Bridge 2: Quantum Mechanics¶

Extension of Quantum Mechanics to Cosmological Scales¶

In ΛCDM, quantum mechanics applies only to elementary particles and high-energy physics. The Universe as a whole follows classical laws.

TMT extends quantum mechanics to cosmological scales via the concept of temporal distortion:

TMT unifies quantum mechanics and gravitation via the Després-Schrödinger equation:

\[ i\hbar [1 + \tau(x)]^{-1} \frac{\partial\psi}{\partial t} = \left[-\frac{\hbar^2}{2m_{eff}} \nabla^2 + V(x) + mc^2\tau(x)\right] \psi \]

\[ i\hbar [1 + \tau(x)]^{-1} \frac{\partial\psi}{\partial t} \]

Interpretation: Proper time flows differently from cosmic time:

\[ dt_{\text{proper}} = [1 + \tau(x)] \cdot dt_{\text{cosmic}} \]

Larger \(\tau\) → time flows more slowly → particle evolves more slowly.

Term 1 - Modified kinetic energy:

\[ \hat{A}_{\text{kinetic}} = -\frac{\hbar^2}{2m_{eff}} \nabla^2\psi \]

with \(m_{eff} = m_0/\gamma_{\text{Després}}\) where \(\gamma_{\text{Després}} = 1/\sqrt{1 - 2\Phi/c^2 - v^2/c^2}\)

Term 2 - Classical potential (unchanged):

\[ \hat{A}_{\text{potential}} = V(x)\psi \]

Term 3 - Temporal potential (NEW!):

\[ \hat{A}_{\text{temporal}} = mc^2\tau(x)\psi \]

This is the key term: a new potential energy created by temporal distortion itself.

Term	Expression	Effect
\([1+\tau]^{-1}\)	Temporal factor	Quantum clock slowed near masses
\(m_{eff}\)	Effective mass	Mass varies with gravitational distortion
\(mc^2\tau\)	Temporal potential	Energy linked to local temporal distortion

Limit	Condition	Result
Flat space	\(\tau \to 0\)	Recovers standard Schrödinger equation ✓
Classical	\(\hbar \to 0\)	Recovers Hamilton-Jacobi equation ✓
Weak field	\(\tau \ll 1\)	Reproduces Einstein's gravitational redshift ✓

See the Lexicon for complete definitions of \(\tau(x)\), \(\gamma_{\text{Després}}\), and all TMT terms.