Monopoles  typeset 784 Jun 23, 2003  Smith

Charge quantization, the topology and 3-dimensionality of the universe, and abolishing

monopoles

Warren D. Smith ^{∗}

WDSmith@fastmail.fm

14 Jan., 2000

Abstract —

Why do all electrons have the same charge and why do all protons have exactly the opposite charge? Dirac provided a possible answer by proposing the existence of magnetic monopoles.

We propose a simple possible topological explanation of why charge is quantized, involving a tiny permanent magnetic field trapped in the topology of the universe. The idea apparently works for every possible compact 3-manifold topology for the universe except for the 3sphere S^{3} and elliptic 3-geometry. This picture does not need to assume magnetic monopoles exist, and indeed looks incompatible with their existence.

Which topologies for the universe are consistent (or not) with laws of physics? We present an argument that all orientable 3-manifolds should be consistent with a very wide class of possible laws of physics; but almost all n-manifolds for each n ≠ 3 won’t be. This is perhaps a “reason the world is 3 dimensional.”

Next we argue that if point monopoles exist, and Dirac’s wave equation of quantum mechanics in static (scalar and vector) potentials holds, then a contradiction arises. Hence monopoles cannot exist – or: they are not points; or: Dirac’s equation for quantum mechanics is not correct. Meanwhile non-point monopoles lead to other unpalatabilities or contradictions.

Hence there are reasons to prefer my topological explanation of charge quantization to Dirac’s monopole hypothesis, if possible.

(In the event that non-point monopoles do exist, we show how, by very elementary reasoning, to deduce certain of their properties, e.g. their moment of inertia.)

Keywords — Monopoles, Dirac, topology of the universe, dimensionality of the universe, combing hair on manifolds, charge quantization, Hopf fibration, moment of inertia.

∗Temple University Math. Dept. Wachman Hall, 6th floor, 1805 North Broad Street Philadelphia, PA 19122.

1 Review of previous arguments about magnetic monopoles and charge quantization

No magnetic monopoles have ever been found, despite extensive experimental searches [5][11][24][25]^{1}. There are nevertheless three reasons why many people believe (or hope) that monopoles exist:

1. The quantization of charge would be explained (§1.2)^{2}.

1Cabrera [11] found a single monopole candidate event with his superconducting magnetic loop detector in 1982. It was consistent with a Dirac monopole (EQ 4) with Z = 1. However, this observation has never been reproduced despite running much bigger and better detectors for longer, and Schwinger’s theoretical arguments [68] suggest that Z must be a multiple of 4. Hence this candidate event is regarded as dubious. A possible future monopole search might be conducted with huge superconducting loops placed somewhere cold in the outer solar system.

2“Kaluza-Klein theories” [39][41][66][7][80][20] postulate extra “compact” dimensions of space. The simplest K-K theory involves one extra 5th “cylindrical” dimension, which has the same circumference everywhere (of order the Planck length). The electromagnetic potential 4-vector A^{μ} is actually the 5th row of the metric tensor g_{αβ} (except that we impose the demand that g_{55} be a constant).

Supposedly, in K-K theory the Einstein equations obeyed by g_{αβ}, under certain assumptions about “how things look” in the 4D “projection” when we “average over microscopic details in the 5th dimension x^{5}” and assume g_{αβ} does not depend on x^{5}, yield the usual 3+1D Einstein gravity equations as well as the Maxwell electromagnetism equations, at least in vacuum. Electric charge in K-K theory is actually momentum in the 5th dimension. Clockwise motion around the extra dimension corresponds to positive charge; counterclockwise to negative. Charge reversal and parity reversal are the same thing (and it seems to be difficult or impossible to incorporate parity violation into a K-K theory). Charge conservation arises from conservation of momentum. Angular momentum conditions lead to charge quantization automatically [41]. My brief examination of K-K theory suggests that it is quite complicated and the fundamental claims above have not been justified rigorously, nor has a careful examination of the fundamental assumptions behind K-K theory been made (e.g.: which ones are really necessary?).

Certain GUTs also yield charge quantization automatically; thus on p.430 of [13]: “whenever the unification gauge group is simple, charge quantization will follow... because the eigenvalues of a simple nonAbelian group are discrete while those corresponding to the Abelian U(1) group are continuous. For example in SO(3) the eigenvalues of the third component of angular momentum can take on only integer or half integer values while... [for] U(1) symmetry of translation invariance in time... no restriction...” Although I do not under-

2. Maxwell’s equations would get a new symmetry (§1.1).

3. The simplest Grand Unified Theory [27] predicted the existence [64][74][13] of monopoles of mass ∼^{>} 10^{16}GeV/c^{2} ≈ 2 × 10^{−11}kg. After SU(5) GUT was refuted by proton lifetime measurements [9]^{3}, other “supersymmetric GUT” theories and “superstring” theories were devised which continue to predict monopoles, although now of larger masses ∼^{>} 10^{17}GeV/c^{2}. These GUT monopoles are predicted to have at least twice Dirac’s minimal magnetic charge value (EQ 4).

In the present paper, we are going to show how to explain charge quantization without any need for monopoles. We are going to argue for this new explanation of charge quantization, and against monopoles:

Our new explanation removes rationale #1. We’ll show that monopole existence would lead to contradictions in present physical laws – undercutting rationale #3 to the extent that present day physical theories are regarded as correct. Of course if they are regarded as overruled by GUTs (at least in appropriate regimes) my objections too are presumably overruled – but still interesting^{4}.

1.1 Maxwell’s equations

Maxwell’s equations feature some remarkable symmetries (e.g. they are invariant under Lorentz transformations, as well as a much larger, albeit lesser known, continuous group of symmetries discovered by Cunningham [15] in 1910). But they are not symmetric under the interchange of electricity and magnetism. By introducing magnetic monopoles we gain such a symmetry^{5}.

Define the “SIpole,” a new SI unit (pending approval!) of magnetic charge, as follows: “One SIpole is the amount of magnetic charge that would experience a force of 1 Newton if placed in a 1 Tesla magnetic field.” (Note, 1 Coulomb is the amount of electrical charge that would experience a force of 1 Newton in an electrical field of 1 volt/meter. 1 SIpole= 1meter×Coulomb/second.) The Maxwell equations with monopoles then are

∇ · E⃗ = ρ_{e}/ϵ_{0}, ∇ · B⃗ = ρ_{m}μ_{0},  (1)
∇ × B⃗ − c^{−2} ˙E⃗ = μ_{0}J⃗_{e}, ∇ × E⃗ + ˙ B⃗ = μ_{0}J⃗_{m}.

Here E⃗ and B⃗ are the electric and magnetic fields, J⃗_{e} and J⃗_{m} are the current densities for electric and magnetic charges respectively, ρ_{e} and ρ_{m} are the charge densities, μ_{0} = 4π × 10^{−7}Henry/meter is the permittivity of free space, and ϵ_{0}μ_{0} = c^{−2} where c = 299792458meter/second is the speed of light. The Lorentz force experienced by an electric charge q moving with velocity v⃗ is then E⃗q+v⃗×B⃗q. The neo-Lorentz force on a magnetic charge g is B⃗g−c^{−2}v⃗×E⃗g.

1.2 Charge quantization

The argument that the existence of even one magnetic monopole anywhere in the universe would imply (to keep physics self consistent) the quantization of both electric and magnetic charges (thus explaining the facts that all electrons have the same charge, and that protons and electrons have exactly opposite charges) is due to P.A.M.Dirac [18]. There are three ways to look at this argument.

1. An integration first performed by J.J.Thomson [76] shows that the angular momentum

A⃗ = \frac{1}{cμ_{0}} ∫_{−∞}^{∞} ∫_{−∞}^{∞} ∫_{−∞}^{∞} r⃗ × E⃗ × B⃗ d^{3}r⃗  (2)

arising from the magnetic field B⃗ = μ_{0}gr⃗/(4πr^{3}) of a point monopole g and the electric field E⃗ = qr⃗/(4πϵ_{0}r^{3}) of a point charge q, is |A⃗| = qgμ_{0}/(4π), where A⃗ points from the charge toward the monopole, regardless of the distance between them. If one assumes that this angular momentum is Zh̄ where Z is an integer, we conclude that

qg = \frac{h}{μ_{0}} Z.  (3)

Assuming the electric charge quantum is e ≈ 1.602 × 10^{−19} coulomb (i.e. the electron charge)^{6}, the quantum of magnetic charge is

g = \frac{h}{μ_{0}e}Z ≈ 3.29106 × 10^{−9}Z SIpole.  (4)

2. During a flyby of a charge q past a fixed monopole g the charge will be pulled by Lorentz forces in the manner of a “twist,” in such a way that the total angular momentum it acquires is 2qgμ_{0}, regardless of the charge’s velocity, its mass, and the minimum distance between the two [13]. Assuming this increment in angular momentum is quantized in units of h̄ again leads to (EQ 3).

stand this, note that it is demonstrated by a logical pathway that does not ever involve a monopole. Thus, it might be possible to explain charge quantization without either our picture, or monopoles; but if we restrict ourselves to presently well accepted physical theories, then Dirac’s and mine are the only explanations I know of.

3According to the 1999 summary of particle data by the Particle Data Group, the mean proton lifetime for decay into a neutral pion and positron is > 5.5 × 10^{32} years with 90% confidence. Ongoing measurements by the super-Kamiokande detector should increase this bound by an order of magnitude; indeed an unsubstantiated claim is made on the super-K web page that the proton lifetime is now known to exceed 10^{33} years.

4In that case my objections at least serve to focus attention on how major consequences can follow from as yet unknown and speculative physical laws in regimes of the very small. Also, the possibility and perhaps desirability of abolishing monopoles will now give more freedom to the creators of GUTs.

5This gain comes at the cost of invalidating the “PT” symmetry arising from space inversion and time reversal – but the “CPT” symmetry, also incorporating charge reversal for both magnetic and electric charges, would remain valid. The gain is greater than the loss, because actually we also would gain a continuous symmetry allowing not only the interchange (with appropriate unit conversion and sign changes:

E⃗ → cB⃗, cB⃗ → −E⃗, cJ⃗_{e} → −J⃗_{m}, J⃗_{m} → cJ⃗_{e}) of magnetic and electric fields and charges, but in fact any 2D rotation E⃗ → E⃗ cos θ + cB⃗ sin θ, cB⃗ → cB⃗ cos θ − E⃗ sin θ, cq → cq cos θ − g sin θ, g → cq sin θ + g cos θ among them (the interchange arising as the special case when the rotation angle θ is θ = π/2) [37].

6Actually, quarks have charges that are integer multiples of e/3.

This would cause the quantum of magnetic charge to be 3 times larger.

However, it is generally conjectured to be impossible to isolate a free quark (or anything of nonzero “color” or subintegral charge) more than at most .86 × 10^{−15}meter away from other quarks which neutralize these anomalies.

