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Finite quantum electrodynamics and gauge invariance

H. C. OHANIAN(*)

Department of Physics, Rensselaer Polytechnic Institute - Troy, NY 12180, USA (ricevuto il 6 Giugno 1997; approvato il 22 Luglio 1997)

Summary. — It was shown in a previous paper that the divergent radiative corrections of conventional QED become finite when calculated with a modified, smeared Feynman propagator. The smearing of the propagator can be interpreted as a smearing of the light cone over a distance of about a Planck length, caused by quantum fluctuations in the geometry. To achieve gauge invariance, a further modification of conventional QED is proposed involving the insertion of an extra form factor into the electron-photon vertex. The form factor corresponds to an extended charge distribution, with a size of about one Planck length and a mass of about one Planck mass.

PACS 12.20 – Quantum electrodynamics. PACS 11.10.Gh – Renormalization. PACS 11.15 – Gauge field theories.

1. – Introduction

In a previous paper [1], we proposed a modification of the Feynman propagator, which is intended to incorporate in a crude, phenomenological way the effects of quantum fluctuations in the geometry on the Planck scale, L

* 41 .6 310233cm . The proposed modification replaces the conventional scalar Feynman propagator GF(x2) by a “smeared” propagator, G–F(x2) 4



dl f (l) GF(x22 l) 4 2 m 4 p2



dl f (l) K1

(

im

k

x22 l 2 ie

)

k

x2 2 l 2 ie . (1)

Here K1 is the modified Bessel function of order 1 and f (l) is a weighting function which is nonnegative and normalized to 1,

s

dl f (l) 41. The details of the weighting function are unknown; they depend on the unknown details of the quantum fluctuations of the geometry. However, f (l) is expected to be nonzero over a range from l 40 to

l CL *2

D 0 , and all kinds of particles are assumed to have the same weighting function.

(*) Current address: P.O. Box 370, Charlotte, VT 05445, USA.

The smearing of the propagator can be interpreted as a smearing of the light cone over a timelike “light shell” extending from x2_{4 0 to x}2_{C L *}2[2].

In momentum space, the smeared scalar propagator is

G–F(p) 4



d4x eip Q xG – F(x2) 42i



dl f (l) kl K1[2i

k

l(p22 m21 ie)]

k

p2 2 m21 ie (2)

and the smeared propagator for spin 1 O2 is

S–F(p) 4 (p

4

1 m) G – F(p) . (3) For Nx2 N c L *2and for Np2

N b L *224 M *24 ( 1 .2 3 1019GeV )2_{, the smeared propagators} in coordinate space and in momentum space approach the conventional Feynman propagators. In consequence, the smeared propagator propagates low-momentum wavefunctions ( p0bM_*) in the normal way. But the smeared propagators suppress high-momentum wavefunctions ( p0D M *) and prevent their propagation within distances C L *.

The smeared propagator in coordinate space either has no singularity on the light cone or only a mild singularity, depending on the details of f (l). In momentum space, at large values of Np2

N , the smeared propagator approaches zero more strongly than the Feynman propagator. For example, if the weighting function has the simple form

f (l) Pln_e2lM *

2

, with, say, n D1, then the smeared propagator involves a Whittaker function [ 3 , 4 ], G–F(p) P exp [ (2p2 1 m2_{) O8M *}2] p2_{2 m}2_{1 ie} W21 2 n , 1 O2[ (2p 2 1 m2_{2 ie) O4 M *}2] (4)

and, for large Np2N , it has the asymptotic behavior G–FK 1 O( p2)2 1n. This asymptotic behavior is valid throughout the complex plane, except for real positive values of p2_(by a Wick rotation, p2 _{can always be made negative in integrations over internal lines in} Feynman diagrams, and therefore the asymptotic behavior for positive p2 _is irrelevant).

As shown in [1], the quenching of singularities in the propagator leads to finite results for all the conventionally ultraviolet-divergent radiative corrections in QED. The unknown details of the weighting function f (l) are largely irrelevant, since these details are hidden in the (finite) renormalizations of charge, mass and wavefunctions. Thus, the smeared propagators provide a “realistic” elimination of the divergences in QED, instead of the “formalistic” amputation of divergences that has been the fashion in field theory for the last fifty years.

The calculations in [1] have a defect in that the vacuum polarization tensor calculated from the smeared propagator is not gauge invariant, that is, the electron current induced in the vacuum is not conserved. Such a nonconservation of the electron current is expected on physical grounds: on the Planck scale, electrons can be absorbed and emitted by miniature black holes of mass M

*, and such processes generate fluctuations in the electron current. When the polarization is calculated from only the electron current (as usually done in QED), gauge invariance will be violated.

