Documentation

The basic text on Quantum Gravity and New Physics had been edited in 2010 still as a print-book in German language („Weltbild … nach Vereinigung aller Kräfte der Natur …”, ISBN 978-3-00-030847-6).

Extracts including consequent onward developments, then, had been presented at various universities from 2011 to 2015 in the course of the annual spring conferences held by the German Physical Society (DPG), sections T (particle physics), GR (Gravity and Relativity), MP (Mathematical foundations of physics), and AGPhil (working group on the philosophy of physics). Their numerous manuscripts, predominantly in English, can be looked up in the internet (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.

These research reports on the common base of particle physics and cosmology, including their confluence giving “Quantum Gravity” and a “Grand Unified Theory” (GUT), had been put forward to formulate a preliminary expert survey in 2013 (“The World Formula, …”, see www.q-grav.com).

Still further actualized, it took shape as a coherent top-down view of our world. In characteristic parts, however, where these reports would have asked too much of the rapid reader, they are more paying tribute to the requirements of a popular-science presentation. The respective e-books are (English versions): 1st edition: “New Physics, Dawn of Cognition” (2013), 2nd edition: “Flow of Time, Quantum Gravity giving New Physics”(2014).

All the same, the author’s endeavour had been not to drop the technical context completely – at least, it survived in terms of commentary headwords and special boxes. This procedure proved good by the striking success of more than 20.000 e-books sold within just 2 years.

Meanwhile, the interior of a Black Hole and its connecting relations across the event horizon are calculated. By indicating ways of experimental verifications of predictions made by Quantum Gravity, the state of the art has reached such a degree of maturity that this 3rd e-book edition may risk the step towards supplementing it to a veritable textbook on Quantum Gravity – the first in the world – embedded in a Grand Unification of all forces of nature. The picture is rounded by deriving the fine-structure constant at its correct value.

The substantial extension of this pioneering work far beyond its precursory editions will be turning upside down considerably deeper cognitions on the nature of things. This is not surprising because Quantum Gravity, the first time in history, is unifying Einstein’s General Relativity with Planck’ quanta. And the Grand Unification, by still including Gell-Mann’s quark model, is showing up as the “Theory of Everything” (ToE) string fans had been looking for in vain so long.

For the rapid reader just interested in a survey, the main body of mathematics is banned into extra boxes, with the respective background knowledge shifted into 12 specialized appendices. By that segregation, this “3rd edition” is satisfying the requirements of an elementary self-instruction manual (when including the appendix) as well as some survey representation for the interested layman (when omitting the appendix).

The original title in German reads “ToE; Neue Physik; Unsere Welt erklärt durch die Quantengravitation. Weltweit 1. Lehrbuch zur QG” (2016).

Special attention is directed to ensure that statements made here are not based on the ivory-towered fantasticalities “beyond” whatsoever as they are customary in this branch. The statements met here are fully in accordance with solid (present-day) experiments and based on consistent mathematics!

 

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, as far as their abstract origins are concerned, are not well understood until to-day.

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 a physical statement in order to be verifiable by measurement. For, a living organism like a human, due to its limited span of life, 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 (Einstein’s General Relativity included) 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 to prove 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 oranges”.)

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 has 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. (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 uncritically have taken over crucial parts of those bad features of the aged quantum field theories – for the expert let me just mention the “variation principle” (Leibnitz, Bernouilli, 400 years ago) with its arbitrariness of parameters in their “path integrals” and “Lagrangians”, e.g., once designed for point mechanics. But is it wise to apply great-great-grandpa’s magic cooking formulas to a world of quanta?

There are typical structures which even the “Standard Models” do not catch at all – not to mention their incompetence of explaining them. Among them, there are notions like “Big Bang”, “Black Hole”, “cosmological constant”, “cosmic inflation”, “Dark Energy”, “Dark Matter”, “parity violation”, “flavour physics”, “broken quantum numbers”, “non-valence part of an elementary particle”, “virtual state”, “quark confinement”, etc., etc. But even such every-day notions like “mass”, “length”, “time”, “dimension”, “charge” are among them.

Everybody is internally conscious of the disability of the “Standard Models” laboriously patched up since decades, which ought to be given to the garbage long since. However, nobody knows by what to replace them – the great helplessness everywhere.

 

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 any more be considered 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, the other way round, they are hoping that, in nature, there are existing structures – still to be uncovered – which will correspond 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 up to the 1960s had been considered as the typical prototype of a confuse, unserious doctrine of incompetence, demonstrating what not is physics. – How times are changing; half a century of stagnation is giving rise to queer practices!

