Fractal Dimensions and Scaling Laws in the Interstellar Medium: a new Field Theory approach

We develop a field theoretical approach to the cold interstellar medium (ISM). We show that a non-relativistic self-gravitating gas in thermal equilibrium with variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field \phi({\vec x}) with exponential self interaction.We analyze this field theory perturbatively and non-perturbatively through the renormalization group approach.We show scaling behaviour(critical) for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation Delta M(R) \sim R^{d_H} for the mass on a region of size R, and Delta v \sim R^q for the velocity dispersion where q = (d_H -1)/2. For the density-density correlations we find a power-law behaviour for large distances \sim|{\vec r_1} -{\vec r_2}|^{2 d_H -6}. The fractal dimension d_H turns to be related with the critical exponent nu of the correlation length by d_H = 1/nu. The renormalization group approach for a single component scalar field in three dimensions states that the long- distance critical behaviour is governed by the (non-perturbative) Ising fixed point. The corresponding values of the scaling exponents are nu = 0.631..., d_H = 1.585... and q = 0.293.... Mean field theory yields for the scaling exponents nu =1/2, d_H = 2 and q = 1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4 \leq d_H \leq 2, 0.3 \leq q \leq 0.6.As typical in critical phenomena, the scaling behaviour and critical exponents of the ISM can be obtained without dwelling into the dynamical (time-dependent) behaviour. The relevant r\^ole of self- gravity is stressed by the authors in a Letter to Nature, September 5, 1996.