Spin Current in <i>BF</i> Theory

• Main Author: Malik Almatwi
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Physics
• Subjects: <i>BF</i> theory; local Lorentz symmetry; local Lorentz currents
• Online Access: https://www.mdpi.com/2624-8174/3/2/29

In this paper, a current that is called spin current and corresponds to the variation of the matter action in <i>BF</i> theory with respect to the spin connection <i>A</i> which takes values in Lie algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">so</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>, in self-dual formalism is introduced. For keeping the 2-form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>B</mi><mi>i</mi></msup></semantics></math></inline-formula> constraint (covariant derivation) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><msup><mi>B</mi><mi>i</mi></msup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> satisfied, it is suggested adding a new term to the <i>BF</i> Lagrangian using a new field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>ψ</mi><mi>i</mi></msup></semantics></math></inline-formula>, which can be used for calculating the spin current. The equations of motion are derived and the solutions are dicussed. It is shown that the solutions of the equations do not require a specific metric on the 4-manifold <i>M</i>, and one just needs to know the symmetry of the system and the information about the spin current. Finally, the solutions for spherically and cylindrically symmetric systems are found.