a t constant a. Le t the d is t inc t principM saddle points for fixed r be (z,, z~, ..., z , ; n•k} .

For sui tably res t r ic ted a, any cont inuat ion along the pa th of steepest descent at constant ~ f rom zi(a) is equivalent to a cont inua t ion along a pa th lying ent i re ly in V (i.e. one pa th can be deformed into the o ther wi thout cros- sing A#\B)~ provided the only poin t on the pa th of steepest descent f rom z~ or which (z, a) lies in A# is z~ itself.

Any subregion of JOg, where this holds, is what was denoted by D**. In conclusion, for ~ e D** the k principal saddle points are identified, the

coefficients ~(a), A(a) arc defined and regular and the Levinson t ransform (4.2) is defined.

Define ne x t D as the connected region D1 c_ j~ c_ D** with the p rope r ty tha t , for a e / ) , the integrMs fg(z, a ) e x p [N](z, ~)] dz exist when the con-

tour ~ consists of ~ finite par t of any pa th of steepest descent f rom ~ principal saddle point .

I f a e / ) , the deformed contour ~ can be t r u n c a t e d - - w i t h a locally uniform exponent ia l ly small e r ror - - , to consist only of paths of steepest descent de- scending f rom certain principal saddle points for a (at least) finite descent.

Denote the l a t t e r by ~(a). Let

I' a) = f g(z, exp dz ~,(a)

for a e / ) ; t hen [ ! - I'[ has a locally uniform, exponent ia l ly small bound. Notice t ha t ~(~) is not necessari ly a fixed contour, so I'(/~, a) is not neces-

sarily a regular funct ion of a. Since all subsidiary saddle points are ei ther i r re levant or subdominant ,

y(a) can be defined so tha t no subsidiary saddle point lies on it. Moreover, / ) _c D# and the t rans format ion u ~ u(z, a) can be defined in a

neighbourhood of ~(a) in such a way as to consist only of paths of steepest descent f rom principal saddle points ; and, since ~(~) passes through no sub- sidi~ry saddle points, a regular inverse t ransformat ion z ~ z(u, ~) m ay be defined for each individuM branch of y(a) descending f rom a principM saddle point.

1 6 8 ~. ~rv , L a n d M. RASETTI

We have then

(4.12) I'(N, ~) = exp [NA(a)]fG(u, ~) exp [Nq~(u, ~(r162 du , 0'(=)

where C'(a) is the image of y(~) under the mapping z --> u and G(u, a) is defined and regular in a neighbourhood of C'(r m a y be assumed to be given once g(z, ~) is given.

l~inally one can possibly extend C'(r162 to infini ty along paths of steepest descent of qS(u, ~(a)).

I n general, the terminal valleys of this new contour (say C(a)) are not uniquely defined, because of some arbitrariness in the definition of C'(=).

One may, however, dissect /3 into subregions /)1,/38, ..., in which C(a) can be defined with fixed te rminal valleys. In part icular, one of such regions, say /~o, contains no.

Final ly, for each a ' E 1) there exist a positive number No(~'), a neighborhood V(=') and ~ sequence of l imited bounds M~(~'), m ---- 0, 1, 2, ..., such tha t (~s)

(4.~3) exp [-- NA(~)] I'(N, ~) =

= x ) +

For N > Xo(~') and ~ e V(~'),

n--1 IR~(iv, =) l< M~(~') N-'~+'' Z -~-"+"~"+~'1~;~'(1"(~), X)l, (4.14)

where

with

X = X ( N , {X)~--(Xl, X 2 , . . . , X1r )

j : l ~ . . . ~ k - - 1 ,

F(r162 the image of C(a) under the t ransformat ion u -~ N-1/ck+~)~, has l imits at infinity.

Equat ion (4.13) finally defines, in the sense specified by (4.14), the asymp- totic expression of I' we were looking for.

In (4.13), (4.14)

.l'l=)

are generalized Airy functions, equal to N~,+~)/(k+x)f(--r [q~(GX)] de. /~(=)

(le) N. BL]~IST]~IN: J. Math. Mech., 17, 533 (1967).

