Narrow Gaussian wave-packets of stationary WKB mode functions

Abstract: As a self-contained part in the project about ridge-lines of wave-packets in quantum cosmology, we present the study of wave-packets composed of WKB mode functions.

Starting from a stationary wave equation that incorporates the Wheeler–DeWitt equation in quantum cosmology, we construct its WKB solutions that contain good quantum numbers. We then build a wave-packet by smearing out the quantum numbers with a Gaussian amplitude, and show that the result 'peaks around' the corresponding classical trajectory.

Implications to the singularity avoidance are to be discussed.

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Testing the volume average regularization of the Wheeler–DeWitt equation

Abstract: One of the most important canonical approaches to quantum gravity is through the Wheeler–DeWitt equation. However, the theory, being a second order differential equation of a functional, produces delta functions and derivatives of delta functions which lead to infinities. Thus a regularization of these terms is needed. In this thesis we will discuss the approach of Justin C. Feng for the regularization of the Wheeler–DeWitt equation, the volume average regularization (arXiv:1802.08576).

For this we will build the ground work to arrive at the Wheeler–DeWitt equation as well as go through some mathematical foundations of functional calculus.

The main idea is to perform an integral average of the second functional derivative part of the Wheeler–DeWitt equation over a finite volume.

This is justified when we consider quantum general relativity as a low-energy effective field theory of the full theory of quantum gravity.

This regularization will lead to an approximate solution of the Wheeler–DeWitt equation for the low-curvature, long-distance limit.

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Towards an emerged classical universe from wave-packets in quantum cosmology

Abstract: I present a progress report of the central part of my thesis project, which is about ridge-lines of wave-packets in quantum cosmology.

In the Wheeler–DeWitt approach of quantum cosmology, it has long been argued that behind a classical universe is a sharply peaked wave-packet, in analogy to wave-packets in traditional quantum mechanics. However, there has so far been no mathematical description of such a “peak”.

I argue that such a description is necessary. Then I derive two distinct such descriptions, namely the contour and stream approaches, and discuss their pros and cons. Possible ways to solve their problems is presented in the end.

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Quantum Gravitational Effects for Non-minimal Density Fields

Abstract: A common rescaling performed in inflationary models is of the inflaton scalar field $\phi$ to a scalar density field $\chi$ through $\phi=a^{-6\xi}\chi$. We show that for the $\chi$-field, a Mukhanov–Sasaki-like variable can be achieved which preserves the gauge invariance of the formalism. We discuss the semi-classical Born–Oppenheimer type of approximation to show that quantum-gravitational-corrections to the power spectrum of the inflationary scalar perturbations is invariant under such a field redefinition. We conclude that the $\phi$-field and the $\chi$-field can be used interchangeably.

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Sugawara model with canonical spin and canonical energy-momentum currents

Abstract: We set up the Sugawara construction for the canonical currents of spin and energy-momentum. The canonical spin currents turn out to be linear and the canonical energy-momentum currents bilinear in the vector-axial vector currents. One can interpret our ansatz using two approaches. The first being the standard approach which expresses the Lorentz (spin) connection in terms of the Yang–Mills gauge connection. The other method is the canonical approach leading to a teleparallel model of gravity at the level of strong-interactions. Furthermore, we calculate the anomaly terms in the spin current commutators for spinor fields. These anomalies arise solely due to the interaction of chiral Weyl fermions with $U(1)$ gauge fields on a background Riemann–Cartan spacetime. This results in the breakdown of the Lorentz algebra at the quantum level; the quantum corrections being proportional to the fine structure constant $\alpha$ and the quantized product of electric and magnetic charges. Also, as another application of current algebra we discuss how second derivatives of delta functions arising from the geometry part could potentially be cancelled by the Schwinger terms present in the ETCRs of the symmetric energy-momentum currents.

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Lorentzian Quintessential Inflation

Abstract: From the assumption that the slow roll parameter ε has a Lorentzian form as a function of the e-folds number N, a successful model of a quintessential inflation is obtained. The form corresponds to the vacuum energy both in the inflationary and in the dark energy epochs. The form satisfies the condition to climb from small values of to 1 at the end of the inflationary epoch. At the late universe ε becomes small again and this leads to the Dark Energy epoch. The observables that the models predicts fits with the latest Planck data: r ~ 10^-3 , n_s ≈ 0.965. Naturally a large dimensionless factor that exponentially amplifies the inflationary scale and exponentially suppresses the dark energy scale appears, producing a sort of cosmological see saw mechanism. We find the corresponding scalar Quintessential Inflationary potential with two flat regions - one inflationary and one as a dark energy with slow roll behavior. (arXiv:2004.00339)

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A collapse model for dichotomous measurements in quantum mechanics: reanimating Schrödinger's vision

Abstract: The motivation for this theory comes from two modes of thinking. First, taking seriously the old idea by the founding fathers of quantum mechanics (QM) that the randomness of outcomes in QM is due to the influence of the measurement apparatus. The second mode is the realization that Born's probabilistic interpretation of QM, which destroyed Schrödinger's beautiful ideas, is avoidable if Schrödinger's equation is modified suitably, at the same time restoring determinism and ontological transparency.

