Application of the Tamm-Dancoff-method to the Lee-model

Nuclear Physics 5 (1958) 1 9 5 - - 2 0 1 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publishor

APPLICATION

OF T H E T A M M - D A N C O F F - M E T H O D LEE-MODEL

TO THE

W. H E I S E N B E R G

Max-Planck-Institut /iir Physik, G6ttingen R e c e i v e d 20 S e p t e m b e r 1957 The n e w T a m m - D a n c o I f - m e t h o d is a p p l i e d t o t h e Lee-model, a f t e r t h e r e n o r m a l i z a t i o n for t h e s p e c i a l case of t h e ' d i p o l e - g h o s t ' h a d b e e n c a r r i e d out, The T a m m - D a n c o f f m e t h o d l e a d s t o t h e e x a c t solutions. C o n t r a v a r i a n t r e p r e s e n t a t i o n s of t h e s t a t e v e c t o r s b y m e a n s of t h e r e n o r m a l i z e d o p e r a t o r s can be f o u n d o n l y if t h e s e o p e r a t o r s a re s p r e a d o u t o v e r f i n i t e t i m e i n t e r v a l s , i.e. if i n t e g r a l s of t h e o p e r a t o r s o v e r f i n i t e t i m e i n t e r v a l s are a p p l i e d to t h e s t a t e v e c t o r of t h e v a c u u m .

Abstract:

The new method of quantisation which had been used for a non linear spinor equation 1) has recently been analyzed in connection with the Leemodel 2). The actual calculation of the stationary states and the S-matrix has in the case of the non linear spinor equation been carried out in a first approximation b y means of the new Tamm-Dancoff method, While the ordinary SchrSdinger formalism has been used for the Lee-model. In order to get a close connection between the two mathematical schemes it seems instructive to repeat the calculations for the Lee-model b y means of the Tamm-Dancoff method, i.e. just b y that method which had been used for the non linear spinor equation, and to compare the results with those obtained from the Schr6dinger equation• The notation will be taken from the paper on the Lee-model*). Just as in the case of the spinor equation the calculation shall be performed without using any process of renormalization. Therefore we have to start with the renormalized wave equations which contain only finite constants and are the analogues of tlae non linear spinor equation• The wave equations [ref. 2), eq. (62 a--d)] are written b y means of the field operators ~Ovr(X, t), ~ON(X, t) and a(x, t), referring to the V, N and O-particles, and b y the quantities 9(x, t) and H(x, t) related to a(x, t): t

t

g fto dt Z(t--t

t

)~0vr(X, t')=--2i~r*g

f:oo [a(x, t)a*(x, t') --s(t--t

t

)]~0vr(X, t')dt'

- ~ V ~ a ( x , t)~(x, -oo); • aw~(x) -

OH (x)

at

= - z l g ~ / 2 a*(x)~Ovr(X);

a ~ =A~ (x)--m2~ (x) +

_ _d

e

t

i~g

Oa)

(lb)

~g

~-~ [~*,(x)~N(x) +~* (X)~v,(X)] ;

(lc)

3-' e-,.olx-x'l +k t p * t t - [Yvr(X )yN(X )--V,N(X )Yvr(X )]. (ld)

Ix-x'l

195

196

W. HEISENBERG

s(t--t')

and

x(t--t')

are defined b y (oJ =

V'mo*+k 2)

s(t--t') =
where

h+(z)

=

fo ° kgdk o~ e ~ , , _ o ;

(2)

2~d-~

is a function characteristic for the Lee-model [ref. z), eqs. (15),

(16) and (29)1.

IN+z0

The followiI~g calculations will be confined to the sector w+I,-ll0J" Let q}> be a state vector in this sector. Then we can define the T-functions ZN(Xltl, • • ", x, tz; tN) = <01Ta(xltl) " ' " a(xz t,)~0~(tN)l#>, Zv(Xl t l , ' " ", x , - i t,-1; tv) = <0[Ta(x 1 tl) • • • a (xz_ 1 tz--1)~JVr(tV)[~>,

(3)

or in m o m e n t u m space: ZN(kltl, .. ., k , t , ; tN) = < 0 i T a ( k l t l ) " " " a(k~tz)wN(tN)l~b}, ~v(k~ t ~ , ' - ", kz-1 t,_l ; tv) = <0[Ta (k~ tl) • " " a(k,_l t,-1)~ow(tv)I~>.