Let us, hence, 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 a subject not only to falsification but also to verification; this 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 "group theory" to “group pestilence”, and Pauli jumped onto that trend.

“Learning during all one’s life” at those times, already, had been a unilateral postulate by the elder people directed towards the younger ones, but not towards senior graduates. In professional life, tired after the stress of a long education, it is all too human to face more progressive, novel demands with negative scepticism instead of enthusiastically trying to find out promising points of attachment with respect to some better, new physics – as it had been usual in more productive periods. Where only did the eager expectations of theoreticians ooze away?!

Instead, technically innocent freeloaders are setting hopes in profiting from boosting those instinctive defence mechanisms of their seniors by trying to act as a resonance box of their masters, trumpeting forth their most absurd statements. For the realization of veritably novel findings we, hence, are depending on the creative impulse of descendent generations who are not putting up with being deposited on a sooty steam-engine yard of a railway museum.

On the other hand, we shall observe that group theory, 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.

This book especially addresses the younger students. Off with great-grandpa’s antiquated fug; the future is calling, let us shape it!

 

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, have 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 some final position r”, 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” of both signs in the upper label are called “creation operator” and “destruction operator”, respectively. (For details see the appendices until “Creator and Destroyer”.) Special linear combinations A(r”,r’) and A(r’,r”) of these elements of the matrix A which 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 are meeting in nature!

Now, actual fundamental physics did not yet officially realize that modern physics is more than just 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 will have to be introduced. No, those principles will have to be adjusted instead of just fitting in 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. Such an arbitrary pathway could happen to end at another lattice position allowed for the transposed atom. Usually, however, it will end up at any position between such allowed positions. (Only when hitting an “allowed” position, we are speaking of some respective “eigenvalue” condition.)

Such an intermediate position caught between two stools, however, does not correspond to some useful integration pathway, not even in classical physics. It rather will reflect some interpolated value out of a superposition of “neighbouring” permutations inside the crystal in a statistical sense!

Example for a case where only the natural numbers 1,2,3,… might be admitted as (linear) measuring values. With probability in percent, the “odd” value 2.8 will be reproduced by the superposition

 

2.8 = 2 x 20% + 3 x 80% .

 

Different combinations by different measuring values with correspondingly different percentages would be doing it as well. Only when hitting an “admissible” (“eigenvalue”) position, the indication of one single measuring value at 100% will do it. This, precisely, is the aspect of “probability” in a quantum-type physics. There is nothing mysterious, as certain people like to ascribe to it because of certain mathematical “theorems” (Bell’s theorem, e.g.) whose basic, special assumptions are not really satisfied physically (their restriction to Special Relativity, e.g., or to a continuous, i.e., to an unphysical Base, or similar things).

 

Why are there Dimensions, Why Forces?

We observed 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 eight. (For comparison remember that the complex plane is a 2-dimensional field x = a+ib of r-numbers a and b.) 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, i.e., a single n to a pair (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. This procedure may be repeated:

 

 

Those 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. 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 Unified Theory (GUT)! But let us proceed step by step (in the following chapters).

For non-mathematicians, octonion arithmetic might look 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 elements of phosphorus and oxygen: When put together, they immediately will catch fire. When bottling them into separate tanks, however, then both elements might be stored side by side, and nothing bad will happen.

Let us compare octonions with the 3 dimensions of space: Who is troubled if we measure a rectangle by multiplying 50 yards by 50 yards instead of converting this product into the measuring unit of “acres” (corresponding to our octonions)?? So what!

Let us, hence, 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 against 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. A dimension is its class number. Forces, however, will show up in a certain sense to represent the results of permutations, i.e., they are characterizing the effects of statistics (probability) on specific types of permutations. The following chapters are giving details.

 

Quantum Gravity

Survey: 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 be shown to 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-tuples onto a common base (as opposite variances of some common substructure), then we obtain a consistent Quantum Gravity, already, 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 or to correspond with things outside of it. 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 the sky (giving “vacuum polarization”). Thus, in Quantum Gravity, a vacuum remains empty, indeed – or it is no vacuum.

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, do not even scrape at its surface. After considerable computer time in grand style, they are making up a fuss about having found some approximate, qualitative evidence that a Big Bang might not be singular. – So what: Quantum Gravity is reproducing this result without any effort, exactly, quantitatively, and in a mathematically closed form. (It is a second-order “Casimir” operator.)

