Algebraic Modelling of Quantum Mechanical Equations in the Finite- and Infinite-Dimensional Hilbert Spaces

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Algebraic Modelling of Quantum Mechanical Equations in the Finite- and Infinite-Dimensional Hilbert Spaces Algebraic modelling of quantum mechanical equations in the finite- and infinite-dimensional Hilbert spaces Megill, Norman Dwight Doctoral thesis / Disertacija 2011 Degree Grantor / Ustanova koja je dodijelila akademski / stručni stupanj: University of Zagreb, Faculty of Science / Sveučilište u Zagrebu, Prirodoslovno-matematički fakultet Permanent link / Trajna poveznica: https://urn.nsk.hr/urn:nbn:hr:217:788633 Rights / Prava: In copyright Download date / Datum preuzimanja: 2021-09-26 Repository / Repozitorij: Repository of Faculty of Science - University of Zagreb PHYSICS DEPARTMENT OF THE FACULTY OF SCIENCE Norman Dwight Megill ALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES DOCTORAL THESIS Zagreb, 2011 FIZICKIˇ ODSJEK PRIRODOSLOVNO-MATEMATICKOGˇ FAKULTETA Norman Dwight Megill ALGEBARSKO MODELIRANJE KVANTNO-MEHANICKIHˇ JEDNADŽBI U KONACNOˇ I BESKONACNOˇ DIMENZIONALNIM HILBERTOVIM PROSTORIMA DOKTORSKI RAD Zagreb, 2011 PHYSICS DEPARTMENT OF THE FACULTY OF SCIENCE Norman Dwight Megill ALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES DOCTORAL THESIS Supervisor: Prof. Mladen Paviciˇ c´ Zagreb, 2011 FIZICKIˇ ODSJEK PRIRODOSLOVNO-MATEMATICKOGˇ FAKULTETA Norman Dwight Megill ALGEBARSKO MODELIRANJE KVANTNO-MEHANICKIHˇ JEDNADŽBI U KONACNOˇ I BESKONACNOˇ DIMENZIONALNIM HILBERTOVIM PROSTORIMA DOKTORSKI RAD Mentor: Prof. Mladen Paviciˇ c´ Zagreb, 2011 Committees This thesis, entitled Algebraic Modelling of Quantum Mechanical Equations in the Finite- and Infinite-Dimensional Hilbert Spaces and written by Norman Dwight Megill, has been approved for the Department of Physics of the Faculty of Science of the University of Zagreb by the following committee: Prof. Hrvoje Šikic´ (Department of Mathematics, Faculty of Science, University of Zagreb), Prof. Mladen Paviciˇ c´ (Chair of Physics, Faculty of Civil Engineering, University of Zagreb), Prof. Barry C. Sanders (Institute for Quantum Information Science, Uni- versity of Calgary, Canada), Prof. Aleksa Bjeliš (Department of Physics, Faculty of Science, University of Zagreb), and Prof. Mirko Primc (Department of Mathematics, Faculty of Science, University of Zagreb). The final copy of this thesis has been examined by the signatories, and they found that both the content and the form meet acceptable presentation standards for a scholarly work in the discipline of mathematical quantum physics. The Council of the Department of Physics accepted their opinion on November 8, 2011. The committee for the defense, approved by the Council of the Department of Physics, consisted of Prof. Hrvoje Šikic´ (Department of Mathematics, Faculty of Science, University of Zagreb), Prof. Mladen Paviciˇ c´ (Chair of Physics, Faculty of Civil Engineering, University of Zagreb), Prof. Slobodan Bosanac (Division of Physical Chemistry, Institute Ruder Boškovic,´ Zagreb), Prof. Denis Sunko (Department of Physics, Faculty of Science, University of Zagreb), and Prof. Mirko Primc (Department of Mathematics, Faculty of Science, University