Author: Carlo Andrea Gonano Co-author: Prof. Riccardo Enrico Zich

1. Magnetic monopoles and Maxwell Equations in N-D • ICEAA 2013, • September 9-13, Turin, Italy Author: Carlo Andrea Gonano Co-author: Prof. Riccardo Enrico Zich Politecnico di Milano, Italy

2. Contents “Classic” Maxwell equations Magnetic monopoles and currents The beautiful Dirac’s Symmetrization Cross product’s difficulties and N-D extension Curl in 3-D & N-D extension Revising magnetic field and monopoles Symmetrized Maxwell eq.s in N-D Remarks and conclusions Questions and Extras C.A. Gonano, R.E. Zich

3. Maxwell Equations • In differential form Maxwell Equations can be written as: for electric field E for magnetic field B • Divergence and curl equations for both E and B IT LOOKS AS A QUITE SYMMETRIC SET, BUT… No “magnetic charge”and no “magnetic current”! • Introducing magnetic monopoles and currents, “symmetric” Maxwell Eq.s would look: C.A. Gonano, R.E. Zich

4. Magnetic monopòles BUT WHAT IS A MAGNETIC MONOPOLE? • Let consider field E: its “monopòles” are the electric charges, isolable and observable • A magnet generates a field B and its poles are called North and South: can they be isolated? • Problem: breaking a magnet you will not obtain two magnetic monopòles, but two magnets! • This difference between fields E and B has be known for a long time… HOWEVER, WHY THE DIVERGENCE OF B SHOULD BE ALWAYS ZERO ? C.A. Gonano, R.E. Zich

5. Dirac symmetrization • In 1931 Paul A. Dirac, starting from a quantistic approach,symmetrizes Maxwell eq.s adding magneticmonopòles and currents Symmetrized Maxwell Equations SI units, [rm]= [C/(m/s) ] convention The generalized EM force per unit of volume is: In absence of magneticmonopòles and currents we get back the “classic” Maxwell eq.s and EM force C.A. Gonano, R.E. Zich

6. Arrays vs vectors • The set of eq.s can be written in a more compact form, defining arrays: anti-symm. matrix field array charge array flux array nabla array • Symmetrized Maxwell Equations will look so: divergence eq.s curl eq.s • The generalized EM force per unit of volume will be: ACTUALLY AN ELEGANT SYMMETRIC FORMULATION… C.A. Gonano, R.E. Zich

7. Duality transformation • The set of eq.s by Dirac isinvariantunder the duality transformation field array charge array flux array • Symmetrized Maxwell Equations are left unchanged in form and the generalized EM force is exactly the same: C.A. Gonano, R.E. Zich

8. Consequences In Dirac’s formulation the E and B fields are treated in the same way, so some consequences arise… • Conservation for electricand magnetic charge figure from Wikipedia • As a moving electric charge induces a rotational Bfield, so a moving magnetic charge would induce a rotational Efield C.A. Gonano, R.E. Zich

9. No experimental evidence SO A BEAUTIFUL FORMULATION, BUT… Nowaday, isolated “magnetic charges” have never been observed! • In the last decades, many experiments have been brought on to detect them, without results • On december 2009 at CERN started the Monopole and Exotics Detector At the LHC (MoEDAL) Let’s stop for a moment and try to change our perspective: Can we really treat Eand B fields in the same way? Are we sure that E and B can be summed together? C.A. Gonano, R.E. Zich

10. A Socratic problem First of all… WHAT ARE ELECTRIC AND MAGNETIC FIELDS? • Electric field E(definition): • Magnetic field B has not such an explicit definition… • We can measure a magnetic force and express it as: • For charge Q with speed v, holds: Note the presence of cross-product: it often appears in Maxwell’s, together with the curl. Which is their role? C.A. Gonano, R.E. Zich

11. The “classic” cross product FIRST OF ALL… WHAT’S “CROSS PRODUCT”? • Operation with two vectors a and b • In 3-D, it is variously interpreted… Vector or oriented area? • Gibbs -“cross product”: • it is considered a vector … well, it has a magnitude and a direction! • Grassmann -“wedge product”: • it is the oriented areaof the parallelogram between a and b Identical analitic definition in 3-D: C.A. Gonano, R.E. Zich

