Physics Subtly Implies no Geometry for Space

[Reiku]

There is more I see, you said I pulled a hamiltonian from no where in the Heisenberg equation. Well, no, not really.

If you want to know where I had been heading with that, is that there is such a thing as a ''fundamental Hamiltonian'' which describes high energy states. Obviously it was fundamental because, as you yourself said, ''didn't you notice the hbar in there?''

Markoupoulou remarks in her own work

''It should now be clear that there are two possible notions of time: the time related to the g00 component of the metric describing the geometry at low energy and the time parameter in the fundamental microscopic Hamiltonian.''

So why should I not plug in a Hamiltonian in there? It was there to describe individual particles anywhere, world lines in fact to be more precise.

 

[Reiku]

In an approach according to a paper I linked previously in the past, it was customary as a reparamaterization of the Wheeler de Witt equation to reconfigure time as $$\chi = \tau$$, so the matter field was now acting as a clock.

The Total Hamiltonian for a universe is given by the WDW-Hamiltonian

$$H_{WDW}=H_T=H_{\phi, h_{\mu \nu}} - H_0$$ [1]

where $$h_{\mu \nu}$$ is a metric perturbation. The equation of interest is

$$|\psi> = |\phi_a> |\chi_{\phi, h_{\mu \nu}}>$$ [2]

Using a proceedure of seperation of variables, we obtain

$$(H+V(a)|\psi_a> = E|\psi>$$ [3]

and

$$(H + V(\chi)_{\phi h_{\mu \nu}})|\chi_{\phi, h_{\mu \nu}}> = E|\chi_{\phi h_{\mu \nu}}>$$ [4]

The potential terms in these equations are equal to

$$V(a) = a^2-g^2a^4$$ [5]

and

$$V(\chi) = \chi^2 - g^2\chi^4$$ [6]

Going back to eq. [4] we have

$$(H+(\chi^2-g^2\chi^4))|\chi_{\phi, h_{\mu \nu}}> = E|\chi_{\phi h_{\mu \nu}}>$$ [7]

Because of continuity of eigenvalues, one should state that as $$(\chi_{\phi, h_{\mu \nu}}) \rightarrow \phi$$ then the interaction term $$(\chi^2-g^2c^4) \rightarrow 0$$ goes to zero.

A potential solution to this equation is

$$|\psi> = \phi + \frac{1}{E - H_0} \cdot (\chi^2- g^2\chi^4)|\psi>$$ [8]

where $$\cdot$$ is multiplication. Whilst this is a potential solution, a problem exists. A serious problem in the form of a singularity exists in $$(E-H_0)$$ since $$E$$ is an eigenvalue of $$H_0$$.

There is one distinct way of making this singularity to normally disappear. One way is to make the denominator slightly complex $$i\epsilon$$. Instead, I am going to use a different approach.

To avoid the singular existance of this mathematical object, we are going to substitute $$E$$ for a result of the Schwartzchild Metric

$$Mc^2 - \frac{Gm^2}{2R}$$ [9]

This changes our equation in such a drastic way, it rids us of the singularity in our equation. No longer does $$(E-H_0)$$ hold, because the null energy condition states that as $$M=0$$ all we are left with is the metric.

So we have a correspondance between $$(Mc^2 - \frac{Gm^2}{2R})$$ and $$H_0$$ in which if you imply a nullified energy condition you obtain a solution to avoid the singularity problem.

Interestingly, it does not only imply that our matter field $$\chi$$ no longer has any mass, but it also means we have no radiation fields either.

Let's look at the original Hamiltonian in a different way, now accounting for systems of many particles if one desired but for simplicity we will assume particles $$i$$ and $$j$$.

For $$N$$-particles (2 in our case), we can have

$$\{ \sum_{i} H(i) + \sum \frac{1}{r_{ij}} \} = E \psi$$

where $$r_{ij}$$ calculates the distance between $$i$$ and $$j$$. If we represent this in a Hilbert Space, the configuration space looks like

$$H= \sum_i H_i + \sum_{k \in I} h_k$$

where $$h_k$$ is a hermitian operator. $$I$$ is the set of interactions $$\{i,j\} \equiv k$$ and naturally can form a direct tensor Hilbert space $$\mathcal{H}_3 = \mathcal{H}_1 \otimes \mathcal{H}_2$$.

This is already beginning to be like a theory which deals with no geometry. Indeed, a Hilbert Space are often called ''points'' which describe abstractly the configuration space. This is the similar approach Markoupoulou attempts to make in her Graphity model. She takes Hilbert Spaces and describes their interactions in such a model without resorting to geometry.

The equation $$H= \sum_i H_i + \sum_{k \in I} h_k$$ is actually an equation Markoupoulou uses very very early on in her paper on quantum graphity. It is a simple approach, and has similar undertones to $$\{ \sum_{i} H(i) + \sum \frac{1}{r_{ij}} \} = E \psi$$ where $$r_{ij}$$ calculates the distance between $$i$$ and $$j$$.

Let us first of all, describe the interaction $$k = \{ij \}$$ where $$i$$ and $$j$$ are our particle system. (Can be thought of as a configuration space). Let us now state that the interaction is determined by a potential governing a force between the two particles.

$$V= \sum^{N-1}_{i=1} \sum^{N}_{i+1} g(r_{ij})$$

The calculation of the interaction forces on all $$N$$-particles due to pairwise interactions involves a maximum of $$\frac{N(N-1)}{2}$$ contributions. In markoupoulou's work, $$K_N$$ is the complete graph on$$N$$ vertices, i.e., the graph in which there is one edge connecting
every pair of vertices, so that there is a total of $$N(N- 1)=2$$ edges and each vertex has degree $$N- 1$$ [1]. So what I have done is explicitely describe that there is a force between two particles in my case, (I have defined the set of interactions) and in Markoupoulou's we can see that her total space state of the system is $$\mathcal{H} = \otimes \frac{N(N -1)}{2} \mathcal{H}_{ab}$$.

