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August 1988
LIDS-P- 1807
LATTICE APPROXIMATION IN THE STOCHASTIC QUANTIZATION
OF (04)2 FIELDS 1
by
Vivek S. Borkar
Sanjoy K. Mitter
1The research of the second author was supported in part by the U.S. Army
Research Office, Contract No. DAAL03-86-K-0171 (Center for Intelligent Control
Systems, M.I.T.), and the Air Force Office of Scientific Research AFOSR-850227.
-c-f-- St
5"
,
;
+0t
zher_-:!
'V e
.,!
Z-Z
-sre
rie_
to appear in Proceedings, Meezing on Stochastic Partial
Differential Eauations and Applications II, Trento, Italy, 2/88.
LATTICE APPROXLMATION IN. .THE-STOCHASTIC
1
_ELDS
QUANT.IZATIONOd-0,5.( .4-),-.-F
Vivek S. Borkar
Institute for Fundamental Research
P. 0. Box 1234, Bangalore, India
Tata
(TIFR)
Sanjoy K. Mitter
Department of Electrical Engineering and Computer Science
Laboratory for Information and Decision Systems (LIDS)
Center for !ntellicent Control Systems
- 'MassachusettsInstitute of Technology
Canridce, MA. 02139
U.S.A.
-
-I.
INTRODUCTION
The Parisi-Wu program of.stochastic cquantization
struction of a
carried out for a
K.K.itter in
proves
-
finite volume
[6].
These
(eC4)
related limit
This program was
measure by G.
imi_ theorem.
theorem, viz.,
that
rigorously
Jona-Lasinio and
results .aere extended in
a finite to infinite volume
is to prove a
involves con-
stochastic process which has a prescribed Euclidean cuan---
turn field measure as its invariant measure.
P.
[8]
[2],
The
of the
which also
aim of tis note
finite dimensi-
onal processes obtained by stochastic cuantization of the lattice
fields to their continuum limit,
__ The proof imitates
that
i.e.,
the
(c:)
2
of the limit theorem of
though the technical details differ.
Note
process
[2]
that
of
[2],
(c&)
2
[6].
in broad terms,
this limit theorem can
also be construed as an alternative construction
of the
(¢)
process
- in finite vol-,me.
The next section recalls
the
fi ite
volue
(4)
=rocess.
Section
2
____
IT
srumzarizes
the relevant
facts
the
(Y4) field from Sections 9.5
aSut the lattice approxim ation to
and 9.6 of
[4].
Section IV
proves
the lim.it theorem.
--
iT:e research of the second author was suoore=in- -art -y the U.S.
Army Research Office, Contract NO. A.__LDo3-86-K-0!17
(Center for
Intelliaent Control Systems,
assa cuse_-s Institu-e of Technology),
and-by
the AFrorce O0f of Scie-fi=Rescarch,
Contract
o.
AFOSR-85-0227.
--
II.
THE
PROCESS-
(~4)
2
2
2
Let ICR
be a fi i-teb-rectangte,-whichirfor simplicity, we take to
be the unit cube
3
x =
(xeS
(x- [0-J<L:,
1
2
1
:xnt:¢e]j .hLet A denote
-
K
J
-byth&Ebasis ek (x) = 2
kk,
2
2
k)
= n7r,
2'
+ k2.
Fcr aR,
1
(k2)'
=
n> 1, let H
2
denote the Hilbert space obtained by completing
,.& to the inner product
< f-, g >
the Dirichlet
2-
Laslace oyerator on A.
is diagoal 'd
sin (kx-) sin (k x), x= (x, X), k3 =
11
22
1
2
i = 1, 2 .
In fact, -A e= k 2
where k-k
2
-D (A)
with respec
--
<f, ek> <g, ek>
_k3
_.
where < ,-> is the L scalar product. Topologize Q=UHa by the countable:
_,family
of semi.ntor~s 11*n= <'. > ani- Q'=UHvia duality.
Let C =
1)-
(-A +
C(-, -)
its integral kernel, C" its a-th
" operator power, and pC the centered Gaussian measure cn 'H
with covariance C [2], [6].
Let denote the Wick ordering with respect to
C (see [4], Ch. 38 for a definition).
