Preface

My original intention had been to face the striking success of my “New Physics – Dawn of Cognition” by publishing just a second, extended edition. Then, however, I felt that priorities had shifted that drastically meanwhile that a more specific title seemed to be more adequate.

Nevertheless, the main body of this book will effectively be identical with the old one. The philosophical input situation, however, changed dramatically, still becoming even deeper intelligible. Again, it is the English translation of a research report in German language on the common foundations of particle physics and cosmology, together with their consistent unification to a ”Quantum Gravity“ and to a „Grand Unification Theory” (GUT).

This edition, again, in parts, is an extract – here and there: word for word – out of a more technical report (‘Die “Weltformel”’ (2013)/‘The “World Formula“’) of the same author (www.q-grav.com -> Summary View). But it is an extended update edition.

In characteristic parts, where that report would have asked too much of the rapid reader, this extract is paying more tribute to the requirements of a popular-science presentation. All the same, the author’s endeavour had been not to drop the technical context completely – at least, it will survive in terms of commentary headwords and special boxes.

The main report, ‘The “World Formula“‘, is based on the author’s lectures given at diverse universities, starting in 2011 in the frame of the annual DPG (Deutsche Physikalische Gesellschaft) spring conferences in their sections T (particle physics), GR (gravity and relativity), MP (the mathematical foundations of physics) and AGPhil (working group on the philosophy of physics). Its numerous manuscripts (predominantly written in English) are found in the internet as well, see www.q-grav.com -> Lecture Notes. Their “Abstracts” are published, in addition, under Verhandl. DPG (VI), starting with volume 46 (2011), see www.dpg-physik.de.

At their appointed dates, these lectures represented the actualized research states on a unified field theory as originally registered in terms of the author’s preliminary German print-book ‘Weltbild nach Vereinheitlichung aller Kräfte der Natur im 3. Jahrtausend‘ (2010), registered under the ISBN 978-3-00-030847-6. By such punctually anticipated representations I checked that the statements made are not lively fantasy “beyond” whatsoever but fully in accord with (present-day) experiment.

 

The Old Physics at Its End

This is a story deeply splitting the community of physicists into opponent camps. With his “World Formula”, Einstein once had coined a notion which now stands for failed trials to include electromagnetism into his 1915 concept of General Relativity, which is the geometrization of gravity.

Meanwhile, the number of forces has increased by still adding nuclear forces. It turned out that the dynamics of all those “internal” forces in addition to gravity can be roughly described by Schrödinger’s wave mechanics, which is a particular aspect of quantum theory.

The (”chiral”) interactions of those “internal” nuclear forces seem to follow comparable (“gauge”) structures, too – although, in terms of the “Standard Model”, those (gauge) structures are not well understood until to-day, as far as their abstract origins are concerned.

And quantum theory – the other component of modern physics – is based on Planck’s discovery in 1900 that nature does not behave in a continuous way but is showing up in discrete steps. This, however, is a stringent consequence of physical statements in order to be verifiable by measurements.

For, due to its limited span of life, a living organism like a human cannot count up to infinity. Hence, infinities are unphysical, i.e., not measurable; everything must stay finite in physics. Even an elementary particle cannot be accelerated beyond all bounds, its energy must remain delimited.

As a non-rational, “continuous” number only can be reproduced by an infinite series of rational numbers (decimal digits, e.g.), non-rational numbers are not denumerable, either. Rational numbers, however, are countable. Hence, fundamental physics will have to deal with finite sets of rational numbers only, and not with their limiting values, either.

In consideration of its continuous treatment of space and time, classical physics – including Einstein’s General Relativity – in this sense is “unphysical”, too. Thus, it necessarily will have to be “discreticised”, or “quantized”, which is the modern way of formulating it.

Only, since one century, gravity, i.e., Einstein’s General Relativity, stubbornly refused to “cooperate” with Planck’s quantum theory – and, v.v.: those “internal” forces denied any cooperation with General Relativity, likewise.

