Relativistic probability densities for location
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Bibliographic record
Abstract
Abstract Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ ( x , t ) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes <?CDATA $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ψ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>exp</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">i</mml:mi> <mml:msub> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="bold-italic">k</mml:mi> </mml:mrow> </mml:msub> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> yield normalized functions ψ ( x , t ) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t ) = ( ℏ /2i)[ ψ + ( x , t )∂ μ ψ ( x , t ) − ∂ μ ψ + ( x , t ) · ψ ( x , t )]. However, in the case of bosonic fields, the wave packets ψ ( x , t ) are nonlocally related to the corresponding relativistic quantum fields ϕ ( x , t ), and therefore the canonical local energy-momentum densities <?CDATA ${ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <?CDATA ${ \mathcal P }({\boldsymbol{x}},t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> differ from ℘ μ ( x , t ) and appear nonlocal in terms of the wave packets ψ ( x , t ). We examine the relation between the canonical energy density <?CDATA ${ \mathcal H }({\boldsymbol{x}},t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , the canonical charge density ϱ ( x , t ), the energy pseudo-density <?CDATA $\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mo>˜</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mrow> <mml:mo>℘</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , and the Born density ∣ ψ ( <jats:ita
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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