What happens to conservation laws if the spatial variable is complex?

Question

This is more of a conceptual question. Normally a conservation law will look something like
$$frac{partial j}{partial t}+frac{partial F}{partial x}=0tag{1}$$
where $x$ is typically a real-valued coordinate, or even $nabla$ if we have a space with a few dimensions in it. It's then pretty easy to define an integral over $-infty<x<infty$ or over the surface, etc. which is a conserved quantity.
But let's say we're working in a two-dimensional space now, and we can change our coordinates from $(x,y)$ to $z=x+iy.$ If we wind up with an equation of the form
$$frac{partial j}{partial t}+frac{partial G}{partial z}=0,tag{2}$$
does this give us conservation laws as well? It seems that it is difficult to define any integral over a surface which isn't trivial, because only poles of a complex function will contribute to a nonzero integral. But obviously, many functions in physics won't have poles - a two-dimensional fluid can often be described with a complex velocity, for example, and this is going to be analytic in most cases. We could think of charges and the like as being poles, but it's much harder to see how we get anything like momentum, angular momentum, and so on.
What can we say about the conserved quantities of the system when the variable is complex now instead of real?

Qmechanic · Accepted Answer

For each continuity equation
$$ sum_{mu = 0}^nfrac{partial J^{mu}_a}{partial x^{mu}}~=~0, tag{A}$$
one can define a conserved quantity
$$ Q_a(t)~:=~int_V ! d^nx~ J^0_a(vec{x}, t). tag{B}$$

OP's example:  Let $J^{mu}=J^{mu}_1 +i J^{mu}_2$ be a complex current, and introduce a complex space coordinate $z=x^1+ix^2$, i.e. $n=2$. OP's continuity equation (2) is of the form  (A) if we use Dolbeault derivatives
$$ frac{partial}{partial z}~=~frac{1}{2}left(frac{partial}{partial x^1}-ifrac{partial}{partial x^2} right), qquadfrac{partial}{partial bar{z}}~=~frac{1}{2}left(frac{partial}{partial x^1}+ifrac{partial}{partial x^2} right).tag{C}$$

What happens to conservation laws if the spatial variable is complex?

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