So for now, what I really require is a bit of imagination. A certain matter field in the low energy epoch could satisfy a flux of time. But what we will find at the high energy state, is a place where my matter field vanishes, including the geometrical space - time is a consequence of real bradyonic particles. The universe could have a time, if summing each of these particles up and allowing them to have clocks, in both my vision and Julian Barbours, then you can say only geometric time exists, since in spin networks, space exists but not spacetime.

[1] http://fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

 

[khan]

[1] http://fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

Thanks for the interesting links Reiku.

Keep up the good work :bravo:

 

[Reiku]

The equation $$H= \sum_i H_i + \sum_{k \in I} h_k$$ is actually an equation Markoupoulou uses very very early on in her paper on quantum graphity. It is a simple approach, and has similar undertones to $$\{ \sum_{i} H(i) + \sum \frac{1}{r_{ij}} \} = E \psi$$ where $$r_{ij}$$ calculates the distance between $$i$$ and $$j$$.

Let us first of all, describe the interaction $$k = \{ij \}$$ where $$i$$ and $$j$$ are our particle system. (Can be thought of as a configuration space). Let us now state that the interaction is determined by a potential governing a force between the two particles.

$$V= \sum^{N-1}_{i=1} \sum^{N}_{i+1} g(r_{ij})$$

The calculation of the interaction forces on all $$N$$-particles due to pairwise interactions involves a maximum of $$\frac{N(N-1)}{2}$$ contributions. In markoupoulou's work, $$K_N$$ is the complete graph on$$N$$ vertices, i.e., the graph in which there is one edge connecting
every pair of vertices, so that there is a total of $$N(N- 1)=2$$ edges and each vertex has degree $$N- 1$$ [1]. So what I have done is explicitely describe that there is a force between two particles in my case, (I have defined the set of interactions) and in Markoupoulou's we can see that her total space state of the system is $$\mathcal{H} = \otimes \frac{N(N -1)}{2} \mathcal{H}_{ab}$$.

So for now, what I really require is a bit of imagination. A certain matter field in the low energy epoch could satisfy a flux of time. But what we will find at the high energy state, is a place where my matter field vanishes, including the geometrical space - time is a consequence of real bradyonic particles. The universe could have a time, if summing each of these particles up and allowing them to have clocks, in both my vision and Julian Barbours, then you can say only geometric time exists, since in spin networks, space exists but not spacetime.

[1] http://fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

Now I made mention before that $$r_{ij}$$ was in fact a metric - well actually, that was not quite true. It is what mathematicians define as a semi-metric. Let's see why

Let $$X$$ be a set of pair points $$i$$ and $$j$$ one gets the distance between them by $$ij$$.

Interestingly this all comes back to the Triangle Inequality. In fact, a semi-metric is allowed to violate the Triangle Inequality. So only certain solutions for the metric will allow a complete theory of spin networks in a quantum loop gravity framework (coupled with Geometrogenesis), since spin network (which is at the core of LQG) relies on pin-pointing the states of particles obeying the triangle inequality ie. a must be less than or equal to b+c, b less than or equal to a+c, and c less than or equal to a+b.

 

[Reiku]

Now I made mention before that $$r_{ij}$$ was in fact a metric - well actually, that was not quite true. It is what mathematicians define as a semi-metric. Let's see why

Let $$X$$ be a set of pair points $$i$$ and $$j$$ one gets the distance between them by $$ij$$.

Interestingly this all comes back to the Triangle Inequality. In fact, a semi-metric is allowed to violate the Triangle Inequality. So only certain solutions for the metric will allow a complete theory of spin networks in a quantum loop gravity framework (coupled with Geometrogenesis), since spin network (which is at the core of LQG) relies on pin-pointing the states of particles obeying the triangle inequality ie. a must be less than or equal to b+c, b less than or equal to a+c, and c less than or equal to a+b.

Now, in fotini's model, if $$G$$ is some graph in this spin network space of states. A Hamiltonian operator $$H$$ will assign an energy $$E(G) = <\Psi_G|H|\Psi_G>$$ to this graph.

The reason why fotini can describe it this way (she doesn't mention this) along with a few other things I have brought up so far, if $$A(G)$$ are adjacent vertices and $$E(G)$$ is the set of the edges, then it satisfies

$$(i,j) \in E(G)$$

Each vertex represents a spin-state on the graph. Thus we will begin to see Fotini's approach, that to each vertex $$i \in A(G)$$ there can be associated a Hilbert Space

$$H_G = \otimes_{i \in A(G)} \mathcal{H}_i = \mathbf{C}^{2}^{\otimes N}$$

Which takes on remarkable similarities to Fotini's equation describing the total space state. Indeed, the equation above describes the total number of vertices in our Hilbert Space.

Thus, we come back to a familiar garl. The distance between two particles $$i$$ and $$j$$ are vertices $$i,j \in A(G)$$ in fact apply to the least action principle; it will define a graph geodesic between these two points.

And this is why her model works.

 

[khan]

Interestingly this all comes back to the Triangle Inequality. In fact, a semi-metric is allowed to violate the Triangle Inequality. So only certain solutions for the metric will allow a complete theory of spin networks in a quantum loop gravity framework (coupled with Geometrogenesis), since spin network (which is at the core of LQG) relies on pin-pointing the states of particles obeying the triangle inequality ie. a must be less than or equal to b+c, b less than or equal to a+c, and c less than or equal to a+b.

Minkowski space has a reverse triangle inequality.

http://en.wikipedia.org/wiki/Minkowski_space#Reversed_triangle_inequality


Some scientists believe that in order to truly understand and resolve the twins paradox, the general theory of relativity is needed.

http://www.youtube.com/watch?feature=player_embedded&v=jlJNsRZ4WxI

5:15 of this video

 

[Chipz]

Now, in fotini's model, if $$G$$ is some graph in this spin network space of states. A Hamiltonian operator $$H$$ will assign an energy $$E(G) = <\Psi_G|H|\Psi_G>$$ to this graph.

