A. C. de la Torre
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.
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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.
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.
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…
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…
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 mann…
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.
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.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.
Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
The complementarity between signaling and randomness in any communicated resource that can simulate singlet statistics is generalized by relaxing the assumption of free will in the choice of measurement settings. We show how to construct an ontological extension for quantum mechanics (QM) through the oblivious embedding of a sound simulation protocol in a Newtonian spacetime. Minkowski or other intermediate spacetimes are ruled out as the locus of the embedding by virtue of hidden influence inequalities. The complementarity transferred from a simulation to the extension unifies a number of res…
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.
Synchronization and Fundamental Time: a Connection between Relativity and Quantum Mechanics
本文利用引力电磁类比，将Louis de Broglie关于旋转系同步与角动量量子化的方法推广至弯曲时空，描述旋转源周围的弱引力场并给出新解释。
Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance & Continuity, at the foundation of those axiomatic reconstructions maps to "no preferred reference frame" (NPRF, aka "the relativity principle") as it pertains to the invariant measurement of Planck's constant h for Stern-Gerlach (SG) spin measure…
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,…
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…
In the normal presentation of the EPR problem a comparison is made between the (weak) Copenhagen interpretation of quantum mechanics which seems to suggest that at times action at a distance may take place, and the hidden parameter interpretation which must satisfy Bell's inequality, in contradiction to the predictions of quantum mechanics. In this paper, we consider a relativistic approach to the paradox. However, the frame of reference under consideration is not the usual Lorenz frame but rather the spin frame of reference which is invariant with respect to the SU(2) group.
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…
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.
The relation between the special relativity and quantum mechanics is discussed. Based on the postulate that space-time inversion is equavalent to particle-antiparticle transformation, the essence of special relativity is explored and the relativistic modification on Stationary Schrödinger Equation is derived.
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…
In this paper we introduce the three main notions of probability used by physicists and discuss how these are to be used when invoking spacelike separated observers in a relativistic format. We discuss a standard EPRB experiment and concentrate upon problems of the interpretation of probabilities. We promote a particularly conservative interpretation of this experiment (which need not invoke an objective notion of collapse) where probabilities are, tentatively, passively Lorentz invariant. We also argue that the Heisenberg picture is preferable in relativistic situations due to a conflict betw…
The EPR paradox dates back to 1935 when Einstein et al., through the use of non commuting operators, proposed that quantum mechanics was not complete in that it suggested a `spooky action at a distance.' Later in 1964 John Bell was able to express the dilemma in a simple inequality involving spin-singlet states. If the inequality were satisfied then Einstein was correct and if it were violated then it favored the quantum mechanics point of view. In what follows, we present two approaches to Bell's inequality, and then offer an interpretation from the viewpoint of quantum mechanics based on the…
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…
Recent experiment by Zhinden et al (Phys. Rev {\bf A} 63 02111, 2001) purports to test compatibility between relativity and quantum mechanics in the classic EPR setting. We argue that relativity has no role in the EPR argument based solely on non-relativistic quantum formalism. It is suggested that this interesting experiment may have significance to address fundamental questions on quantum probability.
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…
Galilean Relativity and the Path Integral Formalism in Quantum Mechanics
针对牛顿力学拉格朗日量不满足伽利略变换不变性、量子力学中伽利略 boost 需波函数非平凡变换的问题，采用费曼路径积分形式，证明量子力学中伽利略对称群的投影表示与拉格朗日量的非不变性等价，为理解经典与量子力学中的对称性表示提供了简洁路径。
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…
Inspired by Bohr's dictum that "physical phenomena are observed relative to different experimental setups", this article investigates the notion of relativity in Bohr's sense, starting from a set of binary elements. The most general form of information coding within such sets requires a description by four-component states. By using Bohr's dictum as a guideline a quantum mechanical description of the set is obtained in the form of a SO(3,2) based spin network. For large (macroscopic) sub-networks a flat-space approximation of SO(3,2) leads to a Poincare symmetrical Hilbert space. The concept o…
The physical world is quantum. However, our description of the quantum physics still relies much on concepts in classical physics and in some cases with `quantized' interpretations. The most important case example is that of spacetime. We examine the picture of the physical space as described by simple, so-called non-relativisitic, quantum mechanics instead of assuming the Newtonian model. The key perspective is that of (relativity) symmetry representation, and the idea that the physical space is to be identified as the configuration space for a free particle. Parallel to the case of the phase…
Clocks play a special role at the interface of general relativity and quantum mechanics. We analyze a clock-interferometry thought experiment and go on to theoretically derive and experimentally test a complementarity relation for quantum clocks in the context of the gravitational time lag. We study this relation in detail and discuss its application to various types of quantum clocks.