The (¢4) measure on HP- is
,^~9~~~~~~~~~~~
-'~~~~2
2
-defined by
cdu
= ex
(-
j
( -exp
dx)
I
/ Z
2.1
-,_
where
Z
=
e
dx) d&
J
<00
See [4], Section 8.6, for details.
Le. 0-<s <1 and
Brovmnian motions.
W(t)
=
L
(-),
ks3,
a collection of^
ide=eeden-
standard
Define
(k-)
This defines an H
k
-
1
-(-
>
-)/2t
-valued Wiener process with covariance C2-
C
[2,
[6]
The ecuation
d¢(t)
(C_-)
-
- + C(t).)+
with init-i
l law B can be shown to hav
tion_ as. az. H.-- v-=lue- -process, defi.ini
-(4) process.
2S
See !12], [6] for details.
dt
a
dW(t)
nie
an erodc
ta
a-e na-y w =process
--
called
[2.2]
-
.
so u-
--
th_
.
.
r-- .-en
-
-
-
tex: ot sezon
-
3
:
azna succeeancr-
pe:
'i
here. .. c2
l;eEve a__:Iorr.:
s-me..Sie tr, .
mc.rc-s
;'_-
1III. LATTICE APPROXIMATION
2
Let A = {2- n, n l}- and pick-6cA.
The finite lattice A. with spacing
*naier!ne=---r^s
m.o-r.,n-.
6=2
6 is defined as follows:, 'Let6Z
{ zzcZ
'2rA,
intA-int
fl 6 2 ,
2
3--A= BAf)6Z
= int-A -UBA= Arl 6Z2 .2
,
(int A.)-is the Hilbert space with
inner product
:
<f,f>
intA
lfxEl ,
xcint A6
6
2 (A6)
_ viewed as a subspace of
-
in direction a by
6'
(6, af)
£2(6Z 2 ), define the forward gradient
On
(x)= 6
'ff(x+du
a ) -f(x) ]
unit vector in the a -th direction for a= 1,2.
is the
The backward gradient
2
a,
where va
is its adjoint with respect to the Z2 (6Z )
inner product.
* .
- Let
-LA6= a 6,11
6,1 +
6,2
a,2
6,2'
Then
)f
(
62 (-4f(x) +
(x) =
6
where the summation is over the nearest neighbours of x.
£2(6Z
-projection
2
) -[2(int A6).
_
f(y)
Let n be the
The Dirichlet difference Laplacian LA
0
is defined as T AI
and agrees with A6 on int A.
6
Choose as a basis on Q
6
{= (x) =
(x)lxE
int A.,
(int A6 )
k =
-_Lemma 3.1 ([4], p. 221) {e
,,2 ...
the
(6-1-1)2 functions
,(6o1 -1)r;
a =1,2.
diagonalize -Ao with
i=l
Also,
<, k
-Lemma
3.2
ding- of--
>
-{'int
([4],
p. 222)
(int At)
2
=1 if k=
0
L
2
, =0
otherwise
6n
The map io
e
+
defines an isometic- i.ed-
(A).
Let HO be the projection operator on L (A) which truncates the
0
2
Fourier series at k /r = 6' , so that
T_,
I6
ek =
>
=k= (k.1 .k)
2
1)< -
a°<ek
<
where
o6-l,
-
dE
enotes the
i=1,2.-
(t
)
o
0
2
0
via the above isometry, i.e., let
consider
C.
(resp.left)
is
--
2
wich when restricted to the latt
ce
LeITma 3.3
-
Mcreover,
([4],
supD-
pp. 222-224)
C
C6
-~~~~~
C (x(6
H(
where
C -
zoints
ni-
on £2 (int
<r(O
)
_'
-. L (A),
the right_ -
Con
an operator onL9 (A)
-
i
we can
,
its kernel
'
--e matrix entries of CO as an ocerator
o
-
-
0-
6
-
sum=_ation over
(-nt A.) as a:n oper-_r
i
ey)
'
('
i-
Then
2
acts on £2 (int.)(resp.L
(.').As
C"(x,y)
-
62)
t
A 6 , coinci
es with
.
A.).
0
as operators
for
at<(r
< (2p ',1).
o
on L (A),
-'
---
e=;n text ot seon= ania suzc=eeo;ng pae--s
nerr. .o
t ive
r-'
a-::ton'
If ~ is a Gaussian field with covariance C,
x
2
maaro;n-
IrsE
:he tr.me.
