Acta Physica Academiae Scientiarum Hungaricae, Tomus 25 (2), pp. 215--226 (1968)

NONLINEAR MODEL IN QUANTUM FIELD THEORY

By

F. MEZEI*

INSTITUTE FOR THEORETICAL PHYSICS, ROLAND EOTu UNIVERSITY, BUDAPEST

(Reeeived 23. I. 1968)

The mass spectrum of a nonlinear real sealar field was sought for using Ritz's varia- tional method. In the case of suitable renormalization 1, 2 or 3 finite values of rest mass were found. The different types of these excitations belong to different inequivalent representations of the field operators.

The p r o g r a m m e of the nonl inear field t h e o r y has a m a x i m a l airo of describing a n u m b e r of e l e m e n t a r y part icles and their in terac t ions b y a single field equat ion. Af ter the f irst s teps, however , enormous m a t h e m a t i c a l diffi- cuhies arise. Ju s t because of these we do no t ye t k n o w whe the r such a prog- r a m m e migh t p rove successful, of whe the r ir is i nadequa te for descr ib ing the world a t all. Effor ts h a v e been made essent ia l ly in two directions. The re have been a t t e m p t s to inves t iga te the con ten t of ah equa t ion which has been regard- ed a s a candida te for the full descript ion. This line was followed b y HEISEr~- BEaG in es tabl ishing bis famous equa t ion [1]. On the o ther hand , the ma the - mat ica l s t ruc tu re of s imple models has been inves t iga ted in o rder to gain exper ience for the f ina l solution.

This app roaeh was successful in f inding simple, exac t ly so lvable models and in giving an a p p r o x i m a t e solut ion to o ther models [2]. These ca lcula t ions yield some insight in to the m a t h e m a t i c a l s t ruc tu re of nonl inear f ie ld theories. The present pape r gives ah account of such ah a t t e m p t .

The nonl inear r ea l scalar field, p rev ious ly inves t iga ted b y GOLDSTOr~E, MAaX and KOTI [3] has been modif ied b y an addi t ional t e r m in the field e qua t ion :

[ ] ~ + t,o ~ ~ + ~,~ ~~ - ~o ~ ~~ = o . (1)

2 where n 0 q0 is the new t e rm , whieh des t roys the ~ -+ - - ~ s y m m e t r y of the ori- ginal model . The solut ion was sought for wi th in the f r a m e w o r k of L a g r a n g i a n formal i s ta and canonicaI quan t iza t ion .

* Present address: Institute for Experimental Physies, Roland Eiitviis University, Budapest.

6 " Acta Physica ~4cademiae Scientiarum Httrtgaricae 251 1958

216

where q 2, ~~, ~~ ate real varameters . We use a sys tem of h = l , c ~ l .

For the momen tum conjugated to ~,

F. M E Z E I

w 1. Canonical quantization

The Lagrangian corresponding to the field equat ion (1) is:

2 '

nnits, in which

0L ~ r - -- #, (2) 9+

we require canonical commuta t ion relations:

[~(r, t), ~v(r', t)] = [~r(r, t), Jr(r', t)] = 0 ,

[Jr(r, t), q(r" t)] = -- i6(r -- r ' ) .

The Hamil tonian and the field m o m e n t u m is given b y

(3)

where the vector index k ~ (kx, ky, kz) runs over the values

ki = 2 ~r~ -~/3 n i n~- = 0, • 1, Q- 2, • i = ~,y, z.

Eq. (2) is equivalent to the commuta t ion relations

[pk , q~'] = - i~k,~"; [q~,p~-'] = [q~, ~~'] = 0

and the hermit ic i ty of ~v and 3r requires

q ~ = q - k , P ~ = P - k -

(7)

Acta Physica .4cademiae Scientiarum lfungaricae 25, 1968

Supposing tha t the field is enclosed in a cube of volume ~2 we represent a n d a b y Fourier expansions:

1 qJ(r, t) -~ ~ ~ qk(t) eiar, (5)

1 a(r, t) - - gT2 "~Pk(t)k e-ikr' (6)

P -- 21 f Q r v q + Vq" ~r) d a r . (4)

1NONLINEAR MODEL IN QUANTUM FIELD THEORY 217

The relations (3)--(7) define a quantized field: pk, qk, H, P are elements of au abstract algebra, and they obey the above relations aceording to the opera- tions defined in this algebra.

A representation of this field means a homomorphism between the pk, ffk algebra elements and q ~k operators acting on a separable Hilbert space JE . A representation is called "proper" ir the set of linear operators aeting on ~Ÿ contains elements, say 1~/and Jb, homomorphic to the algebra elements defined by eqs. ( 3 ) a n d (4). For a "physical" representation some further properties (existence of vacuum state, Lorentz invariance) ate required.

The basic problem is to find proper representation. This is over and above the customary difficulties of solving the eigenvalue problem. Exact represent- ations have been found so lar only for very simple models.

Another difficulty beyond those of quantum mechanics is involved in the following. VoN NEVMA~N'S theorem states [4] that in the case of a finite number of pk, qk variables (i.e. quantum mechanics) all irreducible represent- ations are equivalent, i.e. they can be transformed into eaeh other by unitary transformations. So, if one of these representations is proper, all of them ate proper, and all representations give the same physical results (e.g. eigenvalues). For an infinite number of variables this does not hold.

w 2. Generalized coordinate representation

Let us consider the linear space ~ of the twice differentiable functions of a numerable infinite number of real variables

= ~(Xo , x i . �9 . ) .

We define the following linear operations on ~T~:

Xi ~(x0, x i . . . ) = xi ~(x0, x i . . . ) ,

P j ~ ( x 0 , x l ) - ~~(Xo, X~ } 0x i

Since within each Hilbert space ~ c ~-~, where the scalar produet o fany two elements o f ~ , say O and O' is defined in an appropriate way to realize the usual definition (0, 0") = S 0"0 ' dxodxl. . .)

[ x j , x , ] = [~ , , ~,] = 0

and Xi, P j ate hermitian, it is easy to construct a representation of the eommu- tation relations (7) using ~ / s and P]'s.

.Acta Physiea Academiae Scientiarum Hungaricae 25p 1968

218 F. MEZEI

So we identify the labels of the x i variables with the k vectors of eqs. (5) and (6), and introduce the definitions,

1 q i/3-k) ~~=-~( ~- for k > 0, (8)

q-k-~ q$, /3-k =/~~, qo = ~Ÿ q = / 3 0 , (9)

where k > 0 means that the first nonzero coordinate of k is positive. Using the zu E(O, ~) , OkE(O, 2 ~) (k>0) new real variables defined by the equations

x k = z k c o s o k, X_k=Zk s i n o k,

we have on the basis of eqs. (3), (4), (8) and (9) the algebraic expressions homo- morphie to H and P:

= . ~ i k k > 0

1 ~~ g~o x0 + 1 ~ ~ - ,~~~> + (k-" - - g o ~) z~) - 2 0Xo z 2 2 ~ ~ k

~~o I ! ~ ( i ) zk zl zn e i(+-~ +-e'+-~ 8(4-kil-}-m) + 1/D {6~ k,,,m>O

lx31 - 4 - x 0 ~ z ~ + 3 o ) + (10)

-4---~-2~ (~k,~,m.n> 1 "~(+-) zkZlZmZ'~ei(+-ek+-e'+-~191177

1 .~(+-) zkZtZmei(+-~ -4- l-4-m) -4- -~- ~ XO k,l,m>O

3 x2 ~ . 2 1 4) + T o~-~ +T=o

k>0

O

0~9 k

where A(k 2) is the two dimensiona] Laplace operator in the polar variables zk, �91 and 2: ~-*) means summation over all indicated combinations of signs too, and

~ ( k ) = { 1' if k = 0 , 0, if k : ~ 0 .

