On the definition of renormalized field products

ANNALS

OF PHYSICS

On

105,

the

350-366 (1977)

Definition

of

Renormalized

Field

Products

P. S. COLLECOTT* Department of Applied Mathematics University qf Cambridge, Silcer Street,

and Theoretical Physics, Cambridge CB3 9EW, England

Received June 14, 1976

We study the redefinition of the field products appearing in a Lagrangian and its equations of motion in a Normal Product framework. We propose a method of defining these products, which give the finite Green’s functions, in such a way that the canonical derivation of the equations of motion is preserved. This involves the use of the Wilson Expansion in a Dimensionally Regularized form. As an example a #, $, field theory in four dimensions is fully redefined to the l-loop level.

1.

IN-I-R~DUCTI~N

With the advent of Normal Product Methods in Lagrangian field theory [l-5], there has been renewed interest in the possibility of redefining all the field products in the Lagrangian in such a way that all the Green’s functions of the theory are finite [6-81. In this sensewe have a finite theory. Next we may ask for a redefinition of the field products appearing in the equations of motion so that the Green’s functions with insertions of these products are finite [I, g-101, and in this sensewe have finite equations of motion. Third we may ask how these two notions are connected and endeavor to have a theory in which the redefined equations of motion follow from the redefined Lagrangian following the usual canonical, and formally finite, steps. However although there have been various more or less detailed proposals of how to define these normal products [ 1, 4, lo], there has also been a considerable amount of confusion between the finite products which appear in the Lagrangian, and those in the equations of motion, which are in fact distinct types. It is these problems we wish to shed some light on by, first, general discussion, and second, by working a simple example right through as an illustration. This will involve dealing explicitly with the divergent vacuum bubbles and tadpole diagrams which arise, and showing how these problems are coped with within a normal product framework, and are in fact essential elementsof the complete treatment. Section 2 introduces somenotation and discussesin a general setting the redefinition of Lagrangian products relative to a given free Lagrangian, and of operator insertions in a given Lagrangian theory. In Section 3 we consider the derivation of the equations * Address after October 1, 1976: Max-Planck 40, W. Germany.

Institut fur Physik, Fohringer Ring 6, 8 Munchen 350

Copyright All rights

0 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0003-4916

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351

of motion which forms the link between the two types of redefined products which are considered. This then requires a discussionof the Wilson Expansion which is given in Section 4. Section 5 is devoted to a brief discussionof the relationship to other Normal Product formalisms and to a remark concerning the implementation of gauge invariance of field products. Section 6 gives an illustration of the procedures involved by giving a complete redefinition in the terms outlined of a $, ?3, 9 theory in four dimensions.

2. THE GENERAL SCHEMA We shall consider throughout a general renormalizable field theory defined via a polynomial Lagrangian. The restriction to renormalizable theories is by no means essential, merely convenient. lf, in terms of conterterm renormalization, the theory required an indefinite number of counterterm series,then we would have to define a correspondingly infinite number of renormalized products. We make no restrictions as to the simplicity of the theory, its invariances, or the number of massiveor massless coupling constants. The only criterion is that there is an expression, involving possibly infinite-valued parameters, which generates the finite Green’s functions associated with the theory. Supposewe have a Generic or ClassicalLagrangian “YG = ~(A, ) @$)= c hjcDj ,iJ

(1)

J some index set, in which the @, are the local field monomials, associatedwith each of which, including the usual kinetic terms, is a coupling constant hj . In the usual counterterm or multiplicative renormalization schemeswe proceed to define the coupling constant and wave function renormalizations by specifying the bare quantities AjO= Z,J,

.

QjO = 2J#$ .

(24 (2bJ

We will also write QjO = B[@j], DI; = G[@J and then ~j:., = ZfE to tie up with the later notation. If these redefinitions are merely an intermediate renormalization we can define Zj, = 6j]; + Xj, 2

Xjjq = O(h),

(3aj

zjjn-= 6,, + gjk. )

xj2 = x;,c = O(X).

