Lucious

I am the same person.

And it is probably related to the zero point energy field. Photons of light are often virtual particles that have energy and no mass in the zero point range, so a Tachyon is a photon that can exceed the speed of light but only for short periods. It has been postulated that tachyons can be detected by the emission of Cherenkov radiation but how to do this is uncertain as of yet. Some versions of string theory absolutely insist on their existences. In fact, we can loosely say Tachyons exist today as electrons move through superconductors at superluminal speeds. This is of course due to changing the medium in which the electron moves, but is evidence in itself that tachyonic capabilities are not without the realm of possibility in the terms of the relativistic laws. You know, it wouldn't be so strange to find out that light speed is just some special speed - Tachyon and Bradyon speeds could represent a symmetry in the equations that fit nicely to solve some of the most outstanding problems, such as particle entanglement and interaction at large distances. Something myself have been trying to find solutions to using matrix mechanics, unfortunately my last post was a bit of a mess but some folks helped to clean it up. you know, when [tex]\phi = 0[/tex] at the Mexican hat ground state, it's not really viewed as unstable technically. It's actually a very stable Goldstone Boson fluctuating softly at zero point scales. It's only when energy is added to the Boson do we find a flux of the system and it transposes into a particle of mass: In other words, the Higgs Boson gobbled up the energy of the Higgs Boson and used that energy to create a charge of mass.

Funnily enough, I ommited the factor of 2 and well seen! The reason why was because it was a constant and could be configured under the N-power. The correct original equation which would have been given is [latex]G \equiv \frac{Rc^2}{2m}[/latex] But beware, this is only for a two particle Einstein universe example. Instead, we want to talk about the sum of all the systems. I'm glad you mentioned this because for the condition of a simplified two-particle universe [latex]G \equiv \frac{Rc^2}{2m}[/latex] The gravitational equilibrium is missing an extra term, the Torsion, which would make it part of the Full Poncaire Group [latex]\Omega^2\ r = \frac{Gm}{R^2}[/latex] where [latex]\Omega[/latex] is the Torsion field. The Torsion field may also be quantized [latex]\Omega r^2 m = \hbar[/latex]

Hi! That isn't a set exactly, it is a notation used for any mass to an N-power, or it can be a sum as shown [latex]\sum_{i}^{N} m_i[/latex] - The [latex]N[/latex] just represents any integer which may count the systems inside the universe, this is where set theory comes into it. All the systems inside the universe can be simply denoted as [latex]\mathcal{S}[/latex] where it is the ''set of subsystems.'' This is given ''in'' a system, denoted by [latex]\in \mathcal{U}[/latex] where the last symbol represents the whole, or holistic Newtonian system in which all subsystems gravitationally interact. So the question of when you add all causal dynamics in the universe should equally conserve the overall equilibrium, even if local distortions don't agree with this example. Just as we when we look in all directions of the universe, there is no center. To resolve this, all locations in the universe denoted [latex]\mathcal{U}[/latex] are the center and equally the origin of big bang. Or at least, this is what most cosmologists interpret it as. You could actually do more complex math using causal subsystems, but Fotini Markopoulou has written most of the work concerning what the universe looks like from within. (edited) Just to note, the fact the FR-equation is derived independent of time may reflect its more complex quantized form using the General Relativistic equations which lead to the famous Wheeler de witt equation, which has been interpreted to mean the universe is also time independent. It's obtained by making quantized path integrals on the Einstein field equations which yield a very simple, timeless Schrodinger equation [latex]\psi|H> = 0[/latex] As you can see it is a simplified version of the timeless wavefunction, with no time derivative on the RHS. This may have significance with the timeless derivation of FR-equation of cosmological evolution scales. [latex](\frac{\dot{a}}{a})^2 = \frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2}[/latex] Let's put everything to the RHS [latex]\frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2} = (\frac{\dot{a}}{a})^2[/latex] We introduce a universal wavefunction which is dependent on only the generalized coordinate [latex]q[/latex] [latex](\frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2})\psi(q) = (\frac{\dot{a}}{a})^2 \psi(q)[/latex] Keeping this equation in mind, it is totally time independent and if we really wanted to make an evolution in the equation, all we would really need to do is use the variational principle using another state vector which dictate a variational change in the generalized position [latex]\delta q[/latex] (though a small one). A spacially isortopic metric in the line element can be given as [latex]ds^2 = -A(dx^{2}{1} + dx^{2}{2} + dx^{2}{3}) + Bdx^{2}_{4}[/latex] ref 1. (We will come back to this in a moment) For the small quantities we took in [latex]\delta q[/latex] the displacement of the positions of [all] systems in the universe - again, retaining the Newtonian holistic connection between all gravitating bodies. As the scale factor increases according to the equations so must a gravitational redshift between all systems inside the universe. This can be calculated by taking a new metric I derived a while back as an expression form [latex](1 - 2\frac{Gm}{mc^2} \frac{M}{r} + \frac{GQ^2}{c^4 R^2})^2[/latex] If the overall charge of the universe is zero, then the metric charge will also be conserved [latex]Q=0[/latex] reducing it to the more simpler [latex](1 - 2\frac{Gm}{mc^2} \frac{M}{r})[/latex] Now going back to the line element we have [latex]ds^2 = -A(dx^{2}{1} + dx^{2}{2} + dx^{2}{3}) + Bdx^{2}_{4}[/latex] Einstein showed that this equation represents a definition of inertia [latex]\frac{MA}{\sqrt{B}}[/latex] He wanted this inertia to tend to zero [latex]\frac{MA}{\sqrt{B}} \rightarrow 0[/latex] at a spacial infinity. What I noticed is that this couldn't be true under the accepted Newtonian interaction of inertial particles which are continuously connected by the force of gravity. This meant to me at least, that [latex]\frac{MA}{\sqrt{B}} \rightarrow 0[/latex] couldn't be plausible in the classical sense if Newtons first principle of gravitation is wrong. I doubt many could dispute this. ref. 1 http://www.platonia.com/barbour_modern_cosmology.pdf I'll come back to this soon to finish it. Later I am going to show how Einsteins definition of inertia was wrong, but Newtonian physics had it within tangential relationship to the Machian Universe which Einstein based at least 70% of his work upon. The contradiction was that that Einstein wanted to have inertia dissipate to zero at spatial infinity, included to my objection above, I don't believe that infinity is even a real mathematical object of physical significance. I believe that singularities are in fact a direct non-negotiable breakdown of the physics at hand. Before I go into the depth of that math, I have some stuff I want you to read: It is work by myself explaining Machian physics which may be interesting to people here - but on top of this, I show how General Relativity can allow a holistic model in Machian terms according to Einstein's very own principles. This may take about several reads to understand; The problems are numerous and in this essay, I must tackle these issues efficiently. First of all, we need to define what time is. Newton, the father of classical physics defined time as: Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ...'' And concerning absolute space, he said Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space ... Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another. Thinking time flows equably without any reference to anything external, means that time flows relative to the human being. This ''flow of time'' has been in physics literature for a very long time and the conclusions are startling. And though there is nothing external which we can say is in any reference to time, means that any mathematical we use in our equations, assume immediately without any evidence that there is an external time present without the observer. We of course, do not believe in any absolute space or time in the context Newton implied them. In Newton's picture, the universe was a pure vacuum with no quantum activity. In the quantum picture, time doesn't even have a flow. In relativity, there is no true ''flow to time'' either, because fundamental to it is the relativity of simultaneity which argues all events actually happen side by side... time itself was an illusion. Now, this external, intrinsic flow to the observer, which seems to extend from past to future... isn't actually the way modern scientists think about time at all [1] - If time really existed in the quantum sense it would instead be sharp beginnings and ends, fleeting moments of existence which are not continuous. This is the quantum picture of time. The relativity of space says it is continuous. Minkowski (Einstein's teacher) extended his relativity principle to say space-time is continuous also as a fourth dimension of space. (1) The quantum picture of time at first differed with the relativity of time. But sooner or later, the mistakes would show.... Minkowski space was probably based on faulty principles including the idea that time is continuous or discrete... it turns out that time probably doesn't exist at all (this still leaves open the question of whether space is fundamental, maybe something I will tackle in a future paper) (2). Minkowski when he extended Einstein's idea's made time as a fundamental property of the universe. When Paul Dirac was asked about he thought, he said he was ''inclined to disagree to think that four dimensional unity was fundamental.'' Einstein had a bit of a schizophrenic nature on time concerning ''some'' comments over his career, but I think it's clear Einstein never considered time fundamental within physics. He was definitely more than aware that his very own GR which is hailed to this day through experimentation was essentially timeless. After Minkowski had published his theory on four dimensional space, Einstein was known later in his years to comment that the mathematicians ''butchered his theory'' as he knew it. Before Minkowski's idea shot off, Einstein's theory was still basic in how they were interpreted and Machian relativity almost vanished from the minds of physicists altogether; though, not many know, but Einstein was heavily influenced by the idea's of Mach, one of the founding fathers of relativity. Actually Machian theory was more truer in the general relativistic sense Einstein intended his theory than the four-dimensional case proposed by his teacher Minkowski. To be relativistic, you need to talk about each point in space; but points in space are not really physical, only interactions are. This happens because General Relativity is manifestly Covariant which makes sure that the laws of physics remain the same in every coordinate frame. This allows physicists to use diffeomorphism invariance, a beautiful mathematical consequence in which the universe isn't desccribed by a cosmological ... instead it arises as symmetry of the motion of the theory. In other