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Bell’s theorem refuted: Bell’s 1964:(15) is false

Gordon Watson

Published Jun 1, 2014• 0 citations

Abstract

2014.06.03 – the 25th anniversary of my phone-call re Mermin (1988) – 1989.06.03 Abstract: Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell’s theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell’s theorem: at least one step in his supposedly com-monsense analysis must be false. Using undergraduate maths and logic, we find a mathematical error, a false equality, in Bell (1964). Uncorrected, and therefore continuing, this error undermines all of Bell’s EPR-based analysis and many later variants, rendering them false. We can therefore predict with certainty that all loophole-free EPRB-style experiments will also give the lie to Bell’s theorem. “It is a matter of indifference... whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, [Bell writes] as if λ were a single continuous parameter, ” Bell (1964:195). λ may denote “any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR, ” Bell (2004:242); from his final essay. #1. Re Bell (1964) (which, like other key essays here, is available online; see References): Let the unnumbered equations between Bell’s (14)-(15) be (14a)-(14c). Let a,b, c replace his ~a,~b,~c. Given Bell’s (2004:242) λ specification above (from his final essay), let A(a, λi) and B(b, λ′i) denote the respective outcomes (±1) when Alice and Bob test the i-th particle-pair. #2. NB: Primes (′) distinguish λs in Bob’s domain from those in others. With i = 1, 2,.., N (and j similarly), there is no requirement here that any two particle-pairs should be the same. #3. Let expectation 〈A(a)B(b) 〉 replace Bell’s equivalent term P (~a,~b); etc. Let P (. | Z) denote a probability conditioned on Z: where Z is shorthand for EPRB, the experiment based

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The Bell theorem [103] can be seen in several different ways, as the EPR argument. Initially, Bell invented it as a direct logical continuation of the EPR theorem: the idea is to take the existence of the EPR elements of reality seriously, and to push it further by introducing them explicitly into the mathematics with the notation λ; one then proceeds to study all possible kinds of correlations that can be obtained from the fluctuations of one or several variables λ, making the condition of locality explicit in the mathematics (locality was already useful in the EPR theorem, but not used in equations). The reasoning develops within determinism (considered as proved by the EPR reasoning) and classical probabilities; it studies in a completely general way all kinds of correlations that can be predicted from the fluctuations of some classical common cause in the past – if one prefers, from some random choice concerning the initial state of the system. This leads to the famous inequalities. But subsequent studies have shown that the scope of the Bell theorem is not limited to determinism; for instance, the λ variables may determine the probabilities of the results of future experiments, instead of the results themselves (see Appendix B), without canceling the theorem. We postpone the discussion of the various possible generalizations to §4.2.3. For the moment, we just emphasize that the essential condition for the validity of the Bell theorem is locality: all kinds of fluctuations can be assumed, but their effect must affect physics only locally.

Cambridge University Press eBooks•Aug 30, 2012•0 citations

2014.06.03 – the 25th anniversary of my phone-call re Mermin (1988) – 1989.06.03 Abstract: Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell’s theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell’s theorem: at least one step in his supposedly com-monsense analysis must be false. Using undergraduate maths and logic, we find a mathematical error, a false equality, in Bell (1964). Uncorrected, and therefore continuing, this error undermines all of Bell’s EPR-based analysis and many later variants, rendering them false. We can therefore predict with certainty that all loophole-free EPRB-style experiments will also give the lie to Bell’s theorem. “It is a matter of indifference... whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, [Bell writes] as if λ were a single continuous parameter, ” Bell (1964:195). λ may denote “any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR, ” Bell (2004:242); from his final essay. #1. Re Bell (1964) (which, like other key essays here, is available online; see References): Let the unnumbered equations between Bell’s (14)-(15) be (14a)-(14c). Let a,b, c replace his ~a,~b,~c. Given Bell’s (2004:242) λ specification above (from his final essay), let A(a, λi) and B(b, λ′i) denote the respective outcomes (±1) when Alice and Bob test the i-th particle-pair. #2. NB: Primes (′) distinguish λs in Bob’s domain from those in others. With i = 1, 2,.., N (and j similarly), there is no requirement here that any two particle-pairs should be the same. #3. Let expectation 〈A(a)B(b) 〉 replace Bell’s equivalent term P (~a,~b); etc. Let P (. | Z) denote a probability conditioned on Z: where Z is shorthand for EPRB, the experiment based

EPR (1935) famously argue that additional variables will bring locality and causality to QM's completion; we show that they are right. More famously, Bell (1964) cried 'impossible' against such variables; we give the shortest possible refutation of his theorem. With EPR-based variables -- and no QM -- a thought-experiment delivers common-sense locally-causal accounts of EPRB and GHZ in 3-space. We then find the flaw in Bell's theorem: Bell's 1964:(14a) does not equal Bell's 1964:(14b). Thus, at odds with EPR (and us), Bell's unrealistic theorem and its many variants (eg, Mermin, Peres) miss their mark. In short, mixing common-sense with undergrad math and physics in the classical way so favored by Einstein, we interpret QM locally and realistically. Long may EPR rule OK we say.

