On generalized quantum scattering theory: Rayleigh scattering and Thomson scattering with a minimal length

Highlights

• The scattering quantum theory with a minimal length is necessary to probe various very short-distance phenomena.

• The usual Hamiltonian describing the dynamics of the system is modified to include the effect of minimal length.

• Modified quantities such as modified matrix elements and cross sections are derived using the modified Hamiltonian.

• All modified quantities depend on the deformation parameter α, and the usual quantities are recovered in the limit $\alpha \to 0$.

Abstract

In the quantum mechanical approach we study the minimal length effect on Rayleigh scattering and Thomson scattering. We modify the usual Hamiltonian for the motion of an electron in a quantized electromagnetic radiation field by utilizing deformed operators to get the Hamiltonian with a minimal length. We derive expressions for modified Rayleigh scattering and modified Thomson scattering. Using appropriate experimental inputs we get the upper bounds on the deformation parameter and minimal length. The upper bound on the minimal length is less than the electroweak scale.

Introduction

Scattering has been a major apparatus for the understanding of certain physical phenomena and the advancement in different areas of physics. In this article, we focus on Rayleigh scattering and Thomson scattering. Rayleigh scattering is simply the result of an interaction of radiation with particles of sizes much smaller than the wavelength of radiation. Rayleigh scattering is known to be responsible for damping of acoustic and phonon waves in glasses. Furthermore, Rayleigh scattering plays an important role in fiber optics. Thomson scattering is the result of an interaction of radiation with free charged particles such as free electrons. Thomson scattering has been a principal technique for plasma diagnosis in nuclear fusion reactors such as tokamak. However, initial scattering formulations do not offer any physical results at very short-distances. Fortunately, the scattering theory with a minimal length is of great advantage in the study of various short-distances phenomena. The minimal length which was initially put forward by Heisenberg [1], [2], [3] is a frequent prediction of several modern theories related to quantum gravity. Interestingly, the minimal length leads to the generalized uncertainty principle which is usually defined as follows [4], [5], [6], [7], [8], [9], [10]

where $L_P$ is the Plank length, and $\tilde{\alpha }$ is a positive parameter. In the low-energy limit, Eq. (1) leads to the familiar Heisenberg's uncertainty principle, and in the high-energy limit, Eq. (1) yields a minimal measurable length ${(\mathrm{\Delta }X)}_{MIN}=\tilde{\alpha }{L}_P$.

The minimal length consideration offers many advantages for diverse motives, thus several attempts have been made to rewrite theories with a minimal length, for example, it is widely believed that the incorporation of minimal length is one of ways to get rid of infinities in quantum field theories [11], [12], [13]. It was shown by Kempf, et al. that the minimal length can be derived from a modified Heisenberg algebra [14], [15], [16]. The Kempf algebra is a modified Heisenberg algebra for a D-dimensional quantized space satisfying the modified commutation relations

where $i,j=1,2,...,D$, and α, α' are two positive deformation parameters with identical dimensions. The objects $X^i$, and $P^i$ are the modified position and momentum operators respectively. Eq. (2) along with the Schwartz identity yields the inequality [17]

Ineq. (3) yields the following equation [26]

Some studies of a variety of phenomena in the presence of minimal length have been accomplished, and more recently different quantum mechanical models in the minimal length framework have been subject of intensive investigations [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. Since the incorporation of a minimal length leads to the deformed Hamiltonian, thus the useful physical properties should only be those of a system which is characterized in the deformed space. Stesko et al. showed that the modified positions and momentum operators can be expressed in terms of their corresponding usual operators as follows [35]

$$\begin{array}{c}{X}^i=x^i+\frac{2\alpha -{\alpha }^{\prime }}{4}({\mathbf{p}}^2{x}^i+x^i{\mathbf{p}}^2)\hfill \\ P^i=p^i(1+\frac{{\alpha }^{\prime }}{2}{\mathbf{p}}^2)\hfill \end{array}$$

where $x^i$, and

are position and momentum operators in the usual Heisenberg algebra. The position operators commute to first-order in the linear approximation for ${\alpha }^{\prime }=2\alpha$, and this leads to the new expression for Eq. (2)

Brau [36] proved that the following operators are compatible with Eq. (6), in first-order in α

$$\begin{array}{c}{X}^i=x^i\hfill \\ P^i=p^i(1+\alpha {\mathbf{p}}^2)\hfill \end{array}$$

The organization of the article is the following: In Sec. 2 we modify the usual Hamiltonian describing the motion of charged particle of charge e and mass m to get the Hamiltonian with a minimal length. In Sec. 3 we derive the modified Rayleigh scattering and modified Thomson scattering and using experimental inputs we get the upper bounds on the deformation parameter and minimal length. Finally, we summarize our article in Sec. 4.