of Zagreb). The defense took place on December 14, 2011. iv Acknowledgments I am greatly indebted to Prof. Paviciˇ c,´ without whose inspiration and guidance this thesis would not be possible. I also wish to thank Susan Cass for her encouragement and patience. Na´ci iglu u plastu sijena. v Abstract The Hilbert space of quantum mechanics has a dual representation in lattice theory, called the Hilbert lattice. In addition to offering the potential for new insights, the lattice-theoretical ap- proach may be computationally efficient for certain kinds of quantum mechanics problems, particularly if, in the future, we are able to exploit what may be a “natural” fit with quantum computation. The equations that hold in the Hilbert space lattice representation are not com- pletely known and are poorly understood, although much progress has been made in the last several years. This work contributes to the development of these equations, with special atten- tion to the so-called generalized orthoarguesian equations. Many new results that do not appear in the literature are given, along with their detailed proofs. In addition, possible approaches for work towards answering some remaining open questions are discussed. Keywords Hilbert space, Hilbert lattice, orthoarguesian property, strong state, quantum logic, quantum computation, Godowski equations, orthomodular lattice vi Prošireni sažetak Pozadina. Stanja u kvantnoj mehanici mogu se modelirati kao vektori u Hilbertovom prostoru. Skup zatvorenih podprostora konacnoˇ ili beskonacnoˇ dimenzionalnog Hilbertova prostora clanˇ je klase cesticaˇ koje se zovu Hilbertove rešetke (Hilbert lattice, HL). (Rešetka je djelomicnoˇ ureden skup u kojemu svaka dva clanaˇ imaju najmanju gornju i najvecu´ donju granicu. Ovaj i svi drugi ovdje korišteni termini formalno su definirani u disertaciji). Obratno, moguce´ je izvesti Hilbertov prostor polazeci´ od HL. Zbog ovog dvostrukog odnosa razumijevanje svo- jstava HL-a može dovesti do boljeg razumijevanja svojstava Hilbertova prostora. Osim što nudi mogucnost´ novih uvida, teorijski pristup rešetki može biti racunskiˇ efikasan za neke vrste kvantno mehanickihˇ problema, narocitoˇ ako, u buducnosti,´ budemo mogli koristiti ono što bi mogao biti “prirodno” odgovarajuci´ dio za kvantno racunanje.ˇ Jednadžbe koje u Hilbertovu prostoru podržavaju prikaz rešetke nisu u potpunosti poznate i nedovoljno ih se razumije, pre- mda je tijekom nekoliko posljednjih godina ucinjenˇ veliki napredak. Familija svih HL-ova definirna je (aksiomatizirana) skupom uvjeta prvoga reda koji ukljucu-ˇ ju (egzistencijalne) kvantifikatore. Za odredeni broj uvjeta nultog reda odnosno jednadžbi bez kvantifikatora, može se pokazati da vrijede u svakom HL-u. Najociglednijiˇ od njih su jednadžbe koje definiraju bilo koju rešetku (te posebno svaku orto-rešetku), koje su dio skupa aksioma. Godine 1937. Husimi je otkrio ortomodularni zakon (koji je sada takoder dio HL definicije), koji je bio intenzivno obraden u literaturi o klasi ortomodularnih rešetki (OML), kojih je HL podklasa. Za razliku od uvjeta prvoga reda, jednadžbe nam omogucavaju´ da direktno baratamo ob- jektima u podprostoru Hilbertova prostora i dobijemo vrstu racunskeˇ “algebre” za rad s tim objektima. Jednadžbe su posebno prikladne za efikasne racunskeˇ tehnike. Klasa rešetki defini- rana samo jednadžbama, kao što je OML, naziva se jednadžbenim varijetetom. Klasa HL-a sama po sebi nije jednadžbeni varijetet (za što je dokaz naveden u disertaciji). Usprkos tome, klasa rešetki koju je generirao (tj. koja zadovoljava) skup jednadžbi koje vrijede u HL-u može se proucavatiˇ odvojeno kao superklasa od HL-a i svi rezultati su automatski primjenjivi na HL kao poseban slucaj.