## Born correspondence between my model and quantum mechanics

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Born correspondence between my model and quantum mechanics

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Fred Diether asked me a question about the precise relationship between my model and quantum mechanics. In his words, what he wanted to know was how does my model "complete quantum mechanics." To understand this fully one has to go back to the original paper by EPR and understand their argument within the context of my model, as I have done in this paper. However, it is also possible to understand this more simply through the following Born correspondence. A useful reference is my very first paper where I proposed the Clifford-algebraic model, but here is the precise mathematical correspondence between my model and quantum mechanics:

$\boxed{\,\left<\,{\bf n},\,\lambda\,\left|\,\sigma\cdot{\bf n}\,\right|\,{\bf n},\,\lambda\,\right> \,=\,\lambda\,=\,-\left({\rm I}\cdot{\bf n}\right)\left(\mu\cdot{\bf n}\right),}$

where

${\bf n} \,=\,\text{an ordinary vector},$

$\sigma\,=\,\text{a vector made out of Pauli matrices},$

$\lambda\,=\,\pm 1,$

${\rm I}\,=\,{\bf e}_x \wedge {\bf e}_y \wedge {\bf e}_z\,=\,\text{a trivector},$

$\mu\,=\,\lambda\,{\rm I}\,=\,\text{a pair of trivectors},$

$\mu\cdot{\bf n}\,=\,\text{a pair of spin bivectors},$

$\rm{I}\cdot{\bf n}\,=\,\text{the detector bivector},$

and

$\left|\,{\bf n},\,\lambda\,\right> \,=\, \text{the usual eigen vector of}\;\sigma\cdot{\bf n}\;\text{with the eigen value}\;\lambda.$
Joy Christian
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### Re: Born correspondence between my model and quantum mechani

Joy Christian wrote:***
Fred Diether asked me a question about the precise relationship between my model and quantum mechanics. In his words, what he wanted to know was how does my model "complete quantum mechanics." To understand this fully one has to go back to the original paper by EPR and understand their argument within the context of my model, as I have done in this paper. However, it is also possible to understand this more simply through the following Born correspondence. A useful reference is my very first paper where I proposed the Clifford-algebraic model, but here is the precise mathematical correspondence between my model and quantum mechanics:

$\boxed{\,\left<\,{\bf n},\,\lambda\,\left|\,\sigma\cdot{\bf n}\,\right|\,{\bf n},\,\lambda\,\right> \,=\,\lambda\,=\,-\left({\rm I}\cdot{\bf n}\right)\left(\mu\cdot{\bf n}\right),}$

For the left hand side of the expression what is a demonstration example that gives us the +1 eigen value of lambda?
FrediFizzx
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Posts: 1195
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### Re: Born correspondence between my model and quantum mechani

FrediFizzx wrote:
Joy Christian wrote:***
Fred Diether asked me a question about the precise relationship between my model and quantum mechanics. In his words, what he wanted to know was how does my model "complete quantum mechanics." To understand this fully one has to go back to the original paper by EPR and understand their argument within the context of my model, as I have done in this paper. However, it is also possible to understand this more simply through the following Born correspondence. A useful reference is my very first paper where I proposed the Clifford-algebraic model, but here is the precise mathematical correspondence between my model and quantum mechanics:

$\boxed{\,\left<\,{\bf n},\,\lambda\,\left|\,\sigma\cdot{\bf n}\,\right|\,{\bf n},\,\lambda\,\right> \,=\,\lambda\,=\,-\left({\rm I}\cdot{\bf n}\right)\left(\mu\cdot{\bf n}\right),}$

For the left hand side of the expression what is a demonstration example that gives us the +1 eigen value of lambda?

