Relativistic quantum field theory of stochastic dynamics in the Hilbert space
Abstract
In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into 1 +3 dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a randomnumber action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentzinvariant equation of motion for the state vector or density matrix. We derive the path integral formula of S matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic ϕ^{4} theory. The Lorentz invariance of the random S matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both S matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional ϕ^{4} theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory 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 infinitetemperature 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.
 Publication:

Physical Review D
 Pub Date:
 June 2022
 DOI:
 10.1103/PhysRevD.105.115037
 arXiv:
 arXiv:2112.13996
 Bibcode:
 2022PhRvD.105k5037W
 Keywords:

 Quantum Physics;
 Condensed Matter  Quantum Gases;
 High Energy Physics  Theory
 EPrint:
 32 pages, 12 figures