# Is momentum space "less physical" than position space?

In quantum mechanics and quantum field theory it is specially common to work in both position and momentum space. Passing the theory to momentum space is sometimes crucial, as one usually finds that the Hamiltonian of the system is diagonal in this space. However, it seems that the ultimate objects of the theory are always position-dependent. We are always looking for fields $$\psi({\bf{x}},t)$$, propagators $$C(x-y)$$, Schwinger functions $$\langle 0|\psi(x_{1})\cdots\psi(x_{n})|0\rangle$$ and so on.

Is there any reason for always aiming at position dependent objects, even when calculations are easier in momentum space? Maybe the reason why is because real world experiments usually measure things in position space. If so, does it imply that momentum space is "less physical" in some sense?

• In General Relativity, there is no well defined single "momentum space"... Commented Sep 17, 2022 at 21:46
• Isn't this purely opinion-based, even if opinions will tend to be homogenous? How does one measure physicality, and can you take its time derivative?
– g s
Commented Sep 18, 2022 at 0:00
• Define "physical" ;-) Commented Sep 18, 2022 at 11:26

Your question can be equivalently phrased as whether position space is more physical than momentum space.

In a sense, yes. One of the basic facts about our universe (as far as we can tell) is that it exhibits locality in position space, which informally means that what is going to happen at a particular point in space over an infinitesimal interval of time depends only on the state of the universe in an infinitesimal neighbourhood of that point. Formally, it means the laws of physics are expressible as differential equations (there is a lot of nuance here that I am not qualified to discuss).

You can't have nontrivial laws of physics that are local in both position space and momentum space. I guess if the laws of physics had been local in momentum space, then we would have just called momentum space "position space" instead. Of course, you can imagine hypothetical universes that don't have any locality.

Anyway, that's why position space is special.

Well, to begin with, the universe we happen to live and move around in is position space- and we build & run our experiments in position space too. So if one wanted to compare the results of a calculation performed in momentum space with the results of for example a collider run in position space, we would necessarily have to do that conversion. Until we can figure out how to live and build experiments in momentum space, it will always be less "physical" than position space!

• And yet your body, just like every other physical object, has both a position and a momentum. So arguably we live just as much in momentum space as in position space. Commented Sep 18, 2022 at 10:48
• It seems to me that the interesting properties of collider experiments are in momentum space: Momenta of incoming beams and outgoing particles. Commented Sep 18, 2022 at 21:00
• @IlmariKaronen I am not a physicist, but it seems clear to me that what we think of as physical objects are localized in position space and not in momentum space. Commented Sep 19, 2022 at 7:02
• (Note: at least until we average out over long enough timescales, and I think even this only works if the object itself isn't significantly accelerating, rotating, nonrigid etc.) Commented Sep 19, 2022 at 7:16
• @A_P Ok, but a partial answer to that immediately suggests itself: position in space(time) is closely related to causality, while (four-)momentum is not. Edit: which I think is basically what Brian Bi's accepted answer is saying. Commented Sep 20, 2022 at 18:39

I would concede that position-space is often more "intuitive" or "less abstract" than momentum-space. But as for "more or less physical"--I would use the word "interesting" or "relevant" rather than "physical", and in that case it depends on the problem (especially what we are measuring and how we measure it). As a rough rule of thumb, position-space is more relevant when discussing bound systems (e.g. the orbital clouds of a hydrogen atom) and momentum-space is more relevant for free particles/states (e.g. scattering).

I think they are both equally real. The momentum is derived from the change in position over time. In fact, these changes, if irreducible, can be said to constitute time.

In quantum mechanics the momentum operator is connected with the spatial derivative of the wavefunction and the energy to temporal derivative. You can use the energy to generate translations in time and likewise you can use the momentum operator to generate translations.

Moreover, the time-position representation and the energy-momentum representation are (I'm not sure if this is always the case) Fourier transforms of one another.

So time, energy, momentum, and position are tightly intertwined.