".,4eta Physica .Academiae Scientiarum Hungaricae 25, 1968

NONLINEAR MODEL IN QUANTUM FIELD THEORY 219

w 3. Variational method

According to SCRIrF [5] we shall apply Ritz's variational method to approximate the proper representations with the following reasoning. In the Hilbert space of a "good" representation the eigenfunctions of ~ realize different "local" or "condi t ional" min imum values of the expectation value (~,/2/~v). 0bviously, if we carry out this minimalization in a "larger" (non- separable) Hilbert space, the resulting functions possess a min imum property in a "smaller" (physically reasonable, i.e. separable) Hilbert space contained in the original larger space.

We use the trial functions

T = T0(~o ) / /T~(zk , 0~9. (11) k > 0

Thus a single ~k function containsthe variables xk andx_1~. A sufficient norma- lization eondition reads:

<T01T0> = IT0(~0) t~ d~. = 1,

(12)

<TkIT~> I Tk(zk, dzk . ----;= dOk~ Ok)[2Zk

Since t / and ^ commute with each other, we are looking for their common eigenfunctions. The eigenfunctions of P having the form given by eq. (11) ate

T = T..(~0) [If~(Z~) e -in~~', R>0

where vo, fk ate arbitrary functions, nk ---- 0, -61, -62 . . . . . The corresponding eigenvalue is ~ = 27 knk

k > 0

For such wave functions

< 1 8 #2o 2 3 ~ x 3 ~ ~ > <T]I~/IT>= T~ 2 8% 2 x~ § x~lT §

k-~>0~ T ~ - 1 3 22o z~lT0 > _ + kl - z~ ~-) + ~ - (k 2 - ~~)z~ +

xg 3 2~ <T01x01T0> k>0 -~" <Tk l41TD + ~ ~~#, <~klz~lT~> <T~(z~q +

k, />0

3 22 (13) -F 2 ~2 '~- <,T,~]x~]To> k>0 ~ ' <klz~[Tk> "

Acta Physica Academ�98 Scientiarum Httngaricae 25, 1968

220 F. MEZEI

The auxiliary conditions (12) will be inserted into the minimalization with the Lagrange multipliers E 0, Ek (k > 0): These are equivalent to the following system of coupled differential equations:

where

and

where

( 1 ~2 ) 2 0 x~o -4- V(xo) ~o Eo ~[/o, (14)

: o ~~ 1 2 V (xo) = - - X4o 3V ~ X3o - - ~ # x~ -

. . . . . ~ <~~i41~k>, ~r~ k>0

V O _ x~ 9 ~ <Tk[z~}TD

1 "(2" Vk(zk)}Tk Ek~k , (15) ( - - ~ - ~ ~ ' + =

Vk(Zk) = 3 ~2 o ~ _~ 1 [ o . 3 2~ 8Q T k2-- ~" T - - ~ - <w~I41wk>:§

+ ~ <T~[x~l~0> -- 2 ~2

Eqs. (14) and (15) are identical to the Schr/~dinger equations of the one and two dimensional motion of a free particle of mass unity. V(xo) may lead to one dimensional anharmonic oscillations around two equilibrium positions XA and XB (Fig. 1), whereas Yk(zk) correspond to two dimensional anharmonic oscillations around zk = 0.

/ xA /.-, xs

%/ Fig. 1

Acta Physiea Academiae Seientiarum Hungaricae 25, 1968

J

NONLINEAR MODEL IN QUANTUM FIELD THEORY 221

When solving these equations we must take into account t ha t H is defined by a limit f / = lim af-/K, where ~k is obtained from eq. (10) by summing

only over indices I k I, [ l [, I m ], [ n [ < K. 2 2 This is why the parameters 20,#0 and z2 can be considered as functions

of K. I t is sufficient to require weak convergence, tha t is:

