Schrödinger’s Equation, Geometrodynamics, and Quantum Mechanics

By George Shiber Tuesday, November 15, 2016Friday, October 26, 2018 In Mathematical Physics, Mathematics, Philosophy, Philosophy Of Physics, Philosophy Of Science, Physics Geometry, Quantum Geometry

Without getting into the mathematical aspects of the ‘quantum mechanics = Bayes theory in the complex-number-system program’, I will show that a geometrodynamic derivation of quantum mechanics is possible based on a modified Schrödinger equation whose action-propagator yields an interpretation of ‘particle’ as curvature in spacetime, and relate it to the conflict I derived between the Lindblad quantum-jump equation:

$\begin{array}{l}{\rm{d}}\hat \rho _S^C = - i\left[ {{{\hat H}_S},\hat \rho _S^C} \right]{\rm{d}}t - \\\frac{1}{2}\sum\limits_\mu {{\kappa _\mu }} \left[ {{{\hat L}_\mu },\left[ {{{\hat L}_\mu },\hat \rho _S^C} \right]} \right]{\rm{d}}t + \\\sum\limits_\mu {\sqrt {{\kappa _\mu }} } W\left[ {{{\hat L}_\mu }} \right]\hat \rho _S^C{\rm{d}}{W_\mu }\end{array}$

where:

$\begin{array}{l}W\left[ {\hat L} \right]\hat \rho \equiv \hat L\hat \rho + \hat \rho {{\hat L}^\dagger } - \\\hat \rho {\rm{Tr}}\left\{ {\hat L\hat \rho + \hat \rho {{\hat L}^\dagger }} \right\}\end{array}$

with ${\rm{d}}{W_\mu }$ the Weiner-quantum-increments, and the collapse-action-functional corresponding to the collapse equation, given by:

$\begin{array}{*{20}{l}}{d\left| {{\psi _t}} \right\rangle = \left[ { - \frac{i}{\hbar }} \right.Hdt + \sqrt \lambda \int {{d^3}} x\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}\\{ \cdot d{W_t}(\bar x) - \frac{\lambda }{2}\int {{d^3}} x\left. {{{\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}^2}dt} \right]\left| {{\psi _t}} \right\rangle }\end{array}$

where the collapse-action-functional is:

$\int {\exp \frac{i}{\hbar }} {S_{1,2}}\left[ \Gamma \right]\left[ {d\Gamma } \right] = {K_{{t_2},{t_1}}}\left( {{x_2},{x_1}} \right)$

and ${S_{1,2}}\left[ \Gamma \right]$ is given by:

${S_{1,2}}\left[ \Gamma \right] = \int\limits_{{t_1}}^{{t_2}} {\left( {{T^\Gamma }\left( {\dot X,\dot x} \right) - {U_q}\left( {x,X} \right)h\left( {t - {t_3}} \right) - U\left( {x,t} \right)} \right)} dt$

Let me set up the stage. Geometrodynamically, a particle can be interpreted, non-locally, via the following metric:

${\Im _{\mu \nu }} = {g_{\mu \nu }} + H\left( {T_{\mu \nu }^P} \right){\tau _{\mu \nu }}$

with ${g_{\mu \nu }}$ being the pseudo-Riemannian metric and ${\tau _{\mu \nu }}$ the particle-non-local metric, with $T_{\mu \nu }^P$ the corresponding particle stress-energy tensor and $H$ is a functional characterized by:

$H:T_{\mu \nu }^P \in \Re \to H\left( {T_{\mu \nu }^P} \right) \in {R^ + }$

Hence, particles can be interpreted as spacetime-curvature – and this has a natural Clifford-algebraic lift to quantum fields – and reflects the dominant paradigm summarized by: ‘spacetime = entanglement’, which suggests naturally that superoscillatory functions can be used in their representation since they are functions that track unique wave-like phenomena, where a globally band-limited wave consists of local segments that oscillate faster than its fastest Fourier components.

