Examination of the Einstein energy-momentum relationship suggests that simple unbound forms of matter exist in a four-dimensional Euclidean space. Position, momentum, velocity, and other vector quantities can be expressed as Euclidean four-vectors, with the magnitude of the velocity vector having a constant value, the speed of light. We see that charge may be simply a manifestation of momentum in the new fourth direction, which implies that charge conservation is a form of momentum conservation. The constancy of speed implies that all elementary free particles can be described in the same manner as photons, by means of a wave equation. The resulting wave mechanics (with a few small assumptions) is simply the traditional form of quantum mechanics. If one begins by assuming the wave nature of matter, it is shown that special relativistic results follow simply. Thus we see evidence of a strong connection between relativity and quantum mechanics. Comparisons between the theory presented here and Kaluza-Klein theories reveal some similarities, but also many significant differences between them.
We raise fundamental questions about the very meaning of conservation laws in quantum mechanics and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.
Two questions are suggested as having priority when trying to bring together Quantum Mechanics and General Relativity. Both questions have a scope which goes well beyond Physics, and in particular Quantum Mechanics and General Relativity.
In the posting arXiv:2101.11052v2 an experimental protocol involving two spins is proposed, which should show violation of energy conservation in a quantum experiment. In the present comment an unjustified mathematical approximation leading to that conclusion is pointed out. A detailed analysis will restore energy conservation.
We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an "onion-shell" geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a "band structure" in metric space directly arising from the conservation of the $z$ component of the angular momentum.
The conventionalistic aspects of physical world perception are reviewed with an emphasis on the constancy of the speed of light in relativity theory and the irreversibility of measurements in quantum mechanics. An appendix contains a complete proof of Alexandrov's theorem using mainly methods of affine geometry.
Only the position representation is used in introductory quantum mechanics and the momentum representation is not usually presented until advanced undergraduate courses. To emphasize the relativity of the representations of the abstract formulation of quantum mechanics, two examples of representations related to the operators aX+(1-a)P and (XP+PX)/2 are presented.
Non-relativistic quantum mechanics is shown to emerge from classical mechanics through the requirement of a relativity principle based on special transformations acting on position and momentum uncertainties. These transformations keep the Heisenberg inequalities invariant and form a group. They are related to dilatations of space variables provided the quantum potential is added to the classical Hamiltonian functional. The Schrödinger equation appears to have a nonunitary and nonlinear companion acting in another time variable. Evolution in this time seems related to the state vector reductio…
The present paper shows that Edward Nelson's stochastic mechanics approach for quantum mechanics can be derived from the two classical elastically colliding particles with masses M and m satisfying a collision momentum preserving equation. The properties of the classical elastic momentum collision expression determine the full Edward Nelson energy collision energy for both particles. This classical total energy expression does not require a statistical expectation since no process was defined for the energy and it models the main and incident particle velocities perfectly. Quantum mechanics ca…
We introduce a class of non-local Lagrangians which allow for the variational derivation of non-local conser- vation laws in a self-consistent manner. The formalism developed here generalizes previous approaches, used in the context of $\mathcal{PT}$ symmetric quantum mechanics and optics in the paraxial approximation in a twofold way: firstly it is valid for a larger set of linear symmetry transforms and secondly it enables the derivation of additional non-local conservation laws for general higher dimensional wave mechanical systems.
Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems, both as a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation, to ensure they are always guided to where they are most needed. In this paper, probability conservation for complexified Bohmian trajectories is considered. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wavefunctions of complexified variables. This in turn enables m…
We show that one-dimensional Bohmian mechanics is unique, in that, the Bohm trajectories are the only solutions that conserve total left (or right) probability. In Brandt et al., Phys. Lett. A, 249 (1998) 265--270, they define quantile motion--unique trajectories are solved by assuming that the total probability on each side of the particle is conserved. They argue that the quantile trajectories are identical to the Bohm trajectories. Their argument, however, fails to notice the gauge freedom in the definition of the quantum probability current. Our paper sidesteps this under-determinedness of…
We study the conservation of energy, or lack thereof, when measurements are performed in quantum mechanics. The expectation value of the Hamiltonian of a system can clearly change when wave functions collapse in accordance with the standard textbook (Copenhagen) treatment of quantum measurement, but one might imagine that the change in energy is compensated by the measuring apparatus or environment. We show that this is not true; the change in the energy of a state after measurement can be arbitrarily large, independent of the physical measurement process. In Everettian quantum theory, while t…
Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived `principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gath…
The conflict between the locality of general relativity, reflected in its space-time description, and the non-locality of quantum mechanics, contained in its Hilbert space description, is discussed. Gauge covariant non-local observables that depend on gauge fields and gravity as well as the wave function are used in order to try to understand and minimize this conflict within the frame-work of these two theories. Applications are made to the Aharonov-Bohm effect and its generalizations to non Abelian gauge fields and gravity.
