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In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person by its position, but not its momentum. Two fermions can easily have the same momentum while they cannot hold the same position. Are there particles that cannot have the same momentum while can hold the same position? What causes the breaking of the symmetry of $p$ and $q$?

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This question is too unfocused. Clearly, the nice featrues are something inherent to the mathematical structure, envisioned to describe the nature of trajectories $q(t)$. That structure is being employed just because of its niceness. Everybody can now present his personal rant why the formal system, derived from classical mechanics, entails the possibility for that formulation. "The successor of every number $n$ can be obtained by computing $n+1$. But knowing the prime factors of $n$ doesn't seem to help me finding the factors of $n+1$! What could possibly causes this and what does it imply?" –  NiftyKitty95 Oct 30 '13 at 12:19
    
Actually, I kind of like this question, as the curious Planck-pivoting symmetry between ordinary space (q) and momentum space (p) has always intrigued me. Shinjikun, here's a thought you: In q, size costs nothing energetically, whereas in p size carries a huge energy price tag. That's why in metals the electrons at the top of the Fermi seas (in p space!) have energies comparable to X-rays. So part of your answer is this: There is a profound asymmetry in energetics of p vs. q that makes vast distances possible only in q. The harder part: Why do disturbances favor q over p in quantum mechanics? –  Terry Bollinger Oct 30 '13 at 17:15
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One of the features of Hamiltonian mechanics is the freedom to perform canonical transformations $(q,p)\longrightarrow (Q,P)$. For instance $(q,p)\longrightarrow (Q,P)=(-p,q)$. From this point of view, what we call positions and what we call momenta are merely conventional. –  Qmechanic Oct 30 '13 at 19:33
    
It is incorrect to state that two fermions can share the same momentum. It is impossible for two fermions to be in a state with well-defined position, $\psi(x)=\delta(x-x_0)$, and it is equally impossible for them to share a state with well-defined momentum, $\psi(x)=e^{ipx}$. Fermions are only forbidden from occupying the exact same state: equal wavefunctions in the spatial (and therefore momentum) part, as well as the spin component. –  Emilio Pisanty Nov 3 '13 at 13:06
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3 Answers

up vote 2 down vote accepted

We are not living in the $q$ part of phase space : we indeed live in the full phase space since we're definitely not fixed-momentum objects.

However we give more importance to our position than to our velocity/momentum; then the question gets out of physics into psychology.

In my opinion, part of it may be because of the way we gather knowledge : we fix it on fixed-$q$ supports: books, paper,... and we like to stop things to study them (draw a curve of a movement).

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What could possibly causes this

Abstraction: The configuration and phase spaces of analytical and quantum mechanics may be (but not necessarily so) derived from physical space(-time), but they are not the same thing.

Even though the configuration space of a single-particle system might look like real space, it's endowed with structure (force fields, Hamiltonians, wave functions, ...) that generally is not physically real.

and what does it imply?

That you need to take care not to confuse reality with your model thereof.

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Interesting question.

The presence of matter fields leads to the existence of energy and momentum. Without matter fields, all one has is the background space-time continuum. So momentum and energy must inhabit something, the converse is intuitively false (perfect vacuums exist classically). So naturally, we give priority to position co-ordinates owing to their permanent existence as time flows.

The uncertainty principle is another reason (which is closely intertwined with the above reason). Low uncertainty in momentum implies low uncertainty in energy (for free particles which are the building blocks of nature), which implies that the time interval for observation is arbitrarily high. This is obviously not very intuitive, the time intervals are very short in our daily experience. While this is reason is not really significant in our daily experiences, it is significant to a physicist's intuition.

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