An application of hyperdifferential operators to holomorphic quantization

Ahmad Sumadi, Ahmad Hazazi

Spin quantization has always been an interesting intrinsic feature in quantum mechanics. This thesis discussed the holomorphic polarization method, motivated by geometric quantization, in a new formulation of canonical group quantization on noncotangent bundle phase space to produce spin quantization. The first part focuses on determining the one-dimensional complex projective space CP1 as a compact phase space and a special unitary group of degree two SU(2) as its canonical group that is not in the semi-direct product form. The emergence of the hidden discrete symmetry which is not deducible from the Lie algebraic structure of SU(2) indicates that it is the double-covering group. Thus its global structure is determined through the lifting SU(2) action on the fibre bundle over phase space. The second part focuses on the quantization process with the holomorphic wavefunction determined through the holomorphic local section of the fibre bundle and the natural polarization arises through the unitary irreducible representation of SU(2) that does not follow Mackey’s induced representation theory. From the representation operators, a set of spin angular momentum operators is generated as complex differential operators associated with a connection-type term l from action on holomorphic wavefunctions. Such representation operators’ matrix elements and characters are determined as Jacobi polynomials and its application in describing the single-qubit pure state is discussed. In conclusion, it is shown that the holomorphic polarization naturally emerged in the canonical group quantization on non-cotangent bundle phase space and has its application in quantum information theory which arises geometrically from the quantization problem.

Ahmad Hazazi Ahmad Sumadi, Nurisya Mohd Shah, Umair Abdul Halim +1 more

Abstract We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Möbius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description, offering a natural interpretation of standard quantum gates on the Riemann sphere that is examined through their Möbius action on holomorphic wavefunction. These wavefunctions emerge via a quantization process, with the Riemann sphere serving as the classical phase space of the qubit geometry. We quantize this space using Isham’s canonical group quantization with holomorphic polarization, yielding holomorphic wavefunctions and spin angular momentum operators that recover the standard SU (2) algebra with interesting geometric properties. Such properties reveal how geometric transformations induce quantum logic gates on the Riemann sphere, providing a novel perspective on quantum information processing. This result provides a new direction for exploring quantum computation through Isham’s canonical group quantization and its holomorphic polarization method.

Jorma Louko

A finite-dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic, and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. The relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar.