A recent article at scientific american online, reviews the debate and latest research over the idea of non-locality as being a real quality of reality or simply an apparent one, as a consequence of the mathematical framework used to describe the quantum mechanical theory. It covers the various results revealed by people like Bell and Bohm, who provided elegant results that seemed to exclude locality from being a part of reality. Recent findings seem to coroborate those results and lend strenght to the idea that entangled particles are indeed connected in a "spooky" fashion that is independent of their physical distance. However, to me this idea is missing a big fact regarding the nature of reality. Namely, that our theoretical models of it, QM on the small scale and Special and General Relativity on the large scale are incomplete descriptions of the physical world. We still have not resolved the micro and macro worlds using a single theory that smoothly permutes between the various scale domains.
The last 30 years have been spent trying to find resolution in some way, from using string and dimensional theories, to supposing extensions to the standard model, to entire redefinitions of matter and energy as being one and the same with space time itself. These investigations have yielded zero empirically testable fruit, though many of them succeed in finding resolutions that may work, but none are able to target or isolate the precise resolution that applies to the reality we inhabit, nor are any able to make testable predictions that we can use to confirm those theories as the "right" one. One interesting aspect of reality that leads me to believe that what we don't know about reality (the magic that combines QM and GR) is precisely the missing piece that explains if non locality is true, I think when we find that missing magic we will again restore the idea of "locality" except it will be with respect to the relative positioning of the application of the theory. In fact, the reason I believe this has to do with a critical aspect of reality that special relativity that was mathematical revealed several years before Einstein's theory by Hendrik Lorentz, the contraction factor.
1/ 1 - (v^2/c^2)^1/2
This relationship is derived when we attempt to calculate the distances and times associated with measuring an objects position from two different reference frames. Think about what that means, the contraction factor only comes into play when things are moving relative to one another in space and time. We know that this relative positioning is itself relative. Though it can be said that two identical balls placed on a table are non moving relative to one another, they are at two spatially separated positions and any attempt to get one to be where there other is, implies injecting energy into the system and traversing the position across time. Here is where the possibility for an answer to apparent non-locality may come. Non locality in entangled particles assumes that some dimension of the particle set is linked, in the case of electrons or photons this dimension can be the particle sets spin, but the apparent non locality only exists when we represent the spin as a dimensional parameter and NOT as a spacial dimension itself. The newer theories created to attempt resolution of QM and GM have employed the idea of multiply connected spaces to explain reality. The Kaluza-Klein and Kalabi-Yau models of String theories are examples of these theories. They attempt to resolve QM and GM by supposing the existence of additional physical dimensions, these dimensions are said to be 'folded up' and thus unobserved, but what if , what we currently describe as "dimensionless parameters" are indeed these small curled up dimensions? We know for example that spin appears to have polar response characteristics, can this be merely an apparent truth based on our imprecise description of spin as a dimensionless parameter rather than a tightly curled up dimension? If spin being an intrinsic parameter of subatomic particles (and only subatomic ones) exists in a polar (up/down) relationship among collections of particles, is this relationship "real"? A good way to demonstrate how easily we can lose sight of the true nature of a system is by observing a particle moving from different relative positions. For example imagine a square, along the perimeter of this square a dot circumnavigates clockwise, as viewed from above, this simply looks like this:
It is obvious the dot moves along two dimensions from this position, for half the time it is moving up and down and for the other half it is moving left and right. However, when we change our relative positioning to view the dot so that we are viewing the square edge wize, we see that what was complex motion in two dimensions collapses to a simpler motion in 1. Our description of the motion of the dot from this vantage point however still allows us to define the motion consistently *for the relative positions we are currently able to observe*, this is a key distinction as it explains why our mathematical model of the phenomena can be consistent in reality (as is QM and GM) but simultaneously wrong in some subtle way. In this particular example, observation of the dot now does something rather interesting. Just as we know that spin is a polar characteristic of subatomic particles. The motion of the dot , which previously moved half the time left/right and half the time up down, now only moves left/right and the up/down motion has been replaced by a nodal point at either end where the dot appears to "stop". To a relative observer, there is no way to know that the dot is "moving" in an unseen dimension during that nodal point "stop", it instead looks like it has reached the end of a line and after a time standing still has changed its course but those privy to the other relative position, see indeed that the particle has not stopped, it is still exerting energy in an unseen dimension but because our relative position is not privy to that dimension or the energy of the system, we don't see it. From our point of view the dot energy expended while going around the square is different from the dot energy calculated as viewed from the edge (where a dimension is hidden). If we measure the relative difference in energy between these positions we can calculate a dimension contraction factor, a description of how much energy is contracted when a single system is observed from independent sets of orthogonal dimensions. The key finding of this is that, there is no way to prove that any particular position truly IS what we are observing or if it is some view across a set of orthogonal dimensions , a few of which are not aligned to our dimensional view point (as the square was when viewed from above) thus the energy of the dot when it appears to be standing still when viewed edge wise is lost in the edge wise viewpoint despite the fact we can calculate a valid description of its energy from that viewpoint.
