The VIM3 defines a quantity as a “property of a phenomenon, body, or substance”, leaving the characteristics of the term ‘quantity’, related to the chosen characteristics of the relevant properties. The question is: does necessarily a property also necessarily refer to the possible granularity of a phenomenon, body, or substance? Take, for example, for the quantity “mass”: it does not always have to take into account whether or not a phenomenon, body, or substance is subdivided into discrete entities? It depends of the frame of the analysis and also on the chosen measurement unit. In other cases, like temperature, the macroscopic properties are related to the statistical properties of granular substances like atoms and molecules are, so the present meaning of ‘temperature’ is generally lost at the numerical level where the entity’s statistics become meaningless. Yet another case is quantum physics. The paper illustrates the issue and possible solutions under development. Keywords: continuous; granular; quantity; magnitude; quanta; integer number; real number; counting; function
The VIM3 defines a quantity as a “property of a phenomenon, body, or substance”, leaving the characteristics of the term ‘quantity’, related to the chosen characteristics of the relevant properties. The question is: does necessarily a property also necessarily refer to the possible granularity of a phenomenon, body, or substance? Take, for example, for the quantity “mass”: it does not always have to take into account whether or not a phenomenon, body, or substance is subdivided into discrete entities? It depends of the frame of the analysis and also on the chosen measurement unit. In other cases, like temperature, the macroscopic properties are related to the statistical properties of granular substances like atoms and molecules are, so the present meaning of ‘temperature’ is generally lost at the numerical level where the entity’s statistics become meaningless. Yet another case is quantum physics. The paper illustrates the issue and possible solutions under development. Keywords: continuous; granular; quantity; magnitude; quanta; integer number; real number; counting; function
Discrete and continuous measurements All the examples of measurements so far discussed have in common the fact that the experimental output is either a single number or a finite set of numbers. However, in real measurements the output is often a record of some continuous function of time, from which one can get an understanding of the behavior of the measured quantity during some time interval. This type of measurement is called continuous. Correspondingly, measurements that give a discrete set of numbers can be called discrete. Let us note that measurements are sometimes called continuous if they are made not instantaneously, but over a finite duration of time. Evidently, this is not a reasonable viewpoint; all real measurements last for a finite time. The dividing line between continuous and discrete measurements is in fact the character of the output signal: does it give information about the measured quantity only at some chosen moment of time, or does it permit a continuous monitoring of the quantity's time evolution. The necessity to develop the quantum theory of continuous measurements arose in the 1960s, when devices for making continuous measurements were developed with sensitivities close to the quantum level (masers, parametric amplifiers, optical heterodyne receivers, …).
From the preceding exposition it follows that the quantum statistical representation of an individual atomic system does not imply any renunciation regarding further knowledge about the system. On the contrary the statistical character of quantum physics directly reflects the fundamental non-causal appearance of quantal recordings, and through the discovery of the complementarity of quantal phenomena a complete but necessarily statistical description of the atom has been accomplished. Furthermore quantum physics has afforded a statistical description of macroscopic bodies through their representation as assemblies of huge numbers of identical atomic systems, and, apart from accounting in a more fundamental way for the properties already described in classical physics(the correspondence principle, p. 36), quantum statistics explains several macroscopic phenomena which cannot be understood within the conceptual frame of classical physics only. It must be maintained, however, that when observing the properties of individual atomic objects the macroscopic observational apparatus cannot be represented in terms of atomic constituents because this would destroy the unambiguity of the description. The observational apparatus fulfils its basic function of being the spacetime frame of reference for the quantal recordings precisely through its description in classical physics alone. However, when refraining from the analysis of individual quantal processes it is possible to apply quantum statistics to macroscopic phenomena, that is, the causality of classical physical phenomena can be interpreted in the quantum formalism as the statistical result of the interplay of enormous numbers of atomic processes. It is possible, then, to talk of statistical causality as a fundamental aspect of physical reality.
