Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms of contact topology. Our main result is a proof of local, formal gauge equivalence for a broad class of quantum dynamical systems—just as classical dynamics depends on choices of clocks, local quantum dynamics can be reduced to a problem of studying gauge transformations. We further show how to write quantum correlators in terms of parallel transport and in turn matrix elements for Hilbert bundle gauge transformations, and give the path integral formulation of these results. Finally, we show how to relate topology of the underlying contact manifold to boundary conditions for quantum wave functions.
Abstract The problem with prequantization is that the Hilbert space it constructs is too large to represent the phase space of a physically reasonable quantum system. In the first example in §8.4 (a cotangent bundle), the Hilbert space contained wave functions that depended on momentum as well as position; and in the third (a Coadjoint orbit), the group representation was reducible. We have already touched on a possible solution: we should admit only sections of B that are parallel along a polarization. We shall now look in detail at how this works. It should be stressed, however, that the physical justification is not based on general mathematical results (such as the Borel-Weil theorem), but on the way in which the construction works in particular examples. It generalizes and unifies a number of quantization techniques that, in the past, have not appeared to have any obvious connection with each other and that have sometimes seemed overspecialized with applications only to particular physical systems. Also it should be noted that not all syrnplectic manifolds admit polarizations (Gotay 1987).
The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between L 2 ( R ) and the Bargmann–Hilbert space. We pursue this line of thinking one step further and discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. Two programmes have placed projective geometry at the centre of quantum mechanics, arriving from opposite directions. The Brody–Hughston geometric quantum mechanics programme identifies the space of pure states with the complex projective space
CPn, equipped with the Fubini-Study symplectic form. Our companion programme identifies the 3-qubit discrete phase space with the finite projective space
PG(5,2) over
GF(2), equipped with a binary symplectic form. We show that both are specialisations of a single abstract structure: a projective space over a field
F carrying a non-degenerate alternating bilinear form and a central extension of its symplectic group. The central result concerns the Wigner functor
W, sending each quantum system to its Wigner function. Numerical experiment shows that
W is covariant under the real symplectic subgroup of the Clifford group (CNOT-type gates) but fails for phase gates (CS, S). The obstruction is the metaplectic character
χ:Sp(2n,GF(2))→U(1) satisfying a group 2-cocycle condition. The precise unification is: Brody–Hughston's prequantum
U(1) line bundle on
CPn (the Maslov bundle, required for geometric quantisation) and the metaplectic character
χ on
PG(2n−1,2) (required for discrete Wigner covariance) are the same construction over different base fields — a central extension of the symplectic group by a circle. Three additional connections are established. (i) The Fano orbit decomposition of
PG(5,2) — seven orbits of eight points, one per Fano line — is a purely discrete phenomenon with no continuous analogue in
CPn; magic valence is intrinsically a
GF(2) structure invisible to characteristic-zero geometry. (ii) The 6j symbols governing both programmes —
SU(2) spin networks in the Brody–Hughston setting,
Z23 Fano symbols in ours — are evaluations of the same abstract Racah coefficient for different groups; the Origami ISA is the spin-
21 discrete limit of Penrose's spin-network geometry; Racah (1942), Mac Lane (1963), Penrose (1971), and the present work all evaluate the same Pentagon identity for different groups. (iii) The finite Klein correspondence (
PG(5,2) points
↔ PG(3,2) lines) suggests a discrete twistor interpretation of the magic-valence structure, directly connecting to Hughston's twistor programme. The unifying category
MetaProjSymp(F) is defined, the Wigner functor is shown to be natural on it, and the bridge theorem is proved: Kibble's geometrisation of quantum mechanics, the Brody–Hughston programme, and the discrete Fano programme are all full subcategories of the same abstract construction, distinguished only by the choice of field.