ELTE TTK Tudomanytortenet es Tudomanyfilozofia Tanszek Budapest, Pazmany P. setany 1/A TUDOMANYFILOZOFIA SZEMINARIUM (http://hps.elte.hu) ________________________________________________ 1999. oktober 4. (hetfo) 12:30 6. em. 654. (661 a regi szamozas szerint) T o m a s z P l a c e k Department of Philosophy, Jagiellonian University, Cracow OUTCOMES IN BRANCHING SPACE-TIME (OBST) -AN ANALYSIS OF BELL'S THEOREM- The framework of BRANCHING SPACE-TIME (BST; cf. Belnap 1992, SYNTHESE 92, pp. 385--434) has recently been extended to allow for the introduction of outcomes of events and the analysis of GHZ theorems. (Kowalski & Placek, forthcoming in BRIT. J. PHIL. SCI. and INT. J. THEOR. PHYS.) In BST, space-time and modality are incorporated in the very structure of the models, which consist of a pair $\langle W, \leq \rangle$, where $W$ is a non-empty set weakly ordered by $\leq$, which is interpreted as `causally accessible from.' Maximal upward directed subsets of $W$ are called `histories,' and proper subsets of histories are called `events.' Two events are called `space-like separated' if neither causally precedes the other. `Atomic outcomes' of an event $E$ are those parts of the event's causal future that split in $E$. The main result of Kowalski & Placek is that the family of outcomes of an event forms a Boolean algebra. The paper also proves that in GHZ setups, there is always a common cause (CC) in the sense of Reichenbach if directions are held fixed, but that there is no single COMMOM common cause (cf. Hofer-Szabo et al., forthcoming in BRIT. J. PHIL. SCI.) accounting for the outcomes of incompatible settings. For an analysis of Bell's theorem, I assign probabilities to outcomes by imposing a classical probability measure on the Boolean algebra of the outcomes of each given event. In the derivation of Bell's theorem, I use probability measures of the form $p_{L\alpha \cup R\beta}(Lx \cap Ry)$, $x,y \in \{+,-\}$, where the subscript indicates that the result is an outcome of the event of measuring the spin projections along directions $\alpha$ on the left and $\beta$ on the right. Probabilities for single results on the left or on the right are calculated from these measures, allowing us to express correlations as $p_{L\alpha \cup R\beta}(Lx \cap Ry) \neq p_{L\alpha \cup R\beta}(Lx) \times p_{L\alpha \cup R\beta}(Ry)$. Since correlations between space-like separated results appear disturbing, it is natural to look for an explanation in terms of a CC located in the results' common past. The CC's outcomes divide histories in such a way that actual runs of a correlation experiment are seen as belonging to two or more varieties differentiated by hidden factors. You may think of these hidden factors as restoring the deterministic order. You may also be more modest and require only that the hidden factors restore the causal order, i.e., that in each sub-population, the correlations disappear. Formally, for space-like separated events $E$ and $F$ with correlated outcomes $e$ and $f$, respectively, a CC is an event C preceding both $e$ and $f$, such that for every atomic outcome $\omega_{i}$ of $C$, $$ p_{E\cup F\cup C}(e \cap f|\omega_{i}) = p_{E\cup F\cup C}(e |\omega_{i}) \times p_{E \cup F\cup C}(f|\omega_{i})$$, where $p_{E \cup F \cup C}$ is defined on the enlarged probability space. Now, for any correlated pair $e,f$, we CAN construct mathematically an enlarged probability space containing such a CC. Moreover, for any finite number of correlations we CAN construct a single large probability space containing a set of distinct CCs, each CC taking care of one correlation. However, in the Bell/Clauser-Horne argument, one wants something more: one postulates a single common CC accounting for all the correlated outcomes of $L\alpha\cup R\beta$, $L\alpha\cup R\beta--\prime$, $L\alpha--\prime\cup R\beta$, and $L\alpha--\prime\cup R\beta--\prime$. Given the standard assumptions of locality and `no conspiracy,' which in our framework take the form \begin{equation*} \begin{split} & \forall \alpha, \beta, \varphi, x p_{L\alpha \cup R\beta\cup C}(Lx) = p_{L\alpha \cup R\varphi\cup C}(Lx)\ & \forall \alpha, \beta, \gamma, y p_{L\alpha \cup R\beta\cup C}(Ry) = p_{L\gamma \cup R\beta\cup C}(Ry) \end{split} \tag{LOCALITY} \end{equation*} \begin{equation*} \forall \alpha, \beta, \gamma, \varphi, i p_{L\alpha \cup R\beta\cup C}(\omega_i) = p_{L\gamma \cup R\varphi\cup C}(\omega_i), \tag{NO CONSPIRACY} \end{equation*} we derive the Bell/CH inequalities, which are empirically violated. Thus, there cannot be a common common cause accounting for the Bell/CH correlations. -- Laszlo E. Szabo Department of Theoretical Physics Department of History and Philosophy of Science Eotvos University, Budapest H-1518 Budapest, Pf. 32. Phone: (36-1)2090-555/6671 Fax: (36-1)372-2509 Home: (36-1)200-7318 http://hps.elte.hu/~leszabo