[Dubbelosix (OP)]

It was noted from my friend Matti, that the Einstein tensor when written as is symmetric - I tried to argue a proportionality of terms from the tensor - but we must keep in mind what Matti has said because it is important to remember.

The commutation relationship is (in a usual convention)



Here we have explicitly wrote out the connections as having commutative properties satisfying our desired inequality. Writing the whole commutation out to find the christoffel symbols, (using differential notation) reveals the following and using general indices:







Pulling it out of its differential notation form, what we really have is



It is one that follows the spacetime relationship



An application of the commutator can be found from parallel transport, in which one can obtain the identity





Consider now, the Riemann tensor with torsion. It is basically a commutator acting on some vector field



The full Reimann tensor is



The last two terms in display antisymmetry in the commutators. Again the commutation arises from the derivatives of the connections,



This was the exact identity of the two commutators for derivatives concerned in space and derivatives in time - it's much more simple in the construct than the full Riemann tensor would suggest. The Riemann tensor vanishes, if we are in any coordinate system



Then the Christoffel symbols are zero



and



and so



Clearly, we are looking into cases though in which



In which curvature does not vanish, but has a specific meaning and relationship with spacetime.

So far, what we have learned, it is the last two terms in Riemann tensor is antisymmetric: . We want to deal in spaces, maybe even two dimensional cases that are not locally Euclidean - ie. a vanishing Riemann tensor concerned with a commutation between connections describing the space and time derivatives. Hopefully we have clarified any misunderstanding of the commutators role in this and how they are interpreted as an anti-symmetric part of the Riemann tensor. I will continue the work with new ideas later.

[Dubbelosix (OP)]

The possible non-trivial spacetime uncertainty relationship, predicted by both quantum loop gravity and string theory, can be thought of as an analog of a quantum phase space.

Spacetime non-commutativity is defined by replacing the canonical variables with commutation relationships - it seems also non-trivial that the connections of the gravitation field in question have dimension of and so, it seems very natural to assume maybe gravity will follow the same dynamics on the Planck scale. This application of uncertainty into the equations requires a full interpretation. The preliminary investigation which leads to this idea of some unification between gravity and the quantum structure of spacetime came from an investigation into a quantum interpretation of the Geon particle. This required an interpetation where the geometry could be larger than a specified wavelength imposed by the spacetime uncertainty. These uncertainties in spacetime, which is just a reinterpretation of the usual uncertainty principle between energy and time, can be thought of as corrections on a manifold that are deviating it from the classical world. Using this understanding, nothing seems more natural than to look for such non-trivial spacetime relationships and see how they would (directly) relate to gravity - if they can. This may be a key point of how we may be thinking about it wrongly; as Susskind suggested, though it has been suggested not to be taken as a literal equality, there may already be cases which hint that gravity already has commutation possibilities to describe why these corrections are imposed at a Planck length.

We know what that relationship is, we defined it as taking the form of an antisymmetric tensor



How do we interpret this, without delving into the mathematics too deep this time around?

One interpretation may come from Von Neumann actually suggested that a point in quantum phase space is meaningless because of the uncertainty principle - there may be undertones or preferrably, direct relationships of this with regards to relativity. In a way, general relativity already had concluded from base concepts that points in themselves, are not actually physical. General relativity found a solution by treating not only the interaction of a system but also their worldlines. Notice though, in the Von Neumann universe, this becomes a non-problem, because points in space themselves are simply not physical systems when you appropriately correct them using the commutation.

We can intrepret maybe further, as you converge into the quantum Planck scales, you diverge from the classical theory and must be described by the corrected theory, which does involve a concept of the non-commutation. There are other things we must remain vigilant about such theories: Such as any possible unitary violations that can be understood simply as



After some reading, I found an author https://arxiv.org/pdf/hep-th/0007181v2.pdf who has applied the uncertainty identical to me, but hasn't looked at the theory from the same perspective. They find for the equality term ~



And they identify



Which is a very useful construction to remember. Eventually I might find an author who has investigated it directly as I have.  My idea was based on dimensional grounds only, so we investigated likely only one small corner of this field.

