Local symmetries, anomalies and constrains in Burgers Turbulence

We study stochastic Burgers turbulence without pressure. We first show that the variational derivative of the Burgers equation is dependent on the velocity field, suggesting the existence of an anomaly. The anomaly is created by an operator that is non-self-adjoint. To calculate it correctly, we need to find its square. There are similarities with conformal and chiral two-dimensional field theories, but causality is the key that makes the difference. We calculate the determinant and use two local symmetries to verify the result. By requiring the disappearance of the anomaly, the velocity field is constrained and the theory becomes anomaly-free. These symmetries obey Kolmogorov's second law of self-similarity. One can choose an anomaly-free theory, a partially broken theory, or a fully broken theory by choosing the constraint appropriately. There is an analogy to gauge fixing or vacuum selection which define the local configuration.