words, it's measuring change using diffeomorphisms to describe ''time without time.'' Machian relativity doesn't use time either to describe the evolution of systems, instead as he famously put it ''we arrive at the abstraction of time, from the changes of things.'' Interestingly, Leibniz also made a similar argument, believing that space in itself made no sense, unless speaking about locations. Time in itself made no sense, unless inferred from the relative movement of bodies... But Mach's idea's where already bubbling, he wanted to advocate a holistic relativistic model ''When, accordingly, we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.'' Gravitation and Inertia, p. 387 On this relativistic ''space-scape'' I call it, changes of physical systems in this universe are our definition of time, without it, if systems did not change, there would be no way to define time at all. Here is one way to articulate the problem, if change is the true definition of time and change happens in space then time isn't space itself, it's a measure of disorder in space. The closest thing we have to the definition of time as we understand it, is entropy and this thermodynamic law gives rise the cosmological arrow of time as it is known using popular science buzzwords. (Later I will provide a ''toy'' model towards a theory of everything using entropic gravity, the entropy part being our definition of the measure of change and gravity being described from the entropic laws). To believe there is an order to things which gives rise to a ''direction'' in time is a very wrong picture to adopt within relativity because of simultaneity. The relativity of simultaneity forbids us from absolutely saying every event in the universe can ever be agreed upon. Because of this, it argues that events may as well happen side-by-side. Time appears between moving observers which dilates when thing happen in different frame of references; in a funny sense, it is time dilation that gives rise to time [anime *]. Also, there is no place in the universe we can point to and say ''it all happened there.'' In fact, according to modern theory, the big happened at every point throughout space. So where is this linear view of time, with a definite past to some definite future? It's completely analogous to applying concepts like ''up'' and ''down'' in the universe because there are none. So equally, there is no ''true'' arrow in space for the evolution of things either. There is only a measure of change in a given instant. Not only is Newtonian time linear, but he implies it is external flowing to us. To this day, many many years later, no physicist has ever shown evidence of this. Time is not an observable and so cannot be ''measured'' in the quantum sense; it doesn't have any corresponding Hermitian matrix. One can ''measure'' time in relativity, but it isn't a true quantum measurement, nor should any scientist really believe a clock measures time... it measures a mechanical change and we measure this by the displacements of it's hands. There is also no-non trivial operator for time in physics. So as you can see, so far the definition of time is tricky, but Newtonian time is most likely incorrect thinking that external time exists. There are no problems with understanding a subjective time existing... in fact there are internal gene regulators in the brain which act a theory to why humans have a sense of time at all, many animals have these internal circadian rhythms; so we have a perfectly reasonable biological explanation for the subjective experience of time. The sense Newton gave his flow of time without relation to anything external, probably because there is no way you can physically relate something external with time (the measurement of time is ill-defined in relativity whereas in quantum theory, time isn't an observable). Time is likely emergent, induced from the presence of matter fields; this would mean time isn't fundamental after all and there are many great reasons to think this might be the case we will get into later. With all the definition nonsense out of the way, let's just get to some hard facts about the model of timelessness. Why is it a problem? How does time fall out of physics? ref. (1) -Recent experiments have shown space-time still to be continuous to great degree by measuring the time taken of distant photons. This was to measure how ''grainy'' space time was. (2) - Markopoulou has already offered a theory in which time is fundamental but space isn't. She argues that the quantum theory of gravity will essentially be spaceless. I have already surmised myself that maybe not only time is not fundamental but maybe space isn't either: this could provide a reason to entanglement - systems continue to be ''connected'' because separation is a macroscopic illusion and isn't fundamental for quantum systems allowing intrinsic non-locality. Perhaps the final theory will not only be timeless then, but also spaceless. The trick is to able to describe ''time without time,'' just as General Relativity does. [anime *] http://en.wikipedia.org/wiki/File:Re..._Animation.gif As you can see in this on-line animation, events A, B and C all occur at different times depending on the motion of the observer. Because no one can agree on when an event happens, time appears to be an illusion where the past and future are simply products of the human mind. Our distinction of the past and future is called the psychological arrow of time; there are biological gene regulators which play the role for our perception of time and the different speeds at which perception happens. Note the latter phenomenon is biological, while simultaneity is a relativistic phenomenon. The Time Problem of Physics It is true in fact, that time falls out of physics in several different ways, in this work we will investigate a few of these approaches, one of the most famous is the Wheeler de Witt equation. You see, in the 1960's Wheeler came to de Witt for help constructing a theory of quantum gravity. To do this, de Witt quantized the General Relativistic equations to find what has become familiar as we have shown it before [latex]\psi|H> = 0[/latex] cont. next page Such a simple equation, it is in fact the timeless Schrodinger equation. What he found was that if you apply quantum mechanics to the universe, you find out it doesn't have a fundamental clock! This was an interesting equation for some other reasons as well: Apparently, this ''field of gravity'' was unlike any of the other quantum fields they dealt with in quantum theory. It was manifestly real, the field wasn't a complex one. Fields in quantum mechanics are inherently complex, so this was an unusual feature of quantum theory. But moreover, it showed that gravity wasn't what it seemed. Later I will propose a way in which scientists might look forward to: This will require a better understanding of gravity (in which there is no universal agreement with any theory as thus far) - we will see later though, the problem might be it is not a quantum field any way. Before we close off this small section of this work, I want to explain not to be deceived by the idea that time hasn't been used in relativity, of course, special relativity incorporates it without any troubles. The unification of GR to SR might find a deep philosophical breakdown of space with time. Already, motion in GR is described by no time parameter while in special relativity, moving observers experience time. It turns out there is a solution: the problem lies in the use of ''observer.'' In quantum field theory, an observer can be a particle, doesn't need to be a conscious moving observer measuring the time it take for light to reach one place to another (this is misnomer), you are actually measuring your own time, not the photons. Instead, particles can act as observers and when you apply these principles in a cosmological approach, you do find time disappearing - at least our ability to describe it. We will find out later why this is. The Model Towards the Fully Relativistic Space-Scape When we think about Minkowski space, it treats one of it's four space dimensions as an imaginary dimension to account for time. A popular way to think about space and time, is that they form a single system that stretches perhaps... forever. In many ways, it is a bit like a landscape. Time and space in this unified picture make a landscape for all the events inside the universe. However, we are arguing of course there is no time... we argue time is a measure of change [in] space, it isn't space itself. Treating it as a dimension of space is treated as faulty in this work. Change is what defines time, if there was no change, there would be no way to describe time. Change then will be described in a 'space-scape' which was just a fancy way of saying ''arena'' and the arena of space is where statistical averages (particles) interact and change with respect to each other - configuration space is the arena in which the physicist works. This fully space theory of relativity, is probably more relativistic if you can imagine such a thing, than even Minkowski space time. Minkowski space time assumes there is an external time coordinate, whereas this space theory correlates strongly to Machian Relativity which describes ''time without time''. Around Newtons time, he knew there where three different ways you could describe motion in the universe, 1. With an asbolute space-time background 2. With boundary conditions at spatial infinity 3. By Machian relativity It was considered that 3. was the hardest of the approaches which said motion was relative and holistic (in the sense that motion is driven in a causal manner by all other systems in the universe). Minkowski relativity slices up in space and time using Lorentz boosts and you can go from coordinate frame to another. Machian relativity is quite different indeed, it says that any point is relative to every other point in the universe. Now... points in space themselves are not physical according to modern relativity, only interactions are. It is believed that the laws of physics should be the same in every coordinate frame, similar to the Lorentz boost allowing you to go to one coordinate frame to another, diffeomorphism invariance allows you to shuffle freely between coordinates in GR. The motion then in GR arises as a symmetry of the theory, you don't actually have a time parameter. This is important, because while special relativity describes moving observers and uses a notion of local time, General Relativity actually acts a lot more like Machian Relativity at times. The technical difference between the boost and diffeomorphism, is that the boost preserves a spacetime interval naturally and Diffeomorphisms shuffle coordinates without any reference to time, again, the motion appears as a symmetry of the theory. The addition General Relativity has done with the motion is to be able to say points themselves are not physical, which means Machian Relativity needs to account for this. In Machian relativity, how can points be relative to each other but points are not physical in themselves only their interactions are? While Lorentz transformations preserved the assumption the rules of physics should be the same in every coordinate frame, it is General Covariance that allows General Relativity to assume the laws of physics are the same in every coordinate frame. As it turns out, Einstein solved this problem concerning how to give meaning back to coordinates in space. Basically you get physical points back when you consider the entire world line and their interaction. (Already, you need to think ''holistically'' about a systems