3. Consider [18][13] the Dirac wavefunction of an electron in the presence of a point monopole. Dirac’s quantum relativistic equation is actually inapplicable in this situation because it^{7} involves a “vector potential.”

However, Dirac argued that a smooth vector potential can be created that will work everywhere except on one point on each sphere that encloses the monopole. His approach may be thought of as regarding the monopole as one end of a very long and very thin solenoid, and taking the limit as this solenoid becomes infinitely long and thin (Dirac calls the resulting mathematical object a “string”). The vector potential is singular on the curve describing this solenoid. Dirac then wanted the charge not to “see” the solenoid and wanted the wavefunction to be independent (except for symmetries which do not matter) of the solenoid’s location. He argued this and an assumption of continuity would force the “Aharonov Bohm phase shift” arising for a trajectory enclosing the string, to be an integer multiple of 2π; this led to (EQ 3). The precise argument is given in [13]; for the analogous argument in the present paper’s alternative topology-based charge quantization explanation, see §2.3.

Later Schwinger [68] attempted to formulate a theory of (or at least, some constraints on a theory of) relativistic quantum mechanics which would work in the presence of monopoles. In order to remove “arbitrariness in physical predictions” in relativistically invariant “photon exchange between different source types,” he deduced that Z in (EQ

4) must be a multiple of 4.

2 My alternative: Charge quantization from topologically trapped magnetism

It is possible to make a similar argument, but without any need to assume a monopole exists, by assuming a “topologically trapped” magnetic field in the universe. I will prepare the ground by describing easy-to-understand special cases of my idea in §2.1-2.4, then give a fully-rigorous and general description in §2.5.

Throughout this paper, when I speak of “topology of the universe” I am going to mean 3D spatial topology, regarded as a 3-manifold. Definition: Whenever I say “n-manifold” (including here!) I mean “smooth compact^{8}connected boundaryless Riemannian n-manifold.” Until §5, I will mostly ignore the fact that, in general relativity, space and time are in fact inseparably entangled to form a 3 + 1D Lorentzian manifold. As a partial compromise, I will allow thinking of the Riemannian 3-manifold as slowly varying with time.

2.1 The argument in a 3-Torus universe

A simple reification of our idea is a 3D “flat torus” universe T^{3} = S^{1}×S^{1}×S^{1}, i.e. a Euclidean box with sides L_{1}, L_{2}, L_{3} with “wraparound” periodic boundary conditions. Assume there is a constant magnetic field B⃗ in the L_{3} direction. (Of course, any constant B⃗-field is an exact solution of the vacuum Maxwell equations in this 3-torus.)

Figure 1: Rectangular 3-Torus universe with constant trapped magnetic field B⃗. Consider a charge q moving on trajectory 1 around the universe (and returning to its starting point). Other possible trajectories are 2 and 3.

Assume a charge q moves along some trajectory up through the upper box face (coming back again through the bottom face) eventually returning to the same location it started. The total momentum transferred to the charge by Lorentz forces is qBL_{1} in the L_{2} (transverse) direction. The important thing to notice is that this transverse momentum formula holds regardless of the charge’s mass, its velocity, and its trajectory x⃗(t) (provided it has “unit winding number;” otherwise we have to multiply by an appropriate integer).

7If the B⃗ field is allowed to have nonzero divergence, no such potential exists. The experimental confirmation of the “Aharonov-Bohm effect” tells us that a vector potential really is needed in quantum mechanics, so we have no clue how to proceed without it.

8Throughout this paper I’ll assume a compact (i.e. spatially closed & finite) universe. It is unknown if the universe actually is spatially closed & finite. By “closed” here, I do not mean the same thing as cosmologists mean. I mean what mathematicians mean. Cosmologists mean: rebounding to a “big crunch.” I do not care for my purposes whether the universe crunches or expands forever. What I care about is whether the universe is finite in spatial extent at any particular time (e.g.: the 3D surface S^{3} of a 4D sphere, or a 3-torus S^{1} × S^{1} × S^{1}, or a finite-volume hyperbolic 3-manifold) versus whether it is infinite spatially at any particular time (e.g.: euclidean 3-space R^{3}, or 3D hyperbolic space H^{3}, or an “infinite cylinder” S^{1} × R^{2}). It is

presently totally unknown what the topology of the universe is – the best that can be said [45][70] is there are unreliable statistical indications disfavoring universes with positive overall curvature (e.g. S^{3}).

The question of whether the universe is spatially compact is totally unknown.

Other, bizarre, possibilities are also thinkable: boundaryless 3manifolds of finite volume but nevertheless noncompact [1], i.e. having unboundedly large distances! I’ll now demonstrate how to construct such universes. Consider a 4D convex polyhedron all of whose vertices lie on the 3D surface of a ball in R^{4}. (The simplest example:

a 4-simplex, the convex hull of 5 generic points on a sphere in R^{4}.)

The 3D surface of such a polyhedron is – if the interior of the ball is now regarded (via the “projective,” also called “gnomonic,” model of hyperbolic space) as hyperbolic 4-space – a hyperbolic (i.e. negative constant curvature) 3-manifold of finite 3-volume. But as you approach the vertices of the polyhedron, you get infinitely far from everything else in the hyperbolic metric. (Note, the “edges” of the polyhedron are invisible to the intrinsic surface metric.) The whole theory of the present paper probably would allow many noncompact manifolds (including these) too, but I have not examined that question.

Hence with respect to a fixed observer in the center of the box, the charge has acquired angular momentum increment qBL_{1}^{2}/2. If this is quantized in steps of h̄, then q must be quantized in steps of

2h̄/(BL_{1}^{2}).  (WRONG)  (5)

We have deduced charge quantization.

Actually, though, it is wrong to rely on angular momentum. Angular momentum is not a well defined concept in a compact, not rotationally symmetric universe. For example its sign depends on “which way the observer is looking.” Also, if the observer looks “several times round the universe” he can artificially lengthen the moment arm and get an infinite number of different angular momentum values.

It is better to rely on the fact that momentum itself has a quantal property in a compact universe, namely there have to be an integer number of De Broglie wavelengths λ = h/p on a suitable path round the universe in the direction of that momentum p⃗. With p = qBL_{1}, we find that L_{2}p must be h times an integer, therefore charge must be quantized in steps of

\frac{h}{BL_{1}L_{2}} = \frac{h}{Φ}  (6)

where Φ = BL_{1}L_{2} is the total magnetic flux “round the universe.” Note, if the quantum of charge is assumed to be the electron’s charge e, and L_{1} = L_{2} = 10^{10} lightyears, the the flux round the universe must be an integer multip0le of the “flux quantum” Φ_{0} = h/e ≈ 4.136 × 10^{−15}Weber and the trapped magnetic field of the universe must be (an integer multiple of) B = h/(eL_{1}L_{2}) ≈ 4.6 × 10^{−67} Tesla. Thus, an undetectably small B-field is sufficient to do the job. (Cf. [82] for the best known observational upper bound 3 × 10^{−15} Tesla.)

Note that there is no need for the universe to be a rectangular box – any parallelipiped would do.

Now suppose this universe expands, i.e. L_{1}, L_{2}, and/or L_{3} increase. Are we then going to get a quantum of charge that depends on the time-varying size of the universe? (Bad!) No.

As the L_{k} increase, B must decrease in such a way that Φ = BL_{1}L_{2} remains constant. This is because the number of “lines” of magnetic flux must be fixed – lines cannot be created or destroyed since they must be closed loops and can have no endpoints in the absence of magnetic monopoles. (Each “line” represents a fixed infinitesimal amount of magnetic flux.) More precisely: electric currents can (and will) create new closed B-field loops, but such loops must be homotopic to the identity, i.e. contractible to nothing. Uncontractible loops such as ours are “trapped” and can’t be created or destroyed. Hence the charge quantum (EQ 6) would not be affected.

One could similarly argue that were our torus universe “wider” in some places than in others, BL_{1}L_{2} would still remain invariant no matter which cross section one were on. (Anyhow, it would suffice logically for just one closed trajectory to work, since we could consider moving a charge along a path P until reaching the approved trajectory, using it, then moving back along P^{−1}.)

Indeed we are going to show in §2.5 that the same argument will work in any 3-torus, not just these “standard flat” ones – and also in other topologies besides tori.

2.2 Some physical remarks

Physically, one would certainly expect some topologically trapped magnetism in any universe in which it is topologically possible – i.e. it would be surprising to have none.

My trapped B⃗-field hypothesis seems incompatible with Dirac’s hypothesis monopoles exist, since a monopole could keep accelerating along the B⃗-field loop forever, sucking up energy forever. The “potential energy” associated with the B⃗-field would be nonconservative. It seems that a gas of monopoles in such a universe could keep getting hotter and hotter forever, with conservation of energy being disobeyed. To avoid this, we must abolish monopoles. (Presumably the universe is free of trapped E⃗-field for the same reason.)

2.3 Analogue of Dirac’s “string” argument

In earlier drafts of this paper, I had some confused arguments about charge quantization by this rough mechanism. We have seen that trying to argue about angular momentum runs into difficulties. Trying to argue about ordinary momentum works for the flat 3-torus universe, but gets tricky when one tries to consider general curved 3manifolds. So then I was pulled toward Dirac’s own notion (cf. #3 of §1.2) that the best kind of argument depends on (1) the continuity of physics – under infinitesimal perturbations, observable effects should change infinitesimally – and (2) the Aharonov-Bohm phase-angle shift.

I currently believe that this Dirac-like idea does work and is the clearest and best way to proceeed, but even here there are some traps to snare the unwary. E.g., in the 3-torus example we just discussed, one can explain the B⃗-field as being generated by a “vector potential” A⃗ with ∇×A⃗ = B⃗. Thus the constant horizontal field B⃗ = (1, 0, 0) could be thought of as due to a vector potential A⃗ = (0, 0, y). Note that this A⃗ field necessarily has some discontinuities if we consider it “wrapped” into the flat 3-torus, e.g. we could make it discontinuous on the plane y = L_{1} (which by vertical “wraparound” is the same as the plane y = 0) but this A⃗ works everywhere else. One might then imagine making some kind of charge-quantization argument by considering electron trajectories which cross or go near this discontinuity-plane. But this approach does not work! There are several reasons for that. First, A⃗ = (0, −z, 0) also works to explain B⃗ = (1, 0, 0) = ∇ × A⃗. This has an entirely different natural discontinuity plane (z = L_{1} or z = 0). It seems as though one can always use gauge freedom in A⃗ to make it stay continuous in any neighborhood of whatever electron trajectory one is considering. Second, A⃗ is continuous and well behaved everywhere in the covering space (here R^{3}) of our manifold. These facts seem

DocNumber  4  . 2. 3. 0

to prevent making any argument about discontinuties or singularities in A⃗.

