We use a one-dimensional model system to compare the predictions of two different yardsticks to compute the position of a particle from its quantum field theoretical state. Based on the first yardstick (defined by the Newton-Wigner position operator), the spatial density can be arbitrarily narrow, and its time evolution is superluminal for short time intervals. Furthermore, two spatially distant particles might be able to interact with each other outside the light cone, which is manifested by an asymmetric spreading of the spatial density. The second yardstick (defined by the quantum field operator) does not permit localized states, and the time evolution is subluminal.
We use a one-dimensional model system to compare the predictions of two different yardsticks to compute the position of a particle from its quantum field theoretical state. Based on the first yardstick (defined by the Newton-Wigner position operator), the spatial density can be arbitrarily narrow, and its time evolution is superluminal for short time intervals. Furthermore, two spatially distant particles might be able to interact with each other outside the light cone, which is manifested by an asymmetric spreading of the spatial density. The second yardstick (defined by the quantum field operator) does not permit localized states, and the time evolution is subluminal.
Two related problems in relativistic quantum mechanics, the apparent superluminal propagation of initially localized particles and dependence of spatial localization on the motion of the observer, are analyzed in the context of Dirac’s theory of constraints. A parametrization invariant formulation is obtained by introducing time and energy operators for the relativistic particle and then treating the Klein–Gordon equation as a constraint. The standard, physical Hilbert space is recovered, via integration over proper time, from an augmented Hilbert space wherein time and energy are dynamical variables. It is shown that the Newton–Wigner position operator, being in this description a constant of motion, acts on states in the augmented space. States with strictly positive energy are nonlocal in time; consequently, position measurements receive contributions from states representing the particle’s position at many times. Apparent superluminal propagation is explained by noting that, as the particle is potentially in the past (or future) of the assumed initial place and time of localization, it has time to propagate to distant regions without exceeding the speed of light. An inequality is proven showing the Hegerfeldt paradox to be completely accounted for by the hypotheses of subluminal propagation from a set of initial space–time points determined by the quantum time distribution arising from the positivity of the system’s energy. Spatial localization can nevertheless occur through quantum interference between states representing the particle at different times. The nonlocality of the same system to a moving observer is due to Lorentz rotation of spatial axes out of the interference minimum.
Abstract The Newton-Wigner states and operator are widely accepted to provide an adequate notion of spatial localization of a particle in quantum field theory on a spacelike hypersurface. Replacing the spacelike with a timelike hypersurface, we construct one-particle states of massive Klein-Gordon theory that are localized on the hypersurface in the temporal as well as two spatial directions. This addresses the longstanding problem of a “time operator" in quantum theory. It is made possible by recent advances in quantization on timelike hypersurfaces and the introduction of evanescent particles. As a first application of time-localized states, we consider the time-of-arrival problem. Our results are in accordance with semiclassical expectations of causal propagation of massless and massive particles. As in the Newton-Wigner case, localization is not perfect, but apparent superluminal propagation is exponentially suppressed.
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) eld. However, Heisenberg’s uncertainty principle tells us that a state of denite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indenite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with nite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
The concept of position and its measurement are fundamental to modern navigation technology as well as to theoretical physics. Nevertheless, a consistent notion of position has not been found in modern physics because of apparent inconsistencies between quantum mechanics and relativity. In quantum mechanics the probability distribution for a particle's position is determined by its wave function. A wave function strictly localized in space describes a particle within that locality, and nowhere else. But the quantum equations of motion predict the function will disperse, after any length of time at all, to occupy all space. Therefore the theoretical prediction is that, shortly after detection in a laboratory, a particle may be found anywhere in the universe, including regions requiring faster-than-light travel to reach. This contradicts the theory of relativity, which precludes a body traveling faster than the speed of light. The conflict arises from unexamined assumptions regarding the time of the measurement event and the time of the system under measurement. The prevailing view in physics is that time is only a parameter, having a definite value, describing dynamic changes in physical systems. This view endures even though quantum mechanics recognizes position as a quantum variable with inherent uncertainty, and relativity indicates that time should be placed on the same footing as position. But if time is recognized as a quantum variable with inherent uncertainty, the conceptual problems with position are resolved. The particle has finite probability of being in the past (or future) of the observer, while the quantum mechanical operator describing the position measurement contains a term which extrapolates from the intrinsic time of the particle to the time of the observer. Being potentially well in the past of the measurement event, the particle has time to travel to remote regions without exceeding the speed of light, and the conflict with relativity is resolved. A relativistic quantum theory can therefore be constructed incorporating a well-defined concept of position.
Relativistic quantum theory supports two competing notions of particle locality. Newton--Wigner localization provides a sharp position operator on the one-particle Hilbert space but is not Lorentz covariant and leads to instantaneous spreading for positive-energy states. Foldy--Wouthuysen localization, widely used in precision experiments, yields a robust quasi-classical particle description but is representation-dependent and does not coincide with strict field-theoretic locality. We show that this tension is not a paradox in UV-complete nonlocal quantum field theories whose ultraviolet physics is encoded by an entire-function regulator. In such theories the physically accessible observables form a quasi-local algebra generated by regulated smearings with an intrinsic nonlocality length. Sharp Newton--Wigner projectors are not operationally realistic, and Newton--Wigner and Foldy--Wouthuysen localization become equivalent non-singular notions of locality in the same measurement algebra.
We consider the entanglement dynamics between two Unruh-DeWitt detectors at rest separated at a distance d. This simple model when analyzed properly in quantum field theory shows many interesting facets and helps to dispel some misunderstandings of entanglement dynamics. We find that there is spatial dependence of quantum entanglement in the stable regime due to the phase difference of vacuum fluctuations the two detectors experience, together with the interference of the mutual influences from the backreaction of one detector on the other. When two initially entangled detectors are still outside each other's light cone, the entanglement oscillates in time with an amplitude dependent on spatial separation d. When the two detectors begin to have causal contact, an interference pattern of the relative degree of entanglement (compared to those at spatial infinity) develops a parametric dependence on d. The detectors separated at those d with a stronger relative degree of entanglement enjoy longer disentanglement times. In the cases with weak coupling and large separation, the detectors always disentangle at late times. For sufficiently small d, the two detectors can have residual entanglement even if they initially were in a separable state, while for d a little larger, there could be transient entanglement created by mutual influences. However, we see no evidence of entanglement creation outside the light cone for initially separable states.