It has been suggested that the space-time structure as described by the theory of special relativity is a macroscopic manifestation of a more fundamental quantum structure (pre-geometry). Efforts to quantify this idea have come mainly from the area of abstract quantum logic theory. Here we present a preliminary attempt to develop a quantum formulation of special relativity based on a model that retains some geometric attributes. Our model is Feynman's "checker-board" trajectory for a 1-D relativistic free particle. We use this model to guide us in identifying (1) the quantum version of the postulates of special relativity and (2) the appropriate quantum "coordinates". This model possesses a useful feature that it admits an interpretation both in terms of paths in space-time and in terms of quantum states. Based on the quantum version of the postulates, we derive a transformation rule for velocity. This rule reduces to the Einstein's velocity-addition formula in the macroscopic limit and reveals an interesting aspect of time. The 3-D case, time-dilation effect, and invariant interval are also discussed in term of this new formulation. This is a preliminary investigation; some results are derived, while others are interesting observations at this point.
It has been suggested that the space-time structure as described by the theory of special relativity is a macroscopic manifestation of a more fundamental quantum structure (pre-geometry). Efforts to quantify this idea have come mainly from the area of abstract quantum logic theory. Here we present a preliminary attempt to develop a quantum formulation of special relativity based on a model that retains some geometric attributes. Our model is Feynman's "checker-board" trajectory for a 1-D relativistic free particle. We use this model to guide us in identifying (1) the quantum version of the postulates of special relativity and (2) the appropriate quantum "coordinates". This model possesses a useful feature that it admits an interpretation both in terms of paths in space-time and in terms of quantum states. Based on the quantum version of the postulates, we derive a transformation rule for velocity. This rule reduces to the Einstein's velocity-addition formula in the macroscopic limit and reveals an interesting aspect of time. The 3-D case, time-dilation effect, and invariant interval are also discussed in term of this new formulation. This is a preliminary investigation; some results are derived, while others are interesting observations at this point.
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 technical problem in Feynman’s proposal to formulate quantum mechanics in terms of a sum over histories of a particle is that this sum over histories must be performed in imaginary time rather than in the real time that everyone experiences [1]. In this paper, for the derivation of Lorentz transformation from the postulates of Special theory of Relativity, the concept of De Broglie quantum waves is introduced which leads to the conclusions that the derivation of Lorentz transformation does not need the postulate of constancy of light speed in free space in case the postulate of phase coincidence of De Broglie quantum waves is introduced. Consequently, a new proposal is formulated that combines Quantum mechanics and the special theory of relativity and overcomes the technical problem in Feynman’s proposal.
In Part I we suggested that free particle quantum mechanics seems to follow from special relativity. In this note, we focus on the idea of a rest frame defining a rest mass. In such a frame momentum is zero i.e. one has the 4-vector (0,mo)=(p,E). (x,t), however, is also a 4-vector which is usually assigned to a point in space time. In Part I, however, we considered (x,t)=(0,to) associating p=0 and x=0 and alternatively mo and to. Velocity is usually defined as dx/dt =v, but using the Lorentz transformation (1) (with v set to -v), (x,t)=(0,to) transforms to x’/t’ = v. The relationship between x,p and t,E is further manifested through the Lorentz invariant -Et+px in which E scales t and p, x. This we suggest breaks space into pieces proportional to 1/p and time (independently) into pieces proportional to 1/E We next consider particles “created” at different x points. In other words (0,mo) applies, but the particles are at different x positions. In order to keep the relationship p=0,x=0, one must have an x origin at all particle positions. Then spatial scaling of 1/p does not line up unless the particles are n/p (n=integer) apart. Otherwise the scaling in space is unaligned. For different p, scaling in space is unaligned already with (0,mo) and (0, to). Thus for two particles with the same p, but created at different x values at different times and arriving at x1, t1, if there is no alignment, special relativity suggests these two are distinguishable, while Newtonian mechanics does not distinguish. This distinguishability, however, depends on a physical mechanism i.e. a wavelength. Thus, even without considering different p (momentum) values, two identical momenta should be distinguishable if they are not aligned in terms of p scaling even for the same rest mass mo. This we argue is strictly a result of special relativity and is support for the exp(ipx) form of quantum free particle mechanics.
One of the most profound and difficult problems of physics in the last century is the construction of a simple theory which unites the ideas of quantum mechanics and relativity. The concepts which are used in quantum mechanics related to the measurements of positions, momenta, energy and time, are the same concepts in terms of which the special relativistic postulates are formulated. Hence, it is at least possible to formulate the requirement of special relativistic invariance for quantum theories and to ascertain whether these requirements are satisfied. The fact that the answer is nearer to no than to yes, that quantum mechanics has not yet been fully adjusted to the postulates of the special theory, is perhaps irritating. In the general theory of relativity, the situation is different: the space-time coordinates are only auxiliary quantities to which arbitrary values for every event could be given. Hence, the measurement of position, that is, of the space coordin ates, is certainly not a significant measurement if the postulates of the general theory are adopted: any arbitrary value can be given to the coordinates. The same holds for momenta. The purpose of this brief paper is to make in principle some personal and particular remarks on the fundamental concepts of time and mass. Some comments about the consequences on the relativistic dynamics of quantum systems are also formulated.
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.
One of the concepts of Relativity theory that challenges conventional intuition the most is time dilation and length contraction. Usual approaches for describing relativistic effects in quantum systems merely postulate the consequences of these effects as physical constraints. Here, we propose to rebuild Special Relativity from quantum mechanical considerations. This is done by dropping one of its fundamental postulates: the universality of the speed of light. Lorentz transformations are obtained by a quantum mechanism. We use the fact that quantum states depend on the Galilean reference frame where they are defined. In other words, quantum states outside an observer's Galilean reference frame are distorted. Then, we show in a theorem the existence of time-dependent observables that are sensible to this distortion in such a way that their expectation value is a Lorentz-covariant function of time. We then postulate this mechanism as the source of the phenomena of Special Relativity. As a corollary of the main theorem, we show the existence of a Lorentz covariant momentum and mass operators which yield the relativistic momentum and mass. In this theory, the fundamental limit of the speed of light imposes a transparency condition for faster-than-light particles: they are allowed but they are not observable. The transparency effect could explain dark matter in a more general theory following this quantum formalism.