Real-time path integrals provide a natural formulation of quantum dynamics also in situations where a Hamiltonian description is unavailable or impractical, but their direct evaluation is obstructed by severe phase cancellation from the oscillatory weight.A guiding theme of this thesis is that, after complexifying the integration variables, real-time path integrals are naturally organized by complex saddle points.In favorable cases the original contour can be replaced by steepest-descent manifolds (Lefschetz thimbles) attached to the relevant saddles, providing a controlled saddle-based description of real-time observables.In Part I, we study quantum decoherence in a closed-system real-time path-integral formulation: the total system (subsystem plus environment) evolves unitarily, and the environment is traced out only at the final time to obtain the reduced density matrix.For the Caldeira-Leggett model with Gaussian initial conditions, the discretized effective action becomes a quadratic form, so that the reduced density matrix is characterized by a complex saddle point and the thimble integral reduces to a finite-dimensional Gaussian integral.The complex saddle is obtained by solving a large but sparse linear system, allowing an exact evaluation of the reduced density matrix without relying on the usual master-equation approximations.We then extend this methodology to a three-dimensional cubic lattice toy model by separating center-of-mass and relative coordinates, treating the center-of-mass mode as the subsystem and the relative modes as an effective environment.This enables controlled numerical studies of how the suppression of off-diagonal matrix elements depends on the coupling, inverse temperature, and system size, and allows us to extract the early-time scaling behavior of decoherence diagnostics from the complex-saddle evaluation.In Part II, we investigate spacetime dynamics in the Lorentzian IKKT matrix model.To define a Lorentzian formulation that is not continuously connected to the Euclidean model by the standard contour deformation, we include a Lorentz invariant mass term with coefficient γ, for which the classical equations admit real, smooth, expanding solutions when γ > 0. Since the Lorentz group is noncompact, the path integral can contain a divergent volume from the boost directions; we remove this volume nonperturbatively by implementing a boost "gauge-fixing" that selects the frame minimizing tr(A 0 ) 2 , leading to the condition tr(A 0 A i ) = 0 with the associated Faddeev-Popov determinant.We then apply the complex Langevin method to the resulting complex-weight matrix integral.Time and spatial observables are defined from matrix configurations, with holomorphic time ordering implemented by auxiliary variables, and spatial structure characterized through block observables such as the extent R 2 (t) and the moment-of-inertia tensor.In the bosonic model we find approximately real time and no indication of spontaneous breaking of the spatial symmetry in the i parameter range studied.Including fermions, we control the singular-drift problem induced by the Pfaffian by introducing a fermionic mass regulator m f and an anisotropic mass deformation, which enhances fermionic effects while maintaining numerical stability.In this setup we observe a real, expanding spatial extent and a dynamical dominance of three spatial directions at late times, consistent with a (3 + 1)-dimensional expanding spacetime.We finally discuss the intended limiting procedure-taking the large-N limit first and then approaching the undeformed theory-and comment on future directions, including a comparison with partially supersymmetric deformations that preserve 16 supercharges.
Abstract A real‐time finite‐temperature field theory on the Keldysh contour is used to describe the dynamics of an electron system within a random Hubbard model. The effective, i.e. quenched‐averaged, fermionic action under the functional integral is bosonized by introducing both a matrix field and an exchange field via Hubbard‐Stratonovich transformations. The resulting collective action permits the loop expansion, giving rise to a systematic approximation scheme. The saddle point and Gaussian fluctuations around it are calculated.
Large-
S and large-
N theories (spin value
S and spinor component number
N) are complementary, and sometimes conflicting, approaches to quantum magnetism. While large-
S spin-wave theory captures the correct semiclassical behavior, large-
N theories, on the other hand, emphasize the quantumness of spin fluctuations. In order to evaluate the possibility of the nontrivial recovery of the semiclassical magnetic excitations within a large-
N approach, we compute the large-
S limit of the dynamical spin structure factor of the triangular lattice Heisenberg antiferromagnet within a Schwinger boson spin representation. We demonstrate that, only after the incorporation of Gaussian (
1/N) corrections to the saddle-point ($N=\ensuremath{\infty}$) approximation, we are able to exactly reproduce the linear spin-wave theory results in the large-
S limit. The key observation is that the effect of
1/N corrections is to cancel out exactly the main contribution of the saddle-point solution, while the collective modes (magnons) consist of two spinon bound states arising from the poles of the random phase approximation propagator. This result implies that it is essential to consider the interaction of the spinons with the emergent gauge fields and that the magnon dispersion relation should not be identified with that of the saddle-point spinons.