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 ${\varphi }^4$-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 ${\varphi }^4$-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.

33 pages, 12 figures

References in corpus (15)

• Quantum trajectories and open many-body quantum systems
• Testing the limits of quantum mechanical superpositions
• Lower and Upper Bounds on CSL Parameters from Latent Image Formation and IGM Heating
• Underground test of gravity-related wave function collapse
• Collapse models with non-white noises
• A Relativistic Dynamical Collapse Model
• Testing the foundations of quantum physics in space Interferometric and non-interferometric tests with Large Particles
• A proposal for the experimental detection of CSL induced random walk
• The Point Processes of the GRW Theory of Wave Function Collapse
• Gravity and the Collapse of the Wave Function: a Probe into Diósi-Penrose model
• Relativistically Invariant Markovian Dynamical Collapse Theories Must Employ Nonstandard Degrees of Freedom
• Collapse models with non-white noises II: particle-density coupled noises
• Constraining CSL strength parameter $\lambda$ from standard cosmology and spectral distortions of CMBR
• Stress Tensor for Quantized Random Field and Wave Function Collapse
• Feasibility considerations for free-fall tests of gravitational decoherence

Cited by in corpus (4)

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• Random non-Hermitian Hamiltonian framework for symmetry breaking dynamics
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