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Variations on the Dirac String PHYSICAL REVIEW D 98, 105013 (2018) Variations on the Dirac string Brad Cownden* Department of Physics & Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada † Andrew R. Frey Department of Physics, University of Winnipeg, 515 Portage Avenue, Winnipeg, Manitoba R3B 2E9, Canada and Department of Physics & Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (Received 1 August 2018; published 27 November 2018) Dirac’s original solution of the nontrivial Bianchi identity for magnetic monopoles [1], which redefines the field strength along the Dirac string, diagonalizes the gauge and monopole degrees of freedom. We provide a variant of the Dirac string, which we motivate through a formal expansion of the Bianchi identity. We show how to use our variant prescription to study monopole electrodynamics without reference to a dual potential and provide some applications. DOI: 10.1103/PhysRevD.98.105013 I. INTRODUCTION dual potential must be superposed in a “democratic” formalism (which makes it ill-suited to the quantum Magnetic monopoles, while experimentally elusive [2], mechanics of charges and monopoles). This type of hold a special place in particle physics; they complete approach has been advocated at least since [5]. electric-magnetic duality, provide a reason for charge (iii) As pointed out originally in [6,7], the vector poten- quantization, and arise in the spontaneous breaking of tial is not a globally defined function but a section of many grand unification models. At the same time, there is a a fiber bundle. When the field strength has a non- long tradition of reformulating the description of the trivial Bianchi identity, the potential is defined in at monopole’s interaction with the electromagnetic field since ’ least two coordinate patches; the potentials in the Dirac s original work [1,3]: two patches are related by a gauge transformation in (i) Monopoles arise as solitons of broken gauge theo- the overlap of the two patches. While mathemati- ries; the microscopic description as a semiclassical ’ cally rigorous, treating the potential in this manner field configuration describes the monopole s inter- mixes the gauge and monopole d.o.f. because the actions but requires tracking the full degrees of overlap region moves with the monopole. This freedom (d.o.f.) of that gauge theory. While of formalism also obscures the coupling between the course necessary for processes such as monopole potential and monopole (e.g., [7] switches to a dual creation and annihilation (see e.g., [4]), it may be potential formalism to derive the monopole’s computationally excessive when an effective de- classical equation of motion, while [8] following scription is valid. [9] combined Dirac’s formalism below with the (ii) In the absence of electric charges, the field strength ⋆ ≡ ˜ rigorous gauge patching procedure). can be defined in terms of a dual potential F dA, (iv) The potential in a single coordinate patch extends to which couples to monopoles in the same way the all of spacetime except for a half-line singularity vector potential couples to charges. Of course, if the extending from the monopole (Dirac’s famous string fields of both charges and monopoles are of interest, singularity). In [1], Dirac noted that the electromag- the field strengths from both the potential and the netic field strength can be written in terms of a globally defined potential and an extra term sup- *[email protected] ported on the string singularity. This approach † [email protected] separates the electromagnetic and monopole d.o.f. and elucidates the monopole-field strength coupling. Published by the American Physical Society under the terms of However, it suffers some conceptual difficulties, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to including a singular field strength along the (arbi- the author(s) and the published article’s title, journal citation, trarily chosen) string and a constraint that electric and DOI. Funded by SCOAP3. charges cannot intersect the Dirac string world 2470-0010=2018=98(10)=105013(7) 105013-1 Published by the American Physical Society BRAD COWNDEN and ANDREW R. FREY PHYS. REV. D 98, 105013 (2018) Z sheet. Pragmatically, the semi-infinite string can be μ ¯ ν ¯ μ 4 ð⋆ d ⋆ GÞ ¼ dY ∧ dY ∂νδ ðx − YÞ awkward. NZ In this paper, we present a new hybridization of Dirac’s ¯ 4 ¯ μ string formalism with the rigorous gauge patching pre- ¼ − d½δ ðx − YÞdY scription by allowing the Dirac string to end on an N 1 ˜μ − 1 ˜μ unphysical reference monopole, and we review the deri- ¼ð =gÞ| ð =gÞ|à ð4Þ vation of Dirac’s string prescription from the dual potential M M 1 formalism with an emphasis on the origin of the string. We in terms of the currents of monopoles along and Ã. also advocate for a particular configuration for the string As a result, the field strength defined as F ¼ dA − g ⋆ G and give a simple physical picture for its origin. The fixed automatically solves the Bianchi identity for the dynamical M prescription for the string means that the string world sheet monopole along as long as A includes the potential for a M embedding coordinates depend on the monopole position monopole with reference world line à (with appropriate along the entire world sheet, unlike Dirac’s original gauge patching). formalism; our approach presents computational advan- It is worth pausing now to discuss the configuration of ’ tages in several circumstances. The extension of our results the string. In Dirac s formalism, the string is a dynamical to higher-dimensional monopolelike branes appears in the object with no kinetic term (in modern parlance, a tension- less string), so the world sheet N is completely arbitrary companion paper [10] by one of us. Throughout this paper, 2 we work in four-dimensional Minkowski spacetime for except for the specification of its boundary. Dirac fur- M simplicity. We list our conventions, particularly signs, in thermore locates the worldline à at spatial infinity, so the the Appendix. string becomes semi-infinite. However, we are free to specify the configuration of the string; a simple choice in a given reference frame is to choose the reference II. DIRAC’S STRING VARIABLES M ⃗ monopole to be stationary ( à at constant position XÃ) ⃗ The Maxwell equations and Bianchi identities with and the string to be the straight line from Xà to the magnetic sources are monopole position X⃗ðtÞ at each time. We can build an arbitrary world sheet in this way by then extending a Dirac M M d ⋆ F ¼ −⋆j; dF ¼ −⋆|;˜ ð1Þ string from à to another reference worldline ÃÃ, and so on, and then letting the reference worldlines approach each μ ˜μ other. On the other hand, with a single line segment, the where j is the electric current and | is the magnetic μ current. For a single monopole, the current is world sheet embedding coordinates Y depend on the monopole position at all world sheet points τ, σ unlike Z in Dirac’s formalism. We motivate this prescription for the μ μ 4 μ 3 ⃗ |˜ ¼ g dτ∂τX ðτÞδ ðx − XÞ¼gu ðtÞδ ðx⃗− XÞ; ð2Þ Dirac string configuration in Sec. IV. M It is also instructive to consider the Dirac quantization of charge for different formalisms. With distinct gauge M ’ where g is the monopole charge, is the monopole s patches, quantization enforces the requirement that a τ μ worldline, is the worldline time, and X is the monopole single-value wave function in one coordinate patch remains position. The second equality follows in a fixed reference single valued after the gauge transformation in passing to τ μ frame with a worldline static gauge ¼ t; u is the another patch. In Dirac’s original formalism, there is only a ’ monopole s 4-velocity. single gauge, but the wave function picks up a phase under ’ Dirac s key observation stems from the fact that any motion of the string equal to g times the electric flux conserved current (in Minkowski spacetime) can be written through the surface swept out by the world sheet—the ˜ as the divergence of a two-form, or ⋆| ¼ d ⋆ H for some surface is noncontractible since the charge cannot intersect H. In fact, the field of an isolated monopole provides such the world sheet, as we see below. Since the surface swept an H subject to the additional constraint of the sourceless out by the string is closed (including the point at infinity), Maxwell equations. Instead, following Dirac [1], we define charge quantization is given by the condition that the wave Z function remain single valued if we sweep the string around μν μ ν 4 the charge. When the reference monopole is at a finite G ðxÞ ≡ dY¯ ∧ dY¯ δ ðx − YÞ; ð3Þ N 1An additional sign enters at the last equality from the world where N is the world sheet of a string with boundary sheet orientation as described in the Appendix. ∂N M − M M 2This arbitrariness reflects how a gauge transformation modi- ¼ à ( à is a so-far arbitrary worldline), Yμ τ; σ is the string position at world sheet coordinates fies where the potential of a monopole becomes singular (or more ð Þ precisely the patching required for the potential), so the string can τ σ σ M M ¯ τ∂ σ∂ , with increasing from à to , and d¼d τ þd σ never develop a tension, even quantum mechanically, for an is the world sheet exterior derivative. This has divergence unbroken gauge symmetry. 105013-2 VARIATIONS ON THE DIRAC STRING PHYS. REV. D 98, 105013 (2018) position, both mechanisms for charge quantization are Meanwhile, the electric charge couples to the redefined possible.
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