IntroductionClassical mechanics (CM) is conventionally formulated according to partially contextually complete Block Universe principles: an exophysical observer sitting in some frame of reference looks in on a system under observation (SUO) and assigns truth values to propositions about states of that SUO, but the details of how these truth values are obtained are not given.There is a curious aspect to this scenario: the contextual incompleteness of CM is itself contextual.By this we mean that there are two scenarios, with different general propositional classifications (GPCs). The Experimentalist's PerspectiveFrom a classical experimentalist's perspective, generalized propositions (GPs) in CM take the form P(P , ∅|∅, F ).(6.1)Here P is some proposition of scientific interest, such as "The orbit of Jupiter is an ellipse," and F is relative external context describing the frame of reference used to describe states of the SUO relative to the wider Universe external to that SUO.This form of GP has a GPC of 2 according to the algorithm discussed in Chapter 2.Remark 6.1 Suppose an astronomer used a telescope T to test proposition P .That would not upgrade the GP to one of the form P(P , T |∅, F ) with a GPC of 3. The reason is that the use of a telescope is not essential in this context.Historically, Kepler did not use a telescope to state his laws of planetary motion: he used the data obtained by Tycho using naked eye observations.On the other hand, the observation of the Hubble red shift using telescopes fitted with spectrum analyzers would qualify for such an upgrade.
IntroductionClassical mechanics (CM) is conventionally formulated according to partially contextually complete Block Universe principles: an exophysical observer sitting in some frame of reference looks in on a system under observation (SUO) and assigns truth values to propositions about states of that SUO, but the details of how these truth values are obtained are not given.There is a curious aspect to this scenario: the contextual incompleteness of CM is itself contextual.By this we mean that there are two scenarios, with different general propositional classifications (GPCs). The Experimentalist's PerspectiveFrom a classical experimentalist's perspective, generalized propositions (GPs) in CM take the form P(P , ∅|∅, F ).(6.1)Here P is some proposition of scientific interest, such as "The orbit of Jupiter is an ellipse," and F is relative external context describing the frame of reference used to describe states of the SUO relative to the wider Universe external to that SUO.This form of GP has a GPC of 2 according to the algorithm discussed in Chapter 2.Remark 6.1 Suppose an astronomer used a telescope T to test proposition P .That would not upgrade the GP to one of the form P(P , T |∅, F ) with a GPC of 3. The reason is that the use of a telescope is not essential in this context.Historically, Kepler did not use a telescope to state his laws of planetary motion: he used the data obtained by Tycho using naked eye observations.On the other hand, the observation of the Hubble red shift using telescopes fitted with spectrum analyzers would qualify for such an upgrade.
Introduction In this chapter we apply the concepts presented in Chapter 22 to a topic that has been of much theoretical interest in recent decades: causal set theory (CST). It is interesting and useful to review the status of CST relative to other foundational topics in mathematical physics that have been and remain of prime significance to empirical physics, namely, special relativity (SR), general relativity (GR), and quantum mechanics (QM). Those great frameworks are built on foundations that presuppose the continuity of space and time. Despite this, it is an empirical fact that we are surrounded by discreteness in the form of atoms, so it is natural to see how that discreteness fits in with the continuity of space and time. We have attempted in this book to show that the application of discrete principles to QM can go far beyond the basic Planck quantum of action. We have gone as far as discretizing empirical time, in the form of the stage concept. However, as far as relativity in either of its manifestations is concerned, the case for any form of discreteness has not yet been established empirically, although great efforts are being made in that direction in the program known as quantum gravity . We have particularly in mind the discretization approaches to GR of Regge (Regge, 1961) and Penrose (Penrose, 1971). Regge's approach is a classically based discretization of the Riemannian manifold concept, so has at best a generalized propositional classification (GPC) of two. Penrose's spin network paradigm started as a quantum description of spacetime and has led to much work in the area of spin networks in the hope of unifying GR and QM. However, workers in spin networks and related branches of quantum gravity have focued their attention on the mathematical complexities and hardly any on the role of observers and their apparatus, leading to a GPC classification of quantum gravity as one, that is, as mathematical physics and not physics. Claims made that such programs represent empirical physics have not been empirically substantiated to date; quantum gravity at this time remains empirically vacuous, despite the best efforts of many theorists over many decades.
