McCrossen, Lawrence (1983)
Spin and gauge fields on a lattice.
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Formulating quantum field theories on a lattice provides one way of controlling the divergences that appear in the calculation of physical quantities for these theories. At the same time the formal analogy with statistical mechanics may be exploited, in particular a continuum field theory exists only at a critical point of a statistical mechanical system. Chapter one begins with a review of Wilson's proposal for placing gauge (local) invariant theories on a lattice, whereby quark confinement emerges as a natural consequence. The possibilities for phase transitions and spontaneous symmetry breaking in global and locally invariant theories are discussed. Methods of calculation in lattice theories are introduced, in particular the mean field theory and Monte Carlo methods of integration. In Chapter two those methods are applied to a globally invariant spin theory - the 0(n) generalised Heisenberg model. Details of high and low temperature expansions are also given. Although much is already known about such systems we are able here to check the usefulness of our methods. In addition, the Heisenberg model is to be found at one of the limits of the system in Chapter four. Chapter three is concerned with locally invariant pure gauge theories in four dimensions. Monte Carlo simulations are compared for the abelian U(l) theory and the non-abelian SU(3). In the U(l) case we use a finite scaling argument to suggest a second-order phase transition separating 'Maxwell' and confining regions. In SU(3) the situation is unclear, but is not inconsistent with confinement for all values of the coupling. In chapter four, a two coupling constant model is defined of U(l) gauge fields coupled to n-component complex matter (spin) fields. The action is then invariant to global U(n) transformations as well as local U(l). The model interpolates between pure U(l) gauge theory, a lattice version of the gauge invariant CP n-1 model, and the 0(2n) Heisenberg model. The phase diagram is mapped out in the two coupling constant space and 'masses' are calculated in the various regions.
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Deposited by David Morgan (UBYL020) on
in Royal Holloway Research Online.Last modified on 01-Feb-2017
Digitised in partnership with ProQuest, 2015-2016.
Institution: University of London, Royal Holloway College (United Kingdom).