1 Nature and Physics
Nature is what man and his senses recognise and derive from it. The subjective sensory impressions (“perceptions“) of an individual are called observer positions. Repetitions of such sense-impressions together with derivations are called experiences.
Now, humans are part of nature, and other living beings have other senses and other experiences. It is the task of physics to collect reproducible experiences, to relate them to each other and to eliminate the specifics typical of individual beings from them as far as possible.
Errors and short cuts of the past demand a mathematically clean implementation of measurability and reproducibility. Thus, the reduction of classical transformations to their generators inevitably leads to the absolute conservation of a finite number of invariant quanta. Consistency and completeness, guaranteed by Young tableaus free of symmetry breaks, categorically exclude singularities. The law of great numbers opens up the macrocosm and remedies the classical problems of the measuring process.
The addition of antifermions to the Dirac algebra for fermions gives rise to the U(4,4) of quantum gravity (QG) with event horizon and Big Bang scenario. According to the TCP theorem, the physics inside the black hole is equivalent to ours (with the time arrow reversed). The spin-like rotation around the energy axis drives an endless counter-rotating cycle between CMS time (sin) and heavy mass (cos), which constantly swap roles over eons of years.
1.1 Perception and Logic
A sense of time is widespread – at least for reaction times. The sense of touch, essential for survival, registers positions (relative locations) and (tactile) impulses of structures in the immediate vicinity. Animals usually also possess an optical sense for the remote detection of changes in such structures between two registrations at intervals of a specific reaction time. While reptiles often only perceive changes in the position of structures, the optical sense will be used more generally also to locate them at a distance.
The change of a structure between 2 registrations in the distance of its reaction time helps a living being capable of this to identify objects (pattern recognition). By interpolation, it determines their status of motion (speed and direction). By extrapolation, a transitive logic also leads to foresight, which is useful for its hunting behaviour.
Interpolations and extrapolations are not perceptions, but fictitious additions to them. Due to the existence of reaction times, we should not include motion in the category of perceptions; they are quantities artificially derived from discrete individual perceptions.
Human physics constructs the concept of a mass from momentum and velocity with the help of logic (division). Experience teaches us that the momentum and mass of two objects behave "linearly" (= additively) – but this does not apply to velocity.
The same duality as between linear momentum and non-linear velocity is also provided by physics for the concepts of a (linear) centre-of-mass location and a (non-linear) location: Their quotient again gives rise to (heavy) mass.
The primary terms "velocity" and "location" from the optical sense, thus, turn out to be derived secondary terms from "momentum" and "CMS-location", respectively, each generated by division by their (heavy) mass. Mathematics calls this type of representation by quotient formation a "ray representation" (chapter 4.1). People perceive nature in its ray representation.
1.2 Physics Developing
Classical physics, generally, treats interpolations and extrapolations continuously, as if they were actually measured facts. New physics, on the other hand, works discretely: measuring points m are assigned to a "generator" G as an acting (measuring) operator, which delivers this measured value as its "eigenvalue" only by being applied to a selected physical "(eigen)state" z:
Classically, the discrete spectrum of this finite number of measuring points m is ignored: Instead of arguing with the individual, discrete measuring points m, one classically argues continuously by interpolating their product with (minus the imaginary unit i and) a respective arbitrarily chosen "unit" a (= cm, sec, angle, km, kilo or whatever) to the exponent in a continuous "transformation" T, for which the i only stands as an abstract placeholder for a direction implicitly defined by G. The discrete spectrum of the measuring points m then is used as the basis for the argument:
T again is to be applied to a physical state z (or to a superposition of such). For a rotation T by the angle supplied by applying aG to z, cos projects to the direction of z, sin to the direction perpendicular to it into which the rotation is made. Translation by the distance aG perpendicular to the radial direction given by z interprets the translation as a rotation on the surface of a sphere with a sufficiently large radius |z|, such that its curvature cannot be felt).
This construction makes it possible to transform the successive execution (i.e., the product) of two transformations into the (mathematically simpler) sum of exponents. But the relative position of the discrete, measured values m shifts differently against each other when they are taken to be the exponent of an exponential function (an exponential function is not a straight line!), so that a transformation of overlapping states z to which it is applied normally reshuffles them, i.e. genuinely changes them:
Sometimes it is difficult or even impossible to tell neighbouring eigenvalues m and m+1 apart experimentally. This classical method, hence, may still be quite useful as long as these values are sufficiently close to each other, such that they are subject to the
This applies to the eigenvalues of the spatial operator, for example, whose neighbouring measured values differ only by the tiny fundamental length from gravitation theory. But in the case of spin, already, the experiment shows that measured values m are permissible only as integer or half-integer multiples of Planck's quantum of action (a = h bar); intermediate values do not exist in nature!