The right and maximally simple approach, which avoids all that mess, is to base everything on the following: A⃗-free formulation of the Aharonov-Bohm phase shift principle: Suppose an electron travels along a closed path ∂D that may be regarded as the boundary of a topological disk D. I.e., D is homeomorphic to the set {(x, y) such that x^{2} + y^{2} ≤ 1}, and ∂D is homeomorphic to {(x, y) such that x^{2} + y^{2} = 1}. Then the electron’s wave function will be multiplied by a complex phase e^{iθ} where θ = eΦ/h̄, where e is the electron charge and Φ is the magnetic flux “through the loop,” i.e. crossing D.

Figure 2: Rectangular 2-Torus; magnetic field B⃗ is perpendicular to the plane of the figure. Consider the electron trajectory shown dashed. Regard it as infinitesimally close to the boundary of the rectangle.

The point is that this avoids employing any vector potential A⃗ as in the usual expression θ = (e/h̄) ∫ A⃗ · dℓ⃗.

Now in our flat 3-torus universe example, simply consider the electron trajectory in figure 2. Obviously, this trajectory encloses a topological disk passing flux Φ = L_{1}L_{2}B. Therefore, the Aharonov-Bohm phase shift angle is infinitesimally close to θ = eΦ/h̄. But, also obviously, this trajectory should have Aharonov-Bohm phase shift angle infinitesimally close to zero, because of our assumption of the continuity of physics, and the fact that any phase shift on part I of the trajectory should be exactly canceled by the phase shift experienced on the (infinitesimally nearby, and opposite) part III; and similarly II and IV should cancel. The only way to resolve this contradiction is to demand that θ must be an integer multiple of 2π. This yields our usual quantization condition EQ 6.

2.4 But the argument does not work in a 3-sphere universe

It is possible to make a nontrivial topologically trapped constant B⃗-field on a 3-sphere S^{3} (I.e. a nonzero exact time-independent solution of Maxwell’s vacuum equations). There is a topologically unique way to do this, called the “Hopf fibration” (described below). However, it turns out that the total Lorentz momentum increment experienced by an electron on a closed trajectory in this universe, is always zero – and no interesting deductions can be made by considering the Aharonov-Bohm phase shift and continuity – so neither approach to deducing charge quantization works.

The Hopf fibration ([79] p.103-108; [34][29]) is best described using “quaternions.” Quaternions are like complex numbers except they are 4D instead of 2D because there are 3 square roots of −1 (called i,j,k) not just one root i. Arithmetic: ij = −ji = k, jk = −kj = i, ki = −ik = j, ijk = i^{2} = j^{2} = k^{2} = −1. Quaternion addition is an abelian group, and multiplication of nonzero quaternions is a noncommutative group (indeed {i, j, k, 1} either commute or anticommute), and the distributive law holds. Quaternions Q = a + bi + cj + dk (a,b,c,d, real) have “norm” |Q|^{2} ≡ a^{2} + b^{2} + c^{2} + d^{2} and norms are multiplicative. Multiplication by a fixed unit-norm quaternion performs a rigid rotation on 4-space.

We now are ready to describe the Hopf fibration on the 3-sphere |Q| = 1.

If you rotate a 4-vector by rotating in the (1, i) plane by i (90 degrees) and rotate in the (j, k) plane by i (also 90 degrees), then the resulting vector is orthogonal to the original vector, no matter what that original vector was. (In contrast it is impossible to devise a fixed 3D rotation with this property, since any 3D rotation leaves its “pole” unaffected. Indeed, this is impossible in all odd dimensions by an eigenspace argument, but plainly possible in even dimensions.) Thus if Q is a unit-norm quaternion, then iQ is also, and (iQ) · Q = 0 where · denotes the ordinary real 4-vector inner product.

Geometrically this means iQ is a continuous unit 4-vector field tangent to the 3-sphere |Q| = 1. I.e. you can “comb the hairs on a 3-sphere.” This combing is called the “Hopf fibration^{9}.” It has some remarkable properties, e.g. every flowline of the vector field is a great circle geodesic; any two such circles are linked; any two of the circles are “parallel” in the sense that they are at a constant distance from one another. It is the unique^{10}fibering of S^{3} into S^{1}’s in the sense that any other one on any other S^{3} is diffeomorphic to the Hopf fibration of the standard round S^{3}.

In contrast, you cannot comb the hairs on a basketball (2sphere) [62][22][55]^{11}; but you can of course on a 1-sphere. Hence if the universe were a 3-sphere and had a magnetic field along the Hopf fibration, this magnetic field would be “topologically trapped” and unable to go away. This B⃗- field also is an exact solution of the time independent source free Maxwell equations, indeed essentially the unique one [33][29], so it will stay there.

Unfortunately, if a charge q moves around a great circle geodesic of the universe, say the equator in the (1, i) plane, then the B⃗-field will be everywhere tangent to the

DocNumber  5  . 2. 4. 0

9Of course, we could have used vQ instead of iQ, where v is any unit-norm quaternion with zero real part.

10This uniqueness may be shown with the aid of the fact that any fibration of S^{3} into S^{1}’s must have base space S^{2} (for S^{3} to be simply connected, the base must be also, hence by Poincare’s proof of the 2D version of the generalized Poincare conjecture, it must be S^{2}) and the “classification theorem for fiber bundles” mentioned in section 24.4 of [19]. But there are smooth nowhere zero flows on the 3-sphere with no periodic orbits [44].

11It often is claimed incorrectly that any smooth vector field on a 2-sphere must have at least two zeros. Wrong: one suffices. Consider the lines parallel to the x-axis on the plane; place unit vectors parallel to those lines; scale them by exp(−x^{2} − y^{2}); and stereographically project them up onto the sphere.

trajectory, and hence the charge will acquire momentum increment 0. In other cases such as the great circle in the (1, j) plane the B⃗-field will be everywhere orthogonal to the trajectory, but rotating around it in such a way that the net effect is still zero. The fundamental reason for all the zeros is that every closed trajectory is contractible.

Yet another remarkable property of the Hopf fibration (viewed as a magnetic field) is that it is (up to a constant factor) its own vector potential, i.e. its curl is itself.

All of the above argument concerning S^{3} and the Hopf fibration also applies to 3D elliptic geometry (i.e., S^{3} with antipodal points identified).

2.5 Mathematically rigorous generalization to all topologies

Theorem 1 (Trapped static B⃗-fields solving Maxwell equations exist) Every 3-manifold with uncontractible closed loops has some topologically trapped magnetic field B⃗ solving the time-independent vacuum Maxwell equations.

Proof. Clearly, in any 3-manifold with an uncontractible closed loop one may construct some topologically trapped B⃗-field: simply use a “bundle” of fluxlines going around some smooth uncontractible loop, and zero elsewhere. (We can even make this field smooth, without difficulty, e.g. consider |B⃗| = exp(1/[ϵ^{2} − d^{2}]) where d is the distance to the uncontractible loop, 0 < d < ϵ, and ϵ is sufficiently small.) Assume from now on that the total amount of flux going round said loop, is normalized to 1.

Now among all such fields B⃗, consider^{12} the one with minimum energy ∫ |B⃗|^{2}d^{3}volume.

Any B⃗ obeying the time independent source free Maxwell equations ∇×B⃗ = 0 and ∇·B⃗ = 0, is the gradient (at least locally) of some “potential” field Φ obeying Laplace’s equation ∇^{2}Φ = 0. But minimizing energy ∫ |∇Φ|^{2}d^{3}volume yields solutions of Laplace’s equation. QED.

Theorem 2 (Charge quantization) In any 3-manifold containing uncontractible closed loops, there are connected, but non-simply-connected, 2D submanifolds S. Under the assumption that Aharonov-Bohm phase shifts vary infinitesimally if the electorn’s trajectoryis varied infinitesimally, we conclude that electron charge must be an integer multiple of h/Φ_{S} (or equivalently if the electron charge is regarded as fixed, that Φ_{S} must be an integer multiple of the flux quantum Φ_{0}).

Proof. Consider an electron trajectory corresponding to the boundary ∂D of a topological disk D consisting of all S except for an infinitesimal amount. We find that the amount Φ_{S} of magnetic flux passing through S must be an integer number of flux quanta Φ_{0}, implying that charge must be quantized in steps of h/Φ_{S}. (Should all be obvious by following the preceding argument^{13}). QED.

2.5.1 Some remarks

Remark. Poincare’s conjecture (one of the biggest open problems in topology) states that a compact simply connected 3-manifold is homeomorphic to the 3-sphere.

Generalizations of the Poincare conjecture to n-manifolds have been proven for n = 2 (H.Poincare), n ≥ 5 (S.Smale, E.C.Zeeman 1961), and n = 4 (M.Freedman 1981), but the original n = 3 problem remains open^{14}.

So assuming Poincare’s conjecture, every thinkable compact 3D universe containing topologically trapped magnetism must have quantized charge – except S^{3} (and 3D elliptic geometry)! Remark. “Conformal^{15} transformations” preserve solutions of Maxwell’s equations. Thus a conformal map applied to the electron’s trajectory will yield in the new transformed universe, a new trajectory, still with the same Aharonov-Bohm phase shift angle, and a new magnetic field, still a solution of the time-independent Maxwell equations. Thus, at least as far as conformal transformations of the metric are concerned, the charge quantum in a universe is a topological invariant of that universe. Schoen [67] showed that every Riemannian n-manifold may be conformally transformed to one with constant scalar curvature. There is good reason to believe [69] that our universe is, essentially, a 3-manifold with constant scalar curvature. Remark. Note, it would be possible to have two homotopically inequivalent uncontractible loops (in some topologies,

z^{2} + (x − \frac{(1 − ϵ)x}{\sqrt{x^{2} + y^{2}}} )^{2} + (y − \frac{(1 − ϵ)y}{\sqrt{x^{2} + y^{2}}} )^{2} = ϵ^{2}

12Assuming such a minimum exists should be adequate for physicists. Mathematicians will be quick to observe that this argument, used by Riemann in the mid-1800s and called by him the “Dirichlet principle,” is not rigorous because this existence assumption needs to be justified [57]. However, fortunately, for smooth maps from a compact n-manifold to a manifold with nonpositive curvature (in particular flat space R^{3}) the fact that we may always deform the map into a harmonic, i.e. energy minimizing, map, has been justified [21] by the “heat flow method.”

13 Incidentally, Stokes’s theorem does not necessarily hold (at least, without modification) if the region enclosed by the loop is not topologically equivalent to a disk x^{2} + y^{2} < 1 in R^{2}. (Mentioned, e.g., in [56] box 4.1 page 96, 4c, and is the reason for the use of “star shaped domains” in [51]. As a simple counterexample, consider, instead of the disk {x, y, z|x^{2} + y^{2} < 1, z = 0}, the toroid where the case ϵ → 0^{+} makes the invalidity of Stokes’s theorem completely obvious. (A referee had used Stokes’s theorem in this wrong way to try to come to the [ridiculous] conclusion that the magnetic flux around a 3-torus “must” be zero, thus “invalidating” the present paper!)

