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arXiv:2301.04549

Space-time-hap: a coordinate system for the multiverse and its application to show that free choice is observer-dependent

Ghislain Fourny

Published Jan 11, 2023• 0 citations
arXivHTMLPDF

Abstract

This note proposes a paradigm and coordinate system that extends flat, four-dimensional Minkowski spacetime to a broader framework that identifies an event not only in space and in time, but also in terms of possible world, with a third category of coordinates called ``hap'' modelling contingency and counterfactuals. Semantically, hap is based on a Bohmian-like configuration space, in which initial conditions at a specific moment in time uniquely identify a trajectory. This framework can be used to support reasonings that rigorously distinguish between causal dependencies (through time), statistical dependencies (through space) and counterfactual dependencies (through hap). As an example, we use this framework to show that the assumption of free choice is not absolute, but rather depends on the chosen frame of reference: while Alice may see a choice made freely, which is formally a unilateral, single-coordinate translation in hap, Bob sitting in another reference frame might see this same choice not made freely and observe instead a translation in hap jointly across multiple coordinates, which indicates a counterfactual dependency. The framework is agnostic regarding which flavour of the De-Broglie-Bohm theory is considered, as it only makes general assumptions on the set of trajectories that are commonly accepted in the Bohmian mechanics community. It is also general enough to support reasoning on theories for which the separation between possible worlds might also depend on the frame of reference, i.e., two observers might disagree on whether two spacetimehap events happen in the same world or not.

