Does realism imply locality or vice versa?

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Discussion Overview

The discussion revolves around the relationship between realism and locality in the context of quantum mechanics, particularly through the lens of Bell's inequality and hidden variable theories. Participants explore whether the violation of one necessarily implies the violation of the other, and the implications for interpretations of quantum mechanics.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if realism is violated, locality may also be violated, as seen in the EPR experiment where wave function collapse occurs simultaneously for both observers.
  • Others clarify that not all hidden variable theories are local, suggesting that non-local hidden variables can exist without violating Bell's inequality.
  • There is a proposal that if no hidden variables exist, the absence of realism could imply that locality is also violated, though this remains uncertain.
  • Some participants differentiate between types of locality, such as local realism and signal locality, indicating that the failure of realism does not necessarily negate signal locality.
  • A participant presents a non-realistic theory that claims to explain observed correlations without relying on realism, inviting further examples from others.
  • Concerns are raised about how the collapse of the wave function in the EPR scenario could be reconciled with the preservation of locality if realism is absent.
  • Discussion includes the idea that if outcomes are not predetermined, various scenarios could arise, including non-local interactions or time-symmetric interpretations that preserve locality.
  • Some participants emphasize that dropping realism or locality leads into complex interpretations of quantum mechanics, with no experimental evidence favoring one over the other at this time.

Areas of Agreement / Disagreement

Participants express multiple competing views on the implications of realism and locality, with no consensus reached on whether one necessarily implies the other. The discussion remains unresolved regarding the interplay between these concepts.

Contextual Notes

Participants note that the definitions of realism and locality are crucial to the discussion, and that the implications of Bell's inequality depend on these definitions. There are references to various interpretations of quantum mechanics, which complicate the discussion further.

  • #121
stevendaryl said:
where
[itex]C_\lambda(\alpha, \alpha', \beta, \beta') \equiv X_A(\alpha, \lambda) X_B(\beta, \lambda) + X_A(\alpha', \lambda) X_B(\beta, \lambda) + X_A(\alpha, \lambda) X_B(\beta', \lambda) - X_A(\alpha', \lambda) X_B(\beta', \lambda)[/itex]

It might not be obvious, but since each of the Xs are between +1 and -1, it follows that [itex]C_\lambda(\alpha, \alpha', \beta, \beta')[/itex] must lie in the range [itex][-2, +2][/itex].

Letting
  • [itex]A \equiv X_A(\alpha, \lambda)[/itex]
  • [itex]A' \equiv X_A(\alpha', \lambda)[/itex]
  • [itex]B \equiv X_B(\beta, \lambda)[/itex]
  • [itex]B' \equiv X_B(\beta', \lambda)[/itex]
  • [itex]F(A,A', B, B') \equiv AB + A'B + AB' -A'B'[/itex]
We want to show that [itex]-2 \leq F(A,A',B,B') \leq +2[/itex], under the assumption that all 4 variables lie in the range [itex][-1, +1][/itex].

Let [itex]A_{max}, B_{max}, A'_{max}, B'_{max}[/itex] be a choice of the variables that maximizes [itex]F[/itex]. Then:
  • Either [itex]A_{max} = \pm 1[/itex] or [itex]\frac{\partial F}{\partial A}|_{A = A_{max}, B=B_{max}, A'=A'_{max}, B' = B'_{max}} = 0[/itex]
  • Either [itex]A'_{max} = \pm 1[/itex] or [itex]\frac{\partial F}{\partial A'}|_{A = A_{max}, B=B_{max}, A'=A'_{max}, B' = B'_{max}} = 0[/itex]
Suppose [itex]\frac{\partial F}{\partial A} = 0[/itex]. That implies that [itex]B_{max} + B'_{max} = 0[/itex]. So in this case: [itex]F(A_{max}, A'_{max}, B_{max}, B'_{max}) = A'_{max} (B_{max} - B'_{max}) \leq 2[/itex]

Suppose [itex]\frac{\partial F}{\partial A'} = 0[/itex]. That implies that [itex]B_{max} - B'_{max} = 0[/itex]. So in this case: [itex]F(A_{max}, A'_{max}, B_{max}, B'_{max}) = A_{max} (B_{max} + B'_{max}) \leq 2[/itex]

So we conclude that if [itex]F > 2[/itex], it must be when [itex]A_{max} = \pm 1[/itex] and [itex]A'_{max} = \pm 1[/itex]. We have two cases: they have the same sign, or they have opposite signs.

Suppose [itex]A_{max} = \pm 1[/itex] and [itex]A'_{max} = +A_{max}[/itex]. Then [itex]F(A_{max}, A'_{max}, B_{max}, B'_{max}) = \pm (B_{max} + B'_{max}) \pm (B_{max} - B'_{max}) = \pm 2 B_{max} \leq 2[/itex]

Suppose [itex]A_{max} = \pm 1[/itex] and [itex]A'_{max} = -A_{max}[/itex]. Then [itex]F(A_{max}, A'_{max}, B_{max}, B'_{max}) = \pm (B_{max} + B'_{max}) \mp (B_{max} - B'_{max}) = \pm 2 B'_{max} \leq 2[/itex]

So in all cases, [itex]F(A,A',B, B') \leq 2[/itex]. We can similarly prove [itex]-2 \leq F(A,A',B,B')[/itex]. So we conclude:

[itex]-2 \leq F(A,A',B,B') \leq +2[/itex]
 

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