PATIq INTEGRALS WITH COVLESCENT DOMINANT SADDLE POINTS 169

Moreover, the func t ions /~ ) ( a ) are solutions of the following recursion relations which derive in a s t ra ightforward way f rom a formal expansion of (4.12).

Le t G,,(u, cx), H,,(u, (x), P~)(a) denote sequences, for n ~ 0, 1, 2, ..., of func- t ions regular for u E ~1, a e D~, with

(4.16) Go(u, ~) ----- G(u, r .

Then k--i

(4.17) G.(u, ~) = ~ P~'(cc)u ~ § R . (u , r162 ~(~) )/~u , r=O

(4.18) G,+~(u, a) = -- ~H, (u , a ) /~u .

In (4.13), R,~(N, a) equals

e x p [ - - N A ( ~ ) ] l ' ( N , r ~ (--)~P~(cL)N- ( - - u ) ' e x p [ N q ~ ( u , ~ ) ] d u - - E r , ~::a 0 n ~ o

where E r denotes the in tegrated te rms f rom the end points of/~(a) which are exponent ia l ly small.

I t is, therefore , equal to

(4.19) ~--(m+l) f (~m.~l(U , ~) exp [Nq~(u, ~)] du ,

nam e ly - - modu lo the extra factor N-(m+l~--it has the form of the original in- tegral with G~+I replacing G.

The bound (4.14) can be obta ined in a s t ra ight forward way from (4.19) by t ransforming to the variable $ first, and using the definition (4.]5) of d~ k).

5 . - T h e c a s e k - - - - 2 .

In the case when k ---- 2, the funct ional recursion relat ions (4.17) and (4.18) can be reduced in a s t ra ightforward fashion to simple numerical recursions.

The vector ~ is now one-dimensional, i.e. it has only one component , say ~1--~ ~, and

(5.1) ~ / S u = u ~ -- ~.

We adopt for Go(U) a je t approximat ion, i.e. we assume it is in terpola ted to the required, arbi t rary , degree of accuracy by a polynomial of degree ~, namely

(5.2) Go(u) = ~ gsu 8. B~D

170 r. ~ L and ~. RASETTI

I n w h a t follows for c o m p u t a t i o n a l c o n v e n i e n c e we wil l select v ~- 3m ~- 2,

m>0, and the set {g,; s ---- 0, ..., 3m ~- 2} shall be assumed as given.

,Direct analysis of the recursion relations (4.17) and (4.18) leads to conclude

that, under hypothesis (5.2), unknown functions involved in them are

]) m functions G~(u), each a polynomial in u of degree v -- 3n, n = i , ..., m;

2) m ~- i functions H~(u), each a polynomial in u of degree v -- 3n -- 2,

n = O , . . . , m ;

3) 2(m -}- 1) c o n s t a n t s B(. ~ iO(~ I~, n = 0, .. . , m.

The r ecu r s ion re la t ions t h e m s e l v e s in t he p r e s e n t case read

(5.3)

W e wr i t e

6~.(u) /,co) + ~,, x~nc,,) = ur'~i + (u2 - - ~ ' ~ " ' n = O, . . . , m ;

( ~ . + I ( U ) = - - ~ H n ( U ) / ~ U ,

G~(u) = o , 2~,(u) = o , q > m @ l , _p~o) = p(1) = 0 - - q

y - - ~

(5.4) G.(u) = ~ .,"("b"w , 0 < n < m , a = 0

where, for cons i s t ency w i t h (5.2), g~~ for a l l s f r o m 0 to v. Ana logous ly ,

we set v - - 3 . - - 2

(5.5) H , ( u ) = ~: h?'u', f - -0

w h e r e b y ~ - - 3 . - - 2

(5.6) ~ H , ( u ) / ~ u : ~, thp)u t-a.

B y e n t e r i n g (5.4)-(5.6) i n t o (5.3), b y c o m p a r i n g t he coefficients of equa l powers

of u on b o t h sides, t he l a t t e r t u r n s i n to a set of n u m e r i c a l r ecurs ion r e l a t i o n s :

(5.7)

gy = ~:o,- ~h:,,

g~.) (1) = P ' . - ~ h i " ' ,

g(.~ __ h(.) _ r.h(.) r t - ' 2 o r '

g ( . ) _ h ( . ) V--3.--I -- V--3.--1 '

g(.) i(.) y - - 3 . ~ g%--3" '

~ ( . ) _ _ --(. '{-1) t - - - - g s - a I s ,

2 < r < v - - 3 n - - 2 ,

l < s < v - - 3 n - - 2,

for a l l n f r o m 0 to m.

