The N-quantum approximation in static models

ANNALS

OF PHYSICS:

The

39,

(1966)

k%-‘@7

N-Quantum

Approximation ANTONIO

Rutgers,

The

State

University,

in Static

Models*

YAGNAMENTA~

New

Brunswick,

New

Jersey

The N-quantum approximation (N&A) is a method designed to construct approximate operator solutions to the field equations. We show in detail how this method works and can be made compatible with the renormalization program in the Lee model and for the scalar field interacting with a fixed fermion source. The terminated in-field expansions can still be applied successfully in the presence of bound states, for which one will introduce new infields, or if one deals with resonances. This is discussed on the examples of the V-e-bound state and the unstable V-particle. I. INTRODUCTION

The large numerical value of the coupling constants in strong interaction physics prohibits the application of perturbation theory. If one wants to derive predictions from the field equations which can be compared with experiment one will have to use some nonperturbative method of solution. There are several such attempts in the literature. The early Tamm-Dancoff method1 used bare states as a base in Fock space.It has been improved by the introduction of eigenstates of the total Hamiltonian (2). Other methods which rest on rearrangements of the Born serieshave been used by Jost and Pais (3). While there is no proof of convergence over the entire range of application it has led to quite promising approaches like Weinberg’s (4) quasiparticle method and the discussion of the p-meson given by Alexanian and Wellner (5). The Tamm-Dancoff methods use expansions in terms of states; therefore they separate positive and negative frequencies of the field operators. This makes these methods noncovariant and has limited their applicability to low energy pionnucleon scattering. Greenberg (6) proposed to avoid these difficulties by the formulation of a method which would approximate the Heisenberg fields directly * Based in part on a thesis submitted at the University of Maryland in partial fulfillment of the Ph.D. requirements. For more details we refer to U. of Maryland Tech. Report No. 494, 1965. t Supported in part by the United States Air Force under Contract AFSR 500-64 and by the National Science Foundation. I The original references are Tamm and Dancoff (1). For a discussion of difficulties see (la). 453

454

PAGNAMENTA

in terms of in-fields. Because the in-fields create eigenstates of the total Hamiltonian this method has properties similar to those of the new Tamm-Dancoff method. In each order, for instance, it sums up infinite subsets of Feynman graphs and this lets us hope that it may have a chance to work for strong interactions. It has the advantage of being manifestly covariant which is not the case for Tamm-Dancoff methods. This will encourage the application to relativistic theories. Further it deals with connected graphs only which is appreciated if one tries to solve the integral equations numerically. The main problem with any nonperturbative approximation scheme is to make it compatible with the renormalization program. Greenberg (6) has shown, by comparison with perturbation theory, that one can obtain an integral equation for a finite vertex function in first order of the N-quantum approximation (N&A) for the As-theory. The process of termination of the series is not simple and therefore it may be worthwhile to look at the N&A in some simple models. We show in detail how the NQA works in some static models, specifically in the Lee model (2”) and for the neutral scalar field interacting with a fixed fermion source. In these models we can obtain exact results2 and therefore establish some working experience with the new approach. This is the main purpose of this work. We give a general outline of the idea of the N&A in Section II. In Section III, A we give the equations of motion for the Heisenberg fields in the Lee model. The technique of the N&A is then explained in detail on the example of the NBsector in Section III, B. Here we point out the difficulties with a naive approach and show that Greenberg’s reformulation leads to the exact and therefore finite answers. This method, formulated in graphs, then gives the complete solution of the Vt9/N20 sector in Section III, C (8). The number of in-fields one has to introduce generally depends on the number of stable objects, bound states, and elementary particles present in the theory. The necessary modifications when there existsa VO/N20 bound state are given in Section IV. In Section V we discuss both two and three particle sectors of the Lee model if the V-particle is unstable. Since the solutions of the relevant integral equations have been published earlier (9, 10) and discussed in several places,3 (12) we will not repeat them here. In the Lee model the N&A leads in low orders to exact results for the V/NO and the Vt?/N28 sectors. We therefore add a discussion of the scalar field interacting with a fixed fermion source in Section VI. In this model we can find the exact solutions for the complete Heisenberg fields and show that the N&A gives finite approximations. * Historically it is this method which has actually led to the first complete off-shell soluof t,he V-O sector (8). 3 An integral equation of the V-&type, including the displacement in the denominator, has been solved by a deductive method by Litvincuk (11). Later different methods have been presented by Howard and others (11~~). For a discussion of uniqueness and solution of the homogeneous equation see ref. 12.

tion

N-QUANTUM

II.

APPROXIMATION

THE

N-QUANTUM

IN

STATIC

MODELS

455

APPROXIMATION

Firstly we outline the basic idea of the N-quantum approximation method (NQA). In this we follow Greenberg’s work (6) to which we refer for det’ails. The central mathematical tool of the N&A is the expansion of the Heisenberg field in a complete set of in-fields. This representation, often called Haag expansion (IS), holds if the axiom of asymptotic irreducibility is assumed. For a single neutral and scalar Heisenberg field A (k) which has only one single asymptotic quantum (only one in-field Ai,(k)G,(k)) and no bound states we can write the relativistic expansion

We have given the expansion in momentum space for convenience only. k is a four momentum vector and 6,(k) = 6(k2 - nl”). The double dots on the right indicate that the products of in-fields are normal ordered. The factor l/n! has been inserted because the Ai,( Ic) have Bose symmetry. The unknowns in t’his expansion are the coefficient functions f(“)(kl , . . . , ,&,). These are distributions with support in lcf = mi2 due to the &functions. They are closely related to the so-called r-functions of GLZ (14). Note that the k in the Heisenberg field is not on the mass shell. The principle of the N&A consist,s in terminating the series (1) which is written for t,he renormalized Heisenberg field at some finite order. This ansatz is then substituted into the equations of motion. The exact process of termination has to be consistent with the renormalization program and is in general not simple. It will, for some stat,ic models, be described later. For a relativistic example we refer again to ref. 6. Different normal ordered products of in-fields are independent. Therefore this process will give C-number equations for the expansion coefficients. The solution of these equations will give approximate S(n)‘s. To obtain these coefficients is the main objective of the N&A and the hope is Ohat already low order approximations will lead to useful information on scattering and production amplitudes. We will therefore give here some of the properties of t)hose coefficients and show how they are related to physically interesting yuan tities. The properties of the approximate f (n)‘~ have to be derived from a finite part of the expansion ( 1). Most of these properties are related to special properties of the field, like neutrality and Bose symmetry, and have to be derived for each case. However each term in (1) is manifestly covnriant. Therefore the approxi-

456

PAGNAMENTA

mate f (“j’s are Lorentz

scalars and satisfy

f’“‘(Akl

, - . * ,A/?,) = p)(h)

* * * , k?J.

It may be interesting to note at this point that the Hamiltonian expressed in terms of the in-fields takes on the free field functional form. This follows from the fact that the Hamiltonian is the time translation operator of the theory and is, in general, only true for the exact solution. The physical interpretation of the f”’ is seen most easily on a few examples. Computing (0 ) A(k) ) k 1k 2).m with the help of Eq. 1 and the LSZ (15) reduction formula one sees (6) that f’” (kr , kz) represents the three pieces of the vertex function with two legs (those corresponding to in-fields) on the mass shell and one leg (the one corresponding to the Heisenberg field) off the mass shell. In a similar way one finds that f’“’ gives the connected part only of the elastic scattering amplitude, f’” the connected part only of a production amplitude. We shall come back to these interpretations when we discuss the amplitudes of the Lee model. There has been some independent but related work in the literature. We think of the Symanzik lectures (16) on Greens functions and of Taylors extensive discussion (17) of the field equations. Both these authors work with functions similar to our fen)’ s but which have all their legs off the mass shell. Symanzik mainly discusses the analytic properties of the Greens functions. Taylor gives an interesting analysis of a variety of approximation methods. To our knowledge they do not derive numerical results. At present the only field theoretic discussion in strong interaction physics which has led to good numbers that could be compared with experiment is the one of Alexanian and W ellner (18) of the J = T = 1 sector of the +4-theory. This has now been extended to the scalar dipion (19). The NQA has already been applied to the deuteron by Greenberg and Genolio (29), to relativistic recoil corrections in the hydrogen atom by Greenberg and Woods (20) and is at present being applied to bound states of quarks to form mesons and baryons by Greenberg (21) . III.

THE

LEE

MODEL

In this section we illustrate the N-quantum approximation method on the Lee model (7). This model is nonrelativistic and therefore we shall clearly not get anything from covariance. However the simplicity of the model and the fact that exact solutions are known in the lowest sectors at least makes it tempting to use it as a testing ground for new approximation schemes. After a short description of the model we shall explain and develop the intended procedure in the N§or. The method then leads to the complete solution of the Vo/N20 sector. Because one can apply the reduction formula the in-field expansion yields the connected parts of the amplitudes in, it seems to us, a more

N-QUANTUM

APPROXIMATION

Iii'

STATIC

MODELS

4.57

direct way4s 5 than the states of the Tamm-Dancoff method do. In spite of its simplicity the model and the amplitudes derived in its framework are of some physical interest due to the fact that one can obtain exact solutions which satisfy two and three particle unitarity. A.