The reason why fotini can describe it this way (she doesn't mention this) along with a few other things I have brought up so far, if $$A(G)$$ are adjacent vertices and $$E(G)$$ is the set of the edges, then it satisfies

$$(i,j) \in E(G)$$

Each vertex represents a spin-state on the graph. Thus we will begin to see Fotini's approach, that to each vertex $$i \in A(G)$$ there can be associated a Hilbert Space

$$H_G = \otimes_{i \in A(G)} \mathcal{H}_i = \mathbf{C}^{2}^{\otimes N}$$

Which takes on remarkable similarities to Fotini's equation describing the total space state. Indeed, the equation above describes the total number of vertices in our Hilbert Space.

Thus, we come back to a familiar garl. The distance between two particles $$i$$ and $$j$$ are vertices $$i,j \in A(G)$$ in fact apply to the least action principle; it will define a graph geodesic between these two points.

And this is why her model works.

You truly are a crazy man. You've managed to string together a string of mathematical conjectures together with only tangential relevance...almost as if they were linked together on some page in a web-like fashion in a freely accessible source.

 

[AlphaNumeric]

Let's start with, why you think it involves a geometry?

What about my last answer to this did you not understand? You use the Schwarzchild metric. That's a geometry!

''No, I said that the Hamiltonian (and thus terms in the Hamiltonian, including ) is not a configuration space or points in a configuration space or anything like that.''

But you stated something I stated and said it was wrong. Even if this is not what you implied, you really need to use your imagination AN. If two particles are seperated by a distance r_ij then the two particles must have positions in space somewhere. In a very loose, but still correct statement, that if particles are described by a Hilbert Space as I have demonstrated then we must be implying some kind of configuration space.

You're simply not reading what I say. Or you just don't understand basic meaning of terminology like 'Hamiltonian'.

I know you can describe points in space using a Hilbert space, since you can define the dot product as your inner product. However, you said the Hamiltonian is described as a configuration space. No, th Hamiltonian ACTS on a space, which may sometimes (but not always) be a configuration space.

You're responding to things I never said. Whether it's an attempt to make it seem to the lay person you're retorting my criticisms or you're just plain ignorant I don't know.

Just also to clear up a few loose ends:

''Reiku, do you really think I (or anyone who didn't sleep through high school) believes you did any such calculations? ''

In the paragraph you have stated, I was quite aware of the density relationship . Is there any reason why my own relationship does not hold? If so I will quite willingly forget those relationships. But I can assure you right now that $$x_i$$ is not $$X$$ since $$x_i$$ is the four velocity. The four velocity (summed over all particles given by $$i$$) multiplied by the density gives you your matter field acting like a unit timelike vector. As our original field $$\phi$$ acted on $$x^{\mu} \tau}$$, it created $$\chi$$ the inertial matter field. Thus from

$$\rho = T_{ab}\phi^a\phi^b$$

multiply by the four velocity gives

$$\rho x_i = T_{ab}\phi^a\phi^b (x^{\mu}\tau)$$

The right hand side turns into $$T_{ab} \chi^a \chi^b$$

.... and here I realized I've made a mistake, I've dropped a term on the right hand side

because from there it should be, divide by $$\partial t$$ on both sides and taking partial \rho gives

$$\frac{\partial \rho x_i}{\partial t}= \frac{ T_{ab} \chi^a \chi^b}{\partial t}$$

Which means my assumptions on Heisenberg equation don't apply. Ignore it now AN.

Jesus, where to start. As I said and which you obviously don't understand, your index structure isn't right. Nor does the third expression follow from the second. Nor does your last expression make sense. You don't 'divide by $$\partial t$$', dear god that isn't what differentiation is about! It seems you can't even take partial derivatives properly, you think there's division involved. This just shows how ridiculous your claims of doing work in this stuff is, you can't do things expected of 1st years yet you're claiming to be coming up with solutions to research level quantum cosmology!

You really have something wrong with you if you think you're doing anything other than being massively dishonest. And even if you're aware you're being dishonest the fact you continue to do it, despite being exposed as a hack repeatedly, says something else is wrong with you personality-wise.

Well, considering I never reparamaterized any of my equations and I mentioned it just for the sake of stating some possible solutions to the WDW equation, I will find the paper....


....right http://arxiv.org/PS_cache/hep-th/pdf/9503/9503073v2.pdf

Page 8-11 should suffice.

Firstly you've done what you often do and that's show where you're getting all your equations from. Just as when you linked to that YouTube lecture which contained all the equations you were spouting, right down to dubious notation, you've shown your hand by linking to that paper.

The paper gives the proper context of all the stuff you've been throwing out. Your problem is you don't understand it so you don't know how to give snippets of it in a way which makes sense, hence why your 'results' are all over the place.

The potential is a functional of the scale factor, a not uncommon notion in cosmology because it's a much better and more 'universal' (in some sense) parameter to describe how things vary. In fact that is what the $$t=a$$ thing refers to. In the equations the a is playing the role of a temporal coordinate because it's monotonic increasing so you can do a valid reparametrisation. It isn't that the original potential is a manifest function of time,

Considering how that entire section is about inner products, which are required in the definition of Hilbert spaces, and all of it flows from a Hamiltonian constraint it's a little odd you don't understand which is which and how they relate to one another. Oh wait, no it isn't, you don't understand any of it.

OOOooohhh right. I read back on your posts (end up ignoring half of it because you tend to write so much),

No, you ignore it because you can't respond to the repeated demonstrations you're dishonest and a hack. Clearly from your little snippet replies when you think you can throw something back at me you do it, you don't pass up the chance.

via a series of unjustified non-sequitors, that you end up with a result which doesn't involve geometry. You used the SC metric! You made an explicit reference to a geometry.

