Active and passive transformation

Question

Some active transformations on the system can be seen also as passive transformations, for example the rotation of the system can be seen as the rotation of the observer in the opposite direction. Anyway, other transformations are just active, for example the change in the number of particles in a system. So my question is, what are in general the transformation that can be seen both as passive and active? if it's possible answer with little math, I'm now interested in understanding the concept.

Answer 1

An active transformation of a manifold $M$ is a diffeomorphism $M \rightarrow M$. It actually moves the points of the manifold. This is why its called an active transformation.

This is in contrast to a passive transformation which does not move the points of a manifold. To explain this, first suppose that the manifold is globally chartable to keep things simple, with a chart $u :M \rightarrow U$ where U is an open in $\mathbb{R}^d$ where $d$ is the dimension of the manifold. Then a passive transformation is a diffeomorphism of the charting space, say $\phi: U \rightarrow V$ where $V$ is another open in $\mathbb{R}^d$. This induces a chart $\phi \circ u : V \rightarrow M$. Then a point $p$ of $M$ has coordinates $u(p)$ under the chart $u$ and coordinates $(\phi \circ u)(p)$ under the chart $\phi \circ u$. These two coordinates will differ even though the point $p$ is unchanged. This is why it is called a passive transformation.

Another way of looking at the same phenomena is to consider a chart $M \rightarrow U$ then a diffeomorphism applied on the left, that is on $M$ is an active transformation whilst a diffeomorphism applied to the right, that is on $U$, is a passive transformation.

So my question is, what are in general the transformation that can be seen both as passive and active?

You can apply both an active and passive transformation to give a transformation that has aspects of both.

Answer 2

Let me try to formulate the difference between an active and a passive transformation (this should be consistent with the definition on Wikipedia here):

The active transformation acts only on the coordinates of an object, making the object actively rotating/ deforming/ moving. All of this happens with respect to a chosen basis which remains unchanged during the transformation.

The switch of the perspective - or the change of the basis system - is the passive transformation. However, a change of basis will not only change the basis, but also the coordinates of the objects in order to leave the overall system invariant. Thus, concidering your first example, in the new basis system, the coordinates of the initially rotating object would be constant. The coordinates of any surrounding objects, that stood still before, would rotate in the opposite direction. I would call this rotation in opposite direction again an active rotation with respect to the new basis system (an example of this would be the sun, that is - from our perspective on the rotating earth - moving actively through the sky).

The passive transformation therefore affects the basis vectors as well as the coordinates of the observed objects. In contrast, the active transformation only affects the coordinates of objects.

Within those definitions, I think what you mean by a transformation that is "just active", you really mean a transformation where it doesn't make sense or seems impossible to change the basis system (which would be a passive transformation) in order to counter the active transformation. And really, considering your second example, it is hard to imagine a basis transformation to a basis in which the number of particles is not constant.

Mathematically however, there is no reason not to use such a basis. The underlying concept here is the concept of a vector space (the coordinates of the observed objects) and its dual vector space (the home of the basis vectors). In mathematics, these two spaces are really (in most situations) interchangeable and the metric tensor is the tool to switch the roles of vectors and dual vectors.

So, as a more general answer, I would say that any active transformation could be countered by a passive transformation. It just doesn't help us in some situations.

Final remark on the example with the changing particle number: this is really a thing in quantum mechanics, where the second quantization is also called the "occupation number representation", in which it is possible to start with a state with $n$ particles and create or anihilate particles from there. If I remember correctly, the $n$-particle states are Eigenstates in this basis, which would mean that in other bases the number of particles is not fixed. I thought there was even an uncertainity relation with the number of particles included.