I L NUOVO CIMENTO

VoL. 63 A, N. 1

1 Maggio 1981

Multidimensional Perturbation Theory, the Field Theory Limit and Vacuum Tunnelling. H . J . W . 51i)LLE]~-KIRsTE~

Department o] Physics, University o] Kaiserslantern - 6750 Kaiserslautern, W. Germany (ricevuto il 28 Ottobre 1980)

S u m m a r y . - - A very general multidimensional perturbation technique is developed which embraces the familiar oscillator and W K B approximations as well as their continuation and is, therefore, applicable in weak-coupling (low temperature) cases. After an introductory application in classical mechanics, the method is used for the explicit solution of the multidimensional wave equation of a particle in both its classically allowed and classically forbidden domains. Field-theoretic models arc studied b y introducing a lattice and taking the limit of an infinite number of degrees of freedom. The procedure allows a systematic calculation of higher-order corrections to the leading one-quantum variable.dependent wave functional and its eigenenergies. Finally the amplitude describing the tunneling of field configurations from one vacuum to another is calculated, and the relation between the tunneling amplitude and instanton contributions is discussed.

1. -

Introduction.

S e m i - c l a s s i c a l m e t h o d s h a v e a t t r a c t e d c o n s i d e r a b l e i n t e r e s t r e c e n t l y as a m e a n s for f i n d i n g p a r t i c l e l i k e s t a t e s i n field t h e o r y . T h i s i n t e r e s t h a s g r o w n p a r t i c u l a r l y since i t h a s b e e n r e a l i z e d t h a t i n s t a n t o n s o l u t i o n s c a n p l a y a n i m p o r t a n t role i n t h e n a t u r e of t h e g r o u n d s t a t e of a n o n - A b e l i a n g a u g e t h e o r y . A n e a r l y i n v e s t i g a t i o n i n t h i s d i r e c t i o n w a s u n d e r t a k e n b y DAs~m~ et aL (1),

(1) R. F. DASHEN, B. HASSLACHEI~ and A. N~vv.u: Phys. Rev. D, 10, 4114, 4130, 4138 (1974). 1 -

ll

Nuovo

Oimento

A.

1

9.

I[. J. W. ~r

who explored the applicability of the W K B approximation from a functional integral approach and thus needed a multidimensional generalization of the classic method. I n their work D A s m ~ et a~. (1) used the semi-classical forrealism developed b y GVTZWrLL~ (~) and MASLOV (3), which avoids the explicit introduction of W K B eigenstates. More recently, however, the explicit construction of W K B wave functionals was t a k e n up b y GE]~VAIS and SAKITA (4) and BITA~ and CHA~a (5) in their studies of v a c u u m tunnelling in field theory, and in subsequent papers (6,7) these authors discussed various aspects and applications of their techniques. The methods we shall develop below (based on a multidimensional p e r t u r b a t i o n approach t h a t we suggested recently (s)) are closely related to the procedures employed b y these authors in their first papers (4,5), though much more general However, most of the associated subtleties, such as the distinction between strong- and weak-coupling W K B approximations etc., have already been discussed in the work of Dashen et al. (1). I n the following our main interest is focused on the development of a powerful and systematic multidimensional p e r t u r b a t i o n t h e o r y and its ~pplication (in the limit of an infinite n u m b e r of dimensions) to the problem of v a c u u m tunnelling in a simple model field theory. The m e t h o d - - d e v e l o p e d in successive stages from classical mechanics (sect. 2) to q u a n t u m mechanics (sect. 3) and field t h e o r y (sect. 4 ) ~ i s a direct generalization of the perturbation procedure t h a t we developed (8) and tested (~o) for one-dimensional wave equations and which (due to its systematic formulation) p e r m i t t e d the straightforward computation of such nontrivial properties as the large-order behaviour of the cigenvalue expansion (11). I n each case we assume t h a t some approxim a t e form of the solution is known (e.g. the associated classical solution) and t h e n formulate a systematic procedure for the computation of p e r t u r b a t i o n corrections. Proceeding along parallel lines, we derive in each case pairs of M. CTUTZWILL]~R:J. Math. Phys. (N. Y.), 8, 1979 (1967); 1O, 1004 (1969); 11, 1791 (1970); 12, 343 (1971). (3) V. ]~[ASLOV: Teov. Mat. Eiz., 2, 30 (1970) (English translation: Theor. Math. Phys. (USSI~), 2, 21 (1970)) ; Theory el Disturbances and Asymptotic Methods (Moscow, 1965) ; Thdorie des perturbations et mdthods asymptotiques (Paris, 1972). (4) J.-L. GERVAIS and B. SAKITA: Phys. Rev. D, 16, 3507 (1977). (5) K.M. BI~a~ and S.-J. CHANG: Phys. l~ev. D, 17, 486 (1978); 18, 435 (1978). (e) J.-L. G~.~vAIs and B. SAKITA: Phys. Rev. D, 18, 453 (1978); H. DE VE~A, J.-L. G~VAIS and B. SAKITA: Nuel. Phys. B, 139, 20 (1978); B. SAKI~A: Tunneling phenomena in gauge ]ield theories, CCNY-Report No. HEP-78/20 (1978). (7) S.-J. CHANG: Vacuum tunneZing in Minkowski space, ILL-Report No. (TH)-77-28 (1977), unpublished. (8) H. J. W. M~LLER-KIRSTEN: Phys..Lett. A, 70, 383 (1979). (9) A general discription with a list of all previous publications has been given in H. J. W. MOLLER-KIRST~: Phys. l~ev. D, 22, 1952 (1980). (lo) H. J. W. ~/[0LZER-KIRSTE:~and S. K. Bosv.: J. Math. Phys. (N. Y.), 20, 2471, 1878 (1979). (11) H. J. W. Mi)LLER-KIRsTEN: Phys. Rev. D, 22, 1962 (1980). (2)

M U L T I D I M E N S I O N A L P E R T U R B A T I O N THEORY, T H E F I E L D T H E O R Y L I M I T ETC.

S

oscillatorlike and WKB-like solutions (valid in adjoining domains) together with one and the same equation determining (as appropriate) an eigenvalue or an auxiliary parameter. We also consider the continuation of these solutions in regions of common validity. W e include (with sect. 2) a brief account of our procedure applied to classical mechanics, as this m a y help in understanding the mathematically more intricate cases of q u a n t u m mechanics and field theory. Section 3 treats the same problem as Gv.RVAIS and SAKITA (4), although, we believe, in a more complete and systematic m a n n e r permitting numerous applications in various fields. Finally, in sect. 4, we consider v a c u u m tunnelling in field t h e o r y in the m a n n e r of Bitar and Chang (~). H e r e our starting point is the SchrSdingerlike equation for the density m a t r i x element which we then solve in the weakcoupling (low temperature) domain (this seems to be the case of relevance in field t h e o r y (1)), and we demonstrate t h a t the procedures developed previously for an a r b i t r a r y n u m b e r of dimensions can be generalized to the continuum limit. Various problems associated with this limit (such as the infinite energy of the vacuum) are discussed towards the end. Finally, in sect. 5, we summarize our results.

2 . - Multidimensional perturbation theory in classical mechanics: complete solution of the equation of motion. 2"1. F o r m u l a t i o n . - I n classical mechanics one is primarily interested in r motion of a mass point along a p a t h which is traced out as a function of some p a r a m e t e r t (e.g. time, angle, etc.). The considerations of a p e r t u r b a t i o n t h e o r y make sense if some approximate form of the p a t h is known or easily derivable. Thus the p a t h described b y a charged particle in a synchrotron is approximately circular, and this approximate p a t h can be described b y the co-ordinate r of the particle as a function of the angle 0 of its position on the path, i.e. r(O). Another example is the approximation of the quantum-mechanical motion of a particle b y its classical or most probable escape p a t h as a function of the time variable t, i.e. r(t). The problem of perturbation t h e o r y is the calculation of the deviations v of the exact p a t h R from the approximate p a t h r. This is the problem we shall be concerned with in a later section. I n spite of its a p p a r e n t elementarity we know no reference in the literature in which the considerations given below have been given previously except for the general direction of the reasoning which can be found in the work of Guignard (12). I t will be shown in the following t h a t the solution of this classical problem requires the solution of v e r y complicated equations (1~) G. GUIC~NARD:A general treatment o] resonances in accelerators, CERN-Report 78-11 (1978), unpublished.

4

E . .I. W. I~IIJLLER-KIRSTEN

(equivalent to complicated wave equations), and this complexity is presumably a reason for the n o n p o p u l a r i t y of this subject in texts on classical mechanics. As stated above, we write

R=r+v,

(1)

where r is calculable as in one-dimensional classical mechanics, i.e. from a potential V(r) b y solving

(2)

~o~ + V(r) = Eo,

and iv[ is assumed to be small compared with [r i. F o r convenience we set the mass of the particle equM to one uncl denote differentiation with respect to t b y a dot. /~o, of course, is the appropriate energy. The Hamiltonian of the complete system is

(3) and we write the energy E (4)

E : Eo + E ' ,

E being constant, b u t not Eo, E ' separately, in general. Here, however, E0 can be looked at as the dominant t e r m in an expansion of E. I t is, therefore, also constant, although this is n o t required for the considerations below. Substituting (1) into (3) and expanding V(R) around V(r), we obtain

(5)

~- v

+~.~+v.(Vv)~+~\~-~]

r

+ v(r)

I t is convenient though not essential to choose v such t h a t

(6)

b-i~ + v.(VV)r = O,

b./~--v-b':

i.e.

O.

Then, differentiating (5) with respect to v~, we obtain

(7)

d dv~

~

)

1

~,---- -- V~'(r)v~-- ]: ( n - - l ) ' V,~...;._,(r)% ... %_,. ,~=3 9

E q u a t i o n (7) is the set of equations describing the transverse motion of the particle, i.e. away from the p a t h r(t). I n p~rtieular, we observe t h a t the deviation in any particular direction i is coupled through the potential to the deviations along the other directions. W e thus have a set of coupled and nonlinear second-order differential equations. The solution of these equations is in general possible only with help of p e r t u r b a t i o n theory.

MULTIDIMENSIONAL

PERTUI~BATION THEORY~

THE

FIELD THEORY

LIMIT ]~TC.

Before we proceed, it is convenient to define a Hamiltonian H• for the transverse motion. We write (with p~ ~ 6~) (8)

H • = Ho +

B1,

where (summation over identical indices being understood) 1

1

1~,

H. = ~p~ + ~ V,j (r)v, vs,

(9) n~8

H a m i l t o n ' s canonical equations (10)

pi~t--

@pi

and i 1 TT(n} t .=~ ( n - - 1) ! v~h'"~"-*(r) vh "'" v~._~

then lead back to (7). We will first of all ignore //1. Then later we will develop a perturbation technique to take into account the perturbing H a m i l t o n i a n //1. Our first step, therefore, consists in solving the equation of transverse motion derived from 11o, i.e.

