What exactly is the Parity transformation? Parity in spherical coordinates

Question

When I encounter the parity transformation in physics,I feel often it's not really clear what we are doing and I want to understand it in a more rigorous way, can you help me with that?(I want to see it from the point of view of tensors.)I always hear that the electric field changes it's sign, often denoted as

$$\vec{E}'(\vec{x}')=-\vec{E}(\vec{x})\tag{1}$$

I assume this means the following: there is a vector field $\vec{E}(\vec{r})$ in a euclidian vector-space, $\vec{r}$ is the position vector (both are independent of a coordiante system-> invariant tensors) and we have chosen a cartesian coordinate system, where the {x,y,z} coordinates are measured to be increasing in a certain direction, which gives rise to a local (covariant) basis in every point in space according to

$$\vec{e}_i(x,y,z)=\frac{\partial}{\partial x^i}\vec{r}(x,y,z)\tag{2}$$

(This equation will of course also work in curvilienar coordinate systems, then the basis will change from one point to the other, whereas in affine coordinates like the cartesian they will be constant)

Now the parity transformation is done by what is usually denoted as (x,y,z) -> (-x,-y,-z). I think this means that we now create a second coordinate system, where the {x',y',z'} coordinates are measured as increasing in exactly the opposite direction compared to the unprimed system.This will cause the basis vectors (who are covariant vectors) of the primed system to point in the opposite direction in every point.Now we can decompose the invariant E-field vectors in every point with respect to either the unprimed or the primed basis and compare their contravariant components. And since for every vector $\vec{V}$ we have

$$\vec{V}=V^i \vec{e}_i=V'^i \vec{e}'_i\tag{3}$$

this means that the components have the relationship

$$E'^i=-E^i\tag{4}$$

(the co- and contravariant tensors transform the same way in this case)

My first question would be if this is what physicists mean when they do a parity transformation and say that the E-field changes it's sign? (I think in this case it would be true for any configuration of charges, but let's consider a point charge at the origin).

But then if we didn't start with a cartesian system but with spherical coordinates, and perform a parity transformation, eq. (4) is still supposed to hold (at least if I have understood correctly what wikipedia and other sources mean when they speak about the sign flip, so please correct me if I'm wrong), which implies, because of how (2) relates the coordinate lines to the basis vectors, that in the primed system, the radius is now measured to be increasing towards the origin, which is not true.(Or is it?)

My second question is: how do the coordinate lines of the spherical system look like (how do the primed axes compare to the unprimed?), after the parity transformation, and how can one say that the E-field changes its sign under parity transformation, without having to measure the radius as increasing towards the origin? I mean that r is zero at the origin but becomes negative as we go outward, because this is required to make the $\vec{e}_r$ vectors point to the origin according to (2).

(For a point charge, the only nonzero $E^i$ will be the componets of the $\vec{e}_r$ basis vectors so only the orientation of the radius coordinate matters, doesn't it?)

In the literature I only encounter the case that the angles of the spherical coordinate system are transformed under parity and not the radius and imo this would not result in a sign flip of the E-field.

Answer 1

It turns out that a vector field $\vec F(x, y, z)$ is best thought of as a local directional derivative which we'd write $\mathsf F= \vec F \cdot \nabla$, taking scalar fields $f(x,y,z)$ and producing new scalar fields $g(x,y,z)$ from them. When we start from this place of$$g(x,y,z) = \mathsf F[f](x,y,z) = F_x\frac{\partial f}{\partial x} + F_y\frac{\partial f}{\partial y} + F_z\frac{\partial f}{\partial z}$$ for scalar fields $F_{x,y,z}(x, y, z)$, and we define that $\hat g(x, y, z) = g(-x, -y, -z)$ and $\hat f(x, y, z) = f(-x, -y, -z)$, as the parity transform asks us to perform, we will find that the chain rule tells us that $\partial_x \hat f = -\partial_x f,$ and we get one big minus sign on the right hand side. We can correct for this if we define that $\hat F_{\alpha}(x, y, z) = -F_\alpha(-x, -y, -z).$ So there is both a minus sign "inside" and "outside" the parentheses; the inner one comes from the fact that $\vec F$ needs to be using parity-transformed inputs and the outer one comes from the fact that this reflection actually changed the direction of $\vec F$ itself.

In spherical coordinates (I will take $0\le \theta < 2\pi$ as azimuthal while $0\le \varphi\le\pi$ comes down from the North Pole) both of these ideas get a little complicated; we instead have $$g(r,\theta,\varphi) = \mathsf F[f](r, \theta, \varphi) = F_r \frac{\partial f}{\partial r} + F_\theta \frac1{r\sin\varphi} \frac{\partial f}{\partial \theta} + F_\varphi\frac1{r}\frac{\partial f}{\partial\varphi}.$$Our parity transform needs to map latitudes to opposite latitudes $\varphi\mapsto\pi-\varphi$ and needs to also rotate a point 180 degrees about the globe, $\theta\mapsto\theta + \pi.$ Under this mapping actually $\sin\varphi\mapsto\sin\varphi,$ so there's a sort of "double negative" in this $\sin$ term and we therefore find that: $$\begin{array}{ll} F_r(r,\theta,\varphi) &\mapsto~ {+F_r(r, \theta + \pi, \pi - \varphi)},\\ F_\theta(r,\theta,\varphi) &\mapsto~ {+F_\theta(r, \theta + \pi, \pi - \varphi)},\\ F_\varphi(r,\theta,\varphi) &\mapsto~ {-F_\varphi(r, \theta + \pi, \pi - \varphi)}.\end{array}$$Another way to interpret this is to just think about the unit vectors at a given point in space; remember that when we're using curvilinear coordinates these unit vectors vary over space itself, so unlike $\hat x$, the unit vector in the $x$-direction which is the same at all points in a Cartesian space, the unit vector $\hat r$ changes to point away from the origin at all points in space. In fact one can work out that these unit vectors are:$$\begin{array}{ll} \hat r &=~ {+\hat z}~\cos\varphi + \hat x~ \sin\varphi \cos\theta + \hat y\sin\varphi\sin\theta\\ \hat \theta &=~ {-\hat x}~\sin\theta + \hat y~ \cos\theta\\ \hat \varphi &=~ {-\hat z}~\sin\varphi + \hat x~ \cos\varphi \cos\theta + \hat y~\cos\varphi\sin\theta \end{array}$$

When we reflect space our starting $\hat r$ gets mapped to the reflected $\hat r$; they both are unit vectors pointing away from the origin. Similarly our starting $\hat \theta$ gets mapped to the reflected $\hat\theta$; they both are unit vectors pointing "West". But when we reflect $\hat \varphi$, which points "South" everywhere on the sphere, we get that the reflected vector points "North", which is opposite of its new local-unit-vector. So that component, and only that component, must pick up the minus sign.