- Email: [email protected]
Quantum electrodynamics as a supersymmetric theory of Loops
ANNALS
OF PHYSICS
173, 249-276
Quantum
(1987)
Electrodynamics as a Supersymmetric Theory of Loops* S. G.
l3eparlmenl
Center .for Theorericul of Physics, Mossachuserrs
RAJEEV
Physics, Laboratory ,for Nuclear Science and Insbtute qf Technology. Cambridge, Massachusetts
Received
January
02139
13, 1986
It is shown that the Wilson loop is not a renormalizable operator even in free QED. A modification is suggested and proven to be renormalizable. Feynman’s worldline formulation of QED is generalized to include spinning particles and matter loops, using this modified Wilson loop. QED formulated this way is shown to have a supersymmetry. A superfield formalism is found that describes it. QED is rewritten as a second quantized held theory of loop functionals. ((” 1987 Academic Press. Inc. I.
INTRODUCTION
The problem of understanding quantum chromoldynamics (QCD) in terms of color singlet fields is one of the outstanding challenges in particle physics. Much work was done in this direction [l-5], yet the problem remains unsolved. There are some indications that QCD is equivalent to a theory of strings [2] which approach a free theory as [3-51 N, + m. In this paper, we study the much simpler problem of rewriting quantum electromdynamics (QED) in terms of gauge invariant operators. The main motivation is the hope that some light will be shed on the non-Abelian theory. Of course, one does not expect confinement in QED-at least not for weak couplings. So these gauge invariant operators will not create one-particle states. (In QCD, due to confinement, we expect the one-particle states to be created by gauge invariant operators. ) However, QED is of some interest as a theory in itself. There has been previous work aimed at rewriting QED in terms of gauge invariant operators [6-g]. The basic results of this paper are as follows: 1. We first ask what natural gauge invariant operator is, whose Green’s functions are to be studied. A popular choice is the Wilson loop,’
* This work is supported in part through funds provided by the U.S. Department under contract number DE-AC02-67ER03069. ’ .x“(u). 0 < c < 7’ is a closed path: -p = 1 .. ‘4; we are in Euclidean space [x’(O)
of Energy
(D.O.E.)
=x(r)].
249 OOO3-4916/87
$7.50
Capynght 0 1987 by Acadamc Press, Inc. All rights of reproduction m any form reserved.
250
S. G.
RAJEEV
However, we shall show that even in a free gauge theory, there are severe problems in defining W[x] as a renormalized quantum operator [9-111. 2. In any case, in a theory with spinning matter fields, it is more natural to consider a modified Wilson 10op,~
The (T. F term represents the “precession” of the magnetic moment p. In fact, this operator will be shown to be renormalizable (in an appropriate sense) if the magnetic moment is that of a Dirac particle. 3. x[x] is basically the action of a spinning particle propagating in a loop. But this can also be written in terms of a path integral over Grassman variables [12-131
where
and
This Lagrangian for a charged spinning particle has a supersymmetry (SUSY). In fact, one could say that it is this SUSY that is responsible for the cancellation of the non-renormalizable infinities in 1. To make the SUSY more explicit, we may consider
The objects we study are the n-point function of this operator. 4. We shall see that this SUSY is unbroken by quantization. The Ward identities of this symmetry will be derived. 5. For this purpose, a one-dimensional superlield formalism will be developed. 6. We will integrate over the photon and electron fields of QED and express it entirely as a theory of the operators x[x, $1. This will be an interacting theory of loops, exactly equivalent to QED. This [14, 151 is3 similar to Feynman’s action-at20p,, = t[y,,, y,,]. Y,, are the Dirac matrices, and g is the electric charge. We have used “$’ both as an index and to denote magnetic moment. There should be no confusion from this. 3 Feynman succeeded only in doing scalar electrodynamics this way, and that too without scalar self-interaction and without scalar loops. This “approximation” to QED will be of much use to us in Section II.
QUANTUM
251
ELECTRODYNAMICS
a-distance formalism of electrodynamics. We have now succeeded in introducing spinning particles as well as fermion loops into this formalism. 7. There are some analogies between our theory of loops and string theories. In fact, QED seems to be equivalent to the massless projection of some supersymmetric string theory. However, we will show that our theory of loops has only a finite number of propagating modes, unlike string theories. For non-Abelian theory, there are arguments [3, 51 that there must be an infinite number of propagating modes. 8. We will see from the proof of equivalence that the theory of loops is a renormalizable theory. We will also provide a gauge and Lorentz invariant, supersymmetric regularization. Section II and Appendix A study the question of renormalizability of the Wilson loop in free QED. Section III and Appendix B prove that the modified Wilson loop is multiplicatively renormalizable in free QED. Appendix C develops the superfield formalism to the level required. In Section IV and Appendix D we study QED with fermions and develop the expansion in powers of N,.. In Section V we compare and contrast the theory of loops with string theories. Section VI contains some inevitable speculations.
II. THE WILSON Loop
IN QUANTUM
THEORY
In a pure gauge theory, 9 = $F?,,, a natural gauge invariant
(2.1)
operator is the Wilson 10op,~ (2.2)
The Wilson loop plays a useful role because many other gauge invariant objects are derived from it. For example, consider a spin zero quantum field d(x) coupled to a background electromagnetic field. The following is an example of a gauge invariant expectation value
(2.3) Here, z is a curve that starts at x’ and ends at x.
4.u(t) denotes a closed curve, 0 4 I < T and x(0)=x(T). The symbol 2 will be used for path-ordered exponential. For Abelian classical gauge theory, there is no need for an ordering prescription.
252
S. G.
DA(x, x’) is the scalar propagator path-integral representation
RAJEEV
in the background
A. This has a well-known*’
Here, Q,(x, x’) is the set of all curves ~4t) such that y(O) = x y(T) = x’. This gives a path integral representation
for @[z] (2.5)
Here, y + z denotes the closed curve obtained by combinig y and Z. In this section we study the question of renormalizability of the Wilson x(t) is a differentiable curve and A, is a classical gauge field, there is no with the definition of W. However, if A, is a quantum field, there is no meaning to (2.2) without malization. For example, even d2(.y) is not well defined in a field theory, renormalization. The usual resolution is to define 1211 q?(x) = Yt, _ lifc,(x,
x’) (b(s) qqx’) - c*(x, x’)].
loop. If problem a renorwithout
(2.6 1
If c-number functions (ci, c2) can be found such that the above limit exists, then this is a valid definition. But it is by no means guaranteed that such a function exists. (Typically, renormalization also might mix different operators.) In our case, there is an additional complication. We are not satisfied by W being well defined for smooth curves. In most applications, W appears inside a path integral over .Y, over some range 52 In
S[x]
ep
w2)j.i’wrW[X]~
(2.7)
This means that we must consider W as a functional on the space of all curves that can contribute to this path integral. Unfortunately, this is a bizarre class of curves. They are functions such that, llxll* = j .2’(x) dr
QUANTUM
253
ELECTRODYNAMICS
is finite. It is known that a “typical” function of this class is continuous differentiable [22]. In fact, they are “half-differentiable,” i.e., Yt “-1
but not
Ix(t’) + x(t)1 ,t’-f,l12
exists. Even when A, is a classical background field, W[x] = erg~a~‘U(r) dr may not be well defined [22] as a function of the space of curves 9. Rather, it is a distribution, so that
exists for an appropriate class of test functionals. The class of test functions allowed is determined by physics. A natural choice is F[x] =exp i J,,(t)x,(t)dt.
(2.8 1
i
If we differentiate with respect to J, we will generate the various expectation values <-Y,,,(tl) . -~,,(t,,) >. On the other hand, we may choose J,,( t ) = q1’ ‘6( t - t , ) + kj,2’6( t - tr). . + kjp(
t - t,,).
(2.9)
Then we are calculating (e Al I(rlleik,- r,,(In)>
(2.10)
which is also very useful physically. (See Sect. IV). The quantity I/(kf) .y) = p.\-“1 will be called a “vertex functional” by analogy with string theories. Now let us consider the case where A, is quantized as a free field. We may be tempted to say that W[x] is a renormalizable operator if we can find a function C depending on the U.V. cutoff a il such that
w/J-~1 = Wxl C,,C~l has finite n-point functions in the limit A -+ cc. But, as described earlier, we are really interested in averages of the kind’ S(kitip,s,) 5 G[.x]
= F”
= J 3[x]
211 i’“r9[.y],
L2[y](Ol
so g[x]
simply
W[x]
W[v]lO)
f erk+(‘l) i eip~-v’s~‘. (2.11) i=l ,=I
denotes
the Wiener
measure
over
closed
curves.
254
S. G.
RAJEEV
Let us introduce a length a as an U.V. cutoff. W[x] Wilson loop, = e ita(
W,[x]
will then be replaced by a “bare”
“p”“c,[x],
(2.12)
go(a) being a bare-coupling constant. C,[x] is a sort of wave function renormalization of W. This is a C-number function. Actually through the path integral over x, the vertex operators eik’-‘(‘) might also need a wave function renormalization. Thus we are finally led to the following definition, DEFINITION.
W[x]
will be said to be a renormalizable
operator,
if there are
C,Cxl, ZCal, go(a),suchthat
’
fi
fj
I=1
i=l
V(k,itliI
x/(th))
z(a)
(2.13)
exists in the limit a -+ 0. This definition is somewhat different from the way the renormalizability of a composite operator is usually [21] defined. In a scalar field theory, for example, an operator 6(x) is defined through the functions G(x; x, . ..x.) = (O@(x) q4(x,)...q3(xn)IO). Here, b(x) is the fundamental scalar field. Even for a multiplicatively malizable operator one does not find that 9(x, .. ‘X,)’
renor-
(olo(x,)~~~o(x,)lo)
exist. The latter is more divergent than G. However, our aim is to rewrite the entire theory in terms of Wilson loops. So it seems natural to require that the n-point function of the Wilson loop (similarly averaged) exists. Next, we will study whether the Wilson loop is renormalizable in the sense defined above. For practice, let us first consider smooth curves that do not intersect. The n-point functions %‘oc-Xl. ..x.]
