Abstract An example of a relativistic mechanics of n particles in the two‐dimensional Minkowski space is constructed. The essential feature of the theory is the occurrence of two histories for every physical system.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are significant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benefits of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantummechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic effects become observable, for which nonrelativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.34) must be used. There are, however, more subtle effects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
Up to this point, our study of quantum mechanics has concerned itself with the behaviour of particles that inhabit a Galilean spacetime. For many purposes, in atomic, molecular and condensed matter physics, this theory is quite adequate. We saw in earlier chapters, however, that our actual spacetime has a structure which is much closer to that of the Minkowski spacetime of special relativity and that more general structures must be considered when gravitational phenomena are signicant. From a purely theoretical point of view, it is therefore important to formulate quantum theory in a way which is consistent with these more general spacetimes. The benets of constructing a relativistic quantum theory actually go far beyond the aesthetic satisfaction of making our geometrical and quantum-mechanical reasoning compatible. For one thing, we shall discover that the relativistic theory provides a deeper understanding of spin and the distinction between fermions and bosons, which in the non-relativistic theory appear simply as facts of life that we must strive to accommodate. Also, of course, there are many situations in which relativistic eects become observable, for which non-relativistic theory provides no explanation. The most obvious are high-energy scattering experiments, in which particles acquire kinetic energies comparable with or greater than their rest energies mc2, and the correct 4-momentum (3.33) must be used. There are, however, more subtle eects, such as the spin-orbit coupling that is essential for interpreting atomic spectra, which are also of relativistic origin.
Abstract One serious objection to the whole of quantum theory as we have developed it so far, is that it is inconsistent with the theory of relativity. In order to see how we may overcome this defect, at least as far as the special theory of relativity is concerned, we shall first recall some of its basic features. Shortly after Einstein’s original 1905 papers on the subject, Minkowski realized that the physical ideas could also be understood geometrically, by combining space and time into a single four-dimensional real vector space, M, whose elements we shall call four-vectors.
Relativistic quantum theories are equipped with a background Minkowski spacetime and non‐relativistic quantum theories with a Galilean space‐time. Traditional investigations have distinguished their distinct space‐time structures and have examined ways in which relativistic theories become sufficiently like Galilean theories in a low velocity approximation or limit. A different way to look at their relationship is to see that both kinds of theories are special cases of a certain five‐dimensional generalization involving no limiting procedures or approximations. When one compares them, striking features emerge that bear on philosophical questions, including the ontological status of the wave function and time reversal invariance.
Abstract Quantum mechanics describes multi-electronic systems by using probabilistic theories. All of these theories provide a probabilistic conception of quantum mechanics, with an astonishing accuracy (for instance the Helium atom ground state is computed with more than 20 digits of accuracy) but unfortunately seems to be a bit “disconnected” with macro-scale physics. Indeed, the real observable world is for us a 3D+time Minkowski space but quantum mechanics is expressed in multi-dimensional Hilbert space: which is not intuitively physical but seems to be purely mathematical. In this article we try to construct a relativistic equation that gives an approximately correct description of atomic physics in the spacetime (i.e the “real space”) for 2 electrons systems: i.e a 2-electrons Dirac equation. The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave. For this we start from the Hartree equation, which gives already a spacetime description of each electron wave function. Then, we switch to Hartree-Fock representation, which gives birth to the exchange potential, that can be also expressed in spacetime. By extending the concept of exchange potential to the relativistic 4-exchange: we obtain a relativistic 2-electrons spacetime Dirac-Hartree-Fock (STDHF) equation (i.e in Minkowski space). That equation takes into account relativistic effects but not correlation effects, which is the difficulty of this study. Then we will show the equivalence of the 2-electrons STDHF equation with the Hartree-Fock equation in non-relativistic, stationary and scalar regime. Electron correlation issues will be discussed as well.