Exploring Nonlocal Quantum Gravity: Unraveling the Lorentzian Path Integral and Conformal Instability

Nonlocal quantum gravity, a theory that merges gravitational interaction and quantum mechanics, has been a subject of interest for over five decades. The theory is defined by a Lorentzian path integral, a mathematical concept used in quantum mechanics to calculate the probability of a system transitioning from one state to another. The theory also introduces the Lorentzian path integral of nonlocal quantum gravity along with the functional measure, the Faddeev-Popov sector, and the field correlators. The study also presents four examples of nonlocal quantum gravity – Einstein gravity, Stelle gravity, minimally coupled nonlocal quantum gravity, and nonminimally coupled nonlocal quantum gravity.

What is the Path Integral and Conformal Instability in Nonlocal Quantum Gravity?

The study of nonlocal quantum gravity, a theory that combines gravitational interaction and quantum mechanics, has been a topic of interest for the past five decades. This theory is characterized by asymptotically polynomial nonlocal operators, which are entire functions of the Laplace-Beltrami operator square. These operators do not add extra poles in the propagator while simultaneously taking the form of finite-order polynomials in the ultraviolet (UV).

The theory’s action is defined by a Lorentzian path integral, which is a mathematical concept used in quantum mechanics to calculate the probability of a system transitioning from one state to another. This path integral, along with its perturbative expansion and the quantum effective action, is derived from a purely gravitational action in D-topological dimensions.

The Lagrangian, a function that describes the dynamics of a physical system, is given a concrete name based on four theories: Einstein gravity, Stelle gravity, minimally coupled nonlocal quantum gravity, and nonminimally coupled nonlocal quantum gravity. Each of these theories has its own unique set of constants and parameters.

How is the Lorentzian Path Integral Introduced in Nonlocal Quantum Gravity?

The Lorentzian path integral of nonlocal quantum gravity is introduced along with the functional measure, the Faddeev-Popov sector, and the field correlators. The Faddeev-Popov sector is a method used in quantum field theory to handle gauge fixing, which is the process of selecting a single representative out of each class of physically equivalent configurations. Field correlators, on the other hand, are mathematical objects that describe the correlation between different points in a quantum field.

The Lorentzian path integral is then used to move to perturbation theory, which is a mathematical approach used to find an approximate solution to a problem that cannot be solved exactly. This is done by introducing small perturbations to the system and studying their effects. The Efiimov analytic continuation of scattering amplitudes to Euclidean momenta and back to Lorentzian is also described.

What is the Conformal Instability Problem in the Euclidean Path Integral?

The conformal instability problem in the Euclidean path integral is a challenge that arises in the study of nonlocal quantum gravity. This problem is solved by making suitable gauge choices at the perturbative level. A gauge choice is a method used in physics to deal with redundant degrees of freedom in a physical system.

The conformal instability problem arises when trying to calculate the path integral in Euclidean space, which is a mathematical space that generalizes the ordinary two-dimensional plane to higher dimensions. This problem is solved by making suitable gauge choices at the perturbative level, which helps to ensure the stability of the system.

How are the Four Examples of Nonlocal Quantum Gravity Presented?

The four examples of nonlocal quantum gravity – Einstein gravity, Stelle gravity, minimally coupled nonlocal quantum gravity, and nonminimally coupled nonlocal quantum gravity – are presented in detail. Each of these theories has its own unique set of constants and parameters, which are used to define the Lagrangian of the system.

Einstein gravity is defined by the Ricci scalar and the Riemann tensor, which are mathematical objects used to describe the curvature of spacetime. Stelle gravity, on the other hand, is defined by a set of constants of dimensionality.

Minimally coupled nonlocal quantum gravity is characterized by form factors that depend on the Laplace-Beltrami operator and three energy scales. These form factors are parametrized by two entire functions, which are asymptotically polynomial in the UV and such that they approach zero in the infrared (IR).

Nonminimally coupled nonlocal quantum gravity, on the other hand, is defined by a set of form factors that depend on the Laplace-Beltrami operator and in principle on three energy scales. The form factors can be parametrized by two entire functions, which are asymptotically polynomial in the UV and such that they approach zero in the IR.

What is the Future of Nonlocal Quantum Gravity?

The study of nonlocal quantum gravity is still in its early stages, and there are many questions that remain unanswered. For example, many works have been devoted to Feynman diagrams and scattering amplitudes, but little has been said about their origin from a path integral.

Furthermore, these diagrams and amplitudes are invariably calculated in Euclidean momentum space due to the difficulty or even impossibility to handle a well-defined nonlocal quantum field theory (QFT) exclusively in Lorentzian signature. Therefore, questions may arise on whether the fundamental formulation of nonlocal quantum gravity is based on a Lorentzian or a Euclidean path integral, whether such path integral is convergent, and so on.

The future of nonlocal quantum gravity will likely involve further exploration of these questions, as well as continued development of the theory’s mathematical framework. This will involve a deeper understanding of the Lorentzian path integral, the functional measure, the Faddeev-Popov sector, and the field correlators, as well as the perturbative expansion and the quantum effective action.