= (01 W[x,]
... W[x,]lO)
(2.14)
can be calculated explicitly (in the presence of an U.V. cutoff a) for pure Maxwell theory. It is just a matter of doing a Gaussian path integral over A,
QUANTUM
255
ELECTRODYNAMICS
1
dt dt’ .
Here, we have introduced photon propagator
(2.16)
a length “a” to serve as an U.V. cutoff replacing
the
(More fancy cutoffs like dimensional regularization could have been used, but this would do just as well for our purposes.) Divergences arise whenever x,(t) + xj( I’). This happens whenever (i) (ii)
t-tt’.
Two different curves intersect i#j.
x;(t) = Xj( t’), (iii)
Some curves intersect itself x;(t) = Xj( t’),
t # t’.
The latter two cases do not arise by hypothesis. Let us see if W can be “renormalized” for the case of smooth, non-intersecting curves. This has been studied by Polyakov and the answer seems to be that it is possible. The infinity in $9 arises from the region z + r’ in the first term of (2.5). One may calculate in the neighborhood of t N r’,
Thus, the infinity can be removed by defining W,[x]
= e -(l/*)sm(~)I~dr~igIA~.~i.Pdr
>
where &n w u/a and m. is renormalized to cancel the above infinity. For this to work it was important that there were no infinities from the second term, which represents the interaction between two curves. This happens because we assumed that they would not intersect.
256
S. G.
RAJEEV
If the curves intersect, there is an infinity as can be seen by studying the neighborhood of an intersection. Let x(r) and y(6) be two curves intersecting at T = t; r’ = t’. In this neighborhood
.<(T).j(T’) i(t).j(t’) i I a d7dT’[X(T)--!‘(51)]2+a2-a dTd7’[x(t)(r-f)-j(t)(7’-1’)]~+a’ -c1 i(t).y(t’) In a/E.
i2(t)j2(t)
- (.t(t)yyt)y
So within the class of smooth curves the operator W cannot be given a renormalized definition: there are infinities that cannot be absorbed by redefining W. But this in itself need not be an insurmountable problem. For, we are interested in whether G exists in the limit a + 0 after renormalization. The situation that Y blows up on a certain class of curves need not be a problem if this class does not contribute to the path integral. Now the question of whether the above average exists is the same as whether the one-dimensional “field theory” defined by the action
leads to finite n-point (Ole”” (‘I). eik8r”Jl IO) if m, is appropriately renormalized in the limit a -+ 0. A renormalization of m will (if we go back to the definition of ~o[x, ” x,,]) be absorbed as a redefinition of W. This seems like a very difficult issue at first. But in fact we can reduce this to a well-known problem in ordinary quantum field theory. The action S is the actionat-a-distance form of scalar electrodynamics [ 14J6 In fact, by doing perturbation theory in c( we can recover the Feynman rules. By expanding in tl we see that Feynman diagrams of scalar QED are recovered with two important exceptions. (1) (2)
Only diagrams with exactly n scalar loops appear. There are no scalar self-couplings.
Thus, the problem of renormalizability of the composite operators W[x] has been reduced to the following question: Can all the divergences of this truncated scalar electrodynamics be absorbed into a mass renormalization?’ First of all, since no “virtual” scalar loops appear due to exception (l), there is not charge renormalization. There is a mass renormalization due to the usual selfenergy diagrams. o We are still studying the renormalizability of a certain composite What we are doing is to rephrase the problem as the renormalizability ’ There might also be renormalization of the source, Z(a).
operator in free gauge theory. of a certain interacting theory.
QUANTUM
257
ELECTRODYNAMICS
However, in scalar electrodynamics there is also another divergence, a self interaction of the scalar field, There is no way to remove this divergence except by adding a term
.I62.1
dT, d7,
d4[X,(T,)-S,(t,)]
dT,d7,
I
to the action S. But this means that the operator W are not renormalizable in the sense defined earlier. There are infinities in the n-point functions that cannot be removed by a redefinition of W. This is claim (1) of the Introduction. III.
THE MODIFIED
Since we are ultimately to consider’
WILSON LOOPAND SUPERSYMMETRY
interested in QED with fermionic matter, it seems natural
instead of the Wilson loop. This is the amplitude to propagate around a loop.
for a spinning particle with g = 2,
x is as good as the Wilson loop for heavy particles, becausethe magnetic moment
term is suppressed for such particles. Also, in the limit of very large curves, this term has little effect, the magnetic moment coupling falls of faster.’ However, the cr. F does not have an effect on the ultraviolet properties of the theory. In fact, it improves the U.V. properties so that x is renormahzable whereas W is not. This is proved in Appendix B. Thus, even in a pure gauge theory, we should use x rather than W to get gauge invariant amplitudes. The improved renormalization properties suggests that there is a hidden symmetry in the problem. We shall see next that this is so. There is a hidden supersymmetry in the Green’s function of x. There is a way to describe a spinning particle using Grassman variables rather than spin matrices. Consider the Lagrangian [ 131 (3.2)
Here $J‘ is a real Grassman variable. This is invariant
*m’F=fJ ,” F,,,,; u,‘“=i[Y,,,Y,~l.
9 We thank
B. Svetisky
for discussions
on this point
under
258
S. G.
RAJEEV
as can be verified by straightforward computation; E is a real Grasmann number. To see that this supersymmetric quantum mechanics really describes a spinning particle, let us canonically quantize Y. If g = 0,
a9 y= aGp ill/". So that the canonical commutation
A representation
relations are
of this algebra is obtained on Dirac wave functions qJx)
yr being the Hermitian
Dirac matrices. The Hamiltonian
is”
H= +~(-V*+m;)SxP For the interacting
theories, we get similarly
H=$((iyD)*+m;) =g-D*-o.F+m;]; D being the covariant derivative, D,, = a,, - igA,.
Now we can write x in terms of 9, x[x]==j9[lj]e-% To see this, we use the following expression for eiH’ for small”
T,
e-(.r-.r1)2/4T (x'j31eCHTIx,
lo a/3 are Dirac indices. I’ O(T) denotes relative
order
a)
=
in T. Remaining
16n2T2
x @(x, x’) + O(T),
terms are - TX lirst
term.
See Appendix
D.
QUANTUM
ELECTRODYNAMICS
259
where
the integral being along the straight line
It is more reasonable, in view of SUSY to consider x[x, $1 z e-Jcb’d
without averaging over $. This makes the consequences of SUSY explicit;
4”(c):= CxYl), V(t)1 is a closed curve in a space with four real and four Grassman coordinates. Let Q be the generator of SUSY. Consider the n-point function s
x[(,]...x[S,,]
e~“‘4’JF2~[A]=~~[5,...rn].
These Green functions satisfy the Ward identities of SUSY Q,30=Oo;
j= 1 . ..n.
i.e., - ii+F‘( I,)
These survive quantization because dimensional regularization (for example) preserves this symmetry. We will actually use a different regularization which also preserves it. Now, we can actually calculate q. by integrating out the photon explicitly
where JJ~)=~j~dci;‘(r)S(x-xi(r)) i O +~~T’dr~~“‘(r)~vl’~(z)a,h;[~-xi(~)]. i
O
Thus, go = e-“2J
(J,(~)J,(r,)l(5--‘,)2+.2)d,~.~
= exp - S,C5,1,
260
S. G. RAJEEV
where
The expressions are beginning to get complicated. It would be nice to have a short-hand notation for this. This is provided by the “one-dimensional” superfield formalism’* of Appendix C. Then S[:jl=fx
j(i’D3<:‘dr+m;
jdt
I ,.
We conclude by pointing
out that $, has the form To= axK1-‘xc~,llo) = rtiie
xoC:r.
:,J
Compare this with a free-field theory of a field with expectation-value
zero,
<4(-y)4~) > = D(x, Y) then (e in& ~1). . eE4+(‘“)) = nile-“D(“‘. c/j, So we are beginning to see the connection of QED to a free-field theory. However, this is a free-field theory in an infinite dimensional space. In the next section we elaborate on this connection and also extend it to the interacting theory.
IV. EXPANSION IN POWERS OF IV,Consider QED with N,.“flavors” of electrons. If one looks at the action of string theory (in Sect. V) that is claimed to be equivalent to QED, one sees that N, acts like a coupling constant. This suggests that we should look for an expansion of QED in terms of N, and not IX. Such an expansion is interesting in itself. ‘* Reference
24 contains
a similar
formalism
for left-handed
superlields
in two dimensions.
