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Real time approach to instanton phenomena
Nuclear Physics B142 (1978) 125-147 © North-Holland Pubhshmg Company
REAL TIME APPROACH TO INSTANTON PHENOMENA (II). M u l t i d i m e n s i o n a l p o t e n t i a l w i t h c o n t i n u o u s s y m m e t r y H J. DE V E G A
Laboratotre de Phystque Thdonque et Hautes Energzes *, Untverstt~ l~erre et Marle Curie, Parts, France J.L. G E R V A I S
Laboratotre de Phystque Th~ortque de l'Ecole Normale Sup~neure **, Pans, France B. S A K I T A
The Oty College of the Oty Unwerstty of New York ~, New York, USA Received 10 July 1978
We extend further prewous works on the tunnehng problem of degenerate classical ground states for systems with many degrees of freedom which are based on WKB solutions of the Schroedmger equahon These solutions are exact to the first two orders m #I. The point of the present paper is to deal with the case of potenhals with continuous symmetry assunung that the spectrum of small oscillations around classical vacua has no zero modes besides the ones due to spontaneous symmetry breakmg This continuous symmetry introduces non-trivial difficulties m matching between WKB and harmomcosctUator wave functions, because the mstantons and the classical vacuum leave mvanant different subgroups of the symmetry group, m general Ttus problem is solved here. We deternune the ground-state wave function as well as a general formula for the groundstate energy which is shown to agree with the path-integral computation m the ddutegas approximation The results presented here have a direct apphcahon to field theory which we discuss m a compamon paper.
1. I n t r o d u c t i o n Ttus p a p e r is the t h i r d o f a s e n e s o n t h e WKB a p p r o x i m a t i o n t o t h e S c h r o d l n g e r e q u a t i o n f o r s y s t e m s w i t h m a n y degrees o f f r e e d o m a n d field t h e o r y . In t h e first
* Laboratolre Assoc16 au CNRS, 4, place Jussleu, Tour 16, ler &age, 75230 Pans, Cedex 05 ** Laboratolre propre du CNRS assoct~ ~ l'Ecole Normale Sup~neure et ~tl'Umversltd de PansSud $ Supported m part by National Science Foundation under Grant no. PHY77-03354 and m part by the Research Foundation of the City Umverslty of New York (New York, NY 10031) under Grant no. 11681. 125
126
H.J. De Vega et al. / lnstanton phenomena (11)
paper [1 ], denoted by I, the general WKB ansatz was estabhshed for wave functions associated wath a given classical trajectory In configuration space. In a second paper [2], denoted by II, we computed tunneling effects on the ground state for multidimensional potentials The essential features of the method are as follows. We study stationary states thereby ehmmating the true time of the problem from the start. The WKB eIgenstates of the Hamdtonlan are described by wave functions concentrated In a small neighborhood of the curves which represent the relevant classical solutions in configuration space. They are most easdy expressed in terms o f the local coordinates o f the corresponding region which naturally involves the position q on the classical curves. This parameter q can be piecewise related to the time r o f classical solutions which is thus only used as a geometrical parameter o f configuration space Going to local coordinates along the curves is equivalent to the introduction o f q as a collective coordinate. In serm-classlcal hmlts, its dynamics is found to be mostly onedimensional WKB, while the other degrees of freedom are essentially harmonic '* For tunnehng we use the one lnstanton solution, in order to budd the WKB wave function in the forbidden region. The ground-state wave function IS then constructed explicitly by matching this WKB expression with the harmonic-oscillator wave function around each minimum which would be the ground state in the absence of tunneling In paper II we rederlved In this way the expression for the ground-state energy shift already obtained In the dilute-gas approximation by the imaginary time path integral method. We have thus proved that this latter result is the correct semi-classical approximation This proof, however, was restricted to the case where there is no continuous symmetry and where the spectrum of small oscillations around the absolute minima of the potential has an energy gap In the present paper we remove the first restriction and extend the discussion of paper II to the case of continuous symmetry It IS proved that the dilute-gas approximation is still correct if there is a continuous symmetry, provided the spectrum o f small oscillations around ground states has no zero modes besides those due to spontaneous symmetry breakdown. In a recent paper the gauge problem of field theory was discussed in A0 = 0 canonical formalism, in such a way that time-independent gauge transformations are treated on the same footing as the other symmetries of the system [3] (paper III) This approach is best suited to our purpose. Hence, the present extension o f II Is crucial for field theory applications which we discuss in a companion paper [4]. Since we assume that the theory has an energy gap we still exclude field theories with massless pamcles. As discussed in I, the dilute-gas approximation may break down in this very important case This more involved problem is currently under investigation In sect 2 we establish the general form of the WKB wave function with contlnu* It can be pointed out that our semi-classical wave functmns are exact to the first two orders m h (h -1 and h0), contrary to other discussions m the recent hterature
H J De Vega et al. / lnstanton phenomena (llJ
127
ous symmetry, for the case of interest in field theory, namely when the potential is periodic. In sect 3 we construct the ground-state wave function and compute the ground-state energy shift due to tunneling. The physical picture is as in the first Brillouln zone of a crystal. In sect. 4 we finally show that we obtain the same result for the energy shift as the imaginary time path integral approach In the dilute-gas approximation The p r o o f IS based on the equality between Jost and Fredholm determinants for the scattenng problem associated with small fluctuations. Since we have an energy gap the corresponding potential tends exponentially to ItS limit at infinity, in which case the above equahty certainly holds The situation is not so clear if there IS no energy gap This case IS currently under investigation. The present article is not self-contained. A good knowledge o f papers I, II and III will be assumed in order to keep the length of the discussion within reasonable llrmts 2. WKB wave function
Throughout our chscusslon, we adopt the same notation as in I and II. R represents a p o m t of configuration space with an arbitrary number, N, of components, V(R) is the potential energy In the following we shall only need the expression for the WKB wave function in classically forbidden regions. We shall only consider this case exphcitly and introduce a classical trajectory r ( r ) which satisfies rrr = V V , - ] ,1(r)2 r + V(r)=Eo.
(1)
The r indices mean r derivatives, and obviously Eo < V(r) We further assume that the system IS mvarlant under a connected group ~ of continuous transformations with parameters X u, t~ = 1, ..., k and denote b y R [xl the corresponding transform o f R. It w111be assumed to take the form
(RtX]), = O(X),j(R)I
(2)
Since it has an lnhomogeneous term, a gauge transformation an field theory does not satisfy this relation. As discussed In III and In the companion paper [4], ttus does not make much difference, so in the present discussion we make use o f the above form. If V(R) IS mvarIant under ~ , that is If
V(R) = V(RIXI),
(3)
there exists a continuous set of classical solutions r~rX)l where X are arbitrary constant parameters of ~ * * In order to slmphfy the discussion we assume that ~ Is completely broken by r0") Our discussion can be easily generahzed if needed
H.J De Vega et al / lnstanton phenomena (11)
128
In I we built a WKB solution of the Schrodmger equation 1 H = ~p2 + V(R)
H~b = Eta,
(4)
by perforrmng the canomcal transformation R = r('r) + ~rta(7" ) '0a ,
(5)
rr na = 0 .
(6)
a
The geometrical meaning of this transformation IS that for a given R we define r such that the correspondmg point ~s on the normal to the classical curve at r(r). The r/a's are the components o f R - r ( r ) on a moving frame orthogonal to the curve at r ( r ) This change of variable is only possible m a small neighborhood o f t ( r ) , such that this normal is umque As discussed In I, this procedure IS consistent if the wave function decreases exponennally away from r ( r ) , that as If the phenomenon considered occurs mostly m configuration space In a small tube around r(r). It is clear that this is not true ff there IS a continuous symmetry since all classical curves r l X l ( r ) are on the same footing In this case we modify (5) In order to obtain a good parametrlzatlon o f the neighborhood of the points r~r~l for arbitrary X. This IS a manifold c-ggwith more than one dimension. The generahzanon of (5) IS N-1
R=rlX](r)+ ~
na(X,r)rl a,
(7)
a=k+l
ha(X, r) r[rXl(r) = 0
n a nb=6ab ,
,
ha(X, z) 0rlXl(7") 0X ~
=
0,
(8)
a= 1,..,k
The geometrical meaning is the same as before except that the classical curve r ( r ) has been replaced by the higher-dimensional manifold c ~ . The change o f variable (7) makes sense in a neighborhood o f c ~ and we shall have a consistent approxImation scheme if the wave function is exponentmlly decreasing away from It, as one can check explicitly in the final result In the same way as (5), (7) as basically a simple change o f variable. It can be handled simply, however, only by applying standard h e algebra techniques to the group ~. These were already briefly sketched In III It IS convenient to Introduce the multlpllcanon law f2(X, Y) of the group ~. It is such that for instance (RIXI ])(x21 = R (~(x2, X])l .
(9)
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H..J De Vega et al / Instanton phenomena (II)
The following derlvatwes of ~ are useful
as2~(X'Y)I c/It- 1 ~(X) =
~-1 ~,(y) _
0 Yt~
(10)
Y=O'
O~(X' Y)[ aXO x=o
(11)
The role of ~ and c~ IS to allow the computatxon of derlvatwes of the group element with respect to group parameters Indeed if I s are the infinitesimal generators in any given representation one has
a~(d( e' I ¢xt3) = l l.r ct~ (X) e' IOx# , D~,~(e' S#x#) = e'10 x# tI./ cl?~( X') .
(12)
It follows that if we define Zo¢ = ¢'ff~- l ( x ) ~
l
,
(13)
we have [L., Lt~] : , ~ L . r ,
[L'~a,L~] =, ~ / ~ ' r ,
(14)
IL~, ~1 -- O, where the ~
are the structure constants of the group gwen by
[I~, 1A = ~ I ~ .