In the previous paper, we suggested that the solution of this gauge problem would have to emerge from the as yet unknown details of the quantum fluctuations of the

geometry. Here we propose a simple solution of the gauge problem by an effective modification of the electron-photon vertex, chosen so as to achieve gauge invariance of the polarization tensor. Although such a solution of the gauge problem may seem forced, it has two points it its favor: i) a modification of the vertex is mandated by quantum gravitational effects, which necessarily alter the conventional vertex function

egm into some more complicated function with a momentum dependence; ii) as we will

see in the next section, the proposed modification of the vertex function displays the physical characteristics of “lumps” of size L

* and mass M*, consistent with the qualitative picture of quantum fluctuations in the geometry.

2. – The form factor

The vertex modification can be conveniently formulated in momentum space, by introducing form factors. For a vertex with electron momenta p and p8 and a photon momentum q 4p2p8, the proposed modification is

egm1 e(pm1 p 8m)(p

4

1 p

4

8 ) G(q2₎ M * 2 1 e(p

4

1 p

4

8 ) gm(p

4

1 p

4

8 ) H(q2₎ M * 2 , (5)

where the form factors G(q2

) and H(q2_{) are scalar functions which are to be adjusted} to achieve gauge invariance of the polarization tensor. Since the modification is supposed to represent gravitational corrections of the vertex, the quadratic occurrence of electron momenta in (5) is natural. If, instead, we were to use a form factor involving the momenta linearly, we would find that the contribution to the gauge-noninvariant part of the polarization tensor has the wrong sign for cancellation. Normalization of the electron charge actually requires an alteration of the first term by a factor 1 24m2

(

_{G( 0 ) 1 H(0)}

)

_{OM *}2

; but this is a very small correction, and it is in any case hidden in the overall charge renormalization. To achieve gauge invariance of the polarization tensor, the form factor G suffices, and we provisionally take H 40. But to achieve gauge invariance of the fourth-rank polarization tensor that enters into photon-photon scattering, it will probably be necessary to adopt a nonzero H . The characteristic mass accompanying the form factors in (5) has been chosen as M

* in anticipation of the result (see below) that with this choice of mass, G(q2_{) is of the order} of magnitude of 1.

With the form factor G(q2_{), the vacuum polarization tensor becomes} Pmn(q) 42 e2 ( 2 p)4



d 4 p Tr

{

y

gm1 G(q2₎ M_*2 ( 2 pm2 qm)( 2 p

4

2 q

4

)

z

(p

4

1 m) Q (6)

y

gn1 G(q2) M * 2 ( 2 pn2 qn)( 2 p

4

2 q

4

)

z

(p

4

2 q

4

1 m)

n

G – F(p) G – F(p 2q) .

As in ref. [1], the evaluation of the integrals can performed by the scaling trick of Bjorken and Drell [5] after inserting the exponential representation for the smeared

propagator, G–F(p) 42i



dl f (l)



0 Q dz exp [ilO4z1i(p2 2 m21 ie) z] . (7)

The dominant terms in the resulting gauge-noninvariant part of the polarization tensor are P( 2 )mn(q) 42 ia p gmn



0 1 dz



dl1



dl2f (l1) f (l2) Q (8) Q

{

2 K2(j) h21 3 Q 25 G M * 2 K2(j) j2 h 4 2 3 Q 29 G 2 M * 4 K3(j) j3 h 6

}

1 O(q2) , where j 4

o

l1( 1 2z)1l2z z( 1 2z) [m 2 2 q2z( 1 2z) ] (9) and h 4

k

m2 2 q2z( 1 2z) . (10)

The condition Pmn( 2 )4 0 gives us a quadratic equation for G(q2). For Nq2N b 1 OL *2, the

series representation of the Bessel function leads to the simple approximation: 3 Q 210 G2



_dz



_dl 1



dl2 f (l1) f (l2) M * 6_[l 1Oz 1 l2O( 1 2 z) ]3 2 (11) 2 3 Q 24G



dz



dl1



dl2 f (l1) f (l2) M_*4[l1Oz 1 l2O( 1 2 z) ]2 2 2



dz



dl1



dl2 f (l1) f (l2) M * 2_[l 1Oz 1 l2O( 1 2 z) ] 1 O(q2_{OM *}2) 40 . If f (l) is more or less concentrated at l CL *2

, each of the integrals is of the order of C 1 , and this yields the estimate

G(q2) C11 O(q2_{OM *}2) . (12)

For Nq2

N D M *2, the corresponding estimate is G(q2) C M * 2 q2 . (13)

Furthermore, from the equation for G(q2_{) it is easy to see that the r.m.s. radius a}2 4 6 O G(0) ¯2O¯q2 G(q2) associated with the form factor G(q2_{) has magnitude a CL *.}

These results suggest that the form factor can be attributed to some kind of charged lumps of size L

* and mass M* generated by the quantum fluctuations in the geometry to which the electron couples or in which the electron is trapped. The form factor must be recalculated in each order of a , but the higher-order terms are not expected to change the qualitative behavior of G(q2_).