(3bj

If we are considering a dimensional renormalization we should also share between Xj, , @joa factor pVpdwith p a ‘t Hooft unit of mass [I l] so that the Lagrangian in [v] has massdimension v while the “finite” parametersA,, Qk retain their dimensions as in the “physical” dimension d. The usual procedure is to assignno factor to the

352

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COLLECOTT

masses of the theory, thus defining a p “+d) factor which can be associated with each field of the generic theory, which via the rest of the interaction terms define the p6(u--d) factors associated with each coupling constant. The usual care is needed in the definition of the v-dimensional Fourier transforms. With the definitions (2) we are led to a Bare Lagrangian -u;, = P(hjO , @,o) -~-=1 hi”cDj” When expressed in terms of the original parameters of the Generic gC; it will be known as the Renormalized Lagrangian 2,

(4) Lagrangian

which we also write as

-6 = c hkLk@l , k.1 L = Z’Z

(6) (7)

= 1 + X’ + 2 + X’2 =1+i?,, - YR = h’L@ = A’@ + x.&D,

(8) (9)

thus defining the counterterm Lagrangian 2Zcr . Notice that here we have the Bare and Renormalized Lagrangians as different analytical functions of the Bare and Renormalized variables, respectively. with the same functional value. So far we have merely defined some notation for standard renormalization. Suppose, however, we wish to renormalize by formally replacing the products Qsi in ZG by Normal Products. That is we wish to define a Normalized Lagrangian

where N[Qji] are the Normal Products, so that all the divergences appearing in the field theory in perturbation theory are compensated for by these Normal Products. More formally these products are defined so that to any given order in the coupling constants of the theory the Green’s functions of the theory are finite to that order. Defining these via the Gell-Mann Low series

it is obvious that a priori the Normal Products involved are those of free-field monomials @,O, and are thus defined relative to some definition of how the Lagrangian is to be split into Free and Interacting pieces. What we are really doing is renormalizing

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PRODUCTS

353

operator insertions from the chosen Interaction Lagrangian in the Free theory, and hoping to define a self-consistent set of operators {N[@,O]: Oi E Ycint) which give finite results upon arbitrary simultaneous insertions in the free theory. It is important that we should be clear as to the exact meaning of this. We mean a set of operators, expressible as linear combinations (with divergent coefficients) of the various Generic operators ~j , which give finite results when inserted into the Gell-Mann Low formula (to any given order). No statement is made about the finiteness of the individual terms: j” do,, ... &,& TrpIo(s,) ... ~.Vo(s,~7)A’[@,“]( JI) ... N[@,,,“]( J’.,,), . This, as will be discussed below, is in contrast to other formulations. Hence, since we have that qV = YR = PO, the problem is to rearrange the terms of pR == xk,2 h,Lk@‘l into -Y$ =: xk h,$V[@,.] in such a way that there is credibility in calling N[@,] the Normalized form of GA . The essential criterion for this will be that h,N[Qk] (J?‘,) shall be 0(X,). With this we arrive at a more formal statement of finiteness. Let %,,(A,, 5;) == YCT,(Ao) + xi &N’.[@,] so that .2$ = 2”($, h). Then to any given order n in X the Gell-Mann Low (perturbation) expansion:

will be finite to order n in A. In other words the sum of all the possible Green’s functions with Normal Products up to order n will be finite to order n in the coupling constants. This is easy to see since the whole perturbation expansion is finite order by order by construction using A!$ . By including all insertions up to order n all terms of order n in N[@], and thus of order >n in A. This appears to be as good a statement as one can get. Since YR has arbitrarily high powers of the A (in general) some N[@] must contain some of them in its explicit form. Thus a single insertion of this will involve arbitrarily high powers of some h and cannot be finite to all orders in A, whereas it will by construction be finite to O(A). There are obviously an infinite variety of ways of splitting 3” into a sum of normal products satisfying the above criterion, and since the Gell-Mann Low series only involves the sum, YNint, of these, there is no way immediately of making a canonical choice. This will be done later using the equations of motion of the theory. When we consider equations of motion of a theory we are on a different footing altogether. Using standard naive manipulations on the functional integral of the Generic Lagrangian we are led to the Generic Equations of Motion in Green’s function terms

5951IOjl2-9

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I’.