If it were true that the irregular and unknowable influence of the measurement apparatus causes the collapse of the wave vector, one should be able to predict particular outcomes of measurements if the 'state' of the 'big system' were known exactly, where the 'big system' consists of the physical system under investigation and the measurement apparatus which is used to probe the former. From this point of view, randomness in quantum mechanical measurements literally arises from the practical impossibility to know exactly the state of macroscopic systems.

The theory starts by modifying the usual Schrödinger dynamics by adding a non-linear term in the usual evolution which depends on a 'new' field (we call it the 'A-field') which describes which quantities the big system 'is looking at'. The dynamics is such that the 'discontinuous' collapse is included in the deterministic dynamics, and hence the collapse is here a continuous (albeit perhaps very fast) process. The theory proceeds by defining an equivalence relation on the space of wave vectors, which is supposed to express the inability to exactly know the state of the big system, the latter being a wave vector as in ordinary quantum mechanics. Furthermore, it defines a canonical probability measure on the equivalence classes, which is supposed to quantify the randomness associated with the inability to know the state of the big system. The theory and in particular the deterministic dynamics is designed in such a way that the statistical predictions of ordinary quantum mechanics are reproduced exactly in the case of dichotomous measurements. The symmetry which maps wave vectors to statistically equivalent wave vectors (belonging to the same 'macro-state') is very similar to a local U(1)-symmetry, 'smearing out' phases.

The theory is entirely unique and does not depend on any dimension-less parameters, although the speaker admits that its definition is not particularly 'beautiful' or 'simple' from a mathematical point of view. Also, the theory is fully consistent with special relativity (but not with general relativity) in its present form.

Open questions are whether the theory also works for more general measurements (non-dichotomous) and how the A-field is related to the Hamiltonian and state of the big system. Also, it is an open problem whether the continuous collapse is 'sufficiently discontinuous' so that no unbridgeable deviations to standard QM arise (quantum jumps, e.g.). Also, the theory imposes a very difficult problem in random matrix theory which one would have to solve in order to simulate the dynamics efficiently on a computer.

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Curvature in Discrete Quantum Gravity

Abstract: Causal Dynamical Triangulation (CDT) is a non-perturbative theory of quantum gravity. It is based on a lattice discretization of general relativity, where the background metric should emerge dynamically. This makes it difficult to construct suitable observables in the model. A recent addition to the list of available CDT observables is the Quantum Ricci Curvature (QRC), which allows one to measure effective average curvature on piecewise flat geometries.

In this seminar, I will go over the basics of CDT, and subsequently define the Quantum Ricci Curvature. Furthermore, I will show how it behaves on general triangulated model spaces, and highlight some results in the quantum gravity context. In addition to this, I will explain how to implement the continuum analog of QRC on spaces with isolated singular points of curvature, and how this can provide additional reference points to help us interpret the measurement results of the QRC in a non-perturbative quantum setting.

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Exteriors to bouncing collapse models

Abstract: We construct a large class of spacetimes that are smoothly matched to homogeneous, spherically symmetric clouds of matter. The evolution of the clouds is left arbitrary to allow for the incorporation of modifications by quantum effects, in particular leading to bounces. We further discuss two simple yet illustrative examples of these spacetimes, both in general terms and for a specific form of the bounce, with a focus on horizon behavior and relevant timescales.

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The redefined no-boundary wavefunction (a useful precursor to inflation?)

Abstract: In this talk, I will present the work of Dr. Jean-Luc Lehners and his group (Feldbrugge et al., 2017, arxiv:1703.02076 [hep-th]) on Lorentzian path integrals in quantum cosmology. The use of Picard-Lefschetz theory to deal with the oscillatory integrals that we encounter in Lorentzian quantum cosmology, as introduced by them will be briefly reviewed. Subsequent work by Dr. Lehners' group is an effort to redefine the no-boundary proposal in case of cosmological mini-superspace models (Di Tucci et al., 2019, arxiv:1911.06701 [hep-th]) where finding the appropriate boundary conditions which lead to the correct saddle-points is emphasized. This work will also be demonstrated in the talk.

I will further discuss how the wavefunction obtained using the above approach can be used to constrain parameters of inflationary models along the lines of the analysis performed in Calcagni et al., 2014 (arxiv:1405.6541 [gr-qc]). Exploring whether the no-boundary proposal is a useful precursor to inflation is the central idea of my master thesis.

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