In [N. Friis, New J. Phys. 18, 033014 (2016)] the non-relativistic description of fermions is considered and in particular the role of the parity superselection rule in relation to the characterization of entanglement. An argument based on the spin-statistics connection is presented as a physical motivation for the parity superselection rule. Since the spin statistics connection was derived in the context of relativistic quantum mechanics it is argued that the inclusion of the parity superselection rule is motivated by Lorentz invariance. Based on this it is further argued that fermionic Quant…
We present a spacetime setting for non-relativistic quantum mechanics that deflates "quantum mysteries" and relates non-relativistic quantum mechanics to special relativity. This is achieved by assuming spacetime symmetries are fundamental in a blockworld setting, i.e., by interpreting spacetime relations as fundamental to relata. To justify this Relational Blockworld (RBW), we adopt a result due to G. Kaiser whereby the relativity of simultaneity, stemming from the Poincare algebra of special relativity, is responsible for the canonical commutation relations of non-relativistic quantum mechan…
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.
Starting with a consideration of the implication of Bell inequalities in quantum mechanics, a new quantum postulate is suggested in order to restore classical locality and causality to quantum physics: only the relative coordinates between detected quantum events are valid observables. This postulate supports the EPR view that quantum mechanics is incomplete, while also staying compatible to the Bohr view that nothing exists beyond the quantum. The new postulate follows from a more general principle of quantum relativity, which states that only correlations between experimental detections of q…
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…
We point out a fundamental problem that hinders the quantization of general relativity: quantum mechanics is formulated in terms of systems, typically limited in space but infinitely extended in time, while general relativity is formulated in terms of events, limited both in space and in time. Many of the problems faced while connecting the two theories stem from the difficulty in shoe-horning one formulation into the other. A solution is not presented, but a list of desiderata for a quantum theory based on events is laid out.
The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-G…
In this article, the axioms presented in the first one are reformulated according to the special theory of relativity. Using these axioms, quantum mechanic's relativistic equations are obtained in the presence of electromagnetic fields for both the density function and the probability amplitude. It is shown that, within the present theory's scope, Dirac's second order equation should be considered the fundamental one in spite of the first order equation. A relativistic expression is obtained for the statistical potential. Axioms are again altered and made compatible with the general theory of …
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…
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…
Ever since we have been in the possession of quantum theories without observers, such as Bohmian mechanics or the Ghirardi-Rimini-Weber (GRW) theory of spontaneous wave function collapse, a major challenge in the foundations of quantum mechanics is to devise a relativistic quantum theory without observers. One of the difficulties is to reconcile nonlocality with relativity. I report about recent progress in this direction based on the GRW model: A relativistic version of the model has been devised for the case of N noninteracting (but entangled) particles. A key ingredient was to focus not on …
In this paper we propose an experiment designed to observe a general-relativistic effect on single photon interference. The experiment consists of a folded Mach-Zehnder interferometer, with the arms distributed between a single Earth orbiter and a ground station. By compensating for other degrees of freedom and the motion of the orbiter, this setup aims to detect the influence of general relativistic time dilation on a spatially superposed single photon. The proposal details a payload to measure the required effect, along with an extensive feasibility analysis given current technological capab…
The Logic of Quantum Mechanics Derived from Classical General Relativity
