Question

What's the relationship between uncertainty principle and symplectic groups? Do the symplectic groups mathematically capture anything fundamental about uncertainty principle?

This post imported from StackExchange Physics at 2014-08-12 09:34 (UCT), posted by SE-user Problemania

Comments

Why do you believe there is a relationship?

This post imported from StackExchange Physics at 2014-08-12 09:34 (UCT), posted by SE-user ACuriousMind

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You're not thinking of the Heisenberg group and its formulation in symplectic spaces, are you? See here and here. Symplectic groups are very different from the HG as the latter is nilpotent.

This post imported from StackExchange Physics at 2014-08-12 09:34 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance

Answer 1

Yes, of course, symplectic groups describe generalized situations that reveal the uncertainty principle.

The reason for the relationship is that the symplectic groups are defined by preserving an antisymmetric bilinear invariant, $$ M A M^T = A $$ where $M$ is a matrix included into the symplectic group is the equation holds and $A$ is a non-singular antisymmetric matrix.

Where does the uncertainty principle enter? It enters because $A$ may be understood to be the commutator (or Poisson bracket) of the basic coordinates $x_i,p_i$ on the phase space. If we summarize $N$ coordinates $x_i$ and $N$ coordinates $p_i$ into a $2N$-dimensional space with coordinates $q_m$, their commutators are $$ [q_m,q_n] = A_{mn} $$ with an antisymmetric matrix $A$. Consequently, the symplectic transformations may be defined as the group of all linear transformations mixing $x_i,p_i$, the coordinates of the phase space, that preserve the commutator i.e. all the uncertainty relations between the coordinates $q_m$.

Curved, nonlinear generalizations of these spaces are known as "symplectic manifolds" and nonlinear generalizations of the symplectic transformations above are known as "canonical transformations".

I think it doesn't make sense to talk about this relationship too much beyond the comments above because the relationship is in no way "equivalence". One may say lots of things about related concepts but they're in no way a canonical answer to your question – they don't follow just from the idea of the "relationship" itself. I just wanted to make sure that a relationship between mathematical structures on both sides, especially the antisymmetric matrix, certainly exists.

This post imported from StackExchange Physics at 2014-08-12 09:34 (UCT), posted by SE-user Luboš Motl