# Theoretical Physics

**4 December 2013**

**Time:**15:00 to 16:00

**Location:**EC Stoner SR 8.60

Stefan Weigert (York)

Triples of Pairwise Canonical Observables

Given a quantum particle on a line, its momentum and

position are described by a pair of Hermitean operators

(p, q) which satisfy the canonical commutation relation

(CCR). A third observable r exists which satisfies CCRs

with both position and momentum. The triple (p, q, r) is not

only unique (up to unitary equivalence) but also maximal

in the sense that no four equi-commutant observables

exist. Being invariant under cyclic permutations, the

triple (p, q, r) endows the Heisenberg algebra with a

threefold, largely unexplored symmetry.

I will discuss consequences of the equi-commutant triple

(p,q,r) and its exponentiated cousin, called a Weyl triple.

For example, a generalisation of Heisenberg's uncertainty

relation involving three standard deviations is proposed,

and the non-trivial minimizing state can be found (joint

work with S Kechrimparis). The triples are also relevant

for a general theory of mutually unbiased bases.

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