An open letter to Bellians and the Annals of Physics' Editors re -- it's a proven scientific fact: a violation of local realism has been demonstrated theoretically and experimentally -- after AoP (2016:67). EPR (1935) famously argue that additional variables will bring locality and causality to QM's completion; we show that they are right. Even more famously, Bell (1964) cried ‘impossible' against such variables; we give the shortest possible refutation of his claims. With EPR-based variables (and without QM), an old thought-experiment delivers a commonsensical locally-causal account of EPRB and GHZ in 3-space. We then name the flaw in Bell's theorem – Bell's error -- Bell's 1964:(14a) ≠ Bell's 1964:(14b) under EPRB. Thus, given Bell's (1988:88) gloss on a snippet of von Neumann's work, ‘There's nothing to Bell's theorem -- nor variants like CHSH (1969), Mermin (1990), Peres (1995) -- it's not just flawed, it's silly; not merely false but foolish.' In short, we show that the whole Bellian canon -- with its no-name brand of local realism -- is all of those, and misleading too. Under EPR, mixing common-sense with undergrad math/physics in the classical way so favored by Einstein, we interpret QM locally and realistically. We thus define a new brand of local realism -- CLR, commonsense local realism -- the union of local-causality (no causal influence propagates superluminally) and physical-realism (some physical properties change interactively). Long may EPR rule OK we say.

Adjusting EPR (to accord with Bohr's insight), and accepting Bell's principles (but not his false inferences), we:— (i) complete the QM account of EPR correlations in a classical way; (ii) deliver Bell's hope for a simple constructive model of EPRB; (iii) justify EPR's belief that additional variables would bring locality and causality to QM's completion; (iv) refute key claims that such variables are impossible — including CHSH, Mermin's three-particle always-vs-never variant of GHZ, and this: in the context of Bell's theorem ‘it's a proven scientific fact that a violation of local realism has been demonstrated theoretically and experimentally,' (Annals of Physics Editors, 2016). In short: we refute Bell's theorem and endorse Einstein's locally-causal Lorentz-invariant worldview.

This open letter challenges Annals of Physics' Editors and Bell's supporters on this front: in the context of Bell's theorem -- after AoP (2016:67) -- ‘it's a proven scientific fact that a violation of local realism has been demonstrated theoretically and experimentally.' We show that such claims under the Bellian canon are curtailed by its foundation on a naive realism that is known to be false; ie, under Bohr's old insight (in our terms), a test may disturb the tested system. Further: (i) We define a general all-embracing local realism -- CLR, commonsense local realism -- the union of local-causality (no causal influence propagates superluminally) and physical-realism (some physical properties change interactively). (ii) Under CLR, with EPR-based variables (and without QM), a thought-experiment delivers a local-realistic account of EPRB and GHZ in 3-space. (iii) Under EPR, mixing common-sense with undergrad math/physics in the classical way so favored by Einstein, we interpret QM locally and realistically. (iv) We find the flaw in Bell's theorem: Bell's 1964:(14a) ≠ Bell's 1964:(14b) under EPRB. (v) EPR (1935) famously argue that additional variables will bring locality and causality to QM's completion; we show that they are right. (vi) Even more famously, Bell (1964) cried ‘impossible' against such variables; we give the shortest possible refutation of his claim. (vii) Using Bell's (1988:88) moot gloss on a fragment of von Neumann's work, we conclude: ‘There's nothing to Bell's theorem -- nor Bellian variants like CHSH (1969), Mermin (1990), Peres (1995); nor Bellian endorsements like those by Bricmont, du Sautoy, Goldstein et al., Maudlin, Norsen, Shimony -- it's not just flawed, it's silly; its assumptions nonsense; it's not merely false but foolish' and misleading. (viii) Our results accord with common-sense, QM, Einstein's principles, EPR's belief and Bell's hopes and expectations.

The Bell theorem [103] can be seen in several different ways, as the EPR argument. Initially, Bell invented it as a direct logical continuation of the EPR theorem: the idea is to take the existence of the EPR elements of reality seriously, and to push it further by introducing them explicitly into the mathematics with the notation λ; one then proceeds to study all possible kinds of correlations that can be obtained from the fluctuations of one or several variables λ, making the condition of locality explicit in the mathematics (locality was already useful in the EPR theorem, but not used in equations). The reasoning develops within determinism (considered as proved by the EPR reasoning) and classical probabilities; it studies in a completely general way all kinds of correlations that can be predicted from the fluctuations of some classical common cause in the past – if one prefers, from some random choice concerning the initial state of the system. This leads to the famous inequalities. But subsequent studies have shown that the scope of the Bell theorem is not limited to determinism; for instance, the λ variables may determine the probabilities of the results of future experiments, instead of the results themselves (see Appendix B), without canceling the theorem. We postpone the discussion of the various possible generalizations to §4.2.3. For the moment, we just emphasize that the essential condition for the validity of the Bell theorem is locality: all kinds of fluctuations can be assumed, but their effect must affect physics only locally.