(4)

(The letter T denotes the time-ordered product.) Integrating eq. (la) over x gives (the N or V-particle is at x = 0):

gf;oodt, z(t_t,)~Vw(t,)=_2:~zigf;odt'[a(O , t)a*(O, t')--s(t--t')]~Vw(t') - v'2 a(o, If one multiplies (5) from the left b y a ( k l t ) a ( k ~ t ) . . , a(IG_lt) and

(5)

considers the matrix element of this operator between the states <0 and ~>, one gets an equation between the two v-functions (4). a(O, t) is to be replaced b y

t) =

(kt) d3k.

(6)

In this w a y one derives from (5)

g f :oo dt' z(t--t')Vv(kl t, k2 t, . . ", lG-x t; t') = -- ~ig

f d 3kf~ oodr' *-Xe'°'("-*' z=xX~2Vwcoz vv(kl

$'

" " "'

k H t , kz+~t,

""

.,kt;t')

(7)

1 f d3k z N ( k l t , . . . , k , _ l t , k t ; - o o ) . If one multiplies (lb) from the left b y a(klt) . . . i~

a(IGt)

one gets

z~(kl t," "', k, t, tN),~_ , g

= --2~/n~

,

zv(klt'""k'-xt'k'+'t'""l~t;

1

t) ~¢/~.

(8)

APPLICATION

OF THE

197

TAMM-DANCOFF-METHOD

These two T a m m - D a n c o f f equations should be sufficient to determine the two v-functions (4). Contrary to the case of the spinor equation 1) it is not necessary here to introduce approximations through the process of contraction. Due to the special simplicity of the Lee model the T a m m Dancoff formalism leads to exact solutions. In a state in which there are several 0-particles, b u t no N- or V-particle, the total energy is simply the sum of the energies of the free 0-particles; there is no interaction between the 0-particles. Therefore the time dependence of the v-functions (4) in the time interval needed for (7) can be given without difficulty: if E is the energy of the state q}), we have for ts~tl, t 2 , ' ' ' , t ~ and t v ~ t 1, t,, . . ., t,_ 1: z

rN(klt 1, ..., k , t , ; tN) = r N ( k l , . . . , k , ) e

1 z--1

z--1

(9)

--i ~ (~Z tZ--i(E-- ~1 c ° l ) t v '

rv(kl tx,'" ", k,_l t,_l ; tv) : rv(kl, "'-, k,_l) e

1

where the v-functions on the right h a n d side depend only on the m o m e n t u m variables. Inserting (9) into (7) and (8) leads to z--1

=

gh+(E - ~_, co,)vv(ka,'-., k,_l)

g

~"

*--1

..

G J d s k • vv(kl ' ' ' " k ' - l ' k'+l'

"' k)

z--1

I

'=' 2V'~ozco(E-- Y O~r--o,+i~,) 1

1

l" dak

z--1

i( E-- ~

vN(k ,

"" ", k z _ l ,

k) e

~Or--OJ)(t+oo ~

1

(~, -+ 0)

(10)

and z

( E _ _ ~xc o t ) V N ( k l , . . . , k z ) -

g ~ % /1~ r v ( k a , ' " , k , _ l , k , + l , . . . , k , ) . 2X/'7r,=1

(11)

The second line in (10) needs an explanation since the time --oo from (7) appears in the exponent. One can see from (11), that v ~ ( k l - . , k,) can be written as v,,(k 1 . . . . . k,) = Vo,,(k 1 . . . . . k , ) + ZN(k~ . . . . k , ) ,

(12)

1

where t o ( k 1 , - . . , k,) is an arbitrary solution of z

yoN(k1, • --, k , ) . ( E - - Z r) = 0, 1

and zN(kl . . . . . 1%) is regular at the point E - - ] ~ [ ~Or = 0. W h e n one inserts (12) in the second line of (10), the contribution from

198

w.

HEISENBERG

the term x~(kl . . . . . k,) vanishes, since the correct interpretation of the exponent would be e

1

-~0

(7-->0).

