Talks

Continuous-variable SICPOVMS seem unlikely to exist, for a variety of reasons. But that doesn't rule out the possibility of other 2-designs for the continuous-variable Hilbert space L2(R). In particular, it would be nice if the coherent states -- which form a rather nice 1-design -- could be generalized in some way to get a 2-design comprising *Gaussian* states. So the question is: "Can we build a 2-design out of Gaussian states?". The answer is "No, but in a very surprising way!" Like coherent states, Gaussian states have a natural transitive symmetry group. For coherent states, it's the Heisenberg group. For Gaussian states, it's the affine symplectic group -- the Heisenberg group plus squeezings and rotations. And this group acts irreducibly on the symmetric subspace of L2(R) x L2(R)... which, by Schur's Lemma, implies that the Gaussian states *should* be a 2-design. Yet a very simple explicit calculation shows that they are not! The resolution is fascinating -- it turns out that the "symplectic twirl" involves an integral that does not quite converge, and this provides a loophole out of Schur's Lemma. So, in the end, we: (1) Show that Gaussian 2-designs do not exist, (2) Demonstrate a major stumbling block to *any* symplectic-covariant 2-designs for L2(R), (3) Gain a pretty complete understanding of *one* of the [formerly] mysterious discrepancies between discrete and continuous Hilbert spaces.

If we imagine that the universe is truly eternal, special challenges arise for attempts to solve cosmological fine-tuning problems, especially the low entropy of the early universe. If the space of states is finite, the universe should spend most of its time near equilibrium. If the space of states is infinite, it becomes difficult to understand why our universe was in a particular low-entropy state. I will discuss approaches to addressing this problem in a model-independent fashion.

"A positive cosmological constant allows arbitrarily many different quantum states, but apparently only if there can be big bangs and/or big crunches. Without any big bang or big crunch, the entropy may be limited by the Gibbons-Hawking entropy of pure deSitter, and the matter entropy might even more limited by a value roughly the three-fourths power of the Gibbons-Hawking entropy. A classical analogue of an upper limit on the entropy is the finite canonical measure for nonsingular cosmologies. Inflation by itself does not explain the arrow of time and seems to require that not only a small region have suitable initial conditions. A special quantum state, such as the Symmetric-Bounce one, with a suitable measure, such as volume averaging and Agnesi weighting, appears possible to explain the arrow of time and other observations, such as high order that would not be expected for Boltzmann brain observations, distant stars that would not be expected if we were fluctuations of empty de Sitter spacetime, and positive cosmological constant that appears not to dominate many other measures."

Closed systems never evolve to lower entropy states -- except when they do, which is if one waits a time that is exponential in the entropy change. Thus macroscopic decreases in entropy are 'never' observed. Yet in cosmology there are eternal systems in which downward entropy fluctuations of any magnitude eventually happen. What is the nature of such fluctuations? I will argue that these can be understood in a simple and general way that sheds light on our understanding of various interesting cosmological processes such as bubble nucleation, black/white hole formation, eternal stochastic inflation, Boltzmann brain formation, and other processes.

Our universe may have formed via bubble nucleation in an eternally-inflating background. Furthermore, the background may have a compact dimension---the modulus of which tunnels out of a metastable minimum during bubble nucleation---which subsequently grows to become one of our three large spatial dimensions. We discuss some potential observational signatures of this scenario.

We propose a novel cosmological scenario, in which standard inflation is replaced by an expanding phase with a drastic violation of the Null Energy Condition (NEC): \dot H >> H^2. The model is based on the recently introduced Galileon theories, which allow NEC violating solutions without instabilities. The unperturbed solution describes a Universe that is asymptotically Minkowski in the past, expands with increasing energy density until it exits the regime of validity of the effective field theory and reheats. This solution is a dynamical attractor and the Universe is driven to it, even if it is initially contracting. Adiabatic perturbations turn out to be cosmologically irrelevant. The model, however, suggests a new way to produce a scale invariant spectrum of isocurvature perturbations, which can be later converted to adiabatic: the Galileon is forced by symmetry to couple to the other fields as a dilaton; the effective metric it yields on the NEC violating solution is that of de Sitter space, so that all light scalars will automatically acquire a nearly scale-invariant spectrum of perturbations.

I will review some recent work on infrared issues for scalar fields in exact and quasi de Sitter space. Renewed interest in this topic has been driven by the observational potential for a more accurate determination of statistics of the primordial curvature perturbations, especially non-Gaussianity. Interestingly, the resulting questions are not only relevant for mapping inflationary models to observation but also link directly to more fundamental questions about the initial state, eternal inflation, and the long time dynamics of interacting quantum fields in curved space. Infrared questions provide a precisely calculable way to put pressure on inflation as a rigorous and consistent framework, ready to confront future observations.

We explore simple but novel solutions of general relativity which, classically, approximate cosmologies cycling through an infinite set of ``bounces." These solutions require curvature K=+1, and are supported by a negative cosmological term and matter with -1 < w < -1/3. They can be studied within the regime of validity of general relativity. We argue that quantum mechanically, particle production leads eventually to a departure from the regime of validity of semiclassical general relativity, likely yielding a singular crunch.

I provide a mathematical model of holographic cosmology whose coarse grained description is that of a homogeneous isotropic, flat universe, which makes a transitions from an FRW to an eternal de Sitter regime.&nbsp;Based on this model, I suggest some heuristic ideas which explain the low initial entropy of the universe and may provide a description of an inflationary era with small fluctuations.

I will argue that anthropic reasoning is unnecessary or misleading when the universe/multiverse is small enough that another observer with exactly your memories is unlikely to exist. Instead, one can evaluate theories or make predictions in the standard Bayesian way, based on the conditional probability of something unknown given all that you do know. Things are not so clear when the universe is large enough that all competing theories predict that an observer with your exact memories exists with probability close to one. I will discuss issues that arise in such large or infinite universes, such as "Boltzmann brains", and will argue that pending better understanding of these issues one should be hesitant to draw conclusions different from those that would apply to a small iverse.