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Emerging Magnetic Monopoles Emerging magnetic monopoles Petr Beneˇs IEAP CTU in Prague ATLAS CZ+SK 2019 Workshop Prague, June 24, 2019 Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 1 / 32 Outline Descriptions of magnetic monopole within different physical paradigms: Classical physics −! Quantum mechanics −! Quantum field theory (Maxwell, Dirac, 't Hooft, Polyakov, Cho, Maison) Optionality/emergence of basic physical properties of magnetic monopole within different paradigms: Existence [sic] Mass Magnetic charge Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 2 / 32 Magnetic monopole in classical physics Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 3 / 32 ! Introducing the magnetic charge to the Maxwell's equations is nothing but trivial. Maxwell's equations Maxwell's equations: _ r · E = ρe r × B − E = je r · B = 0 −r × E − B_ = 0 Lorentz force: F = qe(E + v × B) Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 4 / 32 Maxwell's equations Maxwell's equations: _ r · E = ρe r × B − E = je _ r · B = ρm −r × E − B = jm Lorentz force: F = qe(E + v × B)+ qm(B − v × E) ! Introducing the magnetic charge to the Maxwell's equations is nothing but trivial. Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 4 / 32 Point-like magnetic charge (a.k.a. the magnetic monopole) Magnetic monopole (charge qm) is just a point-like magnetic charge density: The charge density 3 ρm = qmδ (r) B qm From the Maxwell's equations follows the \Coulomb's law" for magnetic monopole: 1 r B = qm 4π jrj3 Easy! Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 5 / 32 Classical monopole: Summary Introducing magnetic monopoles into the Maxwell's equations is trivial: It is even so trivial that all of its properties are completely arbitrary: Its existence is not forbidden, nor required either Mass and charge are arbitrary In fact, by allowing magnetic charges the Maxwell's equations obtain a new symmetry, a U(1) duality transformation between electric and magnetic fields/charges/currents: E0 + iB0 = eiαE + iB From classical point of view merely an esthetical curiosity But perhaps a hint that there is something deeper about the magnetic monopoles... ) Classically, it is a mystery why there are no magnetic monopoles in the Nature (At least for esthetically minded people...) Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 6 / 32 Magnetic monopole in quantum mechanics: Dirac monopole Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 7 / 32 Electromagnetic potentials in QM are indispensable But the Nature is not classical: Non-relativistic Schr¨odingerequation for an electrically charged (qe) particle ( ) in an external electromagnetic field: 1 2 @ − (r − iqeA) + qe' = i 2m @t The wave function doesn't couple directly to the elmag fields E, B, but to the elmag potentials ', A We are specificaly interested in behavior of an electrically charged particle in the field of a static magnetic monopole. ) What A corresponds to magnetic monopole? Answer is not trivial... Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 8 / 32 The Dirac string Turns out that: The potential A (such that B = r × A) can be defined only on a simply connected region where magnetic charge density vanishes. Proof by contradiction: 0 6= ρm = r · B = r · (r × A) = 0 3 3 0 6= qm = d x ρm = d x r · B = dS · B = dS · (r × A) = 0 ´ ´ ‚ ‚ ) For a single magnetic monopole qm: A can be defined everywhere except on a line stretching from the monopole position to infinity ! the Dirac string The precise position and shape of the Dirac string is arbitrary and can be moved around by a gauge transformation ) the Dirac string is non-physical Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 9 / 32 Need for two potentials ) At least two A's are necessary to fully describe field of a magnetic monopole. (Analogue of the situation in differential geometry where a single map is typically not sufficient to cover the whole manifold.) For instance, the following two potentials, A+ and A−, can be used: A+ is defined everywhere but on the negative z-axis: qm 1 − cos θ A+ = eφ 4πjrj sin θ A− is defined everywhere but on the positive z-axis: qm −1 − cos θ A− = eφ 4πjrj sin θ By construction, both yield the same magnetic field (r × A+ = r × A−), so that on their overlap they are necessarily related by a gauge transformation. Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 10 / 32 Dirac quantisation condition Back to an electrically charged particle (qe) in the field of a static magnetic monopole (qm): Each of the two potentials A± gives rise to a distinct Schr¨odingerequation: 1 2 − (r − iqeA+) + = i@0 + 2m 1 2 − (r − iqeA−) − = i@0 − 2m Both equations should lead to the same physics: ) The wave functions + and − must be the same up to a phase. However, turns out that this is in general not possible, unless the charges are related as qe qm = 2πn (n 2 Z) ) The Dirac quantisation condition Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 11 / 32 Comments on the Dirac quantisation condition qe qm = 2πn Robust: can be derived also by different methods and holds also for the soliton-type magnetic monopoles (to be discussed in a moment. ) Can be generalized to dyons (particles carrying both the electric and magnetic charge). The basic implication: If there exists at least a single magnetic monopole (with qm) in the Universe, all electric charges qe must be quantized: 2π qe = n qm Indeed: all electric charges in nature are quantized In fact, explanation of the electric charge quantization was one of the original Dirac's motivations for introducing the magnetic monopoles Today we know also other explanations for electric charge quantization (anomaly freedom, grand unified theories, . ) Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 12 / 32 Dirac monopole: Summary The type of magnetic monopole we have just described is known as the Dirac (magnetic) monopole. Like in classical physics, its existence again optional: It is put into the theory by hand (It is not predicted by the theory, nor required either) Also its mass is again arbitrary On the other hand, QM predicts something non-trivial about the magnetic charge: Dirac quantization condition: qe qm = 2πn Its implications (electric charge quantization) make apparent non-existence even more mysterious... Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 13 / 32 Magnetic monopole in QFT Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 14 / 32 No magnetic monopoles in QFT. ? But our world is described by QFT: We want a QFT description of magnetic monopoles as well For \electric monopoles" (all particles known to date) this can be done very easily and naturally ! the notoriously successful QED or SM Why not to simply incorporate magnetic monopoles into QED as electrons? But as it turns out, the magnetic monopole cannot be (easily and naturally) incorporated into the framework of QFT At least as long as one insists on having both electrically and magnetically charged particles in the theory and treating them symmetrically, on the same footing. Technically, the problems are due to singular character of A (the Dirac string). (In fact, it is possible to incorporate monopoles into QED, but for the prize of doubling number of elmag potentials, introducing constraints, having issues with renormalizability, locality... The resulting highly non-elegant Two-Potential QED is sometimes used, but only as an effective description, not as a fundamental theory.) −! There's a way out, but one has to give up the idea that magnetic monopoles are described in the same way as all other particles . Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 15 / 32 A particle in QFT: plane waves In QFT particles can be treated in two rather different ways: 1 By quantizing plane waves solutions of equations of motion The standard machinery taught in introductory QTF courses: Feynman diagrams, perturbation theory, annihilation/creation operators. ! \elementary" particles: quarks, leptons, gauge bosons, Higgs, gravitons, . Perturbative description Genuinely quantum description Alas, for magnetic monopoles this description doesn't work Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 16 / 32 A particle in QFT: topological soliton But there's another way how to describe a particle: 2 As a topological soliton Solution of the classical EOMs with particle-like properties In a way \bound states" of the elementary particles (plane waves) \Emerging": Not visible in the Lagrangian Inherently non-perturbative: Difficult to calculate scattering amplitudes with them Despite the computational difficulties, the \particle-ness" is perhaps more intuitive (Remarkably, the two seemingly different descriptions (plane waves and solitons) are in fact deeply connected, via S-duality) ! The magnetic monopoles! There are two different solitonic descriptions of monopole by 't Hooft and Polyakov (1974) by Cho and Maison (1996) Let's see. Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 17 / 32 't Hooft{Polyakov magnetic monopole Petr Beneˇs (IEAP CTU in Prague) Emerging magnetic monopoles Prague, June 24, 2019 18 / 32 Georgi{Glashow model The prototypical example: SU(2) gauge theory with Higgs field φ in triplet representation: 1 2 1 µν 2 λ 2 2 2 L = (Dµφ) − (F ) − (φ − v ) 2 4 4 If v2 > 0, the vacuum φ2 = v2 breaks spontaneously the symmetry SU(2) down to the \electromagnetic" U(1).
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