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Dfpolis

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In a paper much discussed on this forum (https://www.physicsforums.com/threads/does-the-bell-theorem-assume-reality.964219/),Roderich Tumulka, "The Assumptions of Bell’s Proof" (http://de.arxiv.org/abs/1501.04168) argues that locality entails two specific types of realism, which he calls R1 and R2. He defined these thus:

Here

He gives the following counterfactual argument, which seems unsound on two counts that I will explain.

Why do I think this argument is invalid?

First, the conclusion, R1 is provably false. Consider the same experiment Tumulka has just described, but now have Bob and Alice align their spin detectors at right angles to each other. Then, no matter what spin they each measure, the vector sum of the measured spins cannot be zero. Still, we know that the initial state had zero angular momentum, so the sum of the measured spins is unequal to the initial angular momentum in apparent violation of conservation of angular momentum. We can easily save the conservation law by saying that the detectors contribute to the measured results, so that the measured values are the result of the interaction of the incident quantum with the detector. This violates R1, but not our common sense notions of realistic behavior. It just means that while quantum systems may have definite properties, actual measure numbers are the result of the details of the interaction of those properties with our detectors.

Since Tumulka's conclusion is false, there must be an error in the reasoning that gave rise to it. I see this error in the counterfactual nature of the argument. Somehow the world in which Alice does not perform her experiment must be differnt than the world in which she does. Can this be true without violating locality? Without Alice's experiment affecting the state of the world at Bob's location? I think it can, if Alice's observations can inform her of conditions at Bob's location. Then, if she performs her observation, she is informed, but if she does not, she is not informed.

How could an experiment at Alice's location inform us of conditions at Bob's location? Before I try to answer this, it is important to note that Tumulka has already conceded that it does. Once Alice has measured spin up, she knows that conditions at Bob's location are such that he will measure spin down. Further, it does not matter if she learns of the conditions at Bob's space-time location in a reference frame where she is informed before Bob does his measurement, or after. All that is important is that an observation at Alice's space-time location can inform us of conditions at Bob's.

Part of the answer is that one condition allowing this to happen spreads out from the origin of the experiment. If a spin-0 particle decays into two spin-1/2 particles, then the observation that informs Alice cannot take place before the spin-1/2 particle she observes arrives. So, the region of enablement does not expand faster than the speed of light. Yet, we know that the observed particle cannot carry sufficient information to explain the correlation between Alice's and Bob's observations. So, more is required.

All EPR type experiments involve entanglement via conservation laws. By Noether's theorem, this implicates correlative symmetries. So, might not the "something more" be constraints placed on Alice's and Bob's observations by symmetry? If we lived in a model world in which Bob's observations must mirror Alice's, then Alice would be able to predict Bob's results from hers without a hint of non-locality.

Of course, our world is more complex, but still, it embodies many dynamic symmmetries. One is the anti-symmetry of Fermion wave functions under interchange of coordinates. In Dirac's many-time formulation of relativistic quantum mechanics, this interchange involves the time as well as space coorinates, linking the multi-electron wave function,

linking the wavefunction at all space time points. In principle, this transtemporal symmetry could inform Alice of detector conditions at Bob's location. At the very least, it makes the assumption of detector independence false -- giving hope for a manifest variable interpretation of quantum theory.

I await comments, pro or con.

(R1) Every quantum observable (or at least (a · σ) ⊗IandI⊗ (b · σ) for everyaandb) actually has a definite value even before any attempt to measure it; the measurement reveals that value.

(R2) The outcome of every experiment is pre-determined by some (“hidden”) variableλ.

Here

*I*is the 2×2 unit matrix and*σ*the triple of Pauli matrices.He gives the following counterfactual argument, which seems unsound on two counts that I will explain.