The best way to understand what this can reveal about entangled particles adding another particle to the system:
If we imagine that the dots are located at opposed corners of the square and are rotating with respect to one another such that they are moving at the same velocity around the perimeter, they will always be in opposition as they rotate. Imagining this to continue as we adjust our perspective to view the edge and what we see is very interesting, we have the image of the dumbell with the dots stationary for half the time, and switching positions along the "bar" of the dumbell for the other half of the time. Mathematically the energy can be described in two ways:
a) It can be described as the left dot moving to the center point on the "bar" and then moving back toward the left corner.
b) It can be described as the left dot moving across to the right and the right dot moving to the left, crossing at the center and both taking opposite positions.
The amazing thing is, that even though there are two ways to describe the energy of the dots motion in the edge view, the answer in both cases is the same. However, when viewed in the top view we see not only that we are missing the actual behavior of the "dots" but there are two ways in which the dots can be moving and both have different energies, yet from our perspective they read as the same. This proves that energy is lost, when the view we have is only restricted to specific dimensions that we are able to judge energy flow between as a subset of the ones the energy actually flows between. Note, to calculate the energy the paths need not require that the points move at constant velocity around the perimeter, so long as both circumnavigating points together cover the perimeter, the total energy of their action will be the same no matter what their relative speeds. So the question is begged , is it possible to determine the missing energy without acquiring the alternate view, the answer is short and fast: NO. The dimensional ambiguity of the energy calculation is the proof of this and is summed up by the simple diagram above. Note this is not to say that the energy can not be determined through some other means (such as by realizing that the emergence of an infinite state space is an alarm to a dimensional ambiguity in a theory that must be resolved in order to gain a complete understanding of a system!)
I think non locality/entanglement arises precisely because of this difference in what dimensions we are able to observe the undulations of various phenomena (be they moving rockets through space relative to one another, or measuring entanglement states of particle spins after separating them great distances) the apparent non locality of particles only tells us that something is missing in our description. This is shown beautifully in the next diagram:
Note , how the third diagram has two "notches" taken out of the square. Yet, despite those notches the lower dimensional calculation of energy is unable to determine that 1/4 of the energy of traversal of the particles is lost in those notches. From the lower dimensional reference frame, the energy of the segmented square is identical to the energy of the second square, in fact there is an infinite number of combinations of left moving graphs with different notch configurations that will appear to have the same energy from the lower dimensional vantage point. We can't precisely say which physical graph matches our calculated energy. This inverse relationship between our energy calculation and the path traversals of the particles that produce is precisely the uncertainty principle. Revealing itself as an artifact of the infinite states in the configuration space that are ambiguous in position despite being fixed in energy. Or conversely, we can see them as being fixed in position (from the perspective of the lower dimensional reference frame) but uncertain in energy (from the perspective of the higher one) The diagrams show it clearly. If we want to know the position we must rely on the configuration space which can give us a state that could possibly be the right one of the current particles, if we want to know the energy we can get it again but without a guarantee of where the particle really is. A final diagram sums up the idea:
We see that in the limit, the number of dot and perimeter configurations that sum to the same energy in the lower dimensional reference frame is infinite, just like how the states of a wave function are infinite for the history of a particle. We also see, that the higher dimensional energy is larger than the corresponding lower dimensional energy for the system for all n as n approaches infinity. This would account for the missing energy factor over all states. Another important revelation is that the particles on the perimeter exhibit non locality and entanglement, seeming to "know what the partner" is doing, this would be true no matter what size the perimeter, they would appear to move in lock step in the lower dimension. Now imagine that the two particles are simply exchanging energy in an unseen dimension (as here)
