Wavelet-based field theory and cross-scale correlations

We consider application of continuous wavelet transform to the field theory of the fields phi
_alpha
(x) depending on both the position x and the resolution a. The correlation functions langphi
_alpha
(x)phi
_alpha'
(x')rang for such fields may be finite at x = x'. In case of the gauge fields the nonlocality of the wavelet transform makes the theory more complicated, but the gauge symmetry for each scale is observed. This approach provides a possibility to estimate the causal effect of the large-scale wave envelope to the small scale entities. The same approach can be applied to the Langevin-type stochastic differential equations, treated within the framework of quantum field theory, time series analysis etc.