history, than just a point, might we find you also need to think of the universe holistically as well?) There are other valid reasons to think Machian relativity has to describe the world, as we saw earlier, one of the popular timeless problems is when you quantize GR you get back the Wheeler de Witt equation, which is really just the time-independent Schrodinger equation. Moreover, the probabilities are for complete three-geometries and the values of matter fields on them which involves the complete configuration of the universe: not for values of some metric on some underlying manifold [1]. This means already, we are getting a sense of Machian relativity when we have to think holistically about the universe as a whole. Indeed, it wasn't long until Einstein realized the metric tensor could describe such a system. Now we can see how points in Machian Relativity can be physical; we have to formulate the theory not only with relative positions in Machian Relativity but it also must make world lines relative, to give positions a physical meaning. To say Machian Relativity not only makes the points but world lines physical, is akin to General Relativity dragging matter with the metric under Diffeomorphisms. So how do we write this new definition of Machian Relativity? I believe it is an important realization that position and world lines need to be included into Machian relativity. This isn't about ''replacing'' current General Relativity, it's doubtful any theory will better it for a long time to come. Instead, we want to incorporate a timeless understanding using Machian relativity (which in many ways already satisfies many contentions GR makes), again, this is largely due to Einstein being heavily influenced by Machian relativity. Points as it turns out, can be physical in Machian space just so long as you include their world lines and their interactions. And so here is the model suggested, a worldline holistic ensemble of relative moving systems in which perhaps order doesn't actually matter according to the relativity of simultaneity. Now to the math! To complete what was being spoken about before we came onto all this stuff about Machian theory. Ok... so if we actually can deal with relative positions then what about all our mathematical tools that use time to describe, say action... how do you understand any of that without time? Well, actually, we have quite number of the tools required to describe timelessness already. We already have a timeless Schrodinger equation. There is also a timeless action using only generalized coordinates [latex]\int \mathbf{p} \cdot d\mathbf{q}[/latex] This is also related to momentum [latex]\int mv\ ds = \int \mathbf{p} \cdot d\mathbf{q}[/latex] With there being a timeless action, there is also naturally a timeless path integral that exists in quantum mechanics [1]. The Newtonian mechanics of a universe of N-gravitating particles in Euclidean space is given quite beautifully by Julian Barbour. These particles will have a mass [latex]m_i[/latex] where [latex](i = 1,2... N)[/latex] and [latex]q = (r_i)[/latex] then a metric can be written [latex]< dq | dq > = \sum_i m_i dr_i \cdot dr_i[/latex] The metric is flat and too simple to yield non-trivial dynamics which can be obtained by the conformal factor which Barbour derives an action [latex]I = \int [<q_{\lambda}|q_{\lambda}> V(q)]^{\frac{1}{2}} d\lambda[/latex] You can go on from here to derive the Euler Langrange equations once you define a kinetic energy term [latex]T = \sum_i m_i \frac{dr_i}{d \lambda} \cdot \frac{dr_i}{d \lambda} = <q_{\lambda} | q_{\lambda}>[/latex] It should be explained, that the equation doesn't explicitely depend on [latex]\lambda[/latex] which usually plays the role of time. In fact, as Julian Barbour says, it is much more illuminating to think of it in it's timeless form, in which our kinetic energy absorbs that with a new term [latex]T^{*} = \sum_i m_i dr_i \cdot dr_i[/latex] Note, that [latex]r[/latex] has been playing the role our generalized coordinates, we will now convert to normal convention. The kinetic energy is related to the action with a [latex]\lambda[/latex]-derivative [latex]2T = \mathbf{p} \cdot d\dot{\mathbf{q}}[/latex] We can also absorb that [latex]\lambda[/latex] term again to simply write the timeless action [latex]2T^{*} = \mathbf{p} \cdot d\mathbf{q}[/latex] As explained, Julian Barbour has indeed wrote out a very nice framework for the theory. And it encorporates these holistic, relativistic idea's Mach wanted to think of the universe. Mach was no doubt influenced by the Newtonian idea that all matter in the universe effected all other systems. Mach believed inertia itself was a property induced by all the systems in the heavens, as he once said: ''You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?'' You can feel a bit of this theory in Machian Relativity as well when he argues all systems are relative to each other in the universe, their positions are relative to all other systems. At least in my work, that may only be true if you consider their world lines and their interactions with it. But this subject of inertia brings me to a last discussion which will take us to Julian Barbour's idea which are largely responsible for my interest on the subject. Relativity says there is a problem, if motion is indeed relative then how do you locally define inertia? Well, actually, Julian Barbour has presented many idea's over the years that might help pave a way to an understanding of how to solve the time problem. He has dedicated work to show that Machian Relativity deals with changes of systems where positions are only relative to each other. He defines it by taking a generic solution of the Newtonian many body problem [2] of celestial planets [3]. Instead of giving time to systems described by their various configurations, you can simply give a sequence of the events. He omits the information (he says which is characterized by six numbers) which describes the position and orientation of the system, he does this by specifying the semi-metric [latex]r_{ij}[/latex] - and finally, he omits the scale information (he calls one number) contained in the metric (separation). He says this is best done by normalizing them by the square root of the moment of inertia [latex]I = \sum _{i} M_i \mathbf{x}_{i}^{2} = \frac{1}{M} \sum_{i <j} m_i m_j r^{2}_{ij}, M = \sum_{i} M_i[/latex] ''The resulting information'' he says, ''can be plotted as a curve on phase space.'' But perhaps more importantly he seems to have solved this problem about how one defined inertia within the theory; it keeps all the valuable information that the original Machian theory was based upon and that involving relative positions. In prof. Susskinds lecture he says the scale factor doesn't depend on the position [math]x[/math], but this would also be in disagreement with Newtonian N-body gravitating systems holistically-connected. If this is the case, then the scale factor very much depends on the position of all systems in the universe ref 1. He also claims the Hubble constant doesn't change with position but changes with time. Assuming he is wrong about the dependency of position, then he is in disagreement with a very workable Machian theory involving change and position including those all-important worldines. The hubble constant in the context could very well include all systems, in opposition to Susskinds claim that the hubble constant IS a function of time but NOT a function of which galaxy we are talking about. This may have something to do with only considering localized events rather than considering a holistic picture within General Relativity. more later [EDITED] REF. - http://www.academia.edu/4228885/Is_the_universe_in_equilibrium

Thank you for the corrections. I wrote this up yesterday and it appears my signs got muddled up nastily. Again, thank you to you both, you have written it correctly.

Assuming a trivial sphere model of the universe where there is a unique center (so long as there is equally a unique center in all parts of the interior of the sphere) then there is using Newtons law a gravitational equilibrium with all other gravitating systems [latex]G \equiv \frac{Rc^2}{m^{N}}[/latex] of all [latex]N-states[/latex] of the various systems of mass [latex]m^{N}[/latex] We can take from any center of origion, [latex]x=1[/latex] for simplicity and is effected by how everything else is moving (These would be physical subsystems in set algebra where the universe is a set [latex]\mathcal{S} \in \mathcal{U}[/latex] where [latex]\mathcal{S}[/latex] are the systems, or subsystems and [latex]\mathcal{U}[/latex] is the universe). When [latex]x=1[/latex] its distance from the center at [latex]x=0[/latex] is by definition the scale factor [latex]a[/latex]. It's kinetic energy [latex]\frac{1}{2}\dot{a}^2[/latex] of outward motion is using a standard equation of cosmology is [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^3 \rho G[/latex] for our sphere model. We know it is in an equilibrium so we can replace [latex]G[/latex] with [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^3 \rho \frac{Rc^2}{m^{N}}[/latex] Now divide by the distance which is just [latex]a[/latex] [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} \frac{a^3 \rho (\frac{Rc^2}{m^{N}})}{a}[/latex] Using a quotation from Susskind, this equation under the form of Newton is independant of time. In other words its a timeless model - but it does describe change. It's also constant which we will denote as [latex]\frac{1}{2}\dot{a}^2 - \frac{4 \pi}{3} a^2 \rho \frac{Rc^2}{m^{N}} = \pm k[/latex] We can further mutipy it by 2 to remove the half [latex]\dot{a}^2 - \frac{8 \pi}{3} a^2 \rho \frac{Rc^2}{m^{N}} = \pm k[/latex] To obtain the Hubble constant, we further divide it by [latex]a^2[/latex] [latex](\frac{\dot{a}}{a})^2 = \frac{8 \pi}{3} \rho \frac{Rc^2}{m^{N}} - \frac{k}{a^2}[/latex] Which leads to the well-known F-R equation of cosmology except in the condition of explicit gravitational equilibrium. It almost seems absurd to think of a universe which is in a gravitational equilibrium when we observe inequality of gravitational forces between different bodies of masses. So what could any good come from thinking in this completely strange way? We must remember, no location inside the sphere can be unique, meaning that every point inside of space is the center of the universe. Because of this complication of fact according to modern interpretation, the equilibrium exists as a network of non-trivial points inside the universe. This may explain the almost uniform observations of what has come to be known as the ''flatness problem.'' In each point in space, there is an [latex]x = 0[/latex] with a scale factor [latex]a[/latex] to a point [latex]x=1[/latex] in some three dimensional hypersphere - but in the model of there also being a location [latex](x', x'', x'''... x^{N} = 0)[/latex] points in space that are all metrically connected to every scale factor [latex](a',a'',a'''... a^N)[/latex] separating everything object of mass [latex]m^N[/latex] in space: You could say that when you calculate all the forces it conserves to zero. In other words, every action and negative reaction which has happened in space holistically comes to a big zero - a form of conservation in its own right. So it is possible to have a Robertson-Friedman equation for a gravitational equilibrium between the masses of interactions, so long as we don't care when they happen. Perhaps the relativity of simultaneity backs this up - perhaps no one can truly agree when events actually happen, so there is an obscurity when you think about the interactions in the first.

I apologize, this equation [latex] i \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} [/latex] expressed the appearance of a new matrix which wasn't accounted for initially on the RSHS, it really has the appearance; [latex] i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}_i = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} [/latex] Where [latex]\mathbf{A}_i = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} [/latex] Chirality is a product of the four common gamma matrices including the pseudoscalar, it is given as [latex]\gamma^5[/latex] represents chirality and chiraity explains the ''handedness'' or spin of a system. Another beautiful but still mysterious part of the equations is that the product of the operating matrices create a diagonally-dominent matrix with negative eigenvalues [latex]A_i \cdot A^i[/latex] Indeed, but mine is represented as a dirac basis in the definition of the fifth gamma matrix, chirality in other words. There is a reference to it in wiki http://en.wikipedia.org/wiki/Gamma_matrices#Dirac_basis [latex]\gamma_3 \mathbf{A}^i = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{5}[/latex] [latex]\gamma^3 \mathbf{A}_i = i\gamma_0 \gamma_1 \gamma_2 \gamma_3 = \gamma_{5}[/latex] We must keep in mind, that even though the above tranformation into two Chiraities hold true using the new definitions [math](\gamma_3 \mathbf{A}^i,\gamma^3 \mathbf{A}_i)[/math] a useful known identity for chirality is [latex]\gamma^5 \gamma^5 = \mathbf{1}[/latex] - the only logical question is whether the notation we have used [math]\gamma^5\gamma_5[/math] holds the same as the product [math]\gamma_3 \mathbf{A}^i \cdot \gamma^3 \mathbf{A}_i[/math] Let's work it out [math]\gamma^3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1\\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix}[/math] To find [math]\gamma_3[/math] you simply calculate [math]\gamma_3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}_i = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math] So [math]\gamma^3A_i = [/math] [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math] [math]= \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix}[/math] [math]\gamma_3 A^i = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix}[/math] Now to calculate the final matrix product [latex]\gamma_3 A^i \gamma^3A^i[/latex] [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}[/math] The only way to retain a symmetry of some kind then it seems that we can still retrieve a diagonally-dominant matrix if the order or multipication (for some unknown reason) did not matter. In this case [math]\gamma^3\gamma_3A^iA_i[/math] [math]= \mathbf{I}_4 \cdot \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix} = \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math] The fact we can still get the diagonal negative entries by assuming the order of multiplication doesn't matter needs further justification. when they compute they give a Diagonally Dominant Matrix, if all its diagonal elements are negative, then the real parts of its eigenvalues are negative. These results can be shown from the Gershgorin circle theorem. This further indicates that there are two solutions in which the possible eigenvalues [math]\pm[/math] can take this as a property of Chirality, which should be thought of as a ''handedness'' to particle systems. note: [math]Det[A_i] = Det[\begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}] = \mathbf{1}[/math] [math]Det[A^i] = Det[\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}] = \mathbf{0}[/math] [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}[/math]

Well the new operating matrices [math]A_iA^i[/math] are ''like a new luggage'' we would have to carry on in the equation because when they act with the gamma matrices [math]\gamma^3\gamma_3[/math] they produce the chirality [math]\gamma^5\gamma_5[/math]. The Dirac Basis has non-trivial sign changes to describe up and down spin states, and just attaching this object which is a matrix [math]\gamma^3A_i[/math] onto just one of the mixed states would allow it to tell the system which state it would be in. It's almost like a mathematical Akashic record, some sort of information written into the system during initial entanglement.

My story, the first three parts. TO EXECUTE A MURDER Join the quirky, hands-on Doctor Theo Rea and his left wing theories on psychology as he helps a former friend DI Meadows solve a series of brutal murders in Cambridge. On their doorsteps young men have been showing up dead, mutilated and dumped in public areas. Theo already has a theory on the mind of this serial killer and his friend Alice Meadows (and her team) is more than happy to embrace the help except for her boss; but being able to put this aside, there are other consequences working in the difficult field of criminal profiling as Theo will find out. 1 And so it was, the first day of his lecturing finished acting as a stand-in professor at Cambridge that Doctor Theo Rea welcomed the idea of a much needed and warming meal at a local diner. Thankfully for him, the bus wasn't too crowded... you see, Dr. Rea had a particular quirkiness... an idiosyncracy one might call it. He was himself a socially innept individual with very little people skills. Ironic then, how he worked in a university, but this wasn't his main practice - he was used to working away at his desk at home with a traditional pad and ink quell to work on, none of this ''art of the tech computer stuff'' as he referred to it. His real problem is that he simply couldn't comprehend the computer; even the simplest mobile phone has enough buttons to press his own. This is why he refuses to use them, as odd it might seem. He is possibly one of the greatest psychologists in the country and yet he still cannot manage a mobile phone. This is the great and wonderful Dr. Theo Rea and he was about to enjoy the great and wonderful Monday house steak with fries on the side, all at a reasonable price of course. The diner he was about to arrive in, was well-known to be quiet in the evenings, so he specifically goes after the university so that he can avoid as many people as possible. As he entered the diner, he picked up from the rack one of the newspapers and took a seat at a table nearby. As he opened the newspaper that had folded over, the front headline bore the gruesome title: 2nd Man Mutilated Police Investigating Serial Killer ''It's not a serial killer,'' he murmered. The waitress came over to him and he looked up before she had a chance to ask him, ''just the usual please,'' and the waitress nodded and replied, ''No problem Theo.'' He looked back down at the story and started to read how the man had been subjected to some kind of sexual mutilation with a sharp object... the newspaper wasn't holding back, he thought. Another man only four days back had been found dumped in a sewer... he could remember it well for the amount of attention it recieved, naturally. He could remember the details of that case as well - the same kind of injuries being reported in this latest incident. It was to his surprise, the latest murder occured only a five minute walk from the diner. This latest man had been dumped in a bin apparently. Ever since hearing and reading about the first murder, doctor Theo had alreadt began profiling possible killers in his mind. You see, Theo is in fact a work-a-holic by the definition. No matter whether he is inside the university giving his lectures or in the comfort of his own house, or outside eating in this case, his brain was always active and this murder case was already forming in his head like an intricate Agatha Christie novel. Just as a police officer is trained to think like a criminal, a psychologist is trained to think like anyone- they are trained to think about emotions and other various feelings linked to the wide spectrum of human behaviour. Not only is he a psychologist, he is also a criminal profiler - the criminal profiling came into the equation when he decided to make his PhD thesis on the study of criminal behaviour. He had ordered his usual, house steak and chips, but unusually, he decided not to leave an empty plate, but instead take a quick detour to the location of the murder scene, which was along Maggie Street, behind a well-known local restaurant. Apparantly the body had only been found hours earlier so when he arrived on the scene dressed still in his work clothes, he blended somewhat in with the local reporters in their fine suits and ties. Theo recognized one of the police officers standing next to a forensic tent and decided to approach them. ''Well, it sure has been a while, Alice,'' said Theo, moving towards the officer for a welcoming hug. DI Alice Meadows was an old school friend of Theo's back in the day... and also a school sweetheart whose relationship lasted half a year... can seem like a lifetime in the eyes of a child; Theo was right though, so much time had passed and he hardly recognized her, nor did she recognize him at first. ''Oh...! Theo! How you been?'' DI Meadows replied. ''Not bad, unfortunate we had to meet up under these circumstances...'' he replied. Alice smiled and said, ''What are you doing here? Not that I am not happy to see you or anything,'' showing off her brilliant white teeth, he had forgotted how pleasent she was to the eye - Theo cleared his throat and said on a more serious note, ''Well I did come here for a reason, it was to have a look at the murder scene.'' DI Meadows, looked around in a bewildering manner, ''exactly why would anyone want to come to a murder scene... God, you didn't know the person did you?'' ''No no, nothing like that,'' he said and continued, ''I'm a psychologist now Alice, I've worked on the criminal profiling side of things for a while.'' ''Are you working for