DESCRIPTION

OF THE

J~ODEL,

EQUATIONS

OF MOTION

If one writes out the covariant Yukawa interaction g&,5+ in terms of creation and annihilation operators one finds a sum of eight terms each describing a different process. In order to study renormalization effects T. D. Lee (1) dropped most of these terms and retaining only two of them he wrote a nonrelativistic model Hamiltonian of the form

H = Ho + HI,

(2)

where

Ho = mo V,+Vu + mN+N

+

s

w(k) ak+uk d3k

So the Lee model describes the interaction of three idealized particles called V, N, and 0. He further assumedthe V and N particles to be very heavy so that their kinetic energies are negligible and their energies are just their masses.The &meson described by the operator ak is taken to obey the relativistic energy momentum relation Wk = k2 + pcL2, where p is the massof the 0 and k its momentum. The O-particle is a boson, thus its field operator satisfies the equal time commutation relations [uk , ak!] = [@+ , akf+] = 0 and [uk, &‘+I = 6(k - k’). It is conventional to consider the V and N particles as fermions requiring their Heisenberg operators to obey the anticommutation relations

(Vu, Vu+] = (N, N+) = 1, 4 This simplification is also present in the T-function formulation of Maxon and Curtis (22). This work is presently being extended to include the V-8 sector (M. S. Maxon, private communication). 6 H. M. Fried (26) has given a functional formulation of the Lee model in which the reduction formula is also applied,

458

PAGNAMENTA

all other anticommutators being zero. V and N commute with ak . This distinction into bosons and fermions is purely academic since it has no consequences as long as the interaction gj( w ) is spin independent and as long as one only considers states with at most one heavy particle (fermion) present. This is the case in practically all discussions6 of the Lee model. The above interaction only allows the processes V % N + 8 at each vertex. This can also be expressed by the statement that the two quantities, called baryon number and charge,

are conserved. Here n; denotes the number of the particles of the ith species. The corresponding combinations of number operators B,, = V,+V,

+ N+N

(4)

Qo, = N+N - ak+ak

(5)

commute with the total Hamiltonian. To obtain finite physical quantities (in the point limit) we have to renormalize the Hamiltonian. The renormalined mass of the V, the renormalized Heisenberg field V, and coupling constant g are defined by the relations m0 = m - 6m v, ~Zg,

= 4zv = cg .

Here am, 2, and C are the renormaliza.tion factors for mass, wave function and coupling constant. The renormalized V-field satisfies (I’, v*] = l/Z. The quantities related to the N- and &fields need no renormalization in this model. The Hamiltonian (2) can now be written H = H,’ + H, Ho’ = .&YLV+V + ??lN+N + j

Uk

ak+ak

d3k

(6)

6 The only exception is the work of H. Chew, who finds that there is no essential difference in the NO-amplitude with spin and suggests from the form of the integral equations that a spin dependent interaction will give different singlet and triplet interactions for VO-scat,bering.

N-QUANTUM

HI’ = C where

APPROXIMATION

s

f(a)[V+Nu,

IN

+ a,+N+V]

STATIC

459

MODELS

d31c- BmV+V

we have set f(w)

The Schrodinger

equation

u(w) __ %tii’

___. = (&

for operators - if

(7)

O(t) in second quantized

O(t)

form is

= [H, O(t)]

(3)

and we use the above Hamiltonian to derive the equations of motion for the Heisenberg fields V(t), N(t), and ak( t). Since V and V+ are canonical we get for the V-field, for instance, the equation

We prefer to work with the Fourier transforms V(t)

= lr

e-iEtV(E)

N(t)

= 1-1 emiFtN(F)

m uk(t)= s-92 e-iGtak(G) With

and define dE dF

(9)

dG.

this we find the three field equations Z(E

- m + Gm)V(E)

(F - m)N(F)

= C 1 d3k dGf(w)uk(G)N(E

= C I’ d3k dGf(w)u,c+(G)V(F

(G - w)ul, (G) = Cf(w)

/ dEN+(E)V(E

-

G)

(10)

+ G)

(11)

+G).

(12)

This is a set of coupled nonlinear integral equations for the Heisenberg field operators. We shall now solve them to different orders using the NQA. To do this we first have to define the in-fields. We require that these satisfy the free-field commutation relations

460

PAGNAMENTA

and

all other (anti-) commutators being zero. We have assumedhere that the three in-fields Vi, , Ni, , and ain,k form an asymptotically irreducible set. In other words that each Heisenberg field has a single in-field limit lim V(t) t-r-m

-

Vi*

lim N(t) -

Nin

lim &(t) t---m

ain&

and that there exist no bound states. The single handed arrow denotes the weak limit obtained by taking matrix elements, in the senseof LSZ (15). B. THE NQA IN THE V/NO SECTOR In the lowest sector of the Lee model we will encounter only algebraic equations in the N-quantum approach and the results are well known. It is therefore here that we will give in detail the proceure of the N&A. To first order in this approach we retain only terms with at most two in-fields. We write the ansatz for the Heisenberg fields V(E)

= 6(E - m)Vin + / d%hS’(~)6(E

N(F)

= 6(F - m)Ni, + / d3khk?(w)6(F - m f ti)&,kVin

C&(G)

=

6(G

-

W)Uin,k

+

hy’(~)6(G

+

m

- m - W)Uin,kNin

-

(13)

m)NkVi,.

In writing (13) we have used the selection rules (3) to reduce further the number of terms. We can generalize the quantum numbers B and Q in an obvious way from the states to apply to the field operators. Since they can be derived from a gauge group these quantum numbers 01 a state undergo the same change if the state is acted upon by a Heisenberg field or by its in-field. Therefore we ascribe to our field operators the following pairs of quantum numbers (B, Q) : Vf( 1, 0), fl(l, l), and a’(O, - 1), the negative values to the annihilators. With these the terms that can appear in the in-field expansions (18) are greatly restricted. The selection rules (3) forbid for instance a term like UinVin from appearing in the V-series. We also have dropped all terms which would create or annihilate two fermions because we shall never consider states which contain more than one heavy par-

N-QUANTUM

APPROXIMATION

IN

STATIC

461

VIODELS

title. The three unknown coefficients h?’ , h$’ and h:“, which we have introduced, are equipped with a superscript (1). This indicates the first order approximation (in N&A) and in general we will expect these coefficients to be changed as we go to higher orders. We now substitute the ansatz (13) into the equations of motion (lo)-(12). We multiply out the products on the right hand side and renormal order the in-fields by use of their commutation relations. Then we use the fact that terms with different products of in-fields are independent. This leads to c-number equations for our unknown functions. Equation (10) for instance reads, after normal ordering and dropping terms with more than two in-fields Z61TL’1’6(E - 112)Vi, + Z(E

- I?2 + 6l72”‘)

* s d3khL1’(w)6(E - 11~- w)Uin,kNi, = C j d3h$(w)6(E - m - ti)ain,kNin

+ C j d3khg’(w)f(w)6(E From the coefficients

of Vin we can read off the equation Z&m(l) = C j d3kf(w)h:“(w).

The coefficients way

- m)Vi,.

(15)

of UL,kVi, and @nVin in Eqs. (11) and (12) give in the same hfr”(w)

= +‘Jf!

= j’&“(w). W

We can now determine the coupling constant renormalization C by use of the Watson-Lepore definition (95) of the renormalized coupling constant. Since f(w) contains the renormalized coupling constant (Eq. (7)) we find C=l by taking the residue of IN equation above

at its pole for w = 0. Hence we find from the

h,,r(w) = -jf

= h,(w). W

Substituting

(16)

this in (15) yields zarn

= -

f

&f2(w) -. W

(17)

462

PAGNAMENTA

In the last two expressions we have omitted the superscript (1) because they are actually the exact expressions which is a peculiarity of the Lee model. The naive N-quantum approach, namely substitution of N + 1 terms and keeping N + 1 terms, does however meet some difficulties with renormalization. Let us look for instance at the coefficients of oi,,hNin in Eq. (14). This gives htl’(w)

=

z

f:oL&,,. u

This result does not look very interesting if we regard it in the limit of a local theory. It is well known that in the limit of a point interaction Z6m diverges. We would then have by(u) = 0. In any realistic theory we want the retarded functions to be finite and free of renormalization constants. With the naive approach we evidently did not approximate the renormalized equations. There must be further terms, contractions from hgiher coefficients, which we have missed. We can get these terms in the following way (6). The field equations are bilinear in the operators. Therefore the right hand side will lead in Nth order to expressionscontaining products of in-fields with up to 2N + 1 factors. We can take these to get approximate expressions for the coefficients up to order 2(N + 1). This does not lead to more unknowns than equations because the approximate coefficients from N + 2 to 2(N + 1) are then given by the ones in lower order. Evidently any renormalization of these higher coefficients would have to come from even higher orders and is therefore to be dropped. Considering the conservation laws in our special case we have only to add the two terms CY;“(CO , w’)u~In,ka’m,k/N.1n6(F + w - w’ - m) c13kd3k’ and aa(l)(~, U’)NkNinUin,d(G

- CO’)d3k’

to the old N- and u-expansions respectively. All other coefficients would be zero to first order. This leads to the new equations

(Zw + ZGm)hv(w) = f(u) (0’ - ~)a,&,

+ 1 d3k~(u’)aN(co’,

LO’) = f(w)h,(w’)

(w’ - o)aa(w, w’) = f(w)hv(w’).

w)

(18) (19) (20)

The equations (15), (16) for h, , hN, and Z6m remain unchanged and are therefore not repeated. Equations (IS), ( 19), and (20) lead to finite answers, as we shall see.

