http://gordita-cartoon-por-pics.tubered69.com/?rebekah-belen

maggie faris porn sock fetish porn star napster of lesbian porn porn modeling agencies post pube porn

]]>Your question now is about all possible observables, including continuous observables such as position and momentum. As we know, in quantum mechanics all observables and states of physical systems are represented by Hermitian operators in Hilbert space. Can they also be represented by points of S^3? No, that is not possible. S^3 is too small a space for that. That is why S^7 is needed, as I have explained in the Octonion-like paper. It turns out that the S^7 needed for this purpose is an algebraic representation space of the quaternionic S^3. So, it is also a Clifford-algebraic space.

Now, to represent all observables we have to go back 92 years and study what von Neumann taught us about how to construct a hidden variable theory for all observables (see his book on QM published in 1932). The trick is to reduce all experimental questions into questions with only “yes” or “no” answers. Such binary questions can then be represented as projection operators in Hilbert space, with eigenvalues, say +1 for “yes” and -1 for “no.” For example, for the position observable X of an electron, we can reduce the problem with questions like: “Is the electron located at x = 3 c.m.?” If the answer is “yes”, then we have only the eigenvalue +1 to worry about, and if the answer is “no” then we have only the eigenvalue -1 to worry about, even though the position of an electron is a continuous variable.

O.k., that is within quantum mechanics. But now you can see that I can use the scalar points of S^7, just like those of S^3, to represent the binary eigenvalues +1 and -1 for any observables. Even continuous observables like position and momentum can be reduced to a set of projection operators within quantum mechanics and scalar points, +1 or -1, within S^7.

Remember that the goal of a hidden variable theory is not to construct an efficient practical theory, which we already have, namely quantum mechanics, but to provide a local-realistic understanding of that practical theory, as Einstein wanted. And the scheme I described above and presented in full detail in the Octonion-like paper does just that.

]]>Sorry if I’m bombarding you with question and perhaps misconceptions. I know your math is right, and I understood how the correlations arise because of S3, but I am way more of a visual person and to really understand something I need a mental picture of how it all fits together. ]]>

https://doi.org/10.1007/s10773-024-05639-2

The point at infinity business is only indirectly connected to the problem at hand, which is to understand how the strong correlations observed in the Bell-test experiments arise naturally if the physical space is modeled as S^3 instead of R^3. Topologically, moving from R^3 to S^3 has to do with the point at infinity, but that is just the beginning. Once we arrive at S^3, we recognize that it is best represented by the set of unit quaternions, for various topological and geometrical reasons. Now quaternions and bivectors (the latter being “pure” quaternions, i.e., without the scalar parts) are naturally non-commutative. Moreover, this non-commutativity is free of all the quantum mechanical baggage like operators and projections postulate to extract observed results from the operators. They are non-commutative for purely geometrical reasons, not quantum mechanical reasons. So, we have a happy situation in S^3 in which we can have our cake and eat it too. Namely, we can have non-commutativity, which is responsible for producing strong quantum correlations, but without the unwanted features like non-locality or non-reality.

But what about the point at infinity, you may ask? Well, it is best to leave it behind. It was just a means to an end. The end goal was to understand the strong correlations in a satisfactory way in terms of quaternions and bivectors constituting S^3, and we have done that.

]]>And, yes, it is indeed the point of hidden variables that there actually are sharp values for both position and momentum at the same time, which are hidden from us because of our inability to measure them simultaneously. But these sharp values cannot be any old sharp values even in a hidden variable theory. They have to be specific eigenvalues of the corresponding quantum mechanical operators. This is the fundamental difference between classical theory and a hidden variable theory. Hidden variable theories are not classical theories. They have to reproduce the precise statistical averages predicted by quantum mechanics. And that can be done only by averaging over specific eigenvalues of the corresponding quantum mechanical operators. The mistake in Bell’s argument is thus the assumption that the sharp values of observables like position and momentum add linearly. But sharp values as specific eigenvalues of the corresponding operators do not add linearly.