P A T H I N T ] ~ G R A L S W I T I t C O A L E S C E N T D O Y I I N A ~ T S A D D L E P O I N T 8 171

E q u a t i o n s (5.7) are a s y s t e m of (v ~- 1)(~ ~- 4)/6 : 3(m -[- 1)(m ~- 2)/2 l inear equa t ions in an equal n u m b e r of u n k n o w n s :

(v ~- 1)(v - - 2)/6 ---- 3m(m + 1)/2 for t he g~>'s, n va 0 ,

(~ -~ 1)/3 : m ~- 1 for e~ch of t h e ~P~~ /)~(~ and -o~(") �9

Not ice t h a t on ly t h e h~o ") have been l i s ted a m o n g t he unknowns , in t h a t t he las t set of equa t ions in (5.7) de t e rmines all t h e o the r h~ ), s > l in t e rms o f a ( n + l )

~]8--I "

:Notice also t h a t sy s t em (5.7) is no t homogeneous , because of t he fac tors g~O) en t e r ing p a r t of the equa t ions .

U p o n e l imina t ion of t he ~("~ s > l , and i n t roduc ing t he more conven ien t no t a t i on _P~~ . ~.~)(~ _P~I). - - ~.~ .~o~(")-- ~.~)(~), t he sys t em can be w r i t t e n as

gi"'

(5.s) C'

= O (o) _/:O(*)

(3) ~(n+l)~ 2

g(.) _(.+1) ,, . - 3 . - 1 = - - g ~ - 3 , , - g L ~ - - 3 n - - 3) ,

g(.~ __ _(.+1> ,,v __ 3n -- 2) v - s . - - - - Y~ -s . - s /L

3 ~ r < v - - 3 n - - 2,

for O < n < ( v - - 5 ) / 3 , and

(5.9)

g~o~) O~O) _ $0r

g(~) _ O(U

g~) = 0(3>

We i n t r o d u c e now t h e fo l lowing ansa t z :

(5.]0) [(v+s--sn--~')/2'i]

n ~,a e-- [r/3•, a- - r--3q,

for l<n~[[(~--r)]3], O < r ~ < v - - 3 and

(5.11) Q~> / l~n x" .~,t. $t-1 ~--1

172 F. Ax.~,. and ~. IRAS]~TTI

for 0 < 1 < 2 and O < n < m, where ~x~ denotes the m a x i m u m in teger less than

or equa l t o x. Once more , b y en te r ing (5.10) a n d (5.11) in to (5.8)~ (5.9) and c o m p a r i n g t he

coefficients of equal powers of ~ on b o t h sides~ t h e l a t t e r t u r n in to a v e r y large

s y s t e m in t he u n k n o w n s z~,e, q . . A l e n g t h y a lgebraic process of successive r educ t ions allows one to e l imina tc

~ a n d we are f inal ly left wi th the fo l lowing v e r y compl i ca t ed re- all of t h e z~,~ curs ion re l a t ion for the {q~'~; 1 ---- O, 1, 2; 0 < k < m ; l~<n~< ~ ( ~ ' m . ~ ~ , ~ , where .,%~- ~'~----

= ~(~ - - 3k + 2 - - I)12~ = ~(3(m -- k) -- 1 + 4)/2]],

(5.12)

n + l n ,,yn+2 qk+~ ---- P~+~ § (n § 1) ~ ,

n n .+I

n n + l r~+ 1 ---- p~+~ § nrk ,

with (( in i t ia l ~) condi t ions

(5.13) p~ = s(s § 1),

r~ = s(s + 2) ,

l < s < ~(3~ + 1)/2]],

1 <~ < 113~/2]1,

1 < s < [ [ (3m - - 1) /2 ] ] ,

where we have set

J,O J J,1 J {~J,2 q ~ - ~ q . , p . - - q . , r ; , - - ~ . .

