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JHEP03(2019)177

Published for SISSA by Springer

Received: November 3, 2018

Accepted: March 21, 2019

Published: March 27, 2019

Resolving the Weinberg paradox with topology

John Terning and Christopher B. Verhaaren

Center for Quantum Mathematics and Physics (QMAP),

Department of Physics, University of California, Davis, CA 95616, U.S.A.

E-mail: jterning@gmail.com, cbverhaaren@ucdavis.edu

Abstract: Long ago Weinberg showed, from ﬁrst principles, that the amplitude for a

single photon exchange between an electric current and a magnetic current violates Lorentz

invariance. The obvious conclusion at the time was that monopoles were not allowed

in quantum ﬁeld theory. Since the discovery of topological monopoles there has thus

been a paradox. On the one hand, topological monopoles are constructed in Lorentz

invariant quantum ﬁeld theories, while on the other hand, the low-energy eﬀective theory

for such monopoles will reproduce Weinberg’s result. We examine a toy model where

both electric and magnetic charges are perturbatively coupled and show how soft-photon

resummation for hard scattering exponentiates the Lorentz violating pieces to a phase that

is the covariant form of the Aharonov-Bohm phase due to the Dirac string. The modulus of

the scattering amplitudes (and hence observables) are Lorentz invariant, and when Dirac

charge quantization is imposed the amplitude itself is also Lorentz invariant. For closed

paths there is a topological component of the phase that relates to aspects of 4D topological

quantum ﬁeld theory.

Keywords: Scattering Amplitudes, Solitons Monopoles and Instantons, Eﬀective Field

Theories, Gauge Symmetry

ArXiv ePrint: 1809.05102

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP03(2019)177

JHEP03(2019)177

Contents

1 Introduction 1

2 Linking numbers 2

3 Electron scattering in a monopole ﬁeld 5

4 Perturbative magnetic charge 6

5 Soft photons 9

5.1 Virtual soft photons 11

5.2 Real soft emission 12

5.3 Position space 13

6 More general trajectories 16

7 Conclusion 21

1 Introduction

In a classic paper, Weinberg [1] derived the Einstein and Maxwell equations using pertur-

bation theory simply by considering the exchange of massless spin 2 and spin 1 particles.

He also considered the extension of the Maxwell equations that includes magnetic charges,

but found that the leading perturbative term in the electric-magnetic scattering amplitude

was not Lorentz invariant. This non-Lorentz invariance was also seen from a variety of ap-

proaches by other authors [2–5]. Schwinger [6–8] put forth a nonlocal Hamiltonian theory

with inﬁnite Dirac strings that was formally shown to be Lorentz invariant once Dirac-

Schwinger-Zwanziger [6–8, 10, 11] charge quantization was imposed. However, leading

order perturbative calculations using Schwinger’s theory [12] were also non-Lorentz invari-

ant. Zwanziger came up with a local Lagrangian formulation [3, 4], but the Lagrangian was

not manifestly Lorentz invariant. Again, formal proofs were given [13–15] that Zwanziger’s

approach, in principle, gave Lorentz invariant observables, but in perturbation theory the

amplitudes are again non-Lorentz invariant. In essence, every approach was forced to

include an arbitrary four-vector, referred to here as n

, that, in some gauges, could be

identiﬁed with the direction of the Dirac string. Because the direction of the Dirac string

can be shifted by gauge transformations, this means that the amplitude’s dependence on

indicates a failure of both Lorentz invariance and gauge invariance.

Magnetic charges were mostly ignored until ‘t Hooft and Polyakov [16–19] showed that

breaking a non-Abelian gauge group with no U(1) factors to a subgroup with U(1) factors

Similar to Dirac’s nonlocal Lagrangian formulation [9].

– 1 –

JHEP03(2019)177

produces topological monopoles. Since that time it has been speculated that recovering

manifest Lorentz invariance would require a non-perturbative calculation, and calculations

have been attempted along those lines [20, 21]. The calculations of refs. [13–15] showed

that topological terms could appear in the QED path integral extended to include mag-

netic monopoles.

