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Physical Review A

Interaction of a composite system with the quantized radiation field in an approximately relativistic theory

Kunnat J. Sebastian

Published Jun 1, 1981• 23 citations
Publisher

Abstract

We show that in order to satisfy the condition of relativistic invariance for the one-photon radiative transition amplitudes of a composite system to relative order $\frac{1}{{c}^{2}}$ in an approximately relativistic canonical theory, the position operator in the theory should satisfy a condition which encompasses the world-line condition of Currie, Jordan, and Sudarshan. We also obtain the expression for the $E1$ and the $M1$ single-photon transition amplitudes of the composite system in the center of momentum frame, including the relativistic corrections of leading order. These expressions show that the relativistic corrections are qualitatively different for the electrically charged and neutral composite systems. They also depend, especially in the case of the $M1$ amplitudes, rather sensitively on the binding interaction among the constituent particles. There are also relativistic corrections which depend simultaneously on the recoil momentum of the composite system and on the use of the relativistic center-of-mass variables in the problem and which so far have not been considered in the literature.

DOI
10.1103/physreva.23.2810

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Using a new approach to the problem of the interaction of a composite system with a quantized radiation field, an expression for the two-photon amplitude of a composite system is derived in its center-of-mass frame, correct to order ${v}^{2}$/${c}^{2}$. In the expression for the amplitude there are terms which depend on the recoil of the composite system due to the emission of the photons. These terms which are of order ${v}^{2}$/${c}^{2}$ compared to the nonrelativistic terms were neglected in the previous relativistic treatments of the problem. We also show that the amplitude is relativistically invariant to order ${v}^{2}$/${c}^{2}$, when viewed from another Lorentz frame. We have used the relativistic center-of-mass and internal variables of Krajcik and Foldy [R. A. Krajcik and L. L. Foldy, Phys. Rev. D 10, 1777 (1974)] throughout the treatment.

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We show that in order to satisfy the condition of relativistic invariance for the one-photon radiative transition amplitudes of a composite system to relative order 1c2\frac{1}{{c}^{2}}c21​ in an approximately relativistic canonical theory, the position operator in the theory should satisfy a condition which encompasses the world-line condition of Currie, Jordan, and Sudarshan. We also obtain the expression for the E1E1E1 and the M1M1M1 single-photon transition amplitudes of the composite system in the center of momentum frame, including the relativistic corrections of leading order. These expressions show that the relativistic corrections are qualitatively different for the electrically charged and neutral composite systems. They also depend, especially in the case of the M1M1M1 amplitudes, rather sensitively on the binding interaction among the constituent particles. There are also relativistic corrections which depend simultaneously on the recoil momentum of the composite system and on the use of the relativistic center-of-mass variables in the problem and which so far have not been considered in the literature.
Using a new approach to the problem of the interaction of a composite system with a quantized radiation field, an expression for the two-photon amplitude of a composite system is derived in its center-of-mass frame, correct to order v2{v}^{2}v2/c2{c}^{2}c2. In the expression for the amplitude there are terms which depend on the recoil of the composite system due to the emission of the photons. These terms which are of order v2{v}^{2}v2/c2{c}^{2}c2 compared to the nonrelativistic terms were neglected in the previous relativistic treatments of the problem. We also show that the amplitude is relativistically invariant to order v2{v}^{2}v2/c2{c}^{2}c2, when viewed from another Lorentz frame. We have used the relativistic center-of-mass and internal variables of Krajcik and Foldy [R. A. Krajcik and L. L. Foldy, Phys. Rev. D 10, 1777 (1974)] throughout the treatment.
We are interested in the energy-momentum relation for a moving composite in relativistic quantum mechanics in many-particle Dirac models. For a manifestly covariant model one can apply the Lorentz transform to go from the rest frame to a moving frame to establish an energy-momentum relation of the form (M∗c2)2+c2∣P∣2\sqrt{(M^*c^2)^2+c^2|{\bf P}|^2}(M∗c2)2+c2∣P∣2​ where M∗M^*M∗ is the kinematic mass. However, the many-particle Dirac model is not manifestly covariant, and some other approach is required. We have found a simple approach that allows for a separation of relative and center of mass contributions to the energy. We are able to define the associated kinematic energy and determine the energy-momentum relation. Our result can be expressed as a modified deBroglie relation of the form ℏω(P)= [M∗(P)c2]2+c2∣P∣2 \hbar ω({\bf P}) = ~ \sqrt{[M^*({\bf P}) c^2]^2 + c^2 |{\bf P}|^2} ℏω(P)= [M∗(P)c2]2+c2∣P∣2​ where the kinematic mass M∗M^*M∗ will depend on the total momentum P{\bf P}P for a general noncovariant potential. The prefactor that occurs we associate with a time dilation effect, the existence of which has been discussed previously in the literature.
Abstract We study quantum tunneling of a composite object, which has a dipole or quadrupole moment coupled with quantized (photon or gravitational) radiation field, through a potential barrier. The dipole or quadrupole moment is represented by an oscillator in the relative coordinate of two constituent particles of the object. The center of mass degrees of freedom of the object is not directly coupled with the radiation field. However, we show that, for the object with the oscillator in the excited state, dissipation caused by the radiation field can suppress its quantum tunneling rate in the center of mass degrees of freedom. In addition, when the initial energy of its center of mass motion is similar to that of the excited state of the oscillator, a spatial superposition state of the object prepared by the barrier can decohere due to the radiation field. The main purpose of this article is to investigate how two different interplays (i) among the center of mass, the relative coordinate degrees of freedom and the potential barrier, and (ii) between the relative coordinate degrees of freedom and the radiation field, can affect the quantum tunneling and the creation of the spatial superposition state of the object. Our toy model can give insight into tests of quantum tunneling and quantum superposition of atoms or molecules with its dipole or quadrupole moment coupled with the radiation field.
Atomic high-precision measurements have become a competitive and essential technique for tests of fundamental physics, the Standard Model, and our theory of gravity. It is therefore self-evident that such measurements call for a consistent relativistic description of atoms that eventually originates from quantum field theories like quantum electrodynamics. Most quantum metrological approaches even postulate effective field-theoretical treatments to describe a precision enhancement through techniques like squeezing. However, a consistent derivation of interacting atomic quantum gases from an elementary quantum field theory that includes both the internal structure as well as the center of mass of atoms, has not yet been addressed. We present such a subspace effective field theory for interacting, spin carrying, and possibly charged ensembles of atoms composed of nucleus and electron that form composite bosons called cobosons, where the interaction with light is included in a multipolar description. Relativistic corrections to the energy of a single coboson, light-matter interaction, and the scattering potential between cobosons arise in a consistent and natural manner. In particular, we obtain a relativistic coupling between the coboson’s center-of-mass motion and internal structure encoded by the mass defect. We use these results to derive modified bound-state energies, including the motion of ions, modified scattering potentials, a relativistic extension of the Gross-Pitaevskii equation, and the mass defect applicable to atomic clocks or quantum clock interferometry.