This chapter presents the measurement problem from the point of view of the thermal interpretation of quantum physics. Everything physicists measure is measured in a thermal environment, for which statistical thermodynamics is relevant. The thermal interpretation agrees with how one interprets measurements in thermodynamics, the macroscopic part of quantum physics, derived via statistical mechanics. By its very construction, the thermal interpretation naturally matches the classical properties of our quantum world: The thermal interpretation assigns states-and a realistic interpretation for them-to individual quantum systems, in a way that large quantum systems are naturally described by classical observables. Since measurement devices are large quantum systems, this is the key to solving the measurement problem. We discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. Since quantum physics makes many deterministic predictions, for example regarding observed spectra, but also assertions about probabilities, we distinguish deterministic and statistical measurements. To clarify the meaning of the concept of measurement, we postulate (in Section 11.1) a measurement principle that defines what it means in the thermal interpretation to measure a quantity with a specified uncertainty. This turns the measurement problem from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system. The essence of scientific practice is the reproducibility of measurements, discussed in Section 11.2. Unlike in traditional interpretations, single, nonreproducible observations do not count as measurements, since this would violate the reproducibility of measurements-the essence of scientific practice. As a consequence, the measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation {A} rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. This difference is most conspicuous in the interpretation of single discrete microscopic events. Except in very special circumstances, these are not reproducible. Thus, they have no scientific value in themselves and do not constitute measurement results. Scientific value is, however, in ensembles of such observations, which result in approximate measurements of q-probabilities and q-expectations. Sections 11.3 and 11.4 distinguish deterministic and statistical measurements, depending on whether a single observation is reproducible, and discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. How to measure the q-probabilities of events, described in terms of POVMs, is discussed in Section 11.5. Section 11.6 poses the measurement problem as a problem of the statistical mechanics of measurement devices, and discuss recent work by Allahverdyan, Balian, and Nieuwenhuizen on its resolution. Section 11.7 discusses the problem how, in the thermal interpretation, discrete events can arise, although qexpectations are by their very nature continuous. This is resolved by pointing to work by Breuer and Petruccione on piecewise deterministic processes (PDP). Section 11.8 sheds additional light on the origin of discrete events by an analysis of the role dissipative bistability plays in obtaining discreteness in classical physics.
The argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over ℂ and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity, and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck, following B. Riemann's comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.
The interference of laser photons (λ=532 nm) when they pass through two slits of the same width has been studied. The interference is observed up to the value S≈11 mm between the slits. Based on this, the assumption has been made that all elementary particles are soliton formations of the electromagnetic field. Solitons of classical physics exhibit the properties of a wave and a corpuscle, thus, we have an explanation of the corpuscular-wave nature of elementary particles as a phenomenon of classical physics. For the formation of solitons, the presence of a medium with the properties of dispersion and nonlinearity is necessary, and this contradicts the foundations of SRT. In this regard, a number of problems related to the foundations of SRT have been analyzed. It has been established that SRT does not pass the verification by the correspondence principle, since at low speeds the Lorentz transformation of time does not pass into the Galilean transformation. This means that SRT cannot be a physical theory. The reader’s attention is drawn to the fact that the relationship between the mass of a body and its internal energy, as well as the dependence of the mass of a body on the speed of its motion, do not belong to the problems of the theory of relativity, and therefore, their experimental confirmation cannot be an argument in favor of SRT. It is shown that the idea of the independence of the speed of quanta on the speed of the frame has a direct experimental refutation – the results of measuring the rotation periods of Io (a satellite of Jupiter) taking into account the addition and subtraction of velocities make it possible to obtain results for the speed of light with an accuracy of 1% of the reference value. This is a direct experimental refutation of the SRT.
## Abstract Physics uses continuous mathematics - fields, manifolds, wave functions, differential equations - to make predictions. Physical observations are recorded as discrete events: detector clicks, particle counts, energy transitions. This paper argues that this is not an accident to be explained away, but a structural fact that should be treated as a working principle. The claim: continuous mathematical objects describe distributions over possible outcomes. They are not physical entities occupying space. Discrete events are what physically happens. Confusing these two produces recurring conceptual difficulties—most visibly in the quantum measurement problem and in spacetime singularities. This distinction reframes, but does not solve, several open problems. It does not derive the Born rule, does not specify a concrete discrete causal structure, and does not yield new experimental predictions. Its value is diagnostic: it identifies where existing puzzles are category errors versus where they are genuine open technical problems, and it identifies a specific reframing of how discrete causal models (such as causal set theory) might recover Lorentz-invariant physics. Keywords: continuity, discreteness, possibility, measurement, causality, causal set theory