(15)
The appearance of these two commuting Lie algebra is well-known in particular m the mechanics of the rigid rotator. One (La) Is the generator of rotation m Inertial frame, the other (/~,~) is the generator of rotation in the body-fixed frame. In all practical apphcatlons we have in mind, it is always possible to define the scalar product which appears in (8) in such a way that for arbitrary points R 1, R2 R[ xj • R tx] = RI . R2
(16)
We can realize (8), in this way, by choosing
na(X, 7") = n[aXI ( r ) -
(17)
H.J. De Vega et al. / Instanton phenomena (11)
130
By defimtlon, ff ~,a are the parameters o f an lnfimteslmal element of (~, one has R [xl ~- R - t)~'~ [R, L,~]
(18)
Looking at (7) and (8) one sees that r and 7?a are mvanant under the group. They are parameters of the orbits o f ~ which are in c ~ or m a neighborhood o f it. Because of the lnhomogeneous term, this argument must be shghtly modified for gauge groups The final result is essentially the same however (see Ill and the compamon paper [4]). WLth the choice (17), we can rewrite (7) as
R = r [Xl 0") +111Xl (7") '
(19a)
N-1
11[ X l =
n[Xl(r)rf.
~
(19b)
a=k+l
It follows from (8) and (12) that T11s such that q'rr=O,
q r,a = 0 ,
(20)
OrIXI r,~- ~
(21)
x=o
In the same way as m I and III, a straightforward computation gives k
P=-tV=
~ r!Xl(I-1)o~(Lo-q,o.;)+ a,#=o
;[Xl ,
N-I
;[Xl=
~)
n[aXl(r)-t
.
(22)
a=k+l
We have adopted a compact notation whxch includes all collectwe coordinates on the same footing. When the summation runs from 0, the mdex zero refers to r. Indlces a :/: 0 with a comma denote denvatwes at vamshmg group parameters. Lo is simply - t a / a r and r, o is r r The m a t n x la~ is gwen by I~(r)
= (r,a + r l , a ) • r , ~ ,
a,/3 = 0 ..... k .
Smce we want to deal with mstantons we shall consider a solution r ( r ) of (1) with E o = 0 For r -->+_ooit tends to different minima o f the potential
(23)
H..J De Vega et al. / Instanton phenomena (II)
131
As in II, we assume that the existence o f several classical ground states is connected with a dxscrete symmetry of the system. In II we considered only two minima Here we shall rather deal with a typical situation o f field theory as discussed m III Specifically we assume that ~ lS the connected component o f a larger group o f mvarlance ~ under which the system IS also lnvanant. The d~screte symmetries are the elements o f ~ / ~ We shall denote b y r o the minimum o f V(R) such that hm
r(r) = ro
(24)
For r -> +0% r ( r ) tends to another m l m m u m which is of the form rio~1 where Y, is an element o f ~ not In ~ , 1.e. a discrete symmetry. Clearly we have to take into account all paths rl zn] (r) for arbitrary integer n. They connect all classical vacua of the form rioznl. Additional discrete symmetries can be ignored If, as we assume, there is no mstanton which connects the different members of the correspondmg fanuly o f classical vacua In II we dealt with the case where y2 = 1. Here we rather assume that ~n is different from 1 for all n except zero (lntermedmte cases are of course possible). Hence we have an mfimte set of classical vacua rio~nl connected b y paths r ( r ) I ~nl denoted by r(on) and r ( r ) (n) respectwely where ever needed. By construction r(r)(n) r ~ - 2
r(o' 0 '
r(r)(n) r--S-~+Zr(on+l) .
(25)
As noted In I the time z is not a good parametrlzatlon o f the whole set o f classical paths since it goes an infinite number of times from - - ~ to +o~ as we move along it. One can instead introduce a penodlc function f with period unity such that for o + 0 + and n integer
f(n - o) -+ _oo,
f(n + o) -->+oo,
(26)
and the complete set o f paths m configuration space is parametrlzed b y Rcl(q ) =r(n)(f(q)),
n ~
(27)
By construction it is such that
Rd(q + 1) = R ~ 1(q),
(28)
and it IS thus penodlc up to a E transformation. In the same way as we did in I, ( 1 9 ) - ( 2 3 ) must be extended by using q instead of ~'. We do not do it explicitly since thJs is simply a complication o f notation and since In each region n ~< q ~< n + 1, we can use r instead o f q . Since these regions are related by Z it is enough to deal for the moment with the region n = 0 which was the starting point o f the present discussion. Near ro we have V ( R ) "" ~(R - ro),(R - ro) 1 ~ .
(29)
132
H J. De Vega et a l / Instanton phenomena (II)
The new feature, as compared to II 1S that l f r o IS not mvarlant by ~ , c~ has zero elgenvalues. To slmphfy we only consider the case we shall encounter later on. We assume that the ground state r o breaks a subgroup ~o of the continuous symmetry group with ko parameters. Because of the symmetry we thus have ~ro,ao = 0 ,
ao = I . . . . ko •
(30)
From now on, wherever needed, we label with an index ao the denvatives with respect to parameters of ~0 at the Identity. We shall assume that the potential is such that ~, has no other elgenvector with exgenvalue zero. In this case for r -+ _+oo, r ( r ) tends to ItS limit exponentially. In field theory, tins excludes the case o f a massless particle. Next one substitutes (19) and (22) into H = ~/~ + V. In the same way as for a sohton, one can in principle establish the complete perturbation scheme and Feynman rules. Tins seems, however, to be too complicated to be useful since, contrary to the cases worked out so far as for instance in III, we have a collective coordinate r or q which IS not associated to a symmetry of the system and it cannot be ehminated from the perturbation scheme [5] In the same way as in (I, II) we will rather work to lowest orders in the coupling constant g As usual, th~s senuclasslcal parameter is defined to be such that V(R) = ~
1
~gR),
(31)
where V does not depend on g otherwise. With this definition g2 plays the role o f h The corresponding WKB wave function is such that H~k = E ~ ,
E = E o + E1 ,
~b+- = e~S°(r)X_+,
E o = O(g-2),
So = O ( g - 2 ) ,
E 1 = O(g°),
X_+= O(g°) •
(32)
Since the system is symmetric under ~ , we can dlagonahze at the same time a complete set of observables budt out o f the mfimteslmal generators We shall only treat the case where these later operators vanish at the classical level. This means that the corresponding elgenvalue of quantum operators are only O(g °) at most and are thus not large In the semlclasslcal approximation. The classical equations (1) can be derived from the LagrangIan .12 : - l ( R r ) 2 - V ( R )
(33)
From the Noether theorem one deduces the classical expressions o f lnfimteslmal generators La, ca = - r r " r, a .
(34)
H J De Vega et al. /Instanton phenomena (11)
133
In the particular case considered, one wall introduce classical solutions such that L~, el = 0 .
(35)
In the same way as m ref. [1 ], eq. (32) leads to T
So(r ) = f
dr' rz(r') 2 ,
(37)
0
~+_ = X_+(det g ) - l / 4 ,
(38)
~X_+ = E~IX_+,
(39)
~g = - ~-~r - T I t "
_ _ - -
., 2 o~a2 + ~1 . wab rla 71b ,
(40)
k
Wab = V(a~) -
~
3(na, r " r , ~ ) ( n b , r ' r , v ) g ~v ,
(41)
6, "/=0
where we have expanded the potential V(r + 11) = V(°)(r) + V(al)r7 a + V(a~)rf~ b + . . ,
(42)
and where we have introduced the metric tensor g~#(r)=r,a.r,#,
a,/3 = 0,. , k ,
det g - d e t ( g ~ ) ,
gee = (g-1)o ~
(43)
According to (35), it is such that go~ = gao = 0
(44)
Tlus last equatxon is necessary to derive eqs. (37)-(40). As in I, the crucml point an solving (40) is that xf v (r) satisfies the small fluctuation equation
og = tl, jor,
02 V R =r(7) Utl (r) - aRc ~R /
(45)
a = k+ 1 ..... N-
(46)
the projection Ua(7") = D " na ,
1 ,
HJ. De Vega et al / Instanton phenomena (II)
134
is a solution of the equatmn
(D2)ab ub (r) - W~b ub (r) = O,
(47)
Dab = t~ab a r + I'ab
(48)
Eq. (47) actually holds only if v
r,~r-Vr
r,a=O,
a=O,
,k.
(49)
It is possible to satisfy this con&tlon by appropriate choice of boundary condltmns (see below) because r, a is also a solutmn of (45) and the left member of (49) is 7" independent. I'ab is the usual Chrlstoffel symbol Pab =nb " nat.
(50)
As m I, eq. (47) can be used to introduce creation-annihilation operators such that the most general wave function is easily written down. We shall only consider the case of the quantum ground state which, in tins approximation is the ground state of these creatlon-annlhdatlon operators. As in II, it is convenient to introduce the evolution operators K +-(r) whmh satisfy g~]~ rr ~ U a K b+ ( r ) ,
K - ( r ) r - - , - 2 e+~r,
(51) K+(r) ~
O(E) e - ~ r o ( z ) -1 .
(51)
Tins last hmlt is an obvaous consequence of the symmetry of V under Z. Because of the symmetry, r,a(z) is a solution of (45). For large r one has N-1
r, ot , i . ~ oo m= 0 C ( - ) m f m
et°rnr ,
0t=0 .... k,
N-1 r, Ot . r ~ ~ m =O
C (+)m flint'] e - t ° m z ,
(52)
where we have introduced a complete set of mgenvectors of
~.-)qOem)l = O)mOem)t,
m = 0 ..... N - - 1
(53)
From (30) it follows that f~o Is equal to r, ~o and that ¢O~ovamshes. Condition (49) wxll be satisfied if we introduce the 2 (N - k - 1) solutmns vm
H.J. De Vegaet al /lnstanton phenomena (11)
135
sattsfylng N--1 u 7. ( r ) , - _ =
C(m-)n fn e t°nr ,
,,=o
m=k+l,...,N-1,
N-1
o +re(r)r~+~ ~
t'(+)n ¢[Zl e-tOnr
(54)
n=O
which are hnearly independent from r,a. As in II, we adopt a bracket notation m vector space so that we can wnte
]r, a(r)) = K+_(r) [X~),
a = 0 ..... k,
[o~n(r)) = K_+(z) I×~ ),
m =k+ 1,...,N-
1,
N--1
I×z) = Z~ c(~-)"tf.>, n=O
N-1
ix+>= ~ c(.,+)"lftZ~>.
(55)
n=O
As in I, it Is convement to introduce foUowmg (46) u~"(r)
-
+ • - v?n(r) na(r) = (nalK+_lX,n).
(56)
Before wntlng down the wave function, we have to discuss its dependence on X. Because of (35) and (44) the equations (37)-(40) do not constraint the X dependence of ~Owhich is thus an arbitrary function of group parameters. The possible dependence of ff on X is to be classified as follows. Introduce maximal commuting subalgebras (Cartan subalgebras) {L~), (/~), ~ = 1, ..., p of ( L ~ and { / ~ respectively and the Caslmlr operators (Cm(L)), {em'(/~)). Because of (12) one has
era(L) =
era(L)
,
(57)
and we can dlagonahze at the same Ume {L~), ( / ~ ) and (em(L)). Denote the set of elgenvalues by (M), (~r} and (J). A complete set of function on ~ is obtained from the matrix elements c'9I~), {M-') of all representations of ~ It is obvious from (12) that It has the above exgenvalue properties. As It Is obvious from (22), the Hamfltoman commutes with La but not w i t h / ~ .
136
H J De Vega et a l / lnstanton phenomena (11)
Hence the energy depends In general upon the values o f {,~t} This is not the case In the particular approximation considered The determination of ~b IS now reduced to essentially the same computation as m I and one finds
{j}-+ e~So(z) e ±E°r l o-+. ~a~b {M), {M~ = (det g)'/-----4X/--~----~-e - ~ . a o q '~
£Z~b = ~
,
(58)
N-1 ~ (u ±-l)m(Du±)r~ , m=K+l
(59)
(U±- l)arn Ub+m =t~ab,
(60)
N-1 m=K+l This determines the WKB wave function for r = f ( q ) , 0 ~< q ~< 1. In other regions we can do the same computation again by letting
R = (rIXl) Iznl + ~
(nlXl)IZnl~Ta,
(61)
a
In such a way that the 2; transformation Is represented by q ~ q + 1 Because 2; IS a symmetry, one easdy sees that all terms In ( 5 8 ) - ( 6 0 ) will be the same for any n. Hence ff~q) for arbitrary q is simply the periodic extension o f (58).
3. Computation o f energy shift We shall proceed essentially as In II following the procedure explained m I and III. We shall only consider the quantum mechanical ground state where ~ IS In the trivial representation of q The more general case IS studied In the companion paper in order to discuss quark confinement As explained above, we consider the case where ~ / ~ contains an additive group Z of Integers with elements 2;n The classical vacua break the lnvarlance under this group which wall be restored by tunnehng In our parametrization • acts on the configuration space by q ~ q + n It is well-known that its irreducible umtary representations are the multIphcatIon by a phase factor e m° where 0 Is arbitrary. We conclude that we can define C0 such that
~o(q + n, ~l) = e m° ~o(q, ~)
(62)
F r o m ~ Invarlance, the dynamics o f q IS associated to a periodic potential as In a one-dimensional crystal. C0 IS the corresponding Bloch wave and 0 IS the so called
137
H J De Vega et al / Instanton phenomena (11)
pseudo-momentl2m conjugate to q. As we shall see, the statable ansatz satisfying (62) is C w ~ ( q , n ) = c e' (q}O [ t - ( q , q ) + e '° C+(q, q)] ,
(63)
where {q]- is the Integer part o f q and where c xs a constant to be adjusted below. In the same way as m II, (63) can be shown to hold if one is suffloently away from the minima of V, namely if for arbitrary X and n [R - r(n)[x]l > > 1 .
(64)
On the other hand, If for some X and n IR - r~n)lXl l < < O ( g - l ) ,
(65)
we can approximate V by its quadratic part and build the corresponding harmonic oscillator wave function around the minimum First we discuss the case n = O. The energy wall be found to be E = ] T' r ~ + e ,
(66)
where e is exponentially small Smce e ¢ 0 we are not at the true zero-point energy and the harmonic wave function is o f the form Co + e Cl where Co and Cl are respectwely exponentially decreasing and xncreaslng functions o f the &stance to the minimum In order to deal with them we Introduce a convement parametrlzatlon o f the region defined by (65) with n = 0 Since r o only breaks the subgroup ~ o this is achieved by letting N
R=rlo rl +
~
(67)
e IYl~b ,
b=ko+l
ro,a 0 e b = 0 , where Y c~o, ao = 1 . . . . k o are parameters Ttus change which was used m III for pretation as (7) with c ~ replaced by the by the achon o f ~o on ro It is of course
o f ~o. field theory has the same geometrical Intermanifold C/go of classical vacua obtained best to choose (see (53))
eb = f b
(68)
To lowest order m e, it Is easy to see that Co and Cl should satisfy MOo = 0 ,
MCl = Co,
(69)
138
H.J. De
Vega et al. / Instanton phenomena (11}
If we define
N 1 ~ [¢a)2(~b)2 M=--~-A+ ~ b=ko+l
(..Oh]
(70)
The Laplacmn m the curved coordinates (67) is handled as m III One has, to leading order m g2
N --A ~_g~O~OL~oL~o +
~
b=ko+X
-t
.
(71)
We shall only need ~bo exphcltly. It will be an elgenstate of/~ao with elgenvalue zero. One then finds
~O=I
73" det (to, t~o[ro,30)l- 1/4 exp( 1 ~ ¢a)a~a2) a=ko+l
a=ko+l
(72)
The normahzatlon of ~ko revolves the Jacobmn of the change of variable from R to Y and From (11) it follows that it is gwen by D(R)
O(YXO, ~a)
=
[detao,3o(ro,aolro,tSo)] 1/2 det3o,.ro(Cg~O).
(73)
One can verify that det(n~ °) dY is the lnvanant integration measure over ~o. Hence qJo is normalized to
f d R ~o(R) ~ =
fdr det(C~o° ) =clY o,
(74)
where c1~o is the lnvarlant volume of ~o. From ~ mvanance it follows that for other values o f n one has
~o")(R) = Co(R[ z - " I ) , where 4o is gwen by (72). Next we determine c by matching the WKB wave function to the harmomc wave function in an overlap region where both (64) and (65) hold. In the same way as m II one can check that ~0o matches with the corresponding piece of qJw~m m the region where
IR - r~oXJ I = O ( g - ~ 3 ,
i Z~<7 < §1,
(75)
up to correction terms at most of orderg 1 - ~ . In this region the classical ttme is large. In this way one determines c. The calculation is practically the same as in II.
139
H.£ De Vega et al. / lnstanton phenomena (11)
One finds
N
c=e_So/2U,
det (Xa[X3) VI c°-~-Ll1/4 [-det (ro,~o Iro.~o) a=ko+l
o~,3=0,. , N ,
ao, 3 o = l ,
(76)
,ko,
with +~
SO = f
dr(~) 2
(77)
We finally compute e following the same method as In II multiplying (69) by 4o we obtain
(78)
~oM41 = (40) 2 and integrate both sides reside a domain Bo of configuration space defined by
3X IR-rloXll
1<.~<1
,~
~,
(79)
where [;~] Is an arbitrary element of ~. Geometrically Bo is a portion of configuration space surrounding cr/~o. Since (40) 5 decreases exponentially away from crgo one has
f dR(Co) 2 -~fdR(~o) ~ = q~o
(80)
Bo Moreover using (69) and Gauss' theorem one finds
f dR 4oMffl = - ~I f Bp
da" (~OV~I) -~
(81)
aBp
As in II we multiply (80) and (81) by e and obtain 1
e-
2q30 f dq" [4o~(e41)] aBp
(82)
By construction aBp IS in the overlap region where 4o and e41 match with the corresponding WKB wave functions and there are only two portions of this bound-
HJ. De Vega et al. / lnstanton phenomena (11)
140
ary where the wave function Is non-zero These are characterized by introducing r~ which are the parameters of intersection of the classical path r(r) and r(r) (-0 with aBp Since this boundary is such that condition (75) holds one has + -T p --+ + o o
r ~+ =
f(q~o)
q ~+ -->
,
Iz~ol = - ( 1 - ~) l n g +
-+o
(83)
, i ~ < l
0(1),
The two regions are
R ~- {r(-O(r~) lxl ,
for some X E (~} ,
(84a)
for some X E ~} ,
(84b)
where -~ means that the two members differ by quantities of order gO (see formula (61)) In region (84a) Co and e ~kl match with ff_(q~-) and e t° ¢+(q~) respectwely and it is most appropriate to change variable m the integral (82) by using (61) with n = 0. Similarly, In region (84b) Co and e 41 match with ~+(q~) and e -~° ~b_(q~) The appropriate change of varxable Is given by eq (61) with n -- - 1 Up to exponentially small corrections ~Bo can be replaced by the two regions r = r~ with arbitrary X and rb As in II one finds e=-
2cl)o
(
dtl ~Y/-d~ g ( q ) ~ _ ~ _
L goo(q) 1, f ~q
_e_,Of d F ~
c)5 = f d X
det(Ct~ ~) ,
~b+ ~
f'-=
}
(85)
df @'
(86)
6,7=1 .... k. c)) is the lnvanant volume of the group ~. As in II one can check that the most Important contribution to (85) is obtained by taking the derivative of the e ±so(O terms and one finally gets N-1
--
cos 0 -
Cl)O
[det(u(q~)u(q~)]det~(q~)~2(qo)_l~
e-so
(87)
'
H.J De Vega et al / Instanton phenomena (II) a , / 3 = O ..... k ,
al,fll=ko+l,
141
,k
Finally the bracket can be recast under a form mamfestly independent of q~, namely N--! ~det(x~ 11X.~!) l-I e = 2 __q~ a=ko+l det( Chgmn/rr) ~o c°s0 1 where, as in II,
(G')a/TO1/2
Q4Ymn1S a Wronskaan associated
}
e-S°'
to (47)
4. Comparison with the imaginary time path integral in the dilute-gas approximation In the dilute-gas approximation the ground-state energy shifts are given by [6]
ep l. = 2 cos O r--,~ llm 21T N ( T1 ) i f
C/)R(r)exp[_ / r dT.(SRr t 2 + V(R))], (88) (one mstanton) -T
where 0 ~< 0 < 27r and N(T) is the same functional integral in the zero-mstanton sector Thas contrlbuhon can be calculated by expanding around the classical solution
R(7.) = r [X~] (7. - 7"0) + s [x~l (7. - 7.o),
(89)
where 7.0 and the X a are to be treated as collective coordinates m order to remove the zero modes from the Euchdean quadratic form. As usual, it is enough to take Xa, 7.0 to be single Integration variables and not functions of 7. In this case s satlsties the c o n d m o n T
f
d7. r,a(r - To) s ( z - to) = 0 ,
a=0 .... k
-T At the one-loop level one has k C-DR(r) = "vQ-(T) det ( c ~ ) d7.o 1-I dXC~C-l)s, 1
det [f d7. r,a(r) • r,#(r)] 0
(90)
H.J De Vegaet al / Instanton phenomena (11)
142
Similarly N(T) 1s computed by expandmg around ro, taking into account the zero modes due to the spontaneous breaking of ~o We thus Introduce the corresponding collective coordinates and write
R(r) = rior~°l + ~(r) [ r~ol
(91)
As above we choose the yao independent of r. Then +T
f
drro,~o o ( r ) = 0 ,
ao=l,
,ko,
-T ko
C/)R =X/Jo(T)
det
( 9 ~ o°) 1-I dY ao Q)o,
1 (fl0, 70 ~
Jo(T) =
det
(92)
°tO= 1
[ro,a 0 ro,3o] (27) kO .