A lot of mathematics could be done on this [2]. Mathematically, a "state" is a vector, also known as a "spinor". The number of its vector components is its dimension. Mathematically speaking, a measuring operator G or a transformation T, then, is a square matrix to be applied to this spinor.
From a pair of states – say z” and z – mathematics also can construct a number w, which it calls a scalar product:
(If the order of the two factors is reversed, the scalar product changes to its conjugate-complex value, represented by the superscript asterisk). A scalar product also is called a "probability amplitude"; for, its absolute value |w| squared denotes the (conditional) probability of simultaneously encountering the state z" when a state z is present. The squared absolute value is a measure of the intersection of both:
Now, a transformation U is called "unitary" if it leaves every scalar product (z arbitrary) invariant:
(The superscript + indicates that the preceding matrix U is to be taken conjugate-complex (*) and transposed at the same time). If all matrix elements of U are real, then the matrix U is called "orthogonal" and it is a rotation of the spinor z. Unitary transformations, hence, are complex-extended rotations.
Unitary matrices play a very special role in physics, because, when they are applied to any 2 spinors, their mutual probability to each other always remains unchanged (probability conservation). This is the mathematical form of a fundamental principle of physics, which by definition applies within every thermodynamically "closed" system (speaking: in the "reaction channel"):
Thermodynamically, "closed" reactions, therefore, work strictly unitary. Thus, entanglement is an effect of the reaction channel, for example. As a counterpart, the dynamic channel (see chapter 1.3) describes thermodynamically open systems. Both channels are compatible with each other, but not commensurable (see chapter 2.1.4).
Strictly speaking, probability conservation is not only guaranteed for unitary transformations. Complex- or otherwise-conjugate transformations do likewise, provided that w only multiplies with its conjugate value w* to +1; the superscript asterisk, then, would more generally denote this "otherwise" conjugation:
An example is a "pseudo-unitary" transformation with
Another example is an "octonic" conjugation in the number field of 8-dimensional "octonions". The “internal” interactions could be interpreted as such an 8-fold degeneracy of dynamics (chapter 3).
Classical physics often demands unitarity and pseudo-unitarity at the same time – without investigating why. It then uses infinite-dimensionality to circumvent this contradiction. Or it dispenses with the conservation of probability, altogether.
1.3 Field Theoretical Models
By mathematics, "field theories" are applications of continuous functional analysis (transformations T) to areas that would be better treated discretely by group theory (generators G). That lack of precision, that application of unsuitable methods, via series of erroneous interpretations, has led to a century of stagnation in the basic theories of elementary particles and the cosmos, while, at the same time, experimental findings have reached peak performance. Let's look at this in detail.
For the multiplication of real as well as complex numbers a and b, the commutative law applies without restriction, i.e. their "commutator" [a,b] disappears:
This is no longer true for matrices A and B – the result C can be non-zero (the imaginary factor i is convention):
As said already, the operator part G of a transformation T = exp[–iG] is called a generator in mathematics. Conversely, transformations are "generated" by generators. All generators of a group of transformations called a "Lie algebra" form a set that is self-contained with respect to commutator formation. Classical physics works continuously with transformations T, New Physics discretely with their generators G. A finite number of measurements, however, only allows statements about discrete values; continuous additions are purely arbitrary.
If in T = exp[ –iaG] all units a always are r-numbers, mathematicians speak of a "real Lie algebra", in the case of c-numbers a of a "complex Lie algebra". In the case that all a are r-numbers, the difference between the two types of channels lies in the property of their Lie algebras:
The complex Lie algebra of dynamics arises from the real Lie algebra of the reaction channel by doubling the number of its real parameters a to pairs (b,d), which can also be combined in a complex way:
For imaginary a = id, the above transformation T = T(a) turns to:
Thus, the trigonometric functions cos and sin merge into the hyperbolic functions cosh and sinh, which graphically look completely different.
In the transition from a real Lie algebra to its corresponding complex Lie algebra, not only the number of its generators doubles – G and G'=iG are mathematically independent generators – but also the spinor dimension of the states to which the generators are to be applied.
According to chapter 2, the reaction channel describes statics and the dynamic channel dynamics. Hence, we need a complex Lie algebra.
If we now refer to the independent directions (= Cartesian components) of a spinor – somewhat misleadingly – as "dimensions" as well, we distinguish these dimensions according to whether they belong to the original reaction channel (a or b) or to its complex extension (d). The dimensions are then called:
For the complex Lie algebra of dynamics, the number of space-like and time-like dimensions is equal; they exist in pairs. Unitary transformations link space-like with space-like and time-like with time-like dimensions trigonometrically (cos, sin); for the hyperbolic linking (cosh, sinh) of space-like with time-like dimensions, on the other hand, the non-unitary extensions are needed:
The transformations of the reaction channel are unitary, those of the dynamic channel are called "pseudo-unitary". More generally, mathematics abbreviates a Lie algebra or a group of transformations with n time-like and m space-like dimensions, as well, by
(Dimensions n=0 or m=0 are omitted.)