< ~ l [ / l ~ > = lim <W]H K ~ [(2g(K), q xg(K))I~> K--~

(16)

for all 7 t, ~ E o~U, the Hilbert space of the specified representat ion. Eq. (16) defines q We are interested in large values of 32, bu t the 32 --~ oo l imit does not alter our results at all, so we shall omit ir. We ment ion t h a t this is not inconsistent, since all physical measurements are confined to a finite region of space. But , on the other hand, the 32 --~ oo limit m a y lead to superfluous complications concerning inequivalent representat ions [6].

w 4. Renormalized solution

We approximate the solution of the system (14)--(15) by the ground state eigenfunctions of one dimensional harmonic oscillators for ~o 0 and by those of two dimensional harmonic oscillators for ~ok. The corresponding frequencies ate co(£ ) ~ m (t) and co~ 0, where t refers to the type of the solution XA or XB aceording to Fig. 1. (In what follows t means " A or B ' . )

Thus

<~U01x01~0> = x t ,

1 (17) (~0[x~0]T0> = Xl -~ 2 Co0 (t) '

1 <Tklz~fTk> =

COk (t)

So we obtain the following relations as conditions for the self-consistency of our approximate solution:

1 V(x~ = 2 (c~176 (x~ -- xt)2 + terms of higher order,

Vk(zk) = ~ (c%(t))2z~, -4- terms of higher order.

Acta Physir Academiae Scientiarum Hungari•ae 25, 1968

222 F. MEZEI

Using eqs. (14), (15) and (17), we have:

I Xx 12__2~2# xt I__/.2 = O)o(t) 2 q- k ~ ,

(18)

and xt is subjected to the condit ion V(xt) = ext remum, i.e.

where

f x , l _ u ~ x, _ ~ 2 = 0 ,

/,2 = / , 2 _ 3 202 Q,, v= = a2o " O,,

(19)

(20)

1 1 Q,

The system of equations (18)--(20) does not have a finite solution for non-vanishing 28, u~, since Q is a quadrat ical ly divergent function of K:

•,(K) = - - 1 ~ . 1 1 ~'~/2 f : fo~ k2sinv ~ ~-. ~ dcp dv ~ dk /2 tkl< K eo~) (2 ag) a a -~12 V~ + my

~(~. 2 ~ ~ ~/ K ~ + m ~ _ .~r ~n K + VK--~m~ ( 2 7 - ) ~ mi

So we are forced to introduce the renormalized parameters which m a y be chosen to be real finite numbers :

22 = lim 2z2(K) In K , K--+~

~ 2 = lira (/,2(K) 3 2£ 2 ) K-~- 2 (2 ~)2 '

K 2 u2= l im U2o(K )

K-~- 2 (2 ~r) e V [ ¡

Thus eqs. (18)--(20) take the form for the interest ing limit K - + oo

322~~ -- #2 = m~,

22 ~t a _ / , 2 ~t -- ~2 = 0 ,

3 22 /,2 = ~2 + _ _ m i ;

2 (2 ~r) 2

(21)

.Acta Physica Academiae Scientiarum Hungaricae 25, 1968

~NONLINEAR MODEL IN QUANTUM FIELD THEORY 322

w h e r e

We are in te res ted in the solution mt of this sys tem. The dependenee of mt on a 2,52, ~2, i.e. on the bare p a r a m e t e r s a~(K), 2 2 ~o(K),t*o(K) is shown in Fig. 2. Ÿ 2 and ~2 a te a lways t a k e n as pos i t ive (in the case of the l a t t e r the

75j ~

I I I | l I I

"c~2 0,5 0,5 ~2 :~ ~~ I-~~ Fig. 2

nega t ive sign corresponds only to the t r a n s f o r m a t i o n 9 -+ --9)" On the other hand , for bo th posi t ive and nega t ive values of ~2 we can get r ight solutions, which we have on the r ight and left h a n d sides of the Figure , respect ive ly . "A'" and "B'" refer to the t y p e of the solut ion as denoted in Fig. 1. We could t ake into account the ignored t e rms of the SchrSdinger eqs. (14), (15) b y more accura te calcula t ion of the eigenfunct ions. I t is easy to see, however , t h a t these correct ions do not inf luence eqs. (18)--(20) at all, even in the case of f inite I21. This means t h a t our r enormal ized solution to the va r i a t iona l p rob lem is exact .

1 E.g. To first order in the perturbation the corrections due to the terms 3~~ z~ /8 ate

The contribution of these to Qt is:

o(~~(K) f~o k2dk ) _ ~2o(K)o(mt).

Because of Ÿ ~ 0 for K --~ o0 this equals 0. This holds for higher order corrections too.

Acta Physiea Academiae Scientiarum Hungaricae 25, 1968

224 F. MEZEI

w 5. M a s s s p e c t r u m

Let us consider the wave funct ions

~,,0;...;,~.~~;... = ~.0 (~0 - ~,) H ~~~'~~(~~, Q~), 0%0

(22)

where ~~o is the energy eigenfunct ion of the one dimensional harmonic oscil- la tor belonging to the eigenvalue to(‰ ) (n o + 1/2), the ~v~ k'mk ate eigenfunctions of two dimensional harmonic oscillators belonging to the energy eigenvalues to (0 (nk ~- 1) and the 8IOOk eigenvalue mk (mk = --nk, - -nk A - 2 . . . nk -- 2, nk). Let the connect ion between the to's be given b y to(0 : Vto(ot)2 +k2.

Consider those ~~0;,k,mk;... funct ions for which only a f ini te n u m b e r of n o and nk are different f rom 0. These functions form a basis f o r a (separable) Hi lber t space ~~/~~r), and sinee 2~(K)--~ 0 ir K--~ ~ , the Q value for eaeh ~o "~ of this set is exae t ly the same. Consequent ly , we obtain f rom eqs. (21) the same to(0 = mt for all these functions. Thus we have a sys tem of exact solutions of the var ia t iona l equations, for which

and

where

<~vT~ IHI W'~ ";nk'm~;'"> = ~(‰ ) no + ~:tok') n k -4- E(o ~ , k > 0

2 o -]- ~ to(t)_ ~. , -4- k>o 4/2 >o

+~[ F,~ol xi i~ x ~ l x ' 13 ~~o I x, 141 - 2 I - V ~ - ) - T t v ~ j + T t - ~ j ]

is the inf ini te zero point energy. So we have obta ined in our approx imat ion the relat ivist ic mass-momen-

t u m spec t rum of free particles of mass mt de termined b y eqs. (21). We canno t claim, however , t ha t we have found the eigenstates of f / , of, at least, t h a t our represen ta t ion defined by/~k, ~~'s aet ing on our H~ r) space is a proper one. Al though

<WTo;...;nk,mk t f / [ ~~o;...;nk.m~;...> = ~ n , , , ; . . . ; ~ ,~, , ,~ , ~mk,mk; . . .

holds for the basis functions (in the limit K --~ ~ b y definition), in general < T I H I ~> does not exist for eve ry T , q~ E ~~r).2

2 E.g.: lira [I Ii (K),po;...;o,o;... II 2 ~ < I-I,fo;...o,o;... IHI,po;...;o,o;... > = r

Acta Physica Academiae Scientiarum Hungaricae 25, 1968

NO1NLII'qEAR MODEL IN QUANTUM FIELD THEORY 225

w 6. Conelusions

To the ex ten t de te rmined b y the fo rm (11) of the tr ial funet ions we have found th~ exac t solut ions (22) of the va r i a t iona l equat ions (13), which in tu rn define a r ep resen ta t ion in the Hi lber t space ~~r) . 3 We have seen t h a t given the bare p a r a m e t e r s ~~,#2, ~~ we m a y get 3, 2, 1 or 0 solutions (see Fig. 2), con-

