Comments on Loop Quantum Gravity by Carlo Rovelli Page 5

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In Complex QM I had envisaged black holes and white holes working to balance each other.

I conjecture that if this network is combined with the mathematics of Organisational Methods then the most economic pathways can be found and perhaps the whole system made to conform to desirable parameters.

In Complex QM the white holes tend to return matter from the edges of the universe to its centre and if we could examine a large enough Spin Network we may be able to prove or disprove this.

Article Statement >

As long as we stay within the classical regime, rather than the quantum one, the gravitational field defines a 4D continuum. We can therefore still think of the field as a sort of space–time, albeit one that bends, oscillates and obeys field equations. However, once we bring quantum mechanics into the picture this continuum breaks down. Quantum fields have a granular structure – the electromagnetic field, for example, consists of photons – and they undergo probabilistic fluctuations.

Alex Comment >

This difficulty is removed with Complex QM as the Heisenberg Uncertainty Principle

is invalidated. Probabilities become movements or displacements into complex space.

These fields can not only be thought of as describing space-time but space- complex displacement.

Article Statement >

The conventional mathematical formalism of quantum field theory relies very much on the existence of background space. There are therefore two possible strategies that we can adopt to construct a quantum theory of gravity. One is to undo Einstein’s discovery and to reintroduce a fictitious back-ground space. This can be done by separating the gravitational field into the sum of two components: one component is regarded as a background, while the other is treated as the quantum field. We are then left with a background space that is available for all our calculations, after which we can hope to recover background independence. This is the strategy

adopted by those who do not regard the general-relativistic revolution as fundamental, but as a sort of accident. And this is the strategy adopted in string theory.

Alex Comment >

This concept background space (is the one of the two options I prefer and) can be replaced with complex space except that we do not intend to discard the results from it at the end of the calculations.

Article Statement >

The key input that made the theory work was an old idea from particle physics: the natural variables for describing a Yang–Mills field theory are precisely Faraday’s “lines of force”. A Faraday line can be viewed as an elementary quantum excitation of the field, and in the absence of charges these lines must close on themselves to form loops. Loop quantum gravity is the mathematical description of the quantum gravitational field in terms of these loops. That is, the loops are quantum excitations of the Faraday lines of force of the gravitational field. In low-energy approximations of the theory, these loops appear as gravitons – the fundamental particles that carry the gravitational force.

This is much the same way that phonons appear in solid-state physics. In other words, gravitons are not in the fundamental theory – as one might expect when trying to formulate a theory of quantum gravity – but they describe collective behaviour at large scales.

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Alex Comment >

Why must the fields close in on themselves? I thought it precisely because they had charge and thus polarity they should close. In a Complex Horse Shoe String the uneven cylindrical forces create the flexing and as you balance the cylindrical forces the string closes into a circle or loop.

I believe that Real Horse Shoe Strings can wrap around a real cylinder in much the same way but perhaps need a spherical space to exist upon. So in either case I believe the situation is more complicated than suggested by Loop QG. This is not a natural focus for Loop QG that is more mathematical in its approach. I suppose from a purely mathematical point of view I would agree as it seems to comply with Cauchy’s Theorem. The difficulty I foresee is explaining this within physics.

Article Statement >

The second strategy uses Feynman’s version of quantum field theory, in which the behaviour of a quantum particle can be calculated by summing all the possible classical paths of the particle. Misner suggested that calculations in quantum gravity could be performed by summing over all possible space–times – an idea that was later developed by theorists that included

Steven Hawking at Cambridge University and Jim Hartle at the University of California in Santa Barbara.

Alex Comment >

This is the main drawback with Loop QG at present. The loops have no coherent framework.

By using complex space as a differential plane I provide backbone and a frame to hang Loop QG upon.

 

Alex Comment >

While the diagram shows a five fold branching and not three or four fold branching the elevation does appear to represent a ztar pattern. So I would suggest that the mathematics of Loop QG could replicate the situation in Complex QM. The structures used for Loop QG are also polygonal as I would expect but this does conform to my hyper-geometry (at least as shown here).

Loops on Loops

Alex Comment >

This concept appears a weakness in Loop QG to me and a stronger case seems to be for Complex Loops on Real Loops and vice versa. The ordering of the loops seems to be a persistent problem. In what circumstances would the loops ever act in conjunction? If the answer is never than how can you have a geometry that is truly random and yet have an ordered universe?

Article Statement >

Smolin and I teamed up with Ashtekar to try and understand the physical meaning of the nets of loops that had emerged from the equations. Through various steps we slowly realized that the loops did not describe infinitesimal elements of space as we had first thought, but rather finite elements of space.

Alex Comment >

This was my main objective in rewriting complex mathematics. The concepts of calculus and complex mathematics have not been radically changed since their conception and I saw the need to do so to remove such confusions.

 

Article Statement >

The idea that there cannot be arbitrary small spatial regions can be understood from simple considerations of quantum mechanics and classical general relativity. The uncertainty principle states that in order to observe a small region of space–time we need to concentrate a large amount of energy and momentum. However, general relativity implies that if we concentrate too much energy and momentum in a small region, that region will collapse into a black hole and disappear. Putting in the numbers, we find that the minimum size of such a region is of the order of the Planck length – about 1.6 ×10 –35 m. that had already been studied.

Alex Comment >

I had estimated the size of the differential particle to be approximately between 10^–35 m or 10^–40 m so it is nice to have a formal figure. I agree with most of this comment except that: can we sure about the 'disappearance'? I suggest a wormhole is a better term than a black hole.

I suppose it is hard to explain if there was a single black hole at the time just after the Big Bang then why has not all space and time been consumed?