12. Rotation-vectors in 3-D • Sum and scalar product are operations easy to extend in N-D, while cross product is defined just in 3-D • Cross product is used also to express rotation-vectors in 3-D… • … but they can not be summed with tip-tail rule, unlike common vectors! In fact, rotations don’t sum, because they don’t commute! THEN, ARE THEY “TRUE” VECTORS OR NOT? • Well, it depends on the “vector” definition… • …however, rotation-vectors and those originated by cross-product, are frequently called axial vectors or pseudo-vectors C.A. Gonano, R.E. Zich

13. Alice through the Looking-glass • Let try to position a set of vectors in front of a mirror • “True” vectors, like radii, velocities, forces etc. are simply reflected… • …on the contrary of moments and pseudo-vectors in general! Cross product does not respect reflection rules!!! In fact, at the mirror the specular image of a right hand is a left hand and counterclock-wise appears clock-wise! C.A. Gonano, R.E. Zich

14. Flatland • In “Flatland” (1884) E. A. Abbott describes life and customs of people in a 2-D world • In this universe vectors can be summed together and projected, areas are calculated, rotations are clock-wise or counterclock-wise, reflection • is possible… • …but cross product does not exists …otherwise, 2-D inhabitants should have great fantasy to imagine a 3rd dimension to contain a vector orthogonal to their plane… • …by the way, why use a vector when, in 2-D, a single scalar number is sufficient to describe a force’s moment? With such a definition, this operation respect all algebric properties of cross-product, but the result is a scalar! C.A. Gonano, R.E. Zich

15. 4-D space • Let try to see if it’s possible to construct cross-productp= aΛbin a 4-D space • Each vector has 4 elements; because p is the unknown, we’ll have 4 scalar unknowns • In 3-D we impose that p is perpendicular to vector a and b and that its magnitude is equal to the area between them But these are just 3 equations, while there are 4 unknowns! Problem has 1 Degree of Indetermination! In fact, in 4-D there is an infinity ∞1 of vectors p that satisfy these requirements! SO, CROSS PRODUCT MAYBE EXISTS JUST IN 3-D, OR IT’S NOT A VECTOR… C.A. Gonano, R.E. Zich

16. Vectors vs matrices FROM WHERE START FOR N-D EXTENSION? • Angular velocity: pseudo-vector w, or matrix W? • Let observe the z-component for M = rΛF: • Subscript “z” doesn’t appear neither in force nor in the arm • The moment, rather than “around axis z”,looks to be “from x to y” … • Let’s try to express moment in a “matrix” way: Do you notice something? C.A. Gonano, R.E. Zich

17. X-product extension in N-D • By treating moment M like a matrix, we notice that: • Expressing M as a function of dyadic products between F and r it yields so: • Let’s note that vectors Fand r can have any dimension N ! So, this is the N-D extension for cross-product Moreover, it can be verified that the new operator respects all required algebric properties! Just, the result is no more a vector, but a matrix! C.A. Gonano, R.E. Zich

18. Curl in 3-Dand in N-D WHAT IS CURL? CAN IT BE EXTENDED IN N-D? • Curl is a differential operator in same way analogous to cross product Analitic definition in 3-D: • In 3-D, curl suffers for the same problems of cross product: in fact the result is a pseudo-vector which does not respect reflection rules etc. • The extension in N-D is instantaneous: Curl’s definition in N-D Even in this case it can be verified that the new operator respects all required differential properties! C.A. Gonano, R.E. Zich

19. Magnetic field in N-D • Magnetic field Bis often involved with X-productand curl, so let’s verify if it is a “true” vector or not • Look at Faraday law and Lorentz force equations in 3-D: • We know, from definition, that E, FB and v are “true” vectors • …and, using N-D notation, the previous equations will look: Thus the magnetic field B is a not a vector, but a pseudo-vector, and, in a wider N-D view, it is a matrix or tensor! • The use of B-tensor is not new, but it seems not to be always understood MAYBE THIS IS THE REASON FOR WHICH MAGNETIC MONOPOLES CAN’T BE FOUND! C.A. Gonano, R.E. Zich

20. Divergence and curl for field B SO, HOW TO RE-WRITE MAXWELL EQUATIONS IN N-DIMENSIONS? • In 3-D the divergence of a pseudo-vector is: • In a similar way, in N-D notation stands: • So: 3-indices divergence • It can be easily demonstrated that the 3-D curlof a pseudo-vector correspond to: with In general, curl of a pseudo-vector is a true vector C.A. Gonano, R.E. Zich

