The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Noncommutative Gravitation as a Gauge Theory of Twisted Poincaré Symmetry

Gravitaation kvanttiteorian muotoilu on ollut teoreettisten fyysikkojen tavoitteena kvanttimekaniikan synnystä lähtien. Kvanttimekaniikan soveltaminen korkean energian ilmiöihin yleisen suhteellisuusteorian viitekehyksessä johtaa aika-avaruuden koordinaattien operatiiviseen ei-kommutoivuuteen. Ei-kommutoivia aika-avaruuden geometrioita tavataan myös avointen säikeiden säieteorioiden tietyillä matalan energian rajoilla. Ei-kommutoivan aika-avaruuden gravitaatioteoria voisi olla yhteensopiva kvanttimekaniikan kanssa ja se voisi mahdollistaa erittäin lyhyiden etäisyyksien ja korkeiden energioiden prosessien ei-lokaaliksi uskotun fysiikan kuvauksen, sekä tuottaa yleisen suhteellisuusteorian kanssa yhtenevän teorian pitkillä etäisyyksillä. Tässä työssä tarkastelen gravitaatiota Poincarén symmetrian mittakenttäteoriana ja pyrin yleistämään tämän näkemyksen ei-kommutoiviin aika-avaruuksiin. Ensin esittelen Poincarén symmetrian keskeisen roolin relativistisessa fysiikassa ja sen kuinka klassinen gravitaatioteoria johdetaan Poincarén symmetrian mittakenttäteoriana kommutoivassa aika-avaruudessa. Jatkan esittelemällä ei-kommutoivan aika-avaruuden ja kvanttikenttäteorian muotoilun ei-kommutoivassa aika-avaruudessa. Mittasymmetrioiden lokaalin luonteen vuoksi tarkastelen huolellisesti mittakenttäteorioiden muotoilua ei-kommutoivassa aika-avaruudessa. Erityistä huomiota kiinnitetään näiden teorioiden vääristyneeseen Poincarén symmetriaan, joka on ei-kommutoivan aika-avaruuden omaama uudentyyppinen kvanttisymmetria. Seuraavaksi tarkastelen ei-kommutoivan gravitaatioteorian muotoilun ongelmia ja niihin kirjallisuudessa esitettyjä ratkaisuehdotuksia. Selitän kuinka kaikissa tähänastisissa lähestymistavoissa epäonnistutaan muotoilla kovarianssi yleisten koordinaattimunnosten suhteen, joka on yleisen suhteellisuusteorian kulmakivi. Lopuksi tutkin mahdollisuutta yleistää vääristynyt Poincarén symmetria lokaaliksi mittasymmetriaksi --- gravitaation ei-kommutoivan mittakenttäteorian saavuttamisen toivossa. Osoitan, että tällaista yleistystä ei voida saavuttaa vääristämällä Poincarén symmetriaa kovariantilla twist-elementillä. Näin ollen ei-kommutoivan gravitaation ja vääristyneen Poincarén symmetrian tutkimuksessa tulee jatkossa keskittyä muihin lähestymistapoihin.

Spin and relativistic motion of bound states

The wave functions of moving bound states may be expected to contract in the direction of motion, in analogy to a rigid rod in classical special relativity, when the constituents are at equal (ordinary) time. Indeed, the Lorentz contraction of wave functions is often appealed to in qualitative discussions. However, only few field theory studies exist of equal-time wave functions in motion. In this thesis I use the Bethe-Salpeter formalism to study the wave function of a weakly bound state such as a hydrogen atom or positronium in a general frame. The wave function of the e^-e^+ component of positronium indeed turns out to Lorentz contract both in 1+1 and in 3+1 dimensional quantum electrodynamics, whereas the next-to-leading e^-e^+\gamma Fock component of the 3+1 dimensional theory deviates from classical contraction. The second topic of this thesis concerns single spin asymmetries measured in scattering on polarized bound states. Such spin asymmetries have so far mainly been analyzed using the twist expansion of perturbative QCD. I note that QCD vacuum effects may give rise to a helicity flip in the soft rescattering of the struck quark, and that this would cause a nonvanishing spin asymmetry in \ell p^\uparrow -> \ell' + \pi + X in the Bjorken limit. An analogous asymmetry may arise in p p^\uparrow -> \pi + X from Pomeron-Odderon interference, if the Odderon has a helicity-flip coupling. Finally, I study the possibility that the large single spin asymmetry observed in p p^\uparrow -> \pi(x_F,k_\perp) + X when the pion carries a high momentum fraction x_F of the polarized proton momentum arises from coherent effects involving the entire polarized bound state.

Hyperfine effects in the nuclear magnetic resonance of paramagnetic molecules

Paramagnetic, or open-shell, systems are often encountered in the context of metalloproteins, and they are also an essential part of molecular magnets. Nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for chemical structure elucidation, but for paramagnetic molecules it is substantially more complicated than in the diamagnetic case. Before the present work, the theory of NMR of paramagnetic molecules was limited to spin-1/2 systems and it did not include relativistic corrections to the hyperfine effects. It also was not systematically expandable. --- The theory was first expanded by including hyperfine contributions up to the fourth power in the fine structure constant α. It was then reformulated and its scope widened to allow any spin state in any spatial symmetry. This involved including zero-field splitting effects. In both stages the theory was implemented into a separate analysis program. The different levels of theory were tested by demonstrative density functional calculations on molecules selected to showcase the relative strength of new NMR shielding terms. The theory was also tested in a joint experimental and computational effort to confirm assignment of 11 B signals. The new terms were found to be significant and comparable with the terms in the earlier levels of theory. The leading-order magnetic-field dependence of shielding in paramagnetic systems was formulated. The theory is now systematically expandable, allowing for higher-order field dependence and relativistic contributions. The prevailing experimental view of pseudocontact shift was found to be significantly incomplete, as it only includes specific geometric dependence, which is not present in most of the new terms introduced here. The computational uncertainty in density functional calculations of the Fermi contact hyperfine constant and zero-field splitting tensor sets a limit for quantitative prediction of paramagnetic shielding for now.