]]>.

I brought up the coins because it is a popular way to derive Bell’s inequality P(A, notB) + P(B, not C) >= P(A, not C). It was indeed my suspicion that here bell comes out right simply because the coins do no represent non-commuting quantities. So basically this derivation is completely irrelevant to bell tests.

.

You say “For example, the uncertainty principle dictates that if the position of a free particle is sharply defined then its momentum is not, and vice versa. So, indeed, the possible values of both position and momentum cannot be simultaneously assigned numbers on a real line.” Isn’t that the point of hidden variables though? That there actually is a value for both at the same time, but hidden from us due to the nature of measurement? I’m struggling a little bit here: you derived Bell correlations by assuming the spin is actually pre-defined, as a bivector. ]]>

Fine’s theorem does not say what you say it does. But I know what you are talking about. It says that CHSH inequalities are not only necessary but also sufficient conditions for the probabilities of the pairs of joint measurement results (thought of as pairs of random variables) to be the marginals of the probability of all four random variables involved in the Bell-test experiments. This theorem is not disputed by anyone, as far as I know.

However, Fine has also argued that the assumption of joint probability distributions for non-commuting observables is unreasonable. But if regarded as random variables in a hidden variable theory, all observables, whether commuting or not, have well-defined joint probability distributions. Therefore, Fine’s objection is not accepted by the proponents of Bell’s theorem.

The issue I have raised about the non-additivity of expectation values for hidden variable theories is much more serious. Although related, it is not equivalent to the issue of the existence of a joint probability space for all four pairs of random variables. In particular, the additivity of expectation values implies the existence of a joint probability space, but not vice versa.

There is no problem with either issue in your example of three simultaneous coin tosses because in that case no non-commuting observables are involved. Therefore, for that example, we can indeed define a joint probability space without difficulty.

By contrast, the additivity of expectation values is not respected for non-commuting observables involved in the Bell-test experiments because their values are not sharply defined. For example, the uncertainty principle dictates that if the position of a free particle is sharply defined then its momentum is not, and vice versa. So, indeed, the possible values of both position and momentum cannot be simultaneously assigned numbers on a real line. In a hidden variable theory, values of observables are eigenvalues of the corresponding quantum mechanical operators, and eigenvalues do not add linearly for non-commuting observables. Consequently, the expectation values do not add linearly either.

]]>I have a question concerning the relationship between the validity of additivity of expectation values and fine’s theorem, which states (unless i misunderstood) that all bell theorem says is that there are no joint probability distributions for non commuting observables.

.

Are these two statements equivalent?

.

Does additivity of e.v. imply existence of a joint probability distribution and vice-versa? Or are these completely unrelated? My confusion possibly arises because I didn’t quite get what it means to have a joint probability distribution. Maybe you can give me a simple example involving three simultaneous coin tosses with heads or tails. I’m struggling to see why we can’t define at least counterfactually some probability distribution for the three coins results, even though we are able to toss only two coins.

.

Also, is it correct to say that additivity of e.v. is not respected because what we average over in non-commuting observables are not real quantities (i.e. don’t fit on the real line of numbers)? Like in the case of spin, we’re averaging over limiting scalar points of quaternions. Bell’s inequality assumes spins are vectors, which can be put into a 1:1 correspondence with the real line.

.

Thank you for taking time to address my questions. ]]>

Yes, the reaction of some members of the community to my work is surprising. It seems that Bell’s theorem has become a kind of religion for some people. They get offended if you tell them that it is based on mistaken assumptions and that you can demonstrate that. Instead of people showing interest in my work, I have encountered much abuse, knee-jerk reactions, and ostracization from some members of the Bell community. But slowly some people in the community (like yourself) have begun to take a positive interest in my work. So there is hope that, eventually, people will understand what I have been saying.

]]>