21. Maxwell Equations in N-D • Thus, “classic” Maxwell Equations in N-D become: • EMforce per unit of volume will look: IS IT STILL POSSIBLE TO INTRODUCE MAGNETIC MONOPOLES AND CURRENTS IN N-D? Yes, but “divergence” and “curl” for B have now different meanings! C.A. Gonano, R.E. Zich

22. Symmetric Maxwell’s in N-D • With N-D notation, “divergence” for B is no more a scalar but a 3-tensor, with indices i,j,k completely arbitrary • So, no guarantee that rm assumes a single value: The existence of magnetic monopole cannot be excluded at all, but its interpretation would be quite different from the electric one! • Anyway, magnetic “charge” would conserve: • …but the meaning of EMforce looks quite obscure now: ??? Thus, in N-D the set of “symmetrized” Maxwell and Lorentz equations reveals to be not so “symmetric” C.A. Gonano, R.E. Zich

23. Remarks Maybe the absence of magnetic monopòles is caused not by the lack of experimental devices, but by a “linguistic” problem! • In 3-D notation we have the impression that electric E and magneticB fields are mathematically “similar”, but only the first one is a “true” vector! • In N-D notation instead it becomes clear that B actually is a tensor • Thus we can not treat them “in the same way”! The initial apparent “symmetry” was misleading Pseudo-vectors, like vorticity, momenta, rotation axis, spin etc. are spreadly used in Physics and Engineering. That of “magnetic monopoles” is just an example to show the potentiality of a different “language” valid in every dimension C.A. Gonano, R.E. Zich

24. Conclusions and future tasks • Magnetic monopoles and current would symmetrize Maxwell’s • Symmetrized Maxwell’s and EM force would be so invariant under the duality transf. • In 3-D both X-product and curl produce pseudo-vectors, while with N-D notation they generate matrices • So magnetic field B is not a vector but a matrix • Interpretation of magnetic monopòles changed thanks to a notation invariant in every dimension STILL MUCH TO DO! THIS IS JUST THE BEGINNING C.A. Gonano, R.E. Zich

25. This is just the beginning… QUESTIONS? EXPLANATIONS? EXTRAS? C.A. Gonano, R.E. Zich

26. Extra details • Relativistic symmetrised Maxwell’s • History of “magnetic particles” • Dirac’s analysis • No experimental evidence • History of vector analisys • Uses for cross-product C.A. Gonano, R.E. Zich

27. History of “magnetic particles” WHY THE DIVERGENCE OF B SHOULD BE ALWAYS ZERO ? • In early XIX cent., Gauss and Weber already considered the question • In his “Wirbelbewegung”(1858) H. von Helmholtz calculates the force exterted on a “magnetic particle” by an electric current • In “A Treatise on Electricity and Magnetism” (1873) J. C. Maxwell reports that experimentally magnetic flux F(B) is always zero across a closed surface • In 1894 Pierre Curie defends the possible existence of “magnetic charge” • In 1931 Paul A. Dirac, starting from a quantistic approach,symmetrizes Maxwell eq.s adding magneticmonopòles and currents C.A. Gonano, R.E. Zich

28. Dirac’s analysis • Because of duality transf. , the ratio between magnetic and electric charge can be arbitrarily changed, thus the magnetic charge can be set to zero just if all particles in the universe have the same ratio (c Qm)/Qe • In Quantised Singularities in the Electromagnetic Field (1931), Dirac inteprets magnetic monopole as a nodal singularity at the end of semi-infinite solenoid • In the same paper, Dirac demonstrates that the existence of just one magnetic monopole in the universe would explain the charge quantization Dirac string, picture from http://cds.cern.ch/record/1360999 C.A. Gonano, R.E. Zich

29. No experimental evidence Nowaday, isolated “magnetic charges” have never been observed, though many experiments have been brought on to detect them • In september 2009, Science reported that J. Morris, A. Tennant et al. from the Helmholtz-Zentrum Berlin had detected a quasi-magnetic monopole in spin ice dysprosium titanate (Dy2Ti2O7) • On december 2009 at CERN started the Monopole and Exotics Detector At the LHC (MoEDAL) HOW TO FIND MONOPOLES? A moving magnetic monopole would cause an E field, so it could be detected by measuring the current induced in a conducting ring However, this would be not sufficient to prove their existence! Magnetic current could be solenoidal! C.A. Gonano, R.E. Zich