QUANTUM
261
ELECTRODYNAMICS
NY is simply a parameter that counts the number of electron loops. The zeroth order in N/is the sum over all Feynman diagrams of QED without electron loops. The first order is the sum over all diagrams with exactly one electron loop and so on. For example, the p-fn of QED vanishes to zeroth order, but not to the 1st order in N,-. Such an approximation to the fl-fn has been studied before [18]. If we “integrate-out” the photon field in QED paying no attention to the dressing of the photon Green’s function, we get the zeroth orders approximation to QED. This is the action-at-a-distance QED discussed [ 141 in Section III. By a systematic expansion in powers of Nf we will find this idea of Feynman to reproduce QED exactly, not merely in the N, + 0 limit. Using supersymmetry, we will also be able to extend his ideas to spin 4 particles. Actually, the case of fermionic QED is more self-contained, since all interactions are given by the gauge principle. In scalar QED, there is also the @” integration (necessary for renormalizability) which has to be represented as a contact interaction of the world lines. The idea of the limit N,+ 0 is a well-known technique in the statistical mechanics [20]. There it is used to approximate the average of an exponential by the exponential of the average
The approximation of ignoring electron loops in QED is somewhat analogous to the large N,. limit in QCD. In the large N,. limit, quark loops disappear. But the analogy should not be carried too far, since there is no restriction in QED on the topology of the Feynman graphs. Let us consider a typical two-point function. 1. d’(x,y)=Se “/4)~P,Syr~-n+mo’y(1/~10) y(x) q(JJ) 9[,4] 9[q]. In Appendix D we have shown that the functional integral over q can be converted to an infinite series of worldline integrals: L.
grx]
X
g(G)
,-rK
vcc.
@Id’.
r(O) = li x(Tl=.r’
The point of writing the fermion path integral this way is that the photon integral is a mere Gaussian. In fact, 2 may be written as
262
S. G.
RAJEEV
Here, the measures have been redefined
NOW
we just do the integration
over A, using Appendix C. We have
where the K.E. terms have been absorbed into the measure. Thus, we have the required action-at-a-distance form of fermionic electrodynamics. Note that the zeroth term in N, is what occured in Appendix B. Note also the manifest supersymmetry.
V. We have accomplished
QUANTUM
FIELD
THEORY
OF Loops
our task. We have shown that given
We can calculate any object in QED by a “mere” worldline integral e.g.,
Furthermore,
we have expressed $9 in terms of gob,
To=
I
9[A]
e~“‘4’rF2X[51]...X[Tn].
In fact, So is known explicitly, ~o~ee~,,,S/C5,,5,1~e~~x,.,Dce,.~,1
where DC<, 5’]=
j dr de DY( r, 0) (5 _ c;j2 + a2 Dc”( t’, 6’) dt’ de’.
QUANTUM
263
ELECTRODYNAMICS
The 9 is known as an infinite series in 2&,
Let us now see if we can rewrite these to form a quantum field theory of loops. An object that creates a loop will be an operator-valued functional d[r]. Just as d(x) in scalar field theory creates a particle at X, @[r] creates a closed string 5. Suppose we postulate a “propagator” for 4,
D being as defined earlier. Then, we can verify that
Colek4C511.. Proof:
~e’g~cS~llO)=~~[~,...~,].
By Wick’s theorem, (()~,v+‘WCiJ
%1/o)
- 2 j-‘[;l
=e
Ott.
;‘I J15’1
gCS1
act’3
Now choose
These formal manipulations can be given a precise meaning by approximating a path integral by a multiple integral and taking the limit. Note that reversing the orientation of the loop changes the sign of 4, c&./l
,&Crll=
s
s
Gjc?l cmat-?I.
Thus,
Y[5,-~EJ=~+mJ : ,mm[rIl.. =
(Ole-
-*9[q,](OINf: . &a;,1 S~Cllek6CS11..
I
0
cosg$[q,]:
>3 e’RICCnll()),
where
This looks like the Gell-Mann-Low 595, ,73:2-2
formula in field theory.
... N,: cosg4[rj,]
264
S. G.
RAJEEV
Thus, if we have a field theory in loop space with the propagator
(WErl and interaction
ii-5’110> =N5,5’1
action s, = N,: j cosgdC
we reproduce the series for QED. Now, it seems that we have more than we started with; CJ~[[], being a functional, has an infinite number of fields in it. But we are claiming that this is equivalent to a field theory with a finite number of fields. How is this possible? In fact only a finite number of modes of 4 actually propagate. To see this, let us “diagonalize” the propagator D,
=
s
dq dz d0 de d-c’ D:“‘(q)
I’;&]
V,,,,&].
(This is not diagonal in ~1,v and r, t’. But these indices can also be diagonalized a linear and a Laplace transform, respectively.) The “vertex function” is defined by
by
V,,pRCrl = myr, H)eiy<(r. 0) These functions span a subspace in the space d[t] the modes of d[<] that can be written as
labelled by a A,(q).
Thus only
actually propagate. This is merely a finite number of modes in space-time. O[<, 5’1 cannot be inverted to yield a “kinetic energy” operator, because it has mostly zero eigenvalues. There is no K such that
However, suppose P[c, 5’1 is the projection we can verify that
operator to the “photon”
modes. Then,
QUANTUM
ELECTRODYNAMICS
265
Consider the operator lt=;[dcf+;dE
+ SUSY completion. 62 6x(a + E) 6x(a - E)
Then
m5,
5’1 = per, 4’1.
Thus, within the subspace of propagating modes, R is the “K. E’ operator. Thus, we find the theory in loop space to be not a string theory. There are only a finite number of propagating modes, whereas in string theory there are an infinite number.
VI. CONCLUSIONS We have shown that the correct operator to study in a gauge theory is the modified Wilson loop ~11x1 and not the usual Wilson loop W[x]. In this paper, only the Abelian theory was considered. It will be shown in a forthcoming paper that x[x] and not l+‘[.~] is renormalizable in non-Abelian gauge theories also. A formulation of non-Abelian gauge theories in terms of x[x] will be quite interesting. The field A,(x) of QCD does not appear to have a direct relation to observable particles. But, we conjecture that x[x] creates linear combinations of one-particle states (i.e., “glueballs”). Also, we believe that the projection to an operator that creates a single eigenstate is the average
sD[x]x[s]V,[x]; P’,[.y] are vertex functionals similar to the ones encountered in the text. Of course, nothing in this paper confirms these conjectures. We intend to follow up on these issues in later publications. The study of QED has led us to the correct choice of loop variables. It also has shown us how to include fermionic matter fields. Even though the pure gauge sector of QCD is much more involved, the interaction of gluons to quarks is not too different from that in QED. So we expect Appendix D to go through without major changes. A functional integral representation for the quark Green function in the presence of an external gluon field can be derived using the Lagrangian of Reference 13. It is conceivable that our way of formulating QED will lead to a better understanding of its non-perturbative features.
266
S. G. RAJEEV
APPENDIX A: ACTION-AT-A-DISTANCE ELECTRODYNAMICS FOR SCALAR PARTICLES
We want to show that
(A.11 describes scalar QED except that the following two kinds of Feynman diagrams are absent: 1. scalar loops; 2. scalar self-couplings. We will prove this in the presence of an U.V. cutoff. So U.V. infinities are regulated. One proof is, of course, in Feynman’s papers [ 14). We give a more updated one. A different approach is mentioned in Appendix B. Consider a scalar 2n-pt function, G,= (OIT~*(X,)~(Y~)~*(-K,)~(~~)...~*(.~,~)~(Y,,)IO)
(A.21
in a theory with action s= 4 j A,,(x) qy(. ~,~‘)A,,dxdx’+~j(Dq5)~(~)d.x+~jm~~q~~d~x; k,(x, x’) is the photon inverse propagator
with a cutoff Q. In terms of path integrals
e -(1;2)jAkAd.YdY’e-Il:2J~~)*D~dr~ll;rl~m’l~I’d.~
G=jg[A]
x #*(xl )4(.Yl).. The functional
(A.3)
4*(x,) 4(Y,).
(A.41
integral over Q can be done G=j!&4]e '1
-(1/2,1AKad.~dr’(Det[_02+n12])
(A.51
{D,(-uil-Yj,)D,(xiz-l'jl)D,(xi~-,~j~j.
Now the process of ignoring scalar loops is the same as dropping [Det( -D” + m*)]. (We will call this process “quenching.“) Then Gzue=
s X
CJJ’CA,,-(1/2,jaKad.Yd.r’ 9[Xi]
sy
the
dT,
e- ll/2)11,=,I.~j(,)dr-m2S~d~ei’~,lA,i;(rJdr
(A.61
QUANTUM
267
ELECTRODYNAMICS
Where the well-known path integral representation for the scalar n-pt function in the presence of an external gauge field has been used. Sz is the set of all paths such that {x;(O)] = (x;} (A.7)
Now calculate the path integral over A to get
G=jm i dl;e-“2’rr,/2’j 0
9[x;] e-S[-yl, R
,=I
S being the action defined in (A.l). Thus, the action S yields the n-pt of truncated scalar QED when integrated over the region f2. Now, we know from the renormalization theory of SQED that 1. In fact, 2. 3. 4.
If scalar loops are ignored there is no coupling constant renormalization. there is no correction at all to the photon propagator. There is a mass renormalization. There is a wave function renormalization Z”(a) There is also a divergence associated to the four pt function. So G,,=Z”(a)
G(po(u);x,
. ..x.,)
is not finite; one has to add a 14” term to the action to recover renormalizability. the worldline formalism this is
In
The quantity in the integral leads to a 6 function as a -+ 0. In the text we considered the path integral over closed curves whereas above we considered a path integral over open curves. The difference will show up in the Feynman rules as a modification of the propagator, and of the range of integration over the proper time parameters. Now integrate over all t’s between 271 and 0. To derive the new propagator let us calculate directly from the path integral
(OlekCr(r)-.~(~‘)310) =s (l/2)~~.~i-Z(T)dreik[x(r)~x(r')]
This can be done by Fourier analysis x(t)=uo+
C u,cosZnnf+
n;tO
1 b,sin-
n#O
2X” t T .
268
S. G. RAJEEV
We will spare the reader the agony of detail. The answer is (Ok
ik’C.~l(r)~.~(r’lllO) _e-k’/l-tl,
where
Note that as (t’- t) + 0, f(t’ - t) - It’ - tl. Thus, near the region t’ - t or k’ -+ co, these propagators reduce to the earlier one. So the change in the boundary condition does not change the degree of divergence of the Feynman diagrams.
