Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e. its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called "quantum potential" in this context is briefly discussed.
Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e. its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called "quantum potential" in this context is briefly discussed.
Several somewhat independent concepts are introduced in this chapter. First, we've stated that the wave equation contains information about the particle. How is the information extracted? By using operators, which often include derivative operations. By "operating" on the wave function, we can pull out certain factors, such as the momentum or the energy of the particle. Here, three operators are discussed: the total energy operator in terms of a time-derivative; a more space-based energy operator called the Hamiltonian, which simply represents the kinetic energy plus the potential energy; and the momentum operator. Second, we show that an oscillating wave function does not necessarily mean that the particle position is oscillating, since the position is related to the square modulus of the wave function. Third, the Heisenberg uncertainty principle is introduced using a free particle with specified energy, but with no specified position. Fourth, the idea of "quantization" is introduced. The quantization, or the existence of only certain allowed energies, arises when the particle is not free but is trapped by a potential, such as an electron trapped in an atom. Since trapped particles are very important to scientists, the entire subject is called "quantum" science.
The meaning of the wave function and its evolution are investi-gated. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A dis-crete model of energy-conserved wavefunction collapse is proposed and shown consistent with existing experiments and our macroscopic expe-rience. Besides, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issues of unifying quantum mechanics and relativity.
There are several links between information theory — between »information as such« whatever that might mean — and the foundations of nature, i e, basic physics. Well, since physics a science still evolving (despite propositions that »the end of science« now in sight), some of the links between information and physics are hypothetical. One linkage fairly obvious — thermodynamics depends upon entropy or some measure of order or disorder, and that clearly has something to do with information. The other possible relationship might be between the uncertainty that a key characteristic of quantum mechanics. We can never know everything on the quantum scale; the Heisenberg uncertainty principle tells that the more information we gain in one direction the less we know in another. If we, on that scale, obtain certainty as to one factor, then another will stay entirely undecided — try to affix a position to a quantum particle, and the momentum will be out of bounds entirely. (This the established dogma currently, though there are some deviating ideas.1) Then the question arises: who does the actual measuring, with what — what particles, at what level, are involved in this act of, as it pronounced, forcing the wave function to collapse? This wave function is the indeterminate particle: the various statistics describing the probabilities that the particle (which simultaneously takes on aspects of a wave) may be in one state or another. We will now attempt to discuss these different — or are they really different? — interdependencies, starting with black holes.
The present work deals with the idea of a multi-time wave function, i.e. a wave function with N space-time arguments for N particles. Firstly, a careful derivation of the necessity of multi-time wave functions in relativistic quantum mechanics is given and a general formalism is developed. Secondly, the physical meaning of multi-time wave functions is discussed in connection with realistic relativistic quantum theories, in particular the "Hypersurface Bohm-Dirac" model. Thirdly, a first interacting model for multi-time wave functions of two Dirac particles in 1+1 space-time dimensions is constructed. Interaction is achieved by means of boundary conditions on configuration space-time, a mechanism closely related to zero-range physics. This is remarkable, as a restrictive consistency condition rules out various types of interaction and consequently no rigorous interacting model was known before. Fourthly, the model is extended to more general types of interaction and to the N-particle case. Higher dimensions are also discussed. Finally, the "Two-Body Dirac equations" of constraint theory are placed within the context of the multi-time formalism. In particular, the question of probability conservation is critically discussed, leading to further implications both for fundamental and applied questions.
Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles: positions constitute a configuration space and the corresponding probability distributions constitute a statistical manifold. The dynamics follows from a principle of inference, the method of Maximum Entropy. The concept of time is introduced as a convenient way to keep track of change. A welcome feature is that the entropic dynamics notion of time incorporates a natural distinction between past and future. The statistical manifold is assumed to be a dynamical entity: its curved and evolving geometry determines the evolution of the particles which, in their turn, react back and determine the evolution of the geometry. Imposing that the dynamics conserve energy leads to the Schroedinger equation and to a natural explanation of its linearity, its unitarity, and of the role of complex numbers. The phase of the wave function is explained as a feature of purely statistical origin. There is a quantum analogue to the gravitational equivalence principle.
We have seen that there is an intimate relationship between probability and information. The values we assign to probabilities depend on the information available, and information is a function of probabilities. This connection makes it inevitable that information will be an important concept in any statistical theory, including thermodynamics and, of course, quantum physics. The probabilistic interpretation of quantum theory has probability amplitudes rather than probabilities as the fundamental quantities. This feature, together with the associated superposition principle, is responsible for intrinsically quantum phenomena and gives quantum information theory its distinctive flavour. We shall see that the quantum rules for dynamical evolution and measurement, together with the existence of entangled states, have important implications for quantum information. They also make it possible to perform tasks which are either impractical or impossible within the classical domain. In describing these we shall make extensive use of simple but fundamental ideas in quantum theory. This chapter introduces the mathematical description of quantum physics and the concepts which will be employed in our study of quantum information.