Therefore only the contribution from voN(k1. . . . . k,) remains and one gets instead of (10) *--1

1

F

*--1

h+(E-- ~ o~3 vv(kl , " ' , k,_l)---- ~ J d s k ~ zv(kl ..... k,_1,,_.1k,+1..... k)

1

,=12V%o,,(E-- X ~ , - ~ + i r ) 1

1 ~ dSk g ~ 7 ~ 3 ~ - ~ voN(k1

. . . . .

k~_l, k). (13)

The two equations (11) and (13) become identical with the SehrSdinger /N+,0 ~ namely the equations (120) and (122) equations of the sector w+(,-1)o/, of ref. 2), if one puts (using again the notation of ref. *))

v~(k, . . . . . k,) = ~(k~ . . . . . k,), go vv(k ~. . . . . k,_x) = ~g 9(k 1. . . . . lq_l).

(14)

This result shows that the Tamm-Dancoff formalism leads to the exact solutions in the case of the Lee model and t h a t the z-functions are -except for constant factors -- identical with the Schr6dinger functions. It should be emphasized, however, t h a t the v-functions of the renormalized operators have finite values, while one of the SchrSdinger functions becomes infinite for go -+ 0; further that the square of the single SchrSdinger functions (or v-functions) integrated over the whole space can diverge. The norm of q}>, however, is finite; formally the norm appears in the SchrSdinger formalism as the difference of two divergent terms. In order to derive the norm in the Tamm-Dancoff formalism it is convenient to calculate -- besides the covariant representation of ~> by means of the z-functions -- a contravariant representation in terms of the renormalized operators. For the sake of simplicity we will confine the calculations to the sector (%+0). In ref. *), eq. (TR) the state ~> was expressed as q~> = (--CV'v*+~'N* fo0(k)a*(k) da k){O).

(15)

In t h e process of renormalization o0(k) turns out to be independent of go(go -+ 0), while c varies as go-1. We define the constant b by c ---- --b/g o. If one introduces the renormalized operator ~w----- (go/g)YJv, one gets

qs> = (~o, ~o*~+~o~*f g(k)a*(k)dSk) ,O).

(16)

APPLICATION OF THE TAMM-DANCOFF-METHOD

199

This expression cannot be used for the representation of # after renormalisation, since it diverges in the limit go -+ 0. It is possible, however, to represent ~b) b y means of the time dependent operators W*r(t). ~VN*(0) and a* (k, 0) shall refer to the time t = 0. Then we t r y the expression

where /(t) is different from zero in a finite interval around t = 0. The only condition imposed on /(t) is, that it must lead to the correct ~function, i.e. to the same Tv-function as (15). This condition gives

f

<01 Vr(0) - (0)10> = <01 vr(0) v*r(t)/(t)dtl0>. (IS) go The first expression is simply --b/g, the second can be taken from the commutation relation of ref. 3), eqs. (52) and (49), b

f

1[

2h'"(Eo) etEot+f°°

/(t)dt-~ 3[h" (Eo)] ~ g One can for instance assume /(t) =

ke*otdo~ 1.

Jm 0 2 h ~ o )

) _J

(19)

{/0 for--to<=t~to otherwise.

Then one gets from (19), using ref. 9), eq. (48),

bg=2to] [ [-3h"(Eo'] 2h'"(Eo) (1 ~ If to <<

lIE o and

<<

sinE°t°

Eoto)+fmooo2h+(o)h-(o) kdo~

1/mo, one can

h±@) ~

sinoJto]l (20) Oto I J"

use the asymptotic expression for h±@):

(Oln(OC 2

(1

for

(o >> too,

(21)

m0

where the constant C depends on E o and can be calculated from ref. z), eq. (29). In a first approximation one gets from (20)

bg-- 4t°/ in ~ _

bgln

or

C

/to=-~ tom~o.

(22)

tomo This result shows that the 'area' S/(t)dt = 2]to increases logarithmically, when to goes to zero, as one could expect from eq. (16). The function a(k) in (17) is not identical with ~0(k) in (15); the first term in (17) gives now after replacing (15) b y (17) some contribution to the zrrfunction, which together with t h e contribution from a(k) must give the same result as ~0(k) in (15). This condition leads for a(k) to a(k) = w(k)--

~,f/(t)dt
0)t~>e'Et

(23)

200

W. H E I S E N B E R G

or again for t o << 1/E o and t o << 1~too, using ref. *), eqs. (17) ,(18) and (21),

(e'O'--l)

1

v's~o, gh-(~o)

f Jok'~dk ' 2#s~do;g(o;--o~+ir)h+(o;)h-(o;)

-- /(t)dt

e~'t--1

O' --> o)

(24)

(7 --> 0).