The action of General Relativity is best visualized by the well-known, familiar model of a flat rubber membrane stretching itself horizontally. An object deposited on its surface, by its weight and by the elasticity of the membrane will give rise to some downward depression, there. By this depression, a small marble, then, kicked (in a non-centric way) towards that object will be deviated from its straight run such as if that object and the marble are attracting each other.

The reason for this strange behaviour is traced back to geometry, i.e., to that depression in the membrane. The formerly flat plane, now, is not any more flat but bent downwards in the region where the object is located. Mathematicians are attributing such a surface curvature to some “non-linear” condition, as they call it. (For, “linear” equations are exclusively describing straight lines and flat planes.)

Special Relativity is a subset of General Relativity, acting in flat space-time only. Physically, it is neglecting the acceleration created by mass attraction. This acceleration, however, just is the crucial result of a (here: gravitational) force. Thus, Special Relativity is dropping forces.

On the other hand, the current theories of elementary particles – i.e., the “quantum field theories” – are exclusively working with Special Relativity only. And no successful trial is officially known proving that they are tolerating their extension to General Relativity, while, equally, the (official) theory of gravity does not show up to tolerate wave mechanics, i.e., the superposition principle of waves. This is another indication of Einstein’s General Relativity apparently not to be consistent with Planck’s quantization concept Schrödinger’s wave mechanics is a derivation of.

Briefly, nobody yet is (officially) acknowledged to have combined Planck’s quantum theory with Einstein’s General Relativity in a consistent way.

By interpreting a linear superposition as a contradiction to a non-linear surface, small-minded contemporaries even are trying to persuade us that a unification of Einstein with Planck should be principally impossible. (However, they are comparing “apples with pears”.)

This (false) conclusion is symptomatic. For, we just realized that Special Relativity is cancelling forces. And particle physicists, instead of letting themselves be guided by General-Relativistic ideas, are continuously inventing a wealth of substituting strategies in order to describe interaction forces by circumventing General Relativity.

V.v., a much more promising access would be to extend General Relativity in order to include the “internal” forces, in addition. This, however, is Einstein’s old idea of a “World Formula” which, then, should be excavated – although, due to its well documented failures in the past, this access had to bear a heavy loss of reputation since.

After the detection of nuclear forces, Einstein’s notion of a “World Formula” even had become somewhat ambiguous. On the one hand, it would have to include the consistent combination of Planck’s quantum theory with Einstein’s General Relativity. This, actually, is attributed to a “Quantum Gravity” still to be constructed.

On the other hand, it would have to include the unification of all “internal” forces with each other and with gravity to a “Grand Unification (Theory)” (GUT) of all forces of nature. (The string models are calling it a “Theory of Everything (ToE)”.)

Our so called “Standard Models”, however, (that of particles and that of cosmology) are far from covering any of those targets.

“String/brane“ models are digging even deeper into that dead end of physics because they have taken over crucial parts of those bad features of the aged quantum field theories – let me just mention the “variation principle” (Leibnitz, Bernouilli, 400 years ago) with its “path integrals” and “Lagrangians”, e.g.

 

The New Physics

Theoretical physics is the mapping of (parts of) nature into mathematics. Current “string theories” do not care about nature; hence, string theories cannot be considered any more to belong to the category of “natural” sciences. Even to their protagonists it is unclear what at all they are mapping into mathematics.

String models do not try to reproduce nature, but they are hoping, the other way round, that, in nature, there are existing structures – still to be uncovered – which are corresponding to their models. This cross-over method “beyond the Standard Model” of not asking theory to reproduce nature but of nature to follow theory, still might keep them busy for another couple of hundreds of years more to come.

Hence, let us follow another line of argumentation. As we cannot count up to infinity, a measurement can reproduce a result at most up to the accuracy of some rational number. The total of all our measuring results, then, must be some finite set of rational numbers. This demands physical models of nature basicly to be of an atomistic structure, i.e., “quantized”, in order to stay measurable; and measurability is the key property of physics. Let us designate their “atoms”, here, as “quanta”. By the huge number of quanta available in our universe, most of their structure only can be covered by statistical methods.