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Article Statement >

It was not until about 1994 that Smolin and I really understood what we had stumbled upon, thanks to a calculation that is routinely performed in quantum theory. By quantizing a theory, certain physical quantities take only discrete values, such as the energy levels in the hydrogen atom. Computing these quantized values involves solving the eigenvalue problem for the “operator” that represents a particular physical quantity. We studied the volume of a region of space – or a certain number of loops – which in general relativity is determined by the gravitational field. By solving the eigenvalue problem of the volume operator, we found that the eigenvalues were discrete – that is, there are elementary quanta of volume, or elementary “grains of space”. Furthermore, these quanta of space resided precisely at the nodes of the nets.

Alex Comment >

I cannot be certain of the scales being used here but the possibility of space (mathematically at least) becoming quantised may be due to the limit for the diameter of a differential particle. The centre of any ztar at a smaller scale would lie precisely at the nodes of the nets.

These nets thus provide a mathematical framework for tying two spatial geometries together and that need not necessarily involve complex space, although it does. Spin Networks endorse why my hyper geometry, using polygons, can work effectively.

Article Statement >

Spin Foam

In loop gravity, space is replaced by a spin network and space–time is therefore described by a history of spin networks. This history of spin networks is called “spin foam”, and it has a simple geometrical structure. The history of a point is a line, and the history of a line is a surface. Spin foam is therefore formed by surfaces called faces, which are the histories of the links of the spin network, and lines called edges, which are the histories of the nodes of the spin network.

 

Alex Comment >

I see Spin Foam as a purely mathematical device. Spin Foam could be adapted, perhaps, to describe the history of a particle moving, say, from 4D into 3D. To do this the Spin Foam would not to be projected around a cylinder but a zunnel. I suppose a zylinder or complex cylinder could describe a similar situation in complex space.

The Convergence of Quantum Theories

There are similarities between String Theory and Loop Quantum Gravity (LQG). First of all the obvious fact that both theories start with the idea that the relevant excitations at the Planck scale are one dimensional objects (call them loops or strings).

In Complex QM these strings can be complex and real. They also possess the characteristic or tendency of winding as a helix. I suggest that when these strings or loops attach themselves to a spherical nucleus they probably unwind and it is this unwinding that creates a surface binding force.

The random interlinking of LQG loops is now doubtful. In fact I consider that Complex QM will have a major effect on LQG.

If the helix is considered straight then it is open but perfectly balanced. When it is horse shoe shaped I agree with Stephen Hawking that it would emit a force at its ends. I expect that this is the surface binding force that tends to make the string close around a circle and form a doughnut torus.

The doughnut torus is now compressive around the inner circle and tensile around the outer circle. So another doughnut torus may not be able to fit around the outer circle but it could act

perpendicular to the first. In this way the three main axes can be constructed for a sphere.

If we allow for these torii to cross when their helix cross then a system of loops can be created at intervals equal to the wavelengths for the helix and the sphere is mapped as an Orbitsphere. This model seems to compliment the one I have already constructed for Dr Mills Orbitsphere.

By this means I am showing that QLG loops are not random as a rule but tend to coalesce around a cylinder, torus, or sphere. In this way Classical Quantum Mechanics, Complex Quantum Mechanics, String Theory, and Loop Quantum Gravity are all converging.

Energy Mass Conversion in Strings

The string changes from a linear state and wraps around a cylinder because the string is not composed of mass nor energy but fluctuates between the two states. The process controlling this is basically spherical in real space and probably inversely spherical in complex space.

When the string forms a closed loop then inner circle side can be considered as having the equivalent of a wave with a shorter wavelength than the outer side. It is questionable whether the equivalent wave on the inner circle can also increase its amplitude.

This seems to be an energy to mass system that is operating inversely to what I have previously described for the photon. In the photon the energy to mass system expands and contracts spherically. This creates a complicated picture for this process that will require more study to determine the spread of this process. The inner and outer wavelengths seem fixed restricting the energy to mass process but this should only be the case if the overall system is balanced.

If we add or subtract some energy there is an inbuilt energy gradient that will direct this process, say, in the photon.

I also consider it very likely that these strings can be composed from many smaller strings as long as they share the same wavelength and amplitude. The part that Brownian Motion plays to allow for this mechanism to adjust itself helps to complete a comprehensive mechanism that is translatable between real and complex space.

The only missing situation for these strings is if a smaller closed string comes into proximity with a string fixed around a nucleus. If it does not match the wavelength of the outer side of the fixed string then it may break and be added to the nucleus to upset the energy-mass balance and create a movement of energy to mass in the nucleus however small.

The only reason for the neighbouring string to break seems to be if it aligns itself adjacent to the centre line of the torus and tries to spread around its smallest circumference. This string is not just subject to these forces in one dimension but two and the forces are tensile. At the very least, then, this smaller string must try and deform into an ellipse.

Conclusion

Without revising Loop QM to have some geometric framework it will not be possible to delve into smaller and smaller lattices of Spin Foam. Thus an orderly result will remain elusive until Complex QM or some similar theory is developed.

My approach to gravity in Complex QM is still being formulated. I have some suggestions as how gravity operates but it is difficult to deal with gravity as it is nearly always abstract. The investigation of gravity requires from a perspective offered by in Complex QM a more daunting task. It is necessary to examine how all the new multi–dimensional forces, complex forces, and fractal forces may play their part. A moment’s reflection will show that this is necessarily the case as gravity can be considered as operating in all these domains.

I have made headway with gravity without using Complex QM and I am therefore cautious in making any suggestions that may be unintentionally misleading in the long term. I do however believe that Complex QM can help to see and understand gravity more clearly.

Complex QM has not been seriously challenged since first announcing it in1999.

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