Towards a consistent theory of fields and particles (part I)

The notion that particles are entities embedded in an ambient space is logically flawed. That may be the reason why string theories, as well as other entities-in-an-ambient-space models have failed to produce a consistent and predictive model of the universe.

In the picture that I am going to present, certain stringy objects appear, not as objects embedded in an ambient space, but as 1-dimensional submanifolds arising from a geometric argument of cobordism towards the inside (the elementary particle). The fields in this sub-manifod satisfy Laplace’s equation in dimension 1, as inferred from a reasoned interpretation of Noether’s 2nd theorem, and they are natural candidates to define or chart space-time itself when you consider their analytic continuation to the complex plane. I will also formulate simplifying hypotheses that will prevent the cobordism argument leading us to an infinite regression.

The whole construction is only clarified in precise mathematical terms if we consider polar coordinates, which are more adequate than Cartesian ones to reveal the important information that any field theory carries with it.

As to gauge fields and their sources, as we will see, they will have to be interpreted only as  local charts to map the structure of the total field:

The attempt to extend the validity of what is nothing but local charts, to include all points of one single manifold (ambient/particle) gives rise to a redundancy that is at the origin of all renormalisation problems, disparity of scales in the renormalisation of the scalar field, and ultimately, an excess counting of the degrees of freedom that leads to a surprisingly large value for the cosmological constant.

The nature of the gravitational field, on the contrary, is better understood from similar geometric arguments, but reversed towards cosmological horizons, towards the outside of the elementary particle. Gravity only makes sense as an outside force, of entropic origin, which sits well with well-known previous ideas (‘t Hooft, 1993; Susskind, 1995Maldacena, 1997Bousso, 2002; Verlinde, 2010.)

Does this mean that it is inconsistent to consider gravitational fields up to Earth-size scales? No. It means that, if we do that, the treatment has to be justified upon the possibility of neglecting all gauge fields (charge compensation for EM, confinement for QCD or massive character of the bosons in SU(2)). All contained in so-called “soft-boson theorems”. And, in any case, no matter what resolution in fields we propose, it can never be extended to include the whole of space-time, because we are using illegal charts for dealing with such questions.

Yes, in the view I am going to present, entropic gravity is very much a logical necessity.  I cannot get involved here in defending Verlinde’s theory, which no doubt has its limitations. Suffice it to say that arguments based on neutron interferometry (Motl, 2010) are easily answered by simply considering the previous observation (that gravity only makes sense ultimately towards the outside) and adding that BH entropy of astrophysical origin (Bekenstein, 1973, Bardeen, Carter, Hawking, 1974, Hawking, 1975) does not give a measure of all the relevant entropy for any physical system, as the holographic principle seems to imply, but only a limit, as we will see. A simple way of saying the same is that the holographic principle must be extended to include all gauge sectors plus gravitation, all conforming to a hierarchy in layers of the total entropy requiring a careful, scale by scale, analysis in which gravitation is the last bundle of field lines to be cardinalised, and always towards the outside.

But the paradigm is more general: All radiation degrees of freedom are entropic with respect to their source degrees of freedom.

As gravity has at its source any local clustering of energy, all the gravitational degrees of freedom are entropic with respect to some other fields determining an energy-momentum T_{\mu\nu} interior to the region that is considered as a source for this gravitational field.

About the ambient manifold

The only thing we know about the manifold we live in is that our universe looks 4-dimensional from where we stand. And, where do we stand?: We always stand in the proximity, further or closer, of either a particle or a cluster of them; and we always observe it from inside a cosmological horizon that makes up the sphere of the sky and can be visualised as a 2-sphere at spatial infinity, out towards which our field lines, not having been trapped in finite regions signaling the presence of field sources, polar or axial, are.

