Spacetime and Euclidean Geometry

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the "spacetime Pythagoras theorem".Comment: 11 pages, 9 figures; for a festschrift honoring Michael P. Ryan; v.2: References to related work adde

Relational Hidden Variables and Non-Locality

We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.Comment: 42 pages in journal style. To appear in Studia Logic

Greenberger-Horne-Zeilinger-like proof of Bell's theorem involving observers who do not share a reference frame

Vaidman described how a team of three players, each of them isolated in a remote booth, could use a three-qubit Greenberger-Horne-Zeilinger state to always win a game which would be impossible to always win without quantum resources. However, Vaidman's method requires all three players to share a common reference frame; it does not work if the adversary is allowed to disorientate one player. Here we show how to always win the game, even if the players do not share any reference frame. The introduced method uses a 12-qubit state which is invariant under any transformation $R_a \otimes R_b \otimes R_c$ (where $R_a = U_a \otimes U_a \otimes U_a \otimes U_a$, where $U_j$ is a unitary operation on a single qubit) and requires only single-qubit measurements. A number of further applications of this 12-qubit state are described.Comment: REVTeX4, 6 pages, 1 figur

An Operational Interpretation of Negative Probabilities and No-Signalling Models

Negative probabilities have long been discussed in connection with the foundations of quantum mechanics. We have recently shown that, if signed measures are allowed on the hidden variables, the class of probability models which can be captured by local hidden-variable models are exactly the no-signalling models. However, the question remains of how negative probabilities are to be interpreted. In this paper, we present an operational interpretation of negative probabilities as arising from standard probabilities on signed events. This leads, by virtue of our previous result, to a systematic scheme for simulating arbitrary no-signalling models.Comment: 13 pages, 2 figure

Bell inequalities as constraints on unmeasurable correlations

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations , , , and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation associated to different choices for B. The same analysis applies to the Hardy model and to the "perfect correlations" discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality.Comment: Some modifications have been done in order to improve clarity of presentation and comparison with other approache

Bell's theorem without inequalities and without unspeakable information

A proof of Bell's theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.Comment: REVTeX4, 4 pages, 1 figure; for Asher Peres' Festschrift, to be published in Found. Phy

Translation-invariance of two-dimensional Gibbsian point processes

The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.Comment: 44 pages, 6 figure

Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann's 'no hidden variables' proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense 'like Bohm's theory.' His seminal result provides a positive answer to the question. I argue that Bell's analysis misconstrues von Neumann's argument. What von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the 'beables' of the theory, to use Bell's term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm's theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system.Comment: 8 pages, no figures; for Peter Mittelstaedt Festschrift issue of Foundations of Physic