Gravitational tensor-monopole moment of the hydrogen atom to order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math>

Abstract

We calculate the gravitational tensor-monopole moment of the momentum-current density <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>T</a:mi> <a:mrow> <a:mi>i</a:mi> <a:mi>j</a:mi> </a:mrow> </a:msup> </a:math> in the ground state of the hydrogen atom to order <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi mathvariant="script">O</c:mi> <c:mo stretchy="false">(</c:mo> <c:mi>α</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> in quantum electrodynamics (QED). The result is <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi>τ</h:mi> <h:mi>H</h:mi> </h:msub> <h:mo>/</h:mo> <h:msub> <h:mi>τ</h:mi> <h:mn>0</h:mn> </h:msub> <h:mo>−</h:mo> <h:mn>1</h:mn> <h:mo>=</h:mo> <h:mfrac> <h:mrow> <h:mn>4</h:mn> <h:mi>α</h:mi> </h:mrow> <h:mrow> <h:mn>3</h:mn> <h:mi>π</h:mi> </h:mrow> </h:mfrac> <h:mrow> <h:mo stretchy="false">(</h:mo> <h:mi>ln</h:mi> <h:msup> <h:mi>α</h:mi> <h:mn>2</h:mn> </h:msup> <h:mo>−</h:mo> <h:mn>0.028</h:mn> <h:mo stretchy="false">)</h:mo> </h:mrow> <h:mo>=</h:mo> <h:mo>−</h:mo> <h:mn>3.06</h:mn> <h:mo>×</h:mo> <h:msup> <h:mn>10</h:mn> <h:mrow> <h:mo>−</h:mo> <h:mn>2</h:mn> </h:mrow> </h:msup> </h:math> where <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msub> <l:mi>τ</l:mi> <l:mn>0</l:mn> </l:msub> <l:mo>=</l:mo> <l:msup> <l:mi>ℏ</l:mi> <l:mn>2</l:mn> </l:msup> <l:mo>/</l:mo> <l:mn>4</l:mn> <l:msub> <l:mi>m</l:mi> <l:mi>e</l:mi> </l:msub> </l:math> is the leading-order moment. The physics of the next-to-leading-order correction is similar to that of the famous Lamb shift for energy levels. Published by the American Physical Society 2024

Topics & Concepts