14Prof. Everett Pitcher, former secretary of the Amer. Math. Soc., claims to have a proof of Poincare’s conjecture. He gave a lecture on it at Lehigh University, Bethlehem PA, on 16 October 2002, and he submitted a 40-page paper on it to Trans. of the AMS. According to the New York Times 15 April 2003, page F3, Grigori Perelman of the Steklov Math. Institute, St.Petersburg, claims to have a partlyunpublished proof of Thurston’s geometrization conjecture (which is more general than Poincare) and lectured on it at M.I.T. [61]. I was not at either lecture and I do not know if their proofs are correct. A $10^{6} prize is being offered by the Clay Math. Institute for a proof or disproof.

15Two smooth manifolds are said to be “conformally equivalent”

if, under some coordinatizations, their metric tensors g_{ab} are proportional with a smooth positive multiplicative pure-scalar function as the proportionality factor.

such as a two-holed torus) and this would yield 2 different charge quantization conditions, both of which would have to be satisfied simultaneously by any charge in that universe. If the two quantum step values had an irrational ratio, this should make nonzero charges impossible in that universe. This strongly suggests that our universe does not have two homotopically inequivalent uncontractible loops, which is a very strong constraint on the topology of the universe.

2.6 Which topologies are compatible with laws of physics?

2.6.1 Weak interaction

Both the unidirectionality of time, and the fact that the weak interaction violates parity symmetry, seem to rule out universes which are non-orientable manifolds, such as the Cartesian product of a circle with 2D “elliptic geometry.” (Any particles which had in the past traveled “round the universe” would be chirally reversed. Such particles have not been observed. It perhaps would be worthwhile to study weak interactions of extragalactic cosmic rays to see if any are chirally reversed.)

2.6.2 The combing of hairs

A necessary condition for our charge quantization argument was that it be possible to “comb the hair” in the 3-manifold representing our universe in some manner homotopically inequivalent to doing nothing, i.e. so that the B⃗-field can’t go away.

A survey of foliations is [26]; for vector fields on manifolds see [42][75]. Also possibly of interest is [60].

It is possible to comb the hairs on a closed 2-manifold (more precisely, foliate it smoothly into 1-manifolds)^{16} exactly when it has Euler characteristic χ = 0, i.e., for the torus and the Klein bottle only. This was shown by Hopf

[35] in 1927. See figure 3. Thurston [78] found an extension to n dimensions.

Every smooth open [32] or closed 3-manifold, orientable

[47] or not [85], has a smooth^{17} 2-foliation. (By “smooth” we mean we may demand C^{∞}.) Furthermore [85][77], any smooth plane field is homotopic to a 2-foliation, so that any smooth vector field (perpendicular to the plane field) is homotopic to one transverse to a 2-foliation. Fact: Consequently, you can always “comb the hairs” on any smooth closed 3-manifold.

Hence, the hair combing condition is hardly restrictive. But now let us consider a more interesting condition.

Definition 3 An n-manifold will be called “m-combable” if on it, there can exist m orthonormal smooth vector fields.

Remark. In order for an n-manifold to be m-combable for any particular m, 1 ≤ m ≤ n, it suffices for it to have m everywhere linearly independent nowhere zero smooth vector fields, because by taking appropriate linear combinations (with coefficients which are appropriate smooth functions of position) we may orthonormalize them. Observation. Certainly, the standard 3-torus and the standard 3-sphere are 3-combable. Proof: let the 3-torus be any parallelipiped with periodic boundary conditions. Now simply use vectors parallel to the axes of an (arbitrarily rotated) 3D Cartesian coordinate system. Let the 3-sphere be the unit norm quaternions |Q| = 1, with some arbitrary pre-rotation of the 4D coordinate system. Now use as the 3 mutually orthogonal unit-length vector fields iQ, jQ, and kQ. QED. Remark. The n-sphere is n-combable^{18} exactly [3][40][54][75] when n = 1, 3, 7, which is related to the existence and uniqueness of the complexes, quaternions, and octonions. [Indeed by a construction analogous to Hopf’s one may construct 3 orthonormal vector fields on S^{n} if 4|(n+1) by using quaternions, and 7 orthonormal vector fields on S^{n} if 8|(n + 1) by using octonions.] Let f(n) be the maximum possible number of linearly independent smooth nowhere zero vector fields on a sphere S^{n}. Then f(n) = 2^{c} + 8d − 1 where n + 1 = (2a + 1)2^{b} and b = c + 4d and 0 ≤ c ≤ 3. Note f(n) = 0 if n is even. For odd n, we have:

  n =1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33
f(n)=1 3 1 7 1  3  1  8  1  3  1  7  1  3  1  9  1

The lower bounds are due to constructions by A.Hurwitz [36] and J.Radon [65] who actually showed that there are f(n) orthogonal (n+1)×(n+1) matrices A_{p} with A_{p}^{2} = −I and A_{p}A_{q} + A_{p}A_{q} = 0 for p ≠ q. Then the matching upper bound was proven by J.F.Adams [2] using K-theory, a branch of cohomology theory.

Do any other 3-tori and 3-spheres (besides these particularly geometrically nice ones) work? Yes. Any 3-torus or 3-sphere conformally equivalent to the standard ones is 3-combable. Indeed certain diffeomorphisms more general than conformal ones will work^{19}: it will suffice for n-combability if we can recoordinatize the n-manifold so that its metric tensor g_{αβ} becomes diagonal. The “Cotton-Darboux theorem” [12][14] shows that every 3-manifold is so recoordinatizable, at least in finite patches locally.

Is there going to be any obstacle preventing extending this patch to cover the whole manifold (provided we are transforming to a compatible topology)? There is no obsta-

DocNumber  7  . 2. 6. 2

16An n-manifold Ω will be said to have a “k-foliation” for some k < n if it is a disjoint union of immersed k-dimensional submanifolds.

If each point of Ω lies in a ball diffeomorphic to a ball in R^{n} with the leaves of the foliation intersecting that ball being diffeomorphic to the standard k-dimensional sets with x_{j} constant for each j > k, then the foliation is “smooth.”

17However, for the 3-sphere it is not possible to find an analytic 2-foliation [31].

18n-combable n-manifolds are often called “parallelizable” in the literature. Some at least as ill-advised words, which have sometimes also been employed, to describe the maximum m such that a manifold in m-combable, are its “span” and “rank.” The present paper turns out not to be the first physics paper using notions of n-combability; see [28].

19It has often been claimed incorrectly that a Riemannian metric is conformally flat if and only if its Weyl tensor vanishes. This is true if the dimension n ≥ 4, but when n ≤ 3, the Weyl tensor automatically vanishes but not all 3-manifolds are conformally flat, not even locally [43] (see also [56] p.550 ex.21.22.). All 2-manifolds are locally conformally flat, and so are the standard round n-spheres (by considering the stereographic projection map).

cle: every 3-manifold has an everywhere orthogonal curvilinear coordinate system^{20}:

Theorem 4 (n-combability) All orientable 3-manifolds are 3-combable. The only 2-combable 2-manifolds are 2-tori. For n ≥ 4, generically, n-manifolds are not n-combable (even locally) although certain specific special n-manifolds can be (e.g. flat n-tori S^{1} × S^{1} × ⋯ × S^{1}).

Proof. As we already mentioned Hopf’s [35] and Thurston’s

[79] theorems show that the 2-manifolds are combable precisely if they have χ = 0, i.e. are a torus or Klein bottle. (Smoothly foliatable into 1D manifolds is the same as having a smooth nowhere zero vector field, because you can use the flowlines of the vector field, which by uniqueness and existence theorems for ordinary differential equations, works.) Now, on orientable 2-manifolds, combable implies 2-combable; simply take the second vector field to be orthogonal to the first (rotated 90^{∘}). But in contrast, even the usual flat Klein bottle (in figure 3) is not 2-combable. Proof: The “2-combing” shown in figure 3 is invalid because the vertical vector field would have to flip direction 180^{∘} discontinuously, at the right edge of the figure. If the two vector fields, near the right edge of the figure, were oriented at any other angles besides normal and parallel to that edge, then they would both discontinuously change direction by some nonzero angle as we crossed the edge.

Figure 3: 2-combing of 2-Torus (a) and failed 2-combing for Klein bottle (b). In both figures a valid 1-combing arises by considering only the horizontal dashed lines.

n-Combability is generically false, even locally^{21} when n ≥ 4 due to insufficient “gauge freedom.” Specifically, it is necessary that the metric tensor g_{αβ} be diagonal (in some coordinatization) and there are n degrees of gauge freedom you have in selecting your favorite coordinatization, which is enough to make it diagonal if n ≤ 3, but not if n ≥ 4.

The result that every orientable 3-manifold is 3-combable was shown by Stiefel [71]. QED.^{22} 2.7 Why is the world 3-dimensional?

2.8 3-manifolds are compatible with photons

The connection of theorem 4 to physics is as follows.

Suppose we regard it as desirable for “momentumpolarization eigenstates” of photons to exist in our universe. In flat space these are just plane waves – with equal wavevector k⃗ everywhere, and polarization vectors E⃗ and B⃗ existing everywhere and everywhere perpendicular both to each other and to k⃗.

These properties in curved space are possible only if the universe’s topology is 3-combable! Note, incidentally, that 3-combability is necessary if a momentum-polarization eigenstate “photon” (with any desired momentum direction and 2 orthogonal polarization directions at any particular one point) is to exist. (If one multiplies the 3 orthonormal vector fields by an appropriate fixed orthogonal 3 × 3 matrix, one can make these 3 directions be whatever one wants at any particular point.) Remark. Indeed, arguably n-combable n-manifolds are globally compatible with every physical law we can formulate in flat n-space, since we can make an everywhere orthogonal curvilinear coordinate system for such a manifold, and the physical laws will be able to operate locally everywhere as though they were in a flat space coordinate system. However, n-manifolds not meeting such a condition, although they would be locally compatible with flat space physical laws, might have some global incompatibility – as in the photon eigenstate example above.

We now see that theorem 4 represents a remarkable property of the dimension “3.” Perhaps we can even brand it as a “reason the world is 3-dimensional.” Summary. We have identified a topological property: ncombability. We’ve shown: “Random” n-manifolds are n-combable, and hence compatible (with “probability 1”) with a wide class of thinkable physical laws, if n = 3. However, if n ≥ 2 but n ≠ 3, then random n-manifolds will not be n-combable (with “probability 1”) and hence would yield a self-contradiction in a wide class of thinkable physical laws. Thus, “with probability 1,” physics can only happen^{23} in 3D.