APPENDIX
B: PROOF OF RENORMALIZABILITY
OF x[x]
IN FREE QED
We have
where x(z), x(0) = .x(T) is a closed curve and [Tfl5’= fCYp ?,,I. We will be interested in the expectation values G(k) = j z(a) eh. ““IL(~)
,$k, ‘+‘I%) fI a[xf]
e-
(1/2,S>&
~~lT)dT,-m;(u)
r,
i=I x
03.2)
(“t~[xll~~‘~[Xkl~O).
The task is to show that all the divergences in this that occur as the U.V. cutoff a + 0 can be absorbed into m,(a). There is also in general a “wave fn” renormalization z(a) of eik’ ‘,
K’“’ is the photon kinetic energy operator with a U.V. cutoff a. Consider the simplest case k = 1, n = 0, G(T)=jg[x]
e -(l/2)S,T.~2(r)dr-m~~(OIX[X]10)
G(T)=
e-“‘4)~F2dyGA(T),
or f&4]
QUANTUM
269
ELECTRODYNAMICS
Now, we will see in Appendix D that G,4( T) = j tr(xlePH’IX)
dx
with H= -D’+a.F+mi.
It is convenient to study instead of G(T), its Laplace transform, G(p)= jf
G(T)e-‘12dT
=sg[A]pp(1/2)jF2dr
tr -D’+ga.
1 F+m2
where we have put n12 =mi+p’, tr
1 -D’+o.F+m’
=Ltr m
1 iy.D+m’
Thus, we have related G(p) to the propagator of fermionic QED. If we expand the right-hand side in powers of g, we will recover all the Feynman diagrams of QED, except for those involving “dressing” of the photon Green’s function. Thus, the U.V. divergences in G(T) are the same as those in QED. But we know that in QED all divergences can be removed by mass and wave function renormalization. (There is no charge renormalization, because the photon Green’s functions are not “dressed.“) A similar argument applies to more complicated cases.
APPENDIX
1-D Superfields
Define
Under SUSY,
C
270
S. G.
RAJEEV
SO
Thus,
Q=$+!$ igj,” 1=l.
Then
This is the SUSY algebra. Put
Superfield Interactions 6A, = c3,Ap6x” = -ie[dyA,+“] .-.- - iEB, B, = a,A,$‘.
Thus, A, and B, form the component of a superfield A,(x + ie+) = A,(x) Call A,(x + iO$) = 4, to avoid confusion. Thus, d,, = A,(x)
+ iO$“ayAP
+ iOB,
QUANTUM
271
ELECTRODYNAMICS
so #,Dt” The propagator
= -@$“Il,‘&A,
- i3A$‘+
iA,
I#“.
of 4, is
So the effect of integrating
out A is
APPENDIX
(i) Path Integral Representation
D
of Electron Amplitudes
We will be interested in evaluating functional
integrals such as
or
Here, S=a
I
F*+
I
q,(,~)[y.D+m,)qd.u.
In this section, let us not consider the integration over A. Let it be an external field. Our task is to express these functional integrals over q as path integrals over world lines. We will show that the calculation of any amplitude can be reduced to that of evaluating an integral over world lines such as
tl, t2 are open curves (P(a), ICIP(a)) and the integral is over some appropriate of such curves. Let us consider for example s
9[q]
eJ4cY.D ‘“1”~Y4(x)q(~)=
Ignore the Det for the moment.
+[Det(r.D+m)]b
[-tr
S(x,y)].
class
272
S. G.
S, is the fermion propagator
RAJEEV
in the presence of the gauge fixed A,
(y . D + m,) S,(x, x’) = 6(x, x’) S,(-L -x’) = (Wb’)
4(x)10),.
In fact, S,(x,x’)=(-y.D+m,)A,(x,x’), where A satisfies [ -(y . D)Z + m;] A,(x, x’) = 6(x, x’). Now
Here, the “y. D” term cannot contribute to the trace. A, always involves an even number of y matrices, so tr y. DA, would involve an odd number. So the trace vanishes. Thus, tr S,(x, x’) = m, tr d,(x, 2). The point of the above exercise is to express everything in terms of the “square” of the Dirac operator, A A. This has closer connection with path integral presentations. This, however, is not essential, and we will see in fact that S,(x, x’) can be given a path integral representation directly. But we will deal with A, for the moment. Clearly any number of products of q’s and 4 has a v.e.v. (in the presence of an internal gauge field) which can be written in terms of A, and its derivatives. Now we claim that
Here, 52, is the set of the worldlines such that x(0) =x x(T) = x’. To see this. note first that
with A= -((Y.Dy+m; = -D2-ga.F+mi-.
QUANTUM
273
ELECTRODYNAMICS
Now, for the “propagator” (x’le-“Ix) we find the path integral representation by interpolating states 1x(?,)) at (intermediate) times t;. The expression for
1 .
Thus,
The limit N+ a3, (ti- t,+ ,) + 0 such that N(t,-
t,- ,) + T
leads to the claimed-path integral representation. Next, we write the same-path integral using Grassman Dirac matrices. We claim
Q being an appropriate
Expanding
variables rather than
range. Clearly, it is sufficient to show that
both sides, we need to show that
To evaluate the RHS we need
($“(t’) $‘(t)>= jy] By point splitting,
G2)l$$ d’l//,(t’) l),(t),
274
S. G.
RAJEEV
This is basically a definition of the path integral. Different boundary conditions on the Green function of d/dt lead to different definition of the path integral. Once we know the above formula, the theorem follows using the trace formula for y matrices. Thus we have the integral representation
D = {x, $1 x(0) =x; x(T) =x’; l)(O) = i//(T)). (ii) Path Integral Representation
of the Determinant
Now let us consider the problem of writing the [Det(y . D +m,)lN’ integral. This is quite complicated. We proceed in steps.
as a path
1. Reduce to the Square of the Dirac Operators. Det(y . D + m,) = Det y&y. D + mo) y5 = Det( -y. D + mo). SO
[Det(y.D+m,)lN’=
[Det(y. D+m,)(-y.D+m,)lNf’*
i.e.,
2.
Path integral.
[Det( -7.0)’
+ mi]Nf!2 = exp;lnDet[-(7.D)‘+mi]
Nf =expttrln[-(y.D)*+mz]. Now consider the tr In,
But the limit T+ 0 generally leads to a U.V. singularity. We will assume that the Det is defined with a U.V. cutoff. A convenient cutoff is to do the integral up to T= E. Using earlier results,
QUANTUM
215
ELECTRODYNAMICS
where Q, is the set of closed curves
x(0) = x(T) VW) = $40 3. E.ypansion in powers of IV,. What we need is exp(N//2) Tr In H. We can get it by expanding the exponential
tr In H not
Det(y D + M,) =
This means that we are integrating
over k closed loops. Here, as before,
It is not difficult to figure out what the expansion in N/- means. The power of N, merely counts the number of electron loops. This expansion makes sense even though N,.> 1. (iii)
Worldline Form of Amplitudes
Now we are ready to find any integral over q’s. Suppose we want 1 g[ql
e~~q’~‘r)+Ma’Y~f-rlrq(,~)q(~)=
[Det(y.D+M,)]‘v’S,4(x,y).
We will find this as a formal power series in N,. The zeroth order in N,.contains closed loops. Thus, the above object becomes
no
Next, y denotes closed loops of various periods. The first term is represented by the “tree-level” contribution and the next term is the “one loop” contribution, and so on. ACKNOWLEDGMENTS I have benefitted A. P. Balachandran.
from conversations I am also grateful
with L. Alvarez-Gaumt, for the interest shown
X. Wu, H. Yamagishi, K. Johnson, by K. Huang and C. Hull.
and
276
S. G.
RAJEEV
REFERENCES
1. A. M. POLYAKOV, Gauge fields as rings of glue, Landau Institute preprint. 2. A. A. MIGDAL, Phys. Rep. 102 (1983), 199 and references therein. 3. G. ‘THOOFT. Nucl. Phys. B 72 (1974), 461; 75 (1974), 461. 4. K. WILSON, Phys. Rev. D 101 (1974), 2445. 5. E. WITTEN. Ann. Phys. (N. Y.) 128 (1980), 363; Nucl. Phys. B 160 (1979), 57. 6. S. MANIJELSTAM, Ann. Phys. (N.Y.) 19 (l962), 1; Phys. Rev. 175 (1968), 1589. 7. T. T. Wu AND C. N. YANG, Phys. Rev. D 12 (1975), 3846; C. N. YANG, Phys. Rev. Lert. 33 445.
8. See Section 3 of Reference 2. 9. J. L. GERVAIS AND A. NEVEU, Phys. Letf. B 80 (1979), Phys. B 163 (1980),
225; Nucl.
Phys.
B 153 (1979),
(1974),
445; Nucl.
189.
IO. H. DORN AND E. WILCZOLEK, Z. Phys. C 9 (1981), 49. 11. V. S. CORSENKOAND S. N. VERGELIS,Nucl. Phys. 164 (1990). S-27. 12. F. BOKUI AND R. CASSELBUONI,Phys. Left 938 (1980), 308. 13. L. ALVERAZ-GAUM~. Harvard preprints HUTP-83/A035, 83/A029. 14. R. FEYNMAN, Phys. Rev. 80 (1950). 440, especially Appendix A. 15. R. P. FEYNMAN. Phys. Rev. D 76. 749, 769. 16. M. KAKU AND L. KIKAWA, Phys. Rev. D 10 (1974), 1110, 1823. 17. C. MARSHALL AND P. RAMOND, Nucl. Phys. B 85 (1975), 375414. (We thank C. Huu for these references.) 18. K. JOHNSON, M. BAKER, AND R. WILEY. Phys. Rev. B 136 (1964), 1111; 163 (1967), 1699; M. BAKER AND K. JOHNSON. Phys. Rer. 183 (1969), 1292; Phys. Rev. D 9 (1971), 26, 25; S. L. ALDER AND W. A. BARDEEN.