the police now?'' She asked with some astonishment. ''No, but I was training some of them on the psychology of killers and rapists and how to profile a killer or suspect for a year back there.'' ''Oh I see,'' replied DI Meadows, ''You've done well for yourself.'' Theo raised his hand, making a gesture over her uniform, ''you've done well as well I see!'' Her smile dissipated somewhat and as though she didn't feel as enthusiastic replied, ''yes well... it pays a bill.'' ''Of course.'' Theo replied. A silence hit between them for a moment. Both went to say something at the same time and both immediately stopped to a small but awkard laugh, DI Meadows took control of the conversation, ''You know Theo, if you don't mind, is it possible you could maybe give me some insight into these murders... I assume you know there have been two?'' Theo nodded his head and said, ''Yes, yes, only a few days ago, another man, mutilated according to the papers, was found in a sewer, on the other end of the city, right?'' ''That's right,'' she replied, ''Is he or she a serial killer?'' Theo sighed and said, ''I read the papers... they all kept calling them a serial killer - by definition you need to kill at least three or four people to become a true serial killer.'' Alice confirmed the misinformation by saying, ''Yes, a reporter earlier was going about saying it was some kind of serial killer.'' ''The reporter sounds like a gormless reprobate,'' he said. ''Are they a nutjob... a psychopath,'' asked the detective investigator. The doctor replied, ''I don't know about the reporter, but as for the killer, they are likely a psychopath, but their brain works completely fine, I wouldn't call the psychopath a nut job... certainly not to their face. You wouldn't be able to tell them apart in an identity parade if that is what you are thinking. Pyschopaths, especially the high functioning kind have perfectly sound brains, with one exception, they have no empathy and so the signalling to their frontal cortex is practically non-existent. This doesn't make them a killer, nor does is it make them dangerous in any way. It is a bit like a human being who has had a lobotomy, as gruesome as it sounds.'' ''So why do we have these negative connotations towards them?'' The detective asked sheepishly. ''Because of media portrayal, movies ect.'' The doctor replied and continued, ''Take Norman Bates from the movie 'Psycho.' He actually wasn't a psychopath, by definition the clinical term would be a multiple personality disorder. ''Funny isn't it... one of the biggest, well-known movies titled Psycho isn't actually about a psychopath? Psychopaths have been called social predators by some psychologists, but I don't believe this is always the case. Psychopaths wear a mask in public life, even though they enjoy manipulating people, a lot of their destructive behaviour is done away from social life, behind closed doors if you like. They don't need to be solitary, but they function much better away from social situations - that's why psychologists sometimes preferred the lesser evil of a name, sociopath. I somewhat envy the psychopath...'' The detective scrumpled her face and said, ''you envy them? What could there possibly be to envy them?'' ''Not only do they live to gratify their needs, they have no problem dealing with situations in which emotions can cloud a judgement. About 1 in 100 of the population are psychopaths and they are working all around you... as lawyers, judges, police officers and even the more exotic jobs, such as spies for the military intelligence. All these jobs are perfect for the psychopath; they are perfect because they require a cold unforgiving nature to them, no offense to yourself of course being an officer of the law, but these are the facts - in many of these jobs, empathy has to be trained out of you... if you don't already have empathy, you clearly have a jump start on the rest of the class,'' the doctor asserted, ''why... even psychologists and psychiatrists have been known to be psychopaths, what better way to have an understanding of the potentially criminal mind than to have one for yourself.'' The detective sparked up one of her cigerettes and asked bluntly, ''are you one?'' ''No,'' he quickly replied, '' but I do have semantic pragmatic disorder which is very similar to psychopathy. At one time, it was believed that psychopathy was a semantic issue - and this still exists as a theory today. Not only is psychopathy an antisocial phenomenon, it is also linked to bad development or no development at all, to bold behaviour and little to no shame in acts - all these traits can be found on the autistic spectrum. Even autism makes it hard for the person to understand feelings, or recognize them. The Pyschopath is believed to understand feelings but doesn't care about them... this can debated. It's difficult to know whether a psychopath truly understands feelings or whether they simply don't care, after all, they have never experienced a full empathic emotion so how can they distinguish it to something else? I have been doing a lot of work on this area... I do believe that psychopathy shares too many traits with autism to be ignored... I believe psychopathy is intimitely linked to the autistic spectrum of disorders. Some people say the psychopath simply doesn't care about human behaviour while the autistic child doesn't understand human behaviour, such as body language for instance. I can honestly say this view is somewhat wrong, as I was perfectly fine understanding body language, I just had a delayed learning capacity as a child and I found it difficult understanding emotions.'' ''It must give you some great insight into behaviour having the semantic issue... so what can you tell us about the murderer... based on the investigations?'' Asked the detective. Theo paused for a moment and then said, ''There's quite a lot that could be said. The fact the body was dumped in a public area, a bin which no doubt was going to get used the next day means the killer had no intentions hiding their work - they intended fully that their acts would be discovered and usually this is to cause fear. They want to be recognized, they want their work to be recognized. The fact both murders so far parallel each other in the mutilations and the sexual connections shows that this could very well be a serial killer in the making. Serial killers often mutilate and murder their victims in the same ways, it's all about reaching the same gratification, the same level of excitement they first experienced: But like many users of drugs, getting the original high becomes harder and harder and so for the psychopath it escallates into more murders often within months at a time. The murderer who wants to conceal his work, often choose secluded area's to dump the bodies of their victims, like open fields, forests and even deep lakes or rivers. The fact the murderer in this case is dumping bodies in public area's means that the murderer gets most of their excitement in the aftermath, when the bodies are recovered and the media reports on them - the murderer often relives the experience when people recognize the heinous acts because then, it not only exists in their mind. Recognizing a murder has been committed alone can be exciting to the psychopath. ''And why a bin,'' Theo continued, ''Is a bin a statement in itself? You dump rubbish in bins, was the victim considered a piece of garbage to the murderer? I take it we still don't have an ID on the victim?'' ''Still a John Doe, unfortunately,'' the detective replied stubbing the cigarette she had chocked the life out onto a wall and then throwing it down a drain. ''We could be dealing with victim who was a prostitute. The mutilations towards the genital area hints at sexual frustration concerning the murderer, perhaps linked to some childhood trauma or some adult sexual dysfunction. As for the killer, they are extremely likely to be male, you need some considerable strength to pull off these murders. They are likely white aged between 20 and 35 years of age. Living alone is a likely possibility but that may not be the case.'' The detective appeared a little surprised, ''so we are dealing with a homosexual killer?'' Theo replied sternly, ''Now I never said that, of course it is a possibility. It could be a man who has been abused both physically and mentally as a child and are exerting their revenge on adult males later in life or that they are reflecting their own sexual dysfunction on males, maybe because he was bullied early on in his life by other male peers.'' ''Revenge killings?'' The detective replied. Theo didn't like to label the killings with anything definite for logical reasons and said, ''Psychology isn't a science in my mind - and even if it was, it’s certainly not an exact one. There are many possibilities that could have led someone to kill men in such horrific ways, sexual dysfunction is one, exerting their anger for their own biological flaws on other men is another, jealousy is after all one of the oldest sins. It could be a homosexual hater, the other end of the extreme which could answer for the sexual mutilation.'' 