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Change without time. Relationalism and field quantization

Johannes Simon

Published Jan 1, 2005• 0 citations
Publisher

Abstract

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?

DOI
10.5283/epub.10300

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•Oct 5, 2026•0 citations
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 quantum gravity there is no notion of absolute time. Like all other quantities in the theory, the notion of time has to be introduced "relationally", by studying the behavior of some physical quantities in terms of others chosen as a "clock". We have recently introduced a consistent way of defining time relationally in general relativity. When quantum mechanics is formulated in terms of this new notion of time the resolution of the measurement problem can be implemented via decoherence without the usual pitfalls. The resulting theory has the same experimental results of ordinary quantum mechanics, but every time an event is produced or a measurement happens two alternatives are possible: a) the state collapses; b) the system evolves without changing the state. One therefore has two possible behaviors of the quantum mechanical system and physical observations cannot decide between them, not just as a matter of experimental limitations but as an issue of principle. This first-ever example of fundamental undecidability in physics suggests that nature may behave sometimes as described by one alternative and sometimes as described by another. This in particular may give new vistas on the issue of free will.
The problem of time is a deep paradox in our physical description of the world. According to Aristotle's relational theory, time is a measure of change and does not exist on its own. In contrast, quantum mechanics, just like Newtonian mechanics, is equipped with a master clock that dictates the evolution of a system. This clock is infinitely precise and tacitly supplied free of charge from outside physics. Not only does this absolute time make it notoriously difficult to make a consistent theory of quantum gravity, it is also the underlying problem in establishing the second law. Indeed, contrary to our experience, the Wheeler-deWitt equation --a canonical quantization of general relativity-- predicts a static universe. Similarly, when simply concerned with the dynamics of a closed quantum system, there is no second law because the Von Neumann entropy is invariant under unitary transformations. Here we are mainly concerned with the latter problem and we show that it can be resolved by attributing a minimal amount of resources to the measurement of time. Although there is an absolute time in quantum mechanics, an observer can only establish a time by measuring a clock. For a local measurement, the minimal entropy production is equal to the number of ticks. This lower bound is attained by a black hole.
In the previous chapters, we have made an imaginary travel in a ship which we might call “time” for analyzing how this concept is present in the physical theories which describe our world and the diverse interactions which make possible its existence as we perceive it under our notion of reality. We have traveled from the submicroscopic scale—requiring a quantum description— up to the cosmological dimensions where gravity acquires a crucial role distorting the space-time of the special relativity theory. In this chapter, we discuss some of the most recent publications related to physical theories about time, starting with the cellular automation interpretation of quantum mechanics (QM) formulated by Gerard’t Hooft, where information loss is considered as the cause of the time-reversal asymmetry. Then we continue with the thermal time hypothesis of Carlo Rovelli, who asserts that it is possible to formulate classical mechanics in a way in which the time variable is treated on equal footing as other physical variables, and not considered as a special and independent variable. Afterward, we analyze theories which propose quantization of time, and then continue with the discussion of proposed time operators for non-relativistic and relativistic QM. At the end of that section, we discuss theories related with time in quantum gravity and then we briefly explore in the physics of time travel.
In this chapter we are going to address the famous ‘problem of time’ which has become the headline for all the physical interpretational problems of the mathematical formalism. Roughly speaking the problem of time is that there is none in GR: at least in the spatially compact case without boundaries the Hamiltonian vanishes on the physical, constraint surface. This is physically relevant because we seem to live in a universe with precisely that spatial topology. Since the Hamiltonian generates time translations in any canonical theory we arrive at the conclusion that ‘nothing moves’ in GR, which is in obvious contradiction to experiment. Since there is no time also the usual interpretation of quantum mechanical measurements at given moments of time breaks down. One can fill books about this issue and we will not even try to cover a substantial amount of the existing literature. A superb source of information on these conceptual problems is Carlo Rovelli's book [3]. Rather, what we will do in what follows is to collect various proposals for solutions to the problem of time taken from other authors, especially Rovelli's relational approach to classical and quantum physics and Hartle et al. 's consistent history interpretation, and combine them into a consistent picture. We do not want to suggest that the resulting picture is to be accepted, rather we want to draw attention to the problems involved and to develop a working hypothesis. The discussion on the interpretation of quantum mechanics is very alive and some authors such as Penrose [243] not only propose to alter the interpretational aspect of quantum mechanics but also the mathematical framework.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
The disappearance of external time from canonical quantum gravity has motivated approaches in which observable temporal evolution emerges from an underlying description without a fundamental external time parameter. The Page–Wootters mechanism demonstrates how conditional evolution can arise through correlations between subsystems within a globally stationary quantum description, while relational approaches associated with Carlo Rovelli similarly challenge the primacy of an observer-independent temporal background.This paper compares these approaches with Registration Theory (RT) and identifies a fundamental distinction. RT does not replace fundamental time with a fundamentally static quantum state. Instead, it proposes a pre-temporal ontology in which registration, distinction, successor dependence, retained consequence, and persistence precede physical temporal coordinates.The paper formalizes the distinction between primitive successor structure, retained history, and host-relative temporal representation. It proposes that the physical content of the past consists of consequences of prior registrations that remain distinguishable in present successor structure. On this account, time does not make change possible; persistent distinguishable change makes temporal description possible.A comparison matrix and proof-status audit distinguish established RT structures from derived propositions, conjectured compatibility, and open derivations. The paper identifies the derivation of relativistic temporal structure, Hilbert-space composition, and general quantum entanglement from registration as remaining research problems.The central distinction is summarized as follows: Registration Theory is not fundamentally timeless; it is pre-temporal.