So to summarize, non locality may simply be an artifact of our ignorance of dimensions in which the energy of the system moves in but that we are currently unable to measure or account for or are being accounted for in the wrong way in our equations. In the suitable set of dimensions locality is preserved, thus what is revealed is that, just as Einstein's great insight in 1905 was to realize that the only way that the speed of light c could be a constant for all reference frames was for all time and space to be relative, so to is locality itself relative and the relation of relativity is a function of the additional dimensional perspectives through which energy of the system ("event" doesn't suitably describe it) flows. Interestingly, the wave function of a particles trajectory describes the histories that define the position and times that a particle inhabits. Actual calculations require the collapse of the wave functions history space to the actual solution upon measurement. The ambiguity of measuring the energy for entangled particles may simply be a result of our not knowing the specific dimensions in which the particle energies are undulating, which in a way are precisely what the histories describe. The sum over them, in the limit converges on the answer since by including all possible histories it guarantees that the "right" histories are contained even if our calculation for them is 'wrong' in the higher dimensional space in which they are actually undulating. This can be shown using the energy square by imagining physical constructions of the square and dots that are unique, with a moments playing with the figures we see that we can adjust the "L" along the hidden dimension by infinitesimal amounts indefinitely and each configuration gives the exact same energy result in the edge view. Thus just like the configuration space of a wave function, the energy calculation for symmetrically undulating points viewed edgewise assume different configurations. In our view every state that consists of "dots" after the E1 stage return the exact same energy, while in the true high dimensional space the energy is always larger for all values approaching infinity. The indeterminacy of the particles or dots positions comes from the fact that we only see part of how their energy moves (the part that traverses the dimensions we observe...in this case the x axis) because higher states in the configuration state can be made of arbitrary numbers of dots moving in such a way as to appear to give the same energy value in our reference frame, if we know the energy precisely we can only take a guess at the position, if we know the position precisely from measuring in our frame of reference, we can only guess at which energy state of an infinite amount is the current one. So this one diagram illustrates the configuration space of the wave function, it describes what non locality is, it explains why entanglement occurs and it reveals why certainty in energy measurement yields uncertainty in position and visa versa! All the answers fall out of the fact that we are looking at a higher dimensional state in a lower dimensions. I am partial to Einstein's view as non locality would seem to also violate conservation of energy. If sub atomic particle states can link the energy configuration of disparate particles than the energy of a system can not ever really be said to be confined to a given space. My instinct leads to only one absolute in all this, that dimensional variation leads to measurement anomalies when measuring quantities between dimensions. These anomolies show up as infinities, imaginary values and apparent non locality, over the total dimensional space through which the energy is commuting there is no paradox...that is what my gut tells me. The beautiful simplicity of the diagrams above also harken to another truth that we find often tied to the laws of nature, symmetry. Many beautiful results have come from looking at existing patterns and predicting the existence of new physics based on symmetry. The prediction of the positron by PAM Dirac, the prediction of the quark by Murray Gel-Mann and the prediction of many of the elements of the periodic table were revealed using symmetry. I think what ever the ultimate answer is to resolve these issues , it will fall out of the recognition of a simple and beautiful symmetry.
Another major assumption made in this diagram is that the number of dimensions that separate our reference frame and the reference frame of the particle is one. What if it is two or three? I haven't postulated a similar diagramatic proof of it but my hunch is the separation doesn't matter. Each dimension would contribute a portion of missing energy and would only enrich the configuration space of the wave function. The reason the configuration space emerges has to do with the differing dimensionality not the specific number of dimensional differences between measured reference frame and total reference frame for the energy of the system. Finally, all this is ultimately just conjecture, finding geometrical correlations between physical realities or the mathematical structure of quantum mechanical wave functions doesn't equate to those geometrical correlations being an expression of the reality or those functions. The slippery slope of correlation versus causation that all scientists must heed, true "proof" would require more mathematically rigorous treatment and some method for experimental verification, which I am unable to provide at the moment.
