There is a theoretical evidence that relativistically invariant quantum dynamics at (enough) large space-time scales can result from a cooperative process of two inter-correlated non-relativistic stochastic dynamics, operating at different energy scales. We show that the Euclidean transition amplitude for a relativistic particle is identical to the transition probability of a Brownian particle propagating in a granular space. We discuss the issue of the robustness of the special-relativistic quantum mechanics thus obtained under small changes in the granular-space distribution. Experimental implications for early Universe cosmology are also briefly outlined.
There is a theoretical evidence that relativistically invariant quantum dynamics at (enough) large space-time scales can result from a cooperative process of two inter-correlated non-relativistic stochastic dynamics, operating at different energy scales. We show that the Euclidean transition amplitude for a relativistic particle is identical to the transition probability of a Brownian particle propagating in a granular space. We discuss the issue of the robustness of the special-relativistic quantum mechanics thus obtained under small changes in the granular-space distribution. Experimental implications for early Universe cosmology are also briefly outlined.
We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than the particle's Compton wavelength. Thus, Lorentz symmetry appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength, is taken. We also present the generalizations needed to accommodate in our scheme the doubly special relativistic dynamics. In this way, a previously unsuspected, common stochastic origin of the two frameworks is revealed for the first time. Issues such as generalized commutation relations are also discussed.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
Relativistic quantum mechanics was developed to reconcile the principles of quantum mechanics with Einstein’s theory of relativity. Despite its success in describing high-energy particles, the theory continues to face unresolved conceptual and mathematical difficulties, particularly in relation to the nature of time, causality, and relativistic consistency. In recent years, the idea of extending space-time into the complex domain has emerged as a useful and potentially meaningful approach to these problems. In this work, a theoretical synthesis is presented to examine the role of complex space-time in relativistic quantum mechanics. The study discusses how complex time arises naturally in relativistic wave equations, quantum field theory, and path integral formulations. Established methods such as Wick rotation, complex coordinate transformations, and non-Hermitian but parity–time (PT) symmetric Hamiltonians are reviewed to illustrate how complex space-time contributes to mathematical stability and clearer physical interpretation (Dirac, 1928; Wick, 1954; Bender & Boettcher, 1998). The relevance of complex space-time is further explored in the context of relativistic particle dynamics and quantum cosmology, where it helps address, issues related to vacuum structure, time evolution, and causality (Feynman & Hibbs, 1965; Hartle & Hawking, 1983). The paper suggests that complex space-time should be viewed not only as a computational technique, but as a framework that may reflect deeper features of relativistic quantum phenomena.
In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.
Development of the contemporary theory of physical phenomena in the microcosm is considered to be a result of development of Einstein's ideas on a possibility of the event space modification and on a possibility of stochastic (Brownian) motion of free particles. One considers the space-time modification, based on a new conception of geometry. In the framework of this conception any geometry is obtained as a result of the proper Euclidean geometry deformation. In the framework of this conception it is possible such a space-time geometry, where the free particle motion appears to be stochastic, although the geometry in itself remains to be deterministic. The stochasticity intensity depends on the particle mass. It is small for particle of a large mass, and it is essential for the particle of a small mass. The space-time geometry may be chosen in such a way, that the statistical description of random world lines of particles be equivalent to the quantum description (the Schrödinger equation). At such a description the quantum principles are not used. They can be obtained as a corollaries of the statistical description of nonrelativistic stochastic particles. The nonrelativistic quantum principles may not be used for description of relativistic phenomena.
Abstract We clarify the way in which cosmological perturbations of quantum origin, produced during inflation, assume classical properties. Two features play an important role in this process: First, the dynamics of fluctuations which are presently on large cosmological scales leads to a very peculiar state (highly squeezed) that is indistinguishable, in a precise sense, from a classical stochastic process. This holds for almost all initial quantum states. Second, the process of decoherence by interaction with the environment distinguishes the field amplitude basis as the robust pointer basis. We discuss in detail the interplay between these features and use simple analogies such as the free quantum particle to illustrate the main conceptual issues.