We consider the effect of relativistic boosts on single-particle Gaussian wave packets. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general the coherence decays with the increase of the momentum of the state, as well as the boost applied to it. Employing a basis-independent formulation, we show, however, that coherence may be preserved even for large boosts applied on narrow uncertainty wave packets. Our result is exemplified quantitatively for practically realizable neutron wave functions.
In this paper, we consider a single-particle, spin-momentum entangled state and measure the effect of relativistic boost on quantum coherence. The effect of the relativistic boost on single-particle generalized Gaussian wave packets is studied. The coherence of the wave function as measured by the boosted observer is studied as a function of the momentum and the boost parameter. Using various formulations of coherence, it is shown that in general, the coherence decays with the increase in momentum of the state, as well as the boost applied to it. A more prominent loss of coherence due to relativistic boost is observed for a single-particle electron than that of a neutron. The analysis is carried out with a generalized Gaussian wave packet of the form $\mathcal{N}{p}^{n}exp(\ensuremath{-}\frac{{p}^{2}}{{\ensuremath{\sigma}}^{2}})$, with
n being the ``generalization parameter'' and
N denoting the appropriate normalization constant. We also obtain a range for parameter
n appearing in the wave packet. The upper bound is found to have a dependence on the mass of the particle and the width of the Gaussian wave packet. We have obtained the Frobenius-norm measure of coherence,
l1- and
l2-norm measure of coherence, and relative entropy of coherence for a
(1+1)- and
(3+1)-dimensional analysis. Corresponding to each of the cases, we observe that the
l1-norm measure of coherence is equal to the Frobenius-norm measure of coherence. We have analyzed the scenario for which such a beautiful coincidence can occur. Finally, we have plotted different measures of coherence for the electron as well as the neutron for different values of the width of the wave function $\ensuremath{\sigma}$, boost parameter $\ensuremath{\beta}$, and generalization parameter
n.
In this study, we investigate the effect of the Lorentz transformation on the measures of quantum coherence in an entangled atomic system. Here, we consider the effect of this relativistic boosts on two-particle entangled generalized Gaussian wave packets in two scenarios. In the first scenario, we consider that the relativistic boost affects the one particle and other remains unaffected while in the second scenario, we consider that both the particles are affected by the effect of the relativistic boost. The coherence of the wave function as measured by the boosted observer is studied as a function of the boost parameter and the width of the Gaussian wave packets. Using various formulations of coherence, it is shown that in general the coherence decays with increase in the width of the Gaussian wave packet, higher values of boost parameter, and the number of particles on which boost is applied.
Decay rates in quantum field theory (QFT) are typically calculated assuming the particles are represented by momentum eigenstates (i.e., plane waves). However, strictly speaking, localized free particles should be represented by wave packets. This yields width corrections to the decay rate and to the clock behavior under Lorentz boosts. We calculate the decay rate of a particle of mass
M modeled as a Gaussian wave packet of width
a and centered at zero momentum. We find the decay rate to be ${\mathrm{\ensuremath{\Gamma}}}_{0}[1\ensuremath{-}\frac{3{a}^{2}}{4{M}^{2}}+\mathcal{O}(\frac{{a}^{4}}{{M}^{4}})]$, where ${\mathrm{\ensuremath{\Gamma}}}_{0}$ is the decay rate of the particle at rest treated as a plane wave. The leading correction is then of order
M2a2. We then perform a Lorentz boost of velocity
v on the above Gaussian and find that its decay rate does not decrease exactly by the Lorentz factor $\sqrt{1\ensuremath{-}{v}^{2}}$. There is a correction of order
M2a2v2. Therefore, the decaying wave packet does not act exactly like a typical clock under Lorentz boosts, and we refer to it is a ``WP clock'' (wave packet clock). A WP clock does not move with a single velocity relative to an observer but has a spread in velocities (more specifically, a spread in momenta). Nonetheless, it is best viewed as a single clock as the wave packet represents a one-particle state in QFT. WP clocks do not violate Lorentz symmetry and are not based on new physics; they are a consequence of the combined requirements of special relativity, quantum mechanics, and localized free particles.
Since their first introduction, Quantum Reference Frame (QRF) transformations have been extensively discussed, generalising the covariance of physical laws to the quantum domain. Despite important progress, a formulation of QRF transformations for Lorentz symmetry is still lacking. The present work aims to fill this gap. We first introduce a reformulation of relativistic quantum mechanics independent of any notion of preferred temporal slicing. Based on this, we define transformations that switch between the perspectives of different relativistic QRFs. We introduce a notion of ''quantum Lorentz transformations'' and ''superposition of Lorentz boosts'', acting on the external degrees of freedom of a quantum particle. We analyse two effects, superposition of time dilations and superposition of length contractions, that arise only if the reference frames exhibit both relativistic and quantum-mechanical features. Finally, we discuss how the effects could be observed by measuring the wave-packet extensions from relativistic QRFs.
It is shown that, in the phase-space representation of quantum mechanics, the uncertainty relation can be stated in terms of the integral invariant of Poincar\'e. The uncertainty relation for spreading free wave packets is discussed as an illustrative example. This phase-space approach can be extended to the relativistic regime. It is shown that Lorentz boosts are area-preserving canonical transformations in the phase space of the light-cone variables. The harmonic oscillator is discussed in detail as an illustrative example for the covariant realization of the uncertainty principle.
We derive closed form expressions of weak relativistic corrections to the wave-packet dynamics of the quantum harmonic oscillator within a perturbative framework. General expressions are derived for the leading-order relativistic contributions to wave-packet parameters, such as the time-dependent widths, variances, and uncertainty relations. Specific calculations are performed for unsqueezed, minimum-uncertainty Gaussian wave packets, valid up to leading order in (
1/c2). When applied to electron wave packets, the results indicate that relativistic effects become non-negligible for keV-scale harmonic confinement energies: the deviations in variances reach
0.15%−1.5% for an electron wave packet confined within the
1−10keV energy range. We also show that the standard saturation of the uncertainty relation remains unaffected by the leading-order relativistic effect.