Does realism imply locality or vice versa?

[stevendaryl]

where
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)

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

Letting

• A \equiv X_A(\alpha, \lambda)
• A' \equiv X_A(\alpha', \lambda)
• B \equiv X_B(\beta, \lambda)
• B' \equiv X_B(\beta', \lambda)
• F(A,A', B, B') \equiv AB + A'B + AB' -A'B'

We want to show that -2 \leq F(A,A',B,B') \leq +2, under the assumption that all 4 variables lie in the range [-1, +1].

Let A_{max}, B_{max}, A'_{max}, B'_{max} be a choice of the variables that maximizes F. Then:

• Either A_{max} = \pm 1 or \frac{\partial F}{\partial A}|_{A = A_{max}, B=B_{max}, A'=A'_{max}, B' = B'_{max}} = 0
• Either A'_{max} = \pm 1 or \frac{\partial F}{\partial A'}|_{A = A_{max}, B=B_{max}, A'=A'_{max}, B' = B'_{max}} = 0

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

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

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

Suppose A_{max} = \pm 1 and A'_{max} = +A_{max}. Then 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

Suppose A_{max} = \pm 1 and A'_{max} = -A_{max}. Then 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

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

-2 \leq F(A,A',B,B') \leq +2