Magnetic Monopoles Maxwell ND Gonano

Author: Carlo Andrea GonanoCo-author: Prof. Riccardo Enrico Zich

Politecnico di Milano, Italy

ICEAA 2013, September 9-13, Turin, Italy

Magnetic monopoles andMaxwell Equations in N-

D

Contents1. “Classic” Maxwell equations 2. Magnetic monopoles and currents3. The beautiful Dirac’s Symmetrization

8. Remarks and conclusions 9. Questions and Extras

4. Cross product’s difficulties and N-D extension

5. Curl in 3-D & N-D extension6. Revising magnetic field and

monopoles7. Symmetrized Maxwell eq.s in N-D

2C.A. Gonano, R.E. Zich

Maxwell Equations

No “magnetic charge”and no “magnetic current”!

• In differential form Maxwell Equations can be written as:

• Divergence and curl equations for both E and B

for electric field E

for magnetic field B

IT LOOKS AS A QUITE SYMMETRIC SET, BUT…

• Introducing magnetic monopoles and currents, “symmetric” Maxwell Eq.s would look:


Magnetic monopòlesBUT 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 ?4C.A. Gonano, R.E. Zich

Dirac symmetrization

Symmetrized Maxwell Equations

SI units, [rm]= [C/(m/s) ] convention

In absence of magnetic monopòles and currents we get back the “classic” Maxwell eq.s and EM force

• In 1931 Paul A. Dirac, starting from a quantistic approach, symmetrizes Maxwell eq.s adding magnetic monopòles and currents

The generalized EM force per unit of volume is:


Arrays vs vectors• The set of eq.s can be written in a more compact form, defining arrays:

field array

charge array

flux array

anti-symm. matrix• Symmetrized Maxwell Equations will look so:

• The generalized EM force per unit of volume will be:

divergence eq.s

curl eq.s

ACTUALLY AN ELEGANT SYMMETRIC FORMULATION…6C.A. Gonano, R.E. Zich

nabla array

Duality transformation• The set of eq.s by Dirac is invariant under the duality transformation

• Symmetrized Maxwell Equations are left unchanged in form and the generalized EM force is exactly the same:

field arraycharge arrayflux array


ConsequencesIn Dirac’s formulation the E and B fields are treated

in the same way, sosome consequences arise…

• Conservation for electric and magnetic charge

• As a moving electric charge induces a rotational B field, so a moving magnetic charge would induce a rotational E field

figure from Wikipedia


No experimental evidence

• In the last decades, many experiments have been brought on to detect them, without results

SO A BEAUTIFUL FORMULATION, BUT…

• On december 2009 at CERN started the Monopole and Exotics Detector At the LHC (MoEDAL)

Can we really treat E and B fields in the same way?

Let’s stop for a moment and try to change our perspective:

Nowaday, isolated “magnetic charges” have never been observed!

Are we sure that E and B can be summed together?


A Socratic problemWHAT ARE ELECTRIC AND MAGNETIC FIELDS?

First of all…

• 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?10C.A. Gonano, R.E. Zich

The “classic” cross product• Operation with two vectors a and b

• In 3-D, it is variously interpreted…

FIRST OF ALL… WHAT’S “CROSS PRODUCT”?

Identical analitic definition in 3-D:

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 area of the parallelogram between a and b


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


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!

In fact, at the mirror the specular image of a right hand is a left hand and

counterclock-wise appears clock-wise!

Cross product does not respect reflection rules!!!


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, reflectionis possible…

With such a definition, this operation respect all algebric properties of cross-product, but the result is a

scalar!

…otherwise, 2-D inhabitants should have great fantasy to imagine a 3rd dimension to contain a vector

orthogonal to their plane…

• …but cross product does not exists

• …by the way, why use a vector when, in 2-D, a single scalar number is sufficient to describe a force’s moment?


4-D space• Let try to see if it’s possible to construct cross-product p = a Λ b in 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…


Vectors vs matrices• Angular velocity: pseudo-vector w, or matrix W?

FROM WHERE START FOR N-D EXTENSION?

• Let observe the z-component for M = r Λ F:• 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” …

Do you notice something?

• Let’s try to express moment in a “matrix” way:


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:

So, this is the N-D extension for cross-product Moreover, it can be verified that the new operator respects

all required algebric properties!

• Let’s note that vectors F and r can have any dimension N !

Just, the result is no more a vector, but a matrix!


Curl in 3-D and in N-D

Even in this case it can be verified that the new operator respects all required differential properties!

• Curl is a differential operator in same way analogous to cross product

WHAT IS CURL? CAN IT BE EXTENDED IN N-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:

Analitic definition in 3-D:

Curl’s definition in N-D


Magnetic field in N-D• Magnetic field B is often involved with X-product and 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, F B 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!

MAYBE THIS IS THE REASON FOR WHICH MAGNETIC MONOPOLES CAN’T BE FOUND!

