We derive here the many-body electronic Hamiltonian with full account of the spin in the non-relativistic limit. Starting from the single-electron Dirac equation and the notion of the spinor, we derive the Pauli equation for an electron in the electromagnetic field and its non-relativistic variant. Then the general many-electron case is studied to show how the spinor idea is generalised by building the spinor of the whole system as a direct poroduct of spinors of every electron. Corespondingly, the matrix Hamiltonian of such a system is built, which acts on such spinors. Next, a complete expansion of a general many-body wave function in terms of elementary many-body spinors (Slater determinants) is constructed, and corresponding matrix elements of one- and two-body operators are worked out that appear to be identical to the Slater rules. Then the second quantisation is developed for this fully relativistic case. Finally, the non-relativistic limit is considered in detail for the general many-fermion case and the quantum problem is reformulated in terms of the large components of the spinor only. The derived Hamiltonian contains the spin–orbit based on the total electric field.

We explore several foundational facts about quantum field theories, starting with the well-known conversion of Schrödinger mechanics to a path integral formulation. Path integral accommodates the natural time-ordering of the operator formulation without any tweak, which leads to how the causal Green’s function replaces its classical counterpart and motivates the Wick rotation of the path integral into Euclidean time, setting the stage for the rest of the volume. Along the way, we pay some attention to the real-space version of various Green’s functions. After a brief exploration of fermionic mechanics and the following spinor fields, we turn to the symmetries and the conservation laws. Although we initially follow the standard treatise of the material, with emphasis on the role of gauging either as part of the dynamics or as an auxiliary handle, we point out how a common confusion over Noether’s energy-momentum tensor stems from an ill-conceived implementation of translations. When properly equipped with the Lie derivative, we show how the Noether procedure always reproduces the symmetric energy-momentum tensor from the metric variation and obviates the "improvement" terms.

A hallmark of the covariantly constant metric is how the Riemann curvature can at best rotate tensors, but this important characteristic is lost in the Christoffel symbol. The Maurer-Cartan alternative addresses this more cleanly by introducing a bigger set of variables, the vielbein, which defines an SO-valued connection 1-form and leads us to the curvature 2-form on par with the ordinary Yang-Mills field strength. This also leads us naturally to spinors, which can be defined with the help of the vielbein via the Clifford algebra. A classification of the latter’s spinor representations is given, with emphasis on when and how the Weyl and the Majorana conditions can be imposed. Also explored is how the vielbein and the spinor should transform under diffeomorphisms, whereby the so-called Kosmann lift enters. The latter is relatively unknown to the physics community, yet quite crucial for many corners of physics where spinors play an important role.

Although the geometry background from Chapter 3 to the early part of Chapter 6 is self-contained, we attach this appendix to make contact with the more modern language of differential geometry. Fiber bundles can be seen as an obvious generalization of (co-)tangent bundles and allow us to introduce the notion of connections and curvatures in a more invariant manner via the principal bundle. This, in turn, leads to the frame bundle and the spinor bundle, which were implicitly invoked in Chapter 5. A quick overview of G-structure and holonomy classification is followed by how one must deal with spinors in curved spacetime. Although this last part is not used in this volume, it would become an essential tool in the companion book “Geometric Quantum Field Theories.”

Space and time are not what they seem. Their true nature only becomes clear as particles reach the speeds close to the speed of light where some of the common sense ideas start to break down. Indeed, one of major themes of twentieth century physics is that common sense is not a good guide when we look closely at the universe. In this chapter, we start to understand the true nature of space and time, as encapsulated in Einsteins theory of special relativity. We will see many wonderful things, from time slowing down to the lengths shrinking. There will be stories of twins and trains and elementary particles failing to die.