Branching Space-time Research Articles

Probabilities in the topos approach to branching space-time

There was previously proposed a topos unified view-point both to relativistic and quantum physics [4-5] on the basis of Belnapian branching space-time [3]. In the context of this approach, to any event of the branching space-time there can be associated a local orthocomplemented lattice (logic) of ‘physically reasonable’ propositions made by a localized observer. Elements of these local orthologics are special collections of the so-called Belnapian worlds (maximal upward directed subsets of events) which the local event belongs to. In the ordinary quantum setting, the counterpart of any local orthologic is the lattice of closed subspaces of Hilbert space of the observer’s environment. The present work suggests a natural scheme of inserting truth probabilities of the ‘physically reasonable’ propositions into the topos framework. To this end, the concept of Chu spaces is used. The role of hypothesis for the state of environment is played by an element of local ‘retro-causal’ orthologic. Sufficient condition of functorial character of the law by which Chu space is related to any event is pointed and proved, i.e. a functor is constructed from the branching space-time to the category of all Chu spaces. There presented a theorem on category nature of Bayesian reassessment of hypothesis likelihood made by the local observer.

Does branching explain flow of time or the other way around?

The article discusses the relation between two intuitive properties of time, namely its flow and branching. Both properties are introduced first in an informal way and compared. The conclusion of this informal analysis is that the two properties do not entail each other nor are they in contradiction. In order to verify this, we briefly introduced the branching temporal structures called branching space-time, branching continuation and their versions Minkowski branching structure and branching time with Instants. Two possible ways how to formalize flow of time are given, one based on the definition of flow of time from temporal logics and the other based on relativistic physics. The latter is used to define flow of time with the use of linearly ordered points on worldines while respecting the ontological definiteness given by the difference of the past and the future. This is formalized in each of the branching models and it is concluded by comparing the resulting properties that branching and flow, even in the formal sense, do not entail each other. However, notions connected with flow of time represent a useful basis for semantics of the branching models.

A Generalized Manifold Topology for Branching Space-Times

The logical theory of branching space-times, which provides a relativistic framework for studying objective indeterminism, remains mostly disconnected from discussions of space-time theories in philosophy of physics. Earman has criticized the branching approach and suggested “pruning some branches from branching space-time.” This article identifies the different—order-theoretic versus topological—perspective of both discussions as a reason for certain misunderstandings and tries to remove them. Most important, we give a novel, topological criterion of modal consistency that usefully generalizes an earlier criterion, and we introduce a differential-geometrical version of branching space-times as a non-Hausdorff (generalized) manifold.

Multiverse and topology of time

Modern cosmological theories like theory of eternal chaotic inflation or cosmological model of brane universe, which represent implication of modern string theory, imply existence of ensemble of actual universes which create so-called multivers. Implication of this theories are of the great importance for philosophy of time. This paper concern topology of time in context of existence of the multiverse. We will show that topology of time in such multiverse, whose particular universes would be generated due to eternal chaotic inflation, should be non-standard and that in the case of existence of multiverse only model which include topology of brenching time should be acceptable like real description of multiverse. Moreover, we will show that, the existence of hypothetical entities like Kerr`s rotating black holes, also represent necessary and sufficient condition of branching spacetime in such multiverse.

On A- and B-theoretic elements of branching spacetimes

This paper assesses branching spacetime theories in light of metaphysical considerations concerning time. I present the A, B, and C series in terms of the temporal structure they impose on sets of events, and raise problems for two elements of extant branching spacetime theories—McCall’s ‘branch attrition’, and the ‘no backward branching’ feature of Belnap’s ‘branching space-time’—in terms of their respective A- and B-theoretic nature. I argue that McCall’s presentation of branch attrition can only be coherently formulated on a model with at least two temporal dimensions, and that this results in severing the link between branch attrition and the flow of time. I argue that ‘no backward branching’ prohibits Belnap’s theory from capturing the modal content of indeterministic physical theories, and results in it ascribing to the world a time-asymmetric modal structure that lacks physical justification.

On Minkowskian branching structures

In Belnap's [Branching space-time. Synthese, 92, 385–434. ‘Postprint’ archived at http://philsci-archive.pitt.edu/archive/00001003] theory of branching space-times (BST) Our World's possible histories are thought of as space-times, yet the theory has models in which histories do not resemble relativistic space-times or any other physical space-times. The aim of this paper is to define a certain class of BST models, called ‘Minkowskian Branching Structures’ (MBSs), in which histories are isomorphic to Minkowski space-time. By focusing on these models rather than on general BST models, we hope that one may be able to improve on earlier BST analyses of physical phenomena. Also, introducing MBSs sets the stage for recent discussions about whether or not branching is physically feasible.