That's ok.

You see, we haven't even applied the Hilbert Space or any spin networks. I recall reading that r_ij is also a metric, a special kind of one, but that must have missed you as well. Niether metric are related, as far as I can tell.

'We'? You aren't doing anything but spewing out other people's work. It's plagiarism. You're passing off other people's explanations as your own understanding. You're not doing anything yourself. you're just mangling together any source you can and trying to con people into believing you're doing some of this stuff yourself. Spinning together multiple people's work and passing off their calculations as your own is plagiarism. You can't even get the meaning of a Hilbert space right. Sure, you can quote Wikipedia at me but it's obvious from this thread and others you don't recognise them when you see them, you don't know how they work or how things relate to them. You're still struggling with the relationship of the Hamiltonian to them, despite me explaining it on more than one occasion. And you're certainly not doing anything to do with spin networks. They're considered unpleasant by actual mathematicians, never mind someone with your level of mathematical ability.

To illustrate consider the second bit of the above quote. You say " I recall reading that r_ij is also a metric, a special kind of one, but that must have missed you as well. ". Can't you work it out for yourself? Surely you know the definition of a metric? Can't you check whether it's a metric or not? You're trying to convince people you're dealing with geometry-less quantum cosmology and you can't work out if something is a metric or not. Why don't you give it a shot, confirm or refute your assertion, precisely and clearly.

I am in no doubt you'll fail. You are, for all intents and purposes when it comes to this level of mathematical physics, innumerate.

Now, quantum graphity actually admits metric solutions, the things which involve space, matter and geometry http://arxiv.org/pdf/0801.0861v2.pdf . The approach however, even though I have adopted and Markoupoulou uses, argue that geometry does not fundmantally.

Like I said, if you think you're spending your time wisely or doing something viable, as you imply when you say "I have adopted...", is anything other than utter dishonest you have a personality issue.

In my original representation to you, I used the LS-equation to solve the original energy hamiltonian of the universe with potential.

No, you didn't. Do you think I can't remember the start of the thread? Are you really so desperate to lie you'll resort to this level of dishonesty?

I had no explicitely stated yet that this specific equation would describe no geometry. That is only achieved when you would begin to model your theory with a Hilbert Space.

Speaking as someone with a PhD in non-geometric spaces I can attest to the fact you don't necessarily require a Hilbert space to construct a description of a system which doesn't contain any geometry. You quote someone saying that the notion of 'here' and 'there' no longer applies. That's precisely the type of space I have published work in regards to.

Hilbert spaces can be used and can allow for some very interesting stuff but they aren't essential. But this is somewhat beside the point, given you clearly can't do any of this stuff.

In this space, as Fotini puts it:

''Information before geometry. Having raised the possibility that geometry does not exist at the fundamental level, we now need to find a way to do physics without geometry. This may appear hard because all our physics is done with geometry. But we can use a relational and information theoretic language.''

Again, speaking as someone with experience with non-geometric constructs the quote it not entirely accurate, there are ways of doing physics not only without making reference to an underlying geometry (that's just background independence) but actually having no geometry at all. Information theory didn't need to come into it.

An example is that she considers a finite relational universe with N constituents, a bit like my (summing over all the particles making a field approach)

Summing over particles does not a field approach make. There's a little more to it. You must have skipped over that section of quantum mechanics when you jumped from high school to the Dirac equation.

which she models as a network of N nodes (a,b...=1) with a Hilbert Space $$\mathcal{H}_{ab}$$ attached to each link (ab). It is this model I am trying advocate.

You can advocate her work but to try to present yourself as working on similar stuff and that's why you're an advocate is just ridiculous.

I think it is the correct approach because it sufficiently describes high energy physics, low energy, geometry stuff exists, we know this. It is once you apply the following approaches could we possibly achieve some kind of quantized theory involving no geometry.

Now you could have said all of that without having to be dishonest and pretend you're doing some of this yourself. You could have started a discussion on non-geometric constructs and talked about this person's work. Instead you peppered it with laughable, ridiculous, delusional claims about yourself and supposed work you're doing. What could have been a good honest discussion you instead used as a bandwagon to try to delude your ego.

And before you whine "Oh you just don't like someone else doing work close to your own!" or "You're jealous" or anything else equally laughable if you're so sure you've got a new approach, valid and unknown to the mainstream submit it to a journal. You clearly have the time to write this stuff up. You know sufficient LaTeX to do the algebra yourself. Tell you what, if you write it up on this forum and send it to me via PM I'll compile it using actual LaTeX and send you the completed .tex and .pdf files, all in the appropriate formatting for a relevant journal, like JHEP. Then you can submit it. Come on, you have no excuse if you really think you're onto something. I'm sufficiently confident you'll crash and burn that I'll help you, so you have no excuse like "I don't know how to make .tex files" or "I can't format it the way they want!".

Put up or shut up.

There is more I see, you said I pulled a hamiltonian from no where in the Heisenberg equation. Well, no, not really.

If you want to know where I had been heading with that, is that there is such a thing as a ''fundamental Hamiltonian'' which describes high energy states. Obviously it was fundamental because, as you yourself said, ''didn't you notice the hbar in there?''

What are you on about? The Hamiltonian in the equation you quoted wasn't in any of your previous expressions, so it was a non-sequitor. I know how the equation itself is derived, as I said it's standard bookwork for an introductory course into QM. And having $$\hbar$$ doesn't make something 'fundamental'. Time and again you're just jamming your foot in your mouth.

Markoupoulou remarks in her own work

''It should now be clear that there are two possible notions of time: the time related to the g00 component of the metric describing the geometry at low energy and the time parameter in the fundamental microscopic Hamiltonian.''

So why should I not plug in a Hamiltonian in there? It was there to describe individual particles anywhere, world lines in fact to be more precise.

Wow, you really are desperate. Did you just look through her paper for something to do with a Hamiltonian so you could throw it out and hope it sticks. It was a non-sequitor because you did a bunch of things which made no mention of a Hamiltonian and then you suddenly pull one out of nowhere. Not only that but the equation you produce is sufficiently well known that it's obvious it doesn't follow from what you'd said. Now I can imagine that there's a paper you're copying from which goes into a lot more detail, explained itself properly, includes many other expressions, equations etc and actually does such a derivation. However, as you generally do, you failed to include sufficient things from the paper in your post to make your post coherent. You really need to learn this method of deception doesn't work on people who know physics. It might seem to you like "Wow, look at all those equations. Everyone will think I'm a genius" but to people who understand the equations its clear you're just pulling them from somewhere.

Seriously, any rational person would have learnt after the first half dozen times of being exposed as dishonest in this manner to stop doing it. Instead you carry on. Like I said in a previous post, you'll lie when you think no one will call you on it, showing you're deliberately deceptive. But even more daft you'll lie to me about physics I've corrected you on dozens of times before.

I'd carry on replying to the other post or two of yours where you post more Wiki/ArXiv lifted equations but I'm hungry so I'm going to eat something. You need to really look at yourself and change how you act because you're not all there upstairs if you think you're able to do this stuff or you're taken seriously by anyone who knows any physics or maths. Hopefully your professed belief you could handle an undergrad course with ease is just that, a professed belief and not an actual belief. You need to get a firmer grasp on reality. You're 27 for god sake. Do something constructive with your life.

 

[Reiku]

What about my last answer to this did you not understand? You use the Schwarzchild metric. That's a geometry!

You're simply not reading what I say. Or you just don't understand basic meaning of terminology like 'Hamiltonian'.

I know you can describe points in space using a Hilbert space, since you can define the dot product as your inner product. However, you said the Hamiltonian is described as a configuration space. No, th Hamiltonian ACTS on a space, which may sometimes (but not always) be a configuration space.

You're responding to things I never said. Whether it's an attempt to make it seem to the lay person you're retorting my criticisms or you're just plain ignorant I don't know.

Jesus, where to start. As I said and which you obviously don't understand, your index structure isn't right. Nor does the third expression follow from the second. Nor does your last expression make sense. You don't 'divide by $$\partial t$$', dear god that isn't what differentiation is about! It seems you can't even take partial derivatives properly, you think there's division involved. This just shows how ridiculous your claims of doing work in this stuff is, you can't do things expected of 1st years yet you're claiming to be coming up with solutions to research level quantum cosmology!

You really have something wrong with you if you think you're doing anything other than being massively dishonest. And even if you're aware you're being dishonest the fact you continue to do it, despite being exposed as a hack repeatedly, says something else is wrong with you personality-wise.

Firstly you've done what you often do and that's show where you're getting all your equations from. Just as when you linked to that YouTube lecture which contained all the equations you were spouting, right down to dubious notation, you've shown your hand by linking to that paper.

The paper gives the proper context of all the stuff you've been throwing out. Your problem is you don't understand it so you don't know how to give snippets of it in a way which makes sense, hence why your 'results' are all over the place.

The potential is a functional of the scale factor, a not uncommon notion in cosmology because it's a much better and more 'universal' (in some sense) parameter to describe how things vary. In fact that is what the $$t=a$$ thing refers to. In the equations the a is playing the role of a temporal coordinate because it's monotonic increasing so you can do a valid reparametrisation. It isn't that the original potential is a manifest function of time,

Considering how that entire section is about inner products, which are required in the definition of Hilbert spaces, and all of it flows from a Hamiltonian constraint it's a little odd you don't understand which is which and how they relate to one another. Oh wait, no it isn't, you don't understand any of it.

No, you ignore it because you can't respond to the repeated demonstrations you're dishonest and a hack. Clearly from your little snippet replies when you think you can throw something back at me you do it, you don't pass up the chance.

'We'? You aren't doing anything but spewing out other people's work. It's plagiarism. You're passing off other people's explanations as your own understanding. You're not doing anything yourself. you're just mangling together any source you can and trying to con people into believing you're doing some of this stuff yourself. Spinning together multiple people's work and passing off their calculations as your own is plagiarism. You can't even get the meaning of a Hilbert space right. Sure, you can quote Wikipedia at me but it's obvious from this thread and others you don't recognise them when you see them, you don't know how they work or how things relate to them. You're still struggling with the relationship of the Hamiltonian to them, despite me explaining it on more than one occasion. And you're certainly not doing anything to do with spin networks. They're considered unpleasant by actual mathematicians, never mind someone with your level of mathematical ability.

To illustrate consider the second bit of the above quote. You say " I recall reading that r_ij is also a metric, a special kind of one, but that must have missed you as well. ". Can't you work it out for yourself? Surely you know the definition of a metric? Can't you check whether it's a metric or not? You're trying to convince people you're dealing with geometry-less quantum cosmology and you can't work out if something is a metric or not. Why don't you give it a shot, confirm or refute your assertion, precisely and clearly.

I am in no doubt you'll fail. You are, for all intents and purposes when it comes to this level of mathematical physics, innumerate.

Like I said, if you think you're spending your time wisely or doing something viable, as you imply when you say "I have adopted...", is anything other than utter dishonest you have a personality issue.

No, you didn't. Do you think I can't remember the start of the thread? Are you really so desperate to lie you'll resort to this level of dishonesty?

Speaking as someone with a PhD in non-geometric spaces I can attest to the fact you don't necessarily require a Hilbert space to construct a description of a system which doesn't contain any geometry. You quote someone saying that the notion of 'here' and 'there' no longer applies. That's precisely the type of space I have published work in regards to.