We are going to adopt a simple form to extend the geometric structure from there on, which does not have to be the only one logically consistent, and it is,

The dimension can be increased towards the outside in n-sphere embeddings and decreased towards the inside with at least one n-1-hyperbolic embedding with respect to its exterior n-sphere

I will use this later and I will explain it better later. We go on.

About entities

The previous observations about fields as local charts suggest to consider the charged particle as nothing more than a focal point of the field lines, which in principle could be nothing but a mirage. I will use this word with some leeway, and I have to trust the readers not to interpret anything beyond what is necessary. To me:

An ontic mirage is any geometric o topologic peculiarity whose ultimate referent cannot be solved, as it is presented as a pair of mutual references

Namely, and in contrast with the usual concept, we do not assume that any entity exists of which the mirage is a mirror image. Particles that our modern formalism considers as fermions lack in principle any unambiguous referent, finding themselves in the theory only as mutual references. In other words, a particle “sees” another analogous particle as a point, but that could be the product of the natural local charts to define the interaction, any possible concept of an entity remaining vacant, or not referred at all.

I will also use:

mirage of the field equations is any solution, mathematical term, parametrisation, etc., that can be eliminated by means of a transformation or re-definition validated by the passive symmetry group of my field theory

The mirage of magnetic monopoles

The notion that charges, currents, fields or sources are “electric” or “magnetic” in character is logically flawed. In fact, the notion itself that there is a possible distinction between charges and their currents, or between gauge fields and their sources, is logically flawed too:

Any fundamental theory must be expressed in terms of its invariant concepts

That is a lesson that Einstein taught us, or should have, and apparently we haven not been able to take to heart for decades.

The invariant concepts in Maxwell’s equations are not “electric” nor “magnetic”, but polar field (P) and axial field (A).

Principle I-A (EM):

It is impossible to decide without ambiguity whether a centre of current is an electric axion (dipole) or a magnetic axion (dipole)

Principle I-P (EM):

It is impossible to decide without ambiguity whether a centre of charge is an electric pole (monopole) or a magnetic pole (monopole)

In order to ensure the rigorous compliance of both previous principles, it is necessary to modify Coulomb’s and Biot-Savart’s laws (resp. Lorentz’s) that define the so-called electromagnetic forces with the following:

Principle II-A (EM):

It is impossible to decide without ambiguity which of the following real, uniparametric combinations the force between two axial centres of current is proportional to:

 \left( e^{i\theta} \boldsymbol{j}_e +e^{-i\theta} \boldsymbol{j}_m \right)^* \cdot \left(e^{i\theta} \boldsymbol{j}_e +e^{-i\theta} \boldsymbol{j}_m \right)

Principle II-P (EM):

It is impossible to decide without ambiguity which of the following real, uniparametric combinations the force between two polar centres of charge is proportional to:

 \left( e^{i\theta} q_e +e^{-i\theta} q_m \right)^* \left(e^{i\theta} q_e +e^{-i\theta} q_m \right)

where \theta is a real parameter, q_e is the electric charge and q_m is the magnetic charge in the fully duality symmetric Maxwell equations:

\nabla \cdot \boldsymbol{E} = \rho_e

-\nabla \wedge \boldsymbol{E} = \frac{\partial \boldsymbol{B}}{\partial t} +\boldsymbol{j}_m

\nabla \cdot \boldsymbol{B} = \rho_m

\nabla \wedge \boldsymbol{B} = \frac{\partial \boldsymbol{E}}{\partial t} +\boldsymbol{j}_e

Observation: If we did not introduce principle II in both its A and P forms, we would be in the same vicious circle that modern field theory is, looking for monopoles, axions, etc., that I will assume inexistent, not because their are impossible, but only because the theory does not really require them. They are simply either mirages of the field equations or the particles we already know. This principle is tantamount to redefining the Lorentz force law (generalisation of Coulomb’s and Biot-Savart laws) to:

\boldsymbol{F} = q_m\left( \boldsymbol{E}+\boldsymbol{v} \wedge \boldsymbol{B} \right) + q_e\left( \boldsymbol{B}-\boldsymbol{v} \wedge \boldsymbol{E} \right)

According to the dual symmetry that the theory itself is suggesting us. That is, if the theory itself treats electric and magnetic charges on an equal footing, it is not for me to say that this symmetry be broken by declaring a distinction that is not in the field equations, and so imposing it myself in the prescription for the interaction between particles. Afterwards we may consider generalisations of Lorentz’s law such as I have written it down.