The Topology of Branching Universes

The purpose of this paper is to survey the possible topologies of branching space-times, and, in particular, to refute the popular notion in the literature that a branching space-time requires a non-Hausdorff topology.

Modeling Modal Talk in Quantum Mechanics

In this paper, modal and counterfactual logical connectives are defined in an extended framework of branching space-time (Belnap, N. D. (1992). Branching space-time. Synthese92, 385–434). It is shown that a variety of definitions of the counterfactual can be given. The validity of certain modal statements occurring in quantum mechanics depends on the choice of definition. These considerations can be applied to an analysis of Stapp’s premises LOC1 and LOC2 from his purported proof of non-locality (Stapp, H. P. (1997). Nonlocal character of quantum theory. American Journal of Physics65, 300–304). It is shown that while the validity of LOC1 depends on the choice of the definition of the counterfactual, LOC2 is absolutely invalid.

Stochastic Outcomes in Branching Space-Time: Analysis of Bell's Theorem

The paper extends the framework of outcomes in branching space-time (Kowalski and Placek [1999]) by assigning probabilities to outcomes of events, where these probabilities are interpreted either epistemically or as weighted possibilities. In resulting models I define the notion of common cause of correlated outcomes of a single event, and investigate which setups allow for the introduction of common causes. It turns out that a deterministic common cause can always be introduced, but (surprisingly) only special setups permit the introduction of truly stochastic common causes. I analyse next the Bell-Aspect experiment and derive the Bell-CH inequalities. I observe that we postulate there not a common cause for outcomes of a single event but rather a common common cause that accounts for outcomes of many events, where 'events' mean 'measurements with (different) directions of polarization'. Since the inequalities are violated, I claim that no causal story can be told about the Bell correlations, where causality is subluminal and restricted by screening-off condition. Similarly, given certain intuitive principles, no deterministic story can be told about these correlations.

GHZ Theorems in the Framework of Outcomes in Branching Space-time †

A simple algebraic framework is constructed, in which nonstochastic GHZ-Belltheorems can be analyzed. The framework merges Belnap's outcomes in branchingtime with his branching space-time (BST). We show that an important structurein BST, called the family of outcomes of an event, is a Boolean algebra. We provethat there is no common cause that accounts for the results of GHZ-Bell experimentbut we construct common causes for two other setups.

QM and STR: The Combining of Quantum Mechanics and Relativity Theory

Combining quantum mechanics with special relativity requires (i) that a spacetime representation of quantum states be found; (ii) that such states, represented as extended along equal-time hyperplanes, be invariant when transformed from one frame to another; and (iii) that collapses of states be instantaneous in every frame. These requirements are met using branching spacetime, in which probabilities of outcomes are represented by the numerical proportions of branches on which the outcomes occur. Quantum states of systems are then identified with the probability values, built into spacetime along spacelike hypersurfaces, of all possible outcomes of all possible tests to which the systems can be subjected.

Outcomes in Branching Space-Time and GHZ-Bell Theorems

The paper intends to provide an algebraic framework in which subluminal causation can be analysed. The framework merges Belnap's 'outcomes in branching time' with his 'branching space-time' (BST). it is shown that an important structure in BST, called 'family of outcomes of an event', is a boolean algebra. We define next non-stochastic common cause and analyse GHZ-Bell theorems. We prove that there is no common cause that accounts for results of GHZ-Bell experiment but construct common causes for two other quantum mechanical setups. Finally, we investigate why some setups allow for common causes whereas other setups do not.

A New Non-Hausdorff Spacetime Model for Resolution of the Time Travel Paradoxes

Physicists interested in chronology-violating spacetimes have considered various strategies for avoiding the time travel paradoxes which can arise in those spacetimes. One such strategy is the adoption of a branching (non-Hausdorff) topology for spacetime. Branching spacetime models also arise in other physical contexts, such as the many-worlds interpretation of quantum mechanics. A known objection to the idea of branching spacetime stems from the fact that an observer in such a spacetime may have multiple futures or pasts. In this paper I describe a new non-Hausdorff spacetime model in which each observer has a unique history. This model differs topologically from previous non-Hausdorff spacetime models in that is not aT1-space.

Branching space-time analysis of the GHZ theorem

Greenberger, Horne, Shimony and Zeilinger gave a new version of the Bell theorem without using inequalities (probabilities). Mermin summarized it concisely; but Bohm and Hiley criticized Mermin's proof from contextualists' point of view. Using the Branching Space-time language, in this paper a proof will be given that is free of these difficulties. At the same time we will also clarify the limits of the validity of the theorem when it is taken as a proof that quantum mechanics is not compatible with a deterministic world nor with a world that permits correlated space-related events without a common cause.