Hilbert spaces can be used and can allow for some very interesting stuff but they aren't essential. But this is somewhat beside the point, given you clearly can't do any of this stuff.

Again, speaking as someone with experience with non-geometric constructs the quote it not entirely accurate, there are ways of doing physics not only without making reference to an underlying geometry (that's just background independence) but actually having no geometry at all. Information theory didn't need to come into it.

Summing over particles does not a field approach make. There's a little more to it. You must have skipped over that section of quantum mechanics when you jumped from high school to the Dirac equation.

You can advocate her work but to try to present yourself as working on similar stuff and that's why you're an advocate is just ridiculous.

Now you could have said all of that without having to be dishonest and pretend you're doing some of this yourself. You could have started a discussion on non-geometric constructs and talked about this person's work. Instead you peppered it with laughable, ridiculous, delusional claims about yourself and supposed work you're doing. What could have been a good honest discussion you instead used as a bandwagon to try to delude your ego.

And before you whine "Oh you just don't like someone else doing work close to your own!" or "You're jealous" or anything else equally laughable if you're so sure you've got a new approach, valid and unknown to the mainstream submit it to a journal. You clearly have the time to write this stuff up. You know sufficient LaTeX to do the algebra yourself. Tell you what, if you write it up on this forum and send it to me via PM I'll compile it using actual LaTeX and send you the completed .tex and .pdf files, all in the appropriate formatting for a relevant journal, like JHEP. Then you can submit it. Come on, you have no excuse if you really think you're onto something. I'm sufficiently confident you'll crash and burn that I'll help you, so you have no excuse like "I don't know how to make .tex files" or "I can't format it the way they want!".

Put up or shut up.

What are you on about? The Hamiltonian in the equation you quoted wasn't in any of your previous expressions, so it was a non-sequitor. I know how the equation itself is derived, as I said it's standard bookwork for an introductory course into QM. And having $$\hbar$$ doesn't make something 'fundamental'. Time and again you're just jamming your foot in your mouth.

Wow, you really are desperate. Did you just look through her paper for something to do with a Hamiltonian so you could throw it out and hope it sticks. It was a non-sequitor because you did a bunch of things which made no mention of a Hamiltonian and then you suddenly pull one out of nowhere. Not only that but the equation you produce is sufficiently well known that it's obvious it doesn't follow from what you'd said. Now I can imagine that there's a paper you're copying from which goes into a lot more detail, explained itself properly, includes many other expressions, equations etc and actually does such a derivation. However, as you generally do, you failed to include sufficient things from the paper in your post to make your post coherent. You really need to learn this method of deception doesn't work on people who know physics. It might seem to you like "Wow, look at all those equations. Everyone will think I'm a genius" but to people who understand the equations its clear you're just pulling them from somewhere.

Seriously, any rational person would have learnt after the first half dozen times of being exposed as dishonest in this manner to stop doing it. Instead you carry on. Like I said in a previous post, you'll lie when you think no one will call you on it, showing you're deliberately deceptive. But even more daft you'll lie to me about physics I've corrected you on dozens of times before.

I'd carry on replying to the other post or two of yours where you post more Wiki/ArXiv lifted equations but I'm hungry so I'm going to eat something. You need to really look at yourself and change how you act because you're not all there upstairs if you think you're able to do this stuff or you're taken seriously by anyone who knows any physics or maths. Hopefully your professed belief you could handle an undergrad course with ease is just that, a professed belief and not an actual belief. You need to get a firmer grasp on reality. You're 27 for god sake. Do something constructive with your life.

The remarks you make on the post with the math, it's ok, I'm dropping all that anyway.

Now moving on, yes I did see the metric part of your explanation. You will need to read further that I have answered this queery. Anything you don't understand with any of the following posts which answer this, I am sure you will not hesitate to ask.

Anyway, I had the theory that space was not fundmental independant of Fotini - I based it on all the nullified conditions of energy, time and matter - advocating Fotini's paper is just one example that the approach should be made. Just so happens her simple mathematical model works best in my mind. I will answer the rest tomorrow if there is indeed anything worth replying to.

 

[Reiku]

You truly are a crazy man. You've managed to string together a string of mathematical conjectures together with only tangential relevance...almost as if they were linked together on some page in a web-like fashion in a freely accessible source.

Yes I know.

Crazy? Isn't there a fine line between smart and insanity?

You know, I web these things together because I see their relevances in my head.

 

[Reiku]

AN

In case you missed it, I have took the liberty of (gathering all the parts of the post which answers your question on the metric)

OOOooohhh right. I read back on your posts (end up ignoring half of it because you tend to write so much), that

via a series of unjustified non-sequitors, that you end up with a result which doesn't involve geometry. You used the SC metric! You made an explicit reference to a geometry.

That's ok.

You see, we haven't even applied the Hilbert Space or any spin networks. I recall reading that r_ij is also a metric, a special kind of one, but that must have missed you as well. Niether metric are related, as far as I can tell.

I qoute Markoupoulou

''Just as there are no waves in the
molecular theory, we will likely not find geometric degrees of freedom in the fundamental theory. By analogy with known physics, we should expect that the quantum theory of gravity is not a theory of geometry. I must emphasize that no geometry does not mean discrete or fuzzy geometry. It means that the most primary aspects of geometry, such as the notion of \here" and \there" will cease to make sense. In fact, we have been grappling with no geometry for a while, in the traditional quantum gravity settings.''

Now, quantum graphity actually admits metric solutions, the things which involve space, matter and geometry http://arxiv.org/pdf/0801.0861v2.pdf . The approach however, even though I have adopted and Markoupoulou uses, argue that geometry does not fundmantally.

Keep in mind also, that fundamentally it is argued that geometry does not exist, not that it does not exist at all. The world of high energy physics is related to permutation symmetries, no locality and no subsystems. Low energy states are concerned with geometry and subsystems and the like.

