In this chapter we are going to discuss some aspects touching upon the discretization of time in the context of Einstein's theories of special relativity (SR) and general relativity (GR). Snyder's quantized spacetime At first sight SR seems the wrong theory for temporal discretization. In Newtonian mechanics, time is absolute, which means that simultaneity is absolute: it is the same for all inertial observers. Therefore, if absolute time is discretized for one inertial observer, it is discretized in exactly the same way for all other inertial observers. In that respect, Newtonian discrete time mechanics does not require a preferred inertial frame. Simultaneity is not absolute in SR, however, which suggests that discretizing time will break Lorentz covariance: the use of a preferred inertial frame in which to temporally discretize seems inevitable. This conclusion is based on classical thinking. The example of quantized angular momentum demonstrates that it is possible to reconcile having a continuous parameter space with a discrete spectrum of observable values. Suppose that an observer is conducting a Stern–Gerlach (SG) experiment, firing a beam of electrons through an apparatus containing a strong inhomogeneous magnetic field (Gerlach and Stern, 1922a). In such an experiment, the observer will first fix the orientation of the apparatus relative to the source. Then it will be found that an electron passing through the apparatus will be detected finally in only one of two possible angular momentum states.
In this chapter we are going to discuss some aspects touching upon the discretization of time in the context of Einstein's theories of special relativity (SR) and general relativity (GR). Snyder's quantized spacetime At first sight SR seems the wrong theory for temporal discretization. In Newtonian mechanics, time is absolute, which means that simultaneity is absolute: it is the same for all inertial observers. Therefore, if absolute time is discretized for one inertial observer, it is discretized in exactly the same way for all other inertial observers. In that respect, Newtonian discrete time mechanics does not require a preferred inertial frame. Simultaneity is not absolute in SR, however, which suggests that discretizing time will break Lorentz covariance: the use of a preferred inertial frame in which to temporally discretize seems inevitable. This conclusion is based on classical thinking. The example of quantized angular momentum demonstrates that it is possible to reconcile having a continuous parameter space with a discrete spectrum of observable values. Suppose that an observer is conducting a Stern–Gerlach (SG) experiment, firing a beam of electrons through an apparatus containing a strong inhomogeneous magnetic field (Gerlach and Stern, 1922a). In such an experiment, the observer will first fix the orientation of the apparatus relative to the source. Then it will be found that an electron passing through the apparatus will be detected finally in only one of two possible angular momentum states.
Modern Physics rests on two pillars: General Relativity (GR) and Quantum Mechanics (QM). The former describes gravitation as the continuous geometry of spacetime, based on differential fields. The latter, in turn, operates in a discretized regime, where energy levels, frequencies, and experimental results arise in quantized packets. The tension between both arises because GR assumes absolute continuity of time and space, while QM presents intrinsically discrete phenomena. Historically, the search for a unified theory started from the assumption that one of the two formulations must be modified — either GR should be discretized, or QM should emerge from an underlying continuous theory. In this work, we argue that the conflict is not fundamental: it stems from the traditional interpretation of time in Special Relativity (SR). When analyzed correctly, the proper time of distinct observers does not evolve synchronously, and this produces beating and discretization effects that explain, without additional postulates, many behaviors labeled as "quantum."
