A Numerical and Analytic Investigation of Probability Distributions of the Stress Tensor in Conformal Field Theory

Repeated measurements of the smeared (averaged) stress-energy tensor of a quantum field in the vacuum state produce random values described by a probability distribution. This distribution is related non-trivially to the choice of smearing function. For unitary, positive-energy conformal field theories in 2-dimensions there are two methods available to determine the distribution. One of which uses conformal welding to determine the characteristic function while the other finds the moment generating function. The aim of the thesis is to numerically implement the conformal welding method and perform comparisons with these results to those generated by the method of moments. We employ the method of conformal welding to numerically analyse the effect of different smearing functions on the probability distribution in a 2-dimensional conformal quantum field theory. Several numerical methods were developed to ameliorate the difficulties posed by each method and improve the quality of the data generated. Both methods are compared to demonstrate their individual benefits and pitfalls when calculating probability distributions. An interesting test case is a smearing function given as the product of a Gaussian function and a Lorentzian function. This is analysed numerically by both the welding method and the method of moments. Furthermore some asymptotic results are obtained from the moments method. Importantly, the Gauss-Lorentz test function is the first known example of a positive test function in which the probability distribution does not belong to the family of shifted Gamma distributions. We also use the numerical welding method to analyse the effect of varying parameters of a plateau test function on the associated probability distribution. This analysis gives numerical evidence that the probability distribution associated to a plateau test function averaging the stress-energy tensor depends on both the switch on parameters and the length of the plateau used.