2 Meadows and Theo had made it back into the heart of the action, where the media had gathered into an assembly all talking over each other as they filmed for the News. Theo kept a watchful eye over the crowd and how each one interacted with each other and generally how they were acting. It had been well known in the annals of psychopathy that the killer sometimes returns to watch the police in their various investigations. Though the situation had became tenebrous - the investigation had hit a stand-still and all Meadows has was a profiling that may or may not fit the bill; though the profiling made by Theo Rea would certainly prove useful in the long run if he has his theory correct. After his discussions with DI Meadows, he went back home to contemplate the evidence he had been given. After a more complete forensic update and sitting silently on a chair in his humble home, contemplating the psychopaths mind. He knew his job only too well and his job of psychology was to get the mind dirty - the very act of changing his mind to think like the psychopaths like this, he knew, was dangerous but a necessary evil he had to entertain to unlock the darkest secrets of the mind of a psychopath bent on self-indulgence and hate. He stood up from his silence and started to walk around the room in a disgruntled manner, uttering away at himself, getting into the moment as it where... ''You slash, you cut... the arms, legs, genitalia and the face... Why? What is your motive... Are you indulging in the pleasure? He quickly made a 360 degree and opened his arms as though about to announce something important to an avid listener, ''No... the indulgence is the mutilation, the killing is a gift isn't it? To watch them go through the pain, to then become their savior in death?'' He paced back and forward, re-evaluating the logic of the arguments he made... it was like trying to piece a jigsaw with no clear matching pieces - as is for the same case of an unstable mind, it's messy and without focus. ''Why all the cuts, why are you indulging in the mutilation if only to give them peace...?'' He then realized upon saying this almost immediately, ''ah!! You're not at peace are you? You give them peace as though you are ridding yourself of the guilt. The great bearer of gift forgives all right? ''Why the sewer and the bin? You knew the council was going to work in that section of the sewer the next day. You know those bins would be used by the restaurant the next day... God yes, you had every intention for the bodies to be found, did you choose the sewer and the bin because you thought of the bodies as filthy, disposable objects?'' And of course, he remembered an influx of psychopaths over the many years in his training, most originating from America in which they would refer to the victim as ''it'' rather than an actual human being, they saw them as objects. But there was more to it he believed, ''But they are not 'just' objects are they? You are the angel of death, death is your gift. You make them suffer... and if they scream and if they shout... you don't give them death do you? You mutilate them until you get no indulgence from it anymore... Yes... you indulge until they can no longer respond; response is the key element... no response, no indulgence... no indulgence, no thrill. Then...'' Doctor Rea slapped his hands together and gave off an echo down the hall by the door, ''you kill them. You gave them what they wanted, when you wanted. It's all about control then a delusion that somehow you do them a favor... '' On those words, Theo's home telephone rang. He paced over quite quickly and picked it up, ''Hello?'' ''Theo? It's DI Meadows here, there has been another murder.'' Theo's face smothered in fear, his worse assumption had came true, the killer certainly has turned serial, ''Do you want me to come in to have a look?'' ''If you could, Theo. That would be great... meet me at the station in about an hour, I'm just trying to arrange the coroner to collect the body.'' Theo replied, ''ok, no problem.'' He hung up the phone and started walking about to and fro again, thinking about his theory of this particular killer. He moved rooms into the kitchen to take out a blade from his drawer and went back into the living room when he realized something about the case, the actual acts of the murder itself in all its dark details. He took the blade into his right hand and postured himself into a threatening demeanour. Only a foot or so from his living room single chair, he plunged forward, cutting the air as he delved the cold blade slash by slash into the fabric. One blow after another, he was trying to forget who he was and pretended that he was the killer - his mind was in a deep frenzy between a line of psychosis and hypnosis, he was effectively forgetting who he was and becoming the person he could see in his mind. After he shredded the couch with twenty-one incisions he said, taking some deep calming breaths, ''You cut and cut and cut... Why do we cut people? You want to bleed them dry... oh god... It's the pressure... You're angry.'' He sat back down on his mangled chair carefully placing the knife by a table next to it. He knew, that killing was one thing - people kill people for many reasons and the reason of this killer was most interesting - but only a psychologist would be fascinated by such things. He knew that killing indicated a dark mind, but the multiple mutilations over the body was more than the act of a dark mind, It had a sexual relationship in regards to the wounded genitalia but there was something twisted, something more. The power of the psychologists mind is unique, in some cases, the psychologist knows the mind of the killer even better than themselves. He could see a message being sent by the killer in his actions and it he probably wasn't aware it. He could relate the mutilations to relieving a blood pressure - this pressure was a euphemism of the anger the killer felt. Knowing that it was very possible the killing was anger driven, heightened the theory that the killings had a connection to revenge, hate and maybe even jealousy being a root cause. The sexual dysfunction theory had been amplified by the anger and frustration, the lacerations of the epidermic and organ features showed that whatever caused the anger and frustration had been let out of possibly a long-time bottled malevolent genie. He snapped out of his deep thought and retrieved his jacket and keys to head to the police station. He needed to see the body, he needed more information to rule out any inconsistencies in his theories. The Station, 10:45pm Upon arriving at the station, Theo had noticed DI Meadows was already standing outside, wrapped up in a scarf and a black puffed jacket... clearly out of uniform. Then again, Meadows hadn't expected to be called back in so late, nor did Theo for that matter, which was worrying. After parking his car, he got out and approached her, ''This isn't good.'' ''Death never is,'' Meadows replied with a saddened tone. ''No, I didn't mean that,'' Theo replied, ''I didn't expect a second murder in the same day... And that cannot be a good thing.'' Meadows showed Theo in and took him into a cold storage room where the newly found body was - the body had been covered by a white non-stain sheet and the coroner was doing his various paper work in the background. ''Same circumstances?'' Meadows shook her head, probably in disgust of the crime, ''Theo, it's got worse. Yes, multiple mutilations on the arms, legs, torso, face... but the genitalia, instead of stab wounds, it's been completely severed.'' Theo asked, ''post-mortem?'' ''The evil bastard did it before,'' replied Meadows, carefully taking the sheet away from the body of the victim, ''He was found in a car recycling depot, near the outskirts of the city. The worker in charge was close to crushing the body along with 16,000 tonnes of metal.'' ''What?'' Theo replied confused, ''This is totally different...'' He looked at the state of the body... he could only handle so much of it and turned away rather quickly and continued, ''The last two murders had bodies placed in places which secured they would be found, without any blunder. Why then has our killer put a body in a car crushing machine? There is no guarantee the body would have been found. Also the cutting of the genitals appeared at first to be a tradition of the mind of this killer, almost ritualistic.'' DI Meadows looked confused and asked, ''So what are you saying, this is a different murder?'' ''I don't know,'' Theo honestly remarked, ''To be frank, we either have a copycat killer trying to imitate the recent couple of murders, or the killer is escalating, getting worse.'' ''Escalating?'' Asked Meadows. ''He's a thrill seeker Alice, the mutilations in the first murder, was repeated in the second murder, to imitate that initial thrill. That's why he put the second body in the trash can, it's basically saying he thought his killing was rubbish, its a connection to saying he never got the same thrill. The second murder to him, was pointless... So now the third victim, has had to endure even worse - the cutting of the genitalia, while being fully awake... it's an escalation, he's getting angrier.'' ''So am I,'' Meadows replied, ''I need to catch him Theo. I won't be able to sleep until I do.'' ''I'm a psychologist, not a psychiatrist unfortunately or I'd prescribe you a hypnotic. It's surprising how anyone could sleep after seeing something so gruesome as this,'' Theo said, putting the cover back over the body, to cover what his eyes where seeing, ''I'm going to help you catch whoever this is... if it's the last thing I do.'' 3 ‘’The murderer enjoys not only the abduction, but believes that he is in some way doing their hostages a ’favor.’ – In his mind the only gratification is finding the murdered and mutilated bodies to recover a sense of awe, about himself, what he is capable of doing... almost, reporting it the world. ‘’Yes, whoever we are dealing with, enjoys the luxury of dealing the cards, they don’t like to be a normal card barer, this person wants to be recognized in his efforts and he clearly enjoys all of it, but most importantly, the end of it, his orgasm exists within recognition.’’ ‘’Theo...!’’ DI Meadows grunted at the lowest tone she could go and he quickly looked up. He forgot how this kind of thing can sound to people who have no understanding of criminal psychology, but he was ready to give it a fair shot as he stood remarkably, unperturbed by the presence of the DCI Holland, who was remarkably no more perturbed by what the good doctor was trying to get across, ‘’I know that last part seems strange,’’ he purported and continued with strength, ‘’... But we cannot ignore the sexual element attached to the mutilation of the genital areas. Usually things with a sexual element are at their peak during the final stages of some sexual divulgence, which won’t last. Like all sexual deviations leading to their peak desires often lead to a state of orgasm – his appears to be intimately linked to the aftermath, not only does his peak enjoyment reach the aftermath, it appears to be the aftermath itself.’’ ‘’Hold on please doctor,’’ DCI Holland asked as they stood up from their desk, ‘’Are you able to give us something, with a bit more depth? I have to report back to my Chief who reports this stuff back to someone even higher than the both of us, so what we need, is some facts, some things you know for sure, are not wrong.’’ ‘’Forty lashes if I am?’’ Said Theo, as he realized more than not he was there on his own accord, this had nothing to do with being paid over his assistance, so far. ‘’Put it this way, if you can’t give us accurate information and someone else dies, how would you feel after that?’’ The cold nature of D. Cheif Inspector Holland was as cold as they looked and on that note Theo was about to turn out of the room, when DI Meadows gently grabbed his arm, locking him in a state of solidarity, ‘’I have brought the doctor in today, I believe he can help us on this case.’’ ‘’Ok, but how, DI Meadows,’’ pressed DCI Holland. ‘’He gives us an insight, into the killers mind. I’ve only been in the doctors presence a few times and already I feel like even I am developing a picture of the killer,’’ she looked into Theo’s eyes, ‘’of course, with entirely his help, we can narrate our facts to a male killer, aged between 20-35 perhaps... white in origin he has told me. It’s rare to find black males killing white males unless gang related.’’ ‘’That is not a profiling, that’s guesswork,’’ proclaimed DCI Holland, moving towards the two in a threatening way, but Theo wasn’t fooled by his Gorilla tactics, ‘’I know today, there has to be an alpha male, but it isn’t you,’’ said Theo, moving past the Detective Constable Inspector and towards the window out of which he faced, a large crowd of moving bodies all demanding going about their various lives, he pondered on the mind of a killer who could easily take any one of these helpless souls, ‘’You are right though, what I have given you isn’t a full profile, but it’s the beginning of one and right now, you won’t find better.’’ ‘’Tell me then,’’ pressed the DCI one final time and very slowly, ‘’a fact about this case we can work on!’’ ‘’They want their work noticed and to do this, they need to spread their work to the masses. They do this by taking advantage of what is printed in the news.... Meadows lifted her hand behind Theo’s back as if to assure that he knew what he was doing. Theo continued talking, ‘’The person, whoever responsible, was making sure the bodies where found.’’ ‘’Good!’’ Replied DCI Holland, ‘’something to work on I see.’’ ‘’Oh there’s more, god there’s more. The escalation of murders mean I am missing something, my theory doesn’t need to mean that the killer should escalate; such a mind is deep and the dark places in this mind could take me years to explore,’’ he replied. The DCI shook his head and said, ‘’I am sure you realize that we don’t actually have years, so you will forgive me if I urge you to move along and let my team deal with what we have with a thank you, of course for your assistance.’’ ‘’Wait a minute,’’ said Theo, cutting Meadows off from what she was about to say, no doubt in relation to an objection, ‘’You won’t find him without me. This man doesn’t make mistakes. All I have given you is a possible profile, based on some of the most heinous acts I have seen in a lifetime... Choosing not to listen to me, cannot possibly help your investigations... The mind of this man is a pure psychopath... to find one as disturbed as this one is, is a rare find, far more than 1 in 100.’’ DCI Holland was very professional, but even the doctors words haunted them to the core, the very criminals that walk the street and only the highest threat is considered, a walking psychopath capable of repeating murders just to satisfy some gratification... no one is truly prepared to hear they will maybe have the pleasure of having to stop them before anyone else gets killed: But do so he must and he was beginning to realize that perhaps having Theo work on his team to be not such a bad idea after all. ‘’How long for a full profile, one which will lead us to the killer... would two weeks be enough?’’ The DCI didn’t want to admit defeat, no more than he wanted Theo off the case so he cut right in before anyone could answer, ‘’Good! Report back in two weeks.’’ DCI Holland left the room closing the door with a slight thud, both Theo and Alice looked at each other with an equal sigh of relief. Alice was happy of course, that she didn’t have to unlock the mind of serial killer without the aid of the doctor, while he was happy about a much more important aspect of why he did this work... the experience. ‘’Are you in trouble?’’ ‘’No’’ replied Alice, ‘’at least I don’t think so. Don’t mind him too much, he’s just a bit old school. He knows how a police man should do his job and it never involved psychologists.’’ ‘’New horrors need new tactics Alice,’’ Theo said smiling, ‘’Come let’s get a coffee. I need to get out of here ... it’s too stuffy.’’ ‘’No I can’t, I need to finish some paper work. Why don’t you head home and try and get some sleep if you can, we need you in top shape.’’ A Few Hours Later... As he returned home that evening, he opened a bottle of the finest red wine from his local and conjured in his mind the various grotesque features of the case. He took a large gulp of his wine from his tall glass and said, ‘’you are not scared of being caught. Maybe you want to get caught?’’ He started to boil his kettle, his real first brew of the night – a coffee, none of this de-caff – a strong cuppa so that he would be able to at least bare another hour of consciousness before his sleeping tablet kicked in – he was a pill popping, drinking genius. He sat beside the window, looking into the night sky, puffing away on a cigarette as he watched a rare influx of shooting stars pass by an almost full, bright moon. ‘’Yeah,’’ he mumbled, ‘’Speed had to be your game.’’ He thought about the preparation of where the bodies had been dumped and knew for certain the psychopath could pull off such a crime without any troubles or feelings which might impede their stamina. Throwing bodies into a council sewer, a restaurant bin at least were hardly the places to dump a body without risk of being seen. The odd one out of course, was the body found in the car skip – he feared to admit it aloud to anyone, but he wasn’t going to catch a murderer based only on what he had, true enough, sacrifices could only give way on the proverbial board to lead to mate. He could only hope what most of us could only dream of ... the hope we feel the least amount of pain; but this is what troubled him so. He knew his murderer had to be the most sadistic he had ever contemplated, those only there to revel in the pain of others... to only then to remove the life from a crippled body is unfathomable to the normal mind. But as always, he had to detach himself from the holistic picture – in order to understand the killer, he had to be the killer, at least figuratively-speaking. The more Theo pretended in his mind he was the killer, the more he could try and create a picture of understanding. It’s almost impossible to imagine the things he must pretend in order to reach a state of clarity to produce a theory which can be falsified. He could see a pattern with the way the news had reported on the bodies and the unseen escalation which Dr. Theo feared the most. It was this part of the killer which still eluded him in its grandure; why suddenly have the bodies been mutilated even worse than before, what did this mean? As he tried to contemplate the acts of terror, he fell asleep paradoxically comfortably in his recently-slashed arm chair. He knew he would have had to sleep eventually... he had lecturing in the morning. The Next Day Cambridge University Theo was standing in front of his class, using a slide projector, going through various images on the enlarged screen so everybody could see it. They were ordinary pictures, of what looked like, ordinary men and women. ‘’You’ll maybe see a pattern emerging,’’ said Theo, continuously but slowly passing to the next profile picture. One of his students said, ‘’They all look the same, ordinary people.’’ ‘’Yes, surprising that, but not what I was looking for,’’ Theo said. He switched the projector off and moved forward towards his audience, ‘’Yes, they look normal, but did you notice their ages? They were barely 20-35 years of age each. Did you also notice, most of them where white males?’’ He then started to walk around the room, ‘’I have students in here of all various ethnicities, could all the Asian and African ethnicities remain seated. Can the rest of the white males here please stand up who are between 20-35?’’ A large enough volumn of the men stood up and Theo said, ‘’and here is how statistics works. You people look normal, but here is the thing, a serial killer could be among us today. Even with each passing second, there could be anything up to 30 serial killers in this country alone and we wouldn’t know a thing about it. ‘’Statistically-speaking, I could have a potential serial killer in my lecture room today,’’ Theo ended.

No thoughts?