S. COLLECOTT

a/+ denotes the Euler derivative w.r.t. v: (%/a~) ~ 2,(8/8(3,~)) ... and Q is some field monomial. It is clear that we are studying the effect of an operator insertion of Q(&&/a$) in a Green’s function of the Generic theory, and as usual naively use the reduction formula to get the operator equation of motion Q(aZ,/&$) :m 0. What is obvious but must be stressed is that (with Q +~. I for the moment) the products involved a9G/&$ are not necessarily those of the Lagrangian 2G , and in any case are treated on a different basis. What we demand from redefined, Normal, Lagrangian products is that they form a consistent set producing finite results upon arbitrary simultaneous insertions in the free theory, as explained above. For products Z2G/?+ in the equations of motion we just demand that they give finite results upon a single insertion in a Green’s function of the already defined complete interacting theory. It should be noted, however, that once these inserted products have been redefined that they and the Normal Products of the original theory are a set of consistent products which produce finite results to the level of arbitrary Normal Product and single Insertion Product insertions in the free-field theory. This is just because each set, those with arbitrary Normal Products and either none or one Insertion Product is finite by definition of, respectively, the Normal and Insertion Products. It is thus seen that we are led to consider more general objects, which we call Renormalized Insertion products, and denote

Yi is any local field polynomial and h is a vector in the coupling constant space of the given theory. n is an m-tuple (n, ,..., n,) of natural numbers corresponding to the m Y,‘s under simultaneous consideration. Each Yj is associated with a coupling constant pj ; these are collected into a vector k. {1t”‘[Y(]: i = I,..., nz) are, by definition, a set of operators which give Green’s functions finite to order n in p with up to ni simultaneous insertions of the Z,h’“[Yf] in an already renormalized theory with couplings A. In this sensethey correspond to renormalizing the products Y, up to a level of n, simultaneous insertions. In this language the Normal Products corresponding to Lagrangian products Qi relative to somefree Lagrangian defined by a coupling constant vector A, are given by

with 111equal to the number of monomials in the interaction Lagrangian. Similarly the equations of motion involve products

where eij , eFjare just unit vectors with 1 in the ijth or T&hplace.

RENORMALIZED

FIELD

The obvious way to start computing new Generic Lagrangian

355

PRODUCTS

these renormalized

products

is to define a

Whether or not this is renormalizable we can still define order by order counterterms which will render the Green’s functions of this theory, which are just the Green’s functions of the original theory with insertions, finite. Clearly we may have to expand the set of Y’? considered order by order if the theory is not renormalizable. but in any case we arrive at a series of partially renormalized Lagrangians

where (@,I u {Yi : i E InI is a set of monomials closed under renormalization to order co in A and n in p. There is no reason why the (Oi: and {Yj) should be distinct, and clearly an infinite additional insertion of some Qi with coupling pi would correspond to raising the coupling constant of Qi from hi to hi $ pLi . We wish to rewrite 2??R.ll as y$

r:

2;;;”

::

1

,]jNy@i] j

~~ c

pLiJ;:y$fJ.

iSI”

By exactly the same arguments as above, our first criterion will be

Let us consider

If we use an intermediate renormalization scheme with minimal mass dependence, such as Dimensional Regularization, we know that the terms of Y~$’ will be power series in (h, IA*).Thus we can write

as a Taylor expansion in I*. The notation is Di = ~~(a/&) (zi) and the normal ordering is used since we are really using the operator (q(a/+,) /U,=,J lticUi and hence all pi’s in D,'s should be considered as to the left of all (a/3pj)‘s. Hence

and we wish to express this as: xi ~J~[!P~].