In mathematics, an atomistic structure is dealt with by combinatorics, and statistics is dealt with by the theory of probability. The combination of combinatorics with probability is “group theory”. A typical example of group theory is the property of a “spin” – that “intrinsic” angular momentum where nothing is rotating.

For the majority of physicists, group theory has remained a complete mystery. Even Einstein did not care about it; his General Relativity does not take it into account, spin is foreign to General Relativity. Schrödinger scornfully renamed it “group pestilence”, and Pauli jumped onto that trend.

On the other hand, we shall observe that this underestimated discipline of mathematics widely swept under the carpet so long just is representing the “missing link” between Planck and Einstein; still during the course of the actual century, it might take over the leading position in fundamental physics.

 

Creation and Destruction Revisited

The first ingredient of a “group theory” is combinatorics. Combinatorics alone, without adding a concept of probability, is giving rise to discrete symmetries, which, in physics, found their application to crystal lattices, e.g.

Let us number (equal types of) atoms in a crystal. When subjecting them to some “transformation” (a rotation, a reflection, or whatsoever) after which every (such) atom of this crystal is moved from its original position to the former position of one of the other atoms (or even to its own former position) without leaving any position empty or doubling it, the crystal under consideration does not change its form – although particular atoms (or even all of them) do change their positions (in a 1:1 way).

A discrete transposition of an atom from a former position r’ to a final position r”, as effected by some transformation A, can be expressed as well by first deleting it at its former position r’, followed by recreating it at its final position r”:

 

 

Those secondary operators “a” with the upper plus and minus signs are called “creation operator” and “destruction operator”, respectively. (The transposition matrix A collecting all their pair combinations like the one shown above, then, is defining its “adjoint representation”.) Special linear combinations of these elements of the matrix A which the mathematicians are calling “permutations”, in physics are better known as “generators”: In physics, usually,

 

 

Let me stress, however: These permutations, usually, are no real options but pure “thought experiments” in order to visualize the (crystal-like) ordering structures we find in nature!

Now, actual fundamental physics does not yet officially realize that modern physics is more than warming up some classical principles of centuries gone by which merely are extending some classical formalism of functional analysis once learnt at school. Not just a couple of additional parameters has to be introduced. No, those principles have to be adjusted instead of fitting just some additional ad-hoc parameters as the “Standard Models” are practising it!

The traditional formalism of the “Standard Models” is it to integrate a generator G over some pathway. This pathway could happen to end at another lattice position allowed for the transposed atom. Usually, however, it might end up at any position between such allowed positions. Hitting an “allowed” position will lead to some respective “eigenvalue” condition.

Such an intermediate “position”, however, does not indicate the end of a “pathway” in the sense of classical physics but a mixture of “neighbouring” permutations inside the crystal; and the larger that mixing parameter, the larger the “neighbourhood” effectively involved. Here, the probability aspect of “quantum” physics is entering in terms of apparently “intermediate” positions “smeared out” over neighbouring states pretending some statistical “interpolation”.

 

Why are there Dimensions, Why Forces?

We have seen that physics demands some concept of probability on an atomistic model, and we designated those “atoms” as “quanta”. When introducing parameters in terms of labels, we are defining “classes” (i.e., components, dimensions) of those quanta. But how many classes (dimensions) might be there?

Now, for normalization reasons, probability needs a division operator. Then, however, number theory is teaching us that the highest dimension of a field of numbers tolerating a division operator is 8. (For comparison remember that the complex plane is a 2-dimensional field x = a+ib of r-numbers a and b.) The mathematicians named those 8-dimensional numbers “octonions”.

