Non-perturbative renormalization flow in quantum field theory and statistical physics

Motivated by ideas about quantum gravity, a tremendous amount of effort over the past decade has gone into testing Lorentz invariance in various regimes. This review summarizes both the theoretical frameworks for tests of Lorentz invariance and experimental advances that have made new high precision tests possible. The current constraints on Lorentz violating effects from both terrestrial experiments and astrophysical observations are presented.

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Modern Tests of Lorentz Invariance

We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Solving the 3D Ising Model with the Conformal Bootstrap

Aspects of the functional renormalisation group

We established superfluidity in a two-state mixture of ultracold fermionic atoms with imbalanced state populations. This study relates to the long-standing debate about the nature of the superfluid state in Fermi systems. Indicators for superfluidity were condensates of fermion pairs and vortices in rotating clouds. For strong interactions, near a Feshbach resonance, superfluidity was observed for a broad range of population imbalances. We mapped out the superfluid regime as a function of interaction strength and population imbalance and characterized the quantum phase transition to the normal state, known as the Pauli limit of superfluidity.

Fermionic Superfluidity with Imbalanced Spin Populations

We study noncommutative geometry at the quantum mechanics level by means of a model where noncommutativity of both configuration and momentum spaces is considered. We analyze how this model affects the problem of the two-dimensional gravitational quantum well and use the latest experimental results for the two lowest energy states of neutrons in the Earth's gravitational field to establish an upper bound on the fundamental momentum scale introduced by noncommutativity, namely, $\sqrt{\ensuremath{\eta}}\ensuremath{\lesssim}1\text{ }\mathrm{meV}/c$, a value that can be improved in the future by up to $3$ orders of magnitude. We show that the configuration space noncommutativity has, in leading order, no effect on the problem. We also analyze some features introduced by the model, especially a correction to the presently accepted value of Planck's constant to $1$ part in ${10}^{24}$.

Noncommutative gravitational quantum well

We study the 3d Ising universality class using the functional renormalization group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and antisymmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross correlations of scaling exponents, their dependence on dimensionality, and the numerical convergence of the derivative expansion. Collecting all available data from functional renormalization group studies to date, we estimate that systematic errors are in good agreement with findings from Monte Carlo simulations, ϵ-expansion techniques, and resummed perturbation theory.

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Ising exponents from the functional renormalisation group

We discuss the ``constant speed of sound'' (CSS) parametrization of the equation of state of high-density matter and its application to the field correlator method (FCM) model of quark matter. We show how observational constraints on the maximum mass and typical radius of neutron stars are expressed as constraints on the CSS parameters. We find that the observation of a $2\text{ }\text{ }{\mathrm{M}}_{\ensuremath{\bigodot}}$ star already severely constrains the CSS parameters, and is particularly difficult to accommodate if the squared speed of sound in the high-density phase is assumed to be around $1/3$ or less. We show that the FCM equation of state can be accurately represented by the CSS parametrization, which assumes a sharp transition to a high-density phase with density-independent speed of sound. We display the mapping between the FCM and CSS parameters, and see that FCM only allows equations of state in a restricted subspace of the CSS parameters.

Constraining and applying a generic high-density equation of state

Spontaneous symmetry breaking in noncommutative cutoff $\ensuremath{\lambda}{\ensuremath{\Phi}}^{4}$ theory has been analyzed by using the formalism of the effective action for composite operators in the Hartree-Fock approximation. It turns out that there is no phase transition to a constant vacuum expectation of the field and the broken phase corresponds to a nonuniform background. By considering $〈\ensuremath{\varphi}(x)〉=A\mathrm{cos}(\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{x})$ the generated mass gap depends on the angles among the momenta $\stackrel{\ensuremath{\rightarrow}}{k}$ and $\stackrel{\ensuremath{\rightarrow}}{Q}$ and the noncommutativity parameter $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\theta}}.$ The order of the transition is not easily determinable in our approximation.

Dario Zappalà

Papers

Noncommutative gravitational quantum well

Ising exponents from the functional renormalisation group

Constraining and applying a generic high-density equation of state

Nonuniform symmetry breaking in noncommutative λ Φ 4 theory

Improving the renormalization group approach to the quantum-mechanical double well potential