When considering this heavy mass as a constant, then space-time together with energy-momentum is reproducing Heisenberg’s uncertainty relation. The special mathematics behind that combination (i.e., the “commutator” of space-time with energy-momentum) is defining the “canonical” quantization of good old quantum mechanics – originally once some relic of the ancient variation calculus.

However, when considering heavy mass as an operator as it is, there will result additional terms proportional to some inverse (squared) length. This length will reveal itself to be the radius of the Big Bang region. (In Quantum Gravity, the “Big Bang” is extended, it no more is a point singularity.) And these additional terms just are reproducing the behaviour of Dark Energy. Thus, by inserting the experimental Dark Energy data, we are measuring the Big Bang radius of our universe.

In Quantum Gravity, an elementary particle is following the same equations as our universe as a whole. The difference is its point of observation: a particle is observed from outside, our universe from inside. A particle, hence, is appearing “small” to us, its reactions are fast, at most too fast in order to be registered in all details by experiment.

On the other extreme, our universe is appearing “huge” to us, its variations, usually, are that slow that they hardly are observable, either. The connection of both realms by Quantum Gravity, hence, is offering us the opportunity to study particle reactions in slow motion and the evolution of our universe in time-lapse.

Quantum Gravity does not only reproduce Dark Energy, but also cosmic inflation. And Hubble’s law is derived as some internal detail of Einstein’s cosmological constant artificially split off from it. The latter is shown to correspond to the (inverse) “propagator” of particle physics (where the partition between (squared) heavy mass and Einstein’s cosmological constant is a matter of definition because, for experimental verification, at the time being, only their sum is measured).

The apparent complexity of quantum theories is artificial, not intrinsic. Its origin is our affinity, developed by evolution, to observe space-time Q by its “ray representation” X. When dropping those blinkers, an essentially more distinct picture of the principles governing nature will pop up by considering it by the additive CMS-space-time.

That far, already, this preview on subsequent chapters.

 

The Physics of Actions

Classical physics – Einstein’s classical version of General Relativity included – is the physics of numbers. Quantum physics, however, is the physics of actions. What is the difference?

For real or complex numbers, their ordering within a sum or within a product is irrelevant: a+b = b+a and axb = bxa. Mathematicians say: they are “commuting” with each other, they are “commeasurable”. For actions, the situation is different: they might either commute or not. For an example, let me define two special actions:

In the case BA (mathematicians are reading that from right to left), I could succeed in obtaining the answer because that individual is overrun only subsequently: B is after A. In the inverse case AB, however, that individual is overrun (B) before I had asked my question (A). Then, my chance for getting an answer all the same, will be rather delimited.

The compound reaction AB, thus, will not necessarily be identical with BA. Their degree of conformity is measured by their difference AB–BA, which is the “(minus-)commutator” [A,B] of A with B. For commuting actions (AB=BA), this commutator will be vanishing.

In mathematics, an “action” is represented by an “operator” or, what is the same, by a “transformation”. But let me state a warning: Not every abstract “transformation” formulated by mathematicians also will be a physically executable “action” as well!

On the contrary: The overwhelming multitude of all “actions” defined by mathematicians are not physically executable actions from a state X to a state Y, but just are serving as a sample for a physical comparison of a state X with a state Y. Time increase is an example where two distinguishable states – a starting state and an end state – are compared with each other, without a physically executable procedure existing (except just waiting) which might “perform” such a transformation there and back.

The mathematical background is the following (see details in the appendix on “Groups”): Contrary to the situation in classical physics, permutations, i.e., the small, finite number of “generators”, are representing the fundament of physics – and not their integrated transformations containing those additional, continuous path parameters not observed by experiment!

The problem in classical physics is: To which of our 8 “dimensions” shall we assign its r-number pathway parameters? If just admitting the 2 dimensions of a complex plane – as it is custom in classical physics – we always have the choice between the r-numbers on the real axis and those on the imaginary axis.

In the “Standard Model”, people had the alternative either to fix their decision for the real or for the imaginary axis, and that for every parameter separately. Thus, classical physics decided to assign different types of axes for the 3 space parameters on the one hand, and for the time parameter on the other hand. This had been leading Einstein to the Lorentz transformations of Special Relativity.

All that, however, is the result of interpreting artificial “pathway” parameters which are not even “observed” in experiment. Observed are only the 2 states at the beginning and at the end. Quantum Gravity is free of such pre-settings. In Quantum Gravity, there are both opposing types of systems potentially present simultaneously, which are acting on the identical physical base of “generators” (admitting a complex Lie algebra). These two systems are, e.g.:

The reaction channel, which is conserving probability, simply is ignored in conventional physics. This will drop fundamental properties of physics! This is one of the reasons why the “Standard Model” has to work with coupling constants whose values cannot be calculated from theory but are accessible only by experiment. (For, the pseudo-unitary dynamic channel does not offer a “positive-definite norm” in order to conserve probability.) Matter submerging behind the event horizon, however, will prove the physical coexistence of both channels side by side.