首次从经典广义相对论推导量子力学逻辑，通过将粒子建模为含闭合类时曲线的4-geon，证明测量命题满足正交模格而非布尔逻辑。
In this paper a one to one correspondence is established between space-time metrics of general relativity and the wave equations of quantum mechanics. This is done by first taking the square root of the metric associated with a space and from there, passing directly to a corresponding expression in the dual space. It is shown that in the case of a massless particle, Maxwell's equation for a photon follows while in the case of a particle with mass, Dirac's equation results as a first approximation. Moreover, this one to one correspondence suggests a natural explanation of wave-particle duality.…
In relativity, two simultaneous events at two different places are not simultaneous for observers in different Lorentz frames. In the Einstein-Podolsky-Rosen experiment, two simultaneous measurements are taken at two different places. Would they still be simultaneous to observers in moving frames? It is a difficult question, but it is still possible to study this problem in the microscopic world. In the hydrogen atom, the uncertainty can be considered to be entirely associated with the ground-state. However, is there an uncertainty associated with the time-separation variable between the proto…
We show that the local and deterministic mode of description is not only in conflict with the quantum theory, but also with relativity. We argue that elementary relativistic properties of spacetime lead to the emergence of a non-deterministic quantum-mechanical picture involving quantum superpositions and complex probability amplitudes.
Quantum Mechanics from General Relativity and the Quantum Friedmann Equation
本文证明重新表述第一弗里德曼方程的线性方程是量子宇宙学方程的一阶WKB展开，揭示了广义相对论与量子力学的深层联系，并通过量子尺度因子消除奇点，在辐射主导期与Seiberg-Witten公式对偶。
Relational Quantum Mechanics, Quantum Relativism, and the Iteration of Relativity
本文以关系量子力学为案例，探讨量子相对主义中被忽视的相对性迭代问题，指出关系量子力学承诺无限制迭代原则，该原则虽能保障交互结果的可交流性与一致性，但与关系量子力学的保守解读不兼容，需采用视角事实概念，并反思了相关研究现状与发展方向。
This paper constitutes a background to the paper 'Quantum mechanics as "space-time statistical mechanics"?', arXiv:quant-ph/0501133, presented previously by the author. But it is also a free-standing and self-contained paper. The purpose of this paper is to give the reader an increased and a deeper understanding of the special theory of relativity, and the spacetime ideas lying behind the above mentioned paper. We will here consider, discuss, define, analyse, and explain things such as, e.g., the constancy of the speed of light, synchronization, simultaneity, absolute simultaneity, absolute sp…
The paper has heuristic character. The conceptual frame, based on the assumption of quantum uncertainty only, has been formerly introduced in two papers [S. Tosto, Il Nuovo Cimento B, vol. 111, n.2, 1996 and S. Tosto, Il Nuovo Cimento D, vol. 18, n.12, 1996]. Instead of attempting to increase the accuracy of some existing computational model through a new kind of approximation, these papers acknowledge since the beginning the lack of deterministic information about the local properties of the constituent particles, considered random, unknown and unpredictable and thus ignored in principle. The…
Quantum mechanics appears to contain ghosts from both classical statistical mechanics and special relativity. On one hand, both the Dirac and Schrödinger equations have classical analogs that emerge directly from classical statistical mechanics, unimpeded by major problems of interpretation. On the other hand, the formal analytic continuation that takes these classical equations to the quantum version introduces a velocity dependent phase. However, among classical theories, only in relativistic mechanics does one find path-dependent phase in the form of relativistic time dilation. This paper e…
Gravitation, according to General Relativity, is an attribute of space-time's geometry and hence not a force in the Newtonian sense. This is a consequence of Einstein's equivalence principle, which so far passed all experimental tests with high precision. However, the search for possible violations continues, for they, as the case may be, are expected to point towards more fundamental theoretical extensions of General Relativity. In particular, it is expected that useful insights are gained by studying the interaction between gravitational fields and genuine quantum-mechanical systems, like at…
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