Abstract. This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed.Elliptic boundary-value problems, index problems for closed manifolds and for manifolds with boundary, Bott periodicity and K-theory are then presented. This makes it clear why the Dirac operator is the most fundamental, in the theory of elliptic operators on manifolds. The topic of spectral geometry is developed by studying non-local boundary conditions of the Atiyah-Patodi-Singer type, and heat-kernel asymptotics for operators of Laplace type on manifolds with boundary. The emphasis is put on the functorial method, which studies the behaviour of differential operators, boundary operators and heat-kernel coefficients under conformal rescalings of the background metric. In the second part, a number of relevant physical applications are studied: non-local boundary conditions for massless spin-1/2 fields, massless spin-3/2 potentials on manifolds with boundary, geometric theory of massive spin-3/2 potentials, local boundary conditions in quantum supergravity, quark boundary conditions, one-loop quantum cosmology, conformally covariant operators and Euclidean quantum gravity.
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CHAPTER ONE
DIRAC OPERATOR IN PHYSICS AND MATHEMATICSAbstract. This introductory chapter begins with a review of Dirac's original derivation of the relativistic wave equation for the electron. The emphasis in then put on spin-structures in Lorentzian and Riemannian manifolds, and on the global theory of the Dirac operator, to prepare the ground for the topics studied in the following chapters. 2
The Dirac equationIn the development of theoretical physics, the Dirac operator finds its natural place in the attempt to obtain a relativistic wave equation for the electron.Although we shall be mainly concerned with elliptic operators in the rest of our book, we thus devote this section to Dirac's original argument (Dirac 1928, Dirac 1958. Indeed, it is well known that the relativistic Hamiltonian of a particle of mass m is where the p µ should be regarded as operators: one is familiar with the need to obtain wave equations which are linear in p 0 .To combine relativistic invariance with linearity in p 0 , and to obtain equivalence with Eq. (1.1.4), one looks for a wave equation which is rational and linear in p 0 , p 1 , p 2 , p 3 : 1.6) where α and β are independent of p. Since we are studying the (idealized) case when the electron moves in the absence of electromagnetic field, all space-time points are equivalent, and hence the operator in square brackets in Eq. (1.1.6) is independent of x. This implies in turn that α and β are independent of x, and commute with p and x operators. At a deeper level, α and β make it possible to obtain a relativistic description of the spin of the electron, i.e. an angular momentum which does not result from the translational motion of the electron.At this stage, one can multipl...