Random Walks and the Correlation Length Critical Exponent in Scalar Quantum Field Theory

Details Random Walks and the Correlation Length Critical Exponent in Scalar Quantum Field Theory Random Walks and the Correlation Length Critical Exponent in Scalar Quantum Field Theory

1992-04-08

arxiv.org/abs/hep-lat/9202002v1

Physics

High Energy Physics

High Energy Physics - Lattice

Scientific paper

The distance scale for a quantum field theory is the correlation length $\xi$, which diverges with exponent $\nu$ as the bare mass approaches a critical value. If $t=m^{2}-m_{c}^{2}$, then $\xi=m_{P}^{-1} \sim t^{-\nu}$ as $t \to 0$. The two-point function of a scalar field has a random walk representation. The walk takes place in a background of fluctuations (closed walks) of the field itself. We describe the connection between properties of the walk and of the two-point function. Using the known behavior of the two point function, we deduce that the dimension of the walk is $d_{w}=\phi / \nu$ and that there is a singular relation between $t$ and the energy per unit length of the walk $\theta \sim t^{\phi}$ that is due to the singular behavior of the background at $t=0$. ($\phi$ is a computable crossover exponent.)

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