Canonical group quantization on non-cotangent bundle phase space and its application in quantum information theory
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 quantiza...
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| Format: | Thesis |
| Language: | English |
| Published: |
2024
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| Online Access: | http://psasir.upm.edu.my/id/eprint/118371/ http://psasir.upm.edu.my/id/eprint/118371/1/118371.pdf |
| Summary: | 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. |
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