In a comparison of the principles of special relativity and of quantum mechanics, the former theory is marked by its relative economy and apparent explanatory simplicity. A number of theorists have thus been led to search for a small number of postulates - essentially information theoretic in nature - that would play the role in quantum mechanics that the relativity principle and the light postulate jointly play in Einstein's 1905 special relativity theory. The purpose of the present paper is to resist this idea, at least in so far as it is supposed to reveal the fundamental form of the theory…
Conservation laws are discussed in conjunction with quantum-mechanical indeterminacies of the corresponding observables. The considered examples show that the connections between energy and its indeterminacy may be quite intricate. The indeterminacies of the parts of a composite system may be correlated in such a way that the net value of the considered observable is definite. This may be the case with specific entangled superposition of eigenstates in a composite system. In some cases the fact of energy conservation cannot be convincingly established and generally the concept of conserving ob…
By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $ε$ embedded in $R_3$. In such a system with a small thickness $ε$, the usual two-dimensional conservation law does not hold and we find an anomaly. The anomalous term is obtained by the expansion of $ε$. We find that this term has a Gaussian and mean curvature dependence and can be written as the total divergence of some geometr…
The geometric properties of General Relativity are reconsidered as a particular nonlinear interaction of fields on a flat background where the perceived geometry and coordinates are "physical" entities that are interpolated by a patchwork of observable bodies with a nonintuitive relationship to the underlying fields. This more general notion of gauge in physics opens an important door to put all fields on a similar standing but requires a careful reconsideration of tensors in physics and the conventional wisdom surrounding them. The meaning of the flat background and the induced conserved quan…
Being quantized, conserved Noether symmetry functions are represented by Hermitian operators in the space of solutions of the Schrodinger equation, and their mean values are conserved.
In 1981, Mermin published a now famous paper titled, "Bringing home the atomic world: Quantum mysteries for anybody" that Feynman called, "One of the most beautiful papers in physics that I know." Therein, he presented the "Mermin device" that illustrates the conundrum of quantum entanglement per the Bell spin states for the "general reader." He then challenged the "physicist reader" to explain the way the device works "in terms meaningful to a general reader struggling with the dilemma raised by the device." Herein, we show how "conservation per no preferred reference frame (NPRF)" answers th…
A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.
Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
A sharper formulation is presented for an interpretation of quantum mechanics advocated by author. As an essential element we put forward conservation laws concerning the ontological nature of a variable, and the uncertainties concerning the realisation of states. Quantum mechanics can then be treated as a device that combines statistics with mechanical, deterministic laws, such that uncertainties are passed on from initial states to final states.
In previous papers I expounded non-linear Schrodingerist quantum mechanics as a solution of the Measurement Problem. Here I show that NLQM is compatible with Einstein's theory of General Relativity. The extension to curved space-times presumes adoption of "space-time wavefunctions" (sometimes called "multi-time wavefunctions") and some additional algebraic structure: a "bitensor" supplementing Einstein's metric tensor. This kind of matter may violate the Strong Energy Condition even without a mass term, possibly with implications for the formation of singularities within Black Holes.
Attempting to reconcile general relativity with quantum mechanics is one of the great undertakings of contemporary physics. Here we present how the incompatibility between the two theories arises in the simple thought experiment of preparing a heavy object in a quantum superposition. Following Penrose's analysis of the problem, we determine the requirements on physical parameters to perform experiments where both theories potentially interplay. We use these requirements to compare different systems, focusing on mechanical oscillators which can be coupled to superconducting circuits.
In this paper, starting from vortices we are finally lead to a treatment of Fermions as Kerr-Newman type Black Holes wherein we identify the horizon at the particle's Compton wavelength periphery. A naked singularity is avoided and the singular processes inside the horizon of the Black Hole are identified with Quantum Mechanical effects within the Compton wavelength. Inertial mass, gravitation, electromagnetism and even QCD type interactions emerge from such a description including relative strengths and also other features like the anomalous gyromagnetic ratio, the discreteness of the charge,…
Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equi…
A thought experiment is considered on observation of instantaneous collapse of an extended wave packet. According to relativity of simultaneity, such a collapse being instantaneous in some reference frame must be a lasting process in other frames. But according to quantum mechanics, collapse is instantaneous in any frame. Mathematical structure of quantum mechanics eliminates any contradictions between these two apparently conflicting statements. Here the invariance of quantum-mechanical collapse is shown to follow directly from the Born postulate, without any use of mathematical properties of…
Equilibrium statistical mechanics rests on the assumption of ergodic dynamics of a system modulo the conservation laws of local observables: extremization of entropy immediately gives Gibbs' ensemble (GE) for energy conserving systems and a generalized version of it (GGE) when the number of local conserved quantities (LCQ) is more than one. Through the last decade, statistical mechanics has been extended to describe the late-time behaviour of periodically driven (Floquet) quantum matter starting from a generic state. The structure built on the fundamental assumptions of ergodicity and identifi…
The purpose of this contribution is to provide an introduction for a general physics audience to the recent results of Emile Grgin that unifies quantum mechanics and relativity into the same mathematical structure. This structure is the algebra of quantions, a non-division algebra that is the natural framework for electroweak theory on curved space-time. Similar with quaternions, quantions preserve the core features of associativity and complex conjugation while giving up the unnecessarily historically biased property of division. Lack of division makes possible structural unification with rel…
The central equations in classical general relativity are the Einstein Field Equations, which accurately describe not only the generation of pseudo-Riemannian curvature by matter and radiation manifesting as gravitational effects, but more importantly mass-energy dynamics, evolution and distribution on the space-time manifold. Herein, we introduce a geometric phase in general relativity corresponding to Schwarzschild black hole information content. This quantity appropriately satisfies a local conservation law subject to minimal coupling, with other desirable properties such as the quantizatio…
A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton-Wigner representation we discuss the spectral decomposition of the canonical operators and give a proof of the Parseval-Plancherel relation and the Born rule for linear superposition. We then discuss the representations of pure quantum states and their dual vectors, and construct the Fock space and the associated quantum field theory for Bose-Einstein and Fermi-Dirac statistics.