[Dubbelosix (OP)]

An Approach to Spacetime Triangulation as the Benchmark towards Gravitational Unification

It is well known from Pythagoras' theorem that there exists the spacetime inequality ~







Is it possible to apply a spacetime commutator inside of this inequality? Yes I think so! Or at least, this occurred to me.

For a scalar product defind on a vector space the length of vector is determined by



With some invesigation (see references) a spacetime inequality can indeed satisfy the following relationship



Squaring both sides also yields







from which it follows



which is known as the Cauchy Schwartz inequality which can be thought of as a direct interpretation of a spacetime uncertainty. Another important identity whicch further can be identified from the spacetime relationships is



If you have trained your eye on all my previous work into gravity, this looks like the structure of commutators!

In a Hilbert space, you can define new vectors

\sqrt{| < \Delta X_A^2 > < \Delta X_B^2>|} \geq \frac{1}{2} i < \psi |X_A X_B| \psi > +  i < \psi |X_B X_A| \psi > = \frac{1}{2} < \psi |[X_A, X_B]| \psi >

(system here can't deal with the latex ^^^ translate it for yourself if you can in code


The left hand side calculates the deviation of the derivative from the mean of the derivative, at least, that is how it would be interpreted in the approach we took to the quantization of gravity. We will use these solutions as a benchmark into how to treat this commutivity in spacetime from the connections we solved.


ref http://rocs.hu-berlin.de/qm1415/resources/Lecture_Notes_10_11_12.pdf

[Dubbelosix (OP)]

Not sure if all these equations will show, the place seems sensitive to certain latex equations. If it doesn't work, I will take it out of its latex code so you can translate it.


To give a hint in how to do this unification attempt, we have three key equations,

1.



These are the exact Christoffel symbols of the antisymmetric tensor indices .

2.



This was an equation derived by another author, finding the relationship in a different form argued from quantum mechanics. As you will see in key equation 3. the form has similarities to application of a Hilbert space ~

3.

\sqrt{|<\Delta X_A^2>< \Delta X_B^2>|} \geq \frac{1}{2} i(< \psi|X_AX_B|\psi > + <\psi|X_BX_A|\psi>) = \frac{1}{2} <\psi|[X_A,X_B]|\psi>

Again, this is a Hilbert spacetime commutation relationship of operators which has to translate into the gravitational dynamics dictated by key equation 1.

So let's put it altogether, its just like a jigsaw puzzle now. Implemented the Christoffel symbols in approach 1. into approach 2. we get



In the framework of the Hilbert space it becomes - assuming everything has been done correct, takes the appearance of ~


\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq = \frac{1}{2} i(< \psi|\nabla_i\nabla_j|\psi > + <\psi|\nabla_j\nabla_i|\psi>) = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | R_{ij}| \psi > = \frac{1}{2} < \psi |- [\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]| \psi >


without imaginary number on

[Dubbelosix (OP)]

On Possible Quantum Bianchi Identities


Since we used an antisymmetric object identical to the antisymmetric indices of a Reimann tensor, the Riemann tensor will still be symmetric under an interchange of its first two indices with its last two. We have argued that the Riemann tensor where we use a notation to denote the antisymmetric part . In a valid approach, we have also assumed dynamics satisfying:

   

Bianci idenities are to be studied now - they are related to the vanishing of a Reimann tensor in the sense it is related to the vanishing of the covariant derivative -  the quantum Bianchi identity is to assume there is a quantum, non-zero interpretation of the commutation of two the two connections,



In general relativity, this relationship is usually given as zero - its not difficult to understand, why if we are talking about quantum deviation from a classical theory, why a non-zero theory may be important.

We can argue (maybe) that only points are unphysical and lead to the vanishing of the derivatives
, or even only for classical theory, or both. Of course, the statement

''if it is true in one coordinate system it must be true in any coordinate system,''

Is hard to argue with - but there is more to play with here than just coordinates - by assuming a spacetime relationship at quantum scales by satisfying the spacetime noncommutativity - so scale is an important factor here when talking about gravity, and the spacetime uncertainty needs to be a phenomenon regardless of the coordinate of the system! When we think about Von Neumann operators and phase space, we expect a deviation from the classical way of thinking anyway...