t r a r y to the classical case of 2 of 1 solutions. Our different solutions correspond to different inequ iva len t represen ta t ions of the field opera tors ~ and rr, of course [7]. In the r ep resen ta t ions of the t y p e A or B, <~0[~0[~v> equals XA- -~ - - ~ or XB--~ + ~ , respect ively . B y our app rox ima t ion , which is not a " t o o good" one, as men t ioned in the previous section, we h a v e ob ta ined par t ic les of different masses described b y the same field equat ion. However , these different par t ic les belong to different inequiva len t represen ta t ions . So, in the f r a m e w o r k of the usual q u a n t u m t h e o r y t h e y cannot exist s i m u h a n - eously: the model has no solution conta in ing exci ta t ions cor responding to different solutions of eqs. (21). Ah in t e rp re t a t i on which would give physiea l r ea l i ty to more t h a n one exc i ta t ion descr ibed b y different solut ions of our single field equa t ion lies outside the r ea lm of the present fo rm of q u a n t u m theory .

I t is wor thwhi le ment ion ing t h a t the self-consistent m e t h o d of KAME- rUCHI and UMEZAVA [8], based on the Bogo l jubov t r an s fo rma t ion also leads to eqs. (21), while t h e non-sel f -consis tent a p p r o x i m a t i o n of GOnDSTO~E [3] does not g i re all t e rms in these equat ions . These facts emphas ize the impor t ance of self-consistency.

The au tho r is v e r y much indeb ted to Prof . G. MARX, who sugges ted this work and suppor t ed bis f irst steps. Gra te fu l t hanks are due to Dr . G. KUTI for his cont inuous he lp dur ing the whole work , and to B. PXzMŸ for his k ind help wi th c o m p u t e r calculat ions.

REFERENCES

1. II. P. D• and W. IIXlSE~B~.aG, Z. Naturf., 16a, 726, 1961. 2. E. M. H~.NL~.Y and W. THX~RItr Elementary Quantum Field Theory, New York, 1962. 3. I. GOT.DSTONE, Nuovo Cimento, 19, 154, 1961; G. M~_RX, Acta Phys. Hung., 14, 27, 1962;

G. KuTI and G. MARx, Acta Phys. Hung., 17, 125, 1964; G. M~-Rx and G. KUTI, preprint Budapest, 1964; G. KUTZ and G. MAax, Acta Phys. IIung., 19, 67, 1965.

4, J. v. NEUMANN, Math. Ann., 104, 570, 1931. 5. L. I. SCHIVV, Phys. Rey., 130, 458, 1963. 6. R. tt~_AG and D. KASTLEI~, J. of Math. Phys., 5~ 848, 1964. 7. R. I-IAxG, Brandeis University Summer Institute in Theoretieal Physies, 353, 1960. 8. S. KAMEFUCHI and H. UMEZAVA, NUOVO Cimento, 31, 429, 1964.

a Ir is easy to sbow that our so]ution to the variational problem is exact only within the separable Hilbert space Jf(~).

Acta Physica Academiae Scientiarum Hungaricae 25, 1968

226 F. MEZEI

HEFIHHElglHAS:I MO,!~Eflb B KBAHTOBOITI T E O PH H FIOSI~

O. ME3EH

P e 3 t o M e

B pa£ HcKaac~ MaCC0BbIH cneKTp HeJIHHe~… peanbnoro cKaaapaoro 1JoJ~~a BapHa- I~HOHHbIM MeTOROM PrlTtta. B cJwqae COOTBeTCTBy~OtlIe~I peH0pMa:m3a~HH Ha.~~eHbI KOHeqHble 3HaqeHHa Macc~i noKo~, paBnLm 1, 2 Hall 3. Pa3aHnHbm THnLI ~aHnbIX ~oa6yn<~eHH~ npnHa~- ~e~aT K pa3J~HqHb~M HeaKBHBaJIeHTHblM IIpe~cTaB21eHH~IM onepaTopoB rioJ~~.

Acta Physica Acadamiae Scientiarum Hungaricae 25, 1968