(11)

+

=

o.

E q u a t i o n (11) is still a set of complicated (though linearized) equations; it is of the same degree of complexity as a system of multichannel wave equations. R e c e n t l y we have developed p e r t u r b a t i v e techniques (18) for solving these equations. These methods will not be repeated here. Instead we content ourselves here with the following approximations. We rewrite (11) as (12)

~, + -.vc""~, = _ ~: vc,J,, ~ O(e)

(no summation on the left-hand side). We assume t h a t the off-diagonal terms on the right of (12) can be t r e a t e d p e r t u r b a t i v e l y (this is the assumption of weak channel coupling or weak coupling of degrees of freedom). The solution v~

(13) II. J. ~V. IV[t)LLER-KIRSTENand R. MOLLER: Phys. Rev. D, 20, 2541 (1979).

6

H , J. W. M I ) L L E R - K I R S T E N

of (12) can then be written (13)

v~ : v(~ -[- O(s),

where v(~ i is a solution of (14)

::(o) ~_ V(~). (o) ~ - - 0 "Or i S "Us

"

This is now a second-order differential equation which is of the same degree of complexity as a one-dimensional Schr6dinger equation, since V , (r(t)) is a function of t. This equation can be solved at least perturbatively. Since r(t) is k n o w n from the v e r y beginning, we know the potential V ,I~)( r( )t) as a function of t. I n general it is a continuous function consisting of one or more wiggles, as shown in fig. 1. The solutions of (14) consist of various pieces joined together in regions of c o m m o n validity where one t y p e merges into another. :From the s t u d y of (~simple )~ s t a n d a r d equations such as the Mathieu equation (~4) one

t Fig. 1.

-

The potential V , ,(2) as a function of t.

knows t h a t these pieces are of two t y p e s : a) I n the neighbourhood of an ext r e m u m the solution is expressible in terms of H e r m i t e or parabolic cylinder functions (the variables of these functions are complex in the case of stable extrema, i.e. minima, and real in the case of instability points, i.e. maxima). These functions result from the simple fact t h a t in the neighbourhood of an e x t r e m u m the potential can be a p p r o x i m a t e d b y an harmonic oscillator. b) A w a y from the neighbourhood of an e x t r e m u m the solution is of Green's type, i.e. W K B - l i k e and m a y have to be continued across Stokes discontinuities. 2"2. OseiUatorlike solutions. - We now consider solutions of the first type. I f we expand V i(~) a r o u n d the point t = to (which need n o t even have to be the t point of an extremum), eq. (14) becomes

(15)

-.

-+- Vii (to)

~V~2~, 3-

----ii

I, ",--ii

1 (t-oo,.,

]-'~

0 ,

I )

(la) R. B. DINGLE and H. J. W. ~[ULLER: J. Reine Anyew. Math., 211, 11 (1962).

MULTIDIMENSIONAL

PERTURBATION

THEORY,

THE

FIELD

THEORY

LIMIT

]~TC.

7

where the second superscript on V~ ) implies differentiation with respect to t, and the derivative is t a k e n at t = to. W e now set (16)

h, = { - 2v~7'2'(to)} ,~

and change the independent variable in (15) to (17)

17(2111} \ 9 ti

~|.

~o, = h~ t - to +

The equation then becomes (18)

d

Vi

deo~ +

V~7(t~ + ~

*~

__

'

~ (o7__

.u, - ~=3 n ! ,.,, ~gc~(~

co -]- " "ha ~

]| v~ ~ (o7 .

F o r large values of h~ the right-hand side of this equation is of O(1/h~) compared with the left-hand side and can, therefore, be neglected to a first approximation. I f we set (19)

-

1[ -

h? v ~ ' ( o ) +

IT(2){1)' ' " / ]

1

j - ] q,,

/r

the zeroth-order approximation of eq. (18) is 2 (0)(0}

rl

dd wv~l ~- -~- [2 q ' - - T~J

(20)

2

v(~176 ' -- 0 .

A solution of this equation is the parabolic cylinder function D~(q,_17(os~), i.e. (21)

v(~~176= D~(~,_I)(~).

F o r the complete solution we rewrite (19) as (22)

1

(27

iT(29171 '" I

1

A

where A is a function of q~ ~nd h~ to be determined b y perturba$ion theory. Once A has been determined, eq. (22) can be used to calculate q~ as a function of the potential parameters and h~. We skip the derivation of the complete p e r t u r b a t i o n solution, which is described in detail in ref. (9). Thus, in the context of the present discussion, we restrict ourselves for reasons of b r e v i t y to the dominant contributions. W e write the solution having (21) as its dom-

8

H.

J.

~r. MULLI~I~-KIRST]~lq

inant contribubion: ~ t ~ t , he; ~ t ) . v(~

V(0) ~

This solution is valid in the region around t = to. A second, linearly indep e n d e n t solution in the same domain is obtained b y changing the signs of co and h throughout, as can be seen b y looking at (18) and (20). We write this solution ,

=- v T ) ( q , - - h i ; - - o ~ D .

I t can also be seen from (18) and (20) t h a t ~ further pair of solutions is obtained b y the interchanges (el --> i e o i ,

(23)

q, --> - - q~,

h~ --> - - ih~

qi --> - - q i ,

hi -+ - - i h i .

and ~i

-+

--

io)i ,

The general solution of (19) can, therefore, be written (24)

~/p(0) A (~ --f ~

t

-4- .~(o)~(o) i '

where A(~ and .~(0) are constants. Since we know the solutions of (18), our n e x t step is to calculate the solutions of (11) or (12). W e write these (25)

r

~ Aivi -~ A~vi

(no s u m m a t i o n ) ,

where v~ = v~~ + o(e) and ~ = O~~ Jr O(e), and, as mentioned above, we do not repeat details (~) here of the explicit calculation of the contributions

of o(e). Since we know the solutions r derived from H o , o u r n e x t step is to develop a p e r t u r b a t i o n procedure in order to t a k e into account the perturbing Hamilt o n i a n //1. F o r this reason we introduce the m o m e n t u m ~ defined b y (26)

~ i = Ai~)~ -t- ~ v i

F r o m (25) and (26) we can calculate A~ and A~, i.e.

A,-- ~ (27)

(~,~,--~,~,),

Wiv,,

_~,-

~,]

~ (~, ~ , - ~,~,), W[v,, ~,]

(no summation) .

M U L T I D I M E N S I O N A L PERTURBATION TH:EORY~ T I I E :FIELD THEORY L I M I T ETC.

9

where W is thc constant Wronskian, i.e. W[v~, ~] :

v~ -- i~.

Next we define a Poisson bracket [B, C] by [B, V] =

~,~

~

~

~

.

Then it is readily verified that A{, .~{ satisfy the following relations: [A~, A~] : 0 ,

[.~,, .~]

(28)

= o,

[A,, _~] --

~" w[v,, ~,]

~(ow, using (25) and replacing A~, .4~ by as yet unknown t-dependent functions A~(t), .4~(t), we can re-express H~ of eq. (9), i.e.

in terms of A~(t) and .4{(t), i.e. H , = H~(A,(t), .4,(t), t).

Our procedure is, therefore, to replace the constants A~, A~ in the solution (25) b y t-dependent quantities A~(t), .4~(t) which are such that they reduce to A~, A~ in the limit H~-->0. Thus, replacing A~ by A~(r ~ i , t) and correspondingly A;, we have (3O)

A,(t)

8A,(t) --

8t

+

[A,(t),H•

where we have used ~H.

~i

=

~H• %~"

On the other hand, for He we have (31)

0 --

~cAi(t) c~t

+ [Ai(t), Ho] .

10

H.J.

~v.

MULLER-KIRSTiEN

Subtracting (31) from (30), we obtain d,(t) = [A,(t), H,].

(32)

This relation constitutes a set of equations from which the functions A~(t) can be calculated. A simple manipulation allows us to write (33)

A,(t) = ~ [A,(t), Aj(t)] ~

+ ~ [A,(t), _~r

J

~A~(t ) --

W[v~,

~.;I~(t)

on using (28). We have, therefore, t

A~(t) =

(34)

f

dt

~H1

WIve, ~ ] ~ ( t )

-}- const,

where the constant is our previous, t-independent integration constant A~. Knowing all quantities A~(t) and thus the solutions r we obtain the corresponding solutions ~ involving quantities .~(t) by changing the signs of h~ and ~o~ throughout. The whole problem has then been solved for values of t close to some value to in the vicinity of an extremum of V. (~)(t). 2"3. WKB-like solutions. - We now come to solutions of t y p e b) which are valid in regions of t away from to. F o r this reason we return to eq. (12) and derive a new set of solutions u~ which we write in the same form as (13), i.e. (35)

u~ = uT + O(e).

The contributions of O(e) can again be calculated by perturbation theory, as we have demonstrated elsewhere (~3). The contribution u~~ of course, is a solution of (36)

/~(o) _]_ Vr i

li

~

=-- 0

"

For later convenience our procedure requires the following manipulation. We rewrite (36) as

i~(~ + (V(')tt ~ + (t--to)Vt'~(1)(t ) + h'~(t)}u{~ i ( i i ~ OJ it 0

(37)

O,

where

h4 Q(t) =--V(~)ct~V(2)(t ~ - - ( t - - t O;~V(2~(1)ct il x / t t x'O! ii ~ 01

(38)

"

B y substituting (22) for V(2)lt~ , ~ o,, our eq. (37) becomes

(39)

...,o)

{~ 2

--

I ~(2)u),~,

-.

~ooj +

hiI2(t)

}u ,

=

0.

MULTIDIMENSIONAL PERTURBATION THEORY, THE FIELD THEORY LIMIT ETC.

11

~ e x t we set (40) so t h a t Z(~~ (41)

d:'zi(~

satisfies

92

t 89d'zi(~

h-V +2~h,[~9( ) ] - h i -

X~o5q_

ih) ~2(t)

+

2 [9(t)]~

" 0.