In this paper, we study a toy model in which both electric and magnetic charges

are perturbatively coupled. In this case, the leading soft-photon corrections to a hard-

scattering process can be resummed to all orders in perturbation theory. We show that

this all-order calculation produces Lorentz invariant observables, since the non-Lorentz

invariant (n

dependent) part of the amplitude appears only in a phase. This phase is

4π times product of the electric and magnetic charges times an integral over the particle

paths. For closed paths and string worldsheets this integral is an integer valued topological

linking number. Thus, when Dirac-Schwinger-Zwanziger charge quantization is imposed

the phase is a multiple of 2π, and the amplitude itself is Lorentz invariant. We show that

this topological phase is in fact the string contribution to the Aharonov-Bohm phase [22].

After a brief review of linking numbers, Lorentz violating amplitudes, and the low-energy

eﬀective Lagrangian for perturbative electric and magnetic charges, we present the all-

orders calculation. Finally, we discuss paths that are not closed and Aharonov-Bohm

interference measurements.

2 Linking numbers

Most QED calculations make no mention of topology, and many physicists ﬁnd the jargon

and results unfamiliar. Since a topological linking number plays a prominent role in our

results, we introduce the concept here. This should aid the reader in seeing the topological

hints as they appear in our analysis. Amusingly, linking numbers may also trace their

genesis to Gauss’ study of magnetism, giving a certain poetry to its appearance in the

modern approach to magnetic monopoles.

Gauss recorded his discovery of the linking number in his diary/logbook in 1833, but

the result was not published until 1867 when it was included in his collected work on

electrodynamics [23]. The inclusion of this topological result with his research on electro-

magnetism surprised some, but historians remain convinced that he was led to the linking

number by his work on terrestrial magnetism, and plausible reconstructions of his deriva-

tion have been presented [24]. Given the state of electromagnetic theory in 1833, it would

have taken a Gauss to do it, but in modern language the argument is simple [25]. Gauss

wanted to calculate the work done in moving a magnetic monopole with unit charge on

a closed path C that is wrapped m times by a loop C

carrying a current I. Using the

Biot-Savart law we have



ijk

(x − x

)

|x − x

. (2.1)

Using Stokes’ theorem (circa 1850) for a surface S bounded by C and the Maxwell equations

we can also write:

∇ × B · dS =

J · dS = 4πmI , (2.2)

– 2 –

JHEP03(2019)177

which is just the integral form of Ampere’s law. Combining these two results we ﬁnd

m =

4π



ijk

(x − x

)

|x − x

. (2.3)

Since there was no Levi-Civita symbol in his day, Gauss wrote his formula out in terms of

the components of x and x

, which gave an even more imposing result.

Note that the linking number (2.3) counts the signed crossings of the curve C

with

an arbitrary Stokes surface bounded by C. The direction the monopole is moved along C

and the direction the current ﬂows along C

ﬁxes an orientation on the curves, and the

orientation of the Stokes surface is ﬁxed relative to the orientation of its boundary.

It is generally intractable to directly apply Gauss’ formula to two arbitrary curves, but

since the result is topological, the curves can be deformed to make the calculation simpler.

First, adjust C

to lie in a plane, and then deform C to lie within a small distance |h| above

or below the plane. For small h, Gauss’ integral (2.3) is concentrated in the regions where

the curves almost touch. These regions can also be arranged so that the projection of C

onto the plane is oriented along the positive x-axis and C

is oriented along the positive

y-axis. Taking the “ﬂat knot limit” [25] h → 0, and labelling the crossings by an integer,

one ﬁnds the k-th crossing contributes

lim

h→0

arctan

= lim

h→0

sign(h) ≡

c(k) , (2.4)

to the integral. The crossing number c(k) is positive when C is above C

and negative when

C is below C

. We also need to keep track of the relative orientation of C and C

. If the

upper curve must be rotated counter-clockwise (as in the calculation above) the crossing

number is +1, but if it must be rotated clockwise then there is and extra minus sign [25].

Summing over all crossings we ﬁnd

m =

c(k) . (2.5)

Note that since both curves are closed there is an even number of crossings.

Rewriting Gauss’ result (2.3) as

m =

4π



ijk

∂

|x − x

, (2.6)

and using Stokes’ theorem, the linking number can be rewritten as

m =

4π



`ni

∂



ijk

∂

|x − x

4π



− δ



∂

|x − x

. (2.7)

Since C

has no boundary the second term vanishes and we ﬁnd

m =

(3)

(x − x

) , (2.8)

– 3 –

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