Abstract To look at the problem of entanglement is often one thought of a mystery because it was illogical how two systems could simultaneously know which state to be in when a collapse (or measurement) is made on the system. Using four new mathematical tools [math](\gamma_3,\gamma^3,A_i, A^i)[/math] you can find an internal symmetry between two superpositioned states directly related to their Chirality [math]\gamma^5[/math]. The relationship to the Chirality of a particle was simply [math]\gamma_3 \mathbf{A}^i [/math] or [math]\gamma^3 \mathbf{A}_i[/math] but they became the definition of the four common gamma matrices with the use of a unit psuedoscalar [math]i[/math]. To start off. ...all we needed was two new matrices [math]\gamma^3[/math] and [math]\gamma_3[/math] involving their own operators (that are also matrices) [math]A^i[/math] and [math]A_i[/math]. The author cannot specify as yet what these matrices are, but they could be a number of things we might tackle near the end. The work on describing hidden variables using new matrices has been an idea I had for a while now, but melding the two together, one can find my matrices answering for long equidistant signalling, not by any superluminal effects, but rather the spin is already determined using these new matrix coefficients on the equations. Let's start off. The eigenstates of a joint system [math]S_1 + S_2[/math] are products of the eignestates of their subsystsmes, which can be inolved with an interaction term. The two states are [math]U(S_0 \phi_{+}) = S_{+} \phi_{+}[/math] [math]U(S_0 \phi_{-}) = S_{-} \phi_{-}[/math] The initial state could be said to be in a superpositioning; [math]\alpha_{+} \phi_{+} + \alpha_{-} \phi_{-}[/math] Normally attempts of measurement yield measurable results of entanglement, but no specific way to ensure the eigenstates and their complex coefficients [math]\alpha_{+}\phi_{+}[/math] and [math]\alpha_{-}\phi_{-}[/math] enable any blueprint as to how. To do this, we must ensure a new principle, ''determinism at initial entanglement.'' To do this, we need some new tools, notably with the use of the gamma matrix [math]\gamma^3[/math] and it's sign change [math]\gamma_3[/math]- the answer will also be in a hidden set of ''Pilot Matrices,' which governs the spin at the initial state no matter the distance between the entangled states. The superpositioning can now be written [math]\mathbf{A}^i\alpha_{+} \phi_{+} + \mathbf{A}_i\alpha_{-} \phi_{-}[/math] If you hit this with the four common gamma matrices which give the Dirac basis using a unit pseudoscalar, [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3[/math] yields the appropriate spin designated from a pilot state. Chirality and the Dirac base show too much resemblance, the Dirac base is [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix}[/math] And there are only sign changes attributed to the matrices [math](\gamma^3, \gamma_3)[/math] through the use of the diagonal matrices [math]\mathbf{A}^i = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}[/math] and [math]\mathbf{A}_i = \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math] I can quickly show how these hidden variables play in harmony with other mathematical structures. You can not only derive Chirality, the thing which is the set of spins, but you can also derive fundamental rules: [math]\gamma_3 \mathbf{A}^i = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{5}[/math] [math]\gamma^3 \mathbf{A}_i = i\gamma_0 \gamma_1 \gamma_2 \gamma_3 = \gamma_{5}[/math] and the two corresponding anticommutator relationships are [math]<\ \gamma_3 \mathbf{A}^i, \gamma^{5}\ > = (\gamma^3 \mathbf{A}^i) \gamma^{5} + \gamma^{5}(\gamma^3\mathbf{A}^i)[/math] [math]<\ \gamma^3 \mathbf{A}_i, \gamma_{5}\ > = (\gamma_3 \mathbf{A}_i) \gamma_{5} + \gamma_{5}(\gamma_3\mathbf{A}_i)[/math] The matrices are also related to each other intrinsically it appears [math]\gamma^{3} \gamma^1 \gamma^0 = \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1\\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} = \gamma_{3}[/math] And the product matrix of [math]\gamma^3\gamma_3[/math] turned out to be Hermitian even in using [math]\mathbf{A}^i\mathbf{A}_i[/math] which tended towards a diagonally dominant matrix. For those involving the Chirality formula re-written for our new hidden variable matrices [math]\psi_{R,L} = \frac{1 \pm \gamma^3\gamma_3}{2}\psi_{R,L}[/math] This only holds true naturally because it's eigenvalues depend on whether the diagonal entries are negative, again to do this we must use our operating matrices to dictate it whether it has any [math]\gamma^3\gamma_3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i} \cdot i \gamma_0 \gamma_1 \gamma_2 \gamma_3 \mathbf{A}_{i} = \mathbf{I}_4[/math] And it turns out to be Hermitian. When you separate the left handedness from the right handedness in the equations above you find it's Eigenvalues satisfying [math]\pm 1[/math] because of [math](i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i})^2 = \mathbf{I}_4[/math] This is the same as saying [math](\gamma^5)^2 = \mathbf{I}_4[/math], The term [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}[/math] should also anticommute with the four gamma matrices [math]<\i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}, \gamma^{\mu}> = (i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i}) \gamma^{\mu} + \gamma^{\mu} (i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i})[/math] Take the superpositioning state now by hitting it with the gamma matrices [math]\gamma_3\mathbf{A}^i\alpha_{+} \phi_{+} + \gamma^3\mathbf{A}_i\alpha_{-} \phi_{-}[/math] This is a superpositioning of a chirality using [math]\gamma_3 \mathbf{A}^i = i\gamma^0 \gamma^1 \gamma^2 \gamma^3 = \gamma^{5}[/math] [math]\gamma^3 \mathbf{A}_i = i\gamma_0 \gamma_1 \gamma_2 \gamma_3 = \gamma_{5}[/math] It shows that the chirality (spin) has been determined by both [math]\gamma_3\mathbf{A}^i[/math] and [math]\gamma^3\mathbf{A}_i[/math] Let's write [math]\gamma_3A^i \cdot \alpha_{+} \phi_{+} + \gamma^3\mathbf{A}_i\alpha_{-} \phi_{-}[/math] out in full matrix form for the new entries. [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} \cdot \alpha_{+} \phi_{+} + \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\cdot \alpha_{-} \phi_{-}[/math] Keep in mind, these two matrices are really [math](\gamma_3\mathbf{A}^i, \gamma^3 \mathbf{A}_i)[/math], When you multiply them through you end up with [math]\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \alpha_{+} \phi_{+} + \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \alpha_{-} \phi_{-}[/math] Notice that we end up on both sides of the superposition is the Dirac basis - the only way we could find this connection is through the use of a second new Dirac gamma matrix, namely [math]\gamma_3[/math]. There are some interesting symmetries involving how we use the operating matrices [math](A^i,A_i)[/math], and instead make [math]A_i[/math] (for instance) act on [math]\gamma_3[/math] we get negative eigenvalues: For [math]\gamma_3A^i \cdot \alpha_{+} \phi_{+} + \gamma^3\mathbf{A}_i\alpha_{-} \phi_{-}[/math] written out in full matrix form for swapped operating matrices yields. [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix} \cdot \alpha_{+} \phi_{+} + \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix} \cdot \alpha_{-} \phi_{-}[/math] when it is calculated through you get [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \cdot \alpha_{+} \phi_{+} + \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \cdot \alpha_{-} \phi_{-}[/math] Consider the states being composed of an initial state vector [math]\psi_0[/math]. The interaction [math]U[/math] will transform as [math]\psi = U \psi_0 = \alpha_{+} \phi_{+} + \alpha_{-} \phi_{-}[/math] [math]\psi = \alpha_{+} \chi_{+} \phi_{+} + \alpha_{-} \chi_{-} \phi_{-}[/math] The experiment will detect the joint state [math]\chi_{+} \phi_{+}[/math] with a probability of [math]|\alpha_{+}|^2[/math] and likewise one can make [math]|\alpha_{+}|^2 + |\alpha_{-}|^2 = \mathbf{1}[/math] Without some pilot matrices, the [math]\psi[/math] is said to be in a pure state of the joint system [math]S_1 + S_2[/math] insomuch it can be in either state [math]\chi_{+}\phi_{+}[/math] or [math]\chi_{-}\phi_{-}[/math] Solving the ambiguity of which state it really is in, may depend on subtle sign changes using the tools [math]\gamma^3\gamma_3[/math] and [math]A^iA_i[/math] So to finish off, the spin is already determined, probably during initial entanglement. The mix of states - [math]\gamma_3A^i \cdot \alpha_{+} \phi_{+} + \gamma^3\mathbf{A}_i\alpha_{-} \phi_{-}[/math] was just as example but doesn't say much. Say one such approach would be to find significance in hidden spin flips where one [math]\alpha_{+} \phi_{+}[/math] is denoted by positive matrix entries on the Dirac Basis and the spin down corresponds for negative entries... let's try this. Suggesting we want [math]\alpha_{+} \phi_{+}[/math] to be in an up state and for [math]\alpha_{-} \phi_{-}[/math] to be in a down state, we need to write the gamma matrices and the operating matrices correctly, we need some new representation of the superpositioning, instead we can write [math]\gamma^3A^i \cdot \alpha_{+} \phi_{+} + \gamma_3\mathbf{A}_i\alpha_{-} \phi_{-}[/math] [math]\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \alpha_{+}\phi_{-} + \begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix} \cdot \alpha_{-} \phi_{-}[/math] One can interpret that [math]\alpha_{+}\phi_{-}[/math] has for a particle, a spin up and the matrix [math]\begin{pmatrix} 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \\-1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \end{pmatrix}[/math] would make the state [math]\alpha_{-} \phi_{-}[/math] take on a spin down. But this is only a theoretical example, there might be a better mathematical way of representing this I am yet to find. Though the symmetries are striking like jigsaw pieces filling in an entangled puzzle. index RANDOM THOUGHTS AND ADDITIVES To obtain say the matrix [math]\gamma^3[/math] we would introduce another matrix [math]i \gamma^0 \gamma^1 \gamma^2 \gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0\\0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}[/math] [math]\gamma^3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}^{i} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1\\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix}[/math] To find [math]\gamma_3[/math] you simply calculate [math]\gamma_3 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 \mathbf{A}_i = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\\0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/math] LuciousBass@mail.com