The EPR argument, for the experiment involving two spin-1/2 particles in the singlet state, can be put this way: Suppose that Alice and Bob always choose thez-direction,a = b= (0,0,1). Quantum mechanics then predicts that the outcomes are perfectly anti-correlated,A = −Bwith probability 1. Assume locality. Alice’s experiment takes place in a space-time regionAand Bob’s inBat spacelike separation. There is a Lorentz frame in whichAis finished beforeBbegins; thus, in this frame, there is a time at which Alice’s experiment already has a definite outcome. She can therefore predict Bob’s outcome with certainty, although she cannot transmit this information to Bob before Bob carries out his experiment. Anyway, Bob’s outcome was already fixed on some spacelike hypersurface before his experiment. By locality, his outcome was not influenced by events inA, in particular not by whether Alice did any experiment at all. Thus, the state of affairs inside the past light cone ofB, but beforeBitself, included a fact about the valueBthat Bob will obtain if he carries out a quantum measurement of_{z}I⊗σ. In particular,_{z}Bis a “hidden variable” in the sense that it cannot be read off from_{z}ψ. Since the argument works in the same way for any other directioninstead ofbz, there is a well defined valueBfor every unit vector_{b}, such that if Bob choosesbb_{0}then his outcome will beB_{b}_{0}. Since the argument works in the same way for Alice, also her outcome just reveals the pre-determined valueAfor the particular direction_{a}she chose.a

We see how locality enters the argument, and how (R1), and thus (R2), come out. EPR’s reasoning is sometimes called a paradox, but the part of the reasoning that I just described is really not a paradox but an argument, showing that (L) implies (R1).

Why do I think this argument is invalid?

First, the conclusion, R1 is provably false. Consider the same experiment Tumulka has just described, but now have Bob and Alice align their spin detectors at right angles to each other. Then, no matter what spin they each measure, the vector sum of the measured spins cannot be zero. Still, we know that the initial state had zero angular momentum, so the sum of the measured spins is unequal to the initial angular momentum in apparent violation of conservation of angular momentum. We can easily save the conservation law by saying that the detectors contribute to the measured results, so that the measured values are the result of the interaction of the incident quantum with the detector. This violates R1, but not our common sense notions of realistic behavior. It just means that while quantum systems may have definite properties, actual measure numbers are the result of the details of the interaction of those properties with our detectors.

Since Tumulka's conclusion is false, there must be an error in the reasoning that gave rise to it. I see this error in the counterfactual nature of the argument. Somehow the world in which Alice does not perform her experiment must be differnt than the world in which she does. Can this be true without violating locality? Without Alice's experiment affecting the state of the world at Bob's location? I think it can, if Alice's observations can inform her of conditions at Bob's location. Then, if she performs her observation, she is informed, but if she does not, she is not informed.

How could an experiment at Alice's location inform us of conditions at Bob's location? Before I try to answer this, it is important to note that Tumulka has already conceded that it does. Once Alice has measured spin up, she knows that conditions at Bob's location are such that he will measure spin down. Further, it does not matter if she learns of the conditions at Bob's space-time location in a reference frame where she is informed before Bob does his measurement, or after. All that is important is that an observation at Alice's space-time location can inform us of conditions at Bob's.

Part of the answer is that one condition allowing this to happen spreads out from the origin of the experiment. If a spin-0 particle decays into two spin-1/2 particles, then the observation that informs Alice cannot take place before the spin-1/2 particle she observes arrives. So, the region of enablement does not expand faster than the speed of light. Yet, we know that the observed particle cannot carry sufficient information to explain the correlation between Alice's and Bob's observations. So, more is required.

All EPR type experiments involve entanglement via conservation laws. By Noether's theorem, this implicates correlative symmetries. So, might not the "something more" be constraints placed on Alice's and Bob's observations by symmetry? If we lived in a model world in which Bob's observations must mirror Alice's, then Alice would be able to predict Bob's results from hers without a hint of non-locality.

Of course, our world is more complex, but still, it embodies many dynamic symmmetries. One is the anti-symmetry of Fermion wave functions under interchange of coordinates. In Dirac's many-time formulation of relativistic quantum mechanics, this interchange involves the time as well as space coorinates, linking the multi-electron wave function,

*φ*(**x**_{1,}*t*_{1},*x*_{2},*t*_{2}, ...), by a set of symmetry constraints, e.g.*φ*(**x**_{1,}*t*_{1},*x*_{2},*t*_{2}, ...) = -*φ*(**x**_{2},*t*_{2},**x**_{1,}*t*_{1}, ...)linking the wavefunction at all space time points. In principle, this transtemporal symmetry could inform Alice of detector conditions at Bob's location. At the very least, it makes the assumption of detector independence false -- giving hope for a manifest variable interpretation of quantum theory.

I await comments, pro or con.

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