2.8.1 Other arguments the world “must” be 3D

I’ll quickly summarize previous arguments [73][10][72] for why the universe “had” to be 3D.

Assuming Maxwell’s and Schrödinger’s (or Dirac’s) equations hold, hydrogen has no bound states if n ≥ 4 (atoms implode, indeed with infinite energy release for pointlike electrons and nuclei), but has no unbound states if n ≤ 2. Assuming Newtonian mechanics with r^{−n} force laws hold,

DocNumber  8  . 2. 8. 1

20The reader might want to look at the discussions of orthogonal curvilinear coordinate systems in R^{3} found in [53][58][59].

21A different way to try to prove this would be simply to fall back on the Kervaire-Milnor-Adams results for the n-sphere to prove impossibility in every dimension n ≥ 4 except for n = 7. However, this would only yield a global obstacle – the standard round n-sphere is ncombable locally (for each n ≥ 1) everywhere except at a single point, as may be seen by stereographic projection mapping the surface of that sphere to flat n-space. Presumably, however, the surface of a generic ellipsoid in R^{n+1} is not n-combable even locally.

22Could it be that such topological constraints could also be employed to restrict the possible ways in which “extra dimensions are curled up” in “string theories?”

23Recently “string theories” and “M-brane theory” have been proposed [63], which claim the universe is 9 + 1 (or 10 + 1) dimensional.

Our argument does not rule these out, because they postulate a fixed, rigid metric, namely flat (3 + 1)-space cartesian producted with some particular fixed compact 6D (or 7D) manifold M (for which, there are presently at least about 10^{5} viable candidates). By choosing M to be an atypical, n-combable metric such as the flat n-torus (S^{1})^{n}, our argument, which only applies to generic metrics, is defeated.

planetary orbits are unstable if n ≥ 4 (no orbit which oscillates between a maximum r_{2} and a minimum radius value r_{1} can exist with 0 < r_{1} < r_{2}; the weaker result that circular orbits are unstable if n > 3 is in the undergraduate textbook [30]), but with n ≤ 2 there are no unbound orbits since the escape velocity is infinite. Attempts to generalize the general relativistic Schwarzschild solution to n ≥ 4 space dimensions exhibit the same behavior. Lovelock [49] pointed out that if n+1 ≤ 3+1 there is a unique Einsteinian gravity field equation of order ≤ 2 arising from a scalar Lagrange density depending only on the metric tensor and its first two derivatives. If n ≥ 4, this uniqueness statement no longer holds. Assuming general relativity holds, gravity cannot exist in vacuum if n ≤ 2 since the 4-indexed Riemann tensor would be everywhere zero in vacuum. Mariwalla [52] showed that the TCP theorem for Dirac’s equation would only hold if n were odd; and if we demand the TCP operation be continuously connected to the Lorentz group identity, that forces n ≥ 3; he also says, “An arbitrary geodesic in a curved [Riemannian] space... can be looked upon as the motion of a particle under a force in flat spacetime, and vice versa” only if n + 1 = 4.

Lovelock pointed out that there are various “dimension dependent identities” in differential geometry that hold only in certain dimensions. In particular, Rainich’s identity that underlies “geometrodynamics” (unification of electromagnetism and gravity into a single geometric theory) works in 4D only. The Weyl conformal curvature tensor vanishes in dimensions ≤ 3 only. Misner, Thorne, and Wheeler [56] point out remarkable properties of the Einstein tensor that hold only in 4D. All of classical electromagnetism is based on the properties of the curl and div operators, many of which (since they arise from 4D quaternion algebra) hold only in 3 dimensions.

2.8.2 Some dubious arguments

Philosophers have claimed that since nerves cannot cross in 2D, intelligent observers would be impossible in a 2D universe. (The point being that, while perhaps there are universes of all dimensions, if intelligent observers can only exist in 3D universes, it is no surprise that our universe is 3D.) But that would seem to be refuted by Conway’s construction [8] of “life,” a Turing universal computer in a simple 2-state 2D cellular automaton. An earlier and more complicated Turing universal cellular automaton (29 states at each vertex of a 2D square grid) had been constructed by von Neumann [83]. It was, however, simpler than “life” in the sense that communication only occurred between a site and its 4 nearest neighbors on a square grid, rather than its 8 nearest. Also: optical, audible, or time multiplexed signals can cross through each other.

Another similar biological-philosophical idea: organisms cannot have both a mouth and an anus without being disconnected. But this too seems refuted: there are sea organisms which use the same opening as both a mouth and an anus.

Fluid turbulence could not happen in 2D, which perhaps somehow would mitigate against the evolution of life, but this is hardly convincing.

2.8.3 Comparison with my, more abstract, argument

All of these previous arguments that the world “has to be” 3-dimensional suffer from the flaw that they assume the physical laws in an n-dimensional universe would be the same as here. (Perhaps the inhabitants of a 7D universe are busily arguing that 7 was the only possibility for the same reason.) They also all are aimed at demonstrating the nonexistence of intelligent observers, rather than the logical inconsistency of physics itself – although the arguments for atomic implosion with infinite energy release for the collapse of two oppositely charged point particles, may qualify as an exception.

All the known arguments that 2D universes featuring intelligent life are impossible leave me unconvinced.

It perhaps would be more credible and satisfying if one could argue at a highly abstract level, allowing any of a large class of kinds of physical laws but without specifying what they must be, and deriving a logical contradiction, rather than merely arguing for the unlikelihood of intelligent life. Aside from the atomic implosion arguments, my argument is the only such argument that I know of.

Nevertheless, my argument also is not very satisfying in 2D since it would still permit a 2D torus universe, although all other compact 2-manifolds would be forbidden.

3 Reexamination of Monopoles

In this section I’ll consider a sequence of attacks on the monopole idea. The initial attacks (§3.1-3.2) can be defended against, and we present the defenses. The later attacks (§3.3-3.5) which, e.g., use Dirac’s own equation against him, seem harder to refute unless the defenders rely on new speculative theories of physics in ways that have never been analyzed.

3.1 Curved space

Dirac’s 3 arguments in §1.2 all pertained assuming a Euclidean geometry, i.e. “flat space.” It is not immediately clear any of them will work in curved spaces. However, one could argue that Maxwell’s equations force charge (and magnetic charge) conservation^{24} and hence we could consider transporting an electric and magnetic charge to a flat region of space far from everything else, applying Dirac’s arguments there to see the charges must be quantized, and then transporting them back. This works (at least, to high accuracy) assuming there is anyplace big and flat to go to.

3.2 Point charges versus distributed charges

At first glance, Dirac’s arguments seem to work only if the elementary charges are points. And indeed, all experimental evidence so far is fully consistent with electrons and quarks

DocNumber  9  . 3. 2. 0

24And so do the Einstein-Maxwell equations in curved space.

being point particles, and QED and QCD would have to undergo massive revision, wholly changing and greatly complicating their mathematical structure, if non-point electrons and quarks were allowed^{25}.

But ignore that. Suppose the monopole’s magnetic charge is distributed over a region of nonzero volume. Then Thomson’s integral (EQ 2) would no longer give a result independent of the separation of the two charges. The chargemonopole flyby argument would yield a change in the electron’s angular momentum now dependent on the electron’s “impact parameter” (distance of closest approach) if the electron’s trajectory were allowed to penetrate the monopole. And Dirac’s argument would seem to be totally inapplicable because the divergence of B⃗ would occur throughout a region of nonzero volume, causing Dirac’s wave equation to be inapplicable everywhere in that region, and with no way to get rid of the problem by having a vector potential singular only on a string of measure zero. So in all three cases, the charge quantization argument seems to break down in the presence of non-point charges.

But at second glance, it is again possible to rescue all three arguments. Thomson’s integral would be independent of the separation asymptotically for large separations. Hence charges and monopoles far from one another would be quantized according to (EQ 3). We then again use charge conservation and consider moving them closer together. As we move them closer together, conservation of angular momentum would force the monopole to start spinning. Similarly the flyby argument is rescued by the fact that the monopole will acquire spin if the electron trajectory penetrates it.

Finally, Dirac’s “string” argument still applies for spheres completely enclosing the monopole (the continuity of the wavefunction forces charge quantization). Assuming Dirac’s wave equation holds in this exterior region and some as yet unknown equation holds in the interior region, suffices to force (EQ 3).

3.3 There cannot be both a North pole and South pole (both points) in a universe in which Dirac’s equation holds

Suppose that both a North monopole, and a South monopole, both point particles, exist^{26}. Assume they both are minimally charged (have Z = 1 in EQ 4) since the below argument will only work better otherwise. Transport them to a region of flat space far from everything else.

Now assume Dirac’s wave equation governs their wavefunctions. Due to the symmetry of Maxwell’s revised equations, we may treat these two poles as electric charges of the same masses, but with the value of the fine structure constant α = e^{2}/(4πϵ_{0}h̄c) ≈ 1/137.036 changed [18] to α_{new} ≈ 137.036/4 ≈ 34.259.

Now it is well known that Dirac’s wave equation for positronium is unstable if α > 4. (The full set of exact solutions of the Dirac equation is available in this case.) That is, there will be a nonzero probability amplitude that, during any time interval, the two particles will “fall down the hole” into each other’s singularities. The Hamiltonian operator is not self-adjoint in this case.

Note, if α < 1/N, then the hydrogenic atom (with infinitely massive point nucleus of charge Ne) is stable and such “falling in” is impossible.

Also, Lieb and collaborators [48], showed that under various kinds of “relativistic Schrödinger equations,” if Nα < 2/π and α < 1/94, then “matter” (any configuration of electrons amidst massive point nuclei of maximum charge Ne) is “thermodynamically stable” i.e. obeys a lower bound on its energy, proportional to the number of charges in the system. On the other hand if α > 2.72, matter (even if subintegral nuclear charges are allowed, provided there are enough nuclei) is always unstable and will implode. In later papers of these authors, externally imposed magnetic fields and spin-field interactions were allowed, although the bounds became weaker.

The conclusion we draw from all these results is that the effective value for α for monopole “positronium” is far larger than the bounds of Lieb et al., i.e. the two monopoles would fall into one another.

Two point charges falling into each other’s singularity under Dirac’s wave equation would produce an infinite energy release. This is not the same thing as positron-electron annihilation, which releases only 2m_{e}c^{2} energy. There will always be a finite nonzero probability current “down the hole”, leading, no matter how incredibly small this current is, to an infinite average rate of energy production.

If we regard it as forbidden for there to be a positive probability that the universe will self-destruct at any time, or forbidden for there to be non-self-adjoint Hamiltonians, then we conclude that either

1. Monopoles are not point particles, or

2. There are not both North and South poles in the universe^{27}, or

3. Dirac’s wave equation in flat space is not applicable (and the revised equation exhibits different behavior)^{28}.

DocNumber  10  . 3. 4. 0

25Feynman diagrams would no longer have points as vertices and path integration over many extra degrees of freedom, perhaps an infinite number, would be required.