Historically, the scientific exploration of new phenomena has often been guided by systematic studies of observations, i.e., experimentally verifiable facts, which can be used as the basis to construct the underlying physical laws. As the apex of the investigations, one tries to identify the minimal set of fundamental assumptions – referred to as the axioms – needed to describe correctly the experimental observations. Even though the axioms form the basis to predict the system's behavior completely, they themselves have no rigorous derivation or interpretation. Thus, the axioms must be viewed as the elementary postulates that allow us to formulate a systematic description of the studied system based on well-defined logical reasoning. Even though it might seem unsatisfactory that axioms cannot be “derived,” one has to acknowledge the paramount power of well-postulated axioms to predict even the most exotic effects. As is well known, the theory of classical mechanics can be constructed using only the three Newtonian axioms. On this basis, an infinite variety of phenomena can be explained, ranging from the cyclic planetary motion all the way to the classical chaos. In this book, we are mainly interested in understanding how the axioms of classical and quantum mechanics can be applied to obtain a systematic description for the phenomena of interest. Especially, we want to understand how many-particle systems can be modeled, how quantum features of light emerge, and how these two aspects can be combined and utilized to explore new intriguing phenomena in semiconductor quantum optics.
Abstract The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either ‘true ’ or ‘false’, describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of ‘no go ’ theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. In this respect, the Bub-Clifton ‘uniqueness theorem ’ is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences.
Introduction Throughout this book we have commented on the need to have a wider vision of physics as more than merely developing and solving equations of motion for states of systems under observation (SUOs). It is our premise that any attempt to discuss time needs to address issues such as the status of observers, apparatus, and suchlike, relative to those states of SUOs. This echoes the following quote from Wheeler: Stronger than the anthropic principle is what I might call the participatory principle. According to it we could not even imagine a universe that did not somewhere and for some stretch of time contain observers because the very building materials of the universe are these acts of observer participancy. You wouldn't have the stuff out of which to build the universe otherwise. This participatory principle takes for its foundation the absolutely central point of the quantum: No elementary phenomenon is a phenomenon until it is an observed (or registered) phenomenon. (Wheeler, 1979) We agree with Wheeler to this extent: there seems little, if any, point in speculating about truly unobservable parts of the Universe. If, however, there is a possibility that physical signals or effects from currently unobserved parts of the Universe might one day reach us, then that would in essence make those parts observable, albeit indirectly, and therefore suitable for theoretical analysis and speculation.
Mappings, functions In this book, the term architecture refers to a non-mathematical description of the sequence of events which collectively describes an experiment. Continuous time (CT) classical mechanics (CM) is based on some explicit and implicit assumptions that generate architectures radically different from those describing quantum processes. In the following list, which is not complete by any means, most of these assumptions may appear obvious and overstated, but one or two are subtle and should be brought to light, since they have an enormous impact on the theories which are based on them. Our comments are in square brackets at the end of each item in the list. At any given time, a system under observation (SUO) exists in a unique physical state with absolute properties independent of any observer [a standard classical metaphysical assumption]. The physical state of an SUO at any given time t may be represented by a single element Ψ( t ) called a state of some fixed state space U (or universe) [a reasonable mathematical assumption]. The state Ψ( t ') of an SUO at any time t ' later than t is also an element of U , not necessarily the same state Ψ( t ) at the earlier time [a reasonable classical mathematical assumption]. Observers are exophysical, meaning that they stand outside of U [a nearly universal metaphysical assumption; observers are not discussed in detail in CM or QM simply because there is no comprehensive theory of what constitutes an observer].