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7/5/2025, 7:48:37 AM
>>16707381
Both the integral form and the differential form are the "real equations," they are just different ways of expressing it according to the fundamental theorem of calculus (the derivative being the opposite of the integral and vice versa).
The real interesting part is that all of the laws of electromagnetism arise from the simple fact that "the relativistic Dirac field is invariant for phase shifts."
Summarized: If you take the Lagrangian for the Dirac field, and add a gauge factor (called the A field), then the various cross-combinations of Euler-Lagrange equations for the 3 space-wise dimensions and 1 time dimension, result in all the laws of electromagnetism you see on that table.
This is called "gauge theory" which is a method of identifying an invariant property (or "gauge symmetry") for a force, and trying to rederive its laws by modifying a correcting factor (the "gauge factor").
Electromagnetism was the first of the 4 fundamental forces that we finally achieved an airtight all-explaining gauge theory for, by the 1970s.
The weak and strong forces have their own gauge theories.
We have so far failed to find a (testable) gauge theory that describes the force of gravitation.
This is what scientists mean when they talk about a "Grand Unified Field Theory" or a "Theory of Everything." They are looking for a model of the universe that explains the four fundamental forces, using one large all-encompassing gauge theory.
Both the integral form and the differential form are the "real equations," they are just different ways of expressing it according to the fundamental theorem of calculus (the derivative being the opposite of the integral and vice versa).
The real interesting part is that all of the laws of electromagnetism arise from the simple fact that "the relativistic Dirac field is invariant for phase shifts."
Summarized: If you take the Lagrangian for the Dirac field, and add a gauge factor (called the A field), then the various cross-combinations of Euler-Lagrange equations for the 3 space-wise dimensions and 1 time dimension, result in all the laws of electromagnetism you see on that table.
This is called "gauge theory" which is a method of identifying an invariant property (or "gauge symmetry") for a force, and trying to rederive its laws by modifying a correcting factor (the "gauge factor").
Electromagnetism was the first of the 4 fundamental forces that we finally achieved an airtight all-explaining gauge theory for, by the 1970s.
The weak and strong forces have their own gauge theories.
We have so far failed to find a (testable) gauge theory that describes the force of gravitation.
This is what scientists mean when they talk about a "Grand Unified Field Theory" or a "Theory of Everything." They are looking for a model of the universe that explains the four fundamental forces, using one large all-encompassing gauge theory.
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