Ph~s. Rev. D 4 (1974),
3045.
R. P. FEYXMAN. Phys. Rev. 97 (1955), 660; R. P. FEYNMAN. R. W. HELLEWARTH, C. D. IDDINGS, AND P. M. PLATZMAN, Phys. Rev. 127 (1962), 1004. 20. P. ANDERSON, Les Houches lectures. 21. C. ITZYKSON AND J. B. ZUBER, “Quantum Field Theory.” McGraw-Hill, New York, 1980. 22. L. S. SCHIJLMAN, “Techniques and Applications of Path Integration,” Wiley, New York, 1981. 23. J. SCHWINGER, Phys. Rev. 76 (1949). 749. 24. R. BROOKS, F. MC’HAMMAD, AND S. J. GATES. University of Maryland preprint, September 1985. 19.
OF PHYSICS
173, 249-276
Quantum
(1987)
Electrodynamics as a Supersymmetric Theory of Loops* S. G.
l3eparlmenl
Center .for Theorericul of Physics, Mossachuserrs
RAJEEV
Physics, Laboratory ,for Nuclear Science and Insbtute qf Technology. Cambridge, Massachusetts
Received
January
02139
13, 1986
It is shown that the Wilson loop is not a renormalizable operator even in free QED. A modification is suggested and proven to be renormalizable. Feynman’s worldline formulation of QED is generalized to include spinning particles and matter loops, using this modified Wilson loop. QED formulated this way is shown to have a supersymmetry. A superfield formalism is found that describes it. QED is rewritten as a second quantized held theory of loop functionals. ((” 1987 Academic Press. Inc. I.
INTRODUCTION
The problem of understanding quantum chromoldynamics (QCD) in terms of color singlet fields is one of the outstanding challenges in particle physics. Much work was done in this direction [l-5], yet the problem remains unsolved. There are some indications that QCD is equivalent to a theory of strings [2] which approach a free theory as [3-51 N, + m. In this paper, we study the much simpler problem of rewriting quantum electromdynamics (QED) in terms of gauge invariant operators. The main motivation is the hope that some light will be shed on the non-Abelian theory. Of course, one does not expect confinement in QED-at least not for weak couplings. So these gauge invariant operators will not create one-particle states. (In QCD, due to confinement, we expect the one-particle states to be created by gauge invariant operators. ) However, QED is of some interest as a theory in itself. There has been previous work aimed at rewriting QED in terms of gauge invariant operators [6-g]. The basic results of this paper are as follows: 1. We first ask what natural gauge invariant operator is, whose Green’s functions are to be studied. A popular choice is the Wilson loop,’
* This work is supported in part through funds provided by the U.S. Department under contract number DE-AC02-67ER03069. ’ .x“(u). 0 < c < 7’ is a closed path: -p = 1 .. ‘4; we are in Euclidean space [x’(O)
of Energy
(D.O.E.)
=x(r)].
249 OOO3-4916/87
$7.50
Capynght 0 1987 by Acadamc Press, Inc. All rights of reproduction m any form reserved.
250
S. G.
RAJEEV
However, we shall show that even in a free gauge theory, there are severe problems in defining W[x] as a renormalized quantum operator [9-111. 2. In any case, in a theory with spinning matter fields, it is more natural to consider a modified Wilson 10op,~
The (T. F term represents the “precession” of the magnetic moment p. In fact, this operator will be shown to be renormalizable (in an appropriate sense) if the magnetic moment is that of a Dirac particle. 3. x[x] is basically the action of a spinning particle propagating in a loop. But this can also be written in terms of a path integral over Grassman variables [12-131
where
and
This Lagrangian for a charged spinning particle has a supersymmetry (SUSY). In fact, one could say that it is this SUSY that is responsible for the cancellation of the non-renormalizable infinities in 1. To make the SUSY more explicit, we may consider
The objects we study are the n-point function of this operator. 4. We shall see that this SUSY is unbroken by quantization. The Ward identities of this symmetry will be derived. 5. For this purpose, a one-dimensional superlield formalism will be developed. 6. We will integrate over the photon and electron fields of QED and express it entirely as a theory of the operators x[x, $1. This will be an interacting theory of loops, exactly equivalent to QED. This [14, 151 is3 similar to Feynman’s action-at20p,, = t[y,,, y,,]. Y,, are the Dirac matrices, and g is the electric charge. We have used “$’ both as an index and to denote magnetic moment. There should be no confusion from this. 3 Feynman succeeded only in doing scalar electrodynamics this way, and that too without scalar self-interaction and without scalar loops. This “approximation” to QED will be of much use to us in Section II.
QUANTUM
251
ELECTRODYNAMICS
a-distance formalism of electrodynamics. We have now succeeded in introducing spinning particles as well as fermion loops into this formalism. 7. There are some analogies between our theory of loops and string theories. In fact, QED seems to be equivalent to the massless projection of some supersymmetric string theory. However, we will show that our theory of loops has only a finite number of propagating modes, unlike string theories. For non-Abelian theory, there are arguments [3, 51 that there must be an infinite number of propagating modes. 8. We will see from the proof of equivalence that the theory of loops is a renormalizable theory. We will also provide a gauge and Lorentz invariant, supersymmetric regularization. Section II and Appendix A study the question of renormalizability of the Wilson loop in free QED. Section III and Appendix B prove that the modified Wilson loop is multiplicatively renormalizable in free QED. Appendix C develops the superfield formalism to the level required. In Section IV and Appendix D we study QED with fermions and develop the expansion in powers of N,.. In Section V we compare and contrast the theory of loops with string theories. Section VI contains some inevitable speculations.
II. THE WILSON Loop
IN QUANTUM
THEORY
In a pure gauge theory, 9 = $F?,,, a natural gauge invariant
(2.1)
operator is the Wilson 10op,~ (2.2)
The Wilson loop plays a useful role because many other gauge invariant objects are derived from it. For example, consider a spin zero quantum field d(x) coupled to a background electromagnetic field. The following is an example of a gauge invariant expectation value
(2.3) Here, z is a curve that starts at x’ and ends at x.
4.u(t) denotes a closed curve, 0 4 I < T and x(0)=x(T). The symbol 2 will be used for path-ordered exponential. For Abelian classical gauge theory, there is no need for an ordering prescription.
252
S. G.
DA(x, x’) is the scalar propagator path-integral representation
RAJEEV
in the background
A. This has a well-known*’
Here, Q,(x, x’) is the set of all curves ~4t) such that y(O) = x y(T) = x’. This gives a path integral representation
for @[z] (2.5)
Here, y + z denotes the closed curve obtained by combinig y and Z. In this section we study the question of renormalizability of the Wilson x(t) is a differentiable curve and A, is a classical gauge field, there is no with the definition of W. However, if A, is a quantum field, there is no meaning to (2.2) without malization. For example, even d2(.y) is not well defined in a field theory, renormalization. The usual resolution is to define 1211 q?(x) = Yt, _ lifc,(x,
x’) (b(s) qqx’) - c*(x, x’)].
loop. If problem a renorwithout
(2.6 1
If c-number functions (ci, c2) can be found such that the above limit exists, then this is a valid definition. But it is by no means guaranteed that such a function exists. (Typically, renormalization also might mix different operators.) In our case, there is an additional complication. We are not satisfied by W being well defined for smooth curves. In most applications, W appears inside a path integral over .Y, over some range 52 In
S[x]
ep
w2)j.i’wrW[X]~
(2.7)
This means that we must consider W as a functional on the space of all curves that can contribute to this path integral. Unfortunately, this is a bizarre class of curves. They are functions such that, llxll* = j .2’(x) dr
QUANTUM
253
ELECTRODYNAMICS
is finite. It is known that a “typical” function of this class is continuous differentiable [22]. In fact, they are “half-differentiable,” i.e., Yt “-1
but not
Ix(t’) + x(t)1 ,t’-f,l12
exists. Even when A, is a classical background field, W[x] = erg~a~‘U(r) dr may not be well defined [22] as a function of the space of curves 9. Rather, it is a distribution, so that
exists for an appropriate class of test functionals. The class of test functions allowed is determined by physics. A natural choice is F[x] =exp i J,,(t)x,(t)dt.
(2.8 1
i
If we differentiate with respect to J, we will generate the various expectation values <-Y,,,(tl) . -~,,(t,,) >. On the other hand, we may choose J,,( t ) = q1’ ‘6( t - t , ) + kj,2’6( t - tr). . + kjp(
t - t,,).
(2.9)
Then we are calculating (e Al I(rlleik,- r,,(In)>
(2.10)
which is also very useful physically. (See Sect. IV). The quantity I/(kf) .y) = p.\-“1 will be called a “vertex functional” by analogy with string theories. Now let us consider the case where A, is quantized as a free field. We may be tempted to say that W[x] is a renormalizable operator if we can find a function C depending on the U.V. cutoff a il such that
w/J-~1 = Wxl C,,C~l has finite n-point functions in the limit A -+ cc. But, as described earlier, we are really interested in averages of the kind’ S(kitip,s,) 5 G[.x]
= F”
= J 3[x]
211 i’“r9[.y],
L2[y](Ol
so g[x]
simply
W[x]
W[v]lO)
f erk+(‘l) i eip~-v’s~‘. (2.11) i=l ,=I
denotes
the Wiener
measure
over
closed
curves.
