The Planimeter and Contact Transformation: A Perfect Embodiment of the Weyl-Heisenberg Group and Canonical Transformation's Lost Twin Sister

          @misc{ pirsa_PIRSA:25100166,
            doi = {10.48660/25100166},
            url = {https://pirsa.org/25100166},
            author = {Jackson, Christopher},
            keywords = {Quantum Foundations},
            language = {en},
            title = {The Planimeter and Contact Transformation: A Perfect Embodiment of the Weyl-Heisenberg Group and Canonical Transformation{\textquoteright}s Lost Twin Sister},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {oct},
            note = {PIRSA:25100166 see, \url{https://pirsa.org}}
          }
          

Abstract

Once Heisenberg unlocked the Bohr frequency condition and the canonical commutation relation came out, Quantum Mechanics hit the ground running. In the rush of it all, it’s not clear to me who knew then (and who knows now) that the non-commutativity of phase space displacement is in fact an idea that has been around for at least as long as Jacobi, the father of canonical transformation theory. First conceived of in 1818 and then patented in 1854, the Amsler planimeter is a measuring instrument, known to even Maxwell, that in fact operates on exactly the same commutation relation, hiding in plain sight. Meanwhile, Lie's 1880 theory of transformation groups was also founded on exactly the same structure, what he called the contact element. Who knew? Why aren’t more physicists aware of this? Join me as we explore Poincare’s sudden death, the origins of the Gruppenpest, and Hilbert’s declining health when physicists like Wigner needed him more than ever.