These are some of the bonuses that are almost immediately produced in the context of the theory I am proposing:

  • Vacuum triviality in QFT
  • Almost zero cosmological constant and entropic interpretation of the non-zero value
  • Large N limit of field theories (‘t Hooft, 1974)
  • Absence of monopoles and axions
  • The illusion of inertia (Mach’s principle)
  • Bekenstein-Hawking entropy
  • Ontic mirage of source particles (fermions)
  • Interactions as only beable objects
  • Trivial explanation of confinement for SU\left(3\right) QCD viewed from U\left(3\right)

Vacuum Energy or cosmological constant

As I need quite some space to explain all the ideas involved in the theory I am proposing, as I do not want to leave this entry without offering some of the answers that I have presented as “bonuses,” and as I have already explained why we cannot see any monopoles or axions in the universe, I leave here this very simple calculation of why a number that should be 1, when we estimate it with quantum field theory, it gives 10 raised to the power of 120 times 1.

Where does this excess come from?


Number of times that the number of gauge degrees of freedom exceeds the number of source degrees of freedom in the EM sector (that plays the role of average for all gauge sectors):


Total number of variables for all gauge sectors:

There is a total of 3 gauge sectors, U\left( 1 \right), SU\left(2\right) y SU\left(3\right). Their degrees of freedom are, respectively, 1, 2^2-1=3 and 3^2-1=8. In total:


The overcounting (excess cardinal) will be the excess gauge degrees of freedom with respect to their sources raised to the number of independent degrees of freedom for all gauge sectors. Why?: Because all radiation modes are entropic with respect to their sources. This gives,

\Delta\Lambda=\left( 10^{10}\right)^{12}=10^{120}

This is the value of the entropy of my description. The definition of entropy always requires a description! If I am studying an ideal gas, it is I who is responsible for any variables I may define, make assurances that I can describe it with just a pair of real positive variables, e.g., P, V. Then,

The entropy with respect to that description tells me, through the temperature, how much do the degrees of freedom that I have integrated over, contribute energetically to give credit to a biparametric description of something that actually has about the order of 10^{24} dynamical degrees of freedom.

The adequacy or not of such a description of physical reality rests, of course, on me; and my ability as a physicist to describe the system with an adequate set of variables and the corresponding functional dependences.

What value should it have? Maybe one that excludes the radiation modes, as there are purely entropic. That is:

1^{12} = 1=\Lambda_{\textrm{obs}}

As actually the observed energy density is of one unit (proton) per metre to the fourth.

In plain language: Radiation is entropy; fermions, on the contrary, are non-entropic. Why? Because that’s where I have my source points placed, my fermionic focal points, no matter how much of a mirage they are. Wherever there is a fermion, there is a rendevouz of selectons, which are nothing but scalar topological solitons giving an ontic character to the local interactions between “mirage” fermions.

I have just mentioned the selecton at the end of this entry, in a somewhat enigmatic way, and with this I have to conclude for now. The selecton is a topological, scalar boson, giving ontic character to the interactions between fermions. Its similarity to the term selectron is not arbitrary, as it will be better understood in terms of supersymmetry, which is to be nothing but a useful parametrisation to redefine any hypersurfaces relevant to my problem (from where to where do I consider the chart to be gauge and from where to where fermionic), it is an exact symmetry of Nature, of essentially passive character (redefinible reference, rather than real transformation), and has nothing whatsoever to do with particle multiplets.

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