Quantum dynamical systems are described by means of non-commutative algebras of observables, by means of their time-evolution and by means of the expectation functionals that assign mean values to them.
This paper uses Schwinger's representation of quantum systems of a finite number of degrees of freedom to formulate discrete path integral representations of the dynamics. The representation starts with an observable with a finite number of distinct eigenvalues. Using that observable Schwinger constructs complementary pairs of finite dimensional unitary operators that are finite dimensional analogs of the irreducible Weyl algebra. These can be decomposed into products of irreducible sub-algebras that reduce to q-bit gates for the case of 2N degrees of freedom. In the limit of a large number of discrete degrees of freedom these representations can be used to model quantum systems with continuous degrees of freedom. This limit recovers the continuum Weyl algebra. Path integrals can be formulated in the discrete case. They have the advantage that the number of paths for a finite number of time slices is finite. In this work the path integral is interpreted as the expectation value of a potential functional with respect to a complex probability distribution on the space of paths. An application to scattering from a short-range potential is given. Multi-resolution wavelet bases are used exactly represent a local ϕ4(x) quantum field theory as a theory with an infinite number of discrete local modes. The discrete path integral is illustrated by computing time evolution when this theory is truncated to two coupled modes.
Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutator (OTOC), the time dependence of which reveals salient aspects of the system's dynamics. Here we study the spatially local spin-correlation function, i.e., the expectation value of the spin commutator and the corresponding OTOC of Dirac-Weyl systems in one, two, and three spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. In principle, the problem features two energy scales, the chemical potential and the high-energy cutoff. We find that only the latter is dominant, therefore the time evolution is separated into only two different regions. The spin-correlation function grows linearly with time initially and decays as ${t}^{\ensuremath{-}2}$ for late times. The OTOC reveals a universal t2 initial growth from both the commutator and the variance while its late time decay ${t}^{\ensuremath{-}2}$ originates from the variance of the commutator. This late time decay is identified as a characteristic signature of Dirac-Weyl fermions. These features remain present also at finite temperatures. Our results indicate that Dirac-Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e., interaction and disorder, are analyzed.
This dissertation addresses the problem of constructing true degrees of freedom in quantum mechanical systems with constraints. The method developed relies on assuming the behavior of the system of interest is ``nearly classical', or semiclassical, in a precise quantitative sense. The approximation is formulated as a perturbative hierarchy of the expectation values of quantum observables and their non-linear combinations, such as spreads and correlations. We formulate the constraint conditions and additional quantum gauge transformations that arise directly on the aforementioned quantities. We specialize this framework to several situations of particular interest. The first situation considered is the case of a constraint that commutes with all quantum observables, which is appropriate for quantization of a Lie algebra with a single Casimir polynomial. Through explicit order-by order counting argue that the true degrees of freedom are captured correctly at each order of the approximation. The rest of the models considered are motivated by the canonical approach to quantizing general relativity. The homogeneous sector of general relativity splits into classes of models, according to topology, which are described by a finite number of degrees of freedom and can therefore be quantized as quantum mechanical systems, making our construction directly applicable. In the Hamiltonian formulation the dynamics of these systems is governed by the Hamiltonian constraint, which, in the quantum theory, gives rise to several conceptual and technical issues collectively known as the Problem of Time. One of the aspects of this problem that does not possess a general solution is the dynamical interpretation of the theory. We use our construction, truncated at the leading order in quantum corrections, together with the intuition gained from dealing with constrained systems in classical mechanics, to define local notion of dynamics relative to a chosen configuration clock variable. We consider a class of models, where the chosen clock in not globally valid, eventually leading to singular dynamics. Within these models we construct an explicit transformation between dynamical evolution relative to two distinct clocks, thus providing a consistent local dynamical interpretation of the true degrees of freedom of the quantum theory.