# Probabilities as Field-Energy Fractions ## A deterministic scalar-field account of quantum statistics with a finite-environment test **Anes Palma · August 2025** ### Abstract A source-free scalar energy field $\phi(\mathbf r,t)$ reproduces the Schrödinger equation in the narrow-band limit. We show that the conserved quadratic form $$ \mathcal E[\Phi]=\int d^{3N}r\,|\Phi|^{2} $$ acts as the total field energy in configuration space. During a binary measurement the branch energies $\mathcal E_{\uparrow}$ and $\mathcal E_{\downarrow}$ remain constant and equal the Born probabilities. If the environment that produces decoherence contains only $M$ modes, subsequent recoherence transfers energy between branches and violates standard quantum statistics by a factor $\delta P\!\approx\!e^{-\Lambda(M)t}$. A single-photon interferometer with a tunable cavity reservoir can vary $M$ from $10^{0}$ to $10^{6}$ and detect deviations down to $10^{-3}$. The energy-fraction interpretation is therefore falsifiable with present technology. ### Scalar-field framework The free wave equation $$ \partial_{t}^{2}\phi=c^{2}\nabla^{2}\phi $$ with the ansatz $$ \phi=\Re\!\bigl[\psi(\mathbf r,t)\,e^{-i\omega_{0}t}\bigr],\qquad \epsilon=\frac{|\partial_{t}\psi|}{\omega_{0}}\ll1, $$ reduces to $$ i\partial_{t}\psi=-\frac{c^{2}}{2\omega_{0}}\nabla^{2}\psi . $$ Setting $\hbar=E_{11}/\omega_{11}$ and $m=E_{11}/c^{2}$ gives the Schrödinger form $$ i\hbar\partial_{t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi . $$ For $N$ particles we write $\Phi(\mathbf r_{1},\dots,\mathbf r_{N},t)$. The conserved quantity $$ \mathcal E[\Phi]=\langle\Phi|\Phi\rangle=1 $$ is interpreted as total configuration-space field energy. ### Measurement as branch-energy partition A projective measurement produces $\Phi=\Phi_{\uparrow}+\Phi_{\downarrow}$ with $\langle\Phi_{\uparrow}|\Phi_{\downarrow}\rangle=0$. Because $\mathcal E$ is quadratic, the fractions $$ P(\uparrow)=\langle\Phi_{\uparrow}|\Phi_{\uparrow}\rangle,\qquad P(\downarrow)=\langle\Phi_{\downarrow}|\Phi_{\downarrow}\rangle $$ satisfy $P(\uparrow)+P(\downarrow)=1$ and reproduce the Born rule without extra postulates. ### Finite-environment recoherence Couple the pointer coordinate $Q$ to $M$ environmental oscillators $(x_j,p_j)$ via $$ H_{\text{env}}=Q\sum_{j=1}^{M}g_j x_j . $$ Tracing out the environment yields an off-diagonal decay $$ c(t)=c(0)\,\exp\!\bigl[-\Lambda(M)t\bigr],\qquad \Lambda(M)=\frac{2k_BT}{\hbar^{2}}(\Delta Q)^{2}\sum_{j=1}^{M}m_j g_j^{2}. $$ For finite $M$, coherence revives at $t_{\text{rev}}\sim1/\Lambda$. The branch energy then oscillates as $$ \mathcal E_{\uparrow}(t)=\mathcal E_{\uparrow}(0)+|c(0)|\,e^{-\Lambda t}\sin(\Omega t), $$ giving a probability deviation $\delta P\approx e^{-\Lambda t}$. ### Dynamical derivation of probability drift We model the combined qubit–pointer–bath Hamiltonian as $$ H=H_{\rm S}+H_{\rm E}+H_{\rm I},\qquad H_{\rm S}=\frac{\omega_{q}}{2}\sigma_{z}+\frac{P^{2}}{2M}+V(Q), $$ $$ H_{\rm E}=\sum_{j=1}^{M}\!\Bigl(\frac{p_{j}^{2}}{2m_{j}}+\frac{1}{2}m_{j}\omega_{j}^{2}x_{j}^{2}\Bigr), \qquad H_{\rm I}=Q\sum_{j=1}^{M}g_{j}x_{j}. $$ *Initial state* (after the ideal projective interaction but before bath coupling) $$ \rho(0)=\bigl[\; \alpha\,\lvert\uparrow\rangle\langle\uparrow\rvert\otimes\!\chi(Q-Q_{0}) +\beta\,\lvert\downarrow\rangle\langle\downarrow\rvert\otimes\!\chi(Q+Q_{0}) \bigr]\otimes\rho_{\rm th}, $$ where $\chi$ is a narrow Gaussian pointer packet and $\rho_{\rm th}$ is a thermal bath state. #### Reduced evolution Tracing over the bath (second-order Born–Markov but **without** the continuum limit) gives, in the pointer basis, $$ \rho_{\uparrow\downarrow}(t)=\rho_{\uparrow\downarrow}(0) \,\exp\!