The spacetime relationship



is a model of discretized spacetime - leading to a Planckian spacetime dynamic. The non-zero value of the derivatives implies we have a description of gravity at a quantum scale related to fundamentally to the structure of space.  The vanishing of the Riemann tensor is actually related to the same idea connected to the contraction of the Bianchi identities, which are only zero if they permit a symmetric theory of gravity.

If we think about the vanishing of a metric in terms of the vanishing of an action, the action vanishes because it is invariant under general coordinate transformations - that is, general covariance means there is an invariance of the form of physical laws under any arbitrary differentiable coordinate transformations. But I note again, coordinates in a quantum domain to coordinates in the classical domain, may not change the coordinate but definitely the situation. - notably positions are affected by momentum in phase spaces - something classical space time and the classical objects inside of it are so different.

[Dubbelosix (OP)]

The full Riemann equation (in usual standard form) with LHS  showing commutation in indices, we have





The curvature and torsion is given by





When torsion is non-zero, the Riemann tensor becomes a matrix/object describing the torsion. Though torsion remains a possible way to create a non-vanishing theory of the Riemann curvature, I still prefer a new ''look'' on how we view things - that ''new look'' on things was an argument that quantum domains and classical domains yields different understanding of the physics.... though the statement, ''the laws of physics are true in every coordinate frame'', is correct, I don't dispute this, but the idea that this should transcend into phase space with the same value is hypothesis. In fact, we argue there is a non-vanishing structure to spacetime itself!

For particles that do not have radii, (point particles) we can argue something happens in two ways:

1) Relativity implies as R decreases, the curvature increases and as it approaches zero, approaches infinity: Whether or not a point actually implies infinite curvature is for philosophers, for me, it just implies a non-physical situation. A singularity. You can also argue that self-energies become divergent as well.

2) If on the other hand, we are led to believe that electron particles are not actually pointlike, then this avoids infinite curvature and infinite energy and we save a definition of the electron which must have a radius-structure and therefore a curvature R and a stress energy T and a non-vanishing curvature in Riemann geometry.

[Dubbelosix (OP)]

So... what is the Bianchi identity?

The first term doesnt matter, the indice can be covariant or contravariant. The identity satisfies sign changes in the order of the last three indices



To create the first Bianchi term is easy, contract a definition of the curvature with the metric tensor



Using the same principles, you can form the other three, now all three are, assuming I have done this right










There is a commutation relationship it seems between the metric and the connections. Using the formulation above though, we can see an equivalent form then of the Bianchi identity is



Or simply as



In which the bracket denotes the antisymmetric part of the tensor, which arises from the antisymmetry in and and the definition of is entangled into it.

[Dubbelosix (OP)]

It's also nice to note, this last identity can be seen to be related to the Jacobi triple vector product



In which case, we understand it from the following relationship using basis vectors:

[Dubbelosix (OP)]

Now to teach a little on the notation. If you see you will know this by now as the commutation relation.

Classical it will be zero, but quantum mechanics tells a different story in which the difference is measured by



This shows there are no quantum observables corresponding to the action. Understanding why though this means we have entered the quantum realm is unclear. Even in modern references I have looked for.

Usually when you see something like



It can also be taken as a statement of conservation. Clearly does not equal zero in the quantum domain and the presence of the quantization is not an indication of a breakdown of any conservation law.

Isn't it interesting though, we have been speaking about the non-commutation between space and time - this suggests three possible uncertainty relationships with time by using the three space dimensions.

However, in quantum mechanics, objecfs like commutators are made up of observables. Does this expose a flaw in the understanding of the physics, since time is not an observable? Time doesn't even have a non-trivial operator. So how can time be seen as or like an observable, in some phase space, when it isn't?

This is a good question and I may not have the best of answers, - only than to say, we need to remember, while it seems natural to call it a spacetime uncertainty relationship, its actually just a reinterpretation of the usual energy time relationship.

So while it may seem like we are forced to deal with an object which isn't an observable, technically-speaking, its about the energy-time relationship and the large uncertainties in the momentum of scattered paricles implies that the number of particle states in the short distance of region satisfies



So this is not a true modification but is rather an interpretation, according to Yoneya.

