Canonical Reduction for Dilatonic Gravity in 3+1 Dimensions

What is it about?

From minimal assumptions of analytic continuation and conformal transformations, Mann and Ross had derived a General Relativity (GRT) formulation which introduced the addition of a particle called a 'dilaton' that ensured that a 'd+1' theory would ensure the correct Newtonian limit in 'd' spatial dimensions, as shown by T. Ohta. Herein, we show the '3+1' version of theory, with the right choice of coordinate and gauge conditions in a ADM prescription yields a dilaton field governed by a Logarithmic Schrödinger Equation (logSE). The coefficient of the log term vanishes in the far-field i.e. flat space. Thus, we get part of quantum mechanics from GRT itself. The logSE is also seen in Superfluids and has been proposed as a model for the Higgs field. This paper provides a key piece in a proposed solution to the century-old problem of trying to reconcile General Relativity (GRT) with Quantum Theory.

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Photo by Iván Díaz on Unsplash

Photo by Iván Díaz on Unsplash

Why is it important?

The paper establishes a previously unknown connection between General Relativity and Quantum Theory, in particular the Logarithmic Schrödinger Equation (LogSE) which has applications in Superfluids. The original action with the dilaton has conformal variance and the initial approach overlaps with the preliminary stages of Loop Quantum Gravity. The logarithmic term of LogSE s found to be essential in resolving the puzzle of sound propagation in liquid helium at low temperatures.

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