Links:
http://www.sciam.com/article.cfm?id=was-einstein-wrong-about-relativity
http://en.wikipedia.org/wiki/Length_contraction
The last 30 years have been spent trying to find resolution in some way, from using string and dimensional theories, to supposing extensions to the standard model, to entire redefinitions of matter and energy as being one and the same with space time itself. These investigations have yielded zero empirically testable fruit, though many of them succeed in finding resolutions that may work, but none are able to target or isolate the precise resolution that applies to the reality we inhabit, nor are any able to make testable predictions that we can use to confirm those theories as the "right" one. One interesting aspect of reality that leads me to believe that what we don't know about reality (the magic that combines QM and GR) is precisely the missing piece that explains if non locality is true, I think when we find that missing magic we will again restore the idea of "locality" except it will be with respect to the relative positioning of the application of the theory. In fact, the reason I believe this has to do with a critical aspect of reality that special relativity that was mathematical revealed several years before Einstein's theory by Hendrik Lorentz, the contraction factor.
1/ 1 - (v^2/c^2)^1/2
This relationship is derived when we attempt to calculate the distances and times associated with measuring an objects position from two different reference frames. Think about what that means, the contraction factor only comes into play when things are moving relative to one another in space and time. We know that this relative positioning is itself relative. Though it can be said that two identical balls placed on a table are non moving relative to one another, they are at two spatially separated positions and any attempt to get one to be where there other is, implies injecting energy into the system and traversing the position across time. Here is where the possibility for an answer to apparent non-locality may come. Non locality in entangled particles assumes that some dimension of the particle set is linked, in the case of electrons or photons this dimension can be the particle sets spin, but the apparent non locality only exists when we represent the spin as a dimensional parameter and NOT as a spacial dimension itself. The newer theories created to attempt resolution of QM and GM have employed the idea of multiply connected spaces to explain reality. The Kaluza-Klein and Kalabi-Yau models of String theories are examples of these theories. They attempt to resolve QM and GM by supposing the existence of additional physical dimensions, these dimensions are said to be 'folded up' and thus unobserved, but what if , what we currently describe as "dimensionless parameters" are indeed these small curled up dimensions? We know for example that spin appears to have polar response characteristics, can this be merely an apparent truth based on our imprecise description of spin as a dimensionless parameter rather than a tightly curled up dimension? If spin being an intrinsic parameter of subatomic particles (and only subatomic ones) exists in a polar (up/down) relationship among collections of particles, is this relationship "real"? A good way to demonstrate how easily we can lose sight of the true nature of a system is by observing a particle moving from different relative positions. For example imagine a square, along the perimeter of this square a dot circumnavigates clockwise, as viewed from above, this simply looks like this:
It is obvious the dot moves along two dimensions from this position, for half the time it is moving up and down and for the other half it is moving left and right. However, when we change our relative positioning to view the dot so that we are viewing the square edge wize, we see that what was complex motion in two dimensions collapses to a simpler motion in 1. Our description of the motion of the dot from this vantage point however still allows us to define the motion consistently *for the relative positions we are currently able to observe*, this is a key distinction as it explains why our mathematical model of the phenomena can be consistent in reality (as is QM and GM) but simultaneously wrong in some subtle way. In this particular example, observation of the dot now does something rather interesting. Just as we know that spin is a polar characteristic of subatomic particles. The motion of the dot , which previously moved half the time left/right and half the time up down, now only moves left/right and the up/down motion has been replaced by a nodal point at either end where the dot appears to "stop". To a relative observer, there is no way to know that the dot is "moving" in an unseen dimension during that nodal point "stop", it instead looks like it has reached the end of a line and after a time standing still has changed its course but those privy to the other relative position, see indeed that the particle has not stopped, it is still exerting energy in an unseen dimension but because our relative position is not privy to that dimension or the energy of the system, we don't see it. From our point of view the dot energy expended while going around the square is different from the dot energy calculated as viewed from the edge (where a dimension is hidden). If we measure the relative difference in energy between these positions we can calculate a dimension contraction factor, a description of how much energy is contracted when a single system is observed from independent sets of orthogonal dimensions. The key finding of this is that, there is no way to prove that any particular position truly IS what we are observing or if it is some view across a set of orthogonal dimensions , a few of which are not aligned to our dimensional view point (as the square was when viewed from above) thus the energy of the dot when it appears to be standing still when viewed edge wise is lost in the edge wise viewpoint despite the fact we can calculate a valid description of its energy from that viewpoint.
