I'm trying to learn more about Maxwell equations and stumbled upon an essay by professor Freeman J. Dyson from Princeton. He explained Maxwell theory in a very interesting way.
The modem view of the world that emerged from Maxwell's theory is a world with two layers. The first layer, the layer of the fundamental constituents of the world, consists of fields satisfying simple linear equations. The second layer, the layer of the things that we can directly touch and measure, consists of mechanical stresses and energies and forces. The two layers are connected, because the quantities in the second layer are quadratic or bilinear combinations of the quantities in the first layer. To calculate energies or stresses, you take the square of the electric field-strength or multiply one component of the field by another. The two-layer structure of the world is the basic reason why Maxwell's theory seemed mysterious and difficult. The objects on the first layer, the objects that are truly fundamental, are abstractions not directly accessible to our senses. The objects that we can feel and touch are on the second layer, and their behaviour is only determined indirectly by the equations that operate on the first layer. The two-layer structure of the world implies that the basic processes of nature are hidden from our view.
Another part that I found really interesting was this:
The ultimate importance of the Maxwell theory is far greater than its immediate achievement in explaining and unifying the phenomena of electricity and magnetism. Its ultimate importance is to be the prototype for all the great triumphs oftwentieth-century physics. It is the prototype for Einstein's theories of relativity, for quantum mechanics, for the Yang-Mills theory of generalised gauge invariance, and for the unified theory of fields and particles that is known as the Standard Model of particle physics. All these theories are based on the concept of dynamical fields, introduced by Maxwell in 1865. All of them have the same two-layer structure, separating the world of simple dynamical equations from the world of human observation. All of them embody the same quality of mathematical abstraction that made Maxwell's theory difficult for his contemporaries to grasp. We may hope that a deep understanding of Maxwell's theory will result in dispersal ofthe fog of misunderstanding that still surrounds the interpretation of quantum mechanics. And we may hope that a deep understanding of Maxwell's theory will help to lead the way toward further triumphs of physics in the twenty-first century.
Here is the link for the essay: Why is Maxwell's theory so hard to understand?
Can someone please give me a further explanation of this? Which parts of the equations are the first layer and which one are the second? What does he mean by the mechanical stresses and energies? And how are the layers connected (I didn't get what he said)?