Here’s a paradigmatic instance of a superoscillatory function:

$\begin{array}{l}{f_a}\left( x \right) = {\left( {\cos \left( {x/n} \right) + ia\sin \left( {t/n} \right)} \right)^n}\\ = {\left( {\frac{{1 + a}}{2}{e^{it/n}} + \frac{{1 - a}}{2}{e^{ - it/n}}} \right)^n}\,,a > 1\end{array}$

By a use of the Binomial expansion, we get:

${f_a}\left( x \right) = \sum\limits_{j = 0}^n {{C_j}} \left( {n,a} \right){e^{i\left( {1 - 2j/n} \right)t}}$

with

${C_j}\left( {n,a} \right)$

the coefficient of

${e^{i\left( {1 - 2j/n} \right)t}}$

and the key is the following approximation:

${f_a}\left( x \right) \approx {e^{iat}}$

This is central: for any density mass that creates curvature in spacetime with superoscillatory functions, there corresponds energy that is strictly larger than the one predicted by general relativity for the smaller density-mass.

Let

$\left\{ {f_{{\varepsilon _j}{q_j}\left( {\bar x,t} \right)}^j\left( {\bar x,t} \right)} \right\}_{j = 1}^m$

be a class of superoscillatory particles, with the $t \to it$ -rotation. We thus can derive:

${\Im _{\mu \nu }} = {\alpha _{\mu \nu }} + i{\beta _{\mu \nu }}$

with ${\alpha _{\mu \nu }}$ and ${\beta _{\mu \nu }}$ the real-component metrics.

A Fourier superoscillatory transform allows us to express ${\Im _{\mu \nu }}$ as:

${\Im _{\mu \nu }} = {\lambda _{\mu \nu }}{f_{\varepsilon q\left( {\bar x,t} \right)}}\left( {t,\bar x} \right)$

with ${\lambda _{\mu \nu }}$ the real-part metric.

hence, the superposition of the states of the particle is given by

${\underline \Im _{\mu \nu }} = \left( {\lambda _{\mu \nu }^if_{{\varepsilon _1}{q_1}\left( {\bar x,t} \right)}^1\left( {t,\bar x} \right)\lambda _{\mu \nu }^2f_{{\varepsilon _2}{q_2}\left( {\bar x,t} \right)}^2\left( {t,\bar x} \right)...\lambda _{\mu \nu }^mf_{{\varepsilon _m}{q_m}\left( {\bar x,t} \right)}^m\left( {t,\bar x} \right)\,} \right)$

which is quasi-morphically functionally equivalent to the standard wavefunction: that is, since we can describe waves via:

$\lambda _{\mu \nu }^jf_{{\varepsilon _2}{q_2}\left( {\bar x,t} \right)}^j\left( {t,\bar x} \right)$

the superposition is thus the sum:

$^{wf}Z_{\mu \nu }^{\rm{S}} = \sum\limits_{j = 1}^m {\lambda _{\mu \nu }^j} f_{{\varepsilon _j}{q_j}\left( {\bar x,t} \right)}^j\left( {t,\bar x} \right)$

The geometrodynamical model, due to non-locality, need only factor in spin. Hence, a particle and its spin can be expressed by:

$^{wf}{\tilde Z^{P,}}_{\mu \nu }^{\rm{S}} = \sum\limits_{j = 1}^m {\lambda _{\mu \nu }^j} f_{{\varepsilon _j}{q_j}\left( {\bar x,t} \right)}^j\left( {t,\bar x} \right)$

and the superoscillatory function with respect to time:

$f_{{\varepsilon _j}{q_j}\left( {\bar x,t} \right)}^j\left( {t,\bar x} \right)$

reduces to:

$f_{{\varepsilon _j}}^j\left( {\bar x,t} \right) \approx {e^{ - i{\varepsilon _j}t{q_j}\left( {\bar x,t} \right)}}$

noting that the approximative nature of the relations are due to generalized uncertainty principles.

Define now the matrix norm:

$\begin{array}{l}\left\| {\lambda _{\mu \nu }^j} \right\|: = {\left( {\left\| {_1\lambda _{\mu \nu }^j} \right\|\,\;\left\| {_2\lambda _{\mu \nu }^j} \right\|\;...\;\left\| {_n\lambda _{\mu \nu }^j} \right\|} \right)^T}\\ = {A^j}\left( {\bar x,t} \right)\end{array}$

Then we have for the $j -$ th state:

$\begin{array}{l}\left\| {\Im _{\mu \nu }^j} \right\| = \left\| {\lambda _{\mu \nu }^j} \right\|{e^{ - i{\varepsilon _j}{t_q}\left( {\bar x,t} \right)}} = \\{A^j}\left( {\bar x,t} \right){e^{ - i{\varepsilon _j}{t_q}\left( {\bar x,t} \right)}}\end{array}$

and particle-superposition can be defined by:

$\begin{array}{l}\psi \left( {\bar x,t} \right): = \sum\limits_{j = 1}^m {\left\| {\lambda _{\mu \nu }^j} \right\|} f_{{\varepsilon _j}{q_j}\left( {\bar x,t} \right)}^j\left( {t,\bar x} \right)\\ = \sum\limits_{j = 1}^m {{A^j}\left( {\bar x,t} \right){e^{ - i{\varepsilon _j}t{q_j}\left( {\bar x,t} \right)}}} \end{array}$

and by superoscillatory Fourier transforming, we get that $\psi \left( {\bar x,t} \right)$ is the quantum mechanical wavefunction entailed by Schrödinger’s equation:

$i\hbar {\partial _t}\psi \left( {\bar x,t} \right) = H\psi \left( {\bar x,t} \right)$

Now, to close the approximation gap consistent with Heisenberg’s uncertainty principle, define a corrective operator $\Xi$ and write a modified Schrödinger equation:

$\begin{array}{l}i\hbar {\partial _t}\psi \left( {\bar x,t} \right) = H\psi \left( {\bar x,t} \right) + \\\Xi \psi \left( {\bar x,t} \right) = {H^ * }\psi \left( {\bar x,t} \right)\end{array}$

This modified Schrödinger equation has energy-values and amplitudes-solutions that match the classical Schrödinger equation. Thus, the modifying operator $\Xi$ has eigenvalues for each corresponding recursive iterations of the differential equation, and we have an empirical equivalence expressed as:

$\Xi :\psi \left( {\bar x,t} \right) - \psi {\left( {\bar x,t} \right)_{Empirical}} = 0$

With ${E_j}$ the energy value of $\Xi$ with probability amplitude ${\left| {{A^j}} \right|^2}$ , and for $E_j^{Experimental}$ , its probability amplitude:

${\left| {A_{Experimental}^j} \right|^2}$

is equivalent to the probability amplitude corresponding to the standard wavefunction.

Now, taking the Planck constant into account, we have:

${\xi _j}{E_j} = E_j^{Experimental}{\mkern 1mu} ,{\mkern 1mu} \forall j = 1,2,...,m$

with:

${\xi _j} \approx {q_j}\left( {\bar x,t} \right)\hbar$

so the modification is approximated by:

${q_j}\left( {\bar x,t} \right) \approx \frac{{{\xi _j}}}{\hbar } = \frac{{E_j^{Experimental}}}{{\hbar {E_j}}}$

Solving:

$\begin{array}{l}i\hbar {\partial _t}\psi \left( {\bar x,t} \right) = H\psi \left( {\bar x,t} \right) + \\\Xi \psi \left( {\bar x,t} \right) = {H^ * }\psi \left( {\bar x,t} \right)\end{array}$

entails that the modified Schrödinger equation yields the accurate energy values of the particles, and they can be interpreted geometrodynamically as analytic curvature properties of spacetime.