In my original representation to you, I used the LS-equation to solve the original energy hamiltonian of the universe with potential. I had no explicitely stated yet that this specific equation would describe no geometry. That is only achieved when you would begin to model your theory with a Hilbert Space. In this space, as Fotini puts it:

''Information before geometry. Having raised the possibility that geometry does not exist at the fundamental level, we now need to find a way to do physics without geometry. This may appear hard because all our physics is done with geometry. But we can use a relational and information theoretic language.''

An example is that she considers a finite relational universe with N constituents, a bit like my (summing over all the particles making a field approach) which she models as a network of N nodes (a,b...=1) with a Hilbert Space attached to each link (ab). It is this model I am trying advocate.

I think it is the correct approach because it sufficiently describes high energy physics, low energy, geometry stuff exists, we know this. It is once you apply the following approaches could we possibly achieve some kind of quantized theory involving no geometry.

 

[Reiku]

Now I can't remember off the top of my head which equation I have used uses a metric, but I am almost sure none of my original equations have tackled a theory without geometry simply because we have not set into motion all these required works which will, in the end remove geometry and describe high energy reality.

 

[Reiku]

The equation $$H= \sum_i H_i + \sum_{k \in I} h_k$$ is actually an equation Markoupoulou uses very very early on in her paper on quantum graphity. It is a simple approach, and has similar undertones to $$\{ \sum_{i} H(i) + \sum \frac{1}{r_{ij}} \} = E \psi$$ where $$r_{ij}$$ calculates the distance between $$i$$ and $$j$$.

Let us first of all, describe the interaction $$k = \{ij \}$$ where $$i$$ and $$j$$ are our particle system. (Can be thought of as a configuration space). Let us now state that the interaction is determined by a potential governing a force between the two particles.

$$V= \sum^{N-1}_{i=1} \sum^{N}_{i+1} g(r_{ij})$$

The calculation of the interaction forces on all $$N$$-particles due to pairwise interactions involves a maximum of $$\frac{N(N-1)}{2}$$ contributions. In markoupoulou's work, $$K_N$$ is the complete graph on$$N$$ vertices, i.e., the graph in which there is one edge connecting
every pair of vertices, so that there is a total of $$N(N- 1)=2$$ edges and each vertex has degree $$N- 1$$ [1]. So what I have done is explicitely describe that there is a force between two particles in my case, (I have defined the set of interactions) and in Markoupoulou's we can see that her total space state of the system is $$\mathcal{H} = \otimes \frac{N(N -1)}{2} \mathcal{H}_{ab}$$.

So for now, what I really require is a bit of imagination. A certain matter field in the low energy epoch could satisfy a flux of time. But what we will find at the high energy state, is a place where my matter field vanishes, including the geometrical space - time is a consequence of real bradyonic particles. The universe could have a time, if summing each of these particles up and allowing them to have clocks, in both my vision and Julian Barbours, then you can say only geometric time exists, since in spin networks, space exists but not spacetime.

[1] http://fqxi.org/data/essay-contest-files/Markopoulou_SpaceDNE.pdf

So, last night I was a buisy boy - I was babysitting, but when the kids went to sleep my mind started racing on my model again.

Beginning to Create a Spaceless Model

So, I began from where I started off. I had went and defined a set of interactions by $$k \equiv (i,j)$$. I determined this by stating that the interaction is determined by a potential governing a force between the two particles.

This was described by

$$V = \sum^{N-1}_{i=1} \sum^{N}_{i+1} g(r_{ij})$$

Something I never explained, yet no one asked anyhow, what $$g$$ stood for. It is a coupling on the distance between the two particles $$i$$ and $$j$$, so it measures the force at any given moment in time by a simple integral method. It is constant for certain forces like charge which do not change for particles but the magnitude does proportional to changes in the distance. So a fixed distance implies a constant coupling. Naturally, different particles have different charge values, so the coupling just seemed appropriate, much like how a yukawa coupling measures the different particles masses in Heirarchy model.

The force then between two particles can be given as

$$F_{ij} = - \frac{\partial V (r_{ij})}{\partial r_{ij}} \hat{n}_{ij}$$ where

$$\hat{n}_{ij}$$ is the unit vector. Perhaps as a little interesting snippet, if one concentrates on the right hand side and take the dot product of the unit vector with a Pauli Matrix (designed to account for a two-spin network), then square, you end up with

$$- \frac{\partial^2 V^2 (r_{ij})^2}{\partial^2 r_{ij}^2} (\hat{n}_{ij} \cdot \sigma_{ij})^2 = - \frac{\partial^2 V^2 (r_{ij})^2}{\partial^2 r_{ij}^2} \begin{bmatrix} n_3 & n_{-} \\ n_{+} & -n_3 \end{bmatrix}^2$$

Now, from our equation above, we can define a motion for a single particle. To do this, we require two more equations:

$$M_ia_i = \sum^{N}_{j=1, j \ne i} F_{ij}$$

which describes the motion of particle $$i$$ and for particle $$j$$ we have

$$M_ja_j = \sum^{N}_{i=1, i \ne \j} F_{ij}$$

So how do you involve spin in these equations?

Remember, particles $$i$$ and $$j$$ can be replaced by spin networks in an abstract sense, since there are $$N$$-state amount of particles; in entropy-related equations, the $$N$$-state amount of particles is defined by

$$- \ell og \frac{1}{N} = N$$

Where we clearly have $$(i = 1... N)$$ and $$(j = 1... N)$$. These can take on new appearances, we usually denote with an uparrow-downarrow notation which describes a particle, for instance $$i$$ which is either in an up-state or a down-state of spin.

These become the on-off nodes in Fotini Markoupoulou-Kalamari's graphity/geometrogenesis model.

Now that we have defined the interaction as a force, we can now concentrate on new parameters such as the electric charge force between two particles $$i$$ and $$j$$.