W e can rewrite this equation in the f o r m (42)

-~ ~.,z,.(0)__ - ~2 ~{d2z~ - ~ - ~ + ,t h~X) ~ )

where

(43)

~,

- - - 4iEt2(t)]~

- - 2 ( t - - to) "

rii

,.oJ

k~O)

h)

B y construction Ah~ is at m o s t of 0(0) in h 2 for h~-+ cx~. H e n c e to a first a p p r o x i m a t i o n we can neglect the t e r m s on the r i g h t - h a n d side of (42) a n d write for t h e solution to t h a t order (44)

~(o)(o)

. (o)

where Z(~ tq is the solution of (45)

~ . (o)

i.e. (46)

. ~o) 1 Z,q,---- ~ e x p

_ _ f dt ~

{

q'

-,4, ,oo,+ 2(t-to!V,,(2)(1)(to!/1 o. r(8)cl:v,, h)

h2

JJ'

where we h a v e chosen the overall constant to be one. I t is obvious t h a t simplifications arise if to is chosen such t h a t V~5(1)(to) ~- O. I n order to obtain the complete solution X~~ we again h a v e to resort to p e r t u r b a t i o n theory. The procedure is described in detail in ref. (~,1o) and so will not be r e p e a t e d here. The q u a n t i t y d is again determined as a function of q~ a n d is identical with t h e expression obtained previously in conjunction with our first t y p e of solutions (9,10). E q u a t i o n (22) again serves as the equation f r o m which the auxiliary p a r a m e t e r q~ is to be d e t e r m i n e d (we h a v e no condition here which requires q~ to be an integer). A second, linearly independent solution ~o, is obtained b y t a k i n g the complex conjugate of u (~ i or b y changing t h r o u g h o u t

12

I I . 3". W .

MI')LLER-KIRSTEN

the sign of [~(t)] 89or, equivalently, the signs of q~ and t~, as can be seen from (40) and (41). Since we know the solutions of (39), our next step is to calculate the corresponding solutions of (11) or (12). We write these as (47)

U~ - - B~u~ q- Bdi~

(no summation) ,

where u~ -- u (~ -q- 0(e) and ~ = ~(o),~ O(e) and, as stated earlier, we do not go into details here of the explicit calculation of the contributions of O(e). The procedure for obtaining the complete solution of our problem in regions of t outside the immediate neighbourhood of e x t r e m a of V ~ ( t ) now parallels the steps following eq. (25). W e define m o m e n t a .~ b y

(48) and re-express H~(u~) as H d B ~ , P,~). The effect of H~ on the solutions (47) is to give B~, / ~ a dependence on t; i.e. in the same way as (34) we obtain

(49)

where the constant is our previous, t-independent integration constant B~. Of course from B~ we obtain B~ b y complex conjugation. This t h e n solves the problem in domains away from an extremum. 2"4. Matching. - The complete solution of our problem for all values of t is obtained b y matching together various pieces of the solution, each of which is of one or the other type described above. This is, therefore, the only problem which remains. There are two t y p e s of matchings: first the continuation of a solution of t y p e a) above to a solution of t y p e b) above in a region of common validity and t h e n the continuation of a solution of t y p e b) to another solution of t y p e b) across a Stokes singularity (15), i.e. where ~(t)--~ 0. The second t y p e of matching is well known from W K B t h e o r y (1~,1~). We shall not enter into a detailed description of the matching procedure here. The solutions obtained above are similar to those derived previously (%1o) for several one-dimensional Schr6dinger equations. The procedure of matching, therefore, follows analogous lines, e.g., as described in ref. (0). (15) R. B. DINGLE: Asymptotic Expansions : Their Derivation and Interpretation (London, 1973). (16) N. FROMAN gild P. O. FR(:JMAN: The JWK:B Approximation (Amsterdam, 1965).

M U L T I D I M E N S I O N A L P E R T U R B A T I O N THEORY~ T H E F I E L D THEORY L I M I T ETC.

13

3. - Multidimensional perturbation theory for the Sehriidinger equation. 3"1. F o r m u l a t i o n . - Our objective here is to e x t e n d t h e m e t h o d s developed previously for solving the equation of m o t i o n in classical mechanics to the corresponding p r o b l e m in q u a n t u m mechanics. W e h a v e shown elsewhere (s) t h a t our m e t h o d s can i m m e d i a t e l y be applied to the case of a n o n s e p a r a b l e two-dimensional Schr6dinger equation. I n the following our a p p r o a c h will be essentially the same, although our starting point, m o t i v a t e d b y recent investigations (4,5) of v a c u u m tunnelling and instanton p h e n o m e n a , is s o m e w h a t different. W e f o r m u l a t e t h e p r o b l e m in such a w a y t h a t the Schr6dinger equation corresponding to the classical equation of m o t i o n (this describes the (~m o s t probable escape p a t h 7>(~7) also i n v e s t i g a t e d b y BITAIr a n d CHANG (~)) is s e p a r a t e d off in the first place, and the p e r t u r b a t i v e t r e a t m e n t t h e n allows the c o m p u t a t i o n of the deviation of the particle f r o m this m o s t p r o b a b l e p a t h , i.e. t h e p r o b a b i l i t y of such deviations and hence its tunnelling t h r o u g h appropriate p o t e n t i a l barriers. As in classical mechanics we let t be a p a r a m e t e r which p a r a m e t r i z e s t h e classical p a t h given b y the co-ordinate r = r ( t ) . An element ds of this p a t h is t h e n given b y (50)

(ds) ~ = d r . d r

with ds = (r~) 89dt, where (51)

dr r~ = d--/"

I f we choose t-----s so t h a t r~--l,

we see t h a t rt is a unit vector t a n g e n t i a l to the p a t h at the point r ( t ) . W e can, therefore, define a tangential vector along the p a t h of length dt b y (52)

drt~ n ~ d t r t .

(17) The term comes from T. BANKS, C. M. BENDER and T. T. ~Vg: P h y s . ~ e v . D , 8, 3346 (1973); T. BANKS and C. M. BENDER: P h y s . Rev. D, 8, 3366 (1973), who undertook a detailed investigation of the WKB expansion for single and coupled anharmonic oscillators.

14

~. J. w. MtiLLEn-X~ST~S

~ e x t we define a v e c t o r o r t h o g o n a l to t h e p a t h at r(t) b y

dr• = dr - - g drt~~

(53)

d r being a n a r b i t r a r y v a r i a t i o n , see fig. 2, a n d g is a c o n s t a n t which has to be chosen such t h a t

(54)

d r • drt~ n = 0 ,

i.e.

(55)

g=

r t~ =

1

or

~

gdt

=

r t.dr

r(t)

o

Fig. 2. - The definition of dr.

I n general t h e r e are m a n y (56)

such o r t h o g o n a l vectors.

W e write, therefore,

~r .(t) = ~, na(t)~Ta(t) , tl

where ha(t) are m u t u a l l y o r t h o g o n a l u n i t v e c t o r s such t h a t (57)

na(t)'nb(t) = r

t+

~

n a ( t ) . r ~ = O.

)

Fig. 3. - The point R(t) away from the classical path.

W e n o w consider a p o i n t R(t) slightly off t h e classical p a t h , as s h o w n in fig. 3. T h e co-ordinates of R(t) can be p a r a m e t r i z e d w i t h respect t o r(t); i.e. we write (58)

R(t) = r(t) + A(t)

MULTIDIMENSIONAL

PERTURBATION

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THE

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THEORY

LIMIT

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15

and (59)

A(t) = nort + ~.~lan~(t) . ft

I n general ~o, rt, ~ , ha(t) can all depend on t. T h e c o m p o n e n t ~o has been ignored b y others (~.5). T h e vector R can now be written as (60)

R(t) = r t ( r ( t ) . r , + no) + ~, n,(t)(r(t).n,(t) + ~,) . a

Our n e x t p r o b l e m is to c o m p u t e d / d R , f r o m which we can construct the mom e n t u m canonical to R. F o r this reason we define

d R = R(t + d t ) - R ( t ) .

(61) W e write this expression

(62)

d R = d H r t -t- ~ , d K a n a ( t ) . a

T h e n d H and dKa can be calculated. Using the orthogonality relations of our unit vectors a n d some relations derivable f r o m these, e.g. (63)

r,'rtt

= O,

we find dH dt - - 1 d- ~ ] o t - - l q •

(64)

,

where ~. = ~/~na(t),

(65)

a

and (66)

dKa dt

b

where a subscript t denotes differentiation with respect to t. F o r convenience we set (67)

~]l,

=

~or,.

T h e a b o v e expression for d R shows t h a t (68)

d d d d-R ~--- rt ~ + ~ n~ dK~" T

16

~r. J .

w.

~t[ULLER-KIRSTE~,

Care is necessary in calculating the differential operators of this expression. Hence The operator d / d R acts on a function y~ ~ y~(t, B,(t), ~l• d d//

dt d d H dt

and d ~ d~lt (t) dt -- c~t -]- dt

8

8~,(t)

+

d~l(t) 8 dt ~B~[t)"

The last expression is the generalization of the usual expression for a t o t a l differential operator to the case of a co-ordinate system which changes its orientation with changing t. The expression follows b y T a y l o r expansion. Now, since ~ / ~ ~ r+(~/C%]o), we have (69)

dvh(t)

~

d

~

dt ~h,(t) -- -~ (rt~k(t))" rt ~

= (rtt~(t) "4- rt~ot(t))'rt

= ~ot(t) ~

since rt'rtt = O. Correspondingly we have

dt

c~l• -- dt ~ ~]~(t)n~(t) "

n~(t)~-~b = = ~,(t)

-~- ~-n'at()n'B(t)~(t)~bb,.'

r+.

The differentiation d/dK~ in d / d R involves only variations orthogonal to Hence

(71)

d

d~. 8

( d

dK~ -- dK~ ~ • = ~

)

8

~ nb(t)~?~(t) 9~ n,(t) ~7--~

x?d~b(t) ~

8 ,

~ d K . ~ b -- ~

as can be seen from our expression for dK,/dt, i.e. eq. (66). Hence finally (72)

d --~ = rt ] (3

_ lq•

1

~]o,"

~

9n +

0 +

+

~ nat(t)"nb(t)~a(t)~--~b) --[- ~'+ano(t)c~, ~.b ~- "

I f we assume t h a t all components re, ~/~ are independent of t (as stated b y BITA• and CHArCG(5) b u t not b y GE~VAm and SAKITA (4)), we h a v e

1

d-R = rt l--'cl•

(+

~tt -~ ~.+n.t(t) ash

-{-

MULTIDIMENSIONAL PERTURBATIOtg TIIEORY, THE FIELD

TH:EO~RY L I M I T E T C ,

17

The w o r k of B i t a r and Chang (~) does not contain the t e r m in n.t'nb due to a highly specialized choice of co-ordinates ~l~. T h e first t e r m containing ~/~t, of course, yields the S c h r f d i n g e r f o r m of the familiar classical equation of motion. We h a v e presented these steps here because we find t h e literature (~,~) on this point unclear. I n the following we restrict ourselves for reasons of simplicity to components ~], which are i n d e p e n d e n t of t (though, of course, variable). Our procedure, however, will be so general t h a t the t-dependence of ~?, could also be t a k e n into account. As a f u r t h e r inessential simplification we set ~]o ---- 0. I n writing down the SchrSdinger equation we set the m a s s m = ~ a n d ]~ --: c -= 1, so t h a t (74)

(V(R)--E)~

d~v -- dR s .