(i)
6(x)
(x) for
i= C i
int A 6 defines a Gaussian lattice field with covariance C
The field 0.
real.lizedby a.Gaussian measure on L (Rlintisl)
can b
&1 --
Explicitly, letting
x d-eno-te+-he.Lebesgue
Ij?int A i,
R
int A6,
the above-measure- is given by
15
1 int
.
m!-2!2
*Xr-n-aqS
2
measure on
A
I d86 (x).
x
This is the lattice analog of PC
,now be defined as follows:
-:
(f)
62
Define for f s
: 0^(,)
The lattice analog p6
(int A),
£2
f(x)-
A
XXint
The lattice analog of p can
O
is given by
This7 defines an L (Anvalued Wiener process with covariance Cf
0
Brownian motions.
For O <s <1, define
B (t)
= 62 C&
(X +1)
(l-s)/2
Bk(t) ek(*)
The
£.,
t >0.-
analog -6
of [2.2]
lattice case is
= 1 in the
4 (x):
(1))dp8C
Forhere
the operators act on L (f).b (e) is viewed here as an L (a)d-valued
2
0
2
.ocess.
Hoeve,
leting
e
(t) =
°- 6k(t)ek, [3(2] tansltes into
an ec-uivalent
_rocesses
ents.
stochastic
differenial
ec'a-c.
¢ok(.) with locally Linschitz
ior
many scalar
finriely
(in fact, polynomial)
This ensures the existence o- an arsi
to [3.2] up to an explosin time. etThct
ise
latter
usd
enceorth.
shos
that
se'
ctroabiity
the operators
a
orehe
_v
a
does not explode ase
interret
ation,
enerxpator o
-cdoin, on L
on L iew
().
te
st
an).
By T
.
or!c 2.3 of
$Ttgcu0=Jgu,
is 2n
a_
ismey-ing t-ht
To
-__
L.
.,
(t
2
),
via
e re tain
the
(A)-e
notatueion
will be
uniquet
of [2],
Section 3,solution
y
[3.2]
is
[33s, tne seme hosnonhe
s for
2
__
intertretation
of operzors
pva
ant
on L20(u3).
probability measure
..
gr4
ga
mllg
i
nJv~atnobbl
.::
usedhencfrhAcoptinsiiatoato[2,Scon,::
-=bow~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~sthtte==--.
-nt
is
'on-
.
doces nodescribed
That
ov
iTt,
,
s
ofsimilar toa-s-
associated trnsltion senicroun
Secto
.o
on £2(int)
as onlyt
comutation
the
t
-nres
measure
-
unique strong solution
___proved by a standard a2clication of Khasm.inskii's test 'or
exolosion exactly as in [G], Secticn.3.
dy identifying the vectr
6 (t) 6, dt + dB
5st =rob
can be considered
=as
coeffici-
I-
u
--- egtn tet
o- ss=on c
for ~a(-).
-..sz__..._
In
fact,
ec.rC
-
the resulting
. rL.: le2ve
always be considered with initial, law.
v
IV.
,
,r--t- _ nS::E
process will be ergodic.
need this fact here, so we omit the details.
2
:21:-:
,-
From now on,
e,
-.
We won't
[3.2]
will
_.
THE CONTINUUM LIMIT
This section establishes the main result of this paper, viz., the_
c.onvergence of 6 ( )to the (¢?) 2 process as6 in in
n: the sense of weak
convergence of Q'-valued processes.
Thus we consider 46(-)as aQ'-valued
process and u6 as- a measure.on Q'via the injection of L 2 t) into Q'
From
_
9
-theorem
.6 . 4 ,p. 2 2 8, [4] , it follorws that the finite diensional marginals of the
-collection {1&6(ek),ksB under ~p
converge weakly to the corresponding
* ones under P as
+-O in A.
Since P, 1i are supported on H 1 , it follows
that ii
p~iweakly as probability measures on Q'.
(A proof of the
-former assertion would go as follows:
..morphic to a G
subset of
[0,1]
a compactification of HF1
:
Since H. lis Polish, it is homeo-
whose closure
T-
As a measure on
can be considered
-1',{6
are tight and for
any weak limit point U thereof, its restriction u' to H 'must
yield the-
same finite dimensional marginals for { (e;) ,ksB as p.
Thus u =u' ='.)
As a first step towards proving the continuum limit,
we prove some
tightness results.
_-
-
Let
4..(t) =
2 (t)
aa(t)
2 fCO(s)ts
2-
023
2C =2fJ
t)
.0 4
--
CO
(s):
S:
ds
0 (t)
for t < 0.
Pick t < t 0 in [0,T], - > T> 0.
In what follows, K denotes a
~~~- -1
2
positive constant (not always the same) that may depend on T, but not
on 6. Let fsQ
Lerr,a 4.1
c
-[(
C-
C
(t) (f) at) I < Kit
_[4.
-
-Proof Using Jensen's inequalitv and stationarity of. (-) .
--
E[(Sta
CY
d(t)
6
(f)
dtC)
<
K
1 (0)
C)
I_ -_[c
12
t
()1
one obtains
144
-
tl
Letting A = ci / dt
S
__By Lemrnma
in 6.
--
-;: --
[4],
l
one
&
cC'
a~''i
c'C
9.6.2,
Using
the
e
'.o
tecta-tion on_the rig-.t is bounded by
0
6
p. 227, [4], the second term above is bounded uni formly
ey--man
rap-n calculations, as in Theorem
-.-
has
...
. ,--
,~
11
8~1
.-
l
.--
53,
p
..191,
...
'
-
.P-|d,
r
II c
4CW)<
(*
K
r
-
-
f i':
Now
=1:12fe,
f -C-icf1
~1;C
~2
;
-,
,
(-+
e _ _ >;
(k+1))
IksB
;
.
--The summand on the right can bedom.inatedin absolute value by
K < fek>2 X2
which is summable for fcQ.
k
By the dominated convergence
theorem;
lirm
s f
II-C
-C_ f 112=
1 CfI
implying sup
-Lemma 4.2
0
[4.
1]3 follows.
<.
CE(C(t)
_°
J
:()
QED
dt)c] < Kt -t2
°
-
2
[4.2]
1
,-2
This follows along similar lines.
E[(B ()(f4
-Lemma 4.3
1
) (f)
Proof The lefthand side ecuals
-'=t
2
31 CC (f,f) 2 It
< 3 sp
f
14]
< Kt'--t
<2.
[4.3]
1
2 It, t 12
Ti
C
As in the
t
t
proof-
in
proo
of Lemma 4.1, one can prove
_
-
z±=
lI
6+0
/
Thus sup lCs('-E
Corollary 4.1
f --
~'
fl 2<- and
the claim follows.
QED
(f)14 < Kit 2 -t,]2
E[1.(t z) (f) -(t)
Proof Follows from [3.2]
Lermma 4.4
-O
and [4.1] --[4.3].
The laws of the processes
QD3('),(
[
(C(0,);
Q'))4--valued
'
(')
061
viewed as
[4.4]
02
8). )]
J03
random variables remain tichIt as
.
varies over A.
Proof
By Theorem 3.1 of
of [l ( ) (f) 6
(
[7],
(f)
- )
it suffices to establish the tightness
(f)
) (f)]
g8
(C([0,T]; R)) -valued
random variables
This, however,
d.e
is i
from
te
c'r abrStrary T>
fiom tness of
weakly as a measure. on H '), the est-nmates
criterion of
[13,
on [0,T] as
a0
nd f-Q.
,u4 (since
[4 .1]-[4.4]
6:
and the
p. 95.
QED
Recall that a familv of probability measures on a product of
Polish spaces is tight if and only if its -mages under projection onto
space are.
each factor
-
This implies,
-
6, ()(),
Letting
in vi ew o =e
denote an en; meration of
t [d
f regoing,
C. (e),
) .
) Ce),
'01
{~:;}
along A.
_collection
Lc ::-rc
tr;:C
C
e
===
i
;:
Then fcr an.-
g,...,g
'
e :vorcTr:,i;
-
_
.:
of
:
.
-'
--
C
_--
_-==:=
::=:.
_:;
as
at. the y converg_ in law as
subse
finite li-neair
are ti**t
droppinng to a subsequence
of. A, denoted by A again, we =ay assm.
e
-0
(e),
) .....
()) (e)
..)]
2.....
(C([0,o]; R)) -valued random variables.
-
(
{ei.
