Special relativity is a fascinating and challenging branch of physics. It describes the physics of the high velocity/high energy regime, frequently turning up phenomena that appear paradoxical in view of our everyday experience. In this book we will be quite selective in our presentation of the theory of special relativity: we will concentrate on those features that we consider necessary for the later applications to relativistic quantum chemistry. We do this in good conscience, knowing that there is a vast literature on the subject, catering to a wide range of audiences—from the quite elementary to the very sophisticated. A few examples are listed in the reference list, but a visit to any nearby physics library will provide an ample selection of reading material for those wishing to delve deeper into the matter. In the present chapter we adopt a minimalist approach. We develop some of the basic concepts and formulas of special relativity, building on a rather elementary level of basic physics. The aim is to provide a sufficient foundation for those who want to proceed as quickly as possible to the more quantum chemical parts of the text. In later chapters we will introduce more advanced tools of physics and revisit some of the subjects treated here. The theory of special relativity deals with the description of physical phenomena in frames that move at constant velocity relative to each other. The classroom is one such frame, the car passing at constant speed outside the classroom is another. The trajectory of a ball being thrown up vertically in the car will look quite different whether we describe it relative to the interior of the car or relative to the interior of the classroom. In particular we will be concerned with inertial frames. We define an inertial frame as a frame where spatial relations are Euclidean and where there is a universal time such that free particles move with constant velocities. In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S’ are expressed in terms of the Galilean transformations.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
PROSPECTIVE The concern of this paper will be different from that of most studies of quantum gravity. There will be no discussion of the problem of quantizing the gravitational field equations, hence nothing about the modifications which quantization requires of space-time structure at distances of the order of the Planck length (∼ 10 −33 cm). The emphasis will be rather upon the implications of quantum mechanics for certain general properties of events and processes which occur in the theatre of space-time, specifically with non-locality and nonlinearity . Quantum mechanics is undoubtedly a non-local theory when it treats correlated spatially separated systems. When one examines closely the character of this non-locality, however, one does not find reasons for modifying the causal structure of space-time as described by special or general relativity theory, but rather reasons for refining the concept of an event. The occurrence of definite outcomes of measurements implies that there are processes in nature governed by nonlinear laws. Nonlinearity is very peculiar from the standpoint of quantum mechanics, since it is contrary to the linearity of the time-dependent Schrödinger equation, which is commonly assumed to govern the dynamics of any isolated physical system. Hence it is necessary to inquire whether a rational treatment of the occurrence of outcomes is possible without some modifications of current quantum mechanics. Whatever decision is reached on this question, however, it does not appear to bear directly on space-time structure, but rather on the general character of processes.
The general view is that all fundamental physical laws should be formulated within the framework given by quantum mechanics (QM). In a sense, QM therefore has the character of a metaphysical theory. Consequently, if it is possible to derive QM from more basic principles, these principles should be of general, philosophical nature. Here, we derive the formalism of QM from well-motivated epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. First, an epistemic formalism is developed, using concepts like knowledge and potential knowledge, identifying a physical state
S with the potential knowledge of the physical world. It is demonstrated that QM emerges from this formalism. However, Hilbert spaces, wave functions and probabilities are defined in certain well-defined observational contexts only. This means that the epistemic formalism is broader than QM. In the fundamental layer of description, the physical state
S is a subset of a state space
S={Z}, such that
S always contains many elements
Z. These elements correspond to unattainable states of complete knowledge of the world. The evolution of
S cannot be determined in terms of the individual evolution of the elements
Z, unlike the evolution of an ensemble in classical phase space. The evolution of
S is described in terms of sequential time
n∈N, which is updated according to
n→n+1 each time an event occurs, each time potential knowledge changes. Sequential time
n can be separated from relational time
t, which describes distances between events in space-time. There is an entire space-time associated with each
n, in which
t represents the knowledge at sequential time
n about the temporal relations between present and past events.
In this paper, the authors apply models extracted from the Many-Body Quantum Mechanics to understand how knowledge production is correlated to the innovation potential of a work team. This study is grounded in key assumtpions. First, complexity theory applied to social science suggests that it is of paramount importance to consider elements of non-objectivity and non-determinism in the statistical description of socio-economic phenomena. Second, a typical factor of indeterminacy in the explanation of these phenomena lead to the need to apply the instruments of quantum physics to formally describe social behaviours. In order to experiment the validity of the proposed mathematic model, the research intends to: 1) model nodes and interactions; 2) simulate the network behaviour starting from specific defined models; 3) visualize the macroscopic results emerging during the analysis/simulation phases through a digital representation of the social network.Request access from your librarian to read this chapter's full text.
A computer is a physical device that helps us process information by executing algorithms. An algorithm is a well-defined procedure, with finite description, for realizing an information-processing task. An information-processing task can always be translated into a physical task. When designing complex algorithms and protocols for various information-processing tasks, it is very helpful, perhaps essential, to work with some idealized computing model. However, when studying the true limitations of a computing device, especially for some practical reason, it is important not to forget the relationship between computing and physics. Real computing devices are embodied in a larger and often richer physical reality than is represented by the idealized computing model. Quantum information processing is the result of using the physical reality that quantum theory tells us about for the purposes of performing tasks that were previously thought impossible or infeasible. Devices that perform quantum information processing are known as quantum computers. In this book we examine how quantum computers can be used to solve certain problems more efficiently than can be done with classical computers, and also how this can be done reliably even when there is a possibility for errors to occur. In this first chapter we present some fundamental notions of computation theory and quantum physics that will form the basis for much of what follows. After this brief introduction, we will review the necessary tools from linear algebra in Chapter 2, and detail the framework of quantum mechanics, as relevant to our model of quantum computation, in Chapter 3. In the remainder of the book we examine quantum teleportation, quantum algorithms and quantum error correction in detail. We are often interested in the amount of resources used by a computer to solve a problem, and we refer to this as the complexity of the computation. An important resource for a computer is time. Another resource is space, which refers to the amount of memory used by the computer in performing the computation. We measure the amount of a resource used in a computation for solving a given problem as a function of the length of the input of an instance of that problem.