By Waldyr A Rodrigues Jr and Edmundo Capelas de Oliveira
Springer
The Many Faces of Maxwell, Dirac and Einstein Equations
In theoretical physics, hardly anything is better known than the Einstein, Maxwell and Dirac equations. The Dirac and Maxwell equations (as well as the analogous Yang–Mills equations) form the basis of the modern description of matter via the electrodynamic, weak and strong interactions, while Einstein’s equations of special and general relativity are the foundations of the theory of gravity. Taken together, these three equations cover scales from the subatomic to the large-scale universe, and are the pillars on which the standard models of cosmology and particle physics are built. Although they constitute core information for theoretical physicists, they are rarely, if ever, presented together.
This book aims to remedy the situation by providing a full description of the Dirac, Maxwell and Einstein equations. The authors go further, however, by presenting the equations in several different forms. Their aim is twofold. On one hand, different expressions of these famous formulae may help readers to view a given equation from new and possibly more fruitful perspectives (when the Maxwell equations are written in the form of the Navier–Stokes equations, for instance, they allow a hydrodynamic interpretation of the electrodynamic field). On the other hand, casting different equations in similar forms may shed light on the quest for unification – as happens, for example, when the authors rewrite Maxwell’s equations in Dirac-like form and use this to launch a digression on supersymmetry.
Another feature of the book concerns concepts in differential geometry that are widely used in mathematics but about which there is little knowledge in theoretical physics. An example is the torsion of space–time: general differential manifolds are naturally equipped with a torsion in addition to the well-known curvature, and torsion also enters into the description of Lie algebras, yet the torsional completion of Einstein gravity, for instance, has been investigated very little. In the book, the authors take care of this issue by presenting the most general differential geometry of space–time with curvature and torsion. They then use this to understand conservation laws, more specifically to better grasp the conditions under which these conservation laws may or may not fail. Trivially, a genuine conservation law expresses the fact that a certain quantity is constant over time, but in differential geometry there is no clear and unambiguous way to define an absolute time.
As an additional important point, the book contains a thorough discussion about the role of active transformations for physical fields (to be distinguished from passive transformations, which are simply a change in co-ordinates). Active transformations are fundamental, both to define the transformation properties of specific fields and also to investigate their properties from a purely kinematic point of view without involving field equations. A section is also devoted to exotic or new physical fields, such as the recently introduced “ELKO” field.
Aside from purely mathematical treatments, the book contains useful comments about fundamental principles (such as the equivalence principle) and physical effects (such as the Sagnac effect). The authors also pay attention to clarifying certain erroneous concepts that are widespread in physics, such as assigning a nonzero rest mass to the photon.
In summary, the book is well suited for anyone who has an interest in the differential geometry of twisted–curved space–time manifolds, and who is willing to work on generalisations of gravity, electrodynamics and spinor field theories (including supersymmetry and exotic physics) from a mathematical perspective. Perhaps the only feature that might discourage a potential reader, which the authors themselves acknowledge in the introduction, is the considerable amount of sophisticated formalism and mathematical notation. But this is the price one has to pay for such a vast and comprehensive discussion about the most fundamental tools in theoretical physics.