The best way to understand what this can reveal about entangled particles adding another particle to the system:
If we imagine that the dots are located at opposed corners of the square and are rotating with respect to one another such that they are moving at the same velocity around the perimeter, they will always be in opposition as they rotate. Imagining this to continue as we adjust our perspective to view the edge and what we see is very interesting, we have the image of the dumbell with the dots stationary for half the time, and switching positions along the "bar" of the dumbell for the other half of the time. Mathematically the energy can be described in two ways:
a) It can be described as the left dot moving to the center point on the "bar" and then moving back toward the left corner.
b) It can be described as the left dot moving across to the right and the right dot moving to the left, crossing at the center and both taking opposite positions.
The amazing thing is, that even though there are two ways to describe the energy of the dots motion in the edge view, the answer in both cases is the same. However, when viewed in the top view we see not only that we are missing the actual behavior of the "dots" but there are two ways in which the dots can be moving and both have different energies, yet from our perspective they read as the same. This proves that energy is lost, when the view we have is only restricted to specific dimensions that we are able to judge energy flow between as a subset of the ones the energy actually flows between. Note, to calculate the energy the paths need not require that the points move at constant velocity around the perimeter, so long as both circumnavigating points together cover the perimeter, the total energy of their action will be the same no matter what their relative speeds. So the question is begged , is it possible to determine the missing energy without acquiring the alternate view, the answer is short and fast: NO. The dimensional ambiguity of the energy calculation is the proof of this and is summed up by the simple diagram above. Note this is not to say that the energy can not be determined through some other means (such as by realizing that the emergence of an infinite state space is an alarm to a dimensional ambiguity in a theory that must be resolved in order to gain a complete understanding of a system!)
I think non locality/entanglement arises precisely because of this difference in what dimensions we are able to observe the undulations of various phenomena (be they moving rockets through space relative to one another, or measuring entanglement states of particle spins after separating them great distances) the apparent non locality of particles only tells us that something is missing in our description. This is shown beautifully in the next diagram:
Note , how the third diagram has two "notches" taken out of the square. Yet, despite those notches the lower dimensional calculation of energy is unable to determine that 1/4 of the energy of traversal of the particles is lost in those notches. From the lower dimensional reference frame, the energy of the segmented square is identical to the energy of the second square, in fact there is an infinite number of combinations of left moving graphs with different notch configurations that will appear to have the same energy from the lower dimensional vantage point. We can't precisely say which physical graph matches our calculated energy. This inverse relationship between our energy calculation and the path traversals of the particles that produce is precisely the uncertainty principle. Revealing itself as an artifact of the infinite states in the configuration space that are ambiguous in position despite being fixed in energy. Or conversely, we can see them as being fixed in position (from the perspective of the lower dimensional reference frame) but uncertain in energy (from the perspective of the higher one) The diagrams show it clearly. If we want to know the position we must rely on the configuration space which can give us a state that could possibly be the right one of the current particles, if we want to know the energy we can get it again but without a guarantee of where the particle really is. A final diagram sums up the idea:
We see that in the limit, the number of dot and perimeter configurations that sum to the same energy in the lower dimensional reference frame is infinite, just like how the states of a wave function are infinite for the history of a particle. We also see, that the higher dimensional energy is larger than the corresponding lower dimensional energy for the system for all n as n approaches infinity. This would account for the missing energy factor over all states. Another important revelation is that the particles on the perimeter exhibit non locality and entanglement, seeming to "know what the partner" is doing, this would be true no matter what size the perimeter, they would appear to move in lock step in the lower dimension. Now imagine that the two particles are simply exchanging energy in an unseen dimension (as here)