The Theory of Temporal Relativity proposes a unified theoretical framework that extends Special and General Relativity into the complex domain, integrating aspects of Quantum Mechanics within a single geometrical structure. The theory introduces a generalized Hilbert-space geometry and interprets quantum states as observer-dependent, arising from complex temporal relations between reference frames. Theory of Temporal Relativity is a theoretical framework proposing a unified extension of Special and General Relativity into the complex domain, with the goal of incorporating aspects of Quantum Mechanics within a consistent geometrical model. The theory generalizes Hilbert-space geometry and relativistic invariance to complex coordinates, introducing observer-dependent descriptions of physical quantities and a reinterpretation of quantum states as complex-phase relations between reference frames. Within this model, fundamental constants such as c, G, and h are not strictly universal but can exhibit apparent local variations arising from the observer’s position in the complex spacetime manifold. These variations preserve global covariance and CPT symmetry while reconciling quantum non-locality with relativistic kinematics. The framework predicts experimentally testable effects, including measurable phase shifts in interferometric experiments, possible frequency modulations in non-inertial systems, and small local deviations of physical constants detectable in condensed-matter and gravitational contexts. The supplementary work, “Quantum Uncertainty, Planck’s Constant, and Ontological Insights from Complex Spacetime Geometry and Local Metric Deformations”, develops the principle of Temporal Relativity as a foundational element of the theory. It formalizes the connection between quantum uncertainty, metric deformations, and complex temporal structures, within an alternative logical framework (AU). This extension provides both an ontological interpretation of the wavefunction and a basis for the emergence of quantum indeterminacy from geometric causes. Together, these works present a coherent attempt to unify the relativistic and quantum domains under a single complex-geometric view of spacetime, maintaining mathematical rigor while providing clear experimental pathways for empirical validation.
Thus far I have been primarily concerned with how an interpretation of quantum mechanics might, or might not, be local. This question is traditionally not sharply distinguished from the question of whether quantum mechanics is consistent with the theory of relativity. However, the two questions are indeed quite distinct, and should be recognized as such explicitly. The theory of relativity What does relativity require? As a first approximation, we may make the distinction by noting that the requirements of the theory of relativity are themselves unclear. Minimally, relativity seems to require that there be no way to distinguish one reference frame from another – i.e., that there be no experimental procedure that can determine which of two inertial observers is ‘really’ moving, or more generally that there be no way to discover an observer's absolute velocity. Most authors are willing to find in (special) relativity a stronger requirement: namely, Lorentz-invariance. They say that relativity requires that in fact there is no such thing as absolute velocity. This requirement goes beyond the minimum – it might be that there is an absolute rest frame (so that absolute velocities are given by motion relative to this frame) while there is no way to find it. (As I will discuss in chapter 9, exactly this situation occurs in Bohm's theory.)
Special relativity is a fascinating and challenging branch of physics. It describes the physics of the high velocity/high energy regime, frequently turning up phenomena that appear paradoxical in view of our everyday experience. In this book we will be quite selective in our presentation of the theory of special relativity: we will concentrate on those features that we consider necessary for the later applications to relativistic quantum chemistry. We do this in good conscience, knowing that there is a vast literature on the subject, catering to a wide range of audiences—from the quite elementary to the very sophisticated. A few examples are listed in the reference list, but a visit to any nearby physics library will provide an ample selection of reading material for those wishing to delve deeper into the matter. In the present chapter we adopt a minimalist approach. We develop some of the basic concepts and formulas of special relativity, building on a rather elementary level of basic physics. The aim is to provide a sufficient foundation for those who want to proceed as quickly as possible to the more quantum chemical parts of the text. In later chapters we will introduce more advanced tools of physics and revisit some of the subjects treated here. The theory of special relativity deals with the description of physical phenomena in frames that move at constant velocity relative to each other. The classroom is one such frame, the car passing at constant speed outside the classroom is another. The trajectory of a ball being thrown up vertically in the car will look quite different whether we describe it relative to the interior of the car or relative to the interior of the classroom. In particular we will be concerned with inertial frames. We define an inertial frame as a frame where spatial relations are Euclidean and where there is a universal time such that free particles move with constant velocities. In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S’ are expressed in terms of the Galilean transformations.
The concept of simultaneity is relative in special relativity whereas, it seems to have a definite meaning in quantum mechanics. We propose to use the invariant space-time interval introduced by special relativity as a benchmark for constructing an absolute notion of space-time. We also propose to illustrate that when no measurement is conducted on a quantum system its wave function lives as a wave in the absolute space-time but, when a measurement is to be conducted, we must switch to an ordinary observable frame of reference where the quantum system lives as a particle.
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.