A brief history of the Heisenberg Uncertainty Principle will be given, with an eye towards introducing the (Weyl)-Heisenberg group. This group (which can be viewed as a subgroup of the group of invertible 3 x 3 real matrices, has many interesting applications, and a few will be discussed.
The Heisenberg Uncertainty Principle and the Heisenberg group
Oct. 27, 2020 1pm (Zoom)
H. Peter Gumm (Philipps University Marburg, Germany)
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Motivating our investigation, we begin by demonstrating how all sorts of transition systems studied in Computer Science - automata (deterministic, nondeterministic, probabilistic), Kripke structures, neighbourhood systems (including topological spaces) and many more - can be naturally modeled as instances of the concept of "universal coalgebra". Each of the mentioned cases requires an appropriate Set-functor replacing what we know in universal algebra as "signature".
Various preservation properties of the signature functor F determine the structure theory of the class of all F-coalgebras. In our talk we concentrate on functors parameterized by some universal algebras, be it a complete lattice or a commutative monoid, and mainly on the functor given by constructing for a set X the free -Algebra .
preserves preimages if and only if each -term which is weakly independent at a variable position x is (strongly) independent of x, (in short: implies its own 'derivative' $\Sigma'$). For every permutable variety V(\Sigma) the free algebra functor weakly preserves kernel pairs. For the converse we exhibit a syntactic condition stating that an equation p(x,x,y) = q(x,y,y) holds if and only both p(x,y,z) and q(x,y,z) arise as substitution instances of a common term s(x,y,z,u). We give a geometrical interpretation of this fact and use it to show that if is n-permutable then weakly preserves kernel pairs iff is Mal'cev.