26Indeed, if the universe has a compact topology, it is topologically necessary for the universe to have exact charge (and magnetic charge)

neutrality. This may be shown using the generalization of Gauss’s divergence theorem to manifolds.

27Note: Gauss’s divergence theorem shows that, if the universe is a compact manifold, then it must contain an equal number of + and − charges and an equal number of North and South monopoles; i.e. it is exactly neutral. So in this case we could conclude that a compact universe cannot contain a monopole.

28Note: when I speak here of “Dirac’s equation” I mean his equation for the wavefunction of an electron in a static 4-potential field.

This equation plainly is self-consistent in certain situations, e.g. the “hydrogen atom” (electron in a Coulomb field) where a full set of wellbehaved exact solutions are known [16]. However, Dirac’s equation plainly is not self-consistent in certain other situations, such as with point monopoles, where singularity and infinite energy release occur in finite time. It could be argued that perhaps this is because Dirac’s 1electron equation alone is not the full physics (although even if so, the fact that the very equation Dirac used to argue for monopoles, now is being used against him, still carries considerable clout). Indeed,

3.4 Another reason point monopoles cannot exist

The potential between a point monopole and a point magnetic dipole (such as a spinning electron) would behave proportionally to −r^{−2} for separation r. As is well known (as we’ve just discussed, even −Kr^{−1} potentials are forbidden, if K is a large enough constant), this is too severe a singularity for quantum mechanics to be defined [46]. Again the electron would “fall in” with infinite energy release in finite time^{29}.

3.5 What if monopoles are not points?

I would be perfectly happy just to dismiss the idea of non-point monopoles since they would overthrow QED and QCD.

However, there are physicists who feel that monopoles do exist, and they are not points. (As we saw in §3.2 this is logically conceivable.) They are motivated by GUT and superstring theories. I will now analyze non-point monopoles

– but using comparatively pedestrian physics and without using any speculative GUT and superstring theories; we will see that some rather disagreeable conclusions follow.

A monopole with distributed charge is necessarily held together by some other (non-electromagnetic) force. This force is strong enough to prevent the monopole from ever breaking apart, no matter how hard you whack it (since otherwise we would violate magnetic charge quantization and/or conservation). The weak and strong forces are inadequate to this task^{30}.

(Of course, the monopole could be held together by some yet unknown “fifth force.” That is essentially what is going on in the GUTs. I’ll ignore that.)

The gravitational force is the only one which seems adequate – but only if the monopole is a black hole: A well known theorem [12] in classical general relativity states that it is impossible to pull a black hole apart, i.e. divide its event horizon into two. But this would require (from the Kerr-Newman exact solution [12] of charged black holes, nowadays thought to be unique, and the condition that an event horizon exist) the mass m_{m} of a Dirac-minimal magnetic pole be at least

m_{m} ≥ \frac{2m_{P}}{\sqrt{α}} ≈ 23.41m_{P} ≈ .51milligram  (7)

(i.e.^{31} m_{m}c^{2} ≥ 4.6×10^{10}Joules) where m_{P} = \sqrt{h̄c/G} is the “Planck mass.” .5milligrams is a macroscopic-scale mass. With a mass this large, gravity at Earth’s surface would exert 5 millinewtons. To counteract this would require a huge magnetic field of 1550 Tesla. (Larger charged monopoles, such as Schwinger’s, would have correspondingly larger masses, so the critical B⃗ field would remain unchanged.) We conclude that any thermalized monopoles this massive on our planet must be resting near its center and attempts to find them in accessible iron ore deposits, moon rocks, etc. [24][25][5] are doomed to failure. (Asteroids, which contain magnetic material and have low gravity, might be a better place to look for accessible monopoles.) The Schwarzschild radius of a mass m is 2Gm/c^{2}, which for this minimal mass would be 7.6 × 10^{−34} meters. This is 19000 times larger than the “Bohr radius”

r_{mono Bohr} = \frac{2h̄}{α_{new}m_{m}c} = \frac{8h̄α}{m_{m}c}  (8)

for monopole positronium, which with the mass value in (EQ 7) would be 4.0 × 10^{−38} meters. (The discrepancy would be even larger for Schwinger’s 4× or 12× larger charged monopoles.)^{32} This suggests that any two opposite monopoles could combine into a magnetically neutral state impossible to pull apart. This suggests in turn that it is impossible to create a separated monopole pair out of pure energy (even granting the feasibility of creating energies of this ridiculously large magnitude), which again suggests that monopoles cannot exist, since they could never have been created. (Dirac’s charge quantization arguments in §1.2 don’t work for non-point monopoles only available as inseparable neutral pairs.)^{33}

In conclusion, both point monopoles and non-point monopoles lead to unpalatable contradictions and selfdefeating conclusions. To accept them, one must have considerable faith in the ability of highly speculative and/or future theories of physics to resolve those contradictions. So my alternative hypothesis (§2.1), which can explain charge quantization without need for monopoles, therefore might be preferred by many people.

gravitational and “higher order Feynman diagram” effects might well become important at very small length scales (although it is unclear what effect they have on stability). It is very difficult to make rigorous statements about quantum field theory, so it is presently impossible to say whether it it mathematically self-consistent either with or without monopoles. (And at present, attempts to combine gravity with QFT are completely inadequate.)

29The “exact solution” [81] of the Dirac equation in a magnetic monopole and Coulomb potential is not applicable here since that solution pertained to a spinless (i.e. unrealistic) “monopole” having no dipole moment, i.e. having only r^{−1}, and not r^{−2}, potential singularities.

30Actually, the strong force is supposed to allow unbreakable attractions between quarks in the sense that, if you pull two quarks apart further than some critical distance, a new quark/anti-quark pair will be created by the “elastic strain energy” and one member of the pair will attach itself to each of your two original quarks to neutralize them. However, the quark pair formation mechanism seems irrelevant to preventing single monopole breakup.

31Actually, if the monopole is assumed to have angular momentum h̄/2, its mass would have to be slightly larger even than this bound, which was computed under the assumption of a nonrotating monopole.

Also, Schwinger’s minimal monopoles with 4 times (or 12 times, if we view e/3 as the minimal electric charge, not e) larger magnetic charges, would have 4 or 12 times larger minimal masses.

32Note, we have used a factor of 2 times the usual Bohr radius formula because we have a “positronium-like” system in mind here, rather than an “infinitely massive nucleus.” The alert reader should justifiably object that Bohr’s radius formula may not be applicable here since it was derived nonrelativistically, while here the monopoles have so large charges that relativistic effects should be important.

However, special relativistic effects would in fact decrease the Bohr radius by multiplying it by \sqrt{1 − v^{2}/c^{2}}, where v is the orbital speed, i.e. my argument only becomes more true.

33Datta [17] invented related, but different (and also valid) arguments to argue for the impossibility of creating a monopole North-South pair.

4 The moment of inertia of a distributed monopole – a relation between its charge and mass distributions

A distributed monopole must be a nonrigid object, since rigid objects are impossible in special relativity (the speed of sound would have to exceed c). Hence it must have a spectrum of excited modes of vibration. Since it is impossible to pull the monopole apart without violating magnetic charge quantization, this spectrum must be infinite.

There will also be excitations arising just from spinning the monopole with nonzero angular momentum quantum numbers.

It might seem impossible to reason about the unknown forces holding the monopole together (or even about gravity). Remarkably, we can, by using very elementary physical arguments.

Consider shooting a (pointlike) electron, initially far away from a monopole, through its center, out the other side, and eventually ending up far away in the other direction. By consideration of the electric and magnetic fields before and after (and considering Thomson’s integral EQ 2 and conservation of angular momentum), we see that this process will increase the angular momentum of the monopole by Zh̄.

An alternative process which will accomplish the same thing would be to shoot in a positron from one side and an electron from the other and let them annihilate inside the monopole (see figure 4).

Figure 4: Two alternative processes for spinning up nonpoint monopoles.

Suppose this process (or the alternative one) is applied n times. We will then have an (excited) monopole with angular momentum nh̄. What will its energy be?

Well, we can deduce the energy of the monopole by deducing the energy required to push the electron along its trajectory. The electron will always move parallel to the B⃗ field and hence will not experience any magnetic Lorentz forces. However, if the monopole is rotating, it will have an electric dipole moment. This will cause an electric field directly opposing the motion of the electron.

The electric dipole moment D_{n} of a mass-m_{m} chargegZ (where g here is Dirac’s minimum magnetic charge) monopole will be D_{n} = gZnh̄/(2m_{m}c^{2}) (not counting any intrinsic moment it would have with n = 0 due to spin; this will be negligible in comparison when n is large). This holds both general relativistically and in Dirac’s wave equation. This corresponds^{34} to an energy of D_{n}e⟨R_{g}^{−2}⟩/(πϵ_{0}) we need to supply to the electron, where ⟨R_{g}^{−2}⟩ is the mean squared-reciprocal radius (about the axis of rotation) of the monopole’s magnetic charge distribution.

Actually, instead of D_{n} it would presumably be better to use (D_{n} + D_{n+1})/2 since the dipole moment changes from D_{n} to D_{n+1} during the process. We conclude that the energy E_{n} of excitation of the monopole after n shootthroughs (when it has acquired angular momentum L = nh̄) is

E_{n} = \frac{gZeh̄μ_{0}⟨R_{g}^{−2}⟩}{2πm_{m}} ∑_{k=0}^{n−1}(k + \frac{1}{2}) = \frac{gZeh̄μ_{0}⟨R_{g}^{−2}⟩}{2πm_{m}} \frac{n^{2}}{2}

= \frac{Zn^{2}h̄^{2}⟨R_{g}^{−2}⟩}{2m_{m}}. (9)

In comparison, the quantum energy levels of a nonrelativistic “rigid rotator” with moment of inertia I are

E_{n} = \frac{h̄^{2}}{2I} (n + 1)n.  (10)

The two equations are the same (except for the details of the behavior for small n, which anyway depended on whether the monopole was assumed to have intrinsic spin and how much that affects its dipole moment, which are issues we have not tried to address). We conclude that

The moment of inertia of a mass-m_{m} monopole of Z times Dirac’s minimal magnetic charge is

I_{monopole} = \frac{m_{m}}{Z⟨R_{g}^{−2}⟩}.  (11)

where ⟨R_{g}^{−2}⟩ is the mean squared reciprocal radius of the monopole’s magnetic charge distribution about its axis of rotation. But of course

I_{monopole} = m_{m}⟨R_{m}^{2}⟩  (12)

where ⟨R_{m}^{2}⟩ is the mean squared radius of the monopole’s mass distribution about its axis of rotation, if the monopole behaves enough like a rigid rotator.