• The use of B-tensor is not new, but it seems not to be always understood


Divergence and curl for field B• In 3-D the divergence of a pseudo-vector is:

In general, curl of a pseudo-vector is a true vector

SO, HOW TO RE-WRITE MAXWELL EQUATIONS IN N-DIMENSIONS?

• In a similar way, in N-D notation stands:• So:

3-indices divergence

• It can be easily demonstrated that the 3-D curl of a pseudo-vector correspond to:

with


Maxwell Equations in N-D

IS IT STILL POSSIBLE TO INTRODUCE MAGNETIC MONOPOLES AND CURRENTS IN N-D?

• Thus, “classic” Maxwell Equations in N-D become:

• EM force per unit of volume will look:

Yes, but “divergence” and “curl” for B have now different meanings! 21C.A. Gonano, R.E. Zich

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

Thus, in N-D the set of “symmetrized” Maxwell and Lorentz equations reveals to be not so “symmetric”

• So, no guarantee that rm assumes a single value:

• …but the meaning of EM force looks quite obscure now:

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:

???


Remarks

• In 3-D notation we have the impression that electric E and magnetic B fields are mathematically “similar”, but only the first one is a “true” vector!

Maybe the absence of magnetic monopòles is caused not by the lack of experimental devices, but by a

“linguistic” problem!

• 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


Conclusions and future tasks

• In 3-D both X-product and curl produce pseudo-vectors, while with N-D notation they generate matrices

STILL MUCH TO DO! THIS IS JUST THE BEGINNING

• Magnetic monopoles and current would symmetrize Maxwell’s

• Symmetrized Maxwell’s and EM force would be so invariant under the duality transf.

• So magnetic field B is not a vector but a matrix

• Interpretation of magnetic monopòles changed thanks to a notation invariant in every dimension


This is just the beginning…

QUESTIONS?EXPLANATIONS?

EXTRAS?


Extra details•Relativistic symmetrised Maxwell’s

•History of “magnetic particles”

•Dirac’s analysis•No experimental evidence


•History of vector analisys•Uses for cross-product

History of “magnetic particles”WHY THE DIVERGENCE OF B SHOULD BE ALWAYS

ZERO ?

• 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 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 1931 Paul A. Dirac, starting from a quantistic approach, symmetrizes Maxwell eq.s adding magnetic monopòles and currents


• 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

Dirac’s analysis

• 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

• In Quantised Singularities in the Electromagnetic Field (1931), Dirac inteprets magnetic monopole as a nodal singularity at the end of semi-infinite solenoid




No experimental evidence• 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)

Nowaday, isolated “magnetic charges” have never been observed, though many experiments have been

brought on to detect them

A moving magnetic monopole would cause an E field, so it could be detected by measuring the current induced in a conducting ring

HOW TO FIND MONOPOLES?

However, this would be not sufficient to prove their existence! Magnetic current could be solenoidal!


Relativistic symmetric Maxwell’s• Using relativistic notation, symmetrised 3-D Maxwell’s will look:

How the results could be “translated”?

• Relativistic EM force would be:

• The EM tensor F is defined together with it’s Hodge dual:

Very nice, but it works just in 3-D!In fact, in N-D the Hodge dual cannot be uniquely defined!

Hodge dual


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!


ConsequencesIn Dirac’s formulation the E and B fields are treated

in the same way, somany 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


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.

Volume’s calculus for parallelogramma

with a, b, c lata• In 1840 H.G. Grassmann introduces

“external product” and a wedge Λ as 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

Vol = (a Λ b) Λ c = a Λ (b Λ c)


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

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

p = a x b


Oddities for X-productThough it’s widely used, cross

product presents some“oddities”…

Right-hand rule

• 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:

…AND MANY PARADOXES ARISE!35C.A. Gonano, R.E. Zich

Uses for 3-D cross product• Cross product appears frequently in Physics and Engineering

• Easy-way for volume’s calculus

• Main use: moment’s definition and calculus

But, for which task cross-product is useful?

Moment M for force F applied to an“arm” r

• …if moments are known, we can describe rotations, torsion, stresses, spin and other quantity for different subjectsLarge applications in many fields!


Style exercizes• Calculus of perpendicular component:

• Moments in Rational Mechanics

• Double cross product:

2nd cardinal equation of motus:

Thus, moments become matrices: paradoxes linked to rotation, reflection, 2-D , 4-D etc. for X-product are

resolved 37C.A. Gonano, R.E. Zich

Contacts and references•Institutional e-mail: [email protected]

•Thesis download (in italian):o https://www.politesi.polimi.it/handle/10589/34061?mode=fullohttps://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

mailto:[email protected]

mailto:[email protected]

https://www.politesi.polimi.it/

https://www.politesi.polimi.it/

Magnetic Monopoles Maxwell ND Gonano

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