Understanding Gauge
I consider two usages of the expression “gauge theory.” On one, a gauge theory is a theory with excess structure; on the other, a gauge theory is any theory appropriately related to classical electromagnetism. I make precise one sense in which one formulation of electromagnetism, the paradigmatic gauge theory on both usages, may be understood to have excess structure and then argue that gauge theories on the second usage, including Yang-Mills theory and general relativity, do not generally have excess structure in this sense.
Equivalent and Inequivalent Formulations of Classical Mechanics
In this article, I examine whether or not the Hamiltonian and Lagrangian formulations of classical mechanics are equivalent theories. I do so by applying a standard for equivalence that was recently introduced into philosophy of science by Halvorson ([2012], [2012]) and Weatherall ([2016a]). This case study yields three general philosophical payoffs. The first concerns what a theory is, while the second and third concern how we should interpret what our physical theories say about the world. 1. Introduction2. When Are Two Theories Equivalent?3. Preliminaries on Classical Mechanics 3.1. Hamiltonian mechanics3.2. Lagrangian mechanics4. Are Hamiltonian and Lagrangian Mechanics Equivalent Theories? 4.1. Tangent bundle versus cotangent bundle4.2. Tangent bundle versus symplectic manifold4.3. Lagrangian vector field versus Hamiltonian vector field5. Conclusion Appendix Introduction When Are Two Theories Equivalent? Preliminaries on Classical Mechanics 3.1. Hamiltonian mechanics3.2. Lagrangian mechanics Hamiltonian mechanics Lagrangian mechanics Are Hamiltonian and Lagrangian Mechanics Equivalent Theories? 4.1. Tangent bundle versus cotangent bundle4.2. Tangent bundle versus symplectic manifold4.3. Lagrangian vector field versus Hamiltonian vector field Tangent bundle versus cotangent bundle Tangent bundle versus symplectic manifold Lagrangian vector field versus Hamiltonian vector field Conclusion Appendix
Experimental energy barriers to anions transporting through nanofiltration membranes.
Environmentally relevant contaminants fluoride, chloride, nitrate, and nitrite face Arrhenius energy barriers during transport through nanofiltration (NF) membranes. The energy barriers were quantified using crossflow filtration experiments and were in the range of 7-17 kcal·mol(-1), according to ion type and membrane type (Filmtec NF90 and NF270). Fluoride faced a comparatively high energy barrier for both membranes. This can be explained by the strong hydration energy of fluoride rather than other ion properties such as bare ion radius, fully hydrated radius, Stokes radius, diffusion coefficient, or ion charge. The energy barrier for fluoride decreased with pressure, indicating an impact of directional force on energy barriers. The influence of temperature-induced pore radius variability and viscosity on energy barriers was considered. The novel link between energy barriers and ion properties emphasizes the importance of ion hydration and/or partial dehydration mechanisms in determining transport in NF.