How can momentum and position be combined into a phase space when they have different units? Elaboration of the question: What is the geometrical interpretation of units? As in, a unit of length is a choice of scaling of  the coordinate systems i.e. it is a choice of diffeomorphism, but then what about all other units? If we take the geometrical approach then there are no units, because the masses of fundamental particles are related to the Casimir of the Lorentz group. To sum up, 


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*Are units a geometric quantity?

*Can a manifold have different units on different dimensions? 
EDIT: A more clear explanation of the statement 'unit of length is a choice of diffeomorphism' can be found in Terrence Tao's blog post on gauge theory, where he considers units as a basic example of a gauge theory.
 A: Units are not a geometric quantity or property. Units signify that a given quantity is measured by a given operation. The unit itself is then usually defined by a referential instance of such a measurement. A canonical example is the International prototype kilogram, a platinum-iridium cylinder stored in Paris whose measurement of mass provides the referential SI unit. I recommend to take a look at the list of SI unit definitions to get a more concrete feeling of this. 
A manifold (a "space") is anything that can be, at least locally, mapped to a patch of $\mathbb{R}^n$ where $n$ is a dimension of the space. One way how we perform that mapping is by choosing units (referential instances) $x_0,p_0$ and comparing the magnitude of our phenomenon $x, p$ to obtain relative magnitudes $\xi,\pi$, which are numbers. The procedure by which we arrive to the numbers $\xi,\pi$ is defined strictly operationally, but once we have done so, we write the result of the measurement as $x= \xi x_0, p = \pi p_0$. In this sense $x,p$ do form a space that is mapped, but certainly never fully captured by groups of numbers.
However, we know that our conventional units are not of fundamental importance. Then, if we make a coordinate transformation on the space $\xi,\pi$ into something like $\tilde{R} = \xi^2 + \pi^2,\, \tilde{\varphi} = \arctan(\xi/\pi)$, it is very unlikely that such a unit-dependent definition will be useful for anything. On the other hand, if there is a meaningful scale governing the problem that allows to make the two quantities have the same units, it can often turn out very useful. 
Consider a point mass $m$ with momentum $p$ at position $x$ away from equilibrium on a spring with stiffness $k$. Stiffness has the dimensions mass over time squared and it can be used along with $m$ to make the phase-space coordinates comparable. It then turns out that it is useful to transform to the coordinates on phase space $\varepsilon, \varphi$ defined as
$$\varepsilon E_0 = \frac{p^2}{2 m} + \frac{k x^2}{2}$$
$$\varphi = \arctan\left( \frac{x}{p} \sqrt{\frac{m}{k}}\right)$$
where $E_0$ is a referential energy unit. Then you find out that in these coordinates the equations of motion turn out to be $\dot{\varepsilon} = 0, \, \dot{\varphi} = \sqrt{k/m}$. This is one of the canonical moments where it becomes clear that the concept of a phase space is very useful.
A: Yea, I thought something along the same lines as Javier...how can we use a complex plane if we can't really add imaginary numbers and real numbers since they have different "units"? 
The reason for using a 2-dimensional plane to represent 2 measurements of different units, or a 3-dimensional space to represent 3 measurements of different units, is exactly the fact that they ARE different units. There is no way to scalarly add vectors that have their components in entirely different dimensions; that is, if the position is on the X axis, for example, and the momentum is on the Y axis, the "magnitude" (more like measurement) of the position has no component along the dimension that the momentum is represented in, and the momentum has no component along the direction the position is represented in, since the 2 vectors are drawn in 2 different dimensions, at right degree angles to each other.
Would it make any sense to represent both magnitude and position on the SAME NUMBER LINE?!?!?! 
NO! Because in that case, we could add those 2 vectors and get a bigger one that would represent...what? Nothing, really, since as you said, the vectors have different units.
The only way to add these vectors would be to make the X vector (position) and the Y vector (momentum) each a component of a bigger vector. Note that in this case, the magnitude of this bigger vector, $sqrt(x^2+y^2)$, is nonsense, since as you said, momentum and velocity have different units.
However, there IS a good reason for using this 2D space to represent 2 different qualities of a system.
Say we have this 2D phase-space, and we have 2 vectors, each with an X component (position) and with a Y component (momentum). We then want it to move without changing the momentum. We can then simply add to X to get a greater position, but keep Y the same, to get the same momentum
If the momentum vector had a component in the direction of position, so not at 90 degrees to the position vector, changing the position graphically would automatically change the momentum. But by making each of the different (not the same units) qualities of a system be represented in different dimensions, all at 90 degrees to each other, we can vary whichever quantity we want (position, momentum, temperature, heck, we could have the Z axis representing how happy the system is) we can vary any of them we want without worrying about it changing the magnitude of the others. That is, as we move along the X axis, we don't change the ammount on any of the other axees, but if the different qualities were NOT all represented in different dimensions, then moving along the X axis would cause one of the other qualities to change, and therefor we wouldn't be able to represent every possible state of the system.
(I'm gonna edit my answer later to make it clearer and with diagrams, I'm in a realy big rush right now, sorry!) Hope it was useful! 
