Energy diffusion due to spontaneous localization (SL) for a relativistically fast moving particle is examined. Spontaneous localization is an alternative to standard quantum theory in which quantum state reduction is treated as a random physical process which is incorporated into the Schr\"odinger equation in an observer-independent way. These models make predictions in conflict with standard quantum theory one of which is nonconservation of energy. On the basis of proposed relativistic extensions of SL it is argued that for a single localized particle, nonrelativistic SL should remain valid in the rest frame of the particle. The implication is that relativistic calculations can be performed by transforming nonrelativistic results from the particle rest frame to the frame of an inertial observer. This is demonstrated by considering a relativistic stream of noninteracting particles of cosmological origin and showing how their energy distribution evolves as a result of SL as they traverse the Universe. A solution is presented and the potential for astrophysical observations is discussed.
Energy diffusion due to spontaneous localization (SL) for a relativistically fast moving particle is examined. Spontaneous localization is an alternative to standard quantum theory in which quantum state reduction is treated as a random physical process which is incorporated into the Schr"odinger equation in an observer-independent way. These models make predictions in conflict with standard quantum theory one of which is nonconservation of energy. On the basis of proposed relativistic extensions of SL it is argued that for a single localized particle, nonrelativistic SL should remain valid in the rest frame of the particle. The implication is that relativistic calculations can be performed by transforming nonrelativistic results from the particle rest frame to the frame of an inertial observer. This is demonstrated by considering a relativistic stream of noninteracting particles of cosmological origin and showing how their energy distribution evolves as a result of SL as they traverse the Universe. A solution is presented and the potential for astrophysical observations is discussed.
In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.
In a previous note (1) we considered a rest mass mo at x=0 and t=to and argued this transforms to E’, p’ with x’/t’=v. Thus a single system may be Lorentz transformed to create particles with mo and all velocities. We suggested that all of these particles are on the same footing in a sense. In this note we try to investigate what this “footing” is. In particular, we suggest nonlocality. A particle with rest mass mo, likely occupies some region of space, but is treated as a point (i.e. x=0) which may be the center of mass.The idea of a particle considered as a point is carried over into special relativity i.e motion. In Newtonian mechanics the particle is also considered to be a point. We argue that even though a particle “system” with mo at x=0 and t=to is a point and that after moving there may still be an x’/t’ = v pertaining to one point representing the system, the localization of mo at x=0 may be transformed into a localization in every constant moving frame. This localization seems to be linked to the idea of x and t being independent in -Et+px just as a particle at x=0 and t=to at rest treats x and t as independent. Thus we argue that the localization is the “same footing” of particles with different v and hence momenta mov. The dot product -Et+px = A is -moto for the rest frame which tends to 0 for to→0 . For any frame, t=b/E and x=b/p yields the same A=0. Furthermore x=b/p and t=b/E is a recipe which holds in all frames and describes nonlocality or a “trapped” length associated with the moving particle. This trapped length is not associated with time because x and t are independent. In other words the system moves with a speed of v=x’/t’, but the particle is “trapped” within a wavelength of the order 1/p in the sense that there is a cycling probability as the particle moves through space ignoring time. One does not simply have a point at rest transformed into a point in motion describing all aspects of the moving particle. If one considers x=0 and x=1/p, particles with different p values may follow the same probability distribution associated with energy flow in the region of one wavelength. If one ignores the motion v=x’/t’, then since time is ignored it is as if one has a particle with different probabilities to be at different x points. In a bound quantum state, time is physically dropped and these probabilities are linked to actual densities. For a moving particle (in one direction) one must consider the point motion v=x’/t’, but also kind of periodic trapping of the particle in space (not linked to time). In order to quantify this for a quantum free particle, one may consider the idea of a particle system with a length proportional to 1/p which is consistent with P(x)=constant. In particular, one may take the square root of 1 so to speak to obtain exp(ipx) and exp(-ipx). This also follows from eigenvalue equations of A= -Et+px i.e. -id/dx exp(ipx) = p exp(ipx). It should be noted (as stated above) that exp(ipx) does not involve time. Thus the trapping is in space and may occur over a long period of time if the velocity is very slow i.e. 1/p is very large. In a sense the particle is trapped in a kind of periodic motion in space. There is locality in the sense of position x’/t’=v, but not in the sense of the trapped periodic motion. The nonlocality seems to exist one opposite scales than the usual point velocity. For example, if a particle has a very small velocity then x’/t’=v→0 i.e. one thinks in terms of a very small x’, but 1/p is very large.
In order to address the measurement problem of quantum theory we make the assumption that quantum state reduction should be regarded as a genuine physical process deserving of a dynamical description. Generalizing the nonrelativistic spontaneous localization models of Ghirardi, Rimini, Weber, and Pearle, a relativistic state reduction mechanism is proposed. The mechanism involves nonlinear stochastic modifications to the standard description of unitary state evolution and the introduction of a mediating field to facilitate smearing of quantum field interactions.
Abstract Before examining experimental consequences of CSL, this chapter discusses the spontaneous localization (SL) theory of Ghirardi, Rimini and Weber (GRW). As for CSL, the chapter presents the structure of the theory, and then specializes it to a non-relativistic theory. SL does not add a term to Schrodinger’s equation. Instead, the state vector is suddenly altered by a “hit” (GRW’s ‘spontaneous localization’), that is, multiplication by an operator, followed by renormalization. The hits occur at random times. The chapter shows the specific choice of operator made by GRW for their non-relativistic theory. Emphasized are two aspects of SL that helped the journey to CSL. One is that the collapse is always acting (one need not “turn it on.” The other is the “smeared” character of the GRW operator. The chapter then shows how the density matrix is calculated. Finally, it discusses the numerical values, chosen by GRW, of the two parameters of SL theory, λ and a, that are provisionally adopted for CSL.
We make a critical comparison of relativistic and nonrelativistic classical and quantum mechanics of particles in inertial frames as well of the open problems in particle localization at both levels. The solution of the problems of the relativistic center-of-mass, of the clock synchronization convention needed to define relativistic 3-spaces and of the elimination of the relative times in the relativistic bound states leads to a description with a decoupled nonlocal (nonmeasurable) relativistic center-of-mass and with only relative variables for the particles (single particle subsystems do not exist). We analyze the implications for entanglement of this relativistic spatial nonseparability not existing in nonrelativistic entanglement. Then, we try to reconcile the two visions showing that also at the nonrelativistic level in real experiments only relative variables are measured with their directions determined by the effective mean classical trajectories of particle beams present in the experiment. The existing results about the nonrelativistic and relativistic localization of particles and atoms support the view that detectors only identify effective particles following this type of trajectories: these objects are the phenomenological emergent aspect of the notion of particle defined by means of the Fock spaces of quantum field theory.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.