3. – Other consequences

Besides generating the desirable cancellation in the gauge-noninvariant part of the polarization, the form factor also generates extra undesirable, or unneeded, terms in the gauge-invariant part of the polarization, in the self-energy, and in the vertex function. These extra terms contribute to the renormalizations, but they do not affect the physically interesting part, that is, the conventionally finite part. To see this in general, note that without the form factor, the integration over internal momenta in any S-matrix element with smeared propagators yields a sum of several Bessel functions Kn(j ) Ojn, where j is of the general form j24 (lOz11 lOz21 R) q 82, and q 82 is a quadratic expression involving the external momenta and masses

(

compare eqs. (8) and (9)

)

. Upon integration over l1, l2, R , the series expansion of this sum of Bessel functions takes the schematic form

AkL_*2k1 Ak 22L_*2k 1 21 R 1 A0L_*01 A ln L *21 A 8 ln q 821 A22L_* 2

1 R . (14)

Here the exponent of the leading term is k GD, where D is the conventional “degree of divergence” calculated with 22 for each photon propagator and 21 for each electron propagator, and the coefficients Akare functions of the external momenta and masses.

The terms with negative or zero powers of L

* in the series (14) give the renormalizations, and the logarithmic term P ln q 82 _{gives the physically interesting} contribution (the conventionally finite part). The terms with positive powers of L

* give small corrections which are negligible if the external momenta are small compared with the Planck mass (these terms are of the same order of magnitude as the terms resulting from gravitational interactions among electrons and photons, supplemented by electromagnetic corrections; it is therefore futile to calculate these terms without also considering these gravitational interactions). If we now replace r of the vertices eg by e( p 1p8)( p

4

1 p

4

_{8) OM *}2, the degree of divergence increases to D 12r, but the division by M

* 2 r

restores the condition k GD, and gives a series

AkL_*2k1 Ak 22L_*2k 1 21 R 1 A 8 L *2 rln q 821 R .

(15)

Here the ln q 82term is multiplied by a factor L * 2 r

, which makes it negligible. Hence the only nonnegligible terms in the series are the renormalization terms.

All such extra renormalization terms in the polarization, the vertex function, and the electron self-energy are of order L

*

0_{. This means that the terms of order L} * 0 calculated in the previous paper must be revised, but the terms of order ln 1 OmL *C50 are unchanged. The previous estimate of roughly 10% for the magnitude of all the renormalizations remains unchanged.

Although the form factor G(q2) solves the gauge problem for the second-rank polarization tensor Pmn associated with two external photon lines, it does not solve this

problem for the higher-rank polarization tensors associated with more than two external photon lines. For instance, the fourth-rank polarization tensor associated with

photon-photon scattering involves a product of four weighting functions, each integrated with respect to its argument, and there are (probably) no simple relationships between this integral over four weighting functions and the simpler integral over two weighting functions in eq. (8). To solve the gauge problem for such higher-rank polarization tensors, we need to introduce a further modification, such as the extra form factor H(q2_{) in eq. (5). Since the crucial gauge-noninvariant term in the} fourth-rank polarization is the constant term (with zero photon momenta), only one adjustable constant is required to achieve gauge invariance. This suggests that we take H 4b G, and adjust the constant b. Whether the resulting quartic equation for b has a solution remains an open question.

Gauge invariance of higher-rank polarizations (three-photon scattering, etc.) is guaranteed, except perhaps for small terms of order L

* 2

. If D G21 (a conventionally finite integral), then the series (14) starts with the term A0L_*0, and includes only positive powers. For such an S-matrix element the calculations with the conventional and the smeared propagators agree except for negligible terms of order L

*

2_{, and the} gauge invariance of conventional QED then guarantees that of smeared QED.

R E F E R E N C E S

[1] OHANIANH. C., Phys. Rev. D, 55 (1997) 5140.

[2] Larry Ford has recently brought to my attention calculations of the smearing of the light cone in the context of the linear approximation of the gravitational field, FORDL. H., Phys. Rev. D, 51 (1995) 1692 and FORDL. H. and SVAITERN. F., Phys. Rev. D, 54 (1996) 2640. The results of these calculations are qualitatively different from eq. (1), presumably because eq. (1) hinges on highly nonlinear gravitational contributions, such as the infinite sum of ladder contributions for the gravitational self-energy described in ref. [1].

[3] OBERHETTINGER F., Tables of Bessel Transforms (Springer-Verlag, New York) 1972, p. 163.

[4] WHITTAKER E. T. and WATSON G. N., Modern Analysis (Cambridge University Press, Cambridge) 1958, p. 339.

[5] BJORKENJ. D. and DRELLS. D., Relativistic Quantum Mechanics (McGraw-Hill, New York) 1964, p. 155.