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This is the English version of a research report in German language on the common foundations of particle physics and cosmology, together with their unification to a ”Quantum Gravity“ and to a „Grand Unification Theory (GUT)“.

In parts, this is an extract – here and there: word for word – out of a more comprehensive report (‘Die “Weltformel”’/‘The “World Formula“’) of the same author, see in the internet under

www.q-grav.com -> Summary View.

In other 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 representation. 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.

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), see

ISBN 978-3-00-030847-6.

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. Their (”chiral”) interactions seem to follow comparable (“gauge”) structures, too – although, until to-day, not well understood ones by the “Standard Model” with respect to their origins.

And quantum theory 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, everything must stay finite in physics. Even an elementary particle cannot be accelerated boundlessly, 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 are “unphysical”, too. Thus, they necessarily will have to be “discreticised”, or “quantised”, 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 (gravitational) force. Thus, Special Relativity is cancelling forces.

On the other hand, the current theories of elementary particles – i.e., the “quantum field theories” – are working with Special Relativity. 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 consistently combined Planck’s quantum theory with Einstein’s General Relativity. 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. (They are comparing “apples with pears”.)

This (false) conclusion, however, 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.

After the detection of nuclear forces, Einstein’s notion of a “World Formula” 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” 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 “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. And “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 nature to follow theory, still might keep them busy for a couple of hundreds of years more to come.

By the way sketched in the previous chapter, it can be shown that physical models of nature must basicly 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 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.

Quantum Gravity

From statistics – together with number theory – we derive that nature should manifest itself in terms of powers of eight dimensions. Measurability adds the condition that those powers should be even numbers. 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.

The first factor of 8, an octet of eight dimensions, had been identified to provide Dirac’s four “covariant” plus his four “contravariant” dimensions; our 4-dimensionality of space and time derives from that. Thus, the 4-dimensionality of space-time (and of energy-momentum) is an output of theory. For all other models – Einstein’s General Relativity included – it still is an external input feature!

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.

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 the physicists even before Einstein and Planck already had been well familiar with.

Modern models – like that of “Loop Quantum Theory” et al., 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 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 and quantitatively.

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 (squared) length. This length reveals to be the radius of the Big Bang region. (In Quantum Gravity, the “Big Bang” is extended, it is no point singularity any more.) 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.

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