In Quantum Gravity, the parallel use of both channels is no problem. When dropping the variation principle that characteristic for the “Standard Model”, an expansion of the operators of one of the systems according to operators of the other system will manage that job for both systems (see the headword “Dirac” in appendix “Creator and Destroyer”).

For a better understanding, let me give an example in classical argumentation. Say, the parameters s and t of a state are lying on a circle:

The same equation can be rewritten in a hyperbolic form as

When measuring s in “cm” and t’ in “sec”, some appropriate normalization might give “1 sec = icm” where “i” is the imaginary unit.

Direct measurements, however, are counting r-numbers only – the imaginary “i” is theory (like any equation). In order to formally support the explicit disappearance of such “i” values, “metric” tensors had been introduced into theory contracting the “i” of a “ket” vector with that of a “bra” vector to a (squared) common factor “–1”.

Now, we should expect that the equation of a hyperbola will cover value ranges (s,t) different from those of a circle; just for t=0=t’ they are coinciding. Solution: With t’=t, the C of the hyperbola will have to vary, too. This, however, assumes that by every t of the circle, the form of the hyperbola will vary, in addition. In order to hike on (the periphery of) a fixed circle, there is corresponding a jump from hyperbola to hyperbola! The value range of the diverse hyperbolas covered thereby will be staying finite, however – like on the circle.

In order to catch it better: In classical physics, the “group” is primary; and the special form of its generators forming some “(real) Lie algebra” is depending on it. In Quantum Gravity, the procedure is inversed: Primary are the generators – the groups which can be constructed out of their “complex Lie algebra” are secondary.

Thus, the channel covered by classical physics people are arguing in is preselected. In New Physics, however, this choice remains open (within the embedding 8-dimensional Quantum Gravity): via rediagonalizations, we are free to jump from one system to the other at pleasure – both channels are at disposal!

As in physics everything must be finite in order to stay measurable, the (“compact”) reaction channel will have to be considered as primary and the (formally: “non-compact”) dynamic channel as secondary, derived: The (2x4 = 8-dimensional) reaction channel will define the spectrum of all observations possible, and the (8-dimensional) dynamic channel, then, will be mixing all that together, again (see appendix “Groups”).

(When broken down to the 4-dimensional subspaces, this is giving the formalism of a “complex Lie algebra” in 4 dimensions. By functional analysis, a U(n,n) is nothing else than the integration of the total of all complex Lie algebras being derivable from a U(n).)

Thus, the ontological problem if we originally have to do it with a “compact” group (with the 1-metric) or with a “non-compact” group (with a different metric) as some different variant of its complex “Lie algebra” will be fading away.

This quarrel about nothing is properly pointing out how essentially artificial the endeavour is trying to nail a Lie algebra down to some special variant. Only its generators are of importance – everything beyond merely is a matter of casual, human interpretation, of a fit of the diagonalization direction most adequate to our argumentation.

Well. Nature primarily will start from the existing (superposition of) quantum states present – no matter which channel will be favoured. Both channels just are differing fibres within a common superset of states according to differing criteria: the reaction channel – due to its conservation of probability – will choose its compact version, while the dynamic channel – due to diverging criteria – one of its non-compact ones.

The collecting construction of a complex Lie algebra as a reservoir for both channels, then, is some purely formalistic act serving for mathematical elegance. The slicing of some set into differing “representations” (having no common states), then, in addition, will provide some dispersion of partial structures onto “neighbouring” partial representations in 4 dimensions, excluding parts we cannot address any more by the mathematics left in our own, isolated representation.

This situation will change again when proceeding to that common superstructure in 8 dimensions which both channel variants are (non-commeasurable) subsets of. Passing from one variant to the other, there, proceeds by rechaining substructures of the like we had been meeting with the circle/hyperbola example: one channel will be “expanded” according to the other one.

After having calculated some coupling constant according to some set of parameters in the reaction channel, e.g., we could switch over to the dynamic channel in order to follow its temporal development. From this back and forth, half-times are derivable.

There are numerous examples in particle physics giving “finite-dimensional” pseudo-unitary representations of the non-compact Lorentz group of Special Relativity, indeed. However, I am not willing to dissipate my energy in too many details, here.