So to summarize, non locality may simply be an artifact of our ignorance of dimensions in which the energy of the system moves in but that we are currently unable to measure or account for or are being accounted for in the wrong way in our equations. In the suitable set of dimensions locality is preserved, thus what is revealed is that, just as Einstein's great insight in 1905 was to realize that the only way that the speed of light c could be a constant for all reference frames was for all time and space to be relative, so to is locality itself relative and the relation of relativity is a function of the additional dimensional perspectives through which energy of the system ("event" doesn't suitably describe it) flows. Interestingly, the wave function of a particles trajectory describes the histories that define the position and times that a particle inhabits. Actual calculations require the collapse of the wave functions history space to the actual solution upon measurement. The ambiguity of measuring the energy for entangled particles may simply be a result of our not knowing the specific dimensions in which the particle energies are undulating, which in a way are precisely what the histories describe. The sum over them, in the limit converges on the answer since by including all possible histories it guarantees that the "right" histories are contained even if our calculation for them is 'wrong' in the higher dimensional space in which they are actually undulating. This can be shown using the energy square by imagining physical constructions of the square and dots that are unique, with a moments playing with the figures we see that we can adjust the "L" along the hidden dimension by infinitesimal amounts indefinitely and each configuration gives the exact same energy result in the edge view. Thus just like the configuration space of a wave function, the energy calculation for symmetrically undulating points viewed edgewise assume different configurations. In our view every state that consists of "dots" after the E1 stage return the exact same energy, while in the true high dimensional space the energy is always larger for all values approaching infinity. The indeterminacy of the particles or dots positions comes from the fact that we only see part of how their energy moves (the part that traverses the dimensions we observe...in this case the x axis) because higher states in the configuration state can be made of arbitrary numbers of dots moving in such a way as to appear to give the same energy value in our reference frame, if we know the energy precisely we can only take a guess at the position, if we know the position precisely from measuring in our frame of reference, we can only guess at which energy state of an infinite amount is the current one. So this one diagram illustrates the configuration space of the wave function, it describes what non locality is, it explains why entanglement occurs and it reveals why certainty in energy measurement yields uncertainty in position and visa versa! All the answers fall out of the fact that we are looking at a higher dimensional state in a lower dimensions. I am partial to Einstein's view as non locality would seem to also violate conservation of energy. If sub atomic particle states can link the energy configuration of disparate particles than the energy of a system can not ever really be said to be confined to a given space. My instinct leads to only one absolute in all this, that dimensional variation leads to measurement anomalies when measuring quantities between dimensions. These anomolies show up as infinities, imaginary values and apparent non locality, over the total dimensional space through which the energy is commuting there is no paradox...that is what my gut tells me. The beautiful simplicity of the diagrams above also harken to another truth that we find often tied to the laws of nature, symmetry. Many beautiful results have come from looking at existing patterns and predicting the existence of new physics based on symmetry. The prediction of the positron by PAM Dirac, the prediction of the quark by Murray Gel-Mann and the prediction of many of the elements of the periodic table were revealed using symmetry. I think what ever the ultimate answer is to resolve these issues , it will fall out of the recognition of a simple and beautiful symmetry.
Another major assumption made in this diagram is that the number of dimensions that separate our reference frame and the reference frame of the particle is one. What if it is two or three? I haven't postulated a similar diagramatic proof of it but my hunch is the separation doesn't matter. Each dimension would contribute a portion of missing energy and would only enrich the configuration space of the wave function. The reason the configuration space emerges has to do with the differing dimensionality not the specific number of dimensional differences between measured reference frame and total reference frame for the energy of the system. Finally, all this is ultimately just conjecture, finding geometrical correlations between physical realities or the mathematical structure of quantum mechanical wave functions doesn't equate to those geometrical correlations being an expression of the reality or those functions. The slippery slope of correlation versus causation that all scientists must heed, true "proof" would require more mathematically rigorous treatment and some method for experimental verification, which I am unable to provide at the moment.
Links:
http://www.sciam.com/article.cfm?id=was-einstein-wrong-about-relativity
http://en.wikipedia.org/wiki/Length_contraction
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