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In the theory of electromagnetism in 1+1 spacetime dimensions (one temporal and one spatial coordinate), one can define the 2-potential vector (analogous to the 4-potential vector in 3+1 spacetime dimensions):

$$A_\mu=\left(A_t,A_r\right)\equiv\left(\phi(t,r),A_r(t,r)\right)$$ given those potentials we can express the magnetic and electric fields:

$$\vec{E}(r,t)=-\vec{\nabla}\phi-\frac{\partial\vec{A}}{\partial t}=-\left(\partial_r\phi+\partial_tA_r\right)\hat{r}$$ $$\vec B(r,t)=\vec{\nabla}\times\vec{A}\to0 $$ In the expression of the magnetic field it seems that there is no proper definition for the curl in 1+1 dimensions because no field could encircle a space with 1 spatial dimension.

Now we will consider the case of static potentials which depend only on the spatial coordinate:

$$A_\mu=\left(\phi(r),A_r(r)\right)$$ In this case the magnetic and electric fields would be: $$\vec{E}(r,t)=-\partial_r\phi\hat{r}$$ $$\vec B(r,t)=0 $$

Given that we are in this static limit, is there are constraints on the value of the magnetic "vector" field component $A_r$? is its value have to be zero?, or can it be any arbitrary numerical or even functional expression?

Qmechanic
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Daniel Vainshtein
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1 Answers1

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In 1+1D there is no magnetic field, while the electric field $E=-\partial_x\phi-\partial_tA$ and the magnetic potential $A$ have only 1 component, cf. e.g. my Phys.SE answer here.

The 2-vector gauge potential $A^{\mu}=(\phi,A)$ consists of an electric potential $\phi$ and a magnetic potential $A$. It is susceptible to gauge transformations. E.g. the magnetic potential vanishes in axial gauge $A=0$.

Qmechanic
  • 201,751