Whereas one can conceive of a relational classical mechanics in which absolute space and time do not play a fundamental role, quantum mechanics does not readily admit any such relational formulation.
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?
In classical information theory (represented by Shannon's framework), the study of information transmission typically presupposes a fixed "channel" endowed with absolute attributes. This assumption implicitly holds that the properties of the channel are determinate and observer-independent, remaining unchanged regardless of perspective or the state of the channel itself. However, contemporary physics---particularly the development of quantum mechanics---fundamentally challenges the very notion of "absoluteness." As Carlo Rovelli has emphasized in one of his core arguments, quantum mechanics compels us to abandon the conception of states and properties as "absolute" and "observer-independent." The insistence on such absoluteness is precisely what underlies many of the paradoxes and interpretative difficulties that plague quantum theory. Rovelli extends this general philosophical critique of "absoluteness" consistently into the domain of information theory, thereby provoking a profound re-examination and critique of the classical presupposition of the "absolute channel." From this relational perspective, information should not be conceived as an abstract entity existing independently of physical relations, but rather as an observable measure of relative relations between systems. Accordingly, the reconstruction of classical information theory on the basis of relational ontology---namely, the reconstruction of information theory itself---is not only a demand of theoretical coherence but also a necessary step in responding to the challenge of quantum reality and in building a more universal theory of information.
Classical physics is traditionally characterized as deterministic and founded on the real number continuum, while quantum mechanics is distinguished by its indeterminism and fundamental uncertainty relations. This apparent dichotomy, however, obscures a deeper continuity. We argue that both frameworks embody a common ontological principle: the physical prohibition of absolute precision. In classical physics, this prohibition is implicit, manifesting in information-theoretic bounds, the non-physicality of infinite-precision initial conditions (chaos theory), and the breakdown of determinism in singular configurations (Norton's dome). These phenomena reveal that the classical reliance on mathematical real numbers with infinite information content is an unphysical idealization. Quantum mechanics, by contrast, does not merely reflect this prohibition; it provides its precise quantitative expression. The Heisenberg uncertainty principle σxσp≥ℏ/2 and the non-commutative structure of observables are the exact mathematical formulation of the fact that physical information is fundamentally finite. The Planck constant ℏ is the universal scale that quantifies the minimal phase space volume—the “atom” of information—below which no further refinement is possible. Thus, quantum mechanics is not a separate theory but the completion of classical physics: it supplies the missing quantification of the principle that nature admits no infinite information density, and consequently, no absolutely precise physical quantities.