Confining boundary conditions from dynamical coupling constants

It is shown that it is possible to consistently and gauge invariantly formulate models where the coupling constant is a non-trivial function of a scalar field. In the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> case, the coupling to the gauge field contains a term of the form <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow></msup><mo>+</mo><msup><mrow><mo>∂</mo></mrow><mrow><mi>μ</mi></mrow></msup><mi>B</mi><mo stretchy="false">)</mo></math> where B is an auxiliary field and <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>j</mi></mrow><mrow><mi>μ</mi></mrow></msub></math> is the Dirac current. The scalar field ϕ determines the local value of the coupling of the gauge field to the Dirac particle. The consistency of the equations determines the condition <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>∂</mo></mrow><mrow><mi>μ</mi></mrow></msup><mi>ϕ</mi><msub><mrow><mi>j</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>=</mo><mn>0</mn></math> which implies that the Dirac current cannot have a component in the direction of the gradient of the scalar field. As a consequence, if ϕ has a soliton behaviour, like defining a bubble that connects two vacua, we obtain that the Dirac current cannot have a flux through the wall of the bubble, defining a confinement mechanism where the fermions are kept inside those bags. Consistent models with time dependent fine structure constant can be also constructe