The axioms of a causal structure are reformulated. A natural generalization is suggested for the case when the subset lattice of space-time events is replaced by a lattice coming from the quantum theory.
There has been a body of work deriving the complex Hilbert-space structure of quantum theory from axioms/principles/postulates to deepen our understanding of quantum theory and to reveal ways to go beyond it to resolve foundational issues. Recent progress in incorporating indefinite causal structure into physical theories suggests that a more comprehensive understanding of both quantum theory and the theory beyond it accounts for indefinite causal structure. We formulate a framework of physical theories without assuming definite causal structure and identify postulates that single out the complex Hilbert-space structure. More than one complex Hilbert-space theory is compatible with the postulates, which leaves room for further search for the best among these theories.
CONSTRUCTIBILITY IN QUANTUM MECHANICS D.J. BENDANIEL Cornell University (djb16@Cornell.edu) Ithaca NY, 14853, USA We pursue an approach in which space-time proves to be relational and its differential properties fulfill the strict requirements of Einstein-Weyl causality. Space-time emerges here from a set theoretical foundation for a constructible mathematics. In this theory the Schrödinger equation can be obtained by adjoining a physical postulate of action symmetry in generalized wave phenomena. This result now allows quantum mechanics to be considered conceptually cumulative with prior physics. A set theoretical foundation is proposed in which the axioms are the theory of Zermelo-Frankel (ZF) but without the power set axiom, with the axiom schema of subsets removed from the axioms of regularity and replacement and with an axiom of countable constructibility added. Four arithmetic axioms are also adjoined; these formulae are contained in ZF but must be added here as axioms. All sets of finite natural numbers in this theory are finite and hence definable. The real numbers are countable. We first show that this constructible theory gives polynomial functions of a real variable, which are of bounded variation and locally homeomorphic. Eigenfunctions governing physical fields can then be effectively obtained. By using an integral of the Lagrange density of a field over a compactified space, we produce a nonlinear sigma model. Finally, the Schrödinger equation follows directly from the postulate and a sui generis proof in this theory of the discreteness of the space-like and time-like terms of the model.
This presentation provides a non-technical overview of the notion of quantum superposition of causal structures, of its applications, and of its proposed physical realizations. The conceptual underpinning for these investigations is a view of quantum theory as a new kind of probability theory. At the axiomatic level, the principles of this new kind of probability theory suggest new causal relations that have no analogue in the classical world. These new causal relations are a potential resource for new technologies, including computation and communication technology.
This paper is devoted to event-observational modelling in physics and more generally natural science. The basic entities of such modelling are events and where space-time is the secondary structure for the representation of events. The novelty of our approach is in using a new mathematical picture of events universe. The events observed by an observer are described by a dendrogram, a finite tree. The event dynamics are realized in the dendrogramic configuration space. In a dendrogram, all events are intercoupled via the hierarchic relational structure of the tree. This approach is called Dendrogramic Holographic Theory (DHT). We introduce the causal structure on the dendrogramic space, like the causal structure on the Minkowski space-time. In contrast to the latter, DHT-emergent causality is of a statistical nature. Each dendrogram represents an ensemble of observers with the same relational tree representation of the events they measured/collected. Technically the essence of causal modelling is in encoding dendrograms by real parameters and in this way transitioning to the real space-time. Then we proceed in the framework of information geometry corresponding to Hellinger distance and introduce a kind of light cone in the space of dendrograms. This is a step towards the development of DHT-analog special relativity.
The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition. Causal order is a concept that is engrained in the standard understanding of time, both in classical and quantum mechanics. Oreshkovet al.generalize the standard formalism of quantum theory to a framework with no pre-existing causal order, and find a new class of correlations that have no analogue in the classical world.