No t i ce t h a t condi t ions (5.13) are cons i s ten t w i t h (5.12) as we]] as w i t h t he r e q u i r e m e n t t h a t q~ = p] = r~ = 1 for all a l lowed s (which in t u r n g u a r a n t e e

-{0) s 8 t he cons i s t ency cond i t ion g~ ---- gj for all 0 < j < v). Al l t he var iables q~, p '

t h u s t u r n ou t to be in tegers . The c o m p l e x i t y of (5.12) arises f r o m t he u p p e r l imits , in t h a t t he r ange of

values a l lowed to t he u p p e r i ndex is different for t h e t h r ee var iables . Thus t he re is l i t t l e or no hope t o solve (5.12), in genera l , b y a closed a l g o r i t h m a n d

one shou ld have to resor t t o n u m e r i c a l analys is . I t o w e v e r , in t h e spir i t of t he

je t a p p r o x i m a t i o n (5.2) one m a y a d o p t t he fo l lowing p rocedure :

i) solve t he recurs ion for a n y in teger v' >> v,

ii) d i s regard t he t e r m s of the resu l t ing sequence outs ide t he a l lowed

range of indices.

I n th i s w a y the p r o b l e m of t he u p p e r b o u n d a r y condi t ions is r e m o v e d (at t he pr ice of a mino r loss of prec is ion on t he {q;m~,~; l __ 0 ,1 , 2; O < k < m } )

and one can d e c o u p l e - - w i t h o u t a n y more cons t r a in t s - - (5 .12 ) in to t h r e e re-

PATH INTEGRALS WITH COALESCENT DOMINANT SADDLE POINTS 173

l a t i ons , e a c h i n v o l v i n g a s ingle v a r i a b l e :

(5.14)

(5.15)

(5.16)

~- 2 _.+a .+1 . . " + ~ - - n ( n + )U~ , q~+~ qk+~ -]- (2n + 1 )~+1

.+1 " + 2 ( n ~+~ = + 1)p ,+ , - - (n + 1)(n + 2 ) p " k +'~ , P~+~ P~+~

n " } ' l - - r n n + l ~ ~ ~ + 9 rk+s - - k+3 - - n r ~ + ~ + 3(n + l , r k + 2 - -

- - 3 ( n + 1)(n + 2~-"+8,'~+1 + (n + 1)(n + 2)(n + 3)r~ +4

w i t h o n l y t h e i n i t i a l c o n d i t i o n s q; = p~ = r o = 1 le f t .

N o t i c e t h a t we need to so lve o n l y (5.14) a n d (5.15), in o r d e r to d e t e r m i n e

c o m p l e t e l y t h e p a r a m e t e r s / ~ o ~ a n d p(1) w h i c h e n t e r t h e a s y m p t o t i c f o r m of t h e

i n t e g r a l (4.13).

I n t h e p r e s e n t ease i n d e e d t h e l a t t e r r e a d s

(5.17) exp [-- NA(a)] I'(N, e) =

= 2: (-1) 'N-'~+'~7>(c(~), x) ~ ~:,(~)~v -~ + R,~+~(N, ~1, r=O,l n=O

w h e r e X - - (X~ = X) w i t h X ---- 2Vt}.

T h e r e c u r s i o n r e l a t i o n s (5.14), (5.15) a r e ea s i l y so lved b y i t e r a t i o n if v is

n o t t o o l a rge (see t a b l e I for a l i s t of so lu t ions u p t o v ---- 17 (m ---- 5)).

TABLE I.

a) {q~; 0 < k < 5 ; l < n < ~ ( 1 9 - - 3 k ) / 2 ~ } .