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P. S. COLLECOTT

Reducing this to operator algebra we wish to express :exp c Dj -

I: = 2 :DjJ;(Di):

i

with thef, analytic. This can most easily be cone by using the symmetric solution exp x:j Dj .fjCDi)

=

:

1

Cj Dj

This leads to the simple expressions

and in particular

and also just taking terms to order n

where Trip meansterms up to order n in p. These definitions are perfectly consistent and have the correct finite properties, but are not necessarily the best choice. When one comes to deal with the equations of motion of the theory one is, essentially by functional differentiation w.r.t. only one ?i , dealing with individual terms rather than the sum of all possible insertions, and a wise choice would be one in which there was a correspondence, even equality, between (a/agj) N”o’“[@J and I~~“[(XDJ~~j)]. This will be the subject of Section 3. Meanwhile we notice that the symmetrical decomposition above can also be obtained by treating a term A,,,~~1 ... p,“j ... as

and taking (mJ& ms) A,,&1 -*. Q ... as the conterterm associated with pj!Pj for an insertion of piYi mi times Vi. Notice also that we can write

3. EQUATIONS OF MOTION As we have continually stressedabove, it is the equations of motion which force us to look at single insertions of operators into the Green’s functions, and moreover,

RENORMALIZED

FIELD

357

PRODUCTS

single insertions of operators we would like to be canonically related to those of the Normalized Lagrangian. Given a Generic Lagrangian &” we define our Generic Action S,h[q] = J” Y(;“(.Y) &C and derive the equations of motion by considering:

This gives

which reduces to the operator equation (~S,“[~]/I~~~(.U)) = (a/?&) YG”(x) =z 0 on using the reduction formula. If we consider the renormalization of these steps we are looking for operators corresponding to (&Y,“[~]/&I~(x)) which give finite Green’s functions upon a single insertion as above. Now

so

and we are therefore

seeking operators

Now in the Normalized or Renormalized vations through to arrive at

theoreies we can follow

the same deri-

?vn = i 1 6(x - Xi){ Tc&( x1) . . . cp’i(Xj) . . . y&,)‘, i=l

with the obvious definitions of S,Vx and SR’. Now the third expression is evaluated wholly within the original, Pa,“, theory and is hence finite. Moreover, it is minimally finite if we use a minimal subtraction scheme.Hence so are the two other expressions since the equations have been maintained on renormalization. Thus each expression is just

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P. S. COLLECOTT

in an obvious notation.

Thus we get

It would thus obviously be desirable to have

completing the analogy between the naive Generic theory and the Normalized theory. Let us now consider the derivation of expressions for Z[a@J&&.]. Following the previous section we look at

and consider its counterterms,

This can be counsidered

However

which is best done by considering

as SG”(p?) = Cjpj!Z’,

we know the renormalized

Since we are only working as expected

with

version of this to be

to order $ (one insertion

of a@/&+?~),we must

define

in other words just the counterterms linear in ($&). This clearly involves only the term $k(a_s;P,“/&$,) (&J. The difficulty arises in sorting out terms of the form &Xir)lig which receive contributions from ($Ji) h;-‘his and (&hi) &‘A,“-‘. Assume for the moment that we have a way of dividing any such term

RENORMALIZED

FIELD

Then we have defined some kind of projection

359

PRODUCTS

operators

gi so that

and

Therefore exactly the same projection Lagrangian Products by

operators

CBj will define for us our Normal

VjYR’- = A,N[@,] and give the canonical relationship between Lagrangian products and products in the equations of motion. Thus the procedure will be to find expressions for I[a@j/a$Jy in fact via the Wilson Expansion, thus discover the projections Bj and hence the correct definition of Normal Lagrangian Products.

4.

