This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell's inequalities and their generalization to more than four observables. INTRODUCTION In this paper we are concerned to present a number of theorems about inferences from phenomenological correlations to hidden causes. In other terms, the theorems are mainly theorems about hidden variables. Most of the proofs will not be given but references will be cited where they may be found. To emphasize conceptual matters and to keep technical simplicity in the forefront, we consider only two-valued random variables taking the values ±1. We shall also assume symmetry for these random variables in that their expectations will be zero and thus they will each have a positive variance of one. For emphasis we state: GENERAL ASSUMPTION The phenomenological random variables X 1 , …, X N have possible values ±1, with means E(X i ) = 0, 1 ≤ i ≤ N . We also use the notation X, Y , and Z for phenomenological random variables. We use the notation E(XY) for covariance which for these symmetric random variables is also the same as their correlation ρ( X, Y ).
This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell's inequalities and their generalization to more than four observables. INTRODUCTION In this paper we are concerned to present a number of theorems about inferences from phenomenological correlations to hidden causes. In other terms, the theorems are mainly theorems about hidden variables. Most of the proofs will not be given but references will be cited where they may be found. To emphasize conceptual matters and to keep technical simplicity in the forefront, we consider only two-valued random variables taking the values ±1. We shall also assume symmetry for these random variables in that their expectations will be zero and thus they will each have a positive variance of one. For emphasis we state: GENERAL ASSUMPTION The phenomenological random variables X 1 , …, X N have possible values ±1, with means E(X i ) = 0, 1 ≤ i ≤ N . We also use the notation X, Y , and Z for phenomenological random variables. We use the notation E(XY) for covariance which for these symmetric random variables is also the same as their correlation ρ( X, Y ).
The primary criterion of adequacy of a probabilistic causal analysis is that the causal variable should render the simultaneous phenomenological data conditionally independent. The intuition back of this idea is that the common cause of the phenomena should factor out the observed correlations. So we label the principle the common cause criterion . If we find that the barometric pressure and temperature are both dropping at the same time, we do not think of one as the cause of the other but look for a common dynamical cause within the physical theory of meteorology. If we find fever and headaches positively correlated, we look for a common disease as the source and do not consider one the cause of the other. But we do not want to suggest that satisfaction of this criterion is the end of the search for causes or probabilistic explanations. It does represent a significant and important milestone in any particular investigation. Under another banner the search for common causes in quantum mechanics is the search for hidden variables. A hidden variable that satisfies the common cause criterion provides a satisfactory explanation “in classical terms” of the quantum phenomenon. Much of the earlier discussion of hidden variables in quantum mechanics has centered around the search for deterministic underlying processes, but for some time now the literature has also been concerned with the existence of probabilistic hidden variables. It is a striking and important fact that even probabilistic hidden variables do not always exist when certain intuitive criteria are imposed.
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.
My contribution to this symposium is focused on the retreat from strong conditions of causality that have been forced upon us by quantum mechanics. My intent is to describe in more or less successive stages the retreat from the paradise of deteministic causation. This retreat has taken place through a thicket of quantum-nechanical details. It is ny intention to describe the general principles involved but to refer to the literature for proofs and full technical elaboration, even of matters that are crucial to the conceptual development. The history of the efforts to prove or disprove the possibility of hidden variables begins at least with von Neumann and includes important work by Kochen and Specker and others, but much of the recent analysis has centered around Bell's inequality and related results. In spite of its importance and significance I shall ignore this earlier history and begin with these recent discussions.
INTRODUCTION The main purpose of this chapter is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell's inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of Bell and Wigner can be weakened to conditional statistical independence rather than conditional determinism because determinism follows from conditional independence and the other assumptions that are made about systems of two spin-½ particles. The second direction is to question the attempt of Clauser and Home (1974) to derive a Bell-type inequality for local stochastic theories of hidden variables which use an assumption of conditional statistical independence for observables. The main thrust of our analysis obviously arises from the probabilistic lemma we prove. Roughly speaking, this lemma asserts that if two random variables have strict correlation, that is, the absolute value of the correlation is one, and it is in addition assumed that their expectations are conditionally independent when a third random variable λ is given, then the conditional variance of X and Y given λ is zero. In other words, given the hidden variable λ the observables X and Y are strictly determined. The lemma itself, of course, depends on no assumptions about quantum mechanics. It may be regarded as a limitation on any theories that assume both strict correlation between observables and their conditional independence on the basis of some prior or hidden variable.
I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour locality over completeness. (4) If we prohibit all superluminal causation, then the violation of the Bell inequality teaches us a lesson, besides quantum mechanics' familiar ones that quantities can lack precise values and that pairs of quantities can lack joint probabilities: namely, some pairs of events are not screened off by their common past.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.