Hence ⟨R_{m}^{2}⟩ ⟨R_{g}^{−2}⟩ = 1. We conclude that the mass and charge distributions of the monopole must differ, unless they both are a “cylindrical delta function” at one particular radius value. Due to nonrigidity both these distributions must vary with n, which seems to make the fact that this equality must continue to hold, miraculous and hence a priori unlikely – and also the prospect of an extremely excited monopole (n very large), which would be a particle of very large macroscopic mass and size, seems appalling ^{35}.

DocNumber  12  . 5. 0. 0

34It may help to consider the following analogous problem. Consider a current loop of radius R carrying current i. The magnetic dipole moment of the loop is μ = iπR^{2}. The B field along the axis of the loop is B = \frac{1}{2} μ_{0}iR^{2}/(R^{2} + x^{2})^{3/2} where x is the distance out of the plane of the loop. Thus a magnetic charge g would gain energy g ∫_{−∞}^{+∞} Bdx = μ_{0}ig = μ_{0}μg/(πR^{2}) by flying along this axial line.

We have the exact same problem but with electricity and magnetism interchanged.

35In riposte, it has been argued to me that a rotationally excited

5 Lorentzian manifolds

The present paper has modeled the universe as a Riemannian 3-manifold with separate “absolute time.” But in fact, in Einstein’s theory of general relativity, the universe is a complicated Lorentzian 3 + 1-manifold. This oversimplification is a serious gap in our rigorous reasoning – I believe the most serious one. What effect does this have on our picture? There are several remarks to be made.

0. The purely topological nature of the arguments in §2.3 and 2.5 hopefully should force them to remain valid – although some of the other arguments may not be as easy to repair. (Thus, slow changes in the 3-manifold with absolute time, should still leave everything valid.)

1. At present I think there is no hope that the present picture of charge quantization can be made into a rigorous theorem in Einstein’s (3 + 1)D Lorentzian model. That is because our argument depends on both differential geometry (of the manifold) and quantum mechanics (to get quantization), and no satisfactory theory of “quantum gravity” has yet been advanced combining these two things. Our whole notion of an electron “coming back to the same point it started from” no longer has any meaning. The concept of the “spectrum of the Hamiltonian” of a particle in the universe seems hard to assign a meaning to. The rigorous theorems in this paper have only been obtained by using a 3-manifold with separate absolute time, and that restriction, in the present theoretical climate, appears necessary.

2. Nevertheless, some parts of our argument do work in Einstein’s (3 + 1)D manifolds. In the “Einstein-Maxwell equations” governing electromagnetism and gravity there still is an analogue of the 4-potential A (see [56] EQ 22.19) in charge-free regions of spacetime.

I do not think the presence of, or creation of, “black holes” need cause any damage to our argument: charges which avoid the holes obey our argument, and those which fall into a hole either are irrelevant or become relevant if they continue their journey around the universe by dragging the hole along for the ride – either way, our arguments remain valid.

6 Summary

Dirac had argued that if monopoles exist, then if charge were unquantized, contradictions would result. The easiest such contradiction to understand is the nonquantization of angular momentum, which is experimentally quantized in units of h̄. This was taken as evidence for the existence of monopoles. I am similarly arguing that, even without monopoles, one may get a similar contradiction provided the universe has a suitable topology and contains a “topologically trapped” magnetic field. Amazingly enough, the momentum increment, times L_{2}, experienced by electrons on distance-L_{1} trips round a flat 3-torus L_{1} × L_{2} × L_{3} universe (with magnetic field in the L_{3} direction) does not depend on the precise shape either of the universe, or of its magnetic field, or of the electron’s trajectory, and it is invariant if L_{1}, L_{2}, or L_{3} are slowly altered. Consioderation of De Broglie wavelnegths then forces charge quantization. This may all be generalized by demanding that the Aharonov-Bohm phase shift angle for certain electron trajectories must, modulo 2π, change only infinitesimally when the electron trajectory changes infinitesimally. That generalization works for almost any compact topology for the universe and almost any magnetic field.

More precisely, the argument works for any compact topology with uncontractible closed loops – and under Poincare’s conjecture the only inpermissible topologies are S^{3} and elliptic 3-geometry. Of these, the latter apparently may be logically ruled out by orientability considerations [69], and present experimental evidence seems to mitigate against universes with positive curvature, i.e. against both. Further, my argument works for any magnetic field which contains nonzero amounts of flux “trapped” in uncontractible loops.

Hence, this too would “explain” charge quantization. It is important to realize that no actual electron needs to take any such trip; merely the logical possibility of it is enough to force the necessary inconsistency in the laws of physics. Some readers of preliminary drafts of this paper have felt that it is crazy science fiction. Although this perception is understandable, I’ll now argue that my hypothesis for charge quantization via topology is actually less outrageous and less crazy^{36} than monopoles.

What assumptions does my proposal rest on?

1. The universe has a topology. (Indisputable.)

2. There is a “topologically trapped magnetic field.” (If the topology is such that this is permitted, it would seem surprising if there were no trapped B⃗-field. The onus certainly would seem to be on opponents of the present idea, to think of any reason why there should be none.)

3. Aharonov-Bohm phase shift varies continuously as trajectory varies. (Very hard to dispute. Otherwise physics would admit peculiar discontinuities at arbitrary locations.)

4. Certain theorems about topology work. (Indisputable.)

These are not wild and crazy assumptions. The only questionable one is whether the universe has a suitable topology, but it has been made clear that “most” topologies will work, and there is presently little or no experimental evidence favoring any particular topology for the universe, so nothing experimental mitigates against me.

Now, let us compare with the monopole camp. Monopoles have never been found despite extensive searches. We’ve shown they necessarily would have some extremely peculiar properties. The sole basis for supposing they exist (and the only way they could exist) is highly speculative theories of physics (grand unified theories, string theories) which so far have essentially no experimental support, and some of which, indeed, have been experimentally refuted. It seems to me that postulating entire new theories of all of physics, including at least a dozen new kinds of forces, a menagerie of new particles, and postulations about their behaviors, with merely ideas about “beauty” and “symmetry” and a little suggestive numerology as motivation, is what is “crazy” and “science fiction,” not my suggestion. Of course, one of these theories could well be correct. I am simply arguing that the more conservative choice in light of present experimental evidence, is mine, since it assumes the least new physics.

monopole would quickly decay by emitting e^{+}e^{−} pairs. One may verify that if m_{m} ≫ m_{e} and if the characteristic length scales (such as ⟨R_{g}^{−2}⟩^{−1/2}) of the monopole are smaller than or comparable to its Compton wavelength h/(m_{m}c), then this decay mechanism is energetically highly favored even when n = 1. This might prevent any attempt to spin a monopole up to high rotational quantum numbers.

36Albeit Dirac’s and my proposals can be regarded as spiritually related, except that now “the universe is the monopole.”

7 Acknowledgements

I’d like to thank Alfred S. Goldhaber, William P. Thurston, and Wolfgang Ziller for some helpful correspondence and William Bialek and A.Peter Blicher for useful conversations. (They may not agree with some or all of the conclusions of this paper.)

References

[1] Colin Adams: The Noncompact Hyperbolic 3-Manifold of Minimal Volume, Proc. Amer. Math. Soc. 100,4 (Aug. 1987) 601-606.

[2] J.Frank Adams: Vector fields on spheres, Bull.Amer.Math.Soc. 68 (1962) 39-41; Annals of Math. 75 (1962) 603-632.

[3] J.F.Adams: On the nonexistence of elements of Hopf invariant one, Ann.Math. 72,1 (1960) 20-104; J.F.Adams & M.F.Atiyah: K-theory and the Hopf invariant, Quart.J.Math.Oxford (2 ser) 17 (1966) 31-38.

[4] J.W.Alexander: An Example of a Simply-Connected Surface Bounding a Region which is not Simply-Connected, Proc. Nat. Acad. Sci. USA 10 (1924) 8-10.

[5] L.W.Alvarez et al: Search for magnetic monopoles in the lunar sample, Science 167 (1970) 701-703.

[6] W.Ballmann, G.Thorbergsson, W.Ziller: Existence of closed geodesics on positively curved manifolds, J.Differential Geometry 18,2 (1983) 221-252.

[7] P.G.Bergmann: Introduction to the theory of relativity, Prentice Hall NY 1942.

[8] E.R.Berlekamp, J.H.Conway, R.K.Guy: Winning ways, for your mathematical plays (2 vols), Academic Press 1982.

[9] G.Blewitt, J.M.LoSecco, & 29 others: Experimental limits on the free proton lifetime for two and three body decay modes, Phys.Rev.Lett. 55 (1985) 2114-2117.

[10] W.Büchel: Why is space 3 dimensional?, English version (I.M.Freedman) Amer.J.Physics 37,12 (1969) 1222-1224; original German: Physikalische Blätter 19,12 (1963) 547-549.

[11] B.Cabrera: First positive result from a superconductive detector for moving magnetic monopoles, Phys.Rev.Lett. 48,20 (1982) 1378-1381.

[12] S.Chandrasekhar: Mathematical theory of black holes, Oxford Univ. Press 1983.

[13] T-P.Cheng & L-F.Li: Gauge theory of elementary particle physics, Oxford Univ. Press 1984, reprinted with corrections 1994.

[14] É.Cotton: Sur les variétés à trois dimensions, Ann. Fac. Sci. Toulouse (ser. 2) 1 (Paris 1899) 385-.

[15] E.Cunningham: The principle of relativity in electrodynamics and an extension thereof, PLMS 8 (1910) 77-98

[16] Charles G. Darwin: The Wave Equations of the Electron, Proc. Royal Soc. London A118 (1928) 654-680 (see also Walter Gordon: Zeitschrift Phys. 48 (1928) 11-).

[17] T.Datta: The fine-structure constant, magnetic monopoles, and the Dirac charge-quantization conditions, Lettere Nuovo Cimento(2) 37,2 (19833) 51-54.

[18] P.A.M.Dirac: Quantized singularities in the electromagnetic field, Proc.Royal Soc. London A 133 (1931) 60-72; The theory of magnetic poles, Phys.Rev. 74,7 (1948) 817-830.

[19] B.Dubrovin, A.T.Fomenko, S.P.Novikov: Modern geometry methods and applications, Springer (GTM #104) 1985.

[20] M.J. Duff: Kaluza-Klein Theory in Perspective, in Proceedings Oskar Klein Centenary Symposium (19-21 September 1994) Sponsored by Royal Swedish Academy of Sciences through its Nobel Committee for Physics and by Stockholm University. World Scientific (ISBN 981-02-2332-3). Also available electronically as

hep-th/9410046.

[21] J.Eells Jr. & J.Sampson: Harmonic mapping of Riemannian manifolds, Amer.J.Math. 86 (1964) 109-160.