The left hand side is "incomplete" in the EPR sense. The right hand side is "complete" in the EPR sense. The left hand side only makes a probabilistic statement:

$\sigma\cdot{\bf n}\,\left| \,{\bf n},\,\lambda\, \right> \,=\,\lambda\, \left| \,{\bf n},\,\lambda\, \right> .$

The left hand side cannot give us the +1 eigen value with certainty. Since it is a quantum mechanical side, it only makes a probabilistic statement. It tells us that if we observe the spin along the direction ${\bf n}$, then we would find spin "up" with 50% chance -- i.e., it says that there is 50% chance that we would observe $\lambda\,=\,+1$.
Joy Christian
Research Physicist

Posts: 1705
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom

### Re: Born correspondence between my model and quantum mechani

Joy Christian wrote:
FrediFizzx wrote:
Joy Christian wrote:***
Fred Diether asked me a question about the precise relationship between my model and quantum mechanics. In his words, what he wanted to know was how does my model "complete quantum mechanics." To understand this fully one has to go back to the original paper by EPR and understand their argument within the context of my model, as I have done in this paper. However, it is also possible to understand this more simply through the following Born correspondence. A useful reference is my very first paper where I proposed the Clifford-algebraic model, but here is the precise mathematical correspondence between my model and quantum mechanics:

$\boxed{\,\left<\,{\bf n},\,\lambda\,\left|\,\sigma\cdot{\bf n}\,\right|\,{\bf n},\,\lambda\,\right> \,=\,\lambda\,=\,-\left({\rm I}\cdot{\bf n}\right)\left(\mu\cdot{\bf n}\right),}$

For the left hand side of the expression what is a demonstration example that gives us the +1 eigen value of lambda?

The left hand side is "incomplete" in the EPR sense. The right hand side is "complete" in the EPR sense. The left hand side only makes a probabilistic statement:

$\sigma\cdot{\bf n}\,\left| \,{\bf n},\,\lambda\, \right> \,=\,\lambda\, \left| \,{\bf n},\,\lambda\, \right> .$

The left hand side cannot give us the +1 eigen value with certainty. Since it is a quantum mechanical side, it only makes a probabilistic statement. It tells us that if we observe the spin along the direction ${\bf n}$, then we would find spin "up" with 50% chance -- i.e., it says that there is 50% chance that we would observe $\lambda\,=\,+1$.

Ok, so that is also how the hbar/2 gets wiped out. Got it. And we still have the 50-50 chance in the RHS.
FrediFizzx
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### Re: Born correspondence between my model and quantum mechani

FrediFizzx wrote:Ok, so that is also how the hbar/2 gets wiped out. Got it. And we still have the 50-50 chance in the RHS.

Yes, on both sides we have 50-50 chance. But the probabilities involved in the RHS are not intrinsic to the spin. They are ordinary coin-toss, or epistemic probabilities.
Joy Christian
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### Re: Born correspondence between my model and quantum mechani

Joy Christian wrote:
FrediFizzx wrote:Ok, so that is also how the hbar/2 gets wiped out. Got it. And we still have the 50-50 chance in the RHS.

Yes, on both sides we have 50-50 chance. But the probabilities involved in the RHS are not intrinsic to the spin. They are ordinary coin-toss, or epistemic probabilities.

So the RHS gives us an initial condition and if we knew it then we could predict the outcome. Not so with the LHS. That is great. Thanks for the explanation.
FrediFizzx
Independent Physics Researcher

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### Re: Born correspondence between my model and quantum mechani

By the way, in the notation of this new simplified derivation of the EPR-Bohm correlation, the Born correspondence between my model and quantum mechanics is:

$\boxed{\,\left<\,{\bf n},\,\lambda\,\left|\,\sigma\cdot{\bf n}\,\right|\,{\bf n},\,\lambda\,\right> \,=\,\lambda\,=\,-\,{\bf D}\left({\bf n}\right)\,{\bf L}\left({\bf n},\,\lambda\right)\!.}$
Joy Christian
Research Physicist

Posts: 1705
Joined: Wed Feb 05, 2014 3:49 am
Location: Oxford, United Kingdom