W e e x p a n d the potential assuming ~o ~ 0:

~176

':=~

\ ] - I O~d ~ ~' J

T h e t e r m containing ~ a ~ plays a particular role in our procedure. W e assume t h a t an orthogonal m a t r i x A exists which diagonalizes the coefficient, i.e. is such t h a t

--AT I ~ V \

1

with A A ~ ~- 1. Like GEI~VAIS a n d SAKITA (4) (see the line following their eq. (2.20)), we assume t h a t ( ~ 2 V / ~ ? b ) o is positive definite and hence a l s o ' - - ~ . W e also assume t h a t ~ ~ 0. I n t e r m s of new co-ordinates ~b defined b y (7'7)

~7. ~ ~ A . ~ b , b

we t h e n h a v e

and

The SehrSdinger equation t h e n assumes the following f u n d a m e n t a l f o r m (note 2 -

II

N itovo

Cimento

A.

18

H.

~T. W .

~ULL]~R-KLRSTEN

t h a t for convenience we retain ~ in various terms):

o-- ~e~+...--E

,,,~,

=

+ :E=

q- ~

v,=

1

+ (1--~]•

1 1 1 - - n . ' r + , rt'n+t(t) q- (1--~l•

1 (+~176

3 (Y}•

~V~ ~ ~n+,," nb,+p,,rt,b,) - ~ -+afjb

3 ~ z ' r t t t ~'+natlb'na-~-a

~

-{- (1 __ ~..r~+)~

~a

~-

1 a+~bf

}(+A_~~+) -~

1 _ .~ ~, (n.,cnr 2

-1- (1--~•

--

~

,2 Z n . + . n + A . o + + A + + ~ + a,b,r

0

1

+ (I --~l•

oqa

~2~

~" (n+,~.n~,)(n.+.n~)A.o~o.Ao,,~.A~tA~,,~ , ~ .

aP+bSj~b ~,d,l,g

The derivatives of unit vectors occurring in this equation can be c o m p u t e d from orthogonality relations such as

(so)

n,(t).r~ = 0 ,

i.e.

n~.rt~ + n++'r+ = 0.

I n accordance with our general procedure as developed in numerous previous publications (s-lo), we now derive two distinct types of solutions of this equation which are valid in c o m p l e m e n t a r y and adjoining domains of t. The m e t h o d is formulated in terms of parameters qt and q. connected with the degrees of freedom r(t) and ~,, respectively, which become odd integers in the discrete sector of the spectrum. One t y p e of solution is valid in the neighbourhood of an e x t r e m u m point and involves parabolic cylinder functions, whereas the other type, valid in domains away from an e x t r e m u m point, is WKB-like. W e shall deal with b o t h types of solutions separately and later discuss their matching and continuation.

3"2. Solutions near an extremum o[ the potential: osciUatorlike solutions. I n eq. (79) we expand the denominator of the first t e r m on the right in

MULTIDIMENSIONAL

powers of V l l - r , . (81)

P:ERTU:RBATION THEORY 9 THE

FIELD

THEORY

LIMIT

19

:ETC.

The equation can t h e n be written 82y;

0 = E~f -4- ~ - -

V(r(t))~f -4-

The t e r m s which h a v e not been written out ean a n d will turbations. W e ignore these t e r m s m o m e n t a r i l y . Then equation is separable in all co-ordinates t, ~a. I n order to tions into a m o r e appealing form, we define p a r a m e t e r s ha

be t r e a t e d as l~erthe a p p r o x i m a t e d bring these equaby

~a = ( - da)"

(82) a n d new variables eo~ b y

av

(83)

ma

(no s u m m a t i o n o v e r a ) .

Our equation then becomes

(84)

o= ~

8~v

-4- ~ - F -

Y(r(0)~ -4-

N e x t we write for the lowest-order t e r m of the expansion of (85)

V,~ = ~,(r(t)) l-[ ~ o ( ~ ) 9 a

Our lowest-order equation is now separable with separation constants qa defined b y ~eo~ -4-

qa-

~v~= 0 ,

so t h a t to the same order

(87)

o = E -4- E [ - ~

1(

)}

20

ti. J. "W'. M'OLLE:R-KIRSTEN

We can solve the latter equation by the same method. In this section we discuss the solution in the neighbourhood of some extremum of Y(r(t)). Thus, if r(to) is near such a point, we can writo the above equation

(88)

~v;,

~t2 +

[

1{(

~ + ~

-

~2V(r)'~i ~t~ /,.(t-t~

(OV(r)/~t)t. [2 (_~)~.] -

--~._~ ~ (t--to). \

at- L.]

~,+

...

where (89)

z ' = F , + L | - - [~ q ~1 a

t

1 (av(,,)/~t?,.

h~o +~

v(,(t,)).

oA'~ + 2 (e2v(,.)/et%

If we set

F

(90) (91)

1

h , = 2 ~, ~t, ],.j

and (92)

our equation becomes

(93)

a~,, +

q,-

v, = . ~ ~. ~-~~ : - ] , . ~ g

~

] ~ , + ....

Thus, for sufficiently large values of each of the parameters h~, h~, the lowest-order term of the expansion of % i.e. ~p(o~,is the product of the solutions of equations

(94)

~q,(o,) Fq,(eo,) = 0,

where d 2

(95)

~,(oJ) -- 2 ~

0) 2

+ q

2

and i ~ t, a. We have, therefore (apart from an overall constant),

(96) and ~q, = D89

~p(0) = I ] %~((o~) ~ ~{q,} is a parabolic cylinder function. If the solution ~ is to be

MULTIDIMENSIONAL PERTURBATION

THEORY~ THE F I E L D

THEORY LI~IT

ETC.

21

square integrable as for bound states~ then the dominant contribution (,a) to the normalization integral comes from a trough region of the potential, i.e. apart fl'om contributions of O(1/h~, 1/h2) the normalization integral is the normalization integral of a product of separated parabolic cylinder functions. Proceeding as in the earlier description of our method, we now write the complete eigenvalue equation (97) 1

~

1

(~Y(r)/3

+ v(,.(to)) - A ,

where z] is the correction t e r m due to terms which have been ignored in the derivation of the dominant contribution, i.e. 3 = O(h).

The subscript (q}, of course, stands for some set of numbers q,, q~. We now rewrite the original equation (79) in a form which includes all contributions which h a v e been neglected so far. Thus, using (97) and setting

i

we can write (79) in the f u n d a m e n t a l form

(99) with (100)

_1 .o

A+

.~

1 /a-v(,) I

~-Nr-l,.k~

~=3 ~ \.l-[ a~/d ~ ~'

~,'

/

~"

J

2

(1--~.

a~ ,,,), ~'''" a t a 5

1 (1--~•

a~ ~ "'" a S a ~ '

(xa) R. S. KAUSHALand H. J. W. I~IOLLER-KIRS~EX:J. Math. Phys. (N. Y.), 20, 2233 (1979).

2 9.

II. J. W. MULLER-]KIRSTEN

where the terms which have not been written out explicitly are easily recognized b y looking at (79). Our n e x t step is the calculation of higher-order terms of the p e r t u r b a t i o n series. I t is, therefore, necessary to look more closely at ~(0). First of all we observe t h a t the first approximation of ~, i.e. ~o(~ = ~{qd, leaves uncompensated on the right-hand side of (99) the contribution (101)

ace) =~r

The terms on the right-hand side involve variables (including derivatives with respect to variables) and perturbation parameters. The variable t occurs in functions such as r~, rt~ which can be expanded in rising powers of t - - t o . Factors such as 1 / ( 1 - - ~l~. r , ) can be e x p a n d e d analogously, since 1~1• is assumed to be sufficiently small. Then, b y means of (92), any power of t - to can be re-expressed as (102)

t-

to =

-

The second t e r m on the right is obviously of lower order t h a n the first. We can proceed in a similar way with any t e r m containing a component ~ of ~1• I n this case we first re-express r/, in terms of components ~b of (77) and t h e n we use (83) to re-express ~ as

(103)

1 =

K

o Ao

.

Thus after suitable expansion (if necessary) the variable dependence of any t e r m in ~c0) is of the form

m,n

i,1

U(Dj

where m, n > O and i, j = t, a. Now the operators on the left-hand side of eq. (99) are each multiplied b y a factor h~ of our perturbation parameters (which are assumed to be large). Inspection of t h e terms on the right-hand side of (99) shows t h a t t h e y are at most of order h~ (in agreement with our general procedure). H e n c e also the coefficients C~.;" of (104) are at most of order h~. The first approximation = ~v(~ therefore, leaves uncompensated a contribution ~I~} which can be written (105)

~co) . . . . . .

~ ~o~q,~(o))

MULTIDIMENSIONAL PERTURBATION THEORY, THE FIELD THEORY LIMIT ]ETC.

23

:Next we use the recurrence relations of parabolic cylinder functions, i.e.

(lO6)

~oVp(o~) = (q, q @ 2) ~pq+~(eo)§ (q, q - - 2)~_2((o)

and

007)

d 2 ~ w~(~) = - - (q, q + 2) w,+~(~) + (q, q - - 2) ~_~(~),

where

(lOS)

(q,q§

=1,

(q, q - - 2 ) =

89

The remainder (105) can then be written ~(o)

(109)

~

--2p

Z [{q,, q, + %}],, v+,+~,>,

+,~ =

where a~ ua, are integers: (110)

[{q~, q~ § u~,}]~ = [q~, q, § up,; q~, q, § up,; q~, q2 + ur ...]~

a n d / z serves as an ordering of successive contributions in descending powers of h~. I t should be observed at this point t h a t A has been absorbed in [{q,, q~}],.. To be more specific, we calculate the dominant contributions to M
§ 4 ~ (na~.n,)Aa~.Aba~-~d v2{q,}§ O(h~ = a~bjc~d

+ 4,~,, . . ~ htl~ h~ ( n ' ~ ' n ' ) ~ ' A ~ A b ~ ( ( ~ 1~7 6

yJ{p'}+O(h~

where functions of t such as r , ( t ) have been replaced b y the first t e r m of their expansion around to. With the help of the recurrence relations (108) we can rewrite ~co~ ~o{~,} as a sum over various ~p{~}. Thus, in the case of only two discrete transverse degrees of freedom, i.e. a, b, ... = x, y, we have~ for example,

(112)

d2

-q+(~o,,o:+:.~:+ 89(q:- 1)~,,~ _:,p:) + + ~ ( q ~ - 1)(q~ - a)(~:,_,.~:+:,p: + + (q= - 1) ~o,_,.o _~.o~).