If each particles $$(i,j)$$ is described by a charge $$(q_i,q_j)$$ then the equation describing the electric charge force between two particles is given as

$$F_{q_{ij}} = \sum^{N}_{i,j}\frac{q_i q_j}{r_{ij}}$$

I have more work. Requires a bit more time to latex it all.

 

[Reiku]

Now, in fotini's model, if $$G$$ is some graph in this spin network space of states. A Hamiltonian operator $$H$$ will assign an energy $$E(G) = <\Psi_G|H|\Psi_G>$$ to this graph.

The reason why fotini can describe it this way (she doesn't mention this) along with a few other things I have brought up so far, if $$A(G)$$ are adjacent vertices and $$E(G)$$ is the set of the edges, then it satisfies

$$(i,j) \in E(G)$$

Each vertex represents a spin-state on the graph. Thus we will begin to see Fotini's approach, that to each vertex $$i \in A(G)$$ there can be associated a Hilbert Space

$$H_G = \otimes_{i \in A(G)} \mathcal{H}_i = \mathbf{C}^{2}^{\otimes N}$$

Which takes on remarkable similarities to Fotini's equation describing the total space state. Indeed, the equation above describes the total number of vertices in our Hilbert Space.

Thus, we come back to a familiar garl. The distance between two particles $$i$$ and $$j$$ are vertices $$i,j \in A(G)$$ in fact apply to the least action principle; it will define a graph geodesic between these two points.

And this is why her model works.

Deriving Markoupoulou's Hamiltonian Assiging an Energy to the Graph

I personally don't know if Markoupoulou had officially derived her Hamiltonian which assigns the energy to the Graph, given as $$E(G)$$, but there is a specific way, using a traditional equation

Since each pauli matrices can take on either eigenvalues of $$+1$$ or $$-1$$, remaining hermitian of course (real objects) pertaining to observable quantities, means that we can plug in any of the eigenvalues it permits, which it turns out according to the dimensions of space, there are six eigenvectors all in all.

The energy assigned by the Hamiltonian, is an observable. To derive Fotini's relationship, one can begin with

$$\bar{\mathcal{M}} = E(G) = \sum_i <i|\psi_G> <\psi_G|H|i>$$

Rearranging yields

$$= \sum_i <i|\psi_G|H|i><i|\psi_G>$$

the $$|i><i|$$ disappears since it is the unit matrix which is just unity $$1$$, then we are left with the expectation

$$E(G) = <\psi_G|H|\psi_G>$$

And viola!



I still have more to write, but I need to calculate things a little further before I am bold enough to post it.

 

[Reiku]

My next approach is to apply a spin network, in a unique way. I am going to run it by some people first before I post it. I don't want to give ammo to the bazuka of math that AN lives by. I have not even studied a math course any higher than college. Maybe some will marvel at the fact that I am trying to apply this model to rather (complex) mathematics. I am atleast trying.... or you will end up in AN's camp. A filled camp here at sciforums, where if you have not passed the necessery degree's, why even speak about the subject?

I won't hold back though. Anyone who knows me now well enough will know I won't hold back on a thought or two

 

[Reiku]

Ok... so here it goes...

Spin has close relationships with antisymmetric mathematical properties. An interesting way to describe the antisymmetric properties between two spins in the form of pauli matrices attached to particles $$i$$ and $$j$$ we can describe it as an action on a pair of vectors, taking into assumption the vectors in question are spin vectors.

This is actually a map, taking the form of

$$T_x M \times T_x M \rightarrow \mathcal{R}$$

This map of an action on a pair of vectors. In our case, we will arbitrarily chose these two be Eigenvectors, derived from studying spin along a certain axis. In this case, our eigenvectors will be along the $$x$$ and $$z$$ axes which will always yield the corresponding spin operator.

$$(d \theta \wedge d\phi)(\psi^{+x}_{i}, \psi^{+z}_{j})$$

with an abuse of notation in my eigenvectors.

It is a 2-form (or bivector) which results in

$$=d\theta(\sigma_i)d\phi(\sigma_j) - d\phi(\sigma_j)d\phi(\sigma_i)$$

This is a result where $$\sigma_i$$ and $$\sigma_j$$ do not commute.


The mapping itself can be identified not only with the vector fields, but every differentiable function of that manifold $$M$$ - this itself determined a unique vector field which is generally called the Hamiltonian vector field.


ps... AN please be brutally kind I know next to nothing about symplectic manifolds. I only know what led me to the above. Something is telling me that the points in this post might have some problems. I don't know yet... I'll need to see what the master of math says.

 

[AlphaNumeric]

Now I can't remember off the top of my head which equation I have used uses a metric, but I am almost sure none of my original equations have tackled a theory without geometry simply because we have not set into motion all these required works which will, in the end remove geometry and describe high energy reality.

Are you being deliberately obtuse or are you actually struggling to understanding English? I've told you several times that you explicitly use the SC metric.

ps... AN please be brutally kind I know next to nothing about symplectic manifolds. I only know what led me to the above. Something is telling me that the points in this post might have some problems. I don't know yet... I'll need to see what the master of math says.

I'm not even going to bother. You've obviously just pulling as much bullshit out as you can. You deliberately admit you haven't got any knowledge on this stuff and you go so far as to leave a message on my vistors wall to try to get me to reply.

You're trolling and you're being very obvious about it, just copying massives of expressions from elsewhere. No one thinks you can do any of this mathematics. If you believe you understand it then you have a real honest to god personality issue. I think you have something wrong with you anyway, as no one of sound mind would spend the years and massive effort to lie so often to the same people, each time hoping to convince them.

Like I said, you're 27 and you're spending your time doing this? You need to do something with your life or else you'll be exactly where you are now when you're 40 and you'll have amounted to nothing.

 

[James R]

Moderator note: Reiku has been banned for 2 weeks for trolling, for pretending to knowledge he doesn't have, and for suspected plagiarism.