254
S. G.
RAJEEV
Let us introduce a length a as an U.V. cutoff. W[x] Wilson loop, = e ita(
W,[x]
will then be replaced by a “bare”
“p”“c,[x],
(2.12)
go(a) being a bare-coupling constant. C,[x] is a sort of wave function renormalization of W. This is a C-number function. Actually through the path integral over x, the vertex operators eik’-‘(‘) might also need a wave function renormalization. Thus we are finally led to the following definition, DEFINITION.
W[x]
will be said to be a renormalizable
operator,
if there are
C,Cxl, ZCal, go(a),suchthat
’
fi
fj
I=1
i=l
V(k,itliI
x/(th))
z(a)
(2.13)
exists in the limit a -+ 0. This definition is somewhat different from the way the renormalizability of a composite operator is usually [21] defined. In a scalar field theory, for example, an operator 6(x) is defined through the functions G(x; x, . ..x.) = (O@(x) q4(x,)...q3(xn)IO). Here, b(x) is the fundamental scalar field. Even for a multiplicatively malizable operator one does not find that 9(x, .. ‘X,)’
renor-
(olo(x,)~~~o(x,)lo)
exist. The latter is more divergent than G. However, our aim is to rewrite the entire theory in terms of Wilson loops. So it seems natural to require that the n-point function of the Wilson loop (similarly averaged) exists. Next, we will study whether the Wilson loop is renormalizable in the sense defined above. For practice, let us first consider smooth curves that do not intersect. The n-point functions %‘oc-Xl. ..x.]
= (01 W[x,]
... W[x,]lO)
(2.14)
can be calculated explicitly (in the presence of an U.V. cutoff a) for pure Maxwell theory. It is just a matter of doing a Gaussian path integral over A,
QUANTUM
255
ELECTRODYNAMICS
1
dt dt’ .
Here, we have introduced photon propagator
(2.16)
a length “a” to serve as an U.V. cutoff replacing
the
(More fancy cutoffs like dimensional regularization could have been used, but this would do just as well for our purposes.) Divergences arise whenever x,(t) + xj( I’). This happens whenever (i) (ii)
t-tt’.
Two different curves intersect i#j.
x;(t) = Xj( t’), (iii)
Some curves intersect itself x;(t) = Xj( t’),
t # t’.
The latter two cases do not arise by hypothesis. Let us see if W can be “renormalized” for the case of smooth, non-intersecting curves. This has been studied by Polyakov and the answer seems to be that it is possible. The infinity in $9 arises from the region z + r’ in the first term of (2.5). One may calculate in the neighborhood of t N r’,
Thus, the infinity can be removed by defining W,[x]
= e -(l/*)sm(~)I~dr~igIA~.~i.Pdr
>
where &n w u/a and m. is renormalized to cancel the above infinity. For this to work it was important that there were no infinities from the second term, which represents the interaction between two curves. This happens because we assumed that they would not intersect.
256
S. G.
RAJEEV
If the curves intersect, there is an infinity as can be seen by studying the neighborhood of an intersection. Let x(r) and y(6) be two curves intersecting at T = t; r’ = t’. In this neighborhood
.<(T).j(T’) i(t).j(t’) i I a d7dT’[X(T)--!‘(51)]2+a2-a dTd7’[x(t)(r-f)-j(t)(7’-1’)]~+a’ -c1 i(t).y(t’) In a/E.
i2(t)j2(t)
- (.t(t)yyt)y
So within the class of smooth curves the operator W cannot be given a renormalized definition: there are infinities that cannot be absorbed by redefining W. But this in itself need not be an insurmountable problem. For, we are interested in whether G exists in the limit a + 0 after renormalization. The situation that Y blows up on a certain class of curves need not be a problem if this class does not contribute to the path integral. Now the question of whether the above average exists is the same as whether the one-dimensional “field theory” defined by the action
leads to finite n-point (Ole”” (‘I). eik8r”Jl IO) if m, is appropriately renormalized in the limit a -+ 0. A renormalization of m will (if we go back to the definition of ~o[x, ” x,,]) be absorbed as a redefinition of W. This seems like a very difficult issue at first. But in fact we can reduce this to a well-known problem in ordinary quantum field theory. The action S is the actionat-a-distance form of scalar electrodynamics [ 14J6 In fact, by doing perturbation theory in c( we can recover the Feynman rules. By expanding in tl we see that Feynman diagrams of scalar QED are recovered with two important exceptions. (1) (2)
Only diagrams with exactly n scalar loops appear. There are no scalar self-couplings.
Thus, the problem of renormalizability of the composite operators W[x] has been reduced to the following question: Can all the divergences of this truncated scalar electrodynamics be absorbed into a mass renormalization?’ First of all, since no “virtual” scalar loops appear due to exception (l), there is not charge renormalization. There is a mass renormalization due to the usual selfenergy diagrams. o We are still studying the renormalizability of a certain composite What we are doing is to rephrase the problem as the renormalizability ’ There might also be renormalization of the source, Z(a).
operator in free gauge theory. of a certain interacting theory.
QUANTUM
257
ELECTRODYNAMICS
However, in scalar electrodynamics there is also another divergence, a self interaction of the scalar field, There is no way to remove this divergence except by adding a term
.I62.1
dT, d7,
d4[X,(T,)-S,(t,)]
dT,d7,
I
to the action S. But this means that the operator W are not renormalizable in the sense defined earlier. There are infinities in the n-point functions that cannot be removed by a redefinition of W. This is claim (1) of the Introduction. III.
THE MODIFIED
Since we are ultimately to consider’
WILSON LOOPAND SUPERSYMMETRY
interested in QED with fermionic matter, it seems natural
instead of the Wilson loop. This is the amplitude to propagate around a loop.
for a spinning particle with g = 2,
x is as good as the Wilson loop for heavy particles, becausethe magnetic moment
term is suppressed for such particles. Also, in the limit of very large curves, this term has little effect, the magnetic moment coupling falls of faster.’ However, the cr. F does not have an effect on the ultraviolet properties of the theory. In fact, it improves the U.V. properties so that x is renormahzable whereas W is not. This is proved in Appendix B. Thus, even in a pure gauge theory, we should use x rather than W to get gauge invariant amplitudes. The improved renormalization properties suggests that there is a hidden symmetry in the problem. We shall see next that this is so. There is a hidden supersymmetry in the Green’s function of x. There is a way to describe a spinning particle using Grassman variables rather than spin matrices. Consider the Lagrangian [ 131 (3.2)
Here $J‘ is a real Grassman variable. This is invariant
*m’F=fJ ,” F,,,,; u,‘“=i[Y,,,Y,~l.
9 We thank
B. Svetisky
for discussions
on this point
under
258
S. G.
RAJEEV
as can be verified by straightforward computation; E is a real Grasmann number. To see that this supersymmetric quantum mechanics really describes a spinning particle, let us canonically quantize Y. If g = 0,
a9 y= aGp ill/". So that the canonical commutation
A representation
relations are
of this algebra is obtained on Dirac wave functions qJx)
yr being the Hermitian
Dirac matrices. The Hamiltonian
is”
H= +~(-V*+m;)SxP For the interacting
theories, we get similarly
H=$((iyD)*+m;) =g-D*-o.F+m;]; D being the covariant derivative, D,, = a,, - igA,.
Now we can write x in terms of 9, x[x]==j9[lj]e-% To see this, we use the following expression for eiH’ for small”
T,
e-(.r-.r1)2/4T (x'j31eCHTIx,
lo a/3 are Dirac indices. I’ O(T) denotes relative
order
a)
=
in T. Remaining
16n2T2
x @(x, x’) + O(T),
terms are - TX lirst
term.
See Appendix
D.
QUANTUM
ELECTRODYNAMICS
259
where
the integral being along the straight line
It is more reasonable, in view of SUSY to consider x[x, $1 z e-Jcb’d
without averaging over $. This makes the consequences of SUSY explicit;
4”(c):= CxYl), V(t)1 is a closed curve in a space with four real and four Grassman coordinates. Let Q be the generator of SUSY. Consider the n-point function s
x[(,]...x[S,,]
e~“‘4’JF2~[A]=~~[5,...rn].
These Green functions satisfy the Ward identities of SUSY Q,30=Oo;
j= 1 . ..n.
i.e., - ii+F‘( I,)
These survive quantization because dimensional regularization (for example) preserves this symmetry. We will actually use a different regularization which also preserves it. Now, we can actually calculate q. by integrating out the photon explicitly
where JJ~)=~j~dci;‘(r)S(x-xi(r)) i O +~~T’dr~~“‘(r)~vl’~(z)a,h;[~-xi(~)]. i
O
Thus, go = e-“2J
(J,(~)J,(r,)l(5--‘,)2+.2)d,~.~
= exp - S,C5,1,
260
S. G. RAJEEV
where
The expressions are beginning to get complicated. It would be nice to have a short-hand notation for this. This is provided by the “one-dimensional” superfield formalism’* of Appendix C. Then S[:jl=fx
j(i’D3<:‘dr+m;
jdt
I ,.
We conclude by pointing
out that $, has the form To= axK1-‘xc~,llo) = rtiie
xoC:r.
:,J
Compare this with a free-field theory of a field with expectation-value
zero,
<4(-y)4~) > = D(x, Y) then (e in& ~1). . eE4+(‘“)) = nile-“D(“‘. c/j, So we are beginning to see the connection of QED to a free-field theory. However, this is a free-field theory in an infinite dimensional space. In the next section we elaborate on this connection and also extend it to the interacting theory.