24

II. J, W. MOLLER-KIRSTEN

I n s p e c t i o n of (111) shows t h a t there is no t e r m of O(h) containing ~{q,}. F r o m this we will deduce later t h a t A is a t m o s t of order h ~ Successive iterations are now p e r f o r m e d in the s t a n d a r d w a y of our procedure (s-n). Since

(]13)

~{r

~ ~ q -]- s{%},

where (114) i

a term aV{q,+.+,} in ~(o) ~o(~,} can be cancelled out b y adding to ~r

~{q,.~llql}/(--~{tiq,}), except, of course, when ~{,+,}----0, i.e.

the c o n t r i b u t i o n each u+, = 0. H e n c e

the n e x t contribution ~(~) to ~(o) is given b y

(115)

ygl)= ~

~

[{q,, q++

p=po uq++mp Uq~O

Uq+}]p

y~{~,,+.q,}

- - 8{uq+}

and leaves u n c o m p e n s a t e d , so far, the sum of t e r m s in ~o) which h a v e each

u,, = O. Since ~(o>: y~{+,}leaves u n c o m p e n s a t e d J+{q,},~(~the iteration y~(~)leaves uncompensated (116) Uq~D

The n e x t contribution F (m, therefore, becomes

(117) uqt~O

9~

-~' [{q,+u.,,q,+u+,-]-u',}].'(.++%+.+,}

"

Proceeding in this m a n n e r , we obtain the w a v e function ~ given b y = ~co) § ,p(~) + ....

(118)

Finally, setting equal to zero the sum of t e r m s left u n c o m p e n s a t e d so far, i.e. the t e r m s containing ~o{q,} in ~(o) ~a<}, ~(1> ~{+,}, "", we obtain an equation f r o m which A and hence E is determined, i.e.

[{q,, q +-}-

(119) I+~/zo

/~=/~o U q | ~ 2 / l

--

'~{/Zqt}

/~'=.u o

) I U L T I D I I ~ I E N S I O N A L P E R T U R B A T I O N T H E O R Y , T H E F I E L D T H E O R Y LIMIT :ETC.

25

We have observed before t h a t (111) does not contain a t e r m in v2(~,} of O(h). Setting equal to zero the dominant t e r m of (119) we, therefore, see t h a t A is at most of order h ~ W e have thus obtained explicitly one branch of an eigensolution of the multidimensional Schr6dinger equation together with its associated eigenvaluo. A second, linearly independent solution is obtained b y changing t h r o u g h o u t the signs of each q, and h~. Our expansions are asymptotic in h2i ' and the solutions are valid around ,

~l~ ~ 0

1

.

The discrete eigenvalues associated with the minimum of the potential at t ---- to are given b y (97). The function A (i.e. the contribution of higherorder correction terms) is determined b y eq. (119). Successive terms of this equation (when rearranged in increasing order of our p e r t u r b a t i o n parameters) can be calculated from a simple consideration of all closed paths in a multidimensional Euclidean lattice which start at {q~} and end at {q,}, the order of the contribution increasing with the n u m b e r of intermediate steps. Similarly the wave function can be determined up to any specified order b y considering all paths on the lattice which start at {q~} and end at {q~ -t- u~,}, where u~, is some n u m b e r depending on the order. We thus have a completely systematic procedure for calculating the eigenvalues and eigenfunctions of a multidimensional wave equation. I t is this t y p e of procedure which t h e n permits the investigation of (e.g.) the large-order behaviour of perturbation t h e o r y (~). 3"3. +~olutions away from an extremum o] the potential: the W K B - l i k e solutions. - We now derive a second pair of large-h~ a s y m p t o t i c expansions for the eigenfunctions of the wave equation in the neighbourhood of the eigenfunctions determining the most probable escape of the particle from a trough of the potential. This pair is valid for larger values of It--tel and I%1, where the expansions obtained above are no longer applicable. Of course, the corresponding eigenvalue expansion will be identical with (97) above, although its derivation is completely different (this is, in fact, one of the beauties of our procedure). Our starting point is eq. (79) which we write ~l

(120)

2 2

J

We rewrite this equation in the form (121)

[

(

%o~ 0 ,

26

H.J.W.

MULLiER-KIRSTEN

where Ve ~ V(r(t))oV~ ~ Va(~.) contains only those p a r t s of t h e e x p a n s i o n of V ( r + ~• w h i c h d e p e n d solely on ~ , a n d Z is m a d o up of all r e m a i n i n g terms, i.e. p r o d u c t s of different ~ a n d derivatives. :Next we define t h e following quantities (122)

h~ W , ( x , ) ~-- V,(x,) - - V,(x,o) - - (x," - - x,oj' V a)x'~t ~o,~,

where i --~ t, a a n d x t ~-- t ,

Xa :

and

~a

x~o ~ to,

I f we s u b s t i t u t e these expressions for V t , equation becomes

(123)

# ~

+

x.o = 0 .

V~ into (121) a n d use (97), our

. (1,(/,) + g,(x,))- A + z x,, ~

,p = o,

where ]t(Xi) = ~q~h~ 1 -- hl Wi(x~) --

V,(x,o)

--

(x, --

x,o) V~l)(x~o)

and for i = t , (124)

g~(x~) = --h~

for i -~ a .

o Ab~

W e n o w set (again i : t, a) (125)

,p = ~(t, ~o) l-[ E,[W,(x,)], i

where (126)

,tw,l: o.1

S u b s t i t u t i n g (125) into (123), we o b t a i n

(127)

~o,)~(t, ~.) = ~(o)'q~(t, ~o),

where

(128) (129)

!

d

d.,[W,(x,)] -- 4[W,(x3]~ ~

+ q,--

2(x,--X~o) V~l'lx t ~ io]

h~

2

W2)(x,)

h~ V,(x,o) + [W,(x,)]~

2

g,(x,)

:~r

PERTUI~BATION THEORY,

THE

FIELD

THEORY

LIMIT

ETC.

27

and 2

(130)

d*

Proceeding as in the case of our first pair of solutions, we ignore m o m e n t a r i l y the t e r m s on the right-hand side of (127), since these are of lower order in h 2 The d o m i n a n t contribution to the solution ~v is t h e n

(131)

~(o) =

q~(~,,~= 1] %,,[W,(x,)], i

where (132)

[__[. d*i J4[W~(x~)]~

1 %,[W~(x~)] - - [W,(x,)]~ exp

2 ( x i - - X~o) V~m (X~o) 9 qi-

2(v,(X,o)Tg,(x,))l]

h~

Our n e x t step is the calculation of higher-order t e r m s of the p e r t u r b a t i o n series. The first a p p r o x i m a t i o n of % i.e. @o)= q~{~,), leaves u n c o m p e n s a t e d on the r i g h t - h a n d side of (127) the contribution (133)

a~(o)' ~{~,} -- ~(o~, ~{q~} _

_

9

As in the case of our first pair of solutions we first re-express this q u a n t i t y as sum over various ~o{~,+~q,}. The solutions and eigenvalues t h e n follow as in t the previous case, because eq. (113) holds also for the o p e r a t o r ~{~,}, i.e.

(134)

9/

m {q,+=q~}

--

t

~{qd 2V e{uq,}

where %a

= ~ h~uo .

The procedure for deriving the sum over ~{q,+~q,} is the same as in our previous work (s). The function Wdx~) of eq. (122) has been constructed in such a w a y that Wdx~o) = 0 ,

W('(x~o) = o .

Then, b y using the T a y l o r expansion of W~ a r o u n d X~o, the integral

[w~(**)]~ f4d_

28

m

can be evaluated.

J.

W,

I~ULLER-KIlCST]~N

Inserting the result into ~t

-- exp [ f

(135)

dxi

a.nd reversing the resulting series and squaring, we obtain

(xi - - x~0)~q~ : -

(136)

~ d.~+1~,_(2~+1),

where (137)

d~ = 1

d - - 1 V~*'(x,o) ,

a--

6

V[2'(xio)

' "'"

and we have made use of the fact t h a t Cfq~+a ~q~+b __ Cfqr

(138)

We can deal with derivatives in a similar manner.

Thus

d dx~. ~q,[W(x~)] ---- fq,(x~)(fq,[W(x~)]

(139)

and ]~,(x~) is readily written down from (132). I f we e x p a n d /~,(x~) a r o u n d x;o and use (136), it is clear t h a t a n y derivative of ~q, can be re-expressed as a sum over various ~q,-(25+~). Since the m e t h o d is now clear, we do not enter into a more detailed description of these calculations here, which are in a n y case practically the same as in the one-dimensional case described elsewhere (9,1o). Hence finally vo

--co

(14o) ,a=]~ 0 UqI~2.U (~uo>O) f

[{q, q, +

%}]'.+<.,+uo,>

9

The coefficients [...]~ follow from the calculations just described. The only formal difference between ~I~); and ~ ' ) } of the previous section is t h a t the sum over u~, extends from a finite positive value up to - - c~ (of course, the coefficients a n d functions are now completely different). This difference is due to the infinite n u m b e r of terms of the binomial expansions as compared to the finite n u m b e r of terms of the recurrence relation for parabolic cylinder functions. I t is clear t h a t the procedure for deriving successive contributions ~(1), ~(2) ... of W is now the same as t h a t of the previous section. Thus~ in the same w a y

M U L T I D I M E N S I O N A L PERTURBATION T H E O R Y , T H E F I E L D TItEOI~Y L I ) I I T ETC,

2~

aS before, we obtain the solution (141)

~v =

~0(~ - ~ ~0(1) ~ - .,.

together with ,~ corresponding equation for A (simply replace the bracket coefficients in (119) b y primed brackets). We h a v e verified previously (%~o) t h a t this m e t h o d yields the same expansion for A as the m e t h o d of subsect. 3"2. Finally we can deduce from (125) and (127) t h a t a second, linearly independent solution F possessing the same value of A is obtained b y changing the sign of each W~ or t h e signs of each q~ and h~ throughout. The region of validity of the solutions is found b y demanding t h a t successive contributions ~(o), ~(~) ... of ~ form a rapidly decreasing sequence. One finds t h a t this is the region b e y o n d the neighbourhood of x~o, i.e. b e y o n d oc>O . Finally, in order to permit comparison with the procedures of Gervais and Sakitu if) and Bitar und Chang (5), it m a y be useful to point out the relation between our solutions (as derived in this subsection) and those of the usual W K B method. Looking at (125) and (131) we see t h a t the leading t e r m of the solution y, is x~

(142)

9e x p

[_f. L J~[W,(x,)]~

h*,

q'--

,,,

h~

--

1

if

--~[[. [W,(x,)]iexp h~ [WJx,)]idx~

-

],

if we retain only the d o m i n a n t terms in the argument of the exponential. I n the one-dimensional case, i.e. i = t, this is

1 [q ] Wt(xt) exp h [Wt(xt)]idxt xt

and h~ W,(x~) = V~(x~) - - V~(x,o) - - ( x ~ - - x~o) V~'(x,o)

with V~(x~)= V(r(t)) in our notation.

