Physics as a (unique) nonlinear math which is free of infinities in solutions by Ivan L Zhogin

Jonathan,

my preferred Math is a unique (and non-Lagrangian) variant of "clear" Absolute Parallelism (or primeval AP, as was used by Einstein-Mayer: single field - frame field, irreducibility; no other fields) which solutions are seemingly free of singularities; that's of exceptional rarity. And I try to indicate other interesting features of this math.

It seems you are writing about some other AP (not primeval) where I am not a specialist.

Long-long ago I ran through couple of Shipov's papers (in Izvestiya VUZov; V.R.Kaigorodov once mentioned him) and, frankly, found nothing interesting (as I can remember): that was not clear-AP, plus Lagrangian equations; no mentioning of topological charges and quasicharges; no tracks of compatibility theory; and all.

(BTW, Lagrangian-ness of equations in AP guarantees the presence of an identity but not their compatibility; and some researchers (e.g. Hehl) considered a pathological, incompatible variant of Lagrangian equations of AP.)

Sure, I am unable to discuss wordings and trifles of Shipov, or, say, the present interpretation of (the) Einstein-Cartan theory (which seemingly preserves nothing from AP and Einstein).

Next, I believe that QM works well, so one needs just (instead of disproving the Bell's inequalities) to give an explanation of the superposition principle and a convincing understanding (interpretation) of QM.

You are right that my exposition of this Math is much less elegant than the Math itself (and, frankly, many items relating to the combinatorics of topological (quasi)particles and the scenario of appearance of a SM-like QFT are unclear yet).

Regards,

Ivan

Well, here was a post by Joe Fisher (on 3/23/15 at 16:37pm UTC, wrote Dear Dr. Zhogin, I do not wish to disrespect you or your essay, but I do...).

At present the post miraculously disappeared - nevertheless I'll try to answer:

Dear Joe Fisher, in absence of tracks of any math, I feel as should judge about a piece of poetry (or philology, or philosophy, if you prefer); and that is not an easy deal for me.

You would (could) look at the text of the dialog between Rabindranath Tagore and Albert Einstein. They agreed that, in absence of humans, Apollo Belvedere would not be handsome (the sample of beauty) any more. However Einstein suggested that the concept of truth is of another sort, truth (relating to physics) should be free of any subjectivity.

In some other place Einstein wrote that (approximately, as I remember) "a priori we might expect that events of the visual environment (outward world) are just slightly ordered - like letters in words (or maybe words in phrases, not sure); however our experience unveils much more rigid links between events (or processes), which only approximately can be expressed in terms of cause and effect".

So, I would prefer a math (yes, without infinities!)

Dear James Putnam,

thanks for greetings.

Dear all.

I have looked at the discussion around some essays, and let me say a few words about the speed of light and the Lorentz group (and sure about Minkowski spacetime as well).

Well, it is better to consider the speed of light simply as a conversion factor (say, between seconds and meters, or foots if someone prefers). Nature does not know about seconds and meters (those are human units relating to the period of human's heart pulse and the length of human's extremities), but she can "rotate" time and space coordinates through Lorentz transformations (so, she considers space and time on an equal footing). Physicists frequently place c=1 (then all coordinates of space-time have the same unit of measure). Lorentz-covariant field equations have the maximum speed of (information) propagation equal 1.

It would be interesting to know the motivation of those who prefer Minkowski space to the approach of General Relativity (sure, most "fundamental" physicists, or those who occupy positions aimed at the fundamental questions, consider the GRT as just a low-energy approximation, with just an accidental feature of general covariance).

There are two possible ways to advocate Minkowski space.

(1) This space is sent down "by a decree" as a very simple and symmetrical space.

Well, some people do not like decrees, and some other people will start to ensure that there are even more symmetrical spaces - Finsler spaces (which have hence more grounds for being chosen).

(2) Minkowski space is a solution of the most simple equation for the space metric (field): Rieman_curvature=0.

That looks better. Finslerian partisans have no ideas on field equations and should hide in bushes.

But we have another problem. This simple equation has more solutions than just Minkowski space, and those extra solutions are singular (and there is no means to exclude them). Indeed, written in covariant indices, this equation is linear in its principle terms, so they remain regular when the covariant metric matrix becomes degenerate (remaining finite - a co-singularity;

[math] \det g_{\mu\nu}=0 [/math]

is a codim 1 subspace).

In terms of frame field, h^a_\mu, the equation for Minkowski space looks even more simple (can be linear): [math]h^a{}_{\mu,\nu} - h^a{}_{\nu,\mu} = 0 [/math]

(one can calculate inertial coordinates y^a integrating y^a_{,\mu} = h^a_\mu).

Again, singular solutions are inevitable.

So, we should look for some better a field equation.

And in my essay the existence of such an equation is explained (a unique equation which is free of both co- and (if D=5) contra-singularities is presented).

Dear Vladimir,

thanks for comments.

I've read your essay; it was not simple reading, and I did not catch very much.

An attempt to substantiate all of mathematics seems to me like telling for all Odessa. Mathematics is toov diverse and must meet different demands of the human practice (incorrect inverse problems, computer science, etc). Physics (fundamental) is another deal and should engage some special, dedicated math, especially beautiful and worthy math, and be the only possible choice in a sense.

There is a natural need in parceling the world, the being, into parts, smaller and smaller pieces, down to a point. Following this way we arrive to the concepts of continuum and field theory. Earlier, in natural philosophy, there were an empty space (absolute, and the absolute time) and material points; according to Newton, the body weight was determined by the number of material points in the body.

At present, the Lorentz group is a sort of engineering science, and it is not good idea to consider the time separately (from the space-time).

Absolute parallelism in fact is not a very good name; the better name is just the frame field theory. The name AP is due to the fact that the field equations can be written using a covariant differentiation with asymmetric connection which is "compatible" with the frame (i.e. the frame can be brought through this differentiation, changing Greek indices to Latin ones, and vice-versa). The curvature tensor for such a differentiation (with asymm. connection) is obviously zero.

However there are the metric and the usual covariant differentiation with symmetrical connection (Levi-Civita), for which the curvature tensor (Riemann tensor) is not zero. In my opinion, it is better to use this usual covariant differentiation, because it leads to the energy-momentum tensor, Riemanian geodesic lines, etc.

By the way, any linear combination of two connections is a connection too, so the range of covariant differentiations in AP is really great; but the symmetry of connection is the most important feature, that simplifies matters!

In general, the AP theory is about the fact that the general covariance (the group of coordinate diffeomorphisms) can coexist with the Lorentz group (however, generally, without inertial coordinates).

One can try to say something about AP in philosophical terms (philosophy is supposed to help in understanding the meaning of some math that tries to describe the objective reality, the being).

Well, the Hegel's statement about the identity of the being and not-being can be stated in a bit more concrete form: locally, or rather, at a point, in zero jets, the being is identical to the not-being.

That is, staying in a point, without differentiating, at one point, it is impossible to distinguish the being from the not-being.

In the AP, sure, the not-being means the trivial solution, the Minkowski space-time, where there are absolutely no differences between the points, nothing able to catch the eye. And the being is a solution of general position (of some perfect field equations) where all points are regular (there are no emergent singular points where the rank of the frame matrix drops per unit).

In vacuum General relativity (no fields other than the metric) it would seem that there is a stronger similarity of points: even in the first jets all points are indistinguishable from each other and from a point of the trivial empty space-time.

However, in solutions of the GRT (of general position) singular points do appear, points where the covariant metric is singular (of co-rang 1). And, of course, these singular points are sharply different from regular points (and points of the not-being).

Best regards