\bigl[-\Gamma(M)t\bigr] \,\exp\!\bigl[i\varphi(t)\bigr], $$ $$ \Gamma(M) =\frac{(\Delta Q)^{2}}{2\hbar^{2}}\!\sum_{j=1}^{M}\!\frac{g_{j}^{2}}{m_{j}\omega_{j}} \coth\!\Bigl(\frac{\hbar\omega_{j}}{2k_{B}T}\Bigr)\!, \qquad \varphi(t)=\sum_{j=1}^{M}\frac{g_{j}^{2}}{m_{j}\omega_{j}^{2}} \sin(\omega_{j}t). $$ For a finite bath the phase $\varphi(t)$ periodically re-phases the two branches and converts part of the off-diagonal term back into diagonal population: $$ \delta P(t)=2\,\text{Re} \bigl[\rho_{\uparrow\downarrow}(0)\,e^{-\Gamma t}e^{i\varphi(t)}\bigr]. $$ Taking equal coupling $g_{j}=g$, identical masses $m_{j}=m$, and $\omega_{j}=(j\pi/L)v$ (1-D cavity of length $L$) yields $$ \varphi(t)=\frac{2g^{2}}{m}\sum_{j=1}^{M}\!\frac{\sin(j\omega_{1}t)} {j^{2}\omega_{1}^{2}}\;, \quad \omega_{1}=\frac{\pi v}{L}, $$ which approximates a **saw-tooth revival** with envelope $$ \delta P(t)\approx|\alpha\beta|\,e^{-\Gamma t}\, \frac{\sin\!\bigl(M\omega_{1}t/2\bigr)} {M\sin(\omega_{1}t/2)} . $$ At the first revival $t_{\rm rev}\!=\!2\pi/\omega_{1}$ the sinc prefactor is $\simeq1/M$, giving $$ \delta P_{\rm max}\approx|\alpha\beta|\frac{e^{-\Gamma t_{\rm rev}}}{M} \propto\frac{1}{M}. $$ *Numerical estimate* (cryogenic cavity, parameters from previous section): $M\!=\!150,\; \Gamma t_{\rm rev}\!\approx\!0.1 \Rightarrow \delta P_{\rm max}\!\sim\!2\times10^{-3}$, in line with the $10^{-3}$ target. --- *Interpretation.* Unitary dynamics **does** conserve total probability, but when the bath is finite the revival transfers weight between diagonal and off-diagonal sectors. Because the measurement record is read **before** full recoherence, the observed outcome frequencies drift by $\delta P(t)$ instead of remaining fixed at $|\alpha|^{2}$ and $|\beta|^{2}$. Taking the bath continuum limit ($M\!\to\!\infty$) restores orthodox statistics. ### Experimental proposal | Element | Specification | Purpose | |---|---|---| | Mach–Zehnder interferometer | Single 1550 nm photons, SNSPD readout | Binary outcomes $\uparrow/\downarrow$ | | Tunable cavity reservoir | $Q$ factor $10^{3}$–$10^{6}$ (mode count $M$) | Control $\Lambda(M)$ | | Cryostat | $T=20$ K | Reduce thermal noise, lengthen $t_{\text{rev}}$ | | Optical delay line | Variable $t=0$–$10\,\mu$s | Observe $\delta P(t)$ | Predicted deviation: $\delta P\approx10^{-3}$ for $M\!\sim\!10^{2}$, $\Delta Q=1\,\mu$m. Photon statistics of $10^{7}$ counts reach $10^{-4}$ precision, sufficient to confirm or rule out the effect. ### Implications * Detecting $\delta P$ validates deterministic field-energy probabilities and quantifies environmental decoherence. * A null result beyond $10^{-4}$ rejects the model while leaving orthodox quantum mechanics intact. ### Conclusion Identifying Born weights with conserved field-energy fractions makes a clear, falsifiable prediction: finite environments induce measurable deviations from standard statistics. A tunable-reservoir single-photon experiment can perform the test now, deciding whether deterministic scalar-field physics underlies quantum probability. ### References 1. A. Palma, *Deriving the Schrödinger Equation from Source-Free Maxwell Dynamics*, manuscript, 2025. 2. W. Zurek, "Decoherence, einselection and the quantum origins of the classical", Rev. Mod. Phys. 75, 715 (2003). 3. Y. Aharonov & M. Scully, "Time-reversal quantum-eraser interactions", Phys. 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