Let us collect facts: “Dynamically” proceeding “reactions” are to be handled by “expanding” one system with respect to the other one. The 400 years old variation principle is a non-necessary, drastic restriction of modern approaches to physics. The (“unitary”) reaction channel is defining the physical basis – while the (“pseudo-unitary”) dynamic channel is working as some overlay structure.

Traditionally, Dirac‘s creation and destruction operators are not applied consistently. Originally, the destruction operator on the bra-hand of an ‘internal product’ once served as a technical tool for counting corresponding creation operators on the ket-hand side, only; equal counts on both sides were to contribute to the product, differing counts did not. (Cf. appendix “Representations”.)

Destruction operators, thus, served for decomposing a composed structure on the bra-hand side into „contravariant“ factors, and creation operators did that job for the ket side into „covariant“ factors. The subsequent admission of creation operators on the bra side and of destruction operators on the ket side, then, gave rise to certain insecurity with those rules, however.

For a quick and dirty ”escape strategy“, non-mathematicians went creating their own, badly “simplified” ”mathematics“ (See appendix “Representations”). That „commutator“ logic (“2nd quantization“, “normal ordering“), however, provided an arbitrary, highly inconsistent result ending up with representing the source of the current chaos and non-comprehension in fundamental physics.

For guaranteeing the conservation of probability, mathematicians are teaching us, a transformation should be “unitary”, in addition. Hence, the reaction channel (in the sense of thermodynamics) is describing some “closed system” conserving probability, while the dynamic channel is describing some “open system” where probability is not necessarily guaranteed. Instead, the dynamic channel by construction enables states to stay r-numbered – a property not offered by the reaction channel (cf. appendix “Groups”).

It is the interplay between a “naked” generator on the one hand and the integrated “action” with its arbitrariness and polyvalence on the other hand which is preventing even experienced, callous physicists of reputation from really “understanding” the rules of quantum mechanics.

It is that simple, however: The generators are containing all “fundamental” physics. We are measuring final results (“eigenvalues”, “expectation values”), and not intermediate “pathways”! Those integration pathways included in the “actions”, however, are representing additional trimmings of mathematics, i.e., arbitrary settings which have not been the subject (“eigenvalues”, “expectation values”) of experimental verification. Period!

The dilemma of current physics is that people are mixing up both aspects. Issues of pure mathematics (statistics) are misinterpreted as physics. All is cooked up to a thermodynamic, tough pulp. Mathematical formalisms having nothing to do with physics, finally, are taken for physics, ending up with the “Standard Model” or even with the string models. Beware of restarting that game!

(The reader should not be troubled by “imaginary eigenvalues”: r-numbered ones just are the result of “Hermitean” operators – and “Hermitecity” is a matter of definition!)

Now, What Really Is Space and Time?

There are subsets of the generators of Quantum Gravity which are identical in both channels – in the reaction and in the dynamic channel. These are their common generators, which the “Standard Model” would call “compact” if it would accept them as operators. A still smaller subset of them, up to some r-number factors (which also are defining the measuring units), even is leaving the state unchanged which they are acting on.

These r-numbers are their “measuring values”, and their generators are called “diagonal”. (This denotation offers if such a generator is written down as some matrix acting on a state vector.) Another word for these “measuring values” is “quantum numbers”.

Technically, such a (linear) “diagonal generator” just is counting the number of certain types of quanta the physical (product) state is consisting of, and the different (linear) quantum numbers of one and the same state are distinguishing themselves by the types of quanta they are adding up with a positive sign and those to be added up with a negative sign.

Hence, up to some arbitrary unit factor like “1 cm times 1 gram” or similar constructs arising from some arbitrary normalization, these “quantum numbers” should be multiples of some integer numbers. (In the case of a “spin” value, that normalization, usually, is arbitrarily separating another factor 1/2 off that value, and, in the case of a particle number or of a charge, people often like to split off a factor 1/3, instead.)

As a result, the statement is: Up to (arbitrarily chosen) normalization factors and up to relative signs, there are 4 “diagonal” linear quantum numbers:

 

 

They are counting the 4 types of quanta a physical state is consisting of in the framework of 4-dimensional Quantum Gravity.

 

 

While we have the

Impressum

Verlag: BookRix GmbH & Co. KG

Übersetzung: This is the English version of the German original "ToE; Neue Physik; Unsere Welt, erklärt durch die Quantengravitation; Weltweit 1. Lehrbuch zur QG" released the same day by the same publisher.
Tag der Veröffentlichung: 02.01.2016
ISBN: 978-3-7396-3010-6

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