WILSON

EXPANSIOKS

The Wilson Expansion [ 121 is an expression of the singularity structure of a Generic Lagrangian Product Qi(s). As usually construed [I, 131 it expresses the singularities as divergences in space-time variables as the space-time variables of the fields composing @, become coincident. However recently we have shown [14] how, using Dimensional Regularization in particular, the Wilson Expansion can be cast to exhibit the singularities of the field products in terms of the usual singularities of perturbation theory, here poles in the dimension of space time. Following Zimmermann’s perturbation theory approach [I] we are led to an expression of the form

where Qj runs over all possible operators. It must be stressed that in terms of our definitions it is the Insertion Product which appears on the right-hand side, not our Normal Product, even when @,(.u) is part of a renormalizable Lagrangian and hence only Lagrangian products Qj can appear on the R.H.S. This can be seen by remembering that Zimmermann‘s proof of the Wilson Expansion [ 1, 71 comes by considering diagrams of the form

8; &

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P. S. COLLECOTT

and splitting this in all possible ways into

3; & 5;. The Qi is a distinguished vertex of the diagram in all cases,as must be remembered when associatinga symmetry factor with the diagram, coming from a contribution to

TQiopo ... The @j vertex is manufactured during the splitting of the diagram and is similarly a vertex of: (

TQjoq” ... p” exp i

s

-Y$,,+

>

i.e., it is a once inserted product. Again, if Qj is in Y&“. then the IIQj]‘s above are i.e.,

If”i’[@j],

not even It;‘“[@j] = (i/GA) 2ZRhjhSh,which is the first-order term of N”o’“[@j]. Hence the Wilson Expansion produces linear relations between Generic and (single) Insertion Products. We are interested in the lnsertion Products of field differentials of Lagrangian products since these appear in the equations of motion. It is easy to seethat these form a closed set under Wilson Expansion, that is, in the Wilson Expansion of {(a@,,/&$,): Qi E&) only Insertion Products of (a@j/a$,)‘s appear with divergent coefficients. This is because the coefficients of I[Yj] in the expansion of L?@i/&$kis PP(T(ZCD~/~I$,~) Fj I coming from a diagram

&, &e?. ..’ .g.

If this is overall divergent then so is

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FIELD PRODUCTS

Qi is a Lagrangian vertex, so this is a possible diagram for (,T$,!Pj;, which will require a divergent counterterm in the Lagrangian: $kY’j . Since the theory is renormalizable, this must be one of the original Lagrangian vertices, and hence Yj is of the form a@,/&& for some @j in TC; . Hence we will produce a matrix equation

or

from which we can in fact derive, using

that Qi = W&N[cDj] (+terms independent of &.), from the collection of which we will arrive at

This we can invert as an alternative to inverting the a@Ja& equations for each k. This latter procedure would arrive at a definition of I[a@,/&&] Vi, k, in other words a definition of the projection operators 9i defined above. From these we know the expression for the Normal Products N[@J in terms of the Generic Products by application of 9%ito gR”. Of course once we have the normal Products we can by truncation define the Insertion Products relative to the free theory to any arbitrary order. It should, however, be noted that in this method of defining the Normal Lagrangian Products the split of the Lagrangian into free and interacting parts has become irrelevant. The Normal Products of bilinear operators are always trivial since these are formed by integration from the linear Insertion monomials which are trivial by virtue of their having trivial Wilson Expansions. There are no one particle irreducible graphs with a one-vertex except the trivial graph. Moreover, it is clear that the Insertion Products cannot be sensitive to a split in the Lagrangian since they are insertions into the full theory. Hence the Normal Lagrangian Products defined this way are independent of the split.

5. THE RELATIONSHIP

TO OTHER NORMAL

PRODUCT

FORMULATIONS

We have proposed a choice of expressionsfor renormalized operator insertions in any polynomial Lagrangian field theory, and by extension the definition of the Normal

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S. COLLECOTT

Products of the interaction vertices in the Lagrangian. It remains, however, to make the connection with other approaches to the redefinition of the products. The Zimmermann approach [l-3] to normal products has been extended by Lowenstein [9] to cover the Green’s functions involving several composite operators. This gives rise to Lowenstein’s Differential Vertex Operations (DVO’s) which insert JZ[@J(x) & into a Green’s function, and are shown to correspond to acting by (a/+) lpzOon the Green’s function defined with the Lagrangian 2%“. This corresponds to our definition I[@] = (a/+) I,,=,, 5?$’ above. Lowenstein then iterates this procedure to define the finite Green’s functions with any number of operator insertions by applying the appropriate sequence of (8/2~~) I,,=,, operators. He writes these objects

