Abstract:
Here we shall find the green's function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory.

Abstract:
This Chapter provides an up to date, pedagogical review of some of the most relevant advances in loop quantum cosmology. We review the quantization of homogeneous cosmological models, their singularity resolution and the formulation of effective equations that incorporate the main quantum corrections to the dynamics. We also summarize the theory of quantized metric perturbations propagating in those quantum backgrounds. Finally, we describe how this framework can be applied to obtain a self-consistent extension of the inflationary scenario to incorporate quantum aspects of gravity, and to explore possible phenomenological consequences.

Abstract:
Bone tissue adapts to its mechanical loading environment. We review here the accelerometric measurements with special emphasis on osteogenic exercise. The accelerometric method offers a unique opportunity to assess the intensity of mechanical loadings. We present methods to interpret accelerometric data, reducing it to the daily distributions of magnitude, slope, area, and energy of signal. These features represent the intensity level of physical activities, and were associated with the changes in bone density, bone geometry, physical performance, and metabolism in healthy premenopausal women. Bone adaptations presented a dose- and intensity dependent relationship with impact loading. Changes in hip were threshold dependent, indicating the importance of high-impacts exceeding acceleration of 4 g or slope of 100 g/s as an osteogenic stimulus. The number of impacts needed was 60/day. We also present the daily impact score to describe the osteogenic potential of daily mechanical loading with a single score. The methodology presented here can be used to study musculoskeletal adaptation to exercise in other target groups as well.

Abstract:
We examine the possibility that circles in the cosmic microwave background could be formed by the interaction of a gravitational wave pulse emitted in some pre-big-bang phase of the universe with the last scattering surface. We derive the expected size distribution of such circles, as well as their typical ring width and (for concentric circles) angular separation. We apply these results in particular to conformal cyclic cosmology, ekpyrotic cosmology as well as loop quantum cosmology with and without inflation in order to determine how the predicted geometric properties of these circles would vary from one model to the other, and thus, if detected, could allow us to differentiate between various pre-big-bang cosmological models. We also obtain a relation between the angular ring width and the angular radius of such circles that can be used in order to determine whether or not circles observed in the cosmic microwave background are due to energetic pre-big-bang events.

Abstract:
Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity.

Abstract:
From an observational perspective cosmology is today in excellent shape - advances in instrumentation and data processing have enabled us to study the universe in detail back to when the first galaxies formed, map the fluctuations in the cosmic microwave background which provide a measure of the overall geometry, and reconstruct the thermal history reliably back to at least the primordial nucleosynthesis era. However recent deep studies of the Hubble expansion rate have suggested that the universe is accelerating, driven by some form of `dark' (vacuum) energy. If true, this implies a new energy scale in Nature of order 0.001 eV, well below any known scale of fundamental physics. This has refocussed attention on the notorious cosmological constant problem at the interface of general relativity and quantum field theory. It is possible that the resolution of this situation will require fundamental modifications to our ideas about gravity.

Abstract:
Nuclear observables such as binding energies and cross sections can be directly measured. Other physically useful quantities, such as spectroscopic factors, are related to measured quantities by a convolution whose decomposition is not unique. Can a framework for these nuclear structure `non-observables' be formulated systematically so that they can be extracted from experiment with known uncertainties and calculated with consistent theory? Parton distribution functions in hadrons serve as an illustrative example of how this can be done. A systematic framework is also needed to address questions of interpretation, such as whether short-range correlations are important for nuclear structure.

Abstract:
This paper is devoted to the study and interpretation of the spectral function $\mathbf{A}(\omega, T)$ of the Keldysh nonequilibrium Green's function. The spatial diagonal of the spectral function is often interpreted as a time-dependent local density of states. We show that this object can take negative values implying that a simple probability interpretation as a time-dependent density of states is not possible. The same issue also occurs for the Wigner function $P(x,p)$ where it is solved by taking the uncertainty principle into account. We follow the same path and incorporate the time-energy uncertainty relation to define a convoluted spectral function that allows for a probability interpretation. The usefulness of this quantity as a interpretative tool is demonstrated by visualizing the charge dynamics in a quantum dot coupled to superconducting leads.

Abstract:
We comment on structural properties of the algebras $\mathfrak{A}_{LQG/LQC}$ underlying loop quantum gravity and loop quantum cosmology, especially the representation theory, relating the appearance of the (dynamically induced) superselection structure ($\theta$-sectors) in loop quantum cosmology to recently proposed representations with non-degenerate background geometries in loop quantum gravity with Abelian structure group. To this end, we review and employ the concept of extending a given (observable) algebra with possibly non-trivial centre to a (charged) field algebra with (global) gauge group.We also interpret the results in terms of the geometry of the structure group G. Furthermore, we analyze the Koslowski-Sahlmann representations with non-degenerate background in the case of a non-Abelian structure group. We find that these representations can be interpreted from two different, though related, points view: Either, the standard algebras of loop quantum gravity need to be extended by a (possibly) central term, or the elementary flux vector fields need to acquire a shift related to the (classical) background to make these representations well-defined. Both perspectives are linked by the fact that the background shift is not an automorphism of the algebras, but rather an affine transformation. Finally, we show how similar algebraic mechanisms, which are used to explain the breaking of chiral symmetry and the occurrence of $\theta$-vacua in quantum field theory, extend to loop quantum gravity. Thus, opening a path for the discussion of these questions in loop quantum gravity.

Abstract:
We analyze cosmology assuming unitary quantum mechanics, using a tripartite partition into system, observer and environment degrees of freedom. This generalizes the second law of thermodynamics to "The system's entropy can't decrease unless it interacts with the observer, and it can't increase unless it interacts with the environment." The former follows from the quantum Bayes Theorem we derive. We show that because of the long-range entanglement created by cosmological inflation, the cosmic entropy decreases exponentially rather than linearly with the number of bits of information observed, so that a given observer can reduce entropy by much more than the amount of information her brain can store. Indeed, we argue that as long as inflation has occurred in a non-negligible fraction of the volume, almost all sentient observers will find themselves in a post-inflationary low-entropy Hubble volume, and we humans have no reason to be surprised that we do so as well, which solves the so-called inflationary entropy problem. An arguably worse problem for unitary cosmology involves gamma-ray-burst constraints on the "Big Snap", a fourth cosmic doomsday scenario alongside the "Big Crunch", "Big Chill" and "Big Rip", where an increasingly granular nature of expanding space modifies our life-supporting laws of physics. Our tripartite framework also clarifies when it is valid to make the popular quantum gravity approximation that the Einstein tensor equals the quantum expectation value of the stress-energy tensor, and how problems with recent attempts to explain dark energy as gravitational backreaction from super-horizon scale fluctuations can be understood as a failure of this approximation.