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My research in stochastic quantum field theory explores whether stochastic dynamics can provide a fundamental description of quantum mechanics and address longstanding problems in quantum foundations. In particular, I develop relativistic stochastic and non-Hermitian quantum-field-theoretic models in which spacetime noise can induce wave-function localization. A central goal is to understand whether such mechanisms can lead to a consistent relativistic theory of spontaneous collapse while recovering the Born rule and the standard quantum description in appropriate limits.
Statistical symmetry breaking and emergent colored noise in a stochastic scalar-doublet field theory
We investigate a relativistic stochastic field theory in which a complex scalar doublet is coupled to a complex white-noise source. The action preserves Lorentz and U(1) × SU(2) symmetries at the statistical level, whereas the corresponding Euler–Lagrange equations exhibit symmetry breaking along individual stochastic realizations. Within a gauge-field-free sector introduced to obtain analytical solutions, we show that the scalar doublet undergoes a noise-driven random walk in field space, leading to a finite, time-dependent ensemble average of its magnitude. As an illustrative application, we further investigate the coupling of the stochastic scalar field to fermions through a Yukawa interaction. The scalar-field solution naturally separates into a tail component, which contributes as an effective mass-like term, and a light-cone component, which acts as a colored-noise source that induces a spatially correlated stochastic phase in the fermion wave function. The statistical properties and correlation length of this emergent colored noise are derived analytically within the adopted approximations. The present work provides an exploratory study of statistical symmetry breaking and emergent colored-noise dynamics in a relativistic stochastic scalar-doublet field theory.
Relativistic model of spontaneous wave-function localization induced by nonHermitian colored noise
We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schrödinger dynamics for the state vector. In this framework, Lorentz and spacetime translation symmetries are preserved only in a statistical sense: the probability distribution of the action remains invariant under these transformations. As a result, the theory describes ensembles of quantum-state trajectories whose probability distributions remain invariant under changes of reference frame. As a concrete example, we augment the Dirac action with a purely imaginary term coupling the fermion density operator to a universal colored noise. This noise is constructed by solving the d’Alembert equation with white noise as its source, using a generalized stochastic calculus in 1+3 dimensions. We demonstrate that the colored noise drives stochastic localization of wave packets and derive the localization length analytically. Remarkably, the localization length decreases as the size of the observable universe increases. Our model thus provides a potential framework for relativistic spontaneous wave-function collapse. While establishing consistency with Born’s law remains an open challenge, the present work constitutes a step toward embedding collapse models into a Lorentz-invariant quantum field theory.
Schrodinger-picture formulation of a scalar quantum field driven by white noise
We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler–Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.
Random non-Hermitian Hamiltonian framework for symmetry-breaking dynamics
We propose random non-Hermitian Hamiltonians to model the generic stochastic nonlinear dynamics of a quantum state in Hilbert space. Our approach features an underlying linearity in the dynamical equations, ensuring the applicability of techniques used for solving linear systems. Additionally, it offers the advantage of easily incorporating statistical symmetry, a generalization of explicit symmetry to stochastic processes. To demonstrate the utility of our approach, we apply it to describe real-time dynamics, starting from an initial symmetry-preserving state and evolving into a randomly distributed, symmetry-breaking final state. Our model serves as a quantum framework for the transition process, from disordered states to ordered ones, where symmetry is spontaneously broken.
Random non-Hermitian action theory for stochastic quantum dynamics: from canonical to path integral quantization
We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.
Relativistic quantum field theory of stochastic dynamics in the Hilbert space
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 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, 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 phi-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 phi-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 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.
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