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Rigorous Real-Time Feynman Path Integral for Vector Potentials

we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the $L^2$ transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities

Response to Merritts et al. (2023): The Anthropocene is complex. Defining it is not

Merritts et al. (2023) misrepresent Paul Crutzen’s Anthropocene concept as encompassing all significant anthropogenic impacts, extending back many millennia. Crutzen's definition reflects massively enhanced, much more recent human impacts that transformed the Earth System away from the stability of Holocene conditions. His concept of an epoch (hence the ‘cene’ suffix) is more consistent with the strikingly distinct sedimentary record accumulated since the mid-20th century. Waters et al. (2022) highlighted a Great Acceleration Event Array (GAEA) of stratigraphic event markers that are indeed diverse and complex but also tightly clustered around 1950 CE, allowing ultra-high resolution characterization and correlation of a clearly recognisable Anthropocene chronostratigraphic base. The ‘Anthropocene event’ offered by Merritts et al., following Gibbard et al. (2021, 2022), is a highly nuanced concept that obfuscates the transformative human impact of the chronostratigraphic Anthropocene. Waters et al. (2022) restricted the meaning of the term ‘event’ in geology to conform with usual Quaternary practice and improve its utility. They simultaneously recognized an evidence-based Anthropogenic Modification Episode that is more explicitly defined than the highly interpretive interdisciplinary ‘Anthropocene event’ of Gibbard et al. (2021, 2022). The advance of science is best served through clearly developed concepts supported by tightly circumscribed terminology; indeed, improvements to stratigraphy over recent decades have been achieved through increasingly precise definitions, especially for chronostratigraphic units, and not by retaining vague terminology

Extraordinary human energy consumption and resultant geological impacts beginning around 1950 CE initiated the proposed Anthropocene Epoch

Growth in fundamental drivers—energy use, economic productivity and population—can provide quantitative indications of the proposed boundary between the Holocene Epoch and the Anthropocene. Human energy expenditure in the Anthropocene, ~22 zetajoules (ZJ), exceeds that across the prior 11,700 years of the Holocene (~14.6 ZJ), largely through combustion of fossil fuels. The global warming effect during the Anthropocene is more than an order of magnitude greater still. Global human population, their productivity and energy consumption, and most changes impacting the global environment, are highly correlated. This extraordinary outburst of consumption and productivity demonstrates how the Earth System has departed from its Holocene state since ~1950 CE, forcing abrupt physical, chemical and biological changes to the Earth’s stratigraphic record that can be used to justify the proposal for naming a new epoch—the Anthropocene

Response to Merritts et al. (2023): The Anthropocene is complex. Defining it is not

Merritts et al. (2023) misrepresent Paul Crutzen's Anthropocene concept as encompassing all significant anthropogenic impacts, extending back many millennia. Crutzen's definition reflects massively enhanced, much more recent human impacts that transformed the Earth System away from the stability of Holocene conditions. His concept of an epoch (hence the ‘cene’ suffix) is more consistent with the strikingly distinct sedimentary record accumulated since the mid-20th century. Waters et al. (2022) highlighted a Great Acceleration Event Array (GAEA) of stratigraphic event markers that are indeed diverse and complex but also tightly clustered around 1950 CE, allowing ultra-high resolution characterization and correlation of a clearly recognisable Anthropocene chronostratigraphic base. The ‘Anthropocene event’ offered by Merritts et al., following Gibbard et al. (2021, 2022), is a highly nuanced concept that obfuscates the transformative human impact of the chronostratigraphic Anthropocene. Waters et al. (2022) restricted the meaning of the term ‘event’ in geology to conform with usual Quaternary practice and improve its utility. They simultaneously recognized an evidence-based Anthropogenic Modification Episode that is more explicitly defined than the highly interpretive interdisciplinary ‘Anthropocene event’ of Gibbard et al. (2021, 2022). The advance of science is best served through clearly developed concepts supported by tightly circumscribed terminology; indeed, improvements to stratigraphy over recent decades have been achieved through increasingly precise definitions, especially for chronostratigraphic units, and not by retaining vague terminology