(25)

~o(k)--fl(t)dt(e'~"--l)(ga/~o~'l. ln°~C)-' --

f

l(t)dt

fo "~k '2dk'

e"'--I

g~/~mco'8(o~'--oJ+iT)

w'C in 2 - m0

It follows from (22) and (25) that for values of co << 1/t o the difference between a(k) and ~0(k) is relatively small, while for oJ >> 1/t o the last term in (25) compensates the main part of 9(k). Therefore we conclude from (25)

__a(k) _ , ~(k)

for o~ << to

(26)

1

for o~ >> to

Finally we get for the norm of ~b) from (17)

<~1~)

(¢1 f,t,*=(t)/(t)dt+V;N* (0) f a(k)a* (k, 0)d 3 kl0)

(27)

= ~v* f / ( t ) d t e~*+ f a(k)rN*(k)d 3 k, where E is the energy of the state ~b). For Et o << 1 eq. (27) becomes

to m--~ +

4~k~ dkl~(k)p.

(28)

One sees from (28) that the expression for the norm in the T a m m Dancoff m e t h o d contains only finite terms. B o t h terms depend on the width t o of the time interval in (17) which can be chosen arbitrarily. The norm itself, however, cannot depend on t 0. Actually it follows from ref. ~), eq. (13) and (21), t h a t b 9(k) ~ for o~ >> [El. (29) Therefore the second term in (28) varies as ½lb2l in (l/t0) for small values of t o, and the norm is independent of t o. If one uses ordinary space instead of m o m e n t u m space in the contravariant representation (17), the function 9(x) varies for small radii as const, r-~/, (ref. 2), eq. (45)). The function a(x), however, is rounded off at

APPLICATION OF THE TAMM-DANCOFF-METHOD

201

small radii of the order to, it agrees only for r >> to with 9(x), and the integral fg*(x)a(x)d3x converges. One would be inclined to consider the expression 9*(x)a(x) as the matter density of the stationary state, as has been suggested in an earlier paper 1). But one sees here t h a t this expression depends on the function ](t) which can be arbitrarily chosen within the condition (19). The fact that a well defined state q~) can have a continuous variety of contravariant representations prevents a unique definition of a matter density for this state. In the Lee model the Tamm-Dancoff method leads to exact solutions. This is certainly not true for the non linear spinor equation 2), where one has to introduce approximations by the process of contraction. The difference between the two cases is caused by the fact t h a t in the Lee model only one particle can be created or absorbed -- actually this simplification can be considered as the main feature of the Lee model --, while in the spinor model m a n y particles m a y be produced. Therefore the Lee model does not provide any direct argument from which one could judge the degree of approximation attainable by the Tamm-Dancoff method in the spinor model. But the following comparison m a y be adequate. The Lee model should be compared with the harmonic oscillator; in both cases the TammDancoff method gives exact results, since only one particle (0particle or photon) can ever be created or absorbed. The spinor model, on the other hand, should be compared with the strongly anharmonic oscillator; in both cases several particles can be produced and the Tamm-Dancoff method needs the process of contraction, therefore it is a method of successive approximations. It was this analogy which had been discussed extensively in the first paper 1) on the spinor model, and it had led to the result that the first approximation in the Tamm-Dancoff method m a y give the eigenvalues with an inaccuracy of 10 to 20 %. The calculations of the present paper seem to support this view. References 1) W. Heisenberg, Nachr. G6tt. Akad. Wiss. (1953) 111; Zs. f. Naturf. 9 a (1954) 282; W . Heisenberg, F. Kortel u. H. Mitter, Zs. f. Naturf. 10a (1955) 425; W. Heisenberg, Zs. f. P h y s i k 144 (1956) 1; Nachr. G6tt. Akad. Wiss. (1956) 27; R. Ascoli u. W. Heisenberg, Zs. f. Naturf. 12a (1957) 177 2) W. Heisenberg, Nuclear Physics 4 (1957) 532