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Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Preface | This book develops Non-Abelian Matrix-Condensate Geometrogenesis (NAMCG) as a single derivation chain: from a microscopic matrix order parameter to an emergent Lorentzian metric, massless spin-2 dynamics, covariantly conserved entanglement backreaction, and a set of astrophysical corollaries. The text is written for an... | [] | This book develops Non-Abelian Matrix-Condensate Geometrogenesis
(NAMCG) as a single derivation chain: from a microscopic matrix order
parameter to an emergent Lorentzian metric, massless spin-2 dynamics,
covariantly conserved entanglement backreaction, and a set of
astrophysical corollaries.
The text is written for a... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Notation | Throughout, spacetime indices \(\mu,\nu,\ldots\) are coordinate indices, \(A,B,\ldots\in\{0,1,2,3\}\) are tangent-space Lorentz indices, and \(I,J,\ldots\) are internal gauge indices of the matrix condensate. The metric signature is \((-,+,+,+)\). Repeated indices are summed. {@{}lll@{}} \toprule Symbol & Meaning & Int... | [] | Throughout, spacetime indices \(\mu,\nu,\ldots\) are coordinate indices,
\(A,B,\ldots\in\{0,1,2,3\}\) are tangent-space Lorentz indices, and
\(I,J,\ldots\) are internal gauge indices of the matrix condensate.
The metric signature is \((-,+,+,+)\). Repeated indices are summed.
\begin{center}
\begin{tabular}{@{}lll@{}}
... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | What Geometrogenesis Must Derive | Learning outcomes.: After this chapter the reader should be able to: - state the geometrogenesis problem as a derivation task, not a metaphor; - explain why analog-gravity metrics are not general relativity; - identify the representation content that a microscopic order parameter must carry if massless spin-2 modes are... | [] | \label{ch:geometrogenesis}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item state the geometrogenesis problem as a derivation task,
not a metaphor;
\item explain why analog-gravity metrics are not general relativity;
\item identify the representation... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Mathematical Preliminaries | Learning outcomes.: After this chapter the reader should be able to: - distinguish a primitive metric from a composite vierbein bilinear, and write the conversion between \(g_{\mu\nu}\) and \(e^A_\mu\); - vary the Einstein--Hilbert action to obtain \(G_{\mu\nu}=8\pi G\,T_{\mu\nu}\) and state the contracted Bianchi iden... | [] | \label{ch:preliminaries}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item distinguish a primitive metric from a composite vierbein
bilinear, and write the conversion between \(g_{\mu\nu}\) and
\(e^A_\mu\);
\item vary the Einstein--Hilbert actio... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Matrix Coordinates and the Microscopic Action | Learning outcomes.: After this chapter the reader should be able to: - write the non-commutative coordinate algebra \([\hat X^\mu,\hat X^\nu]=i\Theta^{\mu\nu}\) and state how \(\Theta^{\mu\nu}\) supplies an ultraviolet cutoff; - write both microscopic actions used in the NAMCG notes---the pregeometric matrix action and... | [] | \label{ch:matrix-action}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item write the non-commutative coordinate algebra
\([\hat X^\mu,\hat X^\nu]=i\Theta^{\mu\nu}\) and state how
\(\Theta^{\mu\nu}\) supplies an ultraviolet cutoff;
\item write bo... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Condensation, Vierbein Emergence, and the Seiberg--Witten Map | Learning outcomes.: After this chapter the reader should be able to: - explain why a strictly constant vacuum \(\langle\Phi^I{}_A\rangle=v\delta^I{}_A\) cannot, by itself, produce a non-degenerate vierbein from \(\langle\partial_\mu\Phi\rangle\); - write the polar decomposition \(\Phi=(v+\sigma)\Lambda\) and construct ... | [] | \label{ch:vierbein-emergence}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item explain why a strictly constant vacuum
\(\langle\Phi^I{}_A\rangle=v\delta^I{}_A\) cannot, by itself,
produce a non-degenerate vierbein from
\(\langle\partial_\... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Irreducible Lorentz Decomposition of Fluctuations | Learning outcomes.: After this chapter the reader should be able to: - convert a fluctuation \(\delta\Phi^I{}_A\) to a spacetime tensor using the background frame; - split that tensor into trace, antisymmetric, and symmetric trace-free pieces, and count components; - match the split to the schematic formula used in the... | [] | \label{ch:lorentz-decomposition}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item convert a fluctuation \(\delta\Phi^I{}_A\) to a spacetime
tensor using the background frame;
\item split that tensor into trace, antisymmetric, and symmetric
trac... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Entanglement Equilibrium and the Modular Hamiltonian | Learning outcomes.: After this chapter the reader should be able to: - write the reduced state \(\rho_D\), the modular Hamiltonian \(H_D\), and the von Neumann entropy \(S_D\) of a causal diamond in the matrix condensate; - state Jacobson's entanglement-equilibrium condition \(\delta S_D=0\) and distinguish it from the... | [] | \label{ch:entanglement-equilibrium}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item write the reduced state \(\rho_D\), the modular Hamiltonian
\(H_D\), and the von Neumann entropy \(S_D\) of a causal
diamond in the matrix condensate;
\item st... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Vanishing of the Tensor Mass Gap | Learning outcomes.: After this chapter the reader should be able to: - write the Fierz--Pauli quadratic action for \(T_{\mu\nu}\) and the linearized Einstein operator that remains when \(m_T=0\); - substitute a compactly supported tensor variation into the entanglement Ward identity and see where \(m_T^2\) appears; - s... | [] | \label{ch:massless-tensor}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item write the Fierz--Pauli quadratic action for \(T_{\mu\nu}\)
and the linearized Einstein operator that remains when
\(m_T=0\);
\item substitute a compactly supported tens... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Heat Kernel and the Coefficient Map to \(\Geff\) | Learning outcomes.: After this chapter the reader should be able to: - write the one-loop effective action of condensate fluctuations as a proper-time integral of the heat kernel; - identify which Seeley--DeWitt coefficient multiplies \(R\) and which multiplies the cosmological term; - reproduce the working formula \fr... | [] | \label{ch:heat-kernel}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item write the one-loop effective action of condensate
fluctuations as a proper-time integral of the heat kernel;
\item identify which Seeley--DeWitt coefficient multiplies \(R\)
... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Ultraviolet Operators, Ghosts, and a Cubic Completion | Learning outcomes.: After this chapter the reader should be able to: - read the \(a_2\) density as a list of four-derivative operators and assign each a role (correction versus propagator); - state Ostrogradsky's theorem in the form used for gravitational effective field theory; - explain why quadratic gravity of Stell... | [] | \label{ch:ultraviolet}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item read the \(a_2\) density as a list of four-derivative
operators and assign each a role (correction versus
propagator);
\item state Ostrogradsky's theorem in the form used f... |
Non-Abelian Matrix-Condensate Geometrogenesis | Edward Schipani | Local Lorentz Locking and the Fate of the Goldstones | Learning outcomes.: After this chapter the reader should be able to: - write the simultaneous action of internal gauge transformations and local Lorentz transformations on \(\Phi^I{}_A\); - identify the diagonal subgroup preserved by the polar vacuum and the six broken generators; - exhibit the Stueckelberg replacement... | [] | \label{ch:lorentz-locking}
\paragraph{Learning outcomes.}
After this chapter the reader should be able to:
\begin{enumerate}
\item write the simultaneous action of internal gauge
transformations and local Lorentz transformations on
\(\Phi^I{}_A\);
\item identify the diagonal subgroup preserved by t... |
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