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arXiv:2411.01427

A cognitive basis for physical time

Per Östborn

Published Nov 3, 2024• 0 citations
arXivHTMLPDF

Abstract

The treatment of time in relativity does not conform to that in quantum theory. In the context of quantum gravity this is called "the problem of time". A crucial difference is that time $t$ may be seen as an observable in relativity theory, just like position $x$, whereas in quantum theory $t$ is a parameter, in contrast to the observable $x$. Aiming to resolve the discrepancy, a formalization of time in the spirit of Kant's Copernican revolution is suggested, where it is required that the treatment of time in physics agree with our cognition. This leads to reconsideration of the notions of identity and change of objects, as well as the nature of physical states and their evolution. The formalization has two components: sequential time $n$ and relational time $t$. The evolution of physical states is described in terms of $n$, which is updated each time an event occurs. The role of $t$ is to quantify distances between events in space-time. There is a space-time associated with each $n$, in which $t$ represents the knowledge at time $n$ about temporal distances between present and past events. A universal ordering of events in terms of $n$ can be postulated even though distances $t$ are relativistic. In short, it is argued that time as a sequential flow of events should be separated from time as a measure of distance between events. In physical models, these aspects of time can be expressed as one evolution parameter and one observable, respectively.

DOI
10.48550/arxiv.2411.01427

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The treatment of time in relativity does not conform to that in quantum theory. In the context of quantum gravity this is called "the problem of time". A crucial difference is that time ttt may be seen as an observable in relativity theory, just like position xxx, whereas in quantum theory ttt is a parameter, in contrast to the observable xxx. Aiming to resolve the discrepancy, a formalization of time in the spirit of Kant's Copernican revolution is suggested, where it is required that the treatment of time in physics agree with our cognition. This leads to reconsideration of the notions of identity and change of objects, as well as the nature of physical states and their evolution. The formalization has two components: sequential time nnn and relational time ttt. The evolution of physical states is described in terms of nnn, which is updated each time an event occurs. The role of ttt is to quantify distances between events in space-time. There is a space-time associated with each nnn, in which ttt represents the knowledge at time nnn about temporal distances between present and past events. A universal ordering of events in terms of nnn can be postulated even though distances ttt are relativistic. In short, it is argued that time as a sequential flow of events should be separated from time as a measure of distance between events. In physical models, these aspects of time can be expressed as one evolution parameter and one observable, respectively.
The treatment of time in relativity does not conform to that in quantum theory. To resolve the discrepancy, a formalization of time is introduced in an accompanying paper, starting from the assumption that the treatment of time in physics must agree with our cognition. The formalization has two components: sequential time nnn and relational time ttt. The evolution of physical states is described in terms of nnn. The role of ttt is to quantify distances between events in space-time. There is a space-time associated with each nnn, in which ttt represents the knowledge at time nnn about temporal distances between present and past events. This approach leads to quantum evolution equations expressed in terms of a continuous evolution parameter σσσ, which interpolates between discrete sequential times nnn. Rather than describing the evolution of the world at large, these evolution equations provide probabilites of a set of predefined outcomes in well-defined experimental contexts. When the context is designed to measure spatio-temporal position (x,t)(x,t)(x,t), time ttt becomes an observable with Heisenberg uncertainty ΔtΔtΔt on the same footing as xxx. The corresponding evolution equation attains the same symmetric form as that suggested by Stueckelberg in 1941. When the context is such that the metric of space-time is measured, the corresponding evolution equation may be seen as an expression of quantum gravity. In short, the aim of this paper is to propose a coherent conceptual basis for the treatment of time in evolution equations, in so doing clarifying their meaning and domain of validity.
Relativistically, time ttt is an observable just like position rrr. In quantum theory, ttt is a parameter, in contrast to the observable rrr. This discrepancy suggests that there exists a more elaborate formalization of time, which encapsulates both perspectives. Such a formalization is proposed in this paper. The evolution is described in terms of sequential time n∈Nn\in \mathbf{\mathbb{N}}n∈N, which is updated each time an event occurs. Sequential time nnn is separated from relational time ttt, which describes distances between events in space-time. There is a space-time associated with each nnn, in which ttt represents the knowledge at time nnn about temporal relations. The evolution of the wave function is described in terms of the parameter σσσ that interpolates between sequential times nnn. For a free object we obtain a Stueckelberg equation ddσΨ(r4,σ)=ic2ℏ2⟨ε⟩□Ψ(r4,σ)\frac{d}{dσ}Ψ(r_{4},σ)=\frac{ic^{2}\hbar}{2\langle ε\rangle}\BoxΨ(r_{4},σ)dσd​Ψ(r4​,σ)=2⟨ε⟩ic2ℏ​□Ψ(r4​,σ), where r4=(r,ict)r_{4}=(r,ict)r4​=(r,ict). Here σσσ describes the time mmm passed between the start of the experiment at time nnn and the observation at time n+mn+mn+m. The parametrization is assumed to be natural, meaning that ddσ⟨t⟩=1\frac{d}{dσ}\langle t\rangle=1dσd​⟨t⟩=1, where ⟨t⟩\langle t\rangle⟨t⟩ is the expected temporal distance between the events that define nnn and n+mn+mn+m. The squared rest energy ε02ε_{0}^{2}ε02​ is proportional to the eigenvalue σ~\tildeσσ~ that describes a 'stationary state' Ψ(r4,σ)=ψ(r4,σ~)eiσ~σΨ(r_{4},σ)=ψ(r_{4},\tildeσ)e^{i\tildeσσ}Ψ(r4​,σ)=ψ(r4​,σ~)eiσ~σ. The Dirac equation follows as a `square root' of the stationary state equation from the condition that σ~>0\tildeσ>0σ~>0, which follows from the directed nature of nnn. The formalism thus implies that all observable objects have non-zero rest mass, including elementary fermions. The introduction of nnn releases ttt, so that it can be treated as an observable with uncertainty ΔtΔtΔt.