Proof:

Take a particle moving from $\bar x'$ to $\bar x''$ with the corresponding action:

${S_\hbar }\left[ {\bar x} \right] = \int_0^t {{L_\hbar }} \left( {\bar x\left( t \right)} \right)dt$

By a superoscillatory Fourier transform, the corresponding propagator is:

${\left\langle {\bar x'';T\left| {\bar x';0} \right.} \right\rangle _\hbar } = \int_{\bar x(0) = \bar x'}^{\bar x(T) = \bar x''} {{e^{i{S_\hbar }\left[ {\bar x} \right]}}} Dx \to \infty$

and taking the corresponding Fock-shift:

${S_{{q_1}\left( {\bar x} \right),...,{q_m}\left( {\bar x} \right)}}\left[ {\bar x} \right] = \int_0^t {{L_{{q_1}\left( {\bar x(t)} \right),...,{q_m}\left( {\bar x(t)} \right)}}} \left( {\bar x(t)} \right)dt$

allows the wavefunction propagator to converge as:

${\left\langle {\bar x'';T\left| {\bar x';0} \right.} \right\rangle _{{q_1}(\bar x),...,{q_m}(\bar x)}} = \int_{\bar x(0) = \bar x'}^{\bar x(T) = \bar x''} {{e^{i{S_{_{{q_1}(\bar x),...,{q_m}(\bar x)}}}\left[ {\bar x} \right]}}} Dx < \infty$

Thus we get the Lagrangian:

$L = \bar \psi {\left( {i{\gamma ^\mu }{D_\mu } - {M_m}} \right)_{1/2 - Spin}}\psi$

with ${\gamma ^\mu }$ the Dirac matrices and the Dirac adjoint is $\bar \psi = {\psi ^ + }{\gamma ^0}$ and the gauge covariance derivative is:

${D_\mu } = {\partial _\mu } + ie{A_\mu } + ie{B_\mu }$

Hence, solving:

one can see that a particle is a superosscilatory functional of spacetime with modified evolution operator:

$\left\{ {\begin{array}{*{20}{c}}{\sum\limits_{j = 1}^n {{a_j}} \exp \left( { - i{H_j}t/\hbar } \right)}\\{\sum\limits_{j = 1}^n {{a_j}} \exp \left( { - i{H_j}t/\hbar } \right) \approx \exp \left( { - i{H_j}t/\hbar } \right)}\end{array}} \right.$

Hence, the evolution of the wavefunction reduces to:

$\begin{array}{c}\psi \left( {t,\bar x} \right) = \sum\limits_{j = 1}^n {{a_j}} {e^{ - i{H_j}t/\hbar }}\psi \left( {0,\bar x} \right) \sim \\{e^{ - iHt}}\psi \left( {0,\bar x} \right)\end{array}$

so one can deduce:

$i{\partial _t}\psi \left( {t,\bar x} \right) = \sum\limits_{j = 1}^n {{a_j}} {H_j}{e^{ - i\left( {{H_j} - H} \right)t/\hbar }}\psi \left( {t,\bar x} \right)$

which is quasimorphic to the Schrödinger equation, and taking a $t \to it$ -rotation, superoscillatory functionals replace the Minkowski metric by the Euclidean metric, and so the non-local geometrodynamical model can be approximated to quantum mechanics by mapping the metric into the quantum wavefunction, yielding a geometric interpretation of particles as spacetime curvature, and the particle-superposition:

given the collapse-equation:

resolves the above-mentioned conflict I derived between the Lindblad quantum-jump equation and the collapse action functional:

$\int {\exp \frac{i}{\hbar }} {S_{1,2}}\left[ \Gamma \right]\left[ {d\Gamma } \right] = {K_{{t_2},{t_1}}}\left( {{x_2},{x_1}} \right)$

Going into the mathematics of how this model fits with the: ‘quantum mechanics = Bayes theory in the complex-number-system program’ is for another day.