We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is $r^{-1}$ in Planck units, where $r$ is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states $S_1$ and $S_2$ cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possib…
The Poincaré-Snyder relativity was introduced in an earlier paper of ours as an extended form of Einstein relativity obtained by appropriate limiting setting of the full Quantum Relativity. The latter, with fundamental constants $\hbar$ and $G$ built into the symmetry, is supposed to be the relativity of quantum space-time. Studying the mechanics of Poincaré-Snyder relativity is an important means to get to confront the great challenge of constructing the dynamics of Quantum Relativity. The mechanics will also be of interest on its own, plausibly yielding prediction accessible to experiments. …
In special relativity theory the physical quantities are generally expressed as function of the velocity. In the particular case of an extended object, the factor 1/gamma of Lorentz contraction of its length in the direction of motion is written as a function of the velocity of the extended object. The same happens with the Lorentz factor gamma of time interval dilation of a moving clock. In this paper we show that when gamma is written as a function of the relativistic energy of the considered object the resulting expressions for the length and for the time interval suggest that the contracti…
The paper investigates the physical content of a recently proposed mathematical framework that unifies the standard formalisms of classical mechanics, relativity and quantum theory. In the framework states of a classical particle are identified with Dirac delta functions. The classical space is "made" of these functions and becomes a submanifold in a Hilbert space of states of the particle. The resulting embedding of the classical space into the space of states is highly non-trivial and accounts for numerous deep relations between classical and quantum physics and relativity. One of the most s…
Two-photon states produce enough symmetry needed for Dirac's construction of the two-oscillator system which produces the Lie algebra for the O(3,2) space-time symmetry. This O(3,2) group can be contracted to the inhomogeneous Lorentz group which, according to Dirac, serves as the basic space-time symmetry for quantum mechanics in the Lorentz-covariant world. Since the harmonic oscillator serves as the language of Heisenberg's uncertainty relations, it is right to say that the symmetry of the Lorentz-covariant world, with Einstein's $E = mc^2$, is derivable from Heisenberg's uncertainty relati…
In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. The Dirac equation is derived and shown to be related to the Lie derivative of the momentum along the curve. In addition,the equations of motion are derived from the Hamilton-Jacobi equation associated with the metric and the wave equation associated with the Hamiltonian is then shown not to commute with the Dirac operator. Finally, the Maxw…
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out her…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are show…
We derive fundamental limits on measurements of position, arising from quantum mechanics and classical general relativity. First, we show that any primitive probe or target used in an experiment must be larger than the Planck length, $l_P$. This suggests a Planck-size {\it minimum ball} of uncertainty in any measurement. Next, we study interferometers (such as LIGO) whose precision is much finer than the size of any individual components and hence are not obviously limited by the minimum ball. Nevertheless, we deduce a fundamental limit on their accuracy of order $l_P$. Our results imply a {\i…
The Nelson stochastic mechanics is derived as a consequence of the basic physical principles such as the principle of relativity of observations and the invariance of the action quantum. The unitary group of quantum mechanics is represented as the transformations of the systems of perturbing devices. It is argued that the physical spacetime has a stochastic nature, and that quantum mechanics in Nelson's formulation correctly describes this stochasticity.
This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincaré transformations, and isomorphisms of these spaces that preserve the indefinite metric. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo tran…
It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However…
We review a recent approach to the foundations of quantum mechanics inspired by quantum information theory. The approach is based on a general framework, which allows one to address a large class of physical theories which share basic information-theoretic features. We first illustrate two very primitive features, expressed by the axioms of causality and purity-preservation, which are satisfied by both classical and quantum theory. We then discuss the axiom of purification, which expresses a strong version of the Conservation of Information and captures the core of a vast number of protocols i…