Tn1 N[@jl(yj) yl(-yl)..*P)N(-~N) > and they are defined via a finite part operation acting term by term on the Gel1 Mann Low expansion. However this involves different notions. What has been defined is an operator written s flTj ?I[@,]( yjj ayj , which we shall write as s I”“[@, ; Q2 ;...I x . ) nj dJ>j which gives a finite result upon insertion in a Greens function of the tv1 >... theory, and is in fact the ((a/&)(a/+J ... !,=,)th derivative of exp i J L$$‘,,(.u) A-. In a simple case

That this gives finite Green’s functions is part of the definition of 2;” and to render a Green’s function finite within a subtraction scheme is essentially trivial. What is not trivial is to relate Z[@, ; Cp,] to the product of operators representing @r and Q2 . What our previous proposal would mean is that we would take objects

for some A. An insertion of ~lZ$‘[@I] is finite to order p1 being just ~lZ1O[@l]to that order and similarly for ~+Zti~[@~]. Also an insertion of z+&[@~] Zt$‘[@2] is to order p1p2 just ~I~2Z~~r’[@I]#[@,I and the sum of all possible insertions to the QIQ2 level is just

and is hence finite to this order as above.

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363

If we now go on to consider a theory with invariances, typically a gauge theory, then it is useful to have (intermediately) Renormalized Lagrangian Products R[Qi] so that

and R[@<] = (Z’.&j @j = LijGj in previous notation. According to our first criterion, N[@J = Qi + O(h) this would be a perfectly good Normal Product, but as we have seen,not the useful one as far as the equations of motion are concerned. It is, however, useful when it comes to considering gauge invariance. The gauge invariance of a theory is originally defined by transformations on the bare fields with, in nonabelian theories, a gauge transformation dependent on the bare gauge couplings. In consequencewhen expressedin terms of the Generic fields and coupling constants (or in common parlance the renormalized quantities) the gauge transformation involves the renormalization constants. Hence it is what we have called the Intermediately Renormalized products, which have soaked up the renormalization constants, which have simple gauge transformations. It is in conse quence of this that when, in the manner of Kluberg et a/. [ 151,we seekgauge invariant renormalized operators, we look at, in our language, the eigenvalues of the matrix ZiR defined by Z[@J = z;;R[@j]. This can of course be cast into statementsconcerning the N[Gj].

6. A SIMPLE EXAMPLE -

[@,y3,g)14

We now give an example of the procedure outlined above for redefining the field products of a theory by considering a very simple example which elucidates most of the points. We choose a q4, 93, y field theory in four dimensions, and work to one loop order. The superrenormalizability of two of the couplings obviously helps for simplicity. Thus we have a generic Lagrangian

Our field monomials are thus GO = 1, @‘1= y, @?I= A$, Gp,= (t/3 !) q”, G4 = 0”~. The three types of l-loop divergence, Fig. 1, can be parametrized in terms of one quantity

and for any B let B -= PP(B), B - FP(B). Then the divergences of the three loops

364

P. S. COLLECOTT

a

FE. I.

b

c

The three one-loop divergences in four dimensions

are A and -&A and (&/2) A. Hence we are led to a Renormalized Lagrangian in the standard counterterm fashion.

the respective diagrams being given in Fig. 2. From this we can derive the renormalized equations of motion (a2

+ m2) F = -(&A”(.@4

+ ~m2g(A/a)) q - (g + $g2(A/lx))(1/3!) C$

- (A -t- jgh(A/ol))( l/2!) p” - (h + &+i/a)).

c

d

e

Q 0

FIG. 2. Diagrams contributing to the renormalization constants. (a) 2-point coupling, (b) 4-point coupling, (c) 3-point coupling, (d) 1-point coupling, (e) O-point coupling.