ˇ Jedan važan neriješen problem je pronaci´ sve moguce´ jednadžbene zakone koji vrijede u HL-u. S jacimˇ jednadžbama moguce´ je proucitiˇ više karakteristika HL-a korištenjem samo jednadžbenih varijeteta. Ovdje je kratki pregled napretka postignutog do sada. Trebalo je nekoliko desetljeca´ nakon Husimieva OML zakona da se pronade drugi, a to je bio ortoarguesiev zakon kojega je otkrio Alan Day 1975. vii 1981. g. Godowski je otkrio nezavisnu beskonacnuˇ familiju HL jednadžbi, baziranu na kvantnim probabilistickimˇ stanjima. Te jednadžbe nazivano “Godowski-eve jednadžbe” ili n- Gos. Godine 1986. Mayet je našao algoritam za generiranje veceg´ skupa jednadžbi (nazvan MGEs), koji je takoder utemeljen na stanjima, cijiˇ su podskup bile Godowski-eve jednadžbe), premda se na pocetkuˇ nije znalo da li je ikoja od njih nezavisna od n-Go jednadžbi. Od 2006. do 2009., Megill i Paviciˇ c´ pronašli su nove jednadžbe utemeljene na Mayet-ovu algoritmu za koje se pokazalo da se ne daju izvesti iz Godowski-evih. U 2000. g. Megill i Paviciˇ c´ otkrili su novu familiju jednadžbi koje vrijede u HL-u—general- izirane ortoarguesieve jednadžbe, nazvane nOA zakoni (n 3). OA zakon Alan-a Day-a je ≥ drugi clanˇ ove serije, zakon 4OA, a Greechie/Godowski-eve jednadžbe izvedene iz OA su ek- vivalentne prvome clanu,ˇ zakonu 3OA. Dok je otvoren problem da li se obitelj nOA sastoji od uzastopno jacihˇ jednadžbi, mi smo dokazali (obimnim kompjuterskim traženjem protuprimjera) da su zakoni 3OA, 4OA, 5OA i 6OA uzastopno jaci.ˇ 2011. g. uspjeli smo dokazati da je zakon 7OA jaciˇ od zakona 6OA. Godine 1995. Maria Solèr je dokazala da je dodavanjem dva dodatna HL aksioma, moguce´ iz HL-a izvesti Hilbertov prostor cijeˇ je polje jedno od “klasicnih”ˇ polja kvantne mehanike (realno, kompleksno ili kvaternionsko). Solèr-in teorem upotpunio je dugo neostvareni cilj da se Hilbertov prostor kvantne mehanike izvede iz nekog HL-a pokazujuci´ da su oni dualni. Godine 2006. Mayet je opisao novu obitelj jednadžbi, nazvanu E jednadžbe, utemeljenu na jednom svojstvu Hilbertova prostora koje se naziva vektorski-valuiranim stanjem. Važno je reci´ da te jednadžbe ne vrijede za svako moguce´ polje koje se može dovesti u vezu s Hilbertovim prostorom vec´ samo za ona polja s karakteristikom 0, koja ukljucujuˇ klasicnaˇ polja kvantne mehanike. To nam daje jednadžbeni uvjet koji je u stvari ovisan o (te ih tako djelomicnoˇ i opisuje) Solèr-inim dodatnim uvjetima (prvoga reda) dodanim HL-u. Ovdje cemo´ ukratko sumirati kljucneˇ teme pokrivene u disertaciji koje se odnose na traženje novih HL jednadžbi. Ortomodularne rešetke. Veliki broj uvjeta koji vrijede u OML-u prikupljen je u poglavlju 3, za kasniju uporabu. Oni koji se nisu ranije pojavili u literaturi popraceni´ su detaljnim
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