N-QUANTUM

APPROXIMATION

IN

STATIC

MODELS

463

When we solve these equations we have to remember the boundary condition imposed by the in-field expansions. We want (13) to reduce for t + - Q, weakly to the in-field terms. Using the Lemma of Riemann and Lebesgue in the form iwt 27&(w) lim -K-= t-+x 6J - ir i 0, we see that we have to add +ie in all denominators of the solutions obtained from our C-number equations. We shall call these the retarded solutions. Substitution of the retarded solution of (19) into (18) gives

f “(A WZ + s [ 1 Z6m +

d3k’ u, _ w _ ic

b(w)

= f(w)

(21)

or definmg

we get

Using this in (19) and (20) gives

dw,

d

= fb)fb’)

%(@, w’) = f&Md

(J _ w : iE)H(w,) (w, _ ,l+

ie)H(w,) *

We have omitted the superscript (1) because an inspection shows that there are no more terms coming from higher orders and these are actually the exact solutions. Thus we see that the improved first order N&A leads to exact results in the Lee model. In the equations of motion we have used only one ordering of the Heisenberg fields. For the exact solutions these fields commute or anticommute at equal times. Therefore for an approximation method one may hope to get a better approximation by using a symmetric combination, either 35 the commutator or anticommutator on the right of (lo)-( 12). I n a consistent approximation scheme the commutator of two bose fields or of a bose and a fermi field and the anticommutator of two fermi fields at equal times should be zero to the accuracy of the approximation. Then one might use these to derive identities which can be useful in the process of solution. Here we have exact solutions and we have observed that the order Nu in Eq. (14) leads to simpler equations for the coefficients than in order UN. By taking matrix elements of equal time commuta-

464

PAGNAMENTA

tom of the Heisenberg fields one derives easily relations like h,(w) = h,(w) and &(W’, w) = cYN(W,w’). These relations become more complicated in other cases. We were able to verify them for all coefficients in the V and VI? sectors except for the bound state functions and have found that they hold exactly as one might expect. We will now find the first order approximation to the matrix element for elastic N&scattering. At the same time we observe the close relationship between the expansion coefficients and a physically interesting quantity, the T-matrix. The S-matrix is defined by SNO = ( out, N&l which, by use of the definition f&e =

NOk , in )

of the asymptotic

(25)

O-state, can be written

,“mm( out, N / ukl( t) 1N& , in )ei”‘t.

For a stable particle, a one particle state is the same for out and in. For ak(t) we insert its Fourier transform (9) to get SNO = lim dG(Ni, t-+m I

1CZAR 1Ninain,k)e-i(G-““)t*

Here we have assumed that the in-state is just given by the product of Ni, and oi,. We see that under the integral we have just the matrix element of ak(G) between in-fields. For ak(G) we insert the in-field expansion. Then the matrix element becomes 6(k - k')6(G

- co') + a&o', w)a(G - u).

The b-functions allow us to do the integral over dG and we find SNB - 6(k - k') = limay,(w’, w)e-i’-““t. t++m Here we insert our expression (24) for CLI,(a’, U) to find sN6 _ 6(k _ k’) = -f(s{(y) w

ei(o’--o)t ; . lirn t-i-m w - wJ + ze

The time limit can be evaluated by using the Riemann-Lebesque ing the T-matrix by Sm - 6[k - k') = 2xiS(w - w’)Tm\w)

Lemma.

Defin-

N-QUANTUM

we can write

APPROXIMATION

IN

STATIC

MODELS

465

for it TN@(w) = lim (w’ - w)(Y=(w’, w’-w

w),

(26)

which gives t’he exact relationship between the coefficient CX~of the in-field expansion and the elastic T-matrix. This relation can be generalized to hold for production amplitudes and for the connected parts of the three particle amplitudes. Using it in (24) leads to TN(W)

=

-f$

w

where H(w), at present, is given by (22). Evidently the term Z6n~ in H(w) 110 longer leads t,o a divergence since for w = 0 it is cancelled by the integral. We see that H(w) has a zero at w = 0. Therefore TNo(u) has a pole there and since we already know that f(w) contains renormalized coupling constant we can again use the Watson-Lepore definition (25) to determine Z. Evidently we want the residue of the pole of H-‘c w) at w = 0 to be + 1. This is equivalent to

f

(27)

H(w) [wql = 1

which leads, by use of (23)) to

The two extra terms besides the integral in (22) are just the two subtractions necessary to make H(w) finite, even in the point limit. Defining Im G(w) and using (17) to find

= 4n2f2(w)k(w)

and (28) we can simply H(w)

make common

= wG(w)

(29) denominators

in (21) (30)

where (31) or (32) In (31) we have taken the point of subtraction at w = 0, in (32) at w = CC. In the point limit UC o) = 1 and Im G(o) = g2/4a. k/w, hence Im G(W) is of

466

PAGNAMENTA O*

FIG.

k, in

1. The coefficient hv can be considered

Nin(m)

aa vertex function

with the V off the mass

shell.

order one for large o and we see that the integral in (21) is linearly, the one in (32) logarithmically divergent while the integral in (31) and hence G(w) is finite. From (31) and (32) we see that

G(O) = I, G(m) = 2,

(33)

which we shall need later. The function G(w) has been discussed by Kallen and Pauli (26). It is easy to show that for a nonvanishing cutoff function it can have at most a zero on the negative real axis. This zero is excluded by requiring 0 < 2 < 1, the no-ghost assumption. One can show that this first order N-quantum approximation diagonalizes the Hamiltonian to the extent that any 3-particle interaction (Yukawa interaction) is removed. We can now set up the entire N&A-approach by use of diagrams. This becomes more useful as the complexity of the equations increases. We have seen that our expansion coefficients are given by the connected parts of matrix elements of the Heisenberg field between in-fields. Let us denote this by a square box where single lines denote in-fields, doubled lines the Heisenberg field. For instance

h,(W) = (0 1V 1Ninain) is represented by the diagram of Fig. 1. The other coefficients which we have introduced are given in a similar way (Fig. 2). Each diagram is associated with a a-function expressing over-all energy conservation. The hv for instance has a 6(E - m - w). We can now write Eq. (15) as in Fig. 3. At the vertex on the right there is a coupling constant (here f( w’) ) as well as an energy a-function. We shall interpret those double lines not ending in a black box as renormalized propagators following the scheme: v:

= (E - m)-’

N:

= (F - m)-’

Cl: -====-

= (G - u)-l.

N-QUANTUM

FIG.

APPROXIMATION

2. Graphs

for

the

IN

first

STATIC

order

YODELS

467

coefficients

Z8mx

FIG.

3. The

equation

for

the mass

renormalization

This can be seen by comparison of Eq. (15) with Fig. 3 or Eq. (19) with Fig. 6 below. All the diagrams are to be read from bottom up to obtain the retarded propagators by adding +ie, Ingoing single lines denote in-field creation operators, out going single lines in-field annihilation operators. The other equations can now be written immediately, Figs. 4 and 5. Here we have left the factor 2-l to save repetition of the diagrams. Further we find the relation of Fig. 6 for (YN and ala. Clearly by substituting the diagrams of Fig. 6 into Fig. 5 we find Eq. (21) in terms of graphs. The diagrams have the advantage that they can be written down by inspection using the selection rules (3) and that they then automatically incorporate energy (and momentum) conservation. From Eq. (26) we see that we get the N&litude by restricting the external Heisenberg leg in the graph for CY~(or CX~) onto the mass shell. This holds in general for all the amplitudes we shall discuss later on. c.

THE

~B/N2&SECTOR

We have described the N-quantum approximation and shown that it works in the No-sector of the Lee model. While in the N§or all the results are well known, in the VB/NSO-sector the common off-shell approaches had led to an integral equation which had not been solved at the time this investigation was carried out. However the elastic V&scattering amplitude and the Vt9 -+ N2o production amplitude had been found by on-shell methods (a?‘). These, however, do not determine the Heisenberg fields. In the meantime we have given the entire section of the V§or using the Tamm-Dancoff method elsewhere (10) and the solution of the integral equations which are central to this problem has been reported earlier (8, 9) and then simplified by several authors (11, 11~). The special interest whirh the lie/N20 sector of the Lee model deserves comes

468

PAGNAMENTA

FIQ. 4. Equation

FIG.

FIG.

16 for ha . The same graphs give the equation

5. The full

6. The equation

equation,

coupling

h,

for a~ . The same equation

for h,

and a.

holds for a,

the fact that it represents an exactly soluble three body problem. The classicmethod to deal with such problems is the resonance approximation (isobar models). While this approximation may be good in some cases,it cannot always be used mainly due to the fact that it neglects the intrinsic three body interactions. The three particle sector of the Lee model approximates nature in that the heavy particle is static, but it preserves Bose symmetry in the N2e-states and since the amplitudes are exact consequencesof a hermitean Hamiltonian they satisfy unitarity. Here we shall solve the V§or by use of the N&A. Again we shall find exact results. In the in-field expansions we have to go second order [using the improved approach), which means that we have to keep terms with up to six infields. The conservation laws imply that terms with six in-fields are zero in this order. We find, written symbolically:

from

N-QUANTUM

APPROXIMATION

IN

STATIC

469

MODELS

In this sector we will have three amplitudes, namely, the one for elastic I’@scattering, 176 ---$ N20 production, and elastic N&9-amplitude. Doing a reduction analogous t,o the one in the last section for the N&litude (relations (X?) to (37) ) we can connect these amplitudes to the above expansion coefficients. Depending on which particle we contract we can relate

Ttrb to 0~~or py (i3.5)

PV8,N28to Pv or PN or ta TM The exact relations which

t0

Or Ka .