1 2 3 4 5 6 7 8 9

0 1 1 1 1 1 1 1 1 1 1 1 4 9 16 25 36 49 64 2 4 28 100 260 560 1064 3 28 280 1380 4760 13160 4 280 3640 22960 5 3640 58240

b) {p~; 0 < k < 5 ; l < n < ~ ( 1 8 - - 3k)/2~}.

k n

1 2 3 4 5 6 7 8 9

0 1 1 1 1 1 1 1 1 1 1 2 6 12 20 30 42 56 2 10 52 160 380 770 1400 3 80 600 2 520 7 840 4 880 8680 46480 5 12 320

174 F. A X ~ L & n ~ M. R A S E T T I

I f a ve r y large ~ is required, the solutions can be found in terms of genera t ing funct ions of the form

2M 8M+1

(5.18) t~a'(x, y) : ~ ~ q~"x~y ', I -~ O, 1, k~O n ~ l

where M >> 1 is such tha t v ~ 6M ~- 2. The la t te r obey second-order differential equations in the variable y - -whose

solutions are parametr ized by x. General ly spe~king such equations lead to functions which are singular

at y ~ O . The series w h e r e b y - - b y comparison with (5AS)--~he coefficients q~'~ should

be obta ined have, therefore , to be defined in a sui tably cut complex y-plane. The equutions have the form

- - ~ ( 1 - - y ) F(~)(x, y) ~- g~(x, y)

where e (l) and d a) are known constants , and g,(x, y) is a known funct ion, depending only on the first te rms:

k

(5.20) q~" ---- 1-[ ( 3 j - - (2 - - 1)), q~ t= n(n + 1).

Equa t ion (5.19) is z nonhomogeneous version of )/Ieijer's G-function equation (17). The G-functions, in tu rn , can be wri t ten us a finite linear combinat ion of

generalized hypergeometr ic functions, whose coefficients are known. Thus, in principle, an explici t closed form for the {q~"; 1 ---- 0, 1} can indeed

be wri t ten down.

The la t te r is, however, so complicated to be hard ly competi t ive wi th the numerical results.

Work is in progress on the possibility of expressing the coefficients q~Z in a more usable compact (asymptotic) form.

6 . - C o n c l u s i o n s .

The pa th integral me thod provides the most appropr ia te f ramework for bo th semi-classical quant izat ion and integral representa t ion of par t i t ion and correlat ion functions.

In practice, however, an impor tan t l imita t ion to its applicabil i ty occurs.

(17) A. I~RD~.~LYI, W. I~r and F. 0BERHETTINGER -" Higher Transcendental E~nctions, VoL 1 (McGraw-Hill, New York, N.Y., 1955). Sect. 5.

PATH INTEGRALS W I T H COALESCENT DOMINANT SADDLE POINTS 1 7 5

Essent ia l ly the only suitable m e t h o d for per forming the funct ional in tegrat ion is resor t ing to the s teepest -descent technique. Where the sys t em is controlled by a set of parameters~ the lut ter allows us, in regular eases~ to set up a reeursive scheme such as t h a t adopted in renormal iza t ion g roup calculations~ and es- sential ly provides a convergent procedure for an es t ima te of the integral .

I t is not so, however~ when regions of the space of pa r ame te r s exist~ cor- responding to which several saddle points coalesce~ p roduc ing a singular si- tua t ion .

I n this p a p e r we discussed a proeedure~ designed jus t to deal wi th such pathological ease~ based on the concept of blowing up the cri t ical poin t into a set of separa ted points, still s ta t ionary , and then pe r fo rming the in tegra t ion over a p a t h which is the (possibly disjoint) union of several pa ths of s teepest descent each descending f rom one of these principul saddle points and each character ized b y constant puramete r . The example thorough ly analysed in sect. 5~ corresponding to two coalescent s ta t ionary points~ should be the basis for a detai led discussion of phase t ransi t ions in sys tems isomorphic to a: ~04:

quan tum field theory . Work is in progress along these lines.

~The authors wish to t h a n k Dr. M. C. GUTZWILLE~ for in teres t ing comments and suggestions and Dr. C. AGNES for precious help wi th the reeursion relat ions

of sect. 5.

�9 R I A S S U N T O

I1 metodo delia fuse stazionaria, utilizzato, in generale, per ealeolare integrali oseil- lanti quali gli integrali di cammino di Feynman, ineontra difiieolts di applicazione quando la fuse dell'integrando - - che ~ controllata dalla posizione di un punto nello spazio dei parametri - - presenta una configurazione earatterizzata dal fatto ehe diversi punti a selia dominanti si fondono in un punto solo. Si piopone qui un metodo per true- tare tale situazione singolare e, in partieolare, si analizza in dettaglio il caso di due punti a selia eoaleseenti.

Pe3ioMe He noJIy~leHO.