In this chapter, spacetime is identified as the empirical realm, to be understood in a relational sense. The spacetime construct emerges from the quantum substratum by way of actualized transactions, which establish spacetime events and their structural connections. This process is discussed in terms of causal set theory. The quantum substratum constitutes the reference for “absolute motion,” and rest-mass systems in the substratum define inertial frames. The transactional process breaks time symmetry, thus establishing an arrow of time and shedding light on the Second Law of Thermodynamics. Implications for free will are also discussed.
In this chapter, spacetime is identified as the empirical realm, to be understood in a relational sense. The spacetime construct emerges from the quantum substratum by way of actualized transactions, which establish spacetime events and their structural connections. This process is discussed in terms of causal set theory. The quantum substratum constitutes the reference for “absolute motion,” and rest-mass systems in the substratum define inertial frames. The transactional process breaks time symmetry, thus establishing an arrow of time and shedding light on the Second Law of Thermodynamics. Implications for free will are also discussed.
We consider the manner in which the spacetime manifold emerges from a quantum substratum through the transactional process, in which spacetime events and their connections are established. In this account, there is no background spacetime as is generally assumed in physical theorizing. Instead, the usual notion of a background spacetime is replaced by the quantum substratum, comprising quantum systems with nonvanishing rest mass. Rest mass corresponds to internal periodicities that function as internal clocks defining proper times, and in turn, inertial frames that are not themselves aspects of the spacetime manifold, but are pre-spacetime reference structures. Specific processes in the quantum substratum serve to distinguish absolute from relative motion.
This paper discusses how the transactional interpretation of quantum mechanics can provide for a natural account of the emergence of spacetime events from a quantum substratum. In this account, spacetime is not a substantive manifold that becomes occupied with events; rather, spacetime itself exists only in virtue of specific actualized events. This implies that spacetime is discrete rather than continuous, and that properties attributed to spacetime based on the notion of a continuum are idealizations that do not apply to the real physical world. It is further noted that the transactional picture of the emergence of spacetime can provide the quantum dynamics that underlie the causal set approach as proposed by Sorkin and others.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
We extend Barandes’ indivisible stochastic interpretation of quantum mechanics to address the emergence of classical spacetime and the arrow of time from collective quantum processes. While Barandes (2025) demonstrates that individual quantum sys- tems can be understood as indivisible stochastic processes, we show that when many such systems interact, their collective behavior generates emergent spatial structure through quantum correlations and emergent temporal divisibility through thermody- namic irreversibility. Specifically, we argue that: (1) spatial relations emerge from patterns of quantum entanglement and correlation among multiple actualizations; (2) thermodynamic entropy production in many-body systems creates an irreversible ar- row of time; (3) this irreversibility transforms the fundamentally indivisible quantum process into an apparently divisible classical process; and (4) the time parameter t itself emerges as a bookkeeping device for tracking these irreversible collective changes. This framework provides a unified account of the quantum-to-classical transition, connect- ing quantum foundations, statistical mechanics, and emergent spacetime in a coherent picture where both space and time arise from the collective dynamics of indivisible quantum actualizations.
I take non-locality to be the Michaelson Morley experiment of the early 21st Century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must be emergent from entangled coherent quantum variables and their behaviors. Two immediate consequences are: i. If we start with non-locality, we need not explain non-locality. We must instead explain an of locality and spacetime. ii. There can be no of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationaly local, can be formulated without matter, and where there is no emergence of spacetime. If true, then quantum gravity cannot be a minor alteration of General Relativity, but must demand its deep reformulation. This will lead to: Matter not only deforms spacetime, but creates spacetime. This quantum creation of spacetime consists in: i. Fully non-local entangled coherent quantum variables. ii. The onset of locality via decoherence. iii. A metric in Hilbert Space among entangled quantum variables by the sub-additive von Neumann Entropy between pairs of variables. iv. Mapping from metric distances in Hilbert Space to metric distances in classical spacetime by episodic actualization events. v. Discrete spacetime is the relations among these discrete actualization events. vi. Now is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously. vii. The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. viii. The arrow of time history of these events is recorded in the very structure of the spacetime constructed. ix. Actual Time is a succession of two or more actual events. Relation to Causal Set Theory, faithful Lorentzian manifolds and light cones are discussed.
The nature of time has long been debated in human history and nowadays is considered of central importance for understanding quantum gravity. We focus on and advocate the relational concept of time, which was put forward in the 17th century in opposition to Newton's absolute time, and only in 1990 explored in a quantum mechanical framework by Carlo Rovelli. After a historical introduction the mathematical models of time are carefully analyzed in chapter 1, followed by a discussion of the role of time played in fundamental theories. Using as an example nonrelativistic mechanics, the process of parametrization is explained, leading to a separation of a 'canonical time coordinate' from an arbitrary evolution parameter. The discussion of the role of time in special and general relativity as well as in quantum mechanics shows that more fundamental theories use less structure of time. This is followed by an exposition of the history of the relational concept of time, which negates the existence of an absolute duration and therefore often is called "timeless". Next it is shown how fundamental theories can be formulated and re-interpreted using this concept. We put emphasis on the hitherto neglected connection between relationalism and non-extensibility, while absolute time is shown to be unproblematic in classical mechanics just because of the possibility to extend the system without changing its nature. We conclude chapter 1 with a new axiomatic basis for the construction of time observables based on a simultaneity relation between 'observations', which are treated as a primitive concept and intuitively correspond to measurement events, but without knowing 'when' these events occur. There is no fundamental time observable; any observable qualifies as a time observable, if it allows to separate all instants. Chapter 2 gives a brief account of three main problems connected with time: a) The problem of the arrow of time. This has to be disentangled from the problem of irreversibility: a solution of the latter essentially excludes cyclic motions and is required for a solution of the former, which consists in showing that a fundamental direction between any two non-identical instants is physically meaningful. We give a formal definition of the arrow of time. This classical analysis is followed by a review of the problem of the arrow of time in quantum theory, where the situation becomes more complicated because of indeterminism. The discussion shows that there is no experimental evidence for a fundamental arrow of time, so that no contradiction with the relational concept of time arises. b) The problem of time measurement, which is of particular importance for the relational approach, in which time has no reality except if measured by a clock. In quantum theory useful time operators seem to be possible only within the more general formalism of positive operator valued measures (POVM). Clocks based on an oscillation mechanism do however require phase measurements; quantum phase operators can be defined as certain POVMs. Phase difference operators do also exist in the traditional Hilbert space formalism, if another quantization is used, as is done with relational quantization. c) The problem of quantum gravity is sketched only briefly. Chapter 3 introduces and discusses a model of Rovelli consisting of two oscillators with no external time. In this model one oscillator is considered as a clock and defines a relational time for the other one. In the first section we introduce this model and generalize it to a free massless scalar field in one dimension. We establish the relation between a single field mode and the infrared behaviour of the field through constants of motion. In the second section after a short review of canonical quantization we review Rovelli's quantization and generalize it to the free field. We could not prove the existence of the quantization map, but calculations using computer algebra indicate that the quantization does exist. For an enlarged algebra we are able to prove the nonexistence of a quantization map similar to a proof by Groenewold and van Hove. In chapter 4 we observe that a clock time always requires infinitely many degrees of freedom, and we make the hypothesis that a time observable is given by the infrared behaviour of quantum fields, leading to a classical notion of time when using algebraic quantum mechanics. This does however not solve the main problem of quantum relationalism: Which conditions determine a particular evolution?