-fN

we need and which

are derived in analogy to (25),

(26) are Sve = -(v&t = s(k

1 v&J+ - k’)

- d)Ty&),

+ 2ai6(w

where Tv8(w) We define production

= lim (0’ o’-0

amplitude

w)av(w',

w).

(36)

by

and find

P

VB+N20(~1,

w2)

=

lim

(w

O-W,+WZ

-

ul

-

wdh((Jl

, LJ2,

(37)

W)-

In the A%?&litude we are not interested in the trivial contributions coming from two body scatterings. For the connected part of the amplitude we find T.v2dw1,

wz ,

w3)

= lim (w3 w&+w,+“?03

+

u4

-

Wl

-

(JZhN(u1,

WZ , W3,

W4).

(38)

All Ohat remains is to find the three relevant coefficients (Ye, PN , and yN. These cont’ain the dynamics of the system. We could easily derive equations for them by substitution of the expansions (34) into the equations of motion. We prefer to do this by use of the graphs and then write the analytic form of the equations. We can consider LYEas a V&litude with one leg off the mass shell and pN as a production amplitude with one leg off the mass shell. Then the relevant diagrams are given in Fig. 7. The factors of 2, x come from symmetrization of the identical bosons. Substitution of the second line for BN into the first equation lea,ds to an equation for LYV, Fig. 8.

470

PAGNAMENTA

FIG. 7. The coefficient CXYis the V&elastic amplitude with the V-leg off-shell, PN the VB/N2&production amplitude with the N-leg off-shell. The equations are coupled as shown.

FIG.

8. The separated

integral

equation

for the off-shell

V&litude

(YY

In this form the equation can easily be translated into analytic form. One observes that all the diagrams in the brackets have in common the incoming (renormalized) V-propagator which therefore can be taken as a common factor to the left. The second diagram on the right further factors into a mass renormalization time CX~while the closed loop involving the CC”-boxin the last diagram indicates that we have obtained an integral equation. Following (21) the factors of CY~can be combined into H(wz - ~1) and we write the integral equation

This is the same equation as has been obtained via the Tamm-Dancoff method by K&11&i and Pauli (26). Its solution is standard by now. If we subject G(w) to the no ghost condition then for w2in the physical region (02 > Jo) it has the unique solution (11, 12)

a”(wl, w2) =

f(wMwd wl(wl

-

w-z +

1

4

[ G(w)

2Wl + 1 _ H(w2)A(w2) B(w27 b-5- WI)] C40)

where we have defined A(w) = ; la fqwdi

w’) Im &

(41)

and 1 1 * dW’ = (42) s 0’) ImG(w’)* a P (w’ - w - ie)H(wz , 0) = A( wq) and therefore reduction to the mass shell by

B(w*,w) We note that

B(w2

N-QUANTUM

(36)

APPROXIMATION

IN

STATIC

471

MODELS

gives TV@(W) = f”b) ~. H(w)

1 + ff(w)A(w) 1 - H(w)A(w)

which agrees with the expression found by Amado (37) by dispersion theoretic methods. To find the production amplitude we write out the graphical equation for PNY in Fig. 7. (w -

Wl -

WZ)PN(Wl,

u2,

w)

=

Since we know CX~, (43) g ives us and (37) one sees directly P “0Gv20(W1)

w2)

=

45

$$~(d-h(w2, ON

u)

+

f(w2)aV(ti,

w)].

(43)

. There is no need to write it out. From (43)

lim O+W,+Wr

If(wdadw2,

w)

+

.Hw&v(w~,

w)],

or, substituting (40) and replacing the outgoing meson energy w by the sum w1 + w2 of the incoming energies, the right hand side becomes

1

+

H(WI

+

wz)A(w

+

~2)

Wl (J2

-

G(wl +

02)

Wl ww1

+

4

(WI + w.~)A(wI

+ wd

w2)

+

-l w2,

+

wzB(w1

w2,

Wld].

A short calculation (10) shows that one can combine the terms in the brackets to give G-‘( wl)G-‘( ~2). Then the production amplitude which clearly is symmetric in the energies of the two outgoing mesons reads

f (df (w2) f-‘(wl,

wp)

=

jg(wI)~(w2)

f(w1 + w2> ’ 1 -

H(wl

+

wz)A(w

+

~2)

’

(44)

Equation (38) shows that we can find the connected part of the N2&litude from the coefficient TN. The graphical equation for yN which is the N2& amplitude with the N-leg off the mass shell can be written as in Fig. 9, or in the analytic form, (w3

+

w4

-

wl

-

WZ)YN(Wl,

(J2,

w3,

w4) (45)

=

%[f(w1)Pv(w2,

w3,wd

+

f(W2)P”(Wl,

w3,w4)1,

which shows that it is connected to a different off shell extrapolation pV of the production amplitude. The equation for pV is found to be as in Fig. 10. Here we substitute TN from (58’) and obtain the integral equation, Fig. 11. The coefficient (YN is known, (24). The other terms can be regrouped as earlier and the integral equation for pV can be written

472

PAGNAMENTA

with the N-leg off-shell (yN) is coupled to the Vc9/N20pro-

FIG. 9. The N2&litude duction amplitude as shown.

FIG.

10. The graphical

equation

FIG. 11. The separated integral equation off-shell two body amplitude aN is known.

coupling

pv to 7~

for the off-shell production

amplitude.

The

(46)

1 + -f(w3)dw1, 2

w2) + / d3KfG)f(W1)

w1P+‘(f-“‘:$

ie’

Here we set w2+ w3 = w. and substitute (24) for CW(CO).To get rid of uninteresting factors let us define c$(w~; wz, w3) by rBv(w1,

w-2, u3 )

=

af(w1)I(wl)f(w3)~(wl)

H(w2;H(w3).

(47)

Here again we have suppressedthe variables ~2, w3 which enter C$only as parameters. Equation (46) leads for $(wl) to the integral equation H(wo - WI)+(W) =

H(w2) 01

-

w3 -

+ ie

Hb3) Wl

-

02

-

ie

-- 1 mIm H(co’)~(u’) dw’ lr sP 0’ - wo + w - ic .

(48)

N-QUANTUM

APPROXIMATION

IN

STATIC

473

MODELS

In writing the integral we have again assumed +(a’) to be independent of the angles and have used the definition (29) for Im H(W) . This integral equation has a certain similarity to the V&equation (39) and we could apply the same methods of solution. The homogeneous equations are the same. The inhomogeneous term in (48) introduces a somewhat more complicat’ed structure. Having solved (39) we can easily guess a very good ansatz for t#his equation. From (48) we see that $( ~1) has a pole each at w2 and w3 with residue +l. The factor (w, - wl) in H introduces an over-all pole. We write in analogy to (40) dw1)

=

l WI + ir

wg -

(49)

uz -Ii w -

603 -

+

ie

w1

_

"w"

_

The constant C is found by direct substitution equation (48). We find by a calculation which

i.

+

cwwo,

wo

-

Wl,].

of this ansatz into the integral is analogous to earlier ones

2H(wo) ’ = 1 - H(w,)A(w,,)

’

Clearly to this equation everything said earlier for (39) applies (12) and the solution (49), (50) is unique in the physical region where w3 > 2~. The mass shell limit lim

(wo

-

Wl)P,(~l

, w2 ) w3)

=

Pm+m(wl

) w2)

opwg=w*+w~

gives back the production amplitude and may serve as a check on the calculation. Substituting everything in (45) and taking the mass shell limit, (38) gives the interesting part of the elastic N2&litude. TN2S(W,

(J2,

w3)

=

%y4+w,~~-wI

This becomes using (47),

v(~lMw*,

w3,

w4)

+

f(w2)Pdw,

03,

41.

(49)

;f(w1)S(W2f(W3f(w4) Ir(,3;H(w4) [w4 +w; - wl) .(-+--03

WI

-

w4

Lo4 + Wl

-

w3

CB(

wo,

wo

-

-

w4

01))

02

+

-

w3

w4

+

i3

_

+CB(wo,wo-wr?)

w2

)I .

474

PAGNAMENTA

Here we let w4 + w1 + w2 - w3, w. = w3 + w4 = 01 + w2 . Then the fractions simply combine to give 2wo/wlw2 . Taking C out of the square brackets these become Tiv28(W1,02;w3,wr)

=- 1 S(Wl)f(w.L) 2

H(wJH(w2)

’ 1-

H(m)

f (W3)f

H(wo)A(wo)

H(ws)H(wd

(w4)

.

(51)

Here w4 = wl + w2 - w3 is understood. We have retained w4 because this denotes the energy of the second outgoing meson and in this form the complete Bose symmetry of the above amplitude is manifest. The last factor represents two body interactions in the final state. Due to time reversal invariance we have the same interactions also between the incoming particles, therefore the same factor in front. The most essential part of this three particle amplitude is the factor in the middle. A discussion of its denominator D(w)

= 1 - H(w)A(w)

(52)

shows that it has just the analytic properties one expects of a S-particle amplitude, essentially elastic and inelastic cuts. With this we have obtained all the amplitudes in this sector of the Lee model under the assumption that there exists no V&bound state. We shall see in the next section that for a large enough coupling constant a V&bound state exists. This bound state will however not change the expressions for the amplitudes of this sector because due to energy conservation it will not be accessible from either the VO- or N20-channel. IV.

