Exact solutions of Wheeler–DeWitt & Yamabe Construction

In summary, the article discusses the exact solutions of the Wheeler-DeWitt equation for the full theory of four dimensional gravity with Lorentzian signature. These solutions have Schrödinger wavefunctionals that are based on 3-metrics with constant spatial scalar curvature, indicating the presence of two physical field degrees of freedom. These solutions are also characterized by minimum uncertainty Gaussians and are associated with a rigged Hilbert space. When the regulator is removed, the solutions exhibit 3-dimensional diffeomorphism and local gauge invariance.
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Mar3que
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Annals of Physics
August 2015, Vol.359:80–96, doi:http://dx.doi.org/10.1016/j.aop.2015.04.016

Exact solutions of the Wheeler–DeWitt equation and the Yamabe construction
  • Eyo Eyo Ita III
  • Chopin Soo
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Abstract
Exact solutions of the Wheeler–DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.
 
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Related to Exact solutions of Wheeler–DeWitt & Yamabe Construction

1. What is the Wheeler-DeWitt equation and why is it important in physics?

The Wheeler-DeWitt equation is a fundamental equation in theoretical physics that is used to describe the quantum state of the universe. It was first proposed by physicists John Wheeler and Bryce DeWitt in the 1960s as an attempt to unify the theories of general relativity and quantum mechanics. This equation is important because it provides a framework for understanding the behavior of the universe at the smallest scales, where classical physics breaks down.

2. What are exact solutions of the Wheeler-DeWitt equation and why are they difficult to find?

Exact solutions of the Wheeler-DeWitt equation refer to solutions that satisfy the equation exactly, without any approximations. These solutions are difficult to find because the Wheeler-DeWitt equation is a highly complex, nonlinear, and infinite-dimensional equation. This makes it challenging to solve using traditional mathematical methods and requires advanced techniques such as supersymmetry and string theory.

3. What is the Yamabe construction and how does it relate to the Wheeler-DeWitt equation?

The Yamabe construction is a mathematical technique used to construct exact solutions of the Wheeler-DeWitt equation. It involves finding a conformal transformation of the Wheeler-DeWitt equation that simplifies its form and allows for the exact solutions to be obtained. This technique has been used in various cosmological models to find exact solutions and has also been applied to other areas of physics.

4. Are the exact solutions of the Wheeler-DeWitt equation observable in the real world?

No, the exact solutions of the Wheeler-DeWitt equation are not observable in the real world. They are mathematical solutions that describe the behavior of the universe at the quantum level. However, they can provide insights and predictions about the behavior of the universe, which can be tested through experiments and observations.

5. How do exact solutions of the Wheeler-DeWitt equation contribute to our understanding of the universe?

The exact solutions of the Wheeler-DeWitt equation play an important role in our understanding of the universe by providing a mathematical framework for studying the dynamics of the universe at the quantum level. These solutions can help us make predictions about the behavior of the universe and test different theories, leading to a deeper understanding of the fundamental laws of nature.

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