(93)
1 ~c~0,~30 ~
For large r, r, ao(r ) tends to ro,a o and it follows Immediately from eq (90) and (93) that J(T)_ ~=
1Lln~Jo~
det Otl,/31=O, k o + l . . . . k
[f
_~
drr, al(z).r,~l(r)]
(94)
and m the one-loop approxlmanon formula (88) becomes +~
ffcDsexp[-~ f C~ ~
dr(slMIs)]
_oo
ep.i. = 2 cos 0X/~ e - s ° C))o
+~o
ffcboexp[-~ f
,
(95)
dr(olMoJo)]
where a2
M = - ~r 2 + U(r)'
a2 Mo = --~-r2 + G2
(96)
As in II, we introduce a regulating parameter z to handle the zero-modes and we get +co
ff ~sexp[-1 f
-~
d r ( s l M - zls)]
+~
ffc~oexp[ -1 f _oo
| / ( - z ) k l +1 = V c/)(z)
dr(°lMo - z l ° ) l
'
(97)
H J De Vega et al / Instanton phenomena (II)
143
where CO(z) - det [(114o - z ) - l ( M - z)] is the Fredholm determinant of the operator M The factor ( - z ) gl+l comes from the (kl + 1) zero exgenvalues of M w h l c h are not present m the spectrum of Mo Now as in If, we consider the lost solutions of the one-dimensional scattering problem in the 7--axis with U(T) as a potential They are N × N matrix solutions of
~r 2 + U(r) F+_(z, r) = zF+_(z, 7-)
(98)
with boundary con&tlons
F_(r,z)
=
Here k(z) = - ~ z F _ ( r , z) F+(7-,z)
e-'~r,
F+(7-,z)
=
O(E) e +'~rO(x;) T
(99)
~2 For 7- -+ +~ we have
= T___~+~
:
.., O(E) [e'kr A - ( z ) + e-,k~rF -[Z)l
O(z)T
e-'krA+(z)+e'krF+(z),
(100)
T---> _ _ ~
where F+_(z) are the lost matrices by definmon In particular for z = 0 K_+(7-) = F+ (7-, O)
(101)
As In II it can be shown that the following properties hold
F+(z) = lc-1 (z) F~(z) T k(z) , A+_(z) = _ ~ - 1 (z) A~_(z) af k(z) ,
F+(z)* k F+(z) - A + ( z ) ? kA+(z) = Ic
(102)
In the case we consider, U(7-) tends exponentially to its limit for large 7-, since r(7-) has this property Then one can show that the Fredholm determinant cO (z) equals the determinant of the lost matrix for arbitrary z fi9 (z) = det F+ (z) = det F _ (z).
(103)
Thus eq. (95) can be written as c~ ep.i. = 2 e - S ° x / ~ - c o s 0 op ° 1 / ,
1 ( - 1 ) ~l+l d k1+1 CO(z) (kl + 1) v
dzkl+l
(104)
H J. De Vegaet al / Instanton phenomena (11)
144
In what follows we shall compute the hmlt in eq (104) and we shall show that the value so obtained for e coincides with the result derived m sect 3 by matching o f wave functions. The method is exactly the same as in II, apart from the new features introduced by the symmetry. The Jost matrices act m the hnear space of the elgenvector of ~ . This space can be decomposed as the &rect sum o f two orthogonal subspaces / and 1_ The first will be the space of elgenvectors of ~ with non-zero elgenvector and the second the kernel of ~ . This decomposition exphcltly reads
e,kr
= ( e'k±rF+l±(z)
e*k±rF+±t--(z) ] (105)
\ e,4Z*F+L±(z)
el
*F+W(z)" '
where F ±± is a hnear mapping from _[ t o / , F t-± from .I. to I_ etc. As m II, it will be convement to use, in the s u b s p a c e / , the basis formed by the vectors I~o,) = IX~I),
I~p,+kl) = c~±IX~I),
oq = 0 , k o + 1 . . . . k ,
t = l ..... k l + l
(106) ,
and other ( N - 2 - 2 k l ) vectors which are hnearly independent from them. Then
/ F±±~)
Flt--(z)
c-/)~)= [ d e t ( h , l ) ] - I det [
(107)
FUll)
Ft-J-(z) ) '
where, as usual h u = (~o,1~0]).
(108)
At z = 0, q)(z) has a zero of order k 1 + 1, as follows from eqs. (A.1). Precisely, we choose the basis (106) because m this basis the matrix F+_(O)qhas the columns ! = 1, ..., k l + 1 and the rows t = kl + 2, ..., 2kl + 2 adentlcally zero. Many other elements vamsh In this limit as we see from (A.2). Then xt can be seen by inspection of the full F±(z) matrix that
dzkl+l [z=o O~l,~i~kl ×(kl+l))
det' F+l±(0) d e t h ±± '
(109)
HZ De Vega et aL I lnstanton phenomena (11)
145
where det' F}±(0) is the determinant of order ( N - kl - ko - 1) of the matrix obtained from FI±(0) by deleting the first (kl + 1) columns and the rows kl + 1 ..... 2ki + 2. In derwlng eq. (109) we have used the equations det F+L-L(0) = 1 ,
A~t-'(O) = O,
F~ZL(o) : O,
(1 10)
which are proved m the appendix. The first determinant m eq. (109) can be computed m the same way as in II, eqs. (5), (17), (18), obtaining (see eq. (94))
/
dF~±(O)l ,\_
,~
Thus the imaginary time path integral gives for the width of the first allowed energy band e -so
ep.i. = 2 rr(kl+l)/2
| / det h±± cp 1/det,F~±(0) q~o cos 0 .
(112)
We shall now prove that the expression obtained for e m sect 3 can be recast m this form. From eqs. (56) and (87) we have ow=:
e-So
c~ , . /
det g~l#l(r) det(~±/n)
L-
'
(113)
J
~q,/31 = k o + 1, ..., k. Then by exphcltly writing det [K+(r) K + ( - r ) T] m the basis (106) for large posltwe r it follows using eq. (A 2), (A 8) that det [K+(r) K+T(--r)] - d e t ' F ~ ( 0 ) det (~oaI (r) l ~ holh ( r ) ) , deth± 0~a~,~l~k o
(114)
[tPC~l(7.)) -~ K+ (7") ]XctI ) Moreover one can prove
det nal%)
det gc~lflI (7") det ( ~ 1(T) ] ~ Z 1 [~Ofll(T)) '
(115)
146
H.£ De Vega et al / lnstanton phenomena (11)
hm
[det ga I th (7-)] 2 --
1
.
r - ~ deta l,~l <~o~1(7-)1w± 1¢~1 (T) > de t (tPal (7-) [wi- 1 [~Ofl1 (7")) These two relations are derived in the same way as similar relations used In II Eq (115) holds because ~± has no zero elgenvalues.
Appendix The zero modes can be expressed m terms of K_+(r) and the constant vectors [~>(see eq. (55)). Then, by using eq (100), F_+(0)[X~I> = 0 ,
t~l = 0, k o + 1,
,k,
F± (0) i t - = 0
(A1) (A.2)
The first equation shows that q)(z) has a zero or order kl + 1 at z = 0. If one considers the decomposmon between ± and t_ subspaces (eq (105) together with relations (102) and (A.2)), one finds A~t-(0) = 0 ,
(A.3)
F_+t-t-(0)tF_+ (0)t--t-(0) - A ~ t - t ( 0 ) A~t-(0) = It_
(A.4)
Finally we consider the Jost soluUon for small but non-zero z In this case one has for non-asymptotic r F_+LL(z, 7") = 142(7-)+/X/z-Z(7-) ,
(m 5)
where 7- ~ z - 1 / 2 > > 1 .
Here W(r) and Z(7-) are soluUons of eq. (98) for z = 0, with the following asymptotic behavior W(~') = I , T~--~
Z(7-) = 7-1
(A.6)
T--~--~
By choosing the vectors ro,~o (as in (106)) as a basis ofl_ one can take simply W(r)ao,t~o = [r(r), 3O]ao
(A.7)
Then by using Wronsklan arguments for W(r) and Z ( r ) and eq. (100) it can be
H J. De Vega et aL / Instanton phenomena (11)
147
F_+t-t-(0) + A_+U-(0) = F_+t-t-(0) - A_+L-t-(0)= O A ( E ) - 1 0 ( 2 ; ) ,
(A.8)
shown that
where OA(2;) is the 2; t r a n s f o r m a t i o n In the adjolnt representation. We have also used that ro, cz0 = lItxor 0 ,
0(2;) - 1 Ice0 0 ( • )
= OA(2;)a030 I3o
Then, eq (110) follows from (A 8).
References [1 ] J L. Gervals and B. Saklta, Plays Rev. D16 (1977) 3507. [2] H de Vega, J.L. Gervals and B. SakRa, Nucl. Phys B139 (1978) 20. [3] J L Gervals and B Saklta, Gauge degrees of freedom, external charges, quark confinement cntenon m A 0 = 0 canomcal formahsm, CRy College prepnnt 78/5, Phys Rev. D, to appear. [4] H.J de Vega, J L. Gervals and B. Saklta, Wave functions and energy for vacuum and heavy quarks from WKB Schroedmger equation for massive gauge theories wath mstantons, PAR-LPTHE 78•09, Phys Rev. D, to appear. [5] J L Gervals, Schladmmg lectures 1977, Acta Phys. Austrlaca, Suppl XVIII (1977) 385. [6] S. Coleman, Uses of lnstanton, Ence Lectures 1977, Harvard preprmt (1978).
REAL TIME APPROACH TO INSTANTON PHENOMENA (II). M u l t i d i m e n s i o n a l p o t e n t i a l w i t h c o n t i n u o u s s y m m e t r y H J. DE V E G A
Laboratotre de Phystque Thdonque et Hautes Energzes *, Untverstt~ l~erre et Marle Curie, Parts, France J.L. G E R V A I S
Laboratotre de Phystque Th~ortque de l'Ecole Normale Sup~neure **, Pans, France B. S A K I T A
The Oty College of the Oty Unwerstty of New York ~, New York, USA Received 10 July 1978
We extend further prewous works on the tunnehng problem of degenerate classical ground states for systems with many degrees of freedom which are based on WKB solutions of the Schroedmger equahon These solutions are exact to the first two orders m #I. The point of the present paper is to deal with the case of potenhals with continuous symmetry assunung that the spectrum of small oscillations around classical vacua has no zero modes besides the ones due to spontaneous symmetry breakmg This continuous symmetry introduces non-trivial difficulties m matching between WKB and harmomcosctUator wave functions, because the mstantons and the classical vacuum leave mvanant different subgroups of the symmetry group, m general Ttus problem is solved here. We deternune the ground-state wave function as well as a general formula for the groundstate energy which is shown to agree with the path-integral computation m the ddutegas approximation The results presented here have a direct apphcahon to field theory which we discuss m a compamon paper.
1. I n t r o d u c t i o n Ttus p a p e r is the t h i r d o f a s e n e s o n t h e WKB a p p r o x i m a t i o n t o t h e S c h r o d l n g e r e q u a t i o n f o r s y s t e m s w i t h m a n y degrees o f f r e e d o m a n d field t h e o r y . In t h e first
* Laboratolre Assoc16 au CNRS, 4, place Jussleu, Tour 16, ler &age, 75230 Pans, Cedex 05 ** Laboratolre propre du CNRS assoct~ ~ l'Ecole Normale Sup~neure et ~tl'Umversltd de PansSud $ Supported m part by National Science Foundation under Grant no. PHY77-03354 and m part by the Research Foundation of the City Umverslty of New York (New York, NY 10031) under Grant no. 11681. 125
126
H.J. De Vega et al. / lnstanton phenomena (11)
paper [1 ], denoted by I, the general WKB ansatz was estabhshed for wave functions associated wath a given classical trajectory In configuration space. In a second paper [2], denoted by II, we computed tunneling effects on the ground state for multidimensional potentials The essential features of the method are as follows. We study stationary states thereby ehmmating the true time of the problem from the start. The WKB eIgenstates of the Hamdtonlan are described by wave functions concentrated In a small neighborhood of the curves which represent the relevant classical solutions in configuration space. They are most easdy expressed in terms o f the local coordinates o f the corresponding region which naturally involves the position q on the classical curves. This parameter q can be piecewise related to the time r o f classical solutions which is thus only used as a geometrical parameter o f configuration space Going to local coordinates along the curves is equivalent to the introduction o f q as a collective coordinate. In serm-classlcal hmlts, its dynamics is found to be mostly onedimensional WKB, while the other degrees of freedom are essentially harmonic '* For tunnehng we use the one lnstanton solution, in order to budd the WKB wave function in the forbidden region. The ground-state wave function IS then constructed explicitly by matching this WKB expression with the harmonic-oscillator wave function around each minimum which would be the ground state in the absence of tunneling In paper II we rederlved In this way the expression for the ground-state energy shift already obtained In the dilute-gas approximation by the imaginary time path integral method. We have thus proved that this latter result is the correct semi-classical approximation This proof, however, was restricted to the case where there is no continuous symmetry and where the spectrum of small oscillations around the absolute minima of the potential has an energy gap In the present paper we remove the first restriction and extend the discussion of paper II to the case of continuous symmetry It IS proved that the dilute-gas approximation is still correct if there is a continuous symmetry, provided the spectrum o f small oscillations around ground states has no zero modes besides those due to spontaneous symmetry breakdown. In a recent paper the gauge problem of field theory was discussed in A0 = 0 canonical formalism, in such a way that time-independent gauge transformations are treated on the same footing as the other symmetries of the system [3] (paper III) This approach is best suited to our purpose. Hence, the present extension o f II Is crucial for field theory applications which we discuss in a companion paper [4]. Since we assume that the theory has an energy gap we still exclude field theories with massless pamcles. As discussed in I, the dilute-gas approximation may break down in this very important case This more involved problem is currently under investigation In sect 2 we establish the general form of the WKB wave function with contlnu* It can be pointed out that our semi-classical wave functmns are exact to the first two orders m h (h -1 and h0), contrary to other discussions m the recent hterature
H J De Vega et al. / lnstanton phenomena (llJ
127
ous symmetry, for the case of interest in field theory, namely when the potential is periodic. In sect 3 we construct the ground-state wave function and compute the ground-state energy shift due to tunneling. The physical picture is as in the first Brillouln zone of a crystal. In sect. 4 we finally show that we obtain the same result for the energy shift as the imaginary time path integral approach In the dilute-gas approximation The p r o o f IS based on the equality between Jost and Fredholm determinants for the scattenng problem associated with small fluctuations. Since we have an energy gap the corresponding potential tends exponentially to ItS limit at infinity, in which case the above equahty certainly holds The situation is not so clear if there IS no energy gap This case IS currently under investigation. The present article is not self-contained. A good knowledge o f papers I, II and III will be assumed in order to keep the length of the discussion within reasonable llrmts 2. WKB wave function
Throughout our chscusslon, we adopt the same notation as in I and II. R represents a p o m t of configuration space with an arbitrary number, N, of components, V(R) is the potential energy In the following we shall only need the expression for the WKB wave function in classically forbidden regions. We shall only consider this case exphcitly and introduce a classical trajectory r ( r ) which satisfies rrr = V V , - ] ,1(r)2 r + V(r)=Eo.
(1)
The r indices mean r derivatives, and obviously Eo < V(r) We further assume that the system IS mvarlant under a connected group ~ of continuous transformations with parameters X u, t~ = 1, ..., k and denote b y R [xl the corresponding transform o f R. It w111be assumed to take the form
(RtX]), = O(X),j(R)I
(2)
Since it has an lnhomogeneous term, a gauge transformation an field theory does not satisfy this relation. As discussed In III and In the companion paper [4], ttus does not make much difference, so in the present discussion we make use o f the above form. If V(R) IS mvarIant under ~ , that is If
V(R) = V(RIXI),
(3)
there exists a continuous set of classical solutions r~rX)l where X are arbitrary constant parameters of ~ * * In order to slmphfy the discussion we assume that ~ Is completely broken by r0") Our discussion can be easily generahzed if needed
H.J De Vega et al / lnstanton phenomena (11)
128
In I we built a WKB solution of the Schrodmger equation 1 H = ~p2 + V(R)
H~b = Eta,
(4)
by perforrmng the canomcal transformation R = r('r) + ~rta(7" ) '0a ,
(5)
rr na = 0 .
(6)
a
The geometrical meaning of this transformation IS that for a given R we define r such that the correspondmg point ~s on the normal to the classical curve at r(r). The r/a's are the components o f R - r ( r ) on a moving frame orthogonal to the curve at r ( r ) This change of variable is only possible m a small neighborhood o f t ( r ) , such that this normal is umque As discussed In I, this procedure IS consistent if the wave function decreases exponennally away from r ( r ) , that as If the phenomenon considered occurs mostly m configuration space In a small tube around r(r). It is clear that this is not true ff there IS a continuous symmetry since all classical curves r l X l ( r ) are on the same footing In this case we modify (5) In order to obtain a good parametrlzatlon o f the neighborhood of the points r~r~l for arbitrary X. This IS a manifold c-ggwith more than one dimension. The generahzanon of (5) IS N-1
R=rlX](r)+ ~
na(X,r)rl a,
(7)
a=k+l
ha(X, r) r[rXl(r) = 0
n a nb=6ab ,
,
ha(X, z) 0rlXl(7") 0X ~
=
0,
(8)
a= 1,..,k
The geometrical meaning is the same as before except that the classical curve r ( r ) has been replaced by the higher-dimensional manifold c ~ . The change o f variable (7) makes sense in a neighborhood o f c ~ and we shall have a consistent approxImation scheme if the wave function is exponentmlly decreasing away from It, as one can check explicitly in the final result In the same way as (5), (7) as basically a simple change o f variable. It can be handled simply, however, only by applying standard h e algebra techniques to the group ~. These were already briefly sketched In III It IS convenient to Introduce the multlpllcanon law f2(X, Y) of the group ~. It is such that for instance (RIXI ])(x21 = R (~(x2, X])l .
(9)
129
H..J De Vega et al / Instanton phenomena (II)
The following derlvatwes of ~ are useful
as2~(X'Y)I c/It- 1 ~(X) =
~-1 ~,(y) _
0 Yt~
(10)
Y=O'
O~(X' Y)[ aXO x=o
(11)
The role of ~ and c~ IS to allow the computatxon of derlvatwes of the group element with respect to group parameters Indeed if I s are the infinitesimal generators in any given representation one has
a~(d( e' I ¢xt3) = l l.r ct~ (X) e' IOx# , D~,~(e' S#x#) = e'10 x# tI./ cl?~( X') .
(12)
It follows that if we define Zo¢ = ¢'ff~- l ( x ) ~
l
,
(13)
we have [L., Lt~] : , ~ L . r ,
[L'~a,L~] =, ~ / ~ ' r ,
(14)
IL~, ~1 -- O, where the ~
are the structure constants of the group gwen by
[I~, 1A = ~ I ~ .
(15)
The appearance of these two commuting Lie algebra is well-known in particular m the mechanics of the rigid rotator. One (La) Is the generator of rotation m Inertial frame, the other (/~,~) is the generator of rotation in the body-fixed frame. In all practical apphcatlons we have in mind, it is always possible to define the scalar product which appears in (8) in such a way that for arbitrary points R 1, R2 R[ xj • R tx] = RI . R2
(16)
We can realize (8), in this way, by choosing
na(X, 7") = n[aXI ( r ) -
(17)
H.J. De Vega et al. / Instanton phenomena (11)
130
By defimtlon, ff ~,a are the parameters o f an lnfimteslmal element of (~, one has R [xl ~- R - t)~'~ [R, L,~]
(18)
Looking at (7) and (8) one sees that r and 7?a are mvanant under the group. They are parameters of the orbits o f ~ which are in c ~ or m a neighborhood o f it. Because of the lnhomogeneous term, this argument must be shghtly modified for gauge groups The final result is essentially the same however (see Ill and the compamon paper [4]). WLth the choice (17), we can rewrite (7) as
R = r [Xl 0") +111Xl (7") '
(19a)
N-1
11[ X l =
n[Xl(r)rf.