But

from (97) (if we ignore non-

30

H.J.W.

MULL:ER-KIRSTiEN

leading, transverse contributions) s ~.~ - - Vt(Xto ) _~_ ~1q t h ,

(with Vt(x,o) being the d o m i n a n t t e r m on the r i g h t - h a n d side) so t h a t

043)

h~ Wt(x~) -,-~ V,(x~) - - E + ~ qt h, - - (xt - - Xto)V~'(xt0).

To the d o m i n a n t order this is h~ W t ( x t ) ~ Vt(xt) - - E . Thus, if we discard all nonleading contributions in our solutions, we are left with the well-known W K B result, a n d t h e n h a v e a g r e e m e n t with the work of Gervais and S a k i t a (4) and B i t a r and Chang (5). I t should be noted, however, t h a t our procedure has m u c h greater generality a n d allows (at least in principle or on the c o m p u t e r ) the calculation of a n y n u m b e r of higher-order terms. The nonleading contributions included in our first a p p r o x i m a t i o n are essential for the f o r m u l a t i o n of t h e a p p r o p r i a t e ordering procedure. I n this connection it should be observed t h a t we h a v e only one eigenvalue equation b u t as m a n y auxiliary p a r a m e t e r s qi as dimensions. The eigenvalue equation determines the p a r a m e t e r ql in the nondiscrete sector of t h e s p e c t r u m as a function of E and qa. The auxiliary p a r a m e t e r s of the t r a n s v e r s e m o t i o n (i.e. q~) cannot be r e m o v e d ; t h e y describe the complicated oscillations a r o u n d the classical p a t h (and t h u s a (( binding ~)to t h e p a t h which m a k e s their a p p e a r a n c e v e r y natural).

4. - Multidimensional perturbation theory in the limit of an infinite number of degrees of freedom: application to field theory. 4"1. F o r m u l a t i o n . - Our objective now is to extend the m e t h o d s developed a b o v e to an infinite n u m b e r of dimensions, which is the case encountered in field theory. I n the following we consider the t h e o r y of a scalar field ~(x) defined b y the H a m i l t o n i a n (144)

~(~, ~) =fd~ [16~+ V(~o)],

where ~b = ~q~/~t,

(145)

v(~) = 89

~+ ~

+ ~(~),

/z the mass a n d ql is a potential, e.g. t h e quartic interaction (1/d!)g2q) 4. The fields ~0(x, t) for x e R a span an infinite-dimensional space, x being a con-

MULTIDIMENSIONAL

PERTURBATION

THEORY,

THE

FIELD

THEORY

LIMIT

ETC.

31

tinous three-dimensional field variable. A point in this space is labelled b y the time p a r a m e t e r t and has co-ordinates

{~(x, t)} = ~(xl, ~), ~(x~, t), .... The canonical c o m m u t a t i o n relation satisfied b y the field q~(x, t) is (146)

[~v(x, t), ~b(x', t)] = i~(x -- x').

A convenient representation of r t) in configuration space (which is compatible with (146)) is --i(8/~q~(x, t)). I n this space

(147)

" r=fdx[

+

We are interested in the transition of the system from a state having field configuration ~' at time ~' to a state having field configuration ~ at time t, i.e. in the amplitude

<% t J~', t'>. This amplitude can be shown (19) to have the p a t h integral representation

(148)

<% tko' , t'> =fq)~(t) exp [is(~)],

whore (149) and

I f we replace the integration variable t b y - - i t , we obtain it

~t t

This expression differs from (148) in t h a t the action S has been replaced b y its Euclidean counterpart and the integration is now along an imaginary time axis. (zg) E. S. ABERS and B. W. LEE: Phys. Rep., 9, 1 (1974), sect. 11.

32

H. J, W.

Mi)LL:ER-KIRST:EN

The transition amplitude <~, tiT' , t'} can also be related to the appropriate density matrix 0 as defined in statistical mechanics (~o). The density m a t r i x #(~, r 3, 3') satisfying the SchrSdinger-like equation (~)

(151)

~-~ ~(qJ, ? ' ; v, 3') = - - : / f ~ ( ? , ? ' ; r, r')

(supplemented b y suitable b o u n d a r y conditions to be discussed later) is given b y T

(152a) T'

or, summing over all paths from ~' to q, b y ('-") ep(v)=~

r

(152b) ef(r

7'

This expression differs from (150) only in the range of integration of the integral in the exponent, i.e. in (152) the range of integration extends over real values of t and so provides t h e exponential damping necessary for the existence of the functional integral (9~ being positive definite). With the postulate of Euclidicity (which says t h a t if a well-behaved t h e o r y can be constructed in Euclidean space, t h e n the corresponding t h e o r y in Minkowski space can be obtained b y analytic continuation) we can find (150) b y solving (151). :Now, integral (]52b) corresponds to a sum over all paths in the configuration space of {q~(x, r)} which start at 3 = v' with field configuration ~' and end at 3. The most i m p o r t a n t paths aro~ of course, those for which fdt~(qJ, r is a minimum~ i.e. those values of T which are solutions of the usual equations of motion (derived from ~fdt~f(~, ~b)= 0) with t replaced b y --iT. These so-called Euclidean solutions ~ are well known as instantons. W e see from the above t h a t their contribution to the transition amplitude Q(~, ~'; v, v') can also be obtained b y calculating (in a corresponding sense) the dominant approximation of a solution ~ of eq. (151). Since it is in general difficult to solve the Euclidean equations of motion as a prerequisite for the evaluation of the transition amplitude, it is often easier to search for appropriate solutions of cq. (151). This is the objective of the present investigation.

(2o) R. P. FEYNMAN: Statistical ~feehanics, Chapt. 3 (Reading, Mass., 1972). (21) R. P. F]~Y~MAN: Statistical Mechanics, Chapt. 3 (Reading, Mass., 1972), p. 48. (22) R. P. F ~ r ~ A ~ : Statistical Mechanics, Chapt. 3 (Reading, Mass., 1972), p. 75.

MULTIDIMENSIONAL PERTURBATION TIIEORY, THE FIELD TI-YEORY LIMIT ETC.

33

In (151) we set (~3) (153a)

~)(qJ, (p'; v, z') = e x p [ - - E ~r

- - ~ ' )],f~)(q~) w *~ ( ~ ' ) ,

or

(153b)

P(~v, T'; ~, v') ----- ~ exp [-- E{q}(z--

)]y~{q)(~)~{,}(~),

{q}

where the sum extends over all eigenstates labelled {q}. This substitution removes the explicit v-dependence of ~ without affecting the implicit dependence provided b y ~ = r v). The eigenfunctional ~0(?) corresponding to an eigenvalue E of the total energy of the system then satisfies (if we ignore indices)

(154)

( ~ - - E) V'(~) = o,

g,q>,

- ~ ~--~ + v(~) w(~) = E~(~) and ~o(~v) has the usual probability interpretation. I t should be noted t h a t the ground state yields the dominant contribution to (153). I n particular, it yields the only effective contribution in the limit v--> c~ (since v ~ 1 / k T , this corresponds to the low-temperature limit in statistical mechanics). We now consider the ordinary three-dimensional space to be a cubic lattice of Z 3 ~ ( L / a ) ~ points. Here a is the lattice constant and L = Z a is the length of one edge of the world-cube. I n this lattice space the field ~v is defined only at the lattice points n ~ - x / a where it is characterized b y ~(n) (and the conjugate m o m e n t m n b y ~t(n)). Some rules for transcribing formulae from the continuum to the lattice and v i c e v e r s a are (24) q~(x, t)

(156)

+--) ~ rf(n, t) ~ 1a qn(t),

~ 5 ( x - x') ,-~ 1 ~,,,,,,

f

dx

~

a ~ ~, n

(~a) R. F. FEYNMAN: S t a t i s t i c a l M e c h a n i c s , C h a p t . 3 ( R e a d i n g , Mass., 1972), p. 49. See also D. 0LIVId, S. SCIUTO a n d R. J. CR~.WTtIER: R i v . N u o v o C i m e n t o , 2, No. 3, 47 (1979).

(~-~) S. K. BosE, A. JAns, H. J. W. Mt~LLER-KIRSTENand N. VAIIEDI: P h y s . 13, 1704 (1976). 3 - 11 Nuovo Cimenlo A.

Rev. D,

3~

H.J.W.

MULL:ER-KIRSTEN

but 1

~(x, t) ~

h~

in order to satisfy (146). The Schr6dinger-type equation (155) then assumes the form ' 2 a~ ~

+ v

~o(~o) = E~(~o).

Writing ~ as the vector having components T~ and

we can rewrite the equation in the form of eq. (74), i.e. (159)

( - ~1 ~~-~ 2 + v(~)) ~o(~)= #~(~).

So far we have not imposed any restrictions on the vector ~ , which can be any vector in the field configuration space. A particular field configuration is provided b y the solution ~0 of the corresponding classicM equation of motion (obtained from (159) b y the substitution --i(~/~r --+ ~ ~ ~ / ~ z ) , i.e. 1 ~ 2

(160)

~-~o-t- V(~o) ---- go

or (in Newtonian form) 1 d - -2 dr

(~)

= <5o =

d V(~o), dcp0

where we have replaced # b y d~o. I t is evident t h a t ~ plays the role of time for a classical t r a j e c t o r y with energy #o and potential V . Of course, here we are dealing with stationary-state functionals y~(T), so t h a t ~ does not have the meaning of the true physical time. I n conformity with our approach in previous sections we assume t h a t the classical solution r of eq. (160) (i.e. qJo(x, 3)) is known (eq. (160) permits direct integration), and we wish to calculate the eigenfunetional ~(~) in the neighborhood of r = ~o- Proceeding as in sect. 3, we define the unit vector dcpo/ds, where ds is an infinitesimal clement of the classical path, and choose as a m a t t e r of convenience ds = dr. Then (161)

~

--r

M U L T I D I M E N S I O N A L P E R T U R B & T I O N THEORY~ T H E F I E L D T I t E O R Y LIMIT ETC.

and r is a unit vector tangential to the t r a j e c t o r y at r we see t h a t for this p a t h

~

F r o m (160) and (161)

#o - v(~o) = 89

We h a v e thus chosen oR0 in such a w a y t h a t Cpo~ is always nonvanishing on a n d t a n g e n t i a l to the p a t h , a n d such t h a t it does not h a v e c o m p o n e n t s perpendicular to the path. This p a t h , which is traced out b y r for v a r y i n g 37 is called the (
q~('r) = ~o('r) § A('r),

i.e.

qJ(x,r) = (po(X, 3) § A(x, 7:)

a n d e x p a n d the deviation A(~) in t e r m s of a complete s y s t e m of m u t u a l l y orthogonal unit vectors u at r i.e. (163)

~(3) = ~(~) § ~•

where ~,1 (v) = ~ocPor

and

~1(3) = ~ n a ( 3 ) .