[Dubbelosix (OP)]

The first term is the first cyclic form of the Riemann tensor is



which has been well-established now it has antisymmetric indices. It also has to be true that the following commutation properties hold



That is the time deriative anti-commutes with the rest of the space derivatives.



Is a form of the Bianchi identity. It may be a good question to ask, whether the first and last Riemann terms have commutation between the metric term at the end. We'll work that out in time. We seem to have ''partially'' created a quantum Bianchi identity by understanding the commutation written in the form



So question is, what notation do we use to express commutation on the first and last cyclic Riemann terms? This serves as a lead forward.

[jeffreyH]

Those darn symbols and their affine connections!
http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html

[Dubbelosix (OP)]

Those darn symbols and their affine connections!
http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html

Indeed, I just checked back as well and found two wrong indices, in wrong places! best to check everything.

[Dubbelosix (OP)]

The first term is the first cyclic form of the Riemann tensor is



which has been well-established now it has antisymmetric indices. It also has to be true that the following commutation properties hold



That is the time deriative anti-commutes with the rest of the space derivatives.



Is a form of the Bianchi identity. It may be a good question to ask, whether the first and last Riemann terms have commutation between the metric term at the end. We'll work that out in time. We seem to have ''partially'' created a quantum Bianchi identity by understanding the commutation written in the form



So question is, what notation do we use to express commutation on the first and last cyclic Riemann terms? This serves as a lead forward.

Let's make a clearing in the forest with some work. I have now written out the full form of the Bianchi identities. Doing so, I was hoping to see if it reveals anything.










Again, this needs to satisfy a Bianchi identity







which we recognize as one of the forms of the spacetime uncertainty locked in equation 2. In the last Cyclic Riemann tensor, is a and indice - note this is an edited correction for right form, check next post



Without the metric, this is just



and the brackets and the indices inside of them denotes the antisymmetric parts which involves and no other space derivative interestingly, at least, from the first definition. Notice then also, when we worked out the Bianchi identities in full form, we could see commutation relations between the time and the space indice -
 




Maybe there is a connection there, maybe there isn't - but its nice to write them down and identify some of the commutation relationships leading to those quantum Bianchi identities.

[Dubbelosix (OP)]

It seems I have done this last Bianchi identity incorrectly ... when you do it right (I mixed up the last indice wrong) then you do not get commutation with and - another good reason why double checking your indices is a good idea. The form this Bianchi identity takes is





So be mindful of this mistake and that it has been fixed.

[Dubbelosix (OP)]

It seems I have done this last Bianchi identity incorrectly ... when you do it right (I mixed up the last indice wrong) then you do not get commutation with and - another good reason why double checking your indices is a good idea. The form this Bianchi identity takes is





So be mindful of this mistake and that it has been fixed.

The full corrected form is

[Dubbelosix (OP)]

Now we can move on to the three important identities we looked at and they will give a quantized look at the identities.










You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indices



The order of the commutation with respect to indices we have looked at have revealed their commutation in the last two indices.

[Dubbelosix (OP)]

Now we can move on to the three important identities we looked at and they will give a quantized look at the identities.










You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indices



The order of the commutation with respect to indices we have looked at have revealed their commutation in the last two indices.

Considering, that from the first quantum Bianchi identity we created, with extra indices, it is common to construct a 1/2-tensor





Which we identified much earlier on is an object which describes the curvature and even implies a torsion. Is it possible to construct similar tensors from the cyclic quantum Bianchi identities?

[Dubbelosix (OP)]

Next week, I will also write out the permutations of the cyclic Bianchi identity to identify the space-time commutations.

[Dubbelosix (OP)]

To take a break from those Christoffel symbols, I have been busy working on the interpretation of the physics in a Hilbert space. My first realisation came from recent discussions on the Bohr orbits and how they do not contain acceleration. This is because the electron is not locked in a classical orbit. Classical theory removed true orbits, when a quantum wave function is considered, the orbit is replaced with an orbital which has no acceleration - which means the electron has an irregular release of radiation that is often considered spontaneous - according to Bohr, only the ''special orbits'' gave up radiation - the superpositioning existing as a cloud round the nucleus removes any acceleration in the system.

