Saturday, July 5, 2014

generalizaton of the Hartle-Hawking no boundary proposal and the Perfect Cosmological Principle

Excerpts from The Revolution of Matter

Writing the Euclidean Hamiltonian as a function of a scalar field and allowing the total energy of Euclidean space to be zero, where the dynamical variables momentum and energy are dimensionless in $\mathbb{R}^4$, and accordingly any wave equation is a purely mathematical object.

$$\mathcal{H}_E (\phi) =\mathbb{T} (\phi) -\mathbb{U} (\phi) =0$$

Here $\mathbb{U} (\phi)$  is the geometric potential for the entire universe of  $\mathbb {R^4} (\phi)$, and since  $\mathbb { R^{1,3}}$ is a subspace of  $\mathbb{R}^4$ any global change in $\mathbb{U} (\phi)$  causes a global change in $\mathbb {R^{1,3}}$. Under the premise that no physical matter existed before the Big Bang at $\mathbb{U}_i$, then matter is absent in  $\mathbb {R^{1,3}}$ and the dynamical variables T, V are zero,

$$\mathcal{H}_M(\mathbb{R}^{1,3})=T+V -G =0$$

$$T=V=G=0$$

Therefore initially $\mathbb {R^{1,3}}$ is as flat as $\mathbb{R}^4$ . Since  $\mathcal{H}_E (\phi)$ = 0, this implies -and most importantly - there is no center or boundaries to  $\mathbb{U}$, for as the momentum and energy tends to zero so the Heisenberg uncertainties in position and time also tend to infinity,

And this is the key idea, applying the Heisenberg uncertainties to the Euclidean and Minkowskian spaces as a whole states the total uncertainty in space and time tends to infinity,

$$X_i\simeq X_i\pm \text{\delta X}_i\simeq X_i\pm \infty$$

So the boundary to any point is at infinity, which is a restatement of the Hartle-Hawking no boundary proposal. Yet adding the maximal uncertainty to the boundary itself,

$$X_i\simeq X_{\max }\pm \text{\delta X}_{\max }$$

implies the boundary simultaneously coincides with every other point within this model universe and has the remarkable result that every point within the potential behaves as a boundary of the potential.

In effect, at $\mathbb{U}_i$ where starting from a homogeneous and isotropic $\mathbb{R}^4$ and then applying the Heisenberg Uncertainty principle to $\mathcal{H}_E$ as $\mathbb{R}^4$ evolves smoothly under the Wick Transformation into $\mathbb {R^{1,3}}$  to the $\mathcal{H}_M$ results in every point in $\mathbb {R^{1,3}}$ being bounded and adjacent to every other point in $\mathbb {R^{1,3}}$ , therefore at the Big Bang every point in $\mathbb {R^{1,3}}$  is homogeneous and isotropically identical to every other point which is of course the Perfect Cosmological Principle.

The maximal uncertainty in the position of the boundary also places each point $x_i$ at infinity, this contradiction is resolved by noting the statistical nature of Heisenberg's Uncertainty principle, it follows that any particles reflecting off the boundary do so with a statistical expectation and therefore the boundary has a probability of position - the boundary is simultaneously at infinity and every point within the potential.

This is a generalization of the Hartle-Hawking no boundary proposal, strictly speaking the uncertainty in the boundary has it everywhere as opposed to placing the boundary at infinity, accordingly I call this the generalized Hartle-Hawking no boundary proposal (gHH).

Hartle-Hawking proposal $X_i\to \infty$
generalized Hartle-Hawking proposal $X_i \to (0,\infty)$

This has three important ideas
1: A universe with zero total energy which is necessary for evolving a universe out of nothing,
2: each point in spacetime is in contact with every other point, which leads to the Perfect Cosmological Principle,
3: the Hartle-Hawking proposal of the universe where having no beginning which removes the problem of a primal cause for the universe.

In succeeding posts to this blog I'm intending to discuss other features of my paper The Revolution of Matter.