Category Archives: Quantum Mechanics

A Brief History of Quantum Gravity

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459px-Einstein_patentofficeProblems cannot be solved by thinking the way we thought when we created them. – Albert Einstein

The incompatibility of quantum mechanics and gravity is seen as a big issue by all physicists. The Planck length

contains the gravitational constant G and Planck’s quantum h, and therefore it is the scale at which “quantum effects of gravity” are supposed to become important. But that is all what physicists have found out about quantum gravity. No theory exists, let alone any evidence of an observable effect. You’ll find the topic somewhat more elaborated in Hawking’s book A Briefer History of Time. The chapter about quantum gravity is comprised of 21 pages, of which almost 20 pages are devoted to repeating gravitation and quantum theory. Andrzej Staruszkiewicz, editor of a renowned physics journal, commented on this topic:

It is tempting to assume that when so many people write about “quantum gravity”, they must know what they are writing about. Nevertheless, everyone will agree that there is not a single physical phenomenon whose explanation would call for “quantum gravity”.

The fact that no theory for quantum gravity exists does not preclude the existence of numerous experts of quantum gravity. According to the science historian Federico di Trocchio, such a „second category of experts“ consists of those whose knowledge will become immediately obsolete once the riddles scientists are studying have been understood. They make their living on the problems that are being tackled unsuccessfully, or, like some string theorists, make ridicuous claims of having explained „the existence of gravity“.

There is simply no theory that combines general relativity with quantum theory. All theoretical recipes cooked up until now have failed, as for instance the so-called ADM formalism, a reformulation of Einstein’s equations of general relativity. It has gotten nowhere, but it is nevertheless considered a bible leading the way. Another great couturier of theories is Abhay Ashtekar, who regularly summarizes the accomplishments of the Loop Quantum Gravity he fathered. His résumé:“it is interesting”.Actually, it is interesting for 30 years now. Not exactly news anymore. But I fear that we will have a long wait for such a Theory of Everything that eventually unifies quantum gravity and general relativity. Success just comes in different flavors.

Theoretical physicists seem to have deployed a number of assumptions that block deeper reflection. For instance, there is the belief in an unalterable gravitational constant G. Solely for this reason, all theoretical attempts trudge through the Planck length’s eye of a needle and come across as trying to push a door which is labeled “pull”.

The essence of the problem – and I think the only thing worth worrying about – is just this: The ratio of the electric and gravitational force of a proton and an electron that form a hydrogen atom is a huge number and nobody knows where it comes from. Period. The only idea with respect to this riddle came from Paul Dirac, the Large Number Hypothesis. – more about that later. Where does that number, 1039 , come from? Beware wannabe unifiers: either you explain it or you had better shut up.

(with quotes from „Bankrupting Physics“ and „The Higgs Fake“)

Why did Nature Invent Spin?

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I think this issue receives too little attention. Usually, it is said that spin is a consequence of the Dirac equation and thus, something that follows necessarily from relativity and quantum mechanics. Let us have a brief look at the argument. Schrödinger’s non-relativistic equation is .

The momentum operator p = m v is and thus, the term on the left-hand side of Schrödinger’s equation is derived simply from the kinetic energy  ½ mv2. It is interesting that none of the successful predictions of Schrödinger’s equation for the hydrogen atom make specific reference to the nature of the electron (for which the wave function gives a probability that it will be found in a certain state). They refer only to the kinetic energy; irrespective of the type of wave-natured particle that orbits the nucleus (in fact, it also works for muonic atoms).

Dirac used the correct special relativistic term for energy E = and replaced Schrödinger’s term. However, there was no explicit justification for switching from the kinetic energy to the total energy of the particle. This conceptual problem was overshadowed somehow by the mathematical problem arising from the Delta operator in the square root, to which Dirac found an ingenious solution using the matrices named after him. The algebra of Dirac matrices transpired to be a description of spin. In the following, the opinion spread that spin was a consequence of putting relativistic energy into the basic equations of quantum mechanics.

However, the initial problem of the missing equivalence of kinetic and total energy persisted. Dirac was also disappointed that he could not deduce any concrete properties for the electron from his equation. The retrospective narrative is that the positron, undiscovered in 1928, was a ‘prediction’ of Dirac’s theory, but Dirac had rather sought to explain the huge mass relation of the proton and electron, which is 1836.15. In fact, in his latter days, Dirac distanced himself a little from his earlier findings and according to his biographer Helge Kragh, he was “disposed to give up everything for what he had become famous”.

Let us adopt another perspective regarding the nature of spin, one which is related to the properties of three-dimensional space; the world we perceive (those who perceive more dimensions should see a doctor). The group of rotations SO(3) obviously must have some significance, but its topology is a little intricate. It lacks a property called ‘simple connectedness’ because the paths in SO(3) may not be contracted. Objects connected to a fixed point with a ribbon must twist 720 degrees, not just 360 degrees, in order to perform a full rotation that leaves the ribbon untwisted (see the visualization here). It seems that nature has a predilection for the generalized rotations called SU(2), which are simpler mathematically and have a surprising feature; they represent precisely the electron’s spin – you need to perform a double twist of 720 degrees, rather than just 360 degrees, to get into the original position. However, there are no Dirac matrices and thus, I think there is an open problem. It seems that the properties of three-dimensional space alone are sufficient to cause spin to emerge – no relativity or quantum mechanics are needed. To put it another way, a direct understanding of quantum mechanics from the geometrical properties of space, if there is one, is still missing.