(In principle ~0~ ~a can still be functions of v~ b u t for simplicity we ignore this dependence here.) We use the same n o t a t i o n as in sect. 3 in order to facilitate an easy transcription f r o m our earlier results. Thus ~/~r of eq. (159) can i m m e d i a t e l y be t a k e n over f r o m d/dR (eq. (72)) and V(cp) f r o m V(R) (eq. (75)). Again we h a v e two types of solutions: one t y p e in t e r m s of oscillatorlike parabolic cylinder functions which is valid in the neighbourhood of an e x t r e m u m of V(cp), and a n o t h e r t y p e in t e r m s of W K B - l i k e exponential functions~ which is valid in domains a w a y f r o m an e x t r e m u m of V(cp) a n d on either side of a classical turning point. These domains are indicated in fig. 4. A t the turning

J v(~p~

E'

/

Fig. 4. - The domains of our solutions: in domains I, I ~, I" .... the solutions are oseillatorlike, in domains II, I I ' , ... they are WKB-like. T, T' are turning points, E, E' are extrema.

36

H.J.

YV. Mi]LL]~R-KIRSTEN

points T , T ' , ... the WKB-like solution in a classically allowed domain (IIa, II'~, ...) has to be matched to the WKB-like solution in the adjacent classically forbidden domain (IIb, I I ' , ...). At the points P, P ' the oscillatorlike solution has to be m a t c h e d to the appropriate WKB-like solution. We still have to specify the b o u n d a r y conditions mentioned after (151). This could be done as follows. We m a y be interested in initial and final field configurations (z'o > 3o)

corresponding to successive minima of V. Then, according to (162),

A(3o) = o ,

a(3'o)= o.

4"2. ,~olutions near a n e x t r e m u m o/ the potential: oscillatorlike solutions. In general the potential Y(~) will exhibit a n u m b e r of wiggles when plotted as a function of ~. The wave functionals ~v(~) in the neighbourhood of an ext r e m u m of V(~) can immediately be written down from the solutions of subsect. 3"2. Transcribing the results, we have (to leading order and apart from an overall multiplicative constant) (164)

~0(r

~ ~t;(~ =

I-I ~,(r i=r,a

where ~ ( t o ) = D~(~_.)(to) and

and (oa, ha are defined similarly (see (82), (83) and expansion (75) of the potential). The eigenvalue E is correspondingly given b y (97). We repeat: if 3 z 3o is a point close to the minimum of V(tflo), (8*V(tpo)/~r~),. is positive and hence h~ is real; if zo is close to a m a x i m u m of V(q~o), h~ is imaginary. I n the latter case the (~eigenvalue ~ equation is an equation determining the auxiliary p a r a m e t e r q~. With the assumption made earlier, t h a t (c-b (tp)/c~7, c~b,~o is positive definite, each h ~ is real and hence each q. (exactly or approximately) an odd integer (describing the ~ binding ~ of the pseudoparticle to the classical or most probable escape path). I n order to be able to establish the connection between the two types of solutions we obtained (and also in order to be able to compare these with the solutions derived b y others(4,~)), we substitute for Fr the well-known

M U L T I D I S I E N S I O N A L PERTURBATION

THEORY~ THE F I E L D T H E O R Y L I M I T ETC,

37

asymptotic expansion

y,~(co) = e x p - - i ~

(16.5)

r176

1

8eo~

a n d r e t a i n (for simplicity) only the m o s t d o m i n a n t t e r m . derives) (166)

-~- . . ~

T h e n (ignoring first

~p(~o) ~ ~/'~(r ; q,, q~, h~, h~) -~ th2t-~'l~h _ _ e x p [ - - ~h~(~ ~ ~ - - re) 2] {h(v - - v~ t(~-', l-I e x p [ - - ~~,,~, ~~, ~(~-~) . a

As n o t e d earlier, the linearly i n d e p e n d e n t solution is

v,i(q~; --q,, --qo, - - h ~;~ * --h~)

*

I~ will be seen b e l o w t h a t b e h a v i o u r (166) of t h e w a v e f u n c t i o n a l is similar to t h a t of t h e c o r r e s p o n d i n g W K B - l i k e solution in t h e d o m a i n w h e r e t h e y overlap. Since (see (91)) (167)

--~ V ~ ( T ) - V~(ro) --~ V ~ ( v ) - - E \~

IV o

( t a k i n g (~V/~v)~.----0, for s i m p l i c i t y ) , we c a n r e w r i t e (166)

(168)

v~(~o)

[V~(T) - - E]~(1-~0 e x p a

(ho~.)

N e x t we r e w r i t e this expression in t h e c o n t i n u u m f o r m u l a t i o n . F o r this p u r p o s e we consider first some n o n t r i v i M e x a m p l e s illustrating the c o n v e r s i o n f r o m the lattice to the c o n t i n u u m a n d vice versa. Consider the p o t e n t i a l (75). I n t h e c o n t i n u u m f o r m u l a t i o n we h a v e a c c o r d i n g to (156) the c o r r e s p o n d e n c e

~=~a~(x),

~/~.~a~/~(x),

so t h a t ~V

(169) S i m i l a r l y (77) requires (170)

A.b --> a 3 A ( x , y ) ,

~b --->a ~ ( y ) ,

38

II. J . W . M 1 ] L L E R - K I R S T E N

so t h a t

~(y)=fAT(y, x)V(x)d x . Finally, from (78) we conclude t h a t (171a) with

e~ --> a2 O2(y)

(see (158))

(171b)

V(b) --> a V ( y ) .

Further, from (82) we conclude t h a t h 4b -~ a 2M ( y )

(171c) E q u a t i o n (76) then reads (172)

fd x d y A T ( t o ,

(

~v

~

A(x, z)

x) k~l(~(y)]~o(y,~),v~

)

1

= - - ~ 02(to)cS(to--z).

~ow, in transcribing (168) we are faced with the problem of specifying the q u a n t u m numbers qo. F o r simplicity we consider here only those configurations which have all transverse deviations in their ground states, i.e. each q, = 1. This is, presumably, also the physically more relevant case. Various kinds of excited states would be described b y q(y), i.e. q would be 1 at some points, but 2, 3, ... at other points. Restricting ourselves to the transverse ground states implies t h a t ~ 89 appearing in (97), i.e. in d~ = a E (recall a

t h a t g~ stands for the lattice eigenvalue), is subject to the correspondence

~h~

l f l ~ abe(y) d y

-~

1f

= 2a 2- h2(y) d y ,

whereas 1 (d

1

3 ~V

3

J

\

y)}2=

c,~(X)lo jhd(y)

so t h a t (see (97)) (173)

1

1 [~V~ 2

V~)]o

. . . .

MULTIDIMENSIONAL

P~ERTURBATION

TIt]~OtCY,

TtIE

FIELD

TI-I]~OI~Y L I M I T

ETC.

~

Thus in the infinite-dimensional limit (i.e. lattice constant a - + 0) the eigenvalue expansion contains a divergent term. This is, of course, w h a t one would expect because (1/2a3)fdyh~(y) is simply the sum of the zero-point energies of the effective oscillators at e~ch point in space. This t e r m m u s t , therefore, be associated with the v a c u u m and thus m u s t be s u b t r a c t e d out. Transcribing similarly the w a v e functional (168) into the c o n t i n u u m formulation, we obtain (174)

~v(~v) _ [Vd~) - - E] '(x-r

exp

r [V~(v) - - E] t 9

.exp[--~fdxdz~(x)~(x, z, ~)~(z)] where

_C2(x, z, v) =

~fdyA(x,y){-- q2(y)}~AT(y, z)

(for each q, = 1 in (168) the second factor in the d e n o m i n a t o r of (168) is simply 1). B y construction Q~(y) is negative (recall t h a t ~] is negative if q~ is an integer), so t h a t Y2(x, z, 3) is positive definite. I t follows t h a t the w a v e functional dies off rapidly in the directions a w a y from the classical p a t h (for which E > V), i.e. p a t h s far a w a y f r o m the classical p a t h are strongly damped. 4"3. Solutions away Item an extremam o] the potential: the WKB-like solutions. - I n m u c h the same w a y as in the preceding section the solutions can i m m e d i a t e l y be t a k e n over from subsect. 3"3. Thus (to le~ding order) xi

(175) i=v,a x~

x~

.... [W~(x~)]~exp h~ [W,(x,)]idx,

exp - - q ~ j i [ w ~ , ) ] i j

with W~, x~ as defined b y (122) (h~W~(x~) is p r e d o m i n u n t l y V(cpo(v))--V(~o(ro)) --~ ~ ( ~ - - zo)2h~ --~ V(~o(~)) - - E and correspondingly h~ W,(xa) ~a 1 2 2 ~2 tr ----~OaXa = 14x,~a]. We can rewrite (175) as (176)

F(r

[V(~o('c)) - - E ] i(~+~o exp exp

[?

"~[V(~o(T.)) - - El+ 9

[h:f[Wa(x~)]'dxo--q.fdx./4[W.(x.)]q [ I [Wo(x~ a

40

H.J.W.

MULLER-KIRSTEN

The dominant part of this expression is

(177)

a

comparing (177) with y , ~ ( r (168) with the replacements

of the preceding section,

i.e.

qr --> - - q~ ,

qo : - - - q . ,

[ V ~ - - EJ~" -~. [ V r - - E ] ~" ,

h~ -~- - - h~ ,

we see t h a t both solutions are the same (in their common domain of validity) except for a multiplying factor involving q~, h~. Thus

(178)

~v~(~, q~, h~) = a(q~, h~)~(q~,--q,,--h~)

with (179)

~,(q, h~) = - - (-- 2)~(-- h~) u'~-" 17[ (-- h~) u~176247 I ] (2~) 9 a

a

I t m a y be worth pointing out that~ due to the generality of our approach and the similarity of construction of our solutions, we have no difficulties with the factors xo multiplying the exponential in (177). The amplitudes derived b y GERVAIS and SAKITA (4) and BITA]r and CHANG (~) differ in this point; GERVAIS and SAKITA (4) have a factor (det u) ! in their formula (2.42) corresponding to the factor YI (x~)l(~-q~) in our formula (168)--whereas BITAa and C H A ~ (0 do not have this factor. Of course, for each q~ ~ 1 (ground state) the factor disappears in this specific solution (leaving the v-dependent factor, in agreement with BITAIr and CHANG (s)), but it becomes crucial in the associated linearly independent solution (q~--~ 1 in (177)). 4"4. The tunnelling amplitude. - I n the preceding sections we developed a m e t h o d for calculating sets of linearly independent wave functionals in the neighbourhood of either a classical solution (in a classically allowed domain) or a (~most probable escape p a t h ~ (in a classically forbidden domain). Like BITAIr and CHA~G (5) we now consider the tunnelling of field configurations from one classically allowed domain around v0 (say I, as in fig. 4) to an adjacent classically allowed domain around v: (I"). The wave functionals have to be

MULTIDIMENSIONAL

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41

chosen such t h a t they are periodic in domains I and I" of fig. 4 but daniped in domain I'. We write the desired wave functional in domain I Tx(~~ , q~, h~) and require this to be continued across the turning point (in the variable 3) into an exponentially decreasing wave functional y,~,(~) in domain I'. Then (with constants A, B, ...)

