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For now I will only give you an overview of the ideas involved and show you how you should interpret the idea of a "local realistic theory" that cannot exist at the microscopic scale. Once you've read it, and if you feel you need more mathematical rigor to be convinced, then I will draw you step by step the proof of Bell's inequality (it is not the only one ...

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Bell's argument makes very weak assumptions about the behavior of the two particles (which is why it's interesting). In effect, the particles are black boxes that take an angle as input and produce a spin direction as output. There is no restriction on how they choose the spin direction; there could be a source of true randomness in there, or a human being ...

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I think you misunderstood the significance of could for a classical theory. The text below the picture you took from Wikipedia says: "Many other possibilities exist for the classical correlation subject to these side conditions", so classicality does not imply linearity. It does, however, rule out the cosine, by the following (slightly heuristic) argument: ...

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I can answer your question regarding the eigenvalue equation of the reduced density matrix. This is basically Helmholtz integral equation of the first kind. This is an eigenvalue of equation for a continuous variable. You can do the quadrature approximation of the integral and get the matrix eigenvalue equation which is just an approximation due to the ...

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You seem to think of entanglement as a property of individual particles. That is not true: Let $\mathcal{H}$ be the Hilbert space of states of a single particle (electron, photon, whatever, doesn't matter). Then the space of states of two particles is given by the tensor product $\mathcal{H}\otimes\mathcal{H}$, and the space of states of $N$ particles is ...

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Yes, we can have entanglement between different degree of freedom of same particle or system. That is known as ''hybrid entanglement'' and that is experimentally demonstrated also. http://arxiv.org/pdf/1007.1322v1.pdf

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A state like $\frac{1}{\sqrt{2}}(a^\dagger_+(\vec p)a^\dagger_+(-\vec p) + a^\dagger_-(\vec p)a^\dagger_-(-\vec p))|0\rangle$ would be an example. It is both entangled in spin and entangled in momenta.

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(1) Spacetime isn't all in the mind. Spacetime is just the gravitational field and it has degrees of freedom independent of your brain, so it's not all in your mind. You can't make an even happen on Tuesday just by thinking it happened on Tuesday. (2) As commented by others, you can't ride a photon. Also, the way to figure out whether something is real is ...

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Give a physically distinguishable definition of "out there" vs. "in the mind" and we can try to discuss this further. As jinawee comments, there are no frames of reference that move with the speed of light, since the photon we "ride on" would have no speed at all by definition of a comoving reference frame, and that contradicts the constancy of the speed of ...

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