The general view is that all fundamental physical laws should be formulated within the framework given by quantum mechanics (QM). In a sense, QM therefore has the character of a metaphysical theory. Consequently, if it is possible to derive QM from more basic principles, these principles should be of general, philosophical nature. Here, we derive the formalism of QM from well-motivated epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. First, an epistemic formalism is developed, using concepts like knowledge and potential knowledge, identifying a physical state SSS with the potential knowledge of the physical world. It is demonstrated that QM emerges from this formalism. However, Hilbert spaces, wave functions and probabilities are defined in certain well-defined observational contexts only. This means that the epistemic formalism is broader than QM. In the fundamental layer of description, the physical state SSS is a subset of a state space S={Z}\mathcal{S}=\{Z\}S={Z}, such that SSS always contains many elements ZZZ. These elements correspond to unattainable states of complete knowledge of the world. The evolution of SSS cannot be determined in terms of the individual evolution of the elements ZZZ, unlike the evolution of an ensemble in classical phase space. The evolution of SSS is described in terms of sequential time n∈Nn\in \mathbf{\mathbb{N}}n∈N, which is updated according to n→n+1n\rightarrow n+1n→n+1 each time an event occurs, each time potential knowledge changes. Sequential time nnn can be separated from relational time ttt, which describes distances between events in space-time. There is an entire space-time associated with each nnn, in which ttt represents the knowledge at sequential time nnn about the temporal relations between present and past events.
The concept of position and its measurement are fundamental to modern navigation technology as well as to theoretical physics. Nevertheless, a consistent notion of position has not been found in modern physics because of apparent inconsistencies between quantum mechanics and relativity. In quantum mechanics the probability distribution for a particle's position is determined by its wave function. A wave function strictly localized in space describes a particle within that locality, and nowhere else. But the quantum equations of motion predict the function will disperse, after any length of time at all, to occupy all space. Therefore the theoretical prediction is that, shortly after detection in a laboratory, a particle may be found anywhere in the universe, including regions requiring faster-than-light travel to reach. This contradicts the theory of relativity, which precludes a body traveling faster than the speed of light. The conflict arises from unexamined assumptions regarding the time of the measurement event and the time of the system under measurement. The prevailing view in physics is that time is only a parameter, having a definite value, describing dynamic changes in physical systems. This view endures even though quantum mechanics recognizes position as a quantum variable with inherent uncertainty, and relativity indicates that time should be placed on the same footing as position. But if time is recognized as a quantum variable with inherent uncertainty, the conceptual problems with position are resolved. The particle has finite probability of being in the past (or future) of the observer, while the quantum mechanical operator describing the position measurement contains a term which extrapolates from the intrinsic time of the particle to the time of the observer. Being potentially well in the past of the measurement event, the particle has time to travel to remote regions without exceeding the speed of light, and the conflict with relativity is resolved. A relativistic quantum theory can therefore be constructed incorporating a well-defined concept of position.
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?
Quantum mechanics has provided philosophers of science with many counterintuitive insights and interpretive puzzles, but little has been written about the role that time plays in the theory. One reason for this is the celebrated argument of Wolfgang Pauli against the inclusion of time as an observable of the theory, which has been seen as a demonstration that time may only enter the theory as a classical parameter. Against this orthodoxy I argue that there are good reasons to expect certain kinds of 'time observables' to find a representation within quantum theory, including clock operators (which provide the means to measure the passage of time) and event time operators, which provide predictions for the time at which a particular event occurs, such as the appearance of a dot on a luminescent screen. I contend that these time operators deserve full status as observables of the theory, and on reflection provide a uniquely compelling reason to expand the set of observables allowed by the standard formalism of quantum mechanics. In addition, I provide a novel association of event time operators with conditional probabilities, and propose a temporally extended form of quantum theory to better accommodate the time of an event as an observable quantity. This leads to a proposal to interpret quantum theory within an event ontology, inspired by Bertrand Russell's Analysis of Matter. On this basis I mount a defense of Russell's relational theory of time against a recent attack.