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PRODUCTS

Hence we need to know about operator insertions: I[‘U], I[?], I[$$], 1[(1/3!) y3], K12~l = GW%)l an d one way of stating the problem is that we need to know how to write these equations of motion in the form of the Generic equations of motion (an2 + nz”) Z[y] = -gI[(1/3!) This is not obvious from the renormalized Wilson Expansion. Using the formula

$1 - hl[&$] equations

CDL= Z[@[] :- 1 PP
-

/71[1].

of motion,

and we need the

I[@,]

we are led to 21=~I f[Q], 9‘ = I[yl,

m/J = m2y1. Putting this in matrix form we get Q Y 2

J

u/3 %p3

1 il”pl ;y,' aY Ii = Inverting

0 ~nz'(A/n) 0I

-&7"( A/a) -&q,qoc) 00 1

-A(‘+!) l-fg(&u) 0

I-$g(&X) 00

0 0I

(1;3Y)p3 m2P,

this to l-loop order is trivial and gives II

I

I IIII ;y,’

E

-&m*(‘qa)

gqLqa)

1+ ; g(A/n)

(l/3 9p3

0

&m2(A/cl,)

A(J/,1)

91 cl”y

01

01

0

0 0 x 1+ ; g( A/S%) 0 0 011

Y $p” (1/3!)g” c12y

366

P. S. COLLECOTT

It will be noticed that theseresults could, in this example, have been obtained directly by differentiating 2& w.r.t. h, m2, h, respectively. This is merely becauseall the products from the equations of motion happen to be already present in the Lagrangian, and to consider the effect of the insertion we are effectively looking at the first-order effect upon increasing the relevant coupling constant. For example, inserting ~(1/3!) y3 we look for the order r-term in the renormalized Lagrangian in terms of ;\ $- E, which is the differential w.r.t. h at l = 0. Now we can “functionally” integrate to find the Normal Products

samematrix =i

:i I 1.

As a check we can add up -- xi hjN[@J and note that we do get the full Renormalized Lagrangian. However this procedure has told us, for instance, how to apportion the term #gh(A/ol)( l/3 !) y 3 in between the two Normal Products to which it contributes in such a way that we have a canonical derivation of the equations of motion.

REFERENCES I. W. ZIMMERMANN, Renormalisation of composite field operators, irz ‘*Lectures in Elementary Particles and Quantum Field Theory,” 1970 Brandeis Summer Institute in Theoretical Physics, (S. Deser et al., Eds.), Vol. 1, MIT Press, Cambridge, Mass., 1971. 2. W. ZIMAIERMANN, Ann. Physics 77 (1973), 536. 3. W. ZIMMERMANN, Ann. Physics 77 (1973), 570. 4. J. C. COLLINS, Nucl. Phys. B 92 (1975), 417. 5. P. BREITENLOHNERAND D. MAISON, Munchen Preprint MPI-PAE/PTh 25. 6. F. JEGERLEHNERAND B. SCHROER, Nucl. Phys. B68 (1974), 461. 7. P. S. COLLECOTT AEU‘D J. C. COLLINS, Nucl. Phys. B93 (1975). 217. 8. J. LOWENSTEIN, M. WEINSTEIN, AND W. ZIMMERMANN, Phys. Rev. D 10 (1974), 2500. 9. J. LOWENSTEIN, Phys. Rev. D 4 (1971), 2281. 10. J. LOWENSTEJN, Comm. Moth. Phys. 24 (1971), I. 11. G. 'THOOFT AND M. VELTMAN, Nucl. Phys. B61 (1973), 469. 12. K. WILSON, Cornell Report, 1964; and Phys. Rea. 179 (1969), 1499. 13. K. WILSON AND W. ZIMMERMANN, Comm. Math. Phys. 24 (1971), 87. 14. P. S. COLLECOTT, Cambridge Preprint DAMTP 75/20, submitted to Ann. Physics. 15. M. KLUBERG-STERN AND J. S. ZUBER, Saclay Preprint DPh-T/74-56.