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 $n$ and relational time $t$. The evolution of physical states is described in terms of $n$. 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. This approach leads to quantum evolution equations expressed in terms of a continuous evolution parameter $σ$, which interpolates between discrete sequential times $n$. 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 $(x,t)$, time $t$ becomes an observable with Heisenberg uncertainty $Δt$ on the same footing as $x$. 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.
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 n and relational time t. The evolution of physical states is described in terms of n. 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. This approach leads to quantum evolution equations expressed in terms of a continuous evolution parameter σ, which interpolates between discrete sequential times n. 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), time t becomes an observable with Heisenberg uncertainty Δt on the same footing as x. 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.
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.
Relativistically, time t is an observable just like position r. In quantum theory, t is a parameter, in contrast to the observable r. 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∈N, which is updated each time an event occurs. Sequential time n is separated from relational time t, which describes 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 relations. The evolution of the wave function is described in terms of the parameter σ that interpolates between sequential times n. For a free object we obtain a Stueckelberg equation dσdΨ(r4,σ)=2⟨ε⟩ic2ℏ□Ψ(r4,σ), where r4=(r,ict). Here σ describes the time m passed between the start of the experiment at time n and the observation at time n+m. The parametrization is assumed to be natural, meaning that dσd⟨t⟩=1, where ⟨t⟩ is the expected temporal distance between the events that define n and n+m. The squared rest energy ε02 is proportional to the eigenvalue σ~ that describes a 'stationary state' 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, which follows from the directed nature of n. The formalism thus implies that all observable objects have non-zero rest mass, including elementary fermions. The introduction of n releases t, so that it can be treated as an observable with uncertainty Δt.
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 S 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 S is a subset of a state space S={Z}, such that S always contains many elements Z. These elements correspond to unattainable states of complete knowledge of the world. The evolution of S cannot be determined in terms of the individual evolution of the elements Z, unlike the evolution of an ensemble in classical phase space. The evolution of S is described in terms of sequential time n∈N, which is updated according to n→n+1 each time an event occurs, each time potential knowledge changes. Sequential time n can be separated from relational time t, which describes distances between events in space-time. There is an entire space-time associated with each n, in which t represents the knowledge at sequential time n about the temporal relations between present and past events.
Conceptual Problems in Quantum Cosmology The application of quantum theory to cosmology raises a number of conceptual questions, such as the role of the quantum-mechanical notion of “observer” or the absence of a time variable in the Wheeler–DeWitt equation. I point out that a relational formulation of quantum mechanics, and more in general the observation that evolution is always relational, provides a coherent solution to this tangle of problems. A number of confusing issues appear when we try to apply quantum mechanics to cosmology. Quantum mechanics, for instance, is generally formulated in terms of an observer making measurements on a system. In a laboratory experiment, it is easy to identify the system and the observer: but what is the observer in quantum cosmology? Is it part of the same universe described by the cosmological theory, or should we think of it as external to the universe? Furthermore, the basic quantum dynamical equation describing a system including gravity is not the Schrödinger equation, but rather the Wheeler–DeWitt equation, which has no time parameter: is this related to the absence of an observer external to the universe? How do we describe the quantum dynamics of the universe without a time variable in the dynamical equation and without an observer external to the universe? I suggest that clarity on these issues can be obtained by simply recognizing the relational nature of quantum mechanics and more in general the relational nature of physical evolution. In the context of quantum theory, this nature is emphasized by the so-called relational interpretation of quantum mechanics (Rovelli, 1996). The relational nature of evolution, on the other hand, has been pointed out by the partial-observable formulation of general covariant dynamics (Rovelli, 2002). A central observation is that there is a common confusion between two different meanings of the expression “cosmology”. This is discussed below, in Section 15.1.1. A considerable amount of the difficulties mentioned above stems, I argue, from this confusion. The discussion clarifies on the role of the quantum mechanical observer in cosmology, which is considered in Section 15.1.2. In Section 15.2, I briefly describe the relational interpretation of quantum mechanics and some related aspects of quantum theory.
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.
In its usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turn, again gives time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by noninstantaneous measurements are described by ``spacetime smeared states.'' The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space.