In a similar theory, a superpositioning of a mass acts like a static gravity. Wave function in Hilbert space allows a superpositioning of geometry!! This approach requires that there exists a form of a static Schrodinger equation. The lack of any acceleration for a geometrical theory is important for the pre-collapse state. Stationary Schrodinger states do exist. Consider a wave with derivatives in space and derivatives in time,
 


The probability is



You can identify the relationship:



The location of the particle, and its mass, will tend around the absolute square value (collapse). One reason to see this as a stationary solution, is because for the very fact it is time-dependent. You can create an operator:



Which is an expectation value and has a domain and has used the time evolution operator . In which





The Problem of Time Dependence

A stationary theory of the gravity does lead to a problem if we consider there is a special non-trivial spacetime relationship (which hints at a time dependence). In which case, we expect there to be a relationship of the form

(unable to put into dirac notation due to site sensitivity to latex)



(some notation latex here does not like)

This physically doesn't make sense, because the dependence of time invites that the geometry is changing with time, when by a restriction of physics, it has to be static.

The purely static case of the expectation value of



(in which X is a function associated to spatial derivatives). would reduce to just the spatial derivatives



The static solutions are nothing but the time-independent Schrodinger equation (!), which is just



A suitable attempt would be to look for time-independent Hamiltonians



We know what . This gives simply



If the wave function is normalized, the last terms come to . That is, the expectation is just the rest energy of the system of the stationary system.

Consider the square of the operator:



This will give



Which leaves the square of the energy. If we understand this as an operator of the form then we have unearthed an important feature about the static system. The difference of the square of the energy yields there is no uncertainty in the system



Notice we have uncovered there is no uncertainty in the geometry of spacetime (if this previous solution holds) for a static theory.


So we have some important questions.

1. Is quantum gravity time-dependent?

2. Do superpositioning of particle states give rise to stationary systems?


To understand why the first question is important, is because if we are arguing for a non-trivial spacetime solution then the geometry is necessarily time-dependent (which may in hindsight appear like an odd thing to say), but the issue of time is a non-trivial one itself - some argue that it doesn't exist.

The static field depends though on a situation: The static theory depends that there is a perfect equilibrium in the system. However, it must also be noted that the particles mass is not quite evenly distributed. In fact, the actual mass of the system will tend around the square of the absolute value of its wave function, so we could argue that there cannot be a true equilibrium.

Can we interpret the square of its wave function as the center of mass in which small variances occur? It would be interesting if gravity coupled to a superpositioning principle for matter may lead to a gravitationally-induced collapse of the system (an idea no less) that Penrose himself has peddled since around 1995.

I found an interesting article that seems closely related to our discussions:

''Even though there is an essential uncertainty in the
energy of the superposition of different space–time geometries,
this kind of energy uncertainty is different in nature from the
energy uncertainty of unstable particles or unstable states in
quantum mechanics (Gao, 2010).''

http://doi.org.ololo.sci-hub.bz/10.1016/j.shpsb.2013.03.001

The article further says:

''The former results from the ill-definedness
of the time-translation operator for the superposed
space–time geometries, while the latter exists in a definite space–
time background, and there is a well-defined time-translation
operator for the unstable states.''


We too covered a problem when transposing our theory of gravity to a Hilbert space, that a stationary theory of superpositioning won't hold for the spacetime uncertainty principle. We have argued also though, that maybe there is no true perfect equilibrium in the system.

Certainly, for the static case of a wave function around a nucleus, it cannot exist as a true static system because the atom has  to give up its energy eventually (unless subjected to a Zeno effect). Though Bohr has attributed it to a special orbit, this is very vague explanation of an otherwise, interesting phenomenon. Most people generally believe for instance, the radiation of an electron to a lower orbital is a spontaneous phenomenon. In all my investigations of quantum mechanics over the years, nothing in physics has made me convinced anything is by chance. There will be an underlying physical reason why an atom will give up its energy eventually, a phenomenon we know as quantum decay.

[Dubbelosix (OP)]

I've taken some of the equations out of their code because of the sensitivity at the site. Won't display properly.