The unexpected plays an important but little-understood role in physics. We are working to understand how the unexpected functions in theory and in experiment. This work leads to novel theoretical and instrumental structures of “two-clock physics”, with implications for the scientific method. The path to these structures that express the unexpected draws on three roots: 1. Physicists write strings of symbols. Quantum theory implies that physicists act as agents who make guesses, expressed as strings of symbols, in the face of unpredictability far more drastic than quantum uncertainty. Assimilating unpredictability as pervasive in theory and experiment opens new avenues to investigation. 2. No two clocks tick alike. We formulate a concept network of transmission of symbols among agents, without assuming any spacetime or any global time variable, but requiring that agents adjust local clock rates. This structure offers alternatives to time and distance adapted to a variety of cases and is an important element of what we call two-clock physics. 3. Sometimes a flipped coin lands on edge. Agents must rely on guesses to steer clock rates to achieve a logical synchronization that avoids confusion of one symbol with another.
The unexpected plays an important but little-understood role in physics. We are working to understand how the unexpected functions in theory and in experiment. This work leads to novel theoretical and instrumental structures of “two-clock physics”, with implications for the scientific method. The path to these structures that express the unexpected draws on three roots: 1. Physicists write strings of symbols. Quantum theory implies that physicists act as agents who make guesses, expressed as strings of symbols, in the face of unpredictability far more drastic than quantum uncertainty. Assimilating unpredictability as pervasive in theory and experiment opens new avenues to investigation. 2. No two clocks tick alike. We formulate a concept network of transmission of symbols among agents, without assuming any spacetime or any global time variable, but requiring that agents adjust local clock rates. This structure offers alternatives to time and distance adapted to a variety of cases and is an important element of what we call two-clock physics. 3. Sometimes a flipped coin lands on edge. Agents must rely on guesses to steer clock rates to achieve a logical synchronization that avoids confusion of one symbol with another.
Curiously overlooked in physics is its dependence on the transmission of numbers. For example the transmission of numerical clock readings is implicit in the concept of a coordinate system. The transmission of numbers and other logical distinctions is often achieved over a computer-mediated communications network in the face of an unpredictable environment. By unpredictable we mean something stronger than the spread of probabilities over given possible outcomes, namely an opening to unforeseeable possibilities. Unpredictability, until now overlooked in theoretical physics, makes the transmission of numbers interesting. Based on recent proofs within quantum theory that provide a theoretical foundation to unpredictability, here we show how regularities in physics rest on a background of channels over which numbers are transmitted. As is known to engineers of digital communications, numerical transmissions depend on coordination reminiscent of the cycle of throwing and catching by players tossing a ball back and forth. In digital communications, the players are computers, and the required coordination involves unpredictably adjusting "live clocks" that step these computers through phases of a cycle. We show how this phasing, which we call `logical synchronization,' constrains number-carrying networks, and, if a spacetime manifold in invoked, put "stripes" on spacetime. Via its logically synchronized channels, a network of live clocks serves as a reference against which to locate events. Such a network in any case underpins a coordinate frame, and in some cases the direct use of a network can be tailored to investigate an unpredictable environment. Examples include explorations of gravitational variations near Earth.
The basic equations of physics involve a time variable t and are invariant under the transformation t → −t. This invariance at first sight appears to impose time reversibility as a principle of physics, in conflict with thermodynamics. But equations written on the blackboard are not the whole story in physics. In prior work we sharpened a distinction obscured in today’s theoretical physics, the distinction between obtaining evidence from experiments on the laboratory bench and explaining that evidence in mathematical symbols on the blackboard. The sharp distinction rests on a proof within the mathematics of quantum theory that no amount of evidence, represented in quantum theory in terms of probabilities, can uniquely determine its explanation in terms of wave functions and linear operators. Building on the proof we show here a role in physics for unpredictable symbol-handling agents acting both at the blackboard and at the workbench, communicating back and forth by means of transmitted symbols. Because of their unpredictability, symbol-handling agents introduce a heretofore overlooked source of irreversibility into physics, even when the equations they write on the blackboard are invariant under t → −t. Widening the scope of descriptions admissible to physics to include the agents and the symbols that link theory to experiments opens up a new source of time-irreversibility in physics.
Preoccupied with measurement, physics has neglected the need, before anything can be measured, to recognize what it is that is to be measured. The recognition of symbols employs a known physical mechanism. The elemental mechanism-a damped inverted pendulum joined by a driven adjustable pendulum (in effect a clock)-both recognizes a binary distinction and records a single bit. Referred to by engineers as a "clocked flip-flop," this paired-pendulum mechanism pervades scientific investigation. It shapes evidence by imposing discrete phases of allowable leeway in clock readings; and it generates a mathematical form of evidence that neither assumes a geometry nor assumes quantum states, and so separates statements of evidence from further assumptions required to explain that evidence, whether the explanations are made in quantum terms or in terms of general relativity. Cleansed of unnecessary assumptions, these expressions of evidence form a platform on which to consider the working together of general relativity and quantum theory as explanatory language for evidence from clock networks, such as the Global Positioning System. Quantum theory puts Planck's constant into explanations of the required timing leeway, while explanations of leeway also draw on the theory of general relativity, prompting the question: does Planck's constant in the timing leeway put the long known tension between quantum theory and general relativity in a new light?
In earlier papers we showed unpredictability beyond quantum uncertainty in atomic clocks, ensuing from a proven gap between given evidence and explanations of that evidence. Here we reconceive a clock, not as an isolated entity, but as enmeshed in a self-adjusting communications network adapted to one or another particular investigation, in contact with an unpredictable environment. From the practical uses of clocks, we abstract a clock enlivened with the computational capacity of a Turing machine, modified to transmit and to receive numerical communications. Such “live clocks” phase the steps of their computations to mesh with the arrival of transmitted numbers. We lift this phasing, known in digital communications, to a principle of logical synchronization, distinct from the synchronization defined by Einstein in special relativity. Logical synchronization elevates digital communication to a topic in physics, including applications to biology. One explores how feedback loops in clocking affect numerical signaling among entities functioning in the face of unpredictable influences, making the influences themselves into subjects of investigation. The formulation of communications networks in terms of live clocks extends information theory by expressing the need to actively maintain communications channels, and potentially, to create or drop them. We show how networks of live clocks are presupposed by the concept of coordinates in a spacetime. A network serves as an organizing principle, even when the concept of the rigid body that anchors a special-relativistic coordinate system is inapplicable, as is the case, for example, in a generic curved spacetime.
This paper contributes to a burgeoning area of investigation, the ambiguity inherent in mathematics and the implications for physics of this ambiguity. To display the mathematical form of equations of quantum theory used to describe experiments, we make explicit the knobs by which the devices of an experiment are arranged and adjusted. A quantum description comes in two parts: (1) a statement of results of an experiment, expressed by probabilities of detections as functions of knob settings, and (2) an explanation of how we think these results come about, expressed by linear operators, also as functions of knob settings. Because quantum mechanics separates the two parts of any description, it is known that between the statements of results and the explanations lurks a logical gap: given any statement of results one has a choice of explanations. Here we work out some consequences of this openness to choice. We show how quantum theory as mathematical language in which to describe experiments necessarily involves multiple descriptions: multiple explanations of a given result, as well as multiple statements of results and multiple arrangement of knobs. Appreciating these multiplicities resolves what otherwise is a confusion in the concept of invariance. Implications of multiplicity of description for the security of quantum key distribution are noted.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation: