{"id":{"repo_id":"bradford","oai_identifier":"oai:bradscholars.brad.ac.uk:10454/14522"},"canonical_url":"https://search.dev.ndltd.org/etd/bradford/oai:bradscholars.brad.ac.uk:10454/14522","repository":{"repo_id":"bradford","name":"University of Bradford","base_url":"https://bradscholars.brad.ac.uk/oai/request"},"display":{"title":"Bi-fractional transforms in phase space","abstract":"The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.","abstract_html":"The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.","abstract_has_math":false,"creators":["Agyo, Sanfo D."],"institution":"University of Bradford","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Vourdas, Apostolos"],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016","date_published":"2016","updated_at":"2026-07-24T01:14:32Z","subjects":["Phase space methods; Coherent states; Bi-fractional coherent states; Bi-fractional Wigner function; Bi-fractional P−function; Bi-fractional Q−function; Bi-fractional Moyal star product; Bi-fractional Berezin formalism"],"languages":["en"],"rights":["\"Creative
The University of Bradford theses are licenced under a Creative Commons Licence."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10454/14522","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Vourdas, Apostolos"]},{"key":"dc:creator","label":"Author","values":["Agyo, Sanfo D."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-01-16T12:46:00Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-01-16T12:46:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2016"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Faculty of Engineering and Informatics, School of Electrical Engineering and Computer Science"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Bradford"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Phase space methods; Coherent states; Bi-fractional coherent states; Bi-fractional Wigner function; Bi-fractional P−function; Bi-fractional Q−function; Bi-fractional Moyal star product; Bi-fractional Berezin formalism"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["\"Creative
The University of Bradford theses are licenced under a Creative Commons Licence."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10454/14522"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed."]},{"key":"dc:title","label":"Title","values":["Bi-fractional transforms in phase space"]}]}],"canonical_facts":{"dc:contributor.advisor":["Vourdas, Apostolos"],"dc:creator":["Agyo, Sanfo D."],"dc:date.accessioned":["2018-01-16T12:46:00Z"],"dc:date.available":["2018-01-16T12:46:00Z"],"dc:date.issued":["2016"],"dc:description.abstract":["The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed."],"dc:identifier.uri":["http://hdl.handle.net/10454/14522"],"dc:language.iso":["en"],"dc:publisher.department":["Faculty of Engineering and Informatics, School of Electrical Engineering and Computer Science"],"dc:publisher.institution":["University of Bradford"],"dc:rights":["\"Creative
The University of Bradford theses are licenced under a Creative Commons Licence."],"dc:subject":["Phase space methods; Coherent states; Bi-fractional coherent states; Bi-fractional Wigner function; Bi-fractional P−function; Bi-fractional Q−function; Bi-fractional Moyal star product; Bi-fractional Berezin formalism"],"dc:title":["Bi-fractional transforms in phase space"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-24T01:14:32Z"}