{t
·,,
1
,...,}
combinaions
rC
r.-
of
[0,]
of
-
=--d a
, the
.
-
t
, .e
n Tir1
'.,
7f-r~rr.=
a
__'
O__.F-
z
cl(tj)(gj),
joint laws of
L-collection f
E[!
6
Eli
2
E[[E
we have
_
-
e
;
Consider a
<... .
.
....
_gj
_r.
, _ _4.5]
[.
-a)2
I.]
(f
-r
-j -gj
..
f4.6]
)!2
-
) (fj -gj) [2] < MI] C 1
(tj
*'
converge.
<j < t
(tt.) (fj
)I.<Mifj-j:-.
t)
(tj)(fj -gj) ] < MIl CI
Er!¢53 (tj)(fj
M s'
Using the kind of estimates used in the
proofs of Lemnas 4-.1-4-.3,
2
_
i
< i < 4,
in Q.
...
,,
-
)/2
(f j
[4 8]
As 6O0
in A,
'for a suitable constant M depending on max (t ,...,tk).
-the right-hand sides of [4.6] - [4.8] converge to the correspondina quanSince cj can be obtained by suitably trun-
-tities with C replacing C6 .
cating thne Foui er series of f
le_, each of these limiting expres-
in
[4.5] can be made smaller than any pre-
-sions and the righthand side of
-- scribed n >O uniformly in 1 <j <k by a suitable choice of {gj}.
can be made smaller
[4.5] - [4.8]
-:,follows that the righthand sides of
It
than any prescribed Ti-+0 uniformly in 6cA and 1 <j <k by a suitable
g;j.
of
'choice
be an enumeration of fi nite linear combinations of {ei
Let {h2z
By a well-known theorem of Skorohod
with rational coefficients.
p. 9),
we can construct on some probability space random variables
6sA,
:Xij,
'
Y.ij
law with
-in
i
<4, 1 < j <k, Z > 1,
1 <i
random variables
joint law of
['i(tj)
and E[IX
Yij
i0
(tj) (h 0 ) } for each fi-ed 6 and Xij
struct on it
[Z6 ij
{XijQ}
such that
3v augmenting this probabilitv space, if
A.
-of
([5],
a-s
i
as 6
necessary, we may con-
s above, such that the
(~,i,j)
(h)
(t)
(hj,
(fj),t
...
]
agrees with that
s
Yij
Si-ce Xcij
-or each 6, i,j.
(h)[] ican be bounded uniformly in 6 'for each
3
Xij2'
I].
Zasij
arees
-
E[L 6
=
t)
]
i,j,Z by estim,a-;es analogous to [4.5]-
[4.3], we have
E[Xc
j6i
-Yij i 2 ]0
- 0, we can On the other hand, given
as 6 0 in A for each i,j,r.
pick Z(j), 1 < j < k, such that setting gj')= n
[A.5] - [4.8] makes
+
_.
-allthe Z_-anti
]!ira :[
_t
-
es on the righthand side
j
-^Zj1-
there less than n.
, i
< 2_
A
,a.
rrt ('j ) -r.....
= 2r
Thus Z 6 . converge in mean sauare fc- each i,j as 6
(C([0,o]; Q')
denote its
thn
l .i
-'i
) --v a l u ed
nlim-_-.
2in 32]
law
--
(f
-X
sA
£
o,
int laws of*_ ^ '(tjt)
that the
5 3-. [7, [7,
now im-les that
Thus
r
, (-)]
''
random variabscs.
as. "'
'
along an aapro*rz
It f1llows
,;Thosrse.
<k} conve
Cj <k
i< i< < ',< 1 <_j
(,
Z21re'
0 in A.
e
con-erve
['' (-),
t
.I
nenceo
Orth).
s=-n=ecue=e,
as
(] )
,
-
( )
By
(
)
7
taing -
:.::.-.
-- _
2 (t) =
._
:.
-, ._......
¢1 (o) +
Theorem 4.1
-
f-tr
-
ci(t)
j
i=2
2 (*) is
-
;
-
as-
14.
91
the__(g) 2 process.
Proof We prove the theore --bv
dent
a- -iter
of
Let fQ..
f
.......
_-~-Y,.
- in ~
ter
of- [4.9.
[4.9].