Among the transformations of type U or O, there also is the unit matrix. If we delete the unit matrix from the set of all generators (mathematically this means: all remaining generators have the "determinant" =+1 in matrix form), then all the above designations are still preceded by an S = "special":
Standard physics often projects onto subsystems of lower dimension. Corresponding systems, thus, automatically are incomplete. Einstein's 4-dimensional Special Relativity, e.g., transforms according to an SO(1,3). Via the "Dirac algebra" of his 16 gamma matrices, Dirac embedded Einstein's incomplete system in a U(2,2) (chapter 2.1.1), in which the number of space-like and time-like dimensions, now, is identical again (=2). With this extension, Feynman celebrated a grandiose success in 1949 (Feynman graphs, QED = quantum electrodynamics).
1.4 Quanta and the Cosmos
Unfortunately, Feynman's model of the so-called "2nd quantisation" contains two gross mathematical mistakes concerning its commutators:
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He does not distinguish Schrödinger's quantum mechanical picture from that of Heisenberg.
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For fermions, he uses the plus instead of the minus commutator ("2nd quantisation").
We shall return to point 1 at the end of chapter 2.1.1. To point 2: The two commutator types differ by a sign:
This made Feynman's model mathematically inconsistent despite its huge success in the press. This became noticeable in terms of singularities. According to the motto "If you do not accept it, I shall apply force", those infinities are artificially fixed through "renormalisations" by applying "infinity minus infinity = finite". But curing symptoms does not solve problems!
However, the physical background to this genuine error had been Feynman's misunderstanding of the Pauli principle, which states: "2 identical fermions must not coincide in all their quantum numbers." Feynman "simplified" this to "2 identical fermions must be antisymmetric to each other." Feynman's form, thus, is sufficient for the Pauli principle – but it is not necessary! (For, Pauli’s principle follows from the shell model (chapter 3.2.3), and not from antisymmetry.
Such a confusion between necessary and sufficient should become symptomatic for the decades after the world wars for the purpose of "simplifying" fundamental physics (of quantum field theories). Mathematically, the plus-commutator is irrelevant! Lie algebras categorically require the minus commutator. The correction of this mistake would easily be possible. But nobody cares about it.
Dirac's pseudo-unitary SU(2,2) is equivalent ("locally isomorphic") to the pseudo-orthogonal SO(2,4), which had already become known under the name "conformal group" between the two world wars. Einstein's SO(1,3) of special relativity is merely a subset of it (chapter 2.1.1):
Now, we had mentioned the ray representation in chapter 1.2; after that, via the ray representation, man uses division in his observation of nature. But r-numbers are 1-dimensional, c-numbers 2-dimensional. According to number theory (keyword "octonions"), however, ("irreducible") numbers supporting division may be at most 8-dimensional.
If we consider Dirac's spinors as such "numbers", then the Dirac formalism is based on a fundamental spinor pair of 4 dimensions each: a fundamental fermion and its corresponding antifermion. Together this makes 4+4 = 8 dimensions – with 2+2 of them time-like and another 2+2 space-like. By rearranging the order of its dimensions, physics as a whole can be interpreted as if it were actually the 8-dimensional extension of the U(4) of a real Lie algebra to the U(4,4) of a complex Lie algebra.
Now, particle physicists tend to bundle their spinors into tensors (= multiple vectors). Vector labels bundle into tensor labels accordingly. The application of a matrix to a vector or tensor merely jumbles up its labelled components without, however, taking their contents into account or even changing them.
As long as we do not multiply components (or divide them by each other), only their labels remain relevant. Octonions determine that exactly 8 types of them exist, but their value multiplication rules as components of a tensor, vector, or spinor remain irrelevant.
If we look at nature purely to be some tensor representation, the values of its components remain black boxes for us. We only know that its (normalised) basic representation q occurs in 8 variants and can be linearly combined from 8 types of special, normalised q1 to q8 – contents unknown. We call these q "quanta", their 8 fixed types q1 to q8 "quantum types".
(The string-brane models are not tensor models, by the way, because they also use the multiplication rules of the values of their components).
