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The Localizability of Particles in De Sitter Space by K

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The Localizability of Particles in De Sitter Space by K Proc. Camb. Phil. Soc. (1971), 70, 283 283 With 2 text-figures POPS 70-31 Printed in Great Britain The localizability of particles in de Sitter space BY K. C. HANNABUSS Mathematical Institute, Oxford [Received 23 July 1970) Abstract. Motivated by the Iwasawa decomposition and its geometrical interpreta- tion, two new decompositions of the de Sitter group are obtained. The first is applied to construct the representations of the de Sitter group in a form immediately com- parable with those of the Poincare group. In particular they act on functions over an hyperboloid like the momentum hyperboloid of the Poincare group, although they require both positive and negative mass shells of that hyperboloid. Using the second decomposition it is shown that the representations of the de Sitter group are localizable in the sense of Mackey and Wightman. Position operators are exhibited. Introduction. De Sitter space is one possible model for the basic geometry of space- time when local irregularities are smoothed out. It provides, amongst other things, for a cosmological red-shift. A description of its basic properties and structure can be found in Robertson and Noonan(io), Chapter 16. We briefly summarize the salient features. The geometry of de Sitter space can be realized as the intrinsic geometry of the hyperboloid in a five-dimensional space whose metric is given by ds2 = dx\ - dx\ ~ dx\ - dx\ - dx\. There is some technical advantage to be gained by refusing to identify de Sitter space with the hyperboloid directly, but rather considering points of de Sitter space to be pairs of diametrically opposite points of the hyperboloid. This makes de Sitter space an imaginary Lobachevsky space with an exterior hyperbolic geometry. Lobachevsky geometries are in fact sufficiently tractable that no simplicity is sacrificed, and some understanding is gained, if we adopt a more general/-dimensional model. This is most easily described if we introduce the pseudo-Euclidean inner product of (/+ l)-vectors, a = (a0,a^, ...,af), andb = {bo,bv ...,bf), f a.b = aobo— 2 «;&3- j=i Then we have the hyperboloid, x.x + 1 = 0, and the metric ds2 = dx.dz. The /-dimensional imaginary Lobachevsky geometry is then obtained by identifying the antipodal points, x and — x. (Although each point of Lobachevsky space corresponds 284 K. C. HANNABTTSS to such a pair of points on the hyperboloid, we shall usually, for the sake of brevity, pick just one of the two to represent it.) Some of the useful geometrical properties of Lobachevsky spaces are to be found in Gel'fand, Graev and Vilenkin(5), though we shall describe those which we need as we go along. One further comment about notation should be made before proceeding. The middle (/— 1) coordinates of each (/+ l)-vector will be treated in a 'coordinate-free' way so that it is convenient to gather them together into a single (/— l)-vector. We shall therefore write a typical (/+ l)-vector as a — (ao,a,af). Later, when we start to use matrices we shall write them in a corresponding block form. We now briefly summarize the salient features of de Sitter space. (Here, as always, it is to be borne in mind that the physics will refer to the special case where / = 4, although the geometry is valid more generally.) De Sitter's original separation of space-time into space and time was effected by arranging that the world-line of a certain observer should be the geodesic {(sht,O,cht):teR} and that space as seen by him at time t should be the intersection of the hyperboloid with the hyperplane ch (t) x0 = sh (t) xf. This gave to space a spherical geometry. Since then a better physical interpretation of the model has been found. The world-line of the observer is still as it was, but space at time t is taken to be the intersection of the hyperboloid with the hyperplane xo + xf = e*. This intersection, which for historical reasons is called a horosphere, can be parametrized as Horospheres are the equivalent in hyperbolic geometry of hyperplanes in Euclidean geometry. Their intrinsic geometry is Euclidean, as is suggested by the use of the Euclidean parameter, r. In general, any. horosphere can be described by some (/+ 1)- vector, q, which is null in the sense that q. q = 0. In fact the horosphere can be given by the equation q.x = 1. We see that our particular spatial horospheres are described by q(t) = (e-^O, -e~*). In Lobachevsky spaces the pseudo-Riemannian metric can easily be integrated to give the hyperbolic distance between two points. One can also, by analogy with the distance of