Byc ensen's inequality and stationarity, E[I j
(s) (C f) ds |2
|¢(s) (Ccf.) ds|]
<1 tE[I
(o)(C
)
<
K_-1C-f C2
_____
..
he rig.thadna side
ends to zero as o6- 0
employed in the proof of Lemma 4.1.
-
J t¢,
(t) (fS)=2
(s) :
[j:
t E
d
()
(C 6
ds
,-
analocous to those 2above.
i. (-),
(°6
s) : (C
0
c
f) ds]
as in
6o
O in
A, by a-ruments
Hence
(s) :O (CO .f)
I,
(Cf)d)
s)
a
f) ds
(0): (CC
(()(-,
|. tg()
0
6
SLilarly
[
_<
Thus
- 1.15i.
6+0
t foll ows t-t
-.
2,
y arguments similar to those
ds) = ((1
(f))
(),-2d(t)
-.
=
-
___
Let_ a > o in
_Et
<
.
-
A.
Then
-
Et[
*
.i.l.
*(f::
(4,)
(S),
(C
s):
(0)
:
ds
(C
s):
(.C:
f)
.,
(C
[4.10
f) ds
(C
-,s
° /,
is
..t(.(=:
by
Ji
.
* -
(ts)
$ e)
*
-hst(.
e
:,
1d
a Suitab.;1
p, 228, [4].
vThus
<)]
C()
f.
_.
0-,- 0
~)
:f)ds)
O
>O un io rmiy in o6 as o6 O, by virtue of (9.6.9),
-he richqthand side c f [4.01]
e-l..als
-1_J.
d) (s).0~(C
: -
w-tere- e-
(C
_.
(s).
f-o
(C )
ds)
-- ecrn text
-The
Tcsfezo=nc anr
sucree-m
,F
-c;Z;..2
z
nerc.=.
-. 4,ma
rr
cir_ ts
*
.-
above limit equals
1
t
1-C
(S): (C
f) ds),
.,-,
a
Thus
)3
(t) (f)
=
1
t
r.;
-(s):
:--
(C
f)
....
ds a-s-
U
Finally,
it is easy to check that Q4 ()
covariance C-.
Thus
satisfies
s)(-)
uniqueness in law of this equation
-clude that
6
[3.2] with initial law yp.
(proved in
[2], Section IV),
By the
we conQED
(
') converge in law to
random variables as 6
be a Wiener process with
(~ )2 process.
1 (-)isthe
.-Corrollary 4.2
will
-0
4 (-) as C([0,]; Q'-valued
in A, as defined originally.
-Proof A careful look at the foregoing shows that any subsequence of A
-will have a further subsequence along which the above convergence holds.
QED
ACKNOWLEDGEMENTS
.r
work was done while both of us were at the Scuola Normale
-This
Superiore, Pisa.
Vivek S. Borkar would like to thank C.I.R.M.,
Italy,
for travel support, and the Scuola Normale Superiore for financial
support, which made this visit possible.
REFERENCES
[1]
P. Billingsley.
Convergence of Probability Measures;
& Sons, New York, 1968).
[2]
V.
S. Borkar, R.
T.
Chari and S.
K.
Mitter.
(John Wiley
"Stochastic quanti-
zation of field theory in finite and infinite volume."
appear in J.
Funct.
-
To
Anal.
[3]
M.
Fukushima and D. W. Stroock.
"Reversibility of solutions to
martingale problems."
To appear in Seinaires de Probabili4's,
Strasbourg.
Glirmm and A. Jaffe.
Quantum Physics: A Functicnal
nteralPoint of View, 2nd. ed.; (Springer-Verlag, 1987).
[4]
-
J.
-[5]
N.
Ikeda and
S. Watanabe.
Stochastic Differential'ETuations
and Diffusion Processes; (North-Holland Publishing Cormany/
Kodansha, 1981).
[6]
G.
Jona-Lasinio and P. K. Mitter.
of field theory"; Comm. Mat-h.
[7]
I.
itoma.
"Tightness of probabilities in C([0,1],' ), D([0,i],
-- )
"; Annals of Prob., 11 (1983), 989-999.
,= [8]
G.
"On the stochastic auantization
Phys., 101 (1985), 409-436-.
-
P-=risi and Y. S. Wu.
"Perturbation theory without
-ing"; Scientifica- Sinica-, 24 (1981), 483-496.
cauge fix-