Coherent States, Gaussian Quantum Foam, and the Emergence of Spacetime

The quantisation of Gaussian Quantum Foam, where spacetime emerges as the distributional limit of homotopic, globally hyperbolic k-spacetimes, is explored. This construction naturally admits a Gelfand triple, promoting the shift vector to an operator-valued distribution. The shift vector is quantised as a bosonic field and expanded in an orthonormal Gaussian basis of annihilation and creation operators. By constructing coherent states as eigenstates of the annihilation operator, a direct correspondence between quantum fluctuations of the shift vector and macroscopic inflationary dynamics is established. The expectation value of the shift vector in a coherent state reproduces the classical Gaussian profile, reinforcing the Correspondence Principle at both macroscopic and distributional limits. The expectation value of the local Hubble parameter precisely matches its classical counterpart, demonstrating that inflation arises naturally from quantum fluctuations of the shift vector, without the need for an inflaton field. The projected energy density in a coherent state precisely reproduces its classical counterpart, confirming that Quantum Foam exhibits dipolar characteristics. It does not inject net energy but fluctuates around zero, holding the seeds of inflation, in complete alignment with Wheeler’s conception of Quantum Foam. These findings suggest that classical spacetime emerges from Quantum Foam without requiring modifications to general relativity. The horizon problem is intrinsically resolved within this framework, eliminating the need for an inflaton field. If this formulation withstands further scrutiny, it provides a novel perspective on the unification of quantum mechanics and general relativity.