THE

Be/N28 BOUND

STATE

In the presence of a bound state the original set of in-fields is no longer irreducible. It can be completed by adding a new in-field representing the bound state. It has been shown earlier that there can arise one VO/N20 bound state in the Lee model for strong enough coupling (28, 29, 12). The precise condition for its existence is that the denominator function D(w) (Eq. (52) ) has a zero on the real axis below the elastic threshold. D(a)

= 0,

WB-=cCL.

(53)

It may be noted that this denominator appears in all three amplitudes of the Ve/N2§or and is already present as (exact) Fredholm denominator in the off -shell expressions. To solve the Lee model in the presence of a Ve-bound state via the ISQA we have to complete our set of in-fields by introducing the new and independent in-field Bin corresponding to the V&bound state of mass mB = m + ww. Since

N-QUANTUM

APPROXIMATION

Bin describes a fermion it anticommutes ain,k . Further we require {Bin y BL)

IN

STATIC

MODELS

with Ni, and Vi, and commutes

475

wit,h

= 1

{Bin, Bif} = (Bin 7 BLJ = 0, While by construction the V0 bound state is composite it is evident that the in-field expansion treats it as an elementary particle. Note that we still only have the three Heisenberg fields V, N, and 19.Together with Bin our ring of in-fields is again asymptotically irreducible. The in-field expansions will get some additional terms. Since Bin has the quantum numbers of a Vinain or a Ninainain combination we have to add in second order the four new coefficients c~(w)

di Bin

(54)

bv b’ 9 a”) aLa:nBi,

(55)

fia( U) VLBi,

+ ?a(W, a’) NLdBin

e

(56)

Here a bar denotes a bound state coefficient. These new functions correspond to vertex functions for the, energetically forbidden, processes B ---f V + 8 and B --+ N + 28. They are the Fourier transforms of the bound state wave functions. We can now substitute the completed in-field expansions into the equations of motion (Eqs. (10)-(X?)). In the two sectors, which we have evaluated, the presence of the bound state coefficients cannot alter the old equations. This is a consequence of the conservations laws (3) and of the fact that not both the homogeneousand the inhomogeneousIGllen-Pauli equat’ion can have a solution

FIG. 12. The graphs for the bound stste coefficients dv and 6~ . EY is the off-shell B-V0 vertex, 6~ the B-N20 vertex.

476

PAGNAMENTA

FIQ. 13. The graphs for the equation efficient8 .

for 8, and E‘~, the higher order bound state co-

at the same time (la). We give therefore only the new equations. Denoting Bi, by a heavy line they can be written in the graphs of Fig. 12. The factors 2 and $$ come from Bose symmetry of fiN with respect to the two incoming mesons. We further have the two equations of Fig. 13 for the coefficients & and z,,. We can separate the equations in Fig. 12 which gives for aV the homogeneous integral equation H(WB - w)& (w) = -f(w)

I f(&” b’) 1 dk w’- wg + w’

This is just the homogeneous part of the Kallen-Pauli equation (39). Its solution now is straightforward (12). We find that it only has a nontrivial solution if D ( ws) = 0. Then its solution, which is unique up to a constant, is f(w) ci!y(w) = N ___ . J(w). w - wg

(57)

Here

J(w)= ; lm (w’- wB+d&(wB- w’)Irn& and N is a normalization constant. Now that we have found 8” the other bound state coefficients follow. From Fig. 12 we read off

J(wd + w2The graphical equations of Fig. 13 can be written & (0) = f d3k’h,*(w’)nv (w’) w - WE

1 -

WB

1

J(wz) .

fl -QUANTUM

APPROXIMATION

IN

STATIC

477

MODELS

and ?a (w, 4

- f(u)

= al+

(J

-

The functions (YN and hN are known 2” to find, after some calculation,

WB

d3k’&

s

(IS),

(24).

(bJ’)aN*(w’). We insert them and (57) for

and

f(m)f(w)

62 (w, w> = a+

w

. N

-

WB

(~1

J(we - (&). G (4

Up to now we had no relation to determine the normalization constant N.’ We can in principle find it in the following way. The baryon number of the V&bound state is +l and is given by (Bin

1 Bop

1 Bin)

=

1

where ) B ) = 1Bin > = BL IO). This shows that in the presence of the bound state the baryon number operator (4), if expressed in the in-fields, must get an additional term Bi,BL to read

B,p = ZV+V + N+N

= VLVin + NLNin

We can compute V+V and N+N from the known we are only interested in the coefficients of BkBi, from N+N

and from V+V

+ BLBin.

part of our expansions. Here which become a contribution

sE*(w’)c(w’) d3k’.

With

this the normalization

1 = 2 / d3k’&*(d)&(‘d’)

condition

leads to

+ 2 1 d3k’ d3kn&*(‘d’,

‘/)&,&d’,

‘d”).

Substituting for z’v and ,!jN we can solve for N. This leads to a quite involved expression. It will be recognized as being identical to the expression obtained with the Tamm-Dancoff method (12). We can show that N” # 0 and positive. 7 In the Tamm-Dancoff

approach N would be the norm of the bound state vector.

478

PAGNAMENTA

Therefore the bound state has a nonvanishing norm and since the Hamiltonian is hermitean we can take N > 0. The charge operator Qop (5), if expressed in the in-fields, will undergo a simila’r change. Let us, at first, consider the charge density operator f&,(x)

= N+Ns(x)

- a’(x)a(x).

We can find the charge distribution within the bound state by evaluating the following matrix element by use of our in-field expansions. Actually only the Fourier transforms of the bound state wave functions are needed. (Bin

1 Qop(z)

Here the constant

1 Bin)

=

C6(X)

=

1/3,“(X)

1’ -

s

1 &(X,

X’)

1’ d3Zf.

C is C = 2 /- ( /?,a&‘, w”)

I2 dk’ dk”.

We have evaluated in Appendix D of ref. 12 the spatial dependence of one bound state wave function and shown that it behaves like r-l exp [- ] k, I r] for large distances r. A similar calculation can be done for G and & . Therefore we see that we have a heavy charge sitting at the origin, the N-particle, and an exponentially decreasing cloud of opposite sign around it. The total charge operator is given by (5)

Qop= / d3h,(z) = N+N - /uk+uk d3,t. The Hamiltonian, additional term

if expressed

in the in-fields,

is diagonal and will have the

This is entirely different from adding another Heisenberg field to the Hamiltonian with the quantum numbers of a VB bound state: B -+ V + 0, B -+ NM. Such a Heisenberg field would require the full set of renormalization constants’. It is interesting to note that the homogeneous KP-equation is not a full Bethe-Salpeter equation. It is actually considerably simpler. But in the same way as a Bethe-Salpeter equation does, it reduces to a Schrijdinger equation for the bound state wave function 4(x) in the nonrelativistic limit (31). * Such a model has been studied recently by J. B. Bronzan (8). In spite of similarities the U-state in this model is quite a bit simpler than the V-0 bound state in the Lee model. One verifies easily that there is in addition to the “elementary 77” still the possibility for a strong coupling “dynamical B” in the Bronzan model. See also I. S. Gerstein, Univ. of Pennsylvania, preprint.

N-QUANTUM

In the nonrelativistic

APPROXIMATION

IN

STATIC

MODELS

479

limit we have

and

kB2

Es = P---B---=-

3-L

1kB 1’ 2P

since lc; = - ) k, I2 is negative. Using these relations in the bound state equation tivistic approximation. It reads

We define the bound state wave function

by

& (k2, EB) = 1 e-ik’x+(z) If we insert this into the above equation

which is the Schrodinger the nonlocal potential

d3x.

it reduces to

equation for the bound state wave function.

where K = (2~~ + ~/.AEB + k’2)1’2. This is not a pure exchange potential, particle, the N, is actually at rest. V. LEE

we can take its nonrela-

MODEL

WITH

AN

probably

UNSTABLE

It involves

because the “exchanged”

V-PARTICLE

If, in the ordinary Lee model, we let the physical mass of the V-particle become larger than the mass of the N plus the mass of a B-meson mV

>

mN

+ p

480

PAGNAMENTA

then the V-particle becomes unstable against the decay V + N + 8. Clearly now mV # 11~~.The N§or of this model has been solved by Glaser and Kallen (52). We have used the essential result in the N2§or to show that, here at least, the Peierls mechanism does not work (SS) . The model is given by the same Hamiltonian (6) and one obtains the same equations of motion (lo)-(12) as earlier. From the standpoint of solution by the N&A one has however to observe that the Heisenberg V-field cannot have an in-field. Therefore the Ni, and oin,k form an irreducible set. This is similar to the observation made by Glaser and Kiillen that the 1N0) scattering states form a complete set in the lowest sector. In the in-field expansions (34) we have therefore to cross out all terms containing ainVin . Then it reduces to

(58)

We have set - over the functions to indicate the unstable case. The unstable V-particle will show up as a resonance in the elastic N&cross section. Therefore to fix the parameters of the model we shall first compute it. Defining S- and T-matrices by L3j.g = -(NO1 1Ne)+

= 6(kl - k) + 2aiT,o(w)6(w

we find by using the LSZ reduction

- w’)

formulas

SNe = 6(k - kl) +

lim &( kl , k) e-i(w--wl)t. t++-

This can be written

T,@(W) = lim (WI - w)&((w,

w).

wpw

(59)

The coefficient & can be computed by only going to first order in the in-field expansion (58). We obtain formally the same equations (16)-(20) as in the N§or of the stable case but due to the fact that there is no Vi, we will not have an equation for the mass renormalization Z&n. We can solve these equations formally to find &

(j&,

k)

=

f(dhv(w) w cdl

+

ie

and then b(W)

f(w) = g(w),

(6’30)

N-QUANTUM

where the denominator

APPROXIMATION

function

A(w)

IN

STATIC

481

1MODELS

is now (we put mv - mN = a,.)