IV. EXPANSION IN POWERS OF IV,Consider QED with N,.“flavors” of electrons. If one looks at the action of string theory (in Sect. V) that is claimed to be equivalent to QED, one sees that N, acts like a coupling constant. This suggests that we should look for an expansion of QED in terms of N, and not IX. Such an expansion is interesting in itself. ‘* Reference
24 contains
a similar
formalism
for left-handed
superlields
in two dimensions.
QUANTUM
261
ELECTRODYNAMICS
NY is simply a parameter that counts the number of electron loops. The zeroth order in N/is the sum over all Feynman diagrams of QED without electron loops. The first order is the sum over all diagrams with exactly one electron loop and so on. For example, the p-fn of QED vanishes to zeroth order, but not to the 1st order in N,-. Such an approximation to the fl-fn has been studied before [18]. If we “integrate-out” the photon field in QED paying no attention to the dressing of the photon Green’s function, we get the zeroth orders approximation to QED. This is the action-at-a-distance QED discussed [ 141 in Section III. By a systematic expansion in powers of Nf we will find this idea of Feynman to reproduce QED exactly, not merely in the N, + 0 limit. Using supersymmetry, we will also be able to extend his ideas to spin 4 particles. Actually, the case of fermionic QED is more self-contained, since all interactions are given by the gauge principle. In scalar QED, there is also the @” integration (necessary for renormalizability) which has to be represented as a contact interaction of the world lines. The idea of the limit N,+ 0 is a well-known technique in the statistical mechanics [20]. There it is used to approximate the average of an exponential by the exponential of the average
The approximation of ignoring electron loops in QED is somewhat analogous to the large N,. limit in QCD. In the large N,. limit, quark loops disappear. But the analogy should not be carried too far, since there is no restriction in QED on the topology of the Feynman graphs. Let us consider a typical two-point function. 1. d’(x,y)=Se “/4)~P,Syr~-n+mo’y(1/~10) y(x) q(JJ) 9[,4] 9[q]. In Appendix D we have shown that the functional integral over q can be converted to an infinite series of worldline integrals: L.
grx]
X
g(G)
,-rK
vcc.
@Id’.
r(O) = li x(Tl=.r’
The point of writing the fermion path integral this way is that the photon integral is a mere Gaussian. In fact, 2 may be written as
262
S. G.
RAJEEV
Here, the measures have been redefined
NOW
we just do the integration
over A, using Appendix C. We have
where the K.E. terms have been absorbed into the measure. Thus, we have the required action-at-a-distance form of fermionic electrodynamics. Note that the zeroth term in N, is what occured in Appendix B. Note also the manifest supersymmetry.
V. We have accomplished
QUANTUM
FIELD
THEORY
OF Loops
our task. We have shown that given
We can calculate any object in QED by a “mere” worldline integral e.g.,
Furthermore,
we have expressed $9 in terms of gob,
To=
I
9[A]
e~“‘4’rF2X[51]...X[Tn].
In fact, So is known explicitly, ~o~ee~,,,S/C5,,5,1~e~~x,.,Dce,.~,1
where DC<, 5’]=
j dr de DY( r, 0) (5 _ c;j2 + a2 Dc”( t’, 6’) dt’ de’.
QUANTUM
263
ELECTRODYNAMICS
The 9 is known as an infinite series in 2&,
Let us now see if we can rewrite these to form a quantum field theory of loops. An object that creates a loop will be an operator-valued functional d[r]. Just as d(x) in scalar field theory creates a particle at X, @[r] creates a closed string 5. Suppose we postulate a “propagator” for 4,
D being as defined earlier. Then, we can verify that
Colek4C511.. Proof:
~e’g~cS~llO)=~~[~,...~,].
By Wick’s theorem, (()~,v+‘WCiJ
%1/o)
- 2 j-‘[;l
=e
Ott.
;‘I J15’1
gCS1
act’3
Now choose
These formal manipulations can be given a precise meaning by approximating a path integral by a multiple integral and taking the limit. Note that reversing the orientation of the loop changes the sign of 4, c&./l
,&Crll=
s
s
Gjc?l cmat-?I.
Thus,
Y[5,-~EJ=~+mJ : ,mm[rIl.. =
(Ole-
-*9[q,](OINf: . &a;,1 S~Cllek6CS11..
I
0
cosg$[q,]:
>3 e’RICCnll()),
where
This looks like the Gell-Mann-Low 595, ,73:2-2
formula in field theory.
... N,: cosg4[rj,]
264
S. G.
RAJEEV
Thus, if we have a field theory in loop space with the propagator
(WErl and interaction
ii-5’110> =N5,5’1
action s, = N,: j cosgdC
we reproduce the series for QED. Now, it seems that we have more than we started with; CJ~[[], being a functional, has an infinite number of fields in it. But we are claiming that this is equivalent to a field theory with a finite number of fields. How is this possible? In fact only a finite number of modes of 4 actually propagate. To see this, let us “diagonalize” the propagator D,
=
s
dq dz d0 de d-c’ D:“‘(q)
I’;&]
V,,,,&].
(This is not diagonal in ~1,v and r, t’. But these indices can also be diagonalized a linear and a Laplace transform, respectively.) The “vertex function” is defined by
by
V,,pRCrl = myr, H)eiy<(r. 0) These functions span a subspace in the space d[t] the modes of d[<] that can be written as
labelled by a A,(q).
Thus only
actually propagate. This is merely a finite number of modes in space-time. O[<, 5’1 cannot be inverted to yield a “kinetic energy” operator, because it has mostly zero eigenvalues. There is no K such that
However, suppose P[c, 5’1 is the projection we can verify that
operator to the “photon”
modes. Then,
QUANTUM
ELECTRODYNAMICS
265
Consider the operator lt=;[dcf+;dE
+ SUSY completion. 62 6x(a + E) 6x(a - E)
Then
m5,
5’1 = per, 4’1.
Thus, within the subspace of propagating modes, R is the “K. E’ operator. Thus, we find the theory in loop space to be not a string theory. There are only a finite number of propagating modes, whereas in string theory there are an infinite number.
VI. CONCLUSIONS We have shown that the correct operator to study in a gauge theory is the modified Wilson loop ~11x1 and not the usual Wilson loop W[x]. In this paper, only the Abelian theory was considered. It will be shown in a forthcoming paper that x[x] and not l+‘[.~] is renormalizable in non-Abelian gauge theories also. A formulation of non-Abelian gauge theories in terms of x[x] will be quite interesting. The field A,(x) of QCD does not appear to have a direct relation to observable particles. But, we conjecture that x[x] creates linear combinations of one-particle states (i.e., “glueballs”). Also, we believe that the projection to an operator that creates a single eigenstate is the average
sD[x]x[s]V,[x]; P’,[.y] are vertex functionals similar to the ones encountered in the text. Of course, nothing in this paper confirms these conjectures. We intend to follow up on these issues in later publications. The study of QED has led us to the correct choice of loop variables. It also has shown us how to include fermionic matter fields. Even though the pure gauge sector of QCD is much more involved, the interaction of gluons to quarks is not too different from that in QED. So we expect Appendix D to go through without major changes. A functional integral representation for the quark Green function in the presence of an external gluon field can be derived using the Lagrangian of Reference 13. It is conceivable that our way of formulating QED will lead to a better understanding of its non-perturbative features.
266
S. G. RAJEEV
APPENDIX A: ACTION-AT-A-DISTANCE ELECTRODYNAMICS FOR SCALAR PARTICLES
We want to show that
(A.11 describes scalar QED except that the following two kinds of Feynman diagrams are absent: 1. scalar loops; 2. scalar self-couplings. We will prove this in the presence of an U.V. cutoff. So U.V. infinities are regulated. One proof is, of course, in Feynman’s papers [ 14). We give a more updated one. A different approach is mentioned in Appendix B. Consider a scalar 2n-pt function, G,= (OIT~*(X,)~(Y~)~*(-K,)~(~~)...~*(.~,~)~(Y,,)IO)
(A.21
in a theory with action s= 4 j A,,(x) qy(. ~,~‘)A,,dxdx’+~j(Dq5)~(~)d.x+~jm~~q~~d~x; k,(x, x’) is the photon inverse propagator
with a cutoff Q. In terms of path integrals
e -(1;2)jAkAd.YdY’e-Il:2J~~)*D~dr~ll;rl~m’l~I’d.~
G=jg[A]
x #*(xl )4(.Yl).. The functional
(A.3)
4*(x,) 4(Y,).
(A.41
integral over Q can be done G=j!&4]e '1
-(1/2,1AKad.~dr’(Det[_02+n12])
(A.51
{D,(-uil-Yj,)D,(xiz-l'jl)D,(xi~-,~j~j.
Now the process of ignoring scalar loops is the same as dropping [Det( -D” + m*)]. (We will call this process “quenching.“) Then Gzue=
s X
CJJ’CA,,-(1/2,jaKad.Yd.r’ 9[Xi]
sy
the
dT,
e- ll/2)11,=,I.~j(,)dr-m2S~d~ei’~,lA,i;(rJdr
(A.61
QUANTUM
267
ELECTRODYNAMICS
Where the well-known path integral representation for the scalar n-pt function in the presence of an external gauge field has been used. Sz is the set of all paths such that {x;(O)] = (x;} (A.7)
Now calculate the path integral over A to get
G=jm i dl;e-“2’rr,/2’j 0
9[x;] e-S[-yl, R
,=I
S being the action defined in (A.l). Thus, the action S yields the n-pt of truncated scalar QED when integrated over the region f2. Now, we know from the renormalization theory of SQED that 1. In fact, 2. 3. 4.
