It is postulated there is not a precise static instant in time underlying a dynamical physical process at which the relative position of a body in relative motion or a specific physical magnitude would theoretically be precisely determined. It is concluded it is exactly because of this that time (relative interval as indicated by a clock) and the continuity of a physical process is possible, with there being a necessary trade off of all precisely determined physical values at a time, for their continuity through time. This explanation is also shown to be the correct solution to the motion and infinity paradoxes, excluding the Stadium, originally conceived by the ancient Greek mathematician Zeno of Elea. Quantum Cosmology, Imaginary Time and Chronons are also then discussed, with the latter two appearing to be superseded on a theoretical basis.
It is postulated there is not a precise static instant in time underlying a dynamical physical process at which the relative position of a body in relative motion or a specific physical magnitude would theoretically be precisely determined. It is concluded it is exactly because of this that time (relative interval as indicated by a clock) and the continuity of a physical process is possible, with there being a necessary trade off of all precisely determined physical values at a time, for their continuity through time. This explanation is also shown to be the correct solution to the motion and infinity paradoxes, excluding the Stadium, originally conceived by the ancient Greek mathematician Zeno of Elea. Quantum Cosmology, Imaginary Time and Chronons are also then discussed, with the latter two appearing to be superseded on a theoretical basis.
Zeno of Elea's motion and infinity paradoxes, excluding the Stadium, are stated (1), commented on (2), and their historical proposed solutions then discussed (3). Their correct solution, based on recent conclusions in physics associated with time and classical and quantum mechanics, and in particular, of there being a necessary trade off of all precisely determined physical values at a time (including relative position), for their continuity through time, is then explained (4). This article follows on from another, more physics orientated and widely encompassing paper entitled "Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity" (Lynds, 2003), with its intention being to detail the correct solution to Zeno's paradoxes more fully by presently focusing on them alone. If any difficulties are encountered in understanding any aspects of the physics underpinning the following contents, it is suggested that readers refer to the original paper for a more in depth coverage.
A clock is a dynamical system which passes through a succession of states at constant time intervals. If coupled to another system, it can measure the duration of a physical process and even keep a permanent record of it, such as in a time-of-flight experiment or in observing the lifetime of an unstable atom. A clock can also be used to control the duration of a process, e.g., the precession of a spin in a magnetic field which is turned on and off at prescribed times. This article shows how to construct time-independent Hamiltonians describing these possible uses of a quantum clock. As expected, a good time resolution entails a large energy exchange between the clock and the other system, thereby modifying the evolution of the latter. This evolution may even be halted by using a clock which is too precise (this is the quantum analog of Zeno’s paradox).
When the concept of Zero is analyzed, it is frequently related to another equally deep and complicated concept – Nothing. If in science, Zero is closely associated with mathematics, then Nothing is primarily related to the material world. The question of whether Nothing can be found in the real physical world is often discussed in relation to the vacuum which, at some point in the history of the development of physics, is thought about as an empty space. Nevertheless, theories of modern physics have demonstrated that even in a vacuum, when all the matter is evacuated from a certain region of space, and this region is isolated from ordinary fields – gravitation field, electromagnetic fields etc – there still remain physical fields called vacuum zero fluctuations that can not be removed in principle. These vacuum fields are properties of space. In this chapter I will analyze another physics theory – namely, Quantum Physics – and will consider its less discussed relations to the concept of Nothing. One of the very first Greek philosophers, Thales (624–548 BCE) from Miletus, declared that Something can not appear from Nothing and it can not disappear, turning into Nothing. This idea, in some aspects, was further developed by Zeno from Elea (495–430 BCE) when he created his paradoxes of motion. These paradoxes can be understood as an analysis of the possibility of infinite division of time and space intervals. When we come to Quantum Physics, the very first appearance of the idea of the smallest possible portion of energy, quantum, was stimulated by the analysis of the spectrum of radiation of heated body – black-body radiation. German physicist Max Planck (1858–1947) demonstrated in 1899 that if we assume that there exists a smallest portion of radiation energy that can not be further divided into smaller parts, we arrive at the description of spectrum of the black-body radiation that perfectly matches the observable reality. In contrast, if we assume that radiation energy can be divided an infinite number of times and there does not exist the smallest possible portion, we arrive at the radiation spectrum which is in a sharp contradiction with observable reality. This contradiction is so strong that it has acquired a special term – the ultraviolet catastrophe. Certain parallels in Zeno’s reasoning and approach used in Quantum Physics can clearly be seen and will be discussed. In the conclusion, I will briefly touch on a similar situation in another field of physics when at the other extreme, departure from the idea of the possibility of the infinitely large quantities, this time velocities of motion, gave birth to another theory of modern physics – Einstein’s theory of special relativity.
Description / Abstract English version (recommended for main entry): Time in physics has long been treated as an external, passive parameter.However, such a conception fails to explain why motion is continuous and why the past and future are asymmetric.This paper reformulates time as a complex rotation, in which the imaginary unit i acts as the generator transforming future-pending (fp) states into past-fixed (pf) states.Expressed as T=pf+i fp=eiθ,T=pf+ifp=eiθ, this framework interprets temporal evolution as the rotation of existence itself.By connecting this model to the uncertainty principle[x^,p^]=iℏ[x^,p^]=iℏ,the paper shows that i represents the physical mechanism underlying noncommutativity, continuity, and irreversibility.Applied to Zeno’s paradox, the introduction of a complex phase allows the “arrow” to acquire motion and pierce the target, resolving the classical problem of infinite division.Furthermore, the cumulative, immutable nature of pf-states suggests a blockchain-like temporal structure, explaining why the past cannot be altered and why time’s arrow cannot reverse.This complex-time geometry unifies quantum indeterminacy, temporal continuity, and informational irreversibility within a single conceptual framework. Keywords complex time,imaginary unit, quantum continuity, irreversibility, Zeno’s parado, uncertainty principle, blockchain analogy, time asymmetry
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.
Much of theoretical physics rests on the idealization that a physical system possesses a complete state at each instant of time—a state sufficient to determine all future evolution. We present independent arguments that this idealization is physically unjustified. First, the state at a temporal point is constructed as the limit of states over intervals; the ε-δ definition of a limit explicitly excludes the limit point from its own construction, guaranteeing convergence of the value in the neighborhood of the point while remaining formally silent on what properties the point itself carries. Second, the joint application of quantum mechanics and general relativity guarantees that the limiting process δt → 0 physically self-destructs at the Planck scale; quantum-mechanical clock bounds independently constrain the measurability of time intervals from below; and special relativity independently requires infinite energy to compress proper time to zero—three distinct mechanisms by which established physics prevents the temporal point from being physically realized. Third, quantum field theory has known since the 1940s that physics evaluated at a spacetime point is pathological—a fact encoded in ultraviolet divergences, systematized by the renormalization program, and given physical interpretation by Wilson's demonstration that the divergences reflect the finite domain of validity of any effective description, not a deficiency of any particular calculational method. These lines of evidence converge on a single conclusion: the instantaneous present—a temporal point carrying complete dynamical information—does not exist. Any dynamical framework that presupposes it, including the Chapman-Kolmogorov factorization condition central to the theory of stochastic processes, rests on an assumption that established physics contradicts.