Spin in Quantum Field Theory

I introduce spin in field theory by emphasizing the close connection between quantum field theory and quantum mechanics. First, I show that the spin–statistics connection can be derived in quantum mechanics without relativity or field theory. Then, I discuss path integrals for spin without using spinors. Finally, I show how spin can be quantized in a path–integral approach, without introducing anticommuting variables. Invited lectures at the 43 Internationale Universitätswochen für Theoretische Physik Schladming, Austria, February March 2005 to be published in the proceedings IFUM-844/FT July 2005 Spin in Quantum Field Theory Stefano Forte Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano via Celoria 16, I-10129 Milano, Italy; [email protected] I introduce spin in field theory by emphasizing the close connection between quantum field theory and quantum mechanics. First, I show that the spin– statistics connection can be derived in quantum mechanics without relativity or field theory. Then, I discuss path integrals for spin without using spinors. Finally, I show how spin can be quantized in a path–integral approach, without introducing anticommuting variables. 1 From Quantum Mechanics to Field Theory Even though everybody learns about spin in their childhood in the context of nonrelativistic quantum mechanics, many of the more interesting dynamical features of spin are only introduced in relativistc quantum field theory. In these lectures, which were originally addressed to an audience of (mostly) condensed-matter physicists, I discuss some relevant aspects of spin dynamics in quantum field theory by showing their origin in quantum mechanics. In the first lecture, after a brief discussion of the way spin appears in nonrelativistic (Galilei invariant) or relativistic (Lorentz invariant) dynamics, I show how the spin–statistics connection can be obtained with minimal assumptions in nonrelativistic quantum mechanics, without invoking relativity or field theory. In the second lecture I show how spin can be quantized in a path– integral approach with no need for introducing quantum fields. In the third lecture I discuss the dynamics of relativistic spinning particles and show that its quantization can be described without using anticommuting variables. A fourth lecture was devoted to the quantum breaking of chiral symmetry – the axial anomaly – and its origin in the structure of the spectrum of the Dirac operator, but since this subject is already covered in many classic lectures [1] it will not be covered here. We will see that even though the standard methods of quantum field theory are much more practical for actual calculations, a purely quantum–mechanical approach helps in understanding the meaning of field–theoretic concepts. 2 Spin and Statistics 2.1 The Galilei Group and the Lorentz Group In both relativistic and non-relativistic dynamics we can understand the meaning of quantum numbers in terms of the symmetries of the Hamiltonian and the Lagrangian and associated action. Indeed, the invariance of the Hamiltonian determines the spectrum of physical states: eigenstates of the Hamiltonian are classified by the eigenvalues of operators which commute with it, and this gives the set of observables which are conserved by time evolution. However, the invariance of the dynamics is defined by the invariance of the action. This is bigger than that of the Hamiltonian, because it also involves time–dependent transformations. For example, in a nonrelativistic theory the action must be invariant under Galilei boost: the change between two frames that move at constant velocity with respect to each other. But the Hamitonian in general doesn’t possess this invariance: Galilei boosts obviously change the values of the momenta, and the Hamiltonian in general depends on them. The set of operators which commute with all transformations that leave the action invariant defines the quantum numbers carried by elementary excitations of the system (elementary particles). A nonrelativistic theory must have an action which is invaraint upon the Galilei group. The Galilei transformations, along with the associate quantummechanical operators are [2]: – space translations: xi → xi = xi + ai; Pi = −i∂i – time translation: t → t′ = t+ a; H = i d dt – Galilei boosts: xi → xi = xi + vit; pi → pi +mvi; Ki = −it∂i −mxi – rotations: xi → xi = Rijxj ; Ji = ǫijkx∂k + σi The generator of rotations is the sum of orbital angular momentum and spin. The generators of the Galilei group form the Galilei algebra: [Ji, Jj ] = ǫijkJk; [Pi, Pj ] = 0; [Ki,Kj ] = 0; [Ji, H ] = [Ki, H ] = 0; [ki, H ] = iPi; [Ji, Pj ] = ǫijkPk; [Ji,Kj] = ǫijkKk; [Ki, Pj ] = iMδij . (1) In order to close the algebra it is necessary to introduce a (trivial) mass operator M which commutes with everything else: [M,Pi] = [M,Ki] = [M,Ji] = [M,H ] = 0. (2) The Casimir operators, which commute with all generators, are C1 = M ; C2 = 2MP0−PiPi; C3 = (MJi − ǫijkPjKk) (MJi − ǫilmPlKm) . (3) In terms of quantum-mechanical operators the Casimirs correspond to – C1 = m (mass);