DOI
10.48550/arxiv.2301.04549

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This note proposes a paradigm and coordinate system that extends flat, four-dimensional Minkowski spacetime to a broader framework that identifies an event not only in space and in time, but also in terms of possible world, with a third category of coordinates called ``hap'' modelling contingency and counterfactuals. Semantically, hap is based on a Bohmian-like configuration space, in which initial conditions at a specific moment in time uniquely identify a trajectory. This framework can be used to support reasonings that rigorously distinguish between causal dependencies (through time), statistical dependencies (through space) and counterfactual dependencies (through hap). As an example, we use this framework to show that the assumption of free choice is not absolute, but rather depends on the chosen frame of reference: while Alice may see a choice made freely, which is formally a unilateral, single-coordinate translation in hap, Bob sitting in another reference frame might see this same choice not made freely and observe instead a translation in hap jointly across multiple coordinates, which indicates a counterfactual dependency. The framework is agnostic regarding which flavour of the De-Broglie-Bohm theory is considered, as it only makes general assumptions on the set of trajectories that are commonly accepted in the Bohmian mechanics community. It is also general enough to support reasoning on theories for which the separation between possible worlds might also depend on the frame of reference, i.e., two observers might disagree on whether two spacetimehap events happen in the same world or not.
The study of quantum reference frames (QRFs) is motivated by the idea of taking into account the quantum properties of the reference frames used, explicitly or implicitly, in our description of physical systems. Like classical reference frames, QRFs can be used to define physical quantities relationally. Unlike their classical analogue, they relativise the notions of superposition and entanglement. Here, we explain this feature by examining how configurations or locations are identified across different branches in superposition. We show that, in the presence of symmetries, whether a system is in "the same" or "different" configurations across the branches depends on the choice of QRF. Hence, sameness and difference - and thus superposition and entanglement - lose their absolute meaning. We apply these ideas to the context of semi-classical spacetimes in superposition and use coincidences of four scalar fields to construct a comparison map between spacetime points in the different branches. This reveals that the localisation of an event is frame-dependent. We discuss the implications for indefinite causal order and the locality of interaction and conclude with a generalisation of Einstein's hole argument to the quantum context.
In general relativity, the causal structure between events is dynamical, but it is definite and observer-independent; events are point-like and the membership of an event A in the future or past light-cone of an event B is an observer-independent statement. When events are defined with respect to quantum systems however, nothing guarantees that the causal relationship between A and B is definite. We propose to associate a causal reference frame corresponding to each event, which can be interpreted as an observer-dependent time according to which an observer describes the evolution of quantum systems. In the causal reference frame of one event, this particular event is always localised, but other events can be ‘smeared out’ in the future and in the past. We do not impose a predefined causal order between the events, but only require that descriptions from different reference frames obey a global consistency condition. We show that our new formalism is equivalent to the pure process matrix formalism (Araújo et al 2017 Quantum 1 10). The latter is known to predict certain multipartite correlations, which are incompatible with the assumption of a causal ordering of the events—these correlations violate causal inequalities. We show how the causal reference frame description can be used to gain insight into the question of realisability of such strongly non-causal processes in laboratory experiments. As another application, we use causal reference frames to revisit a thought experiment Zych et al (arXiv: 1708.00248 ) where the gravitational time dilation due to a massive object in a quantum superposition of positions leads to a superposition of the causal ordering of two events.
We provide a possible fully geometric formulation of the core idea of quantum reference frames (QRFs) as it has been applied in the context of gravity, freeing its definition from unnecessary (though convenient) ingredients, such as coordinate systems. Our formulation is based on two main ideas. First, a QRF encodes uncertainty about what is the observer's (and, hence, the measuring apparatus's) perception of time and space at each spacetime point (i.e., event). For this, an observer at an event ppp is modeled, as usual, as a tetrad in the tangent space TpT_pTp​. So a QRF at an event ppp is a complex function on the tetrads at ppp. Second, we use the result that one can specify a metric on a given manifold by stipulating that a basis one assigns at each tangent space is to be a tetrad in the metric one wants to specify. Hence a spacetime, i.e. manifold plus metric, together with a choice of "point of view" on it, is represented by a section of the bundle of bases, understood as taking the basis assigned to each point to be a tetrad. Thus a superposition of spacetimes gets represented as, roughly speaking, an assignment of complex amplitudes to sections of this bundle. A QRF, defined here as the collection of complex amplitudes assigned to bases at events--i.e., a complex function defined on the bundle of bases of the manifold--can describe, in a local way (i.e., attributing the amplitudes to bases at events instead of to whole sections), these superpositions. We believe that this formulation sheds some light on some conceptual aspects and possible extensions of current ideas about QRFs. For instance, thinking in geometric terms makes it clear that the idea of QRFs applied to the gravitational scenarios treated in the literature (beyond linear approximation) lacks predictive power due to arbitrariness which, we argue, can only be resolved by some further input from physics.
Although time is one of our most intuitive physical concepts, its understanding at the fundamental level is still an open question in physics. For instance, time in quantum mechanics and general relativity are two distinct and incompatible entities. While relativity deals with events (points in spacetime), with time being observer dependent and dynamical, quantum mechanics describes physical systems by treating time as an independent parameter. To resolve this conflict, in this work, we extend the classical concept of an event to the quantum domain by defining an event as a transfer of information between physical systems. Then, by describing the universe from the perspective of a certain observer, we introduce quantum states of events with space-time-symmetric wave functions that predict the joint probability distribution of a measurement (observation) at $(t,\stackrel{P\vec}{x})$. Under these circumstances, without assuming collapse, we propose that a well-defined instant of time, like any other observable, arises from a single event, thus being an observer-dependent property. In this manner, we obtain a stationary quantum state written as a sum of a sequence of normalized states of events with increasing memories. As a consequence, a counterfactual asymmetry along this sequence of events gives rise to the flow of time as being successive ``snapshots'' from the observer's perspective. In this proposal, which contrasts strikingly to the view in which time is an illusion, it is the many distinguishable states in which the observer stores information that makes the existence of time possible.
In non relativistic physics it is assumed that both chronological ordering and causal ordering of events (telling whether there exists a causal relationship between two events or not) are absolute, observer independent properties. In relativistic physics on the other hand chronological ordering depends on the observer who assigns space-time coordinates to physical events and only causal ordering is regarded as an observer independent property. In this paper it is shown that quantum theory can be considered as a physical theory in which causal (as well as chronological) ordering of probabilistic events happening in experiments may be regarded as an observer dependent property.
The concept of “objective reality” is addressed, and an ontological model is suggested, in which correlations of events in the configuration space of the wave function are considered invariant with respect to changes of observer. It is suggested that these statements make the best sense when considered from within a fifth-dimensional framework, extrapolated from the four dimensions of spacetime in a direct way. A pair of postulates is then suggested which strengthens two current models of quantum theory into a broader picture, giving a physical interpretation of Macroscopic Quantum Superposition (MQS) states. Relational Quantum Mechanics (RQM) from Rovelli and Consistent Histories (“CH”) from Griffiths are discussed and related to the postulates. By dropping the assumption that unobserved macroscopic events are “in a definite state” determined independently of an observer, a surprising but consistent theory of quantum macroscopic reality is arrived at (with fewer fundamental assumptions about everyday reality) that does not contradict experiment or everyday experience. As a result, a concept herein termed “retroactive event determination” is extended from a quantum principle (in CH) to a macroscopic principle. Macroscopic events that have not been observed by a particular observer are free to be retroactively determined. The feasibility of applying the physics of quantum operators to macro systems is analyzed using the concept of “macro projectors.” Various concerns with this model are addressed, such as solipsism and decoherence of the wave function for macro objects. A discussion is also made of the philosophical context of the ideas suggested. Some experimental ideas are offered.Keywords: Macroscopic quantum superposition—MQS—consistent histories—relational quantum mechanics—time—spacetime—retrocausation—framework—single-framework rule—macro pointers—synchronicity—meaningful coincidence—positivism—observer effect—measurement problem—delayed choice experiment—retroactive event determination—existential behavior—Many-Worlds model