Relativistic quantum field theory of stochastic dynamics in the Hilbert space

Abstract: We develop an action formulation of stochastic dynamics in the Hilbert space. By generalizing the Wiener process into 1+3-dimensional spacetime, we define a Lorentz-invariant random field. By coupling the random to quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of $S$-matrix and the diagrammatic rules for both the stochastic free field theory and stochastic $\phi^4$-theory. The Lorentz invariance of the random $S$-matrix is strictly proved. We then develop a diagrammatic technique for calculating the density matrix. Without interaction, we obtain the exact $S$-matrix and density matrix. With interaction, we prove a simple relation between the density matrices of stochastic and conventional $\phi^4$-theory. Our formalism leads to an ultraviolet divergence which has the similar origin as that in QFT. The divergence is canceled by renormalizing the coupling strength to random field. We prove that the stochastic QFT is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.