[22] M.Eisenberg & R.Guy: A proof of the hairy ball theorem, Amer.Math.Monthly 86,7 (1979) 572-574.

[23] H.Flanders: Differential forms, with applications to the physical sciences, Academic Press, NY 1963. (Mathematics in science and engineering #11.)

[24] R.L.Fleischer,H.R.Hart,I.S.Jacobs,P.B.Price: Search for magnetic monopoles in deep ocean deposits, Phys.Rev. 184 (1969) 1393-1397 [25] R.L.Fleischer, P.B.Price, R.T.Woods: Search for

tracks of massive multiply charged magnetic poles, Phys. Rev. 184 (1969) 1398-1401.

[26] D.Fuks: Foliations, J.Soviet Math. 18,2 (1982) 255-291 [Itogi Nauki i Techniki Algebra Topology Geometry 18 (Moscow 1981) 151-213 in Russian].

[27] H.Georgi & S.L.Glashow: Unity of all elementary particle forces, Phys.Rev.Lett. 32 (1974) 438-441.

[28] R.P.Geroch: Spinor structures of space-time in general relativity I, J.Math’l. Phys. 9,11 (1968) 1739-1744.

[29] H.Gluck, F.Warner, W.Ziller: The geometry of the Hopf fibrations, L’Enseignement Math. 32 (1986) 173-198; Fibrations of spheres by parallel great spheres, and Bergers’ rigidity theorem, Annals Global Analytic Geometry 5,1 (1987) 53-82.

[30] Herbert Goldstein: Classical Mechanics, Addison-Wesley 1980. [Contains extensive discussion of “Bertrand’s theorem,” for which he cites J.Bertrand: Comptes Rendus 77 (1873) 849-853.]

[31] A.Haefliger: Sur les feuilletages analytiques, Comptes Rendus Acad. Sci. Paris 242,25 (1956) 2908-2910; Comment. Math. Helvetica 32 (1958) 249-329; Varietes feuilletees, Ann.Scuola Norm. Sup. Pisa 16 (1962) 367-397;

[32] A.Haefliger: Feuilletages sur les varietes ouvertes, Topology 9,2 (1970) 183-194.

[33] D-S. Han & J-W.Kim: Unit vector fields on spheres, which are harmonic maps, Mathematische Zeitschrift 227,1 (1998) 83-92.

[34] Heinz Hopf: Über die Abbildungen der dreidimendionalen Sphäre auf die Kugelfläche, Mathematische Annalen 104,5 (1931) 637-665.

[35] Heinz Hopf: Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Annalen 96 (1927) 225-250.

[36] A.Hurwitz: Über die Komposition der quadratischen formen, Math.Annalen 88 (1923) 1-25. (Posthumous.) This result had previously been sketched by Hurwitz in: Über die komposition der quadratischen Formen von beliebig vielen Variablen, Nachrichten der Königl. Ges. der Wissen. Göttingen (1898) 309-316.

[37] J.D. Jackson: Classical electrodynamics (3rd ed.), Wiley 1999.

[38] Proofs of the Jordan curve theorem are in:

L.V. Ahlfors: Complex Analysis, McGraw-Hill 1979. I.Fary & E.M.Isenberg: On a converse of the Jordan curve theorem, Amer. Math. Monthly 81 (1974) 636-639. R.Maehara: The Jordan curve theorem via the Brouwer fixed point theorem, Amer. Math. Monthly 91,10 (1984) 641-643.

C.Thomassen: A link between the Jordan curve theorem and the Kuratowski planarity condition, Amer. Math. Monthly 97,3 (1990) 216-218. Helge Tverberg: A proof of the Jordan curve theorem, Bull.London Math. Soc. 12,1 (1980) 34-38. K. Venkatachaliengar: A simple proof of the Jordan curve theorem, Quart. J. Math. Oxford 8 (1937) 241-244. G.T.Whyburn: Topological analysis, Princeton Univ. Press 1958.

[39] T.F.E.Kaluza: Zum Unitätsproblem der Physik (On the Unification Problem of Physics), Sitzungsberichte Preussische Akademie der Wissenschaften 54 (1921) 966-976.

[40] M.Kervaire: Non-parallelizability of the sphere for n > 7, Proc. Nat’l Acad. Sci. USA 44 (1958) 280-283.

[41] Oskar Klein: Quantum Theory and Five-Dimensional Relativity (in German), Zeit. Physik 37 (1926) 895-; Nature 118 (1926) 516-.

[42] J.Korbas & P.Zvengrowski: The vector field problem; a survey with emphasis on specific manifolds, Expositiones Mathematicae 12,1 (1994) 3-20.

[43] K.S.Kulkarni & U.Pinkall (eds.): Conformal geometry, Vieweg Braunschweig 1988.

[44] K.M.Kuperberg: A smooth counterexample to the Seifert conjecture, Ann. of Math. 140 (1994) 723-732. G.Kuperberg: A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996) 70-97; G.Kuperberg & K.M.Kuperberg: Generalized counterexamples to the Seifert conjecture, Ann. of Math.144 (1996), 239-268.

[45] M.Lachiez-Rey, J.M. Luminet: Cosmic topology, Physics Reports 254 (1995) 135-214.

[46] L.D.Landau & E.M.Lifshitz: Quantum mechanics, nonrelativistic theory, Pergamon 1976. (Section 35, “Fall of a particle to the centre.”)

[47] W.B.R.Lickorish: A foliation for 3-manifolds, Annals of Math. 82 (1965) 414-420.

[48] Elliott H. Lieb: Selecta (ed. by W.Thirring) “The stability of matter from atoms to stars,” Springer (second ed.) 1997.

[49] D.Lovelock: The uniqueness of the Einstein field equations in a 4 dimensional space, Arch. Rat. Mech. & Anal. 33 (1969) 54-70.

[50] David Lovelock: Dimensionally dependent identities, Proc. Cambridge Philos. Soc. 68 (1970) 345-350.

[51] D.Lovelock & H.Rund: Tensors, differential forms, and variational principles, J.Wiley 1978; Dover corrected reprint 1989.

DocNumber  15  . 7. 0. 0

[52] K.H.Mariwalla: Dimensionality of space-time, J.Mathematical Physics 12 (1971) 96-99.

[53] W.Miller Jr.: Symmetry and Separation of Variables, Addison-Wesley, London 1977

[54] J.Milnor: Some consequences of a theorem of Bott, Annals Math. 68 (1958) 444-449.

[55] J.Milnor: Analytic proofs of the ‘hairy ball theorem’ and the Brouwer fixed point theorem, Amer.Math.Monthly 85,7 (1978) 521-524.

[56] C.W.Misner, K.S.Thorne, J.A.Wheeeler: Gravitation, Freeman 1973.

[57] A.F. Monna: Dirichlet’s principle: a mathematical comedy of errors and its influence on the development of analysis, Oosthoek, Scheltema & Holkema, Utrecht 1975.

[58] P. Moon & D.E. Spencer: Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions, Springer-Verlag, Berlin 1961.

[59] P.M.Morse & H.Feshbach: Methods of Theoretical Physics, vols I & II, McGraw-Hill 1953.

[60] K.R.Nagarajan & T.Soundararajan: On the module of 1-forms on a differentiable manifold and parallelizable manifolds, Indian J. Math. 41,1 (1999) 115-129.

[61] Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications, 39 pages,

arXiv:math.DG/0211159.

[62] H.Poincaré: Sur les courbes définies par les équations différentielles, J.Math.Pures Appl. (4) 1 (1885) 167-244 (chapter 13).

[63] J. Polchinski: String theory (2 vols), Cambridge University Press 1998.

[64] A.M.Polyakov: Particle spectrum in quantum field theories, JETP Letters 20 (1974) 194-195.

[65] J.Radon: Lineare Scharen Orthogonaler Matrizen, Abh.Math.Sem.Univ.Hamburg 1 (1922) 1-14

[66] A. Salam & J. Strathdee: On Kaluza Klein theory, Annals of Physics 141 (1982) 316-352.

[67] Richard M. Schoen: Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20,2 (1984) 479–495.

[68] J.Schwinger: Magnetic charge and quantum field theory, Phys.Rev. 144 (1966) 1087-1093; Sources and magnetic charge, Phys.Rev. 173 (1968) 1536-1543.

[69] Warren D. Smith: On the shape of the universe,

http://math.temple.edu/∼wds/homepage/unishape.{dvi,ps,pdf}.

[70] Glenn D. Starkman: Topology and cosmology, Classical Quantum Gravity 15,9 (1998) 2529-2538. This is the introduction to a special issue devoted to questions about the geometry and topology of the universe, which is recommended in its entirety.

[71] E.Stiefel: Richtungsfelder und Fernparallelismus in ndimensionalen Mannigfaltigkeiten, Comment. Math. Helv. 8,4 (1936) 305-353.

[72] F.R.Tangherlini: Schwarzschild field in n dimensions and the dimensionality of space problem, Il Nuovo Cimento 27,3 (1963) 636-651.

[73] Max Tegmark: On the dimensionality of spacetime, Classical & Quantum Gravity 14,4 (1997) L69-L75.

[74] G. ’t Hooft: Magnetic monopoles in unified gauge theories, Nucl.Phys. B79 (1974) 276-284.

[75] Emery Thomas: Vector fields on manifolds, Bull.Amer.Math.Soc. 75 (1969) 643-683.

[76] J.J. Thomson: Elements of electricity and magnetism, Cambridge Univ. Press (3rd ed) 1904. (Section 284).

[77] William P. Thurston: A local construction of foliations for 3-manifolds, Proc. Sympos. Pure Math. 27 part 1 (1975) 315-319.

[78] William P. Thurston: Existence of codimension-1 foliations, Ann.Math. (2) 104 (1976) 249-268.

[79] William P. Thurston: Three-dimensional geometry and topology (edited by Silvio Levy), Princeton University Press (mathematical series #35) 1997.

[80] Marie-Antoinette Tonnelat: Les théoriés unitaires de l’électromagnétisme et de gravitation, Gauthier-Villars, Paris 1965.

[81] G.F.Torres del Castillo & L.C.Cortés-Cuautli: Solution of the Dirac equation in the field of a magnetic monopole, J.Math.Phys. 38,6 (1997) 2996-3006.

[82] J.P.Vallee: Intergalactic and galactic magnetic fields – an updated test, Astrophys.Lett. 23 (1983) 85-94.

[83] J. von Neumann & A.W.Burks: Theory of self reproducing cellular automata, Univ. of Illinois Press, Urbana 1966.

[84] S.Weinberg: Gravitation and cosmology, Wiley 1972.

[85] J.W.Wood: Foliations on 3-manifolds, Annals of Math. 89 (1969) 336-358; Foliations of codimension 1, Bull.Amer.Math.Soc. 76,5 (1970) 1107-1111.