Session G15: Quantum Field Theory in a Curved Spacetime

Live

We argue that if from the viewpoint of semiclassical gravity the Universe began with zero size, then there is a preferred initial vacuum state and that if the quantum fields are in any other homogeneous and isotropic state, then before inflation began the Universe, to a good approximation, was radiation dominated. We then study the effects of a pre-inflationary radiation dominated era that smoothly evolves into an inflationary era by numerically solving the mode equation for a massless, minimally coupled scalar field. We model this transition with a combination of classical radiation and a cosmological constant with the Universe asymptotically approaching de Sitter space. The resulting late-time power spectrum is free of enhancements at all scales, oscillating with peaks that never exceed the Bunch-Davies value. Comparisons with previous results in models in which the Universe began with a radiation dominated phase will be made.   

Live

A remarkable fact about the vacuum state of a quantum field theory is that it is entangled across spacelike separated regions. This entanglement is strong enough to violate a Bell inequality and is ultimately responsible for some of the most important predictions of quantum field theory on curved space, such as the Unruh and Hawking effects. In this talk, I will introduce the entanglement harvesting protocol as an operational way to probe vacuum entanglement. This protocol relies on two atoms, modeled by Unruh-DeWitt detectors, that are initially un-entangled. These atoms then interact locally with the field and become entangled. Because the atoms do not interact with one another, any entanglement between the atoms is a result of entanglement that is 'harvested' from the field, and thus quantifying this entanglement serves as a proxy for how entangled the field is across the regions in which the atoms interacted. Using this protocol, I will show that while the local statistics of each atom are unaffected by the presence of a gravitational wave, the entanglement between them depends sensitively on both the amplitude and frequency of the gravitational wave. This suggests that the entanglement signature left by a gravitational wave may be useful in characterizing its properties.   

Live

Fluctuations in the scalar field driving inflation can lead to an increase in the expansion rate, decreasing the Hubble distance. A converging shell of geodesics just inside the old Hubble distance will be outside the new one and thus be carried away by the Hubble flow. This implies defocusing, which requires violation of the Averaged Null Energy Condition (ANEC). However, semiclassical proofs of ANEC suggest that such violations should not exist. In this talk I will argue that this paradox arises from considering the entire quantum state at early times but only a specific quantum state at late times. Examining a late-times fast expanding state and tracing back its evolution, I show how ANEC is in fact obeyed in an eternally inflating spacetime.   

Live

I will describe an experiment and some theory of an expanding, ring-shaped Bose-Einstein condensate. The expansion redshifts and damps long wavelength excitations, as in an expanding universe. After expansion, energy in the radial mode leads to the production of bulk topological excitations---solitons and vortices---driving the production of a large number of azimuthal phonons and, at late times, causing stochastic persistent currents. These complex nonlinear dynamics, fueled by the energy stored coherently in one mode, are reminiscent of a type of "preheating" that may have taken place at the end of inflation. (Based on 10.1103/PhysRevX.8.021021.)   

On Demand

A derivation of the linear response equations for quantum electrodynamics in 1+1 dimensions will be presented for the case of a spatially homogeneous electric field coupled to a quantized scalar field and a quantized fermion field. It will be shown that these equations depend upon the two point correlation function derived from the current-current commutator, $\langle [ j(t,x), j(t', x')] \rangle$. A criterion for the validity of the semiclassical approximation will be given that involves the stability of solutions to the linear response equations. This is an adaptation of criteria previously used for the validity of the semiclassical approximation for gravity and that for the preheating phase of chaotic inflation. A method of finding approximate solutions to the linear response equations will also be given.   

On Demand

Numerical techniques will be discussed that were used to solve the semiclassical backreaction equations for a quantized scalar field and a quantized fermion field interacting with a homogeneous classical electric field in 1+1 dimensions. These include using adiabatic regularization to renormalize the expectation value of the current, $\langle j_\mu \rangle$, and initial conditions for the backreaction and mode equations. Results of numerical calculations will be shown and their implications for the validity of the semiclassical approximation will be discussed.   

On Demand

A method will be discussed which allows for the numerical computation of the semi-classical stress-energy tensor, $\left\langle in \middle| T_{\mu\nu} \middle| in \right\rangle$, associated with a quantized massless minimally coupled scalar field in the region outside the event horizon of a (3+1)D Schwarzschild black hole that forms from the collapse of a null shell. This method is based on the idea that one can expand the in-modes in terms of a complete set of solutions to the mode equation in the exact Schwarzschild geometry. Applying this method, a full numerical computation of the renormalized stress-energy tensor for the (1+1)D case is done and shown to be equal to the known solution. In (3+1)D, the presence of an effective potential in the mode equation causes scattering effects that make the matching more difficult. Tests that check the validity of the 4D matching method will be discussed.   

On Demand

In dimensionally reduced theories with gauge (or diffeomorphism) invariance, consistency of the Kaluza-Klein ansatz requires the introduction of compensator fields. The compensator fields project gauge-variant field fluctuations to their horizontal components, which in turn determine the gauge-invariant moduli space metric that appears in kinetic terms of the lower dimensional theory. The compensator fields must be introduced "by hand" in dimensional reduction (i.e., compactification truncated to zero modes), but arise automatically in the full untruncated theory. For the U(1), Yang-Mills theory, and pure Einstein gravity compactified on an arbitrary compact manifold, we re-express the full untruncated parent theory in lower dimensional language, and identify the compensator fields. One of their known geometric interpretations features prominently. The moduli space of gauge fields (or metrics) on the compact manifold can be regarded as a principal bundle whose fiber is the space of gauge transformations. The compensator fields arise as a repackaging of the connection on this bundle.   

On Demand

The $n$-point functions of (perturbatively-renormalizable) quantum field theories are known to satisfy asymptotic relations called operator product expansions (OPEs) in the limit that all their spacetime points coincide. The coefficients of these expansions are state-independent and contain essential information about the quantum field theory itself. In (flat) Euclidean spacetime, Hollands et al. have derived novel ``flow equations'' which govern how OPE coefficients depend on the QFT's interaction parameters. Although proven to hold order-by-order in perturbation theory, these flow equations have been proposed as a potential avenue for defining OPE coefficients non-perturbatively. However, serious obstacles arise if one attempts to generalize the Hollands flow equations to curved Lorentzian spacetimes in a local and covariant manner. In this talk, I will describe these issues and sketch our resolutions for a solvable toy model: Klein-Gordon theory on curved spacetime with the (squared) mass, $m^2$, treated as an ``interaction parameter''. The strategies I describe for generalizing this toy model's flow relations in a local and covariant way are expected to be applicable, more generally, to QFTs with nonlinear interactions in curved Lorentzian spacetimes.