30. Relativistic symmetric Maxwell’s How the results could be “translated”? • Using relativistic notation, symmetrised 3-D Maxwell’s will look: • The EM tensor F is defined together with it’s Hodge dual: Hodge dual • Relativistic EM force would be: Very nice, but it works just in 3-D! In fact, in N-D the Hodge dual cannot be uniquely defined! C.A. Gonano, R.E. Zich

31. Magnetic monopòles • In differential form, “symmetric” Maxwell Equations would look: BUT WHAT IS A MAGNETIC MONOPOLE? • Let consider field E: its “monopòles” are the electric charges, isolable and observable • A magnet generates a field B and its poles are called North and South: can they be isolated? • Problem: breaking a magnet you will not obtain two magnetic monopòles, but two magnets! C.A. Gonano, R.E. Zich

32. Consequences In Dirac’s formulation the E and B fields are treated in the same way, so many consequences arise… • Conservation for electric and magnetic charge • A moving magnetic monopole would induct an E field • Dirac demonstrated that the existence of just one magnetic monopole in the universe would explain the charge quantization figure from Wikipedia C.A. Gonano, R.E. Zich

33. A bit of History… • In 1773, Lagrange finds cross product analitically in order to calculate volume of tetrahedra… • …but “vector” haven’t been “invented” yet! • In 1799, C. F. Gauss and C. Wessel represent complex numbers like arrows on a plane. Vol = (aΛb) Λc = aΛ (bΛc) Volume’s calculus for parallelogramma with a, b, c lata • In 1840 H.G. Grassmann introduces “external product” and a wedge Λas its symbol • … but for Grassmann the operation’s result is not a “vector”: though, it’s a area, a volume or a signed iper-volume. • Operation acts on many vector at time: • first N-D“extension” Hermann Gunther Grassmann C.A. Gonano, R.E. Zich

34. A bit of History… • In 1843, W. R. Hamilton introduces quaternions to describe rotations in 3-D. • In 1846 Hamilton adopts the terms scalar and vector referring to real and imaginary parts of a quaternion. • The vectorial part of a product between quaternions with null real part is equal to cross product p=axb William Rowan Hamilton • In 1881-84, J. W. Gibbs writes Elements of Vector analysis, where modern vectorial calculus is explained. In 1901 his disciple Edwin B. Wilson publishes Vector Analysis, which had a large diffusion. • Cross product is indicated with a x and is considered a vector • In 1885, O. Heaviside develops a vector system analogous to Gibbs’s one and applies it to electromagnetism Josiah Willard Gibbs C.A. Gonano, R.E. Zich

35. Oddities forX-product Though it’s widely used, cross product presents some“oddities”… • Need for “clock-wise” and “right-hand” concepts • Cross product is not so easy to use: • wrong signs are the most frequent mistake: + or - ? • In practice, you have to memorize long identities like: Right-hand rule • …AND MANY PARADOXES ARISE! C.A. Gonano, R.E. Zich

36. Uses for 3-D cross product But, for which task cross-product is useful? • Cross product appears frequently in Physics and Engineering • Easy-way for volume’s calculus • Main use: moment’s definition and calculus • …if moments are known, we can describe rotations, torsion, stresses, spin and other quantity for different subjects Moment Mforforce Fapplied to an“arm” r Large applications in many fields! C.A. Gonano, R.E. Zich

37. Style exercizes • Calculus of perpendicular component: • Double cross product: • Moments in RationalMechanics 2nd cardinal equation of motus: Thus, moments become matrices: paradoxes linked to rotation, reflection, 2-D , 4-D etc. for X-product are resolved C.A. Gonano, R.E. Zich

38. Contacts and references • Institutional e-mail: carloandrea.gonano@polimi.it • Thesis download (in italian): • https://www.politesi.polimi.it/ • handle/10589/34061?mode=full • https://www.politesi.polimi.it/ • bitstream/10589/34061/1/2011_12_Gonano.pdf • Keywords (ENG): • cross product n-d extension; • cross product; wedge product; curl; rotor; N-D space; magnetic monopoles • Keywords (ITA): • prodotto vettore n-d; • prodotto vettore; rotore; spazio N-D; monopoli magnetici