a point from a plane in Euclidean space, talk of the (minimum) hyperbolic distance of a point from a horosphere. If the point is described by a vector, x, and the horosphere by a vector, q, then this distance is written T(X, q), and it happens that expr(x, q) = \x.q\. For example, the distance of the point from the basic horosphere, #(0), (or x0+x4 = 1), is given by exp T(X, q[0)) = e*. In other words, the distance is here just the time parameter, t. This will prove useful later on. Returning to the physical interpretation, null-vectors imbedded in the hyperboloid are automatically regarded as light rays. At each point of de Sitter space there is a whole cone of these vectors, the light cone. Particles in de Sitter space 285 A word on physical units is also in order here. For convenience we have normalized the speed of light, c, and the radius of the universe, R, to unity. When we come to produce group representations describing quantum mechanics we shall accord to ft the same treatment. These three dimensionally independent quantities suffice to fix all our units. For example, the natural unit of mass is ftjcR. Taking ft ~ 6 x 10~27 erg. sec, c~3x 1010 cm/sec and the conservative estimate R ~ 1027cm (based on red-shift data), then we have for our natural unit of mass something less than 2 x 10~64g. On this scale even an electron has a mass in excess of 1037 units. As far as quantum mechanics is concerned the importance of this cosmolical model lies in the fact that its physical homogeneity and isotropy give it a very high degree of symmetry. It is clear that the group, SO(f, 1), of proper linear transformations leaving invariant the pseudo-Euclidean inner product preserves both the hyperboloid and its metric structure. It is therefore a group of isometries of the Lobachevsky geometry. By general theory (Robertson and Noonan(lO), Chapter 13), it has the highest dimensional possible. In the physically interesting case when / is 4 there are only two other models possessing so large a symmetry group. These are another hyper- bolic model, variously called (3 + 2)-de Sitter space and anti-de Sitter space, and the familiar Minkowski geometry of Special Relativity, with its Poincare" group. For con- venience we shall restrict our attention from the whole group, SO(f, 1), to its con- nected component of the identity, SO0(f, 1). Physically, JS0O(4, 1) is the analogue for de Sitter space of the Poincare group in Special Relativity, and it is usually called the de Sitter group. The general group, SO0(f, 1), is also sometimes given this name (it is also called a generalized Lorentz group by some authors), but in order to distinguish the physical case from the geometrical we shall coin the term Lobachevsky group. If we write the group action on the right then Lobachevsky space can be identified in the usual way with the factor space of right cosets, H'\SO0(f, 1), where H' is the normalizer of a subgroup H isomorpbic to SO0(f— 1,1) (in the case of de Sitter space itself, isomorphic to the Lorentz group). In fact we shall take H to be the subgroup of SO0(f,l) stabilizing the point x(0) = (0,0, 1) of the hyperboloid: H' is then the subgroup fixing x(0) or interchanging it and its antipode. Having this group of symmetries at our disposal we can investigate quantum theory in de Sitter space by constructing invariant wave equations (see Dirac(3)), or irre- ducible representations of the group. (Dixmier(4) gives a complete classification of the irreducible unitary ray representations of the de Sitter group, filling gaps in earner papers. For values of/greater than 4 there are still some lacunae.) One might expect on naive physical grounds that within small regions the cosmo- logical curvature should scarcely be noticeable, and that therefore quantum mechanics in de Sitter space should approximate to a high degree of accuracy the quantum mechanics of Special Relativity. Now, it is in fact known that over short distances and brief periods the electron wave equation in de Sitter space deviates only slightly from the ordinary Dirac equation (3). It is also known that the de Sitter group contracts onto the Poincare" group in the sense of Inonu and Wigner. We shall attempt to show in this paper how one can make a simple direct comparison of those representations of the Poincare and de Sitter groups which describe particles of real mass. 286 K. C. HANNABUSS Although Dixmier used Lie algebraic methods to construct the representations of the de Sitter group, one can also construct its first principal series of irreducible repre- sentations, which seem to be those describing particles with real mass, by a general global procedure, as has been done by Takahashi in (11).
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