(18o) '

o~(--q,,--h~.;

~:o)

+ B(q~, hy)

o~(q~, h~; ~o)

--

~"~A~ C(q~, h~)~,~b%)(~o, q~, h~) ----

~'~' C(q~, h~)o:(q,, h~; v'0)yh,(~o)(~o,- q ~ , - h~)= .

.

.

.

.

.

.

.

.

.

.

.

.

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.

.

.

.

.

.

.

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.

.

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I

,

,

~

~"~-~-~"D(q~, h~) TL,,~,o)(~o,q~, h~), the dots implying a series of steps similar to those written out. Also we have indicated explicitly the extremum (i.e. the dependenee on to, ~ , ...) to which each quantity is assigned. Relations (180) have to be understood in the sense of continuations, i.e. cyhi b (for instance) is the continuation of ~ into the domain IIb. The wave functional Tw then has the same form as ~/~ except t h a t it is multiplied by a damping factor A, and the domain of the variables (in particular the limits of integration of the integrals in the exponents, e.g. ~~dr i V e - - E ] i) is t h a t appropriate to domain II". The damping v0

factor A is the tunnelling amplitude. Apart from a constant factor arising from the various continuations it is given by (see (174)) To

(~8J) To vo

To

since we assumed (at the very beginning) t h a t ~(x) is independent of v. We see, therefore, t h a t - - a s observed previously by BITAR and CHANG (~)-paths away from the most probable escape path in the tunnelling region give only exponentially damped contributions to tile total amplitude and thus justify the leading nature of the classical solution. However, our demonstration goes beyond t h a t of these authors who assumed the validity of the W K B expansion over the entire domain between consecutive turning points, whereas

~2

I t . J . W . ]VIULL:ER-KIRST:EN

- - a s we h a v e s e e n - - t h i s is correct only a s y m p t o t i c a l l y (dominance of the first t e r m of (165)). Moreover, we h a v e o b t a i n e d explicitly the associated eigenenergy f r o m which the eigenvalues associated with the wells s e p a r a t e d ! b y a barrier can be calculated. Thus, if %, % are the values of ~ characterizing two successive minima, we h a v e f r o m (173)

E0 = V(~o(*, ~o)) + 89 h~o--...,

(182)

E o, ---- V(~o(x , V'o)) -]- ~o-q,;h,;- .... The associated eigenfunctionals will be d o m i n a t e d b y their oscillatorlike pieces a p p r o p r i a t e to these wells, which also d o m i n a t e their respective normalization integrals. The tunnelling a m p l i t u d e A (for tunnelling between neighbouring minima) thus describes the coupling of states in the two wells and, since A is strongly damped, this coupling is correspondingly weak. Of course, in the case of s y m m e t r i c wells the eigenvalues are degenerate. I n order to obtain the t o t a l tunnelling a m p l i t u d e Ato~, we h a v e to sum (181) over all p a t h s in the neighbourhood of the m o s t p r o b a b l e escape path. A similar calctflation is involved in calculating the transition a m p l i t u d e P of (152b), i.e. we h a v e to t r a n s f o r m the integral f r o m variables r to variables ~o0, A (see (162)). I t is best to consider first the co-ordinates of sect. 3, i.e. the transf o r m a t i o n defined b y eq. (60) which we can write (183tt)

R(t) =

rtv(t) -- ~_, n~(t)w~(t) a

with (183b)

v(t) :

r" r t ~- ~o,

w.(t) :

r ' n a ~- ~]~.

I f /~(t), i----O, 1, 2, ..., are (e.g.) Cartesian co-ordinates and ei unit vectors along the axes, we h a v e

R~(t) = e~'rtv(t) -~ ~, ei'n~(t)wa(t). a

Then (184)

dR ~_ I] dR~ = J dv l~ dwa a

a

and the Jacobian J is

c~Ro ~Ro ~v ~w~ (185)

"""

J ~~v

~w~

"" "

MULTIDIMENSIONAL

PERTURBATION

TIIEORY,

THE

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LIM1T

ETC.

4~

If, as in our considerations, r(t) denotes a speeific p a t h which is not varied, we have (using (183))

~)a

~Wa

(each dr, being zero). Hence in this case (186)

dR = J dv 1-[ d~]=. a

Applying these considerations to the present field-theoretic example, we a

TO

(187)

Atot ~0

3o))~(~)] on using (181) and observing t h a t J is independent of ~7. The functional integral is easily evaluated yielding

(1.88)

To

Atot= [det (12(x, z, T'o)--Q(x, z, To))] ~

with a suitable renormalization of Ato t in order to remove an infinitely large /

multiplicative factor ~recall t h a t \

9

,,,

( ~ / "

As shown by BITAI~ and CH).rCG(5), the term in the denominator can be transferred as a contribution F to the potential in the numerator, i.e. V o , = V~-[- F . This expression shows, of course, t h a t the overall contribution of the neighbouring paths is to Inodify the potential Vr associated with the classical path.

44

5.

~.

-

J . ~v. M U L L E R - K I R S T E N

Conclusions.

I n the above we h a v e d e m o n s t r a t e d t h a t a v e r y general multidimensional p e r t u r b a t i o n technique can be developed which is a direct generalization of the m e t h o d t h a t we used previously in a large n u m b e r of one-dimensional problems. I n b o t h the classical case and the finite-dimensional q u a n t u m mechanical case we f o r m u l a t e d the technique as a p e r t u r b a t i o n a r o u n d a given p a t h (in the latter case this is the classical or the m o s t p r o b a b l e escape path). I m p o s i n g the constraint t h a t the exact p a t h deviates only in the sense of a p e r t u r b a t i o n from the a p p r o x i m a t e path~ we arrive at a kind of b o u n d - s t a t e problem: the particle is constrained to a t u b e around the classical p a t h , the deviations orthogonal to this p a t h being quantized. W e h a v e f o r m u l a t e d the solutions in terms of an auxiliary p a r a m e t e r qt. I n a classically allowed domain of the particle the w a v e function is multioscillatorlike a n d qt is the q u a n t u m n u m b e r of the degree of freedom t a n g e n t i a l to the classical p a t h ; in a classically forbidden domain, qt is a complicated function of the energy, the potential a n d the q u a n t u m oscillations in transverse directions. I n the last section we h a v e extended our m e t h o d to the case of an infinite n u m b e r of degrees of freedom. As an example we considered the simple model of a scalar field theory. The transition fronl a finite to an infinite n u m b e r of degrees of freedom is m a d e via the introduction of a three-dimensional cubic lattice. We h a v e d e m o n s t r a t e d explicitly t h a t the zero-point q u a n t u m fluctuations at each point in space lead to an infinite energy of the v a c u u m , which, therefore, has to be renormalized. Since our m e t h o d is essentially a weak-coupling p e r t u r b a tion theory (corresponding to the l o w - t e m p e r a t u r e expansion in statistical mechanics) it is particularly suitable for the calculation of the tunnelling amplitude. W e h'~ve verified t h a t deviations of the particle f r o m its m o s t probable escape p a t h lead to an exponential d a m p i n g of the tunnelling amplitude. I n the a b o v e we h a v e not recapitulated a n y results a b o u t instantons. We content ourselves here with a few linking remarks. I t is well known t h a t the one-instanton solution minimizes the Euclidean action ~g~ and thus represents a solution of the equation of m o t i o n in the classically forbidden domain. Inserting this solution into S E and calculating a m p l i t u d e (148), one arrives at an expression which is not identical with the result of sect. 4, e.g. the W K B result. The explanation for this difference is t h a t the complete calculation of sect. 4 takes into account a n y n u m b e r of back-and-forth-tunnelling a t t e m p t s of the particle before it finally penetrates the h u m p of the potential, whereas the one-instanton solution does not. Thus, in order to reproduce the W K B result via instanton calculations, it is necessary to include a n y n u m b e r of i n s t a n t o n - a n t i i n s t a n t o n pairs (2~,25...6).

(25) E. (.;ILDENEIr and A. PATRASCIOIU: Phys. Rev. D, 16, 423 (1977). ('~) C. G. CALLAN, R. DASHEN and D. J. GRoss: Phys. Rer. D, 17, 2717 (1978).

MULTIDIMENSIONAL PERTURBATION TItEORY~ THE FIELD TtIEORY LIMIT ETC.

@

RIASSUNTO

~

(*)

Si s v i l u p p n u n a t e c n i e a m o l t o g e n e r a l e m u l t i d i m e n s i o n a l e della p e r t u r b ~ z i o n e ehe c o m p r e n d e il f a m i l i a r e o s e i l l a t o r e e le a p p r o s s i m a z i o n i W K B cosl c o m e la loro contin u a z i o n e e d ~ perei6 a p p l i c a b i l e n e i casi di &ccoppiamento debole ( b a s s a ~emperat u r a ) . D o p e u n ' a p p l i c a z i o n e i n t r o d u t t i v a n e l l a m e c c a n i c a classiea, si u s a il m e t o d o p e r la soluzione e s p l i c i t a d e t l ' e q u a z i o n e d ' o n d ~ m u l t i d i m e n s i o n a l e di u n a p a r t i e e l l ~ sia nei d o m i n i e l a s s i e a m e n t e p e r m e s s i ehe in quelli e l a s s i e a m e n t e p r o i b i t i . Si s t u d i ~ n o modelli di t e o r i a dei c a m p i i n t r o d u c e n d o u n reticolo e f a c e n d o il l i m i t e di u n n u m e r o infinito di g r a d i di libertY. L a p r o c e d u r a p e r m e t t e u n caleolo s i s t e m a t i e o di correzioni di o r d i n e s u p e r i o r e al h m z i o n ~ l e p r i n e i p a l e d ' o n d a d i p e n d e n t e d a l l a v a r i ~ b i l e a u n q u a n t o e alle sue a u t o e n e r g i e . I n fine si calcola l ' a m p i e z z a c h e d e s e r i v e il t u n n e l l i n g delle eonfigur a z i o n i di c a m p o d~ u n v u o t o a l l ' a l t r o e si d i s c u t e la rel~zione t r a l ' a m p i e z z a di t u n n e l l i n g e i contributi istantoniei.

(*)

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