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We study the influence of geometry of quantum systems underlying space of states on its quantum many-body dynamics. We observe an interplay between dynamical and topological ingredients of quantum non-equilibrium dynamics revealed by the…
We analyze two models of random geometries~: planar hyper-cubic random surfaces and four dimensional simplicial quantum gravity. We show for the hyper-cubic random surface model that a geometrical constraint does not change the critical…
Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical…
We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic…
The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion…
We establish the phase diagram of three--dimensional quantum gravity coupled to Ising matter. We find that in the negative curvature phase of the quantum gravity there is no disordered phase for ferromagnetic Ising matter because the…
In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sj\"{o}qvist et al. We develop a holonomy theory for the latter which we also…
Time-dependent $\mathcal{PT}$-symmetric quantum mechanics is featured by a varying inner-product metric and has stimulated a number of interesting studies beyond conventional quantum mechanics. In this paper, we explore geometric aspects of…
We argue that theories of quantum gravity constructed with the help of (Causal) Dynamical Triangulations have given us the most informative, quantitative models to date of quantum spacetime. Most importantly, these are derived dynamically…
The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and…
We present evidence that a nonperturbative model of quantum gravity defined via Euclidean dynamical triangulations contains a region in parameter space with an extended 4-dimensional geometry when a non-trivial measure term is included in…
We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an $R^2$-term. The phase diagram as a function of the…
Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining…
The model of Lorentzian three-dimensional dynamical triangulations provides a non-perturbative definition of three-dimensional quantum gravity. The theory has two phases: a weak-coupling phase with quantum fluctuations around a…
This thesis is broadly split into two parts. In the first part, simple state sum models for minimally coupled fermion and scalar fields are constructed on a $1$-manifold. The models are independent of the triangulation and give the same…
The quantum mechanics is proved to admit no hidden-variable in 1960s, which means the quantum systems are contextual. Revealing the mathematical structure of quantum mechanics is a significant task. We develop the approach of partial…
One of several possibilities to construct a quantum theory of gravity is employing the Feynman path integral. This approach is plagued by some problems: the integration measure is not uniquely defined, the Einstein-Hilbert action unbounded,…
A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum…
We explore an extended coupling constant space of 4d regularized Euclidean quantum gravity, defined via the formalism of dynamical triangulations. We add a measure term which can also serve as a generalized higher curvature term and…
The relativistic quantum phase space (QPS) formalism extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, thereby providing a unified setting where the uncertainty principle and…