~
(19b)
a=k+l
It follows from (8) and (12) that T11s such that q'rr=O,
q r,a = 0 ,
(20)
OrIXI r,~- ~
(21)
x=o
In the same way as m I and III, a straightforward computation gives k
P=-tV=
~ r!Xl(I-1)o~(Lo-q,o.;)+ a,#=o
;[Xl ,
N-I
;[Xl=
~)
n[aXl(r)-t
.
(22)
a=k+l
We have adopted a compact notation whxch includes all collectwe coordinates on the same footing. When the summation runs from 0, the mdex zero refers to r. Indlces a :/: 0 with a comma denote denvatwes at vamshmg group parameters. Lo is simply - t a / a r and r, o is r r The m a t n x la~ is gwen by I~(r)
= (r,a + r l , a ) • r , ~ ,
a,/3 = 0 ..... k .
Smce we want to deal with mstantons we shall consider a solution r ( r ) of (1) with E o = 0 For r -->+_ooit tends to different minima o f the potential
(23)
H..J De Vega et al. / Instanton phenomena (II)
131
As in II, we assume that the existence o f several classical ground states is connected with a dxscrete symmetry of the system. In II we considered only two minima Here we shall rather deal with a typical situation o f field theory as discussed m III Specifically we assume that ~ lS the connected component o f a larger group o f mvarlance ~ under which the system IS also lnvanant. The d~screte symmetries are the elements o f ~ / ~ We shall denote b y r o the minimum o f V(R) such that hm
r(r) = ro
(24)
For r -> +0% r ( r ) tends to another m l m m u m which is of the form rio~1 where Y, is an element o f ~ not In ~ , 1.e. a discrete symmetry. Clearly we have to take into account all paths rl zn] (r) for arbitrary integer n. They connect all classical vacua of the form rioznl. Additional discrete symmetries can be ignored If, as we assume, there is no mstanton which connects the different members of the correspondmg fanuly o f classical vacua In II we dealt with the case where y2 = 1. Here we rather assume that ~n is different from 1 for all n except zero (lntermedmte cases are of course possible). Hence we have an mfimte set of classical vacua rio~nl connected b y paths r ( r ) I ~nl denoted by r(on) and r ( r ) (n) respectwely where ever needed. By construction r(r)(n) r ~ - 2
r(o' 0 '
r(r)(n) r--S-~+Zr(on+l) .
(25)
As noted In I the time z is not a good parametrlzatlon o f the whole set o f classical paths since it goes an infinite number of times from - - ~ to +o~ as we move along it. One can instead introduce a penodlc function f with period unity such that for o + 0 + and n integer
f(n - o) -+ _oo,
f(n + o) -->+oo,
(26)
and the complete set o f paths m configuration space is parametrlzed b y Rcl(q ) =r(n)(f(q)),
n ~
(27)
By construction it is such that
Rd(q + 1) = R ~ 1(q),
(28)
and it IS thus penodlc up to a E transformation. In the same way as we did in I, ( 1 9 ) - ( 2 3 ) must be extended by using q instead of ~'. We do not do it explicitly since thJs is simply a complication o f notation and since In each region n ~< q ~< n + 1, we can use r instead o f q . Since these regions are related by Z it is enough to deal for the moment with the region n = 0 which was the starting point o f the present discussion. Near ro we have V ( R ) "" ~(R - ro),(R - ro) 1 ~ .
(29)
132
H J. De Vega et a l / Instanton phenomena (II)
The new feature, as compared to II 1S that l f r o IS not mvarlant by ~ , c~ has zero elgenvalues. To slmphfy we only consider the case we shall encounter later on. We assume that the ground state r o breaks a subgroup ~o of the continuous symmetry group with ko parameters. Because of the symmetry we thus have ~ro,ao = 0 ,
ao = I . . . . ko •
(30)
From now on, wherever needed, we label with an index ao the denvatives with respect to parameters of ~0 at the Identity. We shall assume that the potential is such that ~, has no other elgenvector with exgenvalue zero. In this case for r -+ _+oo, r ( r ) tends to ItS limit exponentially. In field theory, tins excludes the case o f a massless particle. Next one substitutes (19) and (22) into H = ~/~ + V. In the same way as for a sohton, one can in principle establish the complete perturbation scheme and Feynman rules. Tins seems, however, to be too complicated to be useful since, contrary to the cases worked out so far as for instance in III, we have a collective coordinate r or q which IS not associated to a symmetry of the system and it cannot be ehminated from the perturbation scheme [5] In the same way as in (I, II) we will rather work to lowest orders in the coupling constant g As usual, th~s senuclasslcal parameter is defined to be such that V(R) = ~
1
~gR),
(31)
where V does not depend on g otherwise. With this definition g2 plays the role o f h The corresponding WKB wave function is such that H~k = E ~ ,
E = E o + E1 ,
~b+- = e~S°(r)X_+,
E o = O(g-2),
So = O ( g - 2 ) ,
E 1 = O(g°),
X_+= O(g°) •
(32)
Since the system is symmetric under ~ , we can dlagonahze at the same time a complete set of observables budt out o f the mfimteslmal generators We shall only treat the case where these later operators vanish at the classical level. This means that the corresponding elgenvalue of quantum operators are only O(g °) at most and are thus not large In the semlclasslcal approximation. The classical equations (1) can be derived from the LagrangIan .12 : - l ( R r ) 2 - V ( R )
(33)
From the Noether theorem one deduces the classical expressions o f lnfimteslmal generators La, ca = - r r " r, a .
(34)
H J De Vega et al. /Instanton phenomena (11)
133
In the particular case considered, one wall introduce classical solutions such that L~, el = 0 .
(35)
In the same way as m ref. [1 ], eq. (32) leads to T
So(r ) = f
dr' rz(r') 2 ,
(37)
0
~+_ = X_+(det g ) - l / 4 ,
(38)
~X_+ = E~IX_+,
(39)
~g = - ~-~r - T I t "
_ _ - -
., 2 o~a2 + ~1 . wab rla 71b ,
(40)
k
Wab = V(a~) -
~
3(na, r " r , ~ ) ( n b , r ' r , v ) g ~v ,
(41)
6, "/=0
where we have expanded the potential V(r + 11) = V(°)(r) + V(al)r7 a + V(a~)rf~ b + . . ,
(42)
and where we have introduced the metric tensor g~#(r)=r,a.r,#,
a,/3 = 0,. , k ,
det g - d e t ( g ~ ) ,
gee = (g-1)o ~
(43)
According to (35), it is such that go~ = gao = 0
(44)
Tlus last equatxon is necessary to derive eqs. (37)-(40). As in I, the crucml point an solving (40) is that xf v (r) satisfies the small fluctuation equation
og = tl, jor,
02 V R =r(7) Utl (r) - aRc ~R /
(45)
a = k+ 1 ..... N-
(46)
the projection Ua(7") = D " na ,
1 ,
HJ. De Vega et al / Instanton phenomena (II)
134
is a solution of the equatmn
(D2)ab ub (r) - W~b ub (r) = O,
(47)
Dab = t~ab a r + I'ab
(48)
Eq. (47) actually holds only if v
r,~r-Vr
r,a=O,
a=O,
,k.
(49)
It is possible to satisfy this con&tlon by appropriate choice of boundary condltmns (see below) because r, a is also a solutmn of (45) and the left member of (49) is 7" independent. I'ab is the usual Chrlstoffel symbol Pab =nb " nat.
(50)
As m I, eq. (47) can be used to introduce creation-annihilation operators such that the most general wave function is easily written down. We shall only consider the case of the quantum ground state which, in tins approximation is the ground state of these creatlon-annlhdatlon operators. As in II, it is convenient to introduce the evolution operators K +-(r) whmh satisfy g~]~ rr ~ U a K b+ ( r ) ,
K - ( r ) r - - , - 2 e+~r,
(51) K+(r) ~
O(E) e - ~ r o ( z ) -1 .
(51)
Tins last hmlt is an obvaous consequence of the symmetry of V under Z. Because of the symmetry, r,a(z) is a solution of (45). For large r one has N-1
r, ot , i . ~ oo m= 0 C ( - ) m f m
et°rnr ,
0t=0 .... k,
N-1 r, Ot . r ~ ~ m =O
C (+)m flint'] e - t ° m z ,
(52)
where we have introduced a complete set of mgenvectors of
~.-)qOem)l = O)mOem)t,
m = 0 ..... N - - 1
(53)
From (30) it follows that f~o Is equal to r, ~o and that ¢O~ovamshes. Condition (49) wxll be satisfied if we introduce the 2 (N - k - 1) solutmns vm
H.J. De Vegaet al /lnstanton phenomena (11)
135
sattsfylng N--1 u 7. ( r ) , - _ =
C(m-)n fn e t°nr ,
,,=o
m=k+l,...,N-1,
N-1
o +re(r)r~+~ ~
t'(+)n ¢[Zl e-tOnr
(54)
n=O
which are hnearly independent from r,a. As in II, we adopt a bracket notation m vector space so that we can wnte
]r, a(r)) = K+_(r) [X~),
a = 0 ..... k,
[o~n(r)) = K_+(z) I×~ ),
m =k+ 1,...,N-
1,
N--1
I×z) = Z~ c(~-)"tf.>, n=O
N-1
ix+>= ~ c(.,+)"lftZ~>.
(55)
n=O
As in I, it Is convement to introduce foUowmg (46) u~"(r)
-
+ • - v?n(r) na(r) = (nalK+_lX,n).
