This work discloses some inconsistencies of quantum mechanics (QM) in two different contexts. On the one hand the equivalence principle in his two formulations is the starting point of all gravitational theories; yet there is no way to make it consistent with the non-relativistic formulation of QM. Moreover, the evaluation of the Schrödinger equation for a particle in an accelerated reference frame sheds light on a trajectory-dependent phase term subject to proper time. On the other hand, the Bargmann theorem unmistakably proves that the Schrödinger equation does not admit any superposition of different masses if the symmetry chosen for the system is the Galilean one. However, this is incompatible with relativity, since mass and energy are equivalent and one can certainly superimpose different energies. Both inconsistencies support the hypothesis that mass and proper time could be treated as variables. In the third chapter we will describe a theory in which the Galilean symmetry is extended by assuming particle mass and proper time as canonically conjugated classical variables, or conjugated observables in the quantum theory. The first three chapters discuss subjects illustrated in several articles of the professor Daniel M. Greenberger, which are listed in the bibliography. Whereas in the last chapter the results of a recent theory on QW, say a discrete model for quantum particle evolution, will be examined. In particular, the striking result is that, in the most natural physical interpretation of QW dynamics, it is not possible to have a constant and invariant mass in any manner. The long term aim of this work is indeed to study the reasons and consequences of a theory of particles with variable mass in the context of wider studies still in progress on QW.
This work discloses some inconsistencies of quantum mechanics (QM) in two different contexts. On the one hand the equivalence principle in his two formulations is the starting point of all gravitational theories; yet there is no way to make it consistent with the non-relativistic formulation of QM. Moreover, the evaluation of the Schrödinger equation for a particle in an accelerated reference frame sheds light on a trajectory-dependent phase term subject to proper time. On the other hand, the Bargmann theorem unmistakably proves that the Schrödinger equation does not admit any superposition of different masses if the symmetry chosen for the system is the Galilean one. However, this is incompatible with relativity, since mass and energy are equivalent and one can certainly superimpose different energies. Both inconsistencies support the hypothesis that mass and proper time could be treated as variables. In the third chapter we will describe a theory in which the Galilean symmetry is extended by assuming particle mass and proper time as canonically conjugated classical variables, or conjugated observables in the quantum theory. The first three chapters discuss subjects illustrated in several articles of the professor Daniel M. Greenberger, which are listed in the bibliography. Whereas in the last chapter the results of a recent theory on QW, say a discrete model for quantum particle evolution, will be examined. In particular, the striking result is that, in the most natural physical interpretation of QW dynamics, it is not possible to have a constant and invariant mass in any manner. The long term aim of this work is indeed to study the reasons and consequences of a theory of particles with variable mass in the context of wider studies still in progress on QW.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
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.
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.