Equations (59) and (60) give the N&litude

Here B(U) is not yet determined completely. We have to find 2 and Z&m first. Now we want the model to describe an observed resonance at w = wr or m, = mN + We. For a stable V-particle we would determine the mass renormalization (only the product Z6m is relevant) such that the propagator function W(W) has a zero at this point. This results in a pole of T(W) at the energy w = mv - ??aN . For a stable V particle this is on the real axis but below the physical region. The only reason that this does not work for an unstable 1 is that B:(W) is nonzero in the entire complex plane. In particular it has a nonvanishing imaginary part on the positive real axis. To have a resonance at w = wTwe need Refi(w,)

= 0.

(63)

Since the integral in H(w) (61) can be written

s w,

f2b’) _ w

_

id

d3k’

=

P

i

[a

Irf’,w:’

do’

+

iI?

where l?(w) = 47r’ImH(w)k(w)

= 4a2f2(w)lc(w),

we find immediately from (63) Z6m = -Pl

O”Im H(w’) dw, p sP w’ - w7

(64)

Here P indicates the principal value integral. One way of defining the wave function renormalization constant is then to require f

Re I?(w) ( o=or = 1,

instea,d of (27), which leads to z = 1 - y lrn ;;,wJ $

(65)

482

PAGNAMENTA

where we have introduced Pf to indicate “Parti fini” which is the regularized derivative of the principal value integral. We assume that p (w’) has enough well defined derivatives such that the parti fini integral exists. From (61) one sees that one can extend the function g(w) into an analytic function in the entire complex w-plane. It has a branch point at (J = p to which we attach a cut along the positive real axis to + 00. The Riemann surface of g(o) has two sheets. One can extend by analytic continuation the function B(o) through the cut onto its second sheet, there called HI’(o). HII is analytic if I’(w) is analytic, which we assume. We have the relation HII

= HI(w)

+ 2G(o).

Because the function g(u) is locally only two sheeted it is not necessary to specify whether the analytic continuation has been made from above or below. From (61) we can write g(w)

= ReA(w)

+ iI’

and this expression is valid on the entire Riemann surface on which we have defined B(o). Now we may try to make a Taylor expansion for I?(w) around the point w = wr in the physical region. We can write

r(w) = r(wr) + and substituting

(W

-

wr)r’(wl)

+

..a

(64) for Z&z in (61) we find

Re g(w) where the subscripts

p indicate that the principal

value defines these poles. Since w -

(w’

_’

w)P

=

(w’

_’

w&

+

(w’

-

WMW’

wr -

W>P

this can be written

Re A(W) = (w - w,>[Z + q lm FLtw2ti.1 + O(W- ~7)‘. Using (65) we find to first nonvanishing

order in real and imaginary

Bcw) = w - wr+ iryw,).

part (66)

This expansion converges not only for real w but actually in a circle on the Riemann surface around w = wT and this will in general include points on Sheet II (see Fig. 14). We see from (66) that

Hyw, - ir,) = 0

,V-QUANTIJM

APPROXIMATION

IN

STATIC

MODELS

483

FIG. 14. The radius of convergence of a Taylor series around w = W, for P(W) includes in general points on sheet II (dotted half circle) if wy > p. The dotted arrow indicates path of zero as the V becomes unstable.

and this leads to the resonance pole on the second sheet of the T-n~atrix.g We may think of this pole, which for a stable V was at w = 0 on the real axis, as having migrated up to 1 and through the cut onto the second sheet as the V became unstable. From (66) further follows that the residue of this pole is +l. This gives a real renormalized coupling constant and is in agreement with the Watson Lepore definition (25). From the definition (9) of the Fourier transform we see that we can recover the Heisenberg field by If(t)

= / eiE”V(E) dE.

For t = 0 we get V(0) = 1 V(E) dE. We can now find the state vector of the unstable V-particle by acting with these fields on the vacuum. I V(l))

= v+u1 I 0).

(67) A convenient measure for the decay of the V-particle is given by rom the in-field expansion we see that only the first term I (V(o) I V(t)) 12- F can contribute in (67), the others would annihilate the vacuum, therefore 1 v(t))

= /

d3kLy(w)e-2’“N+“)tN:

f&k IO )

and using (60)

(V(0) 1V(t)) = 1 d3k I ;:w”; 9 R. E. Peierls (3.4) first made the fundamental poles on the second sheet and resonances.

,2

e--i(mN+u)t.

observation

of the connection

between

484

PAGNAMENTA

FIG. 15. Deformation of the contour for the integral path is on the second sheet.

in the decay amplit)ude. The dotted

The integral can be written - [

dw (Im &-J

ei(mN+o)t

and by use of the cut of a(w) this is equal to the contour integral

Since H(W) has no zero in the lower half plane we can deform the contour until it is along the real axis. Note that we cannot go into the upper half plane because there the exponential term increases very fast. Our integral is therefore

-s

m dw -,me

-;(otmNv)t .

This integral can be further deformed into an integral Ca with one part on the second sheet. Thereby one picks up a contribution of the pole of r’(w) on the second sheet. We therefore find

(V(O) 1V(l))=s,,... +e-i(mN+++t. Over the entire range of Cs we are far away from Weand therefore A(w) is large hence the integrand small. Neglecting Cawe find 1(V(0)

j v(t))

I2 = e-r(WT)t

which is the exponential decay law. Clearly this derivation depends on the terms coming from the integral C3 being small. One can show that they do not have exponential dependence in time, but we do not enter on their relevance here. For later reference we may mention that we find for the coefficient &N &(‘dl.‘d)

=

f h>f

(4

(w - WI + iE>B(c3).

(6s)

N-QUANTUM

APPROXIMATION

IN

STATIC

485

MODELS

With this we leave the NO-sector and turn to the three particle sector of this model. Since there is no stable V-particle there are no V&litudes in this model. The only three body amplitude is the elastic N2&litude, which we define by S N2.9 = -(NO& Neglecting the trivial unscattered we find TN20(

processes

where either one or both B-particles

lim

wlwZw3) =

1Ne~ez)+.

(w3

04+W1+“t-u3

+

w4

-

wl

-

go through

, w2 3 w3 , ~4).

uZ)?N(uI

(69)

Substitution into the equations of motion of the full ansatz (58) shows that YN is coupled to BV and one obtains the same graphical equations as in the stable case (Figs. 9, 10). In the analytic form one has to replace H(U) by p(w). The fact that a(,) has no zero on the first sheet where we do our integrations refleets itself in the solution of the integral equation. If we let Bv

(w

, w2, w3 > =

we find for #(WI) the integral R(wo

-

w&(w)

=

; f(Wl)f(W2V(W3)

equation

f&2) w1 -

(70)

a(w~g;w3)

mJJ3)

+

w3 -

ie

Wl

-

w2

-

ie

-- 1 mIm H(o’)#(w’) dw’ = s P w’ - wo+ w - ie where again (w3

+

wq -

wo

w2 -

(71)

= w2 •k wg. TN is then given by the relation wl)?N(wl

, w!Z , w3,

a4

>

=

5’5fj(wl)/%‘(w2,

w3,

(72)

(J4)

+ f(w2)Pdw1,

w3,

w4)l.

The solution to this integral equation (71) can be obtained by the samemethods as in the stable N2&case. Because B(w, - W) has no zero we have this time only the two poles at w1 = ~2 and w1 = w3 plus the left hand cut. We try, as an ansatz, #(Wl ; w2, w3) =

1 WI -

w2 -

ie

+

wl

-

l

w3

- ie

+

c&AZ,

w3)lo(wo

where IO(W)

=

i

lrn

(w’

_

wlg;wo

_

w’)

Im

j$yj

.

-

Wl),

486

PAGNAMENTA

The function tution to be

C(W, , wg), which

is a constant

in WI , is found by direct substi-

where

(73) The connected part of the T Me-matrix of yN in (72) and can be written at first -

1

is derived as the mass shell limit (69)

f(Wl)f(W2)f(W3)f(Wl

+

w2

-

w3)

Ti,dw~ , wz, cd = - 2 H(Wl)H(w2)H(W3)H(Wl+ w2- w3)[1(w3) + I(Wl + 02 - w3)l. The last factor We find T~2dw1,

wz,

can be simplified

W3)

=

by techniques

-w &WI

+

given in Appendix

f(Wl)f(WZ)f(W3)fbl W2)

’ fl(W,)~(W,)fl(W3)8(Wl

+

a2

-

a31

+

W2

-

1 of ref. 10.