Now Dirac's 4-spinors do not belong to a unitary U(4) but to a pseudo-unitary U(2,2). The classical field theories, as said, re-arrange the 4+4 = 8 spinor components of the above U(4,4) of dynamics, consisting of 4 components to a time-like U(4,0) plus 4 further components to a space-like U(0,4), by coupling, instead, 2 time-like and 2 space-like dimensions each to a 4-dimensional Dirac spinor of the U(2,2) of a fermion and to the U(2,2) of an antifermion, respectively:
If we assume the 4+4 components of Dirac's basic fermion and antifermion, respectively, in terms of their 8 normalised quantum types q1 to q8 for the states z and z', the 8x8 generators (in their representation as a "covariant" row component |qr> of a “ket vector” times a "contravariant" column component <q’s| of a “bra vector”) are as follows
For the 8x8 matrices of dynamics to be applied to it, the re-arrangement from 4+4 to (2+2)+(2+2) components of an 8-spinor (a1, … , a8) means a splitting into 4 quadrants of Dirac's 4x4 gamma matrices in the 4 corners of the bold black cross, where the (a,b) are partially exchanged for the (d,c):
In modified order, the conversion scheme of the 8 dynamic spinor components a1 to a8 of the U(4,4) reads as follows:
The one caesura that separates the two Dirac spinors (fermion and anti-fermion) from each other in the 8-spinor provides 2 crossing caesuras (bold-black) for the 8x8 matrices, which now separate 2x2 = 4 blocks of Dirac's gamma matrices from each other. One of the caesuras is called "event horizon" and separates a black hole beyond our area from ours in the cosmos. The other caesura separates our area after the "Big Bang" from the area "before the Big Bang".
The quadrants #1 (top left) and #3 (bottom right) each form a U(2,2). However, the other two quadrants #2 (top right) and #4 (bottom left) must first be added or subtracted from each other to form the other two U(2,2). But all those 4 U(2,2) are commuting with each other. In order to get from one U(2,2) to one of the other U(2,2) across their boundaries, hence, it is not enough to apply some U(2,2) generator, but we have to apply suitable generators of one of the transgressing U(4) groups: For crossing the Big Bang and the event horizon, we also need the (b,d) and (a,c) generators of a U(4) (like the +iQ0 and –iM0 of chapter 2.1.1 belonging to the orange or grey ranges: iQ0 for passing the Big Bang and – iM0 for passing the event horizon).
Chapter 1.2 had posed the problem that a translation according to the representation method T = exp[-iaG] of classical particle physics should actually be a rotation, i.e., a path on the surface of a sphere. If we measure the generators A, B, and C in the commutator [A,B] = –iC in units n (i.e. G = nG'), then the commutator changes into
According to this "group contraction" from the 1920s, a linear translation can be interpreted (approximately) as a path on the surface of a sphere with a sufficiently large radius n, indeed.
With a scale of human size for G, the units in which G' is to be measured, thus, become vanishingly small! This is an example of how discrete, neighbouring measuring values of G can be brought arbitrarily close together by a suitable choice of scale, such that they can ultimately even become indistinguishable from one another in terms of measurement. (Example: The tiny fundamental length of our 3-dimensional space). Classical physics makes the (unphysical) transition from n to infinity – quantum gravity, however, accepts Planck’s finiteness of physical numbers and omits this limit.
2 Quantum Gravity (QG)
Without the ray representation, i.e., in its additive, "linear" description, fundamental physics becomes simpler. We recognised its basic principles, already:
From the conservation of probability in the reaction channel it follows that products of a particle reaction – even if they are originally generated via the dynamic channel – must always be multiplied with each other as if all factors had come about unitarily: The reaction channel characterises static processes (change of the observer's point of view) by r-numbers, the dynamic channel compares them as motion (dynamics) by c-numbers (chapter 2.2.9).
This clear separation in the description of thermodynamically "closed" vs. "open" systems guarantees the physical coexistence of (static) entanglement and (dynamic) causality despite their mutual incommensurability.
Finiteness (point 1a) requires that all pseudo-unitary representations must remain finite-dimensional, too. (Since classical particle physics does not distinguish between the two channels, it must either choose its pseudo-unitary representations to be infinite-dimensional or violate probability conservation. This circumstance also leads to Feynman's singularities.)
If we now add the ray representation, we still obtain:
(The 4th space dimension is subject to group contraction.) The above U(4,4) forms the basis of quantum gravity (QG).
After these general discussions, let us now turn to their parameterisation [3].
2.1 Dynamics
The 8 dimensions of quantum gravity, U(4,4), can be understood as a 3-fold nesting of 2-spinors (8 = 2**3):
The spinor shape of an 8-quantum, thus, can be expressed with the help of 3 labels with 2 components (up and down) each:
Impressum
Verlag: BookRix GmbH & Co. KG
Texte: © 2021. All rights reserved.
Übersetzung: Free translation after www.DeepL.com/Translator (free version) from the German original "Quanten, Zyklus der Zeit, Galaktische Mauern, Durch den Ereignishorizont – Quantengravitation" (2021).
Tag der Veröffentlichung: 11.09.2021
ISBN: 978-3-7487-9408-0
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