If scalar loops are ignored there is no coupling constant renormalization. there is no correction at all to the photon propagator. There is a mass renormalization. There is a wave function renormalization Z”(a) There is also a divergence associated to the four pt function. So G,,=Z”(a)
G(po(u);x,
. ..x.,)
is not finite; one has to add a 14” term to the action to recover renormalizability. the worldline formalism this is
In
The quantity in the integral leads to a 6 function as a -+ 0. In the text we considered the path integral over closed curves whereas above we considered a path integral over open curves. The difference will show up in the Feynman rules as a modification of the propagator, and of the range of integration over the proper time parameters. Now integrate over all t’s between 271 and 0. To derive the new propagator let us calculate directly from the path integral
(OlekCr(r)-.~(~‘)310) =s (l/2)~~.~i-Z(T)dreik[x(r)~x(r')]
This can be done by Fourier analysis x(t)=uo+
C u,cosZnnf+
n;tO
1 b,sin-
n#O
2X” t T .
268
S. G. RAJEEV
We will spare the reader the agony of detail. The answer is (Ok
ik’C.~l(r)~.~(r’lllO) _e-k’/l-tl,
where
Note that as (t’- t) + 0, f(t’ - t) - It’ - tl. Thus, near the region t’ - t or k’ -+ co, these propagators reduce to the earlier one. So the change in the boundary condition does not change the degree of divergence of the Feynman diagrams.
APPENDIX
B: PROOF OF RENORMALIZABILITY
OF x[x]
IN FREE QED
We have
where x(z), x(0) = .x(T) is a closed curve and [Tfl5’= fCYp ?,,I. We will be interested in the expectation values G(k) = j z(a) eh. ““IL(~)
,$k, ‘+‘I%) fI a[xf]
e-
(1/2,S>&
~~lT)dT,-m;(u)
r,
i=I x
03.2)
(“t~[xll~~‘~[Xkl~O).
The task is to show that all the divergences in this that occur as the U.V. cutoff a + 0 can be absorbed into m,(a). There is also in general a “wave fn” renormalization z(a) of eik’ ‘,
K’“’ is the photon kinetic energy operator with a U.V. cutoff a. Consider the simplest case k = 1, n = 0, G(T)=jg[x]
e -(l/2)S,T.~2(r)dr-m~~(OIX[X]10)
G(T)=
e-“‘4)~F2dyGA(T),
or f&4]
QUANTUM
269
ELECTRODYNAMICS
Now, we will see in Appendix D that G,4( T) = j tr(xlePH’IX)
dx
with H= -D’+a.F+mi.
It is convenient to study instead of G(T), its Laplace transform, G(p)= jf
G(T)e-‘12dT
=sg[A]pp(1/2)jF2dr
tr -D’+ga.
1 F+m2
where we have put n12 =mi+p’, tr
1 -D’+o.F+m’
=Ltr m
1 iy.D+m’
Thus, we have related G(p) to the propagator of fermionic QED. If we expand the right-hand side in powers of g, we will recover all the Feynman diagrams of QED, except for those involving “dressing” of the photon Green’s function. Thus, the U.V. divergences in G(T) are the same as those in QED. But we know that in QED all divergences can be removed by mass and wave function renormalization. (There is no charge renormalization, because the photon Green’s functions are not “dressed.“) A similar argument applies to more complicated cases.
APPENDIX
1-D Superfields
Define
Under SUSY,
C
270
S. G.
RAJEEV
SO
Thus,
Q=$+!$ igj,” 1=l.
Then
This is the SUSY algebra. Put
Superfield Interactions 6A, = c3,Ap6x” = -ie[dyA,+“] .-.- - iEB, B, = a,A,$‘.
Thus, A, and B, form the component of a superfield A,(x + ie+) = A,(x) Call A,(x + iO$) = 4, to avoid confusion. Thus, d,, = A,(x)
+ iO$“ayAP
+ iOB,
QUANTUM
271
ELECTRODYNAMICS
so #,Dt” The propagator
= -@$“Il,‘&A,
- i3A$‘+
iA,
I#“.
of 4, is
So the effect of integrating
out A is
APPENDIX
(i) Path Integral Representation
D
of Electron Amplitudes
We will be interested in evaluating functional
integrals such as
or
Here, S=a
I
F*+
I
q,(,~)[y.D+m,)qd.u.
In this section, let us not consider the integration over A. Let it be an external field. Our task is to express these functional integrals over q as path integrals over world lines. We will show that the calculation of any amplitude can be reduced to that of evaluating an integral over world lines such as
tl, t2 are open curves (P(a), ICIP(a)) and the integral is over some appropriate of such curves. Let us consider for example s
9[q]
eJ4cY.D ‘“1”~Y4(x)q(~)=
Ignore the Det for the moment.
+[Det(r.D+m)]b
[-tr
S(x,y)].
class
272
S. G.
S, is the fermion propagator
RAJEEV
in the presence of the gauge fixed A,
(y . D + m,) S,(x, x’) = 6(x, x’) S,(-L -x’) = (Wb’)
4(x)10),.
In fact, S,(x,x’)=(-y.D+m,)A,(x,x’), where A satisfies [ -(y . D)Z + m;] A,(x, x’) = 6(x, x’). Now
Here, the “y. D” term cannot contribute to the trace. A, always involves an even number of y matrices, so tr y. DA, would involve an odd number. So the trace vanishes. Thus, tr S,(x, x’) = m, tr d,(x, 2). The point of the above exercise is to express everything in terms of the “square” of the Dirac operator, A A. This has closer connection with path integral presentations. This, however, is not essential, and we will see in fact that S,(x, x’) can be given a path integral representation directly. But we will deal with A, for the moment. Clearly any number of products of q’s and 4 has a v.e.v. (in the presence of an internal gauge field) which can be written in terms of A, and its derivatives. Now we claim that
Here, 52, is the set of the worldlines such that x(0) =x x(T) = x’. To see this. note first that
with A= -((Y.Dy+m; = -D2-ga.F+mi-.
QUANTUM
273
ELECTRODYNAMICS
Now, for the “propagator” (x’le-“Ix) we find the path integral representation by interpolating states 1x(?,)) at (intermediate) times t;. The expression for
1 .
Thus,
The limit N+ a3, (ti- t,+ ,) + 0 such that N(t,-
t,- ,) + T
leads to the claimed-path integral representation. Next, we write the same-path integral using Grassman Dirac matrices. We claim
Q being an appropriate
Expanding
variables rather than
range. Clearly, it is sufficient to show that
both sides, we need to show that
To evaluate the RHS we need
($“(t’) $‘(t)>= jy] By point splitting,
G2)l$$ d’l//,(t’) l),(t),
274
S. G.
RAJEEV
This is basically a definition of the path integral. Different boundary conditions on the Green function of d/dt lead to different definition of the path integral. Once we know the above formula, the theorem follows using the trace formula for y matrices. Thus we have the integral representation
D = {x, $1 x(0) =x; x(T) =x’; l)(O) = i//(T)). (ii) Path Integral Representation
of the Determinant
Now let us consider the problem of writing the [Det(y . D +m,)lN’ integral. This is quite complicated. We proceed in steps.
as a path
1. Reduce to the Square of the Dirac Operators. Det(y . D + m,) = Det y&y. D + mo) y5 = Det( -y. D + mo). SO
[Det(y.D+m,)lN’=
[Det(y. D+m,)(-y.D+m,)lNf’*
i.e.,
2.
Path integral.
[Det( -7.0)’
+ mi]Nf!2 = exp;lnDet[-(7.D)‘+mi]
Nf =expttrln[-(y.D)*+mz]. Now consider the tr In,
But the limit T+ 0 generally leads to a U.V. singularity. We will assume that the Det is defined with a U.V. cutoff. A convenient cutoff is to do the integral up to T= E. Using earlier results,
QUANTUM
215
ELECTRODYNAMICS
where Q, is the set of closed curves
x(0) = x(T) VW) = $40 3. E.ypansion in powers of IV,. What we need is exp(N//2) Tr In H. We can get it by expanding the exponential
tr In H not
Det(y D + M,) =
This means that we are integrating
over k closed loops. Here, as before,
It is not difficult to figure out what the expansion in N/- means. The power of N, merely counts the number of electron loops. This expansion makes sense even though N,.> 1. (iii)
Worldline Form of Amplitudes
Now we are ready to find any integral over q’s. Suppose we want 1 g[ql
e~~q’~‘r)+Ma’Y~f-rlrq(,~)q(~)=
[Det(y.D+M,)]‘v’S,4(x,y).
We will find this as a formal power series in N,. The zeroth order in N,.contains closed loops. Thus, the above object becomes
no
Next, y denotes closed loops of various periods. The first term is represented by the “tree-level” contribution and the next term is the “one loop” contribution, and so on. ACKNOWLEDGMENTS I have benefitted A. P. Balachandran.
from conversations I am also grateful
with L. Alvarez-Gaumt, for the interest shown
X. Wu, H. Yamagishi, K. Johnson, by K. Huang and C. Hull.
and
276
S. G.
RAJEEV
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R. P. FEYXMAN. Phys. Rev. 97 (1955), 660; R. P. FEYNMAN. R. W. HELLEWARTH, C. D. IDDINGS, AND P. M. PLATZMAN, Phys. Rev. 127 (1962), 1004. 20. P. ANDERSON, Les Houches lectures. 21. C. ITZYKSON AND J. B. ZUBER, “Quantum Field Theory.” McGraw-Hill, New York, 1980. 22. L. S. SCHIJLMAN, “Techniques and Applications of Path Integration,” Wiley, New York, 1981. 23. J. SCHWINGER, Phys. Rev. 76 (1949). 749. 24. R. BROOKS, F. MC’HAMMAD, AND S. J. GATES. University of Maryland preprint, September 1985. 19.