W3)

*

(74)

Remembering that WI + w2 - wg is the energy of the fourth meson we again observe complete Bose symmetry. The second factor again represents two body interactions in the initial and final state. The essential part of this three body amplitude is the factor A-‘(wl + ~2) and any indication of a bound state pole would have to be a real zero of x(w) below threshold. Since Z?(z) has no pole on the first sheet it satisfies the unsubtracted dispersion relation, for complex w jq-‘(w) = 1. O” AJ’ __ 1 Im a(,‘) * s I apw-w Using this in the expression B(w)

(73) for A(o)

we find

=s,“(Im~~S,ww,+~~-wIm~. P

For w < 2p the denominator in the inner integral never vanishes and therefore in the entire range of a possible bound state x(w) > 0. From this we conclude that there is no iV2kbound state in the Lee model if the V-particle is unstable

(.2Q,1.9). The same conclusion is arrived at by introducing an in-field for a bound state into the infield expansion. For the relevant coefficient CY~one obtains the homo-

N-QUANTUM

APPROXIMATION

IN

STATIC

MODELS

487

geneous integral equation

But this time n(wg - w) has no zero. Therefore there is no pole at w = ws which one should expect for a bound state wave function. Indeed one can show that this equation only has the trivial solution (19). This also shows that the scattering solution is unique. VI. THE

SCALAR

FIELD

Another interesting model is the scalar field interacting with a fixed fermion source which has been discussed by many authors (37-39). In the soluble sectors of the Lee model the N&A gave the exact answers. Here we can also find the exact answers but the N&A equations which are still soluble will only give approximate results, in any finite order. It is interesting to see that the expansion coefficients do not involve any renormalization constants and therefore they remain finite even in the local limit where the unrenormalized quantities diverge. Contrary to the Lee model however there is no scattering in this theory. A. THE

MODEL

We go out from the urenormalized

Hamiltonian

H = H,, + H, (75)

Here wk = ( k2 i- ~‘)l’~ and

where ~0 is the unrenormalized coupling constant, u(w) the Fourier transform of the source function with t’he familiar properties U(k2 = 0) = td(Wk = ,d) = 1 u(k) -+ 0, k --+ CO,“fast enough.”

(76)

The last property holds also for fo(w) which we call the cutoff function, to(w) contains as factor the unrenormalized coupling constant.

488

PAGNAMENTA

The unrenormalized

fields I,+~,

ak

satisfy

{#u, io

the (anti-)

commutation

= 1

[ak, a;,] = 6(k - k’), all others being zero. 9 is a fermion sitting

relations (77)

at the origin, therefore w

and the above Hamiltonian

commutes

= 0 with

(78) the operator

N+ = #+A

(79)

therefore the number of fermions is a constant of motion. We restrict ourselves to the case of a single fermion in each state. As in the Lee model we can carry through the renormalization. Let us define m. = m -

6m

*u = .w

(80)

x0 = A. There is no coupling constant renormalization in this model. The renormalized field # has the anticommutator {k J/+1 = 2-l.

(81)

The Hamiltonian can now be written in renormalized form.

H = Ho’ + H,’ (82)

Heref(w) = HO. By a slight modification of the dressing transformation given by Greenberg and Schweber (38) we find that the Hamiltonian is diagonalized by (Appendix). $/ = Gin : J-4+-=4):

(83) (84)

Here we h%ve defined the operator

A = If%

&n,k d3k.

(85)

N-QUANTUM

APPROXIMATION

489

IN STATIC MODELS

Let us denote by B the C-number

then one finds by taking the vacuum expectation value of (81) z = eB

(87)

and from the diagonalization of H (88) B. EXACT

SOLUTION

OF THE FIELD

EQUATIONS

We introduce the Fourier transforms of the Heisenberg fields by

$0) =s

e-“““+(E)

ak(t) =

s

e+

dE,

(89)

ak(G) dG.

(90)

With these we use again the Heisenberg field equations to derive the equations of motion

(E -

m -I- Gm)$(E) = /d3k dGf(o)[#(E (G -

uhr(G)

- G)ak(G) + ~k+(W(E

= 2fb.o) /” $+(E - G)+(E)

+ G)l

dE.

(91) (93)

We shall now give the complete in-field solution of these field equations. To avoid the writing of all the energy A-functions we give the Heisenberg fields at time t = 0. + = ,E,

/ c& . . . &

d3,, . . . d:, hl,(p, , . . . PZ , ql , *. . ‘&id (93)

and

+ ‘Uin.pl

+ Uin,pl

“.

Gn,ql ” ’

ain.q,

$2

*in

where we have defined the retarded coefficients by hZm(Pl9

. . * PZ 7 Ql y + ’ * qm)

=

(pl

* ’ ’ PI

) 9

) qi

’ ”

qmJ/in)a

y

490

PAGNAMENTA

gtm(k;

PI

1 * *.

PZ 7 ql)

* * * qm)

=

‘$1

. *.

p$in

1 ak

1 Q1 * * * Qm$in)c.

(96)

Here we have used the notation [PI **a Pn) = f.&,p, * * * CL,,, IO)

(97)

and the subscript c means that we only take the connected part of the matrix element excluding any trivial contractions between the two states. The boundary condition that $J has one in-field gin implies ho0 = 1. The above given expansions can be and the equations for the C-number the two general coefficients ht, and express energy conservation we find equations.

(98)

substituted into the equations of motion coefficients can be derived. We pick out gin . Considering the &functions which two coupled sets of nonlinear integral

(wql + . * . + uqn - up1 -

. . . - upi + Gm)hzm(p1 ) . * . pz ) q1

(wql + * * * + wq, - tipI -

- . - - wpl - %)gzm(k;

=

zf(Uk)

,g,

m’!

, . . * am)

PI , * * * Pz , ql, * * * , Qm>

Sd”p,‘...d’pl~h~~~,~(pl;..p,,p,‘...A.;q,...q,)

(100)

o+b=Z c+d=m

The first two terms on the right in the equation for hl, come from #a in the first field equation. The product a+# does not lead to a similar term since due to (78) $L#in+in = 0. This means that gzm cannot precede hl, . So from a+$ we get only terms of the form oin# which gave the last two expressions in (hl,). Similar considerations hold for glm . As in the Lee model the situation is easier presented by graphs where especially the permutations due to Bose symmetry are visualized. See Figs. 16 and 17. Knowing the relations (83) and (84) we expect that this complex set of integral equations actually has a simple solution. We shall assume, as a working hypothesis, that Qzm =OYl#O,

m # 0,

(101)

N-QUANTUM

APPROXIMATION

IN

STATIC

491

MODELS

“c m’l m’=O a+c= I b+d=m e

m

m-l

FIG. 16. The graph

for the integral

equation

17. The graph

for the integral

equation

(99) for km)

k

FIG.

(100) for glm

and only go0# 0. As a first consequence we find that in the equation for hl, (99) d = m’ = c = 0 which reduces this term to s

d3Wbdhzm(pl

, - . . PZ ; ql,

which factors. If we now tentatively 6m =

s

(102)

. . . qm)gooW,

put d3kf(wk)gm(k),

(103)

the remainder of the equation for hl, is solved by hl,(p, , ‘. . PZ; 41 * *. %) = (-1)”

1 f(Pdf(P2)

m

.

UP1

. * * f(4d

wp2. . . %?n

-

(104)

492

PAGNAMENTA

We can now show that the right side of Eq. (100) is zero for any 1 > 0, m > 0 as follows. From the conditions under the sum follows d = m - c hence we can write the product as h:+d ,.Jb+?d ,m-c . For m # 0 there exists to each such product another product with c replaced by c + 1 to read h:+m~+l,ahta+m+c-1

.

While the first product had a factor ( - l)c+m’ this one has a factor ( - l)cfm’+l, otherwise they are the same; therefore their sum is zero. The same holds true for the other pair of indices. For goo(a) we find -(Jk

g&k)

=

zf(tik)

[

1 + /- d3p’h;E0(p’)hdp’) + 2! $ d3p1’d3p~h2*O(p1’p2’)hU)(p1’. p-2’) +

** -

1

(105)

or g&t&)

= -‘%+

ZK,

(106)

Wk

where the constant K is the infinite this is

product on the right. Substituting

(104)

K = e= = z-1

(107)

f(o) = ----p.

(108)

and we find gc&)

This gives for the mass renormalization (103) the exact result (88). The S-matrix for Q - $ scattering is given by x,&G = -(+4x I !ba)+. and the T-matrix,

as obtained from the reduction formula, is Ta+(o) = lim hll(q t-m

or, substituting

w’)e

i(w’-lu) t

(104) and using the Riemann-Lebesgue

lemma,

N-QUANTUM

Since evidently

APPROXIMATION

hn has no singularity

IN

STATIC

403

MODELS

in the right place, we get

TaJ,= 0. In the same way one sees that all higher processes are absent in this model, hence the X-matrix is trivial.

C. THE FIRST ORDER N-QUANTUM SOLUTION It is interesting to see that the 1st order N&A gives reasonable and finite answers. The first order approximation consists in taking only terms in the in-field expansion with at most two in-fields in a product $’

=

#‘in

+

1

d”k:h~:‘(~)#in

ak

=

ain,k

&n.k

+

f

/

d”kh!i’(W)#in

~OO(~)#‘?n’$in

aL,k

*

(109) (110)

Again we have written the t = 0 expansion to save space. One obtains the general Fourier transformed expansion by adding a proper energy &function to each term. The superscript (1) denotes the first order. We see that we have introduced three unknown functions. We might now substitute (109) and (110) directly into the field equations, However since we will only get an approximation, the Heisenberg fields will no longer commute at equal time. One can make sure that these commutation relations hold, at least to the accuracy of the order in which we compute, by properly (anti-j symmetrizing the right hand side of the equations of motion. We replace therefore the right hand side of (91) by f*i[@

+ a$ + a++ + #a+].

(111)

There is no symmetrization needed in the second equation. Now we substitute our expansion and normal order the in-fields. We obtain four nontrivial equations by reading off the coefficients of different in-field expressions.

