Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori
what is meant by a position measurement on a non-commutative space. It was shown recently in
a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical
position measurements consistently on the non-commutative plane. To do this, it is necessary to
introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued
Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position
alone in non-commutative space cannot yield complete information about the quantum state
of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that
it requires knowledge of all orders of positional derivatives through the star product that is used
ubiquitously to map operator multiplication onto function multiplication in non-commutative
systems. It will be shown that there exist several equivalent local descriptions, which are arrived
at via the introduction of additional degrees of freedom. Consequently non-commutative quantum
mechanical position measurements necessarily confront us with some additional structure
which is necessary (in addition to position) to specify quantum states completely. The remainder
of the thesis, based in part on a recent publication (\Noncommutative quantum mechanics
{ a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev,
L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve
investigations into the physical interpretation of these additional degrees of freedom. For
one particular local formulation, the corresponding classical theory will be used to demonstrate
that the concept of extended, structured objects emerges quite naturally and unavoidably there.
This description will be shown to be equivalent to one describing a two-charge harmonically
interacting composite in a strong magnetic eld found by Susskind. It will be argued through
various applications that these notions also extend naturally to the quantum level, and constraints
will be shown to arise there. A further local formulation will be introduced, where the
natural interpretation is that of objects located at a point with a certain angular momentum
about that point. This again enforces the idea of particles that are not point-like. Both local
descriptions are convenient, in that they make explicit the additional structure which is encoded
more subtly in the non-local description. Lastly we shall argue that the additional degrees of
freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in
a gauge-invariant formulation of the theory. / AFRIKAANSE OPSOMMING: As gevolg van die minimum lengteskaal wat deur nie-kommuterende ko ordinate ge nduseer word
is dit nie a priori duidelik wat met 'n posisiemeting op 'n nie-kommutatiewe ruimte bedoel word
nie. Dit is onlangs in 'n artikel deur Scholtz et al. getoon dat dit wel op 'n nie-kommutatiewe
vlak moontlik is om die begrip van kwantummeganiese posisiemetings te herwin. Vir hierdie
doel benodig ons die konsep van swak (nie-projektiewe) metings wat in terme van 'n positief
operator-waardige maat geformuleer word. In hierdie tesis sal ons egter toon dat 'n meting
van slegs die posisie nie volledige inligting oor die kwantumtoestand van 'n deeltjie in 'n niekommutatiewe
ruimte lewer nie. Ons formalisme behels 'n nie-lokale beskrywing waarbinne kennis
oor alle ordes van posisieafgeleides in die sogenaamde sterproduk bevat word. Die sterproduk
is 'n welbekende konstruksie waardeur operatorvermenigvuldiging op funksievermenigvuldiging
afgebeeld kan word. Ons sal toon dat verskeie ekwivalente lokale beskrywings bestaan wat volg
uit die invoer van bykomende vryheidsgrade. Dit beteken dat nie-kommutatiewe posisiemetings
op 'n natuurlike wyse die nodigheid van bykomende strukture uitwys wat noodsaaklik is om
die kwantumtoestand van 'n sisteem volledig te beskryf. Die res van die tesis, wat gedeeltelik
op 'n onlangse publikasie (\Noncommutative quantum mechanics { a perspective on
structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G.
Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) gebaseer is, behels 'n ondersoek
na die siese interpretasie van hierdie bykomende strukture. Ons sal toon dat vir 'n spesi eke
lokale formulering die beeld van objekte met struktuur op 'n natuurlike wyse in die ooreenstemmende
klassieke teorie na vore kom. Hierdie beskrywing is inderdaad ekwivalent aan die van
Susskind wat twee gelaaide deeltjies, gekoppel deur 'n harmoniese interaksie, in 'n sterk magneetveld
behels. Met behulp van verskeie toepassings sal ons toon dat hierdie interpretasie op
'n natuurlike wyse na die kwantummeganiese konteks vertaal waar sekere dwangvoorwaardes na
vore kom. 'n Tweede lokale beskrywing in terme van objekte wat by 'n sekere punt met 'n vaste
hoekmomentum gelokaliseer is sal ook ondersoek word. Binne hierdie konteks sal ons weer deur
die begrip van addisionele struktuur gekonfronteer word. Beide lokale beskrywings is gerie
ik
omdat hulle hierdie bykomende strukture eksplisiet maak, terwyl dit in die nie-lokale beskrywing
deur die sterproduk versteek word. Laastens sal ons toon dat die bykomende vryheidsgrade in
lokale beskrywings ook as ykvryheidsgrade van 'n ykinvariante formulering van die teorie beskou
kan word.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/5298 |
| Date | 12 1900 |
| Creators | Rohwer, Christian M. |
| Contributors | Scholtz, F. G., University of Stellenbosch. Faculty of Science. Dept. of Physics. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 71 p. : ill. |
| Rights | University of Stellenbosch |
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