(56)
Before wntlng down the wave function, we have to discuss its dependence on X. Because of (35) and (44) the equations (37)-(40) do not constraint the X dependence of ~Owhich is thus an arbitrary function of group parameters. The possible dependence of ff on X is to be classified as follows. Introduce maximal commuting subalgebras (Cartan subalgebras) {L~), (/~), ~ = 1, ..., p of ( L ~ and { / ~ respectively and the Caslmlr operators (Cm(L)), {em'(/~)). Because of (12) one has
era(L) =
era(L)
,
(57)
and we can dlagonahze at the same Ume {L~), ( / ~ ) and (em(L)). Denote the set of elgenvalues by (M), (~r} and (J). A complete set of function on ~ is obtained from the matrix elements c'9I~), {M-') of all representations of ~ It is obvious from (12) that It has the above exgenvalue properties. As It Is obvious from (22), the Hamfltoman commutes with La but not w i t h / ~ .
136
H J De Vega et a l / lnstanton phenomena (11)
Hence the energy depends In general upon the values o f {,~t} This is not the case In the particular approximation considered The determination of ~b IS now reduced to essentially the same computation as m I and one finds
{j}-+ e~So(z) e ±E°r l o-+. ~a~b {M), {M~ = (det g)'/-----4X/--~----~-e - ~ . a o q '~
£Z~b = ~
,
(58)
N-1 ~ (u ±-l)m(Du±)r~ , m=K+l
(59)
(U±- l)arn Ub+m =t~ab,
(60)
N-1 m=K+l This determines the WKB wave function for r = f ( q ) , 0 ~< q ~< 1. In other regions we can do the same computation again by letting
R = (rIXl) Iznl + ~
(nlXl)IZnl~Ta,
(61)
a
In such a way that the 2; transformation Is represented by q ~ q + 1 Because 2; IS a symmetry, one easdy sees that all terms In ( 5 8 ) - ( 6 0 ) will be the same for any n. Hence ff~q) for arbitrary q is simply the periodic extension o f (58).
3. Computation o f energy shift We shall proceed essentially as In II following the procedure explained m I and III. We shall only consider the quantum mechanical ground state where ~ IS In the trivial representation of q The more general case IS studied In the companion paper in order to discuss quark confinement As explained above, we consider the case where ~ / ~ contains an additive group Z of Integers with elements 2;n The classical vacua break the lnvarlance under this group which wall be restored by tunnehng In our parametrization • acts on the configuration space by q ~ q + n It is well-known that its irreducible umtary representations are the multIphcatIon by a phase factor e m° where 0 Is arbitrary. We conclude that we can define C0 such that
~o(q + n, ~l) = e m° ~o(q, ~)
(62)
F r o m ~ Invarlance, the dynamics o f q IS associated to a periodic potential as In a one-dimensional crystal. C0 IS the corresponding Bloch wave and 0 IS the so called
137
H J De Vega et al / Instanton phenomena (11)
pseudo-momentl2m conjugate to q. As we shall see, the statable ansatz satisfying (62) is C w ~ ( q , n ) = c e' (q}O [ t - ( q , q ) + e '° C+(q, q)] ,
(63)
where {q]- is the Integer part o f q and where c xs a constant to be adjusted below. In the same way as m II, (63) can be shown to hold if one is suffloently away from the minima of V, namely if for arbitrary X and n [R - r(n)[x]l > > 1 .
(64)
On the other hand, If for some X and n IR - r~n)lXl l < < O ( g - l ) ,
(65)
we can approximate V by its quadratic part and build the corresponding harmonic oscillator wave function around the minimum First we discuss the case n = O. The energy wall be found to be E = ] T' r ~ + e ,
(66)
where e is exponentially small Smce e ¢ 0 we are not at the true zero-point energy and the harmonic wave function is o f the form Co + e Cl where Co and Cl are respectwely exponentially decreasing and xncreaslng functions o f the &stance to the minimum In order to deal with them we Introduce a convement parametrlzatlon o f the region defined by (65) with n = 0 Since r o only breaks the subgroup ~ o this is achieved by letting N
R=rlo rl +
~
(67)
e IYl~b ,
b=ko+l
ro,a 0 e b = 0 , where Y c~o, ao = 1 . . . . k o are parameters Ttus change which was used m III for pretation as (7) with c ~ replaced by the by the achon o f ~o on ro It is of course
o f ~o. field theory has the same geometrical Intermanifold C/go of classical vacua obtained best to choose (see (53))
eb = f b
(68)
To lowest order m e, it Is easy to see that Co and Cl should satisfy MOo = 0 ,
MCl = Co,
(69)
138
H.J. De
Vega et al. / Instanton phenomena (11}
If we define
N 1 ~ [¢a)2(~b)2 M=--~-A+ ~ b=ko+l
(..Oh]
(70)
The Laplacmn m the curved coordinates (67) is handled as m III One has, to leading order m g2
N --A ~_g~O~OL~oL~o +
~
b=ko+X
-t
.
(71)
We shall only need ~bo exphcltly. It will be an elgenstate of/~ao with elgenvalue zero. One then finds
~O=I
73" det (to, t~o[ro,30)l- 1/4 exp( 1 ~ ¢a)a~a2) a=ko+l
a=ko+l
(72)
The normahzatlon of ~ko revolves the Jacobmn of the change of variable from R to Y and From (11) it follows that it is gwen by D(R)
O(YXO, ~a)
=
[detao,3o(ro,aolro,tSo)] 1/2 det3o,.ro(Cg~O).
(73)
One can verify that det(n~ °) dY is the lnvanant integration measure over ~o. Hence qJo is normalized to
f d R ~o(R) ~ =
fdr det(C~o° ) =clY o,
(74)
where c1~o is the lnvarlant volume of ~o. From ~ mvanance it follows that for other values o f n one has
~o")(R) = Co(R[ z - " I ) , where 4o is gwen by (72). Next we determine c by matching the WKB wave function to the harmomc wave function in an overlap region where both (64) and (65) hold. In the same way as m II one can check that ~0o matches with the corresponding piece of qJw~m m the region where
IR - r~oXJ I = O ( g - ~ 3 ,
i Z~<7 < §1,
(75)
up to correction terms at most of orderg 1 - ~ . In this region the classical ttme is large. In this way one determines c. The calculation is practically the same as in II.
139
H.£ De Vega et al. / lnstanton phenomena (11)
One finds
N
c=e_So/2U,
det (Xa[X3) VI c°-~-Ll1/4 [-det (ro,~o Iro.~o) a=ko+l
o~,3=0,. , N ,
ao, 3 o = l ,
(76)
,ko,
with +~
SO = f
dr(~) 2
(77)
We finally compute e following the same method as In II multiplying (69) by 4o we obtain
(78)
~oM41 = (40) 2 and integrate both sides reside a domain Bo of configuration space defined by
3X IR-rloXll
1<.~<1
,~
~,
(79)
where [;~] Is an arbitrary element of ~. Geometrically Bo is a portion of configuration space surrounding cr/~o. Since (40) 5 decreases exponentially away from crgo one has
f dR(Co) 2 -~fdR(~o) ~ = q~o
(80)
Bo Moreover using (69) and Gauss' theorem one finds
f dR 4oMffl = - ~I f Bp
da" (~OV~I) -~
(81)
aBp
As in II we multiply (80) and (81) by e and obtain 1
e-
2q30 f dq" [4o~(e41)] aBp
(82)
By construction aBp IS in the overlap region where 4o and e41 match with the corresponding WKB wave functions and there are only two portions of this bound-
HJ. De Vega et al. / lnstanton phenomena (11)
140
ary where the wave function Is non-zero These are characterized by introducing r~ which are the parameters of intersection of the classical path r(r) and r(r) (-0 with aBp Since this boundary is such that condition (75) holds one has + -T p --+ + o o
r ~+ =
f(q~o)
q ~+ -->
,
Iz~ol = - ( 1 - ~) l n g +
-+o
(83)
, i ~ < l
0(1),
The two regions are
R ~- {r(-O(r~) lxl ,
for some X E (~} ,
(84a)
for some X E ~} ,
(84b)
where -~ means that the two members differ by quantities of order gO (see formula (61)) In region (84a) Co and e ~kl match with ff_(q~-) and e t° ¢+(q~) respectwely and it is most appropriate to change variable m the integral (82) by using (61) with n = 0. Similarly, In region (84b) Co and e 41 match with ~+(q~) and e -~° ~b_(q~) The appropriate change of varxable Is given by eq (61) with n -- - 1 Up to exponentially small corrections ~Bo can be replaced by the two regions r = r~ with arbitrary X and rb As in II one finds e=-
2cl)o
(
dtl ~Y/-d~ g ( q ) ~ _ ~ _
L goo(q) 1, f ~q
_e_,Of d F ~
c)5 = f d X
det(Ct~ ~) ,
~b+ ~
f'-=
}
(85)
df @'
(86)
6,7=1 .... k. c)) is the lnvanant volume of the group ~. As in II one can check that the most Important contribution to (85) is obtained by taking the derivative of the e ±so(O terms and one finally gets N-1
--
cos 0 -
Cl)O
[det(u(q~)u(q~)]det~(q~)~2(qo)_l~
e-so
(87)
'
H.J De Vega et al / Instanton phenomena (II) a , / 3 = O ..... k ,
al,fll=ko+l,
141
,k
Finally the bracket can be recast under a form mamfestly independent of q~, namely N--! ~det(x~ 11X.~!) l-I e = 2 __q~ a=ko+l det( Chgmn/rr) ~o c°s0 1 where, as in II,
(G')a/TO1/2
Q4Ymn1S a Wronskaan associated
}
e-S°'
to (47)
4. Comparison with the imaginary time path integral in the dilute-gas approximation In the dilute-gas approximation the ground-state energy shifts are given by [6]
ep l. = 2 cos O r--,~ llm 21T N ( T1 ) i f
C/)R(r)exp[_ / r dT.(SRr t 2 + V(R))], (88) (one mstanton) -T
where 0 ~< 0 < 27r and N(T) is the same functional integral in the zero-mstanton sector Thas contrlbuhon can be calculated by expanding around the classical solution
R(7.) = r [X~] (7. - 7"0) + s [x~l (7. - 7.o),
(89)
where 7.0 and the X a are to be treated as collective coordinates m order to remove the zero modes from the Euchdean quadratic form. As usual, it is enough to take Xa, 7.0 to be single Integration variables and not functions of 7. In this case s satlsties the c o n d m o n T
f
d7. r,a(r - To) s ( z - to) = 0 ,
a=0 .... k
-T At the one-loop level one has k C-DR(r) = "vQ-(T) det ( c ~ ) d7.o 1-I dXC~C-l)s, 1
det [f d7. r,a(r) • r,#(r)] 0
(90)
H.J De Vegaet al / Instanton phenomena (11)
142
Similarly N(T) 1s computed by expandmg around ro, taking into account the zero modes due to the spontaneous breaking of ~o We thus Introduce the corresponding collective coordinates and write
R(r) = rior~°l + ~(r) [ r~ol
(91)
As above we choose the yao independent of r. Then +T
f
drro,~o o ( r ) = 0 ,
ao=l,
,ko,
-T ko
C/)R =X/Jo(T)
det
( 9 ~ o°) 1-I dY ao Q)o,
1 (fl0, 70 ~
Jo(T) =
det
(92)
°tO= 1
[ro,a 0 ro,3o] (27) kO .
(93)
1 ~c~0,~30 ~
For large r, r, ao(r ) tends to ro,a o and it follows Immediately from eq (90) and (93) that J(T)_ ~=
1Lln~Jo~
det Otl,/31=O, k o + l . . . . k
[f
_~
drr, al(z).r,~l(r)]
(94)
and m the one-loop approxlmanon formula (88) becomes +~
ffcDsexp[-~ f C~ ~
dr(slMIs)]
_oo
ep.i. = 2 cos 0X/~ e - s ° C))o
+~o
ffcboexp[-~ f
,
(95)
dr(olMoJo)]
where a2
M = - ~r 2 + U(r)'
a2 Mo = --~-r2 + G2
(96)
As in II, we introduce a regulating parameter z to handle the zero-modes and we get +co
ff ~sexp[-1 f
-~
d r ( s l M - zls)]
+~
ffc~oexp[ -1 f _oo
| / ( - z ) k l +1 = V c/)(z)
dr(°lMo - z l ° ) l
'
(97)
H J De Vega et al / Instanton phenomena (II)
143
where CO(z) - det [(114o - z ) - l ( M - z)] is the Fredholm determinant of the operator M The factor ( - z ) gl+l comes from the (kl + 1) zero exgenvalues of M w h l c h are not present m the spectrum of Mo Now as in If, we consider the lost solutions of the one-dimensional scattering problem in the 7--axis with U(T) as a potential They are N × N matrix solutions of
~r 2 + U(r) F+_(z, r) = zF+_(z, 7-)
(98)
with boundary con&tlons
F_(r,z)
=
Here k(z) = - ~ z F _ ( r , z) F+(7-,z)
e-'~r,
F+(7-,z)
=
O(E) e +'~rO(x;) T
(99)
~2 For 7- -+ +~ we have
= T___~+~
:
.., O(E) [e'kr A - ( z ) + e-,k~rF -[Z)l
O(z)T
e-'krA+(z)+e'krF+(z),
(100)
T---> _ _ ~
where F+_(z) are the lost matrices by definmon In particular for z = 0 K_+(7-) = F+ (7-, O)
(101)
As In II it can be shown that the following properties hold
F+(z) = lc-1 (z) F~(z) T k(z) , A+_(z) = _ ~ - 1 (z) A~_(z) af k(z) ,
F+(z)* k F+(z) - A + ( z ) ? kA+(z) = Ic
(102)
In the case we consider, U(7-) tends exponentially to its limit for large 7-, since r(7-) has this property Then one can show that the Fredholm determinant cO (z) equals the determinant of the lost matrix for arbitrary z fi9 (z) = det F+ (z) = det F _ (z).
(103)
Thus eq. (95) can be written as c~ ep.i. = 2 e - S ° x / ~ - c o s 0 op ° 1 / ,
1 ( - 1 ) ~l+l d k1+1 CO(z) (kl + 1) v
dzkl+l
(104)
H J. De Vegaet al / Instanton phenomena (11)
144
In what follows we shall compute the hmlt in eq (104) and we shall show that the value so obtained for e coincides with the result derived m sect 3 by matching o f wave functions. The method is exactly the same as in II, apart from the new features introduced by the symmetry. The Jost matrices act m the hnear space of the elgenvector of ~ . This space can be decomposed as the &rect sum o f two orthogonal subspaces / and 1_ The first will be the space of elgenvectors of ~ with non-zero elgenvector and the second the kernel of ~ . This decomposition exphcltly reads
e,kr
= ( e'k±rF+l±(z)
e*k±rF+±t--(z) ] (105)
\ e,4Z*F+L±(z)
el
*F+W(z)" '
where F ±± is a hnear mapping from _[ t o / , F t-± from .I. to I_ etc. As m II, it will be convement to use, in the s u b s p a c e / , the basis formed by the vectors I~o,) = IX~I),
I~p,+kl) = c~±IX~I),
oq = 0 , k o + 1 . . . . k ,
t = l ..... k l + l
(106) ,
and other ( N - 2 - 2 k l ) vectors which are hnearly independent from them. Then
/ F±±~)
Flt--(z)
c-/)~)= [ d e t ( h , l ) ] - I det [
(107)
FUll)
Ft-J-(z) ) '
where, as usual h u = (~o,1~0]).
(108)
At z = 0, q)(z) has a zero of order k 1 + 1, as follows from eqs. (A.1). Precisely, we choose the basis (106) because m this basis the matrix F+_(O)qhas the columns ! = 1, ..., k l + 1 and the rows t = kl + 2, ..., 2kl + 2 adentlcally zero. Many other elements vamsh In this limit as we see from (A.2). Then xt can be seen by inspection of the full F±(z) matrix that
dzkl+l [z=o O~l,~i~kl ×(kl+l))
det' F+l±(0) d e t h ±± '
(109)
HZ De Vega et aL I lnstanton phenomena (11)
145
where det' F}±(0) is the determinant of order ( N - kl - ko - 1) of the matrix obtained from FI±(0) by deleting the first (kl + 1) columns and the rows kl + 1 ..... 2ki + 2. In derwlng eq. (109) we have used the equations det F+L-L(0) = 1 ,
A~t-'(O) = O,
F~ZL(o) : O,
(1 10)
which are proved m the appendix. The first determinant m eq. (109) can be computed m the same way as in II, eqs. (5), (17), (18), obtaining (see eq. (94))
/
dF~±(O)l ,\_
,~
Thus the imaginary time path integral gives for the width of the first allowed energy band e -so
ep.i. = 2 rr(kl+l)/2
| / det h±± cp 1/det,F~±(0) q~o cos 0 .
(112)
We shall now prove that the expression obtained for e m sect 3 can be recast m this form. From eqs. (56) and (87) we have ow=:
e-So
c~ , . /
det g~l#l(r) det(~±/n)
L-
'
(113)
J
~q,/31 = k o + 1, ..., k. Then by exphcltly writing det [K+(r) K + ( - r ) T] m the basis (106) for large posltwe r it follows using eq. (A 2), (A 8) that det [K+(r) K+T(--r)] - d e t ' F ~ ( 0 ) det (~oaI (r) l ~ holh ( r ) ) , deth± 0~a~,~l~k o
(114)
[tPC~l(7.)) -~ K+ (7") ]XctI ) Moreover one can prove
det nal%)
det gc~lflI (7") det ( ~ 1(T) ] ~ Z 1 [~Ofll(T)) '
(115)
146
H.£ De Vega et al / lnstanton phenomena (11)
hm
[det ga I th (7-)] 2 --
1
.
r - ~ deta l,~l <~o~1(7-)1w± 1¢~1 (T) > de t (tPal (7-) [wi- 1 [~Ofl1 (7")) These two relations are derived in the same way as similar relations used In II Eq (115) holds because ~± has no zero elgenvalues.
Appendix The zero modes can be expressed m terms of K_+(r) and the constant vectors [~>(see eq. (55)). Then, by using eq (100), F_+(0)[X~I> = 0 ,
t~l = 0, k o + 1,
,k,
F± (0) i t - = 0
(A1) (A.2)
The first equation shows that q)(z) has a zero or order kl + 1 at z = 0. If one considers the decomposmon between ± and t_ subspaces (eq (105) together with relations (102) and (A.2)), one finds A~t-(0) = 0 ,
(A.3)
F_+t-t-(0)tF_+ (0)t--t-(0) - A ~ t - t ( 0 ) A~t-(0) = It_
(A.4)
Finally we consider the Jost soluUon for small but non-zero z In this case one has for non-asymptotic r F_+LL(z, 7") = 142(7-)+/X/z-Z(7-) ,
(m 5)
where 7- ~ z - 1 / 2 > > 1 .
Here W(r) and Z(7-) are soluUons of eq. (98) for z = 0, with the following asymptotic behavior W(~') = I , T~--~
Z(7-) = 7-1
(A.6)
T--~--~
By choosing the vectors ro,~o (as in (106)) as a basis ofl_ one can take simply W(r)ao,t~o = [r(r), 3O]ao
(A.7)
Then by using Wronsklan arguments for W(r) and Z ( r ) and eq. (100) it can be
H J. De Vega et aL / Instanton phenomena (11)
147
F_+t-t-(0) + A_+U-(0) = F_+t-t-(0) - A_+L-t-(0)= O A ( E ) - 1 0 ( 2 ; ) ,
(A.8)
shown that
where OA(2;) is the 2; t r a n s f o r m a t i o n In the adjolnt representation. We have also used that ro, cz0 = lItxor 0 ,
0(2;) - 1 Ice0 0 ( • )
= OA(2;)a030 I3o
Then, eq (110) follows from (A 8).
References [1 ] J L. Gervals and B. Saklta, Plays Rev. D16 (1977) 3507. [2] H de Vega, J.L. Gervals and B. SakRa, Nucl. Phys B139 (1978) 20. [3] J L Gervals and B Saklta, Gauge degrees of freedom, external charges, quark confinement cntenon m A 0 = 0 canomcal formahsm, CRy College prepnnt 78/5, Phys Rev. D, to appear. [4] H.J de Vega, J L. Gervals and B. Saklta, Wave functions and energy for vacuum and heavy quarks from WKB Schroedmger equation for massive gauge theories wath mstantons, PAR-LPTHE 78•09, Phys Rev. D, to appear. [5] J L Gervals, Schladmmg lectures 1977, Acta Phys. Austrlaca, Suppl XVIII (1977) 385. [6] S. Coleman, Uses of lnstanton, Ence Lectures 1977, Harvard preprmt (1978).