From: Boudewijn Moonen Subject: Re: spin vs. internal dimensions Date: 21 Aug 2000 18:33:26 GMT Newsgroups: sci.physics.research To: Bjorn Wesen Summary: Gauge theory, principal bundles, and representations of Lie groups Bjorn Wesen wrote: > > Hi! I just got confused on an issue. I had thought I understood how spin > arises for non-scalar fields, because of the spatial rotation generator > having both a field-rotating part and a part rotating the field components > in each point. > > But at the same time shouldn't for example the electron spinor field's > components only be affected by the SU(2) transformations (i.e "internal" > degrees of freedom not connected to spatial things). And if the components > are participants in SU(2) rotations (like turning an electron into a > neutrino) why should they be affected by the spatial rotations at all ? > > I guess I'm just missing something trivial here because I've approached the > problem from two directions (at one point learning the gauge-mechanisms, and > at another point learning spin etc) without getting the whole picture. > > -Bjorn No, spin arises from space-time symmetries, not from internal ones. The theory of the Dirac electron is not a gauge theory, and you should not confuse spin with isospin. Spin ultimatively is a manifestation of the representation theory of the Poincare group, isospin a manifestation of the internal SU(2)-symmetry flipping e.g. the proton and neutron. Since, however, similar constructions are involved, this *can* be confusing indeed. So let me clarify my terms. First, some mathematical constructions. Consider a Lie group G and a manifold P upon which G acts freely from the right.. Under weak technical conditions the quotient M := P/G is a manifold. The map p : P --> M is then called a *principal bundle*. Now given a representation r : G --> GL(V) of G on some vector space V, one may form a vector bundle, denoted P[V], or P x_r V, over M as follows: Let G operate on the product P x V from the right by the rule (p,v)g := (pg,r(inv(g))v) for p in P, g in G, where inv(g) = g^{-1} denotes the inverse og g. Then P x_r V := (P x V)/G , the space of orbits under this action. It is called the *vector bundle associated to the representation r". Physically, a *gauge theory* is a theory obtained by quantization (in whatever manner) from a classical field theory given as follows: - the gauge field is a connection in the principal bundle p - the matter fields are sections of various vector bundles associated to appropriate representations of G - the gauge field induces an action of a covariant derivative on the matter fields (called "minimal coupling") which allows to concoct a Langrangian (which is not of interest here). In this situation, when M is considered as an underlying space-time manifold, P is considered as a bundle the fibres of which over a space-time point in M are to parametrize *internal states* of the physical system under consideration. Therefore, G is dubbed the *internal symmetry group*. It is also called the *gauge group*. Both notions are insofar misleading as neither group acts on M nor on the matter fields, the action of G having been divided out; the only thing acting is the true (infinite dimensional) gauge group, usually called \cal{G}: \cal{G} := Aut(P) , the group of bundle automorphisms of p. Now it may happen that there is a second group, K say, which operates on M and has a representation s on V which intertwines the representation r. In this case, K acts on the associated vector bundle P x_r V by vector bundle morphisms, making it a K-vector bundle over the K-space M, and consequently on the matter fields, these being sections of these bundles. In this case, K is called an *external symmetry group*, quite rightly now, since it acts on M (whence externally) and on the matter fields (whence as symmetries). What can make this confusing now is that the action of K on M and V may come also from such a principal bundle which, however, may have nothing to do with p. More precisely, what I have in mind is the folowing. Let H be a Lie group with subgroup K such that H/K = M = P/G. Let s be a representation of K on V. If I then form the associated vector bundle H x_s V, this will be an H-vector bundle, since H operates in a natural way on M = H/K by left translations. These bundles H x_s V are then called *homogeneous vector bundles*. The key mathematical issue is that therefore H operates on the corresponding matter fields (sections of H x_s V), so that, by starting with the representation s of K, we have constructed a (generically infinite dimensional) representation of H on the matter fields. This is called an *induced representation*. This is precisely what is happening in case of the electron. Namely, how does the Dirac electron arise? For this, let M be four-dimensional flat Minkowski space. Let Po be the Poincare group. It is the semidirect product of space-time translations which can be identified with M itself, and the Lorentz group L (so in this example, H = Po and K = L). It is then known (Wigner 1939) that the unitary irreducible representations of the Poincare group are induced representations as above arising from the finite dimensional irreducible representations of L and so are classified by a real number m, the *mass*, and a half-integer s, the *spin* (the latter arises from the irreps of the maximal compact subgroup of L, SU(2)). This is basically how spin arises. If I remeber darkly if one starts to carry out this construction with the basic representation of L, the corresponding representation is not irreducible. The algebraic constraint needed to project out an irreducible representation then Fourier transforms into the Dirac equation. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712