Discreteness and integrality in Conformal Field Theory

Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists $$ t\ge \frac{c}{12} $$ t ≥ c 12 is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions $$ \Delta >\frac{c-1}{12} $$ Δ > c − 1 12 is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

Scientific Realism and Constructive Empiricism

Two conceptions of the contemporary philosophy of science are taken under consideration: scientific realism and constructive empiricism. Scientific realism presupposes 1) the conception of truth as the correspondence of knowledge to reality, 2) the real existence of entities postulated by a theory. The constructive empiricism puts forward the idea of empirical adequacy: science aims to give us the theories which are empirically adequate and acceptance of the theory involves as belief only that it is empirically adequate. To compare methodological resources of these two positions in the philosophy of science the problem of the interpretation of quantum mechanics is involved. As a methodological realization of scientific realism the ensemble interpretation of quantum mechanics is taken under consideration. K.Popper’s version

On the predictability of outliers in ensemble forecasts

In numerical weather prediction, ensembles are used to retrieve probabilistic forecasts of future weather conditions. We consider events where the verification is smaller than the smallest, or larger than the largest ensemble member of a scalar ensemble forecast. These events are called outliers. In a statistically consistent K-member ensemble, outliers should occur with a base rate of 2/(K+1). In operational ensembles this base rate tends to be higher. We study the predictability of outlier events in terms of the Brier Skill Score and find that forecast probabilities can be calculated which are more skillful than the unconditional base rate. This is shown analytically for statistically consistent ensembles. Using logistic regression, forecast probabilities for outlier events in an operational ensemble are calculated. These probabilities exhibit positive skill which is quantitatively similar to the analytical results. Possible causes of these results as well as their consequences for ensemble interpretation are discussed.

COSMOLOGICAL PARTICLE CREATION: FLUCTUATIONS AND AN ENSEMBLE PICTURE

We point out that in the context of quantum fields in time-dependent classical backgrounds, the number of created particles with a given momentum largely deviates about its mean value. Since the corresponding Fourier modes are nonlocal, this deviation shows that the expectation value of the number operator can make sense only in an ensemble of space–times. Using a complete orthonormal family of localized wave packets, we show how an ensemble interpretation can be given to cosmological particle creation in local terms. The reheating process following inflation is re-examined in the light of this construction.