[

wq + 6m -

9; / d3kf(d

-wq + 6m - ?i [

s

(Qa-4

+

R::)*(w)]

e4

=

f(q)

1 -wgoo(o) =Pf(u)K('). d”kfS(w)(gZ’(u)

+ gh?*(w)

hF(q)

= f(q)

(113)

(114) (115)

494

PhGN.4MENTe4

FIG. 18. The graphs to first order N&A for the vertex functions LO and gm . These graphs can be obtained from the ones in Figs. 16 and 17 by restricting the intermediate boson lines to at most one.

where K”’

= 1 + [ d3f&:‘*(q)h&).

(116)

In graphs this means we restrict ourselves to zero or one intermediate boson. We give Eqs. (114) and (115) in graphical form (Fig. 18). In the first equation there is an additional term with go0replaced by g& . One easily verifies that this set is solved by

-h(w) 6m’l’ =

6m.

We eoe that hlo , hoI , go0, and 6m are exact. The only approximation is in Z where we have picked up the first terms of the exponential. We see that the r-functions are finite. The same situation persists in the second order. Clearly there is no scattering. We are led to the conclusion that in each higher order one will get the exact expansion coefficients and a better approximation to Z in this model. Note that we have used here the naive approach. The improved version, where one retains higher order terms, works too. It leads in first order already to the correct expressions for the second order coefficients and gives a better approximation to

N-QUANTUM

APPROXIMATION

IN

STATIC

495

MODELS

2. Since the equations are somewhat lengthy we do not give them here. They are easily derived from the exact ones (99), (100). VII.

CONCLUSIONS

We have shown that the improved form of the N&A works in some static models. It allowed us to obtain finite results which are consistent with renormalization. The very strong selection rules of the Lee model implied that we did obtain linear integral equations for the coefficients. In the second model we had to deal with coupled nonlinear integral equations. This will likely be the case in any realistic field theoretic model unless one makes further approximations. Apart from several applications which are presently investigated’0 we intend to apply the NQA to the static pion-nucleon model. The equations of the N&A can be clombined with the Faddaev-decomposition (40) to yield compact kernels. In potential scattering, where the number of particles is fixed the equations become exact in each order. This may also be valuable to investigate. The results obtained in the different Lee models are used to provide a preliminary discussion of the generation of higher resonances (33). It is possible to augment t’he Lee model in several directions and still keep it soluble or approximately soluble by the given methods” APPENDIX.

DIAGONALIZATION

OF THE FIELD

SCALAR

We show here that the exact solution the scalar field. We remark that :e

A+-A

HAMILTONIAN

diagonalizes A+

:=e

.e

OF

the Hamiltonian

THE

(82) for

-A

hence

Here we can combine the two exponentials in the center by use of the HausdorfY relation which holds for operators if their commutator is a c-number. A A+ ee

=e

A+A+el/2(A,A+]

We normal order the first factor on the right to find e*.e

A+

=eee

A+

A [A,A+]

.

‘0 0. Greenberg (6) did discuss the N&A as applied to rearrangement collisions and a scalar model of the deuteron. Greenberg and Woods (private communication) are using the NQA to compute relativistic recoil corrections in the hydrogen atom. 11 We think of the models proposed by Goldberger and Treiman (35) and the ones of Srivastava (36).

496

PAGNAMENTA

Now [A, A+] =

B from

(82)

and we find

#+lr/

=

eB&$i,

=

Z-‘#L$in.

Now we are ready to transcribe N in terms of in-fields. We substitute in (82).

H = Z(m - Cim)#L$in 2-l

Collecting and ordering we see that there is no interaction since $$ = 0 we have $~#i,#~~ = #L$in and we find

term left. Further

This shows that if we use the value (88) for 6m the Hamiltonian is diagonal in the in-fields with the physical mass m of the J/-field. A similar diagonalization can be executed to low orders in the Lee model. In the No-sector, or first order, it removes all Yukawa interactions from the Hamiltonian, leaving only four and more particle interactions. ACKNOWLEDGMENTS

I would like to thank Prof. 0. W. Greenberg for his valuable advice during the time this work was carried out. I also want to acknowledge the generous support received from the American-Swiss Foundation for Scientific Exchange, without which this work would not have been possible. RECEIVED

February

24, 1966 REFERENCES

I. TAMM, J. Phys. USSR 9,449 (1945); S. M. DANCOFF, Phys. Rev. 18, 382 (1959). J. C. TAYLOR, Phys. Rev. 96, 1438 (1954). D. AMATI, A. STRANGHELLINI, AND B. VITALE, Nuovo Cimento 13,1143 (1958); B. Bosco, S. FUBINI, AND A. STRANGHELLINI, Nucl. Phys. 10, 663 (1959). 3. R. JOST AND A. PAIS, Phys. Rev. 60, 189 (1959). 4. S. WEINBERG, Phys. Rev. 120, 776; 131, 440 (1963). 6. M. WELLNER, Phys. Rev. 132, 1848 (1963); M. ALEXANIAN AND M. WELLNER, Phys. 1. la. 8.

Rev.

137, B155

(1965).

6.

0. W. GREENBERG, private communication (1962); Bull. Am. Phys. Sot. 9, 448 (1964); “Introduction to the N-Quantum Approximation in Quantum Field Theory.” University of Maryland, Tech. Rept. No. 448 (1965); Phys. Rev. 139, B1038 (1965); Bull.

7.

T. D. LEE,

Am.

Phys.

Sot. Phys.

10, 484 (1965). Rev. 96, 1929

(1954).

N-QUANTUM

APPROXIMATION

IN STATIC MODELS

497

8. A. PAGNAMENTA, Bull. Am. Phys. Sot. Ser. II, 9, No. 4,494 (Spring 1964). 9. R.P. KEN~CHAFTAND R.D. AMADO,.J.MU~~. Phys.6, 1340 (1964). 10. A. PAGNAMENTA, J. Math. Phys. 6, 955 (1965). il. G. S. LITVINCUH, Izvest. Akad. Nauk. SSSR. Ser. Math. 26, 871 (1961). lla. J. C. HOWARD, Ann. Irzst. Henri Poincare 2, No. 2, 105 (1965); E. KAZES, J. Math. Phys., in press; C. SOMMERFIELD, J. Math. Phys. 6, 1170 (1965); T. L. TRUEMAN, Phys. Rev. 137, B 1565 (1965); M. T. VAUGHN, Northeastern University Preprint; H. ARAPI, private communication. 12. A. PAGNAMENTA, J. Math. Phys. 7,356 (1966). IS. R. HAAG, Kgl. Danske Videnskab. Selskab., hlat.Pys. Medd. 29, No. 12 (1955), and references quoted there; see also Eq. (3) in ref. f4. 14. V. GLASER, H. LEHMANN, AND W. ZIMMERMANN, Nuovo Cimento 6, 1122 (1957). 16. H. LEHMANN, K. SYMANZIK, AND W. ZIMMERMANN, Nuovo Cimento 1, 205 (1955). 16. K. SYMANZIK, in “Lectures on High Energy Physics,” B. JAKSIC, ed., Federal Nuclear Energy Commission of Yugoslavia, Herzegnovi, 1961. 17. JOHN G. TAYLOR, Nuovo Cimento Suppl. 1, No. 3 (1963). 18. M. ALEXANIAN AND M. WELLNER, Phys. Rev. 137, B155 (1965). 19. M. ALESANIAN AND M. WELLNER, Phys. Rev., in press. 20. 0. S. GREENBERG AND R. GENOLIO; see 0. W. GREENBERG, Bull. Am. Phys. &‘oc. 10, 484 (1965). 21. 0. W. GREENBERG, private communication. 22. M. S. MASON AND R. B. CURTIS, Phys. Rev. 137, B996 (1964). 25. H. M. FRIED, J. Math. Phys., in press. 24. H. CHEW, Phys. Rev. 133, 2756 (1963). 85. K. N. WATSON, Phys. Rev. 76, 1157 (1949). 26. G. K~LL~~N AND W. PAULI, Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 30, No. 7 (1955). 27. R. D. AMADO, Phys. Rev. 122, 697 (1961). 28. M. T. VAUGHN, R. AARON, AND R. D. AMADO, Phys. Rev. 124, 1258 (1961). 29. T. MUTA, Progr. Theoret. Phys. (Kyoto) 33, 666 (1965). SO. J. B. BRONZAN, Phys. Rev. 139, B751 (1965). 31. MASA~I AND KA~AGUCHI, Progr. Theoret. Phys. (Kyoto) 33, 932 (1965). S2. V. GLASER AND G. KHLLEN, Nucl. Phys. 2, 706 (1956), H. ARAHI, Y. MUNAKATA, M. KAWAGUCHI, AND T. GOTO, Progr. Theoret. Phys. (Kyoto) 17, 419 (1957). SS. A. PAGNAMENTA, Bull. Am. Phys. Sot. 10, No. 4,465 (1965). S4. R. E. PEIERLS, Proc. 1964 Glasgow Conf. (London, 1955). 66. M. L. GOLDBERGER AND S. B. TREIMAN, Phys. Rev. 113, 1663 (1959). S6. P. K. SRIVASTAVA, Phys. Rev. 126, 2906 (1962); ibid. 131, 461 (1963). S7. L. VAN HOVE, Physica 16,145 (1952) ; G. WENTZEL, “Quantum Theory of Fields,” Interscience, New York, 1949. S8. O.W. GREENBERG AND S.S. SCHWEBER, Nuovo Cimento 8,378 (1958). 89. R. HAAG AND G. LUZZATO, Nuovo Cimento 13,415 (1959). 40. L. I). FADDEE, Soviet Ph,ys. JETP 12, 1014 (1961).