For the physicist, this means that he has to split the running label n counting our quanta (n = 1, … , N) into a pair of labels: n -> (r;x) – with its first part r denoting the class (r from 1 to 8) and another part x designating its remaining rest of individuality. And this procedure may be repeated:

 

n -> r;x -> r,s;y -> r,s,t;z -> …

 

The labels x,y,z, … designating the remaining rest of individuality, which is not subject to measurements, usually will be dropped. Hence, from statistics – together with number theory – we derive that nature should manifest itself in terms (of powers) of 8 dimensions. And experiment shows that powers higher than 2, actually (!), are not needed. For the actual state of the art, thus, the dimension of our world is fixed to be 8x8=64.

Now, the first factor 8 will be identified to reproduce Quantum Gravity, the second factor 8 the “internal” forces, and both together the Grand Unification Theory (GUT)! But let us proceed step by step.

For non-mathematicians, octonion arithmetic looks rather strange. The actual state of the art with respect to fundamental physics, however, does not need its sophisticated multiplication rules. We only need that octonions occur in 8 dimensions.

Compare it with the chemical elements phosphor and oxygen: When put together, they immediately will catch fire. When safely bottled, however, each of them in its own bottle, then they can be stored peacefully side by side and nothing will happen.

Consider our actual access to fundamental physics as treating octonions in a bottle, with their glass walls, i.e., with our actual mathematics, shielding us from their aggressive multiplication rules. But we do observe that there are 8 different types of “bottles”, and we take this multiplicity into account. If you like, consider it as some first approximation to a physics to be developed in a future lying far ahead.

To cut it short: Dimensions are the result of probability to be normalizable – especially the 4-dimensionality of space-time – while forces will show up as the necessary result of permutations, i.e., as effects of statistics (probability) on specific types of permutations. The following chapters are giving details.

 

Quantum Gravity

The above first factor 8, an octet of eight dimensions, in quantum field theories had been identified to provide Dirac’s four “covariant” plus his four “contravariant” dimensions. For a preview: Our 4-dimensionality of space and time will derive from that. Thus, the 4-dimensionality of space-time (and of energy-momentum) is an output feature of Quantum Gravity, based on probability. For all other models – Einstein’s General Relativity included – it still is an external input feature of unknown origin!

When setting both 4-touples on a common base (as opposite variances of some common substructure), then we already obtain a consistent Quantum Gravity, the fully quantized version of Einstein’s General Relativity, on a fully quantized bent space-time.

Emerging in a mathematically closed form (i.e., not just as some approximation but in an exact form), in Einstein’s terminology, it proves to be fully “background-independent” – i.e., all physics is staying inside the above “membrane”, unable to leave its bent surface. Thus, this Quantum Gravity has taken the great hurdle no model before, after Einstein, has been able to jump over. (“Loop Quantum Gravity” is not fully quantized!)

Its separation of the two types (“co-“ and “contravariant”) of dimensions in its 4+4 = 8-dimensional version demonstrates that, contrary to the situation in the current models of quantum field theory, no quant is getting lost (as it is standard with the “commutators” of “2nd quantization” in the “Standard Model”) and no quant is falling from heaven (giving “vacuum polarization”). Thus, in Quantum Gravity, a vacuum remains empty, indeed!

In Quantum Gravity, the four non-linear space-time components are simple quotients of generating operators with the generator of heavy mass as their common divisor:

 

 

Quantum Gravity is the only (field-theoretic) model having dug out this almost trivial relation physicists even before Einstein and Planck already had been well familiar with. (That Q = MX is the additive CMS-space-time, where CMS stands for “Centre of Mass System”.)

 

 

Modern models – like that of “Loop Quantum Theory” and other models – which are poor trials of approximating just partial structure components of a veritable Quantum Gravity, not even are scraping at its surface. After considerable computer time in grand style, they

Impressum

Verlag: BookRix GmbH & Co. KG

Übersetzung: This is the English version of the German original "Fluss der Zeit", released the same day by the same publisher.
Tag der Veröffentlichung: 14.11.2014
ISBN: 978-3-7368-5584-7

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