On the geometry of the spin-statistics connection in quantum mechanics

The Spin-Statistics theorem states that the statistics of a system of identical particles is determined by their spin: Particles of integer spin are Bosons (i.e. obey Bose-Einstein statistics), whereas particles of half-integer spin are Fermions (i.e. obey Fermi-Dirac statistics). Since the original proof by Fierz and Pauli, it has been known that the connection between Spin and Statistics follows from the general principles of relativistic Quantum Field Theory. In spite of this, there are different approaches to Spin-Statistics and it is not clear whether the theorem holds under assumptions that are different, and even less restrictive, than the usual ones (e.g. Lorentz-covariance). Additionally, in Quantum Mechanics there is a deep relation between indistinguishabilty and the geometry of the configuration space. This is clearly illustrated by Gibbs' paradox. Therefore, for many years efforts have been made in order to find a geometric proof of the connection between Spin and Statistics. Recently, various proposals have been put forward, in which an attempt is made to derive the Spin-Statistics connection from assumptions different from the ones used in the relativistic, quantum field theoretic proofs. Among these, there is the one due to Berry and Robbins (BR), based on the postulation of a certain single-valuedness condition, that has caused a renewed interest in the problem. In the present thesis, we consider the problem of indistinguishability in Quantum Mechanics from a geometric-algebraic point of view. An approach is developed to study configuration spaces Q having a finite fundamental group, that allows us to describe different geometric structures of Q in terms of spaces of functions on the universal cover of Q. In particular, it is shown that the space of complex continuous functions over the universal cover of Q admits a decomposition into C(Q)-submodules, labelled by the irreducible representations of the fundamental group of Q, that can be interpreted as the spaces of sections of certain flat vector bundles over Q. With this technique, various results pertaining to the problem of quantum indistinguishability are reproduced in a clear and systematic way. Our method is also used in order to give a global formulation of the BR construction. As a result of this analysis, it is found that the single-valuedness condition of BR is inconsistent. Additionally, a proposal aiming at establishing the Fermi-Bose alternative, within our approach, is made.

Yoruba nationalist movements, ethnic politics and violence : a creation from historical consciousness and socio-political space in South-western Nigeria /// Arbeitspapiere / Institut für Ethnologie und Afrikastudien ; 105

Since 1900, the Yoruba people of South-western Nigeria have put its ethnic history at work in the construction of its identity in Nigeria. The exercise resulted in the creation of ethno-nationalist movements and the practice of ethnic politics, often expressed through violent attacks on the Nigerian State and some ethnic groups in Nigeria. Relying on mythological attachment to its traditions and subjective creation of cultural pride, the people created a sense of history that established a common interest among different Yoruba sub-groups in form of pan-Yoruba interest which forms the basis for the people’s imagination of nation. Through this, historical consciousness and socio-political space in which Yoruba people are located acted as instrumental forces employed by Yoruba political elites, both at colonial and post-colonial periods to demand for increasing access to political and economic resources in Nigeria. In form of nationalism, nationalist movements and ethnic politics continued in South-western Nigeria since 1900, yet without resulting to actual creation of an independent Yoruba State up to 2009. Through ethnographic data, the part played by history, tradition and modernity is examined in this paper. While it is concluded that ethno-nationalist movement and ethnic politics in Yoruba society are constructive agenda dated back to pre-colonial period, it continues to transform both in structure and function. Thus, Yoruba ethno-nationalist movement and ethnic politics is ambiguous, dynamic and complex, to the extent that it remains a challenge to State actions in Nigeria.