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). This method has the advantage, too, of working equally well for any Lobachevsky group. It relies heavily upon its exploitation of the well-known decomposition theorem of Iwasawa. This tells us that, if in a non-compact semi-simple Lie group, G, we pick a maximal compact subgroup, K, and a suitable Abelian subgroup, A, then there is a nilpotent subgroup, N, normalized by A, such that any group element, g e G, can be written uniquely as kan, with keK, aeA,neN. Using this one forms the centralizer, M, oiA in K. This con- veniently turns out to normalize N, so that MAN is a subgroup of G. One can form irreducible representations (or ray representations) of this subgroup by lifting a character, x> of A to AN, and taking its product with an irreducible unitary ray representation, D, of M. That is (xD){man) = ^(a)Z)(m), for meM,aeA,neN. This (ray) representation can now be induced to give a (ray) representation of G. Applied to a Lobachevsky group this yields precisely the irreducible unitary ray representations of the first principal series. Now this procedure is somewhat reminiscent of that used to construct the real mass representations of the Poincare group (or, indeed, the corresponding representations f of any of the pseudo-Euclidean groups, R .SO0(f—l, 1)), a fact which can more readily be appreciated when we realize what the group MAN looks like in this parti- cular case. For brevity's sake let us call our Lobachevsky group, G, (G(f) when we wish to specify the fact that it is SO0(f, 1)). We shall write its elements as matrices in block form with an (/— 1)-dimensional square matrix in the middle, surrounded by row and column vectors and scalars. There are, of course, constraints on the entries in order that the matrices lie in G. In fact if T denotes transposition, and if the matrix of the pseudo-Euclidean inner product is /I 0
where 1 is the (/— 1)-dimensional identity matrix, then a condition for A to be in G is ATQA = Q. We notice, incidentally, that Q itself is in G if/ is even. (Q automatically satisfies the above condition, but is an improper transformation if/is odd.) For K we may take I/1 0 0\ 0 a b\: c )' T r l\o c d) and for A r /ch(t) 0 sh(fsh(f))\ a{t) = 1 0 teR 0 \shm 0 chm Particles in de Sitter space 287
Then n(r) =
M is easily identified as a o o\ 0 a 0 1: ae80(f-l) lo o i/ For the de Sitter group, / = 4, we have M ^ #0(3), the rotation group. Taking the matrices to act on the right of the five-vectors describing points in de Sitter space (in accordance with our previously stated convention), the orbit of the point x(0) = (0,0,1) under A is {(sh(t), 0, ch(<)): teB}, that is precisely the world-line of our observer. The orbit of a typical point on this line under the action of N is 3 re* ): rei? }, which is the horosphere forming space at that instant. So A is the group of time translations, N a group of motions in space, and MAN comprises both of these together with the rotations. It is from just such a subgroup of the Poincare" group that we induce its real mass representations. However, there the similarity appears to stop, because the Poincare" group, being the product of the Lorentz group, $0O(3,1), with the space-time translations, realizes the induced representations on functions on a sheet of the hyperboloid, #0O(3, l)/$0(3), whereas the de Sitter group splits in the Iwasawa decomposition as #0(4) AN, (K ^ #0(4)), and its functions are defined on #0(4)/#0(3), a three-sphere. 1. Decompositions. In order to try to circumvent this difficulty let us investigate the geometrical content of the Iwasawa decomposition of G (= G(f)). One can form the right coset space, K\G, on which G acts. This ,is one sheet of the two-sheeted hyperboloid, x.x = 1, or, alternatively, it can be thought of as /-dimensional real Lobachevsky space. In this we have the A -orbit of a point (for example, (1,0,0) has A -orbit {(ch.(t), 0, sh (<))}), and iV^-orbits, or horospheres (like {{ch.(f) + \\r\2et,ret, sh (t) — \ \r\2 e*)}), just as in an imaginary Lobachevsky space. Now it happens that in this space every point lies on just one of the horospheres of the kind given in the parentheses above. So a general point can be written in the form for suitable r and t. This point can be carried back to the point (ch (t), 0, sh (t)) by the action of w(r)-1. If this is followed by an application of ait)-1 then the point is dragged all the way back to the base point, (1,0,0). This procedure can be applied in particular to the point (1,0,0) g, where g is a group element, to get t)-i = (1,0,0) for certain t and r. Since 1 a(t)-x fixes (1,0,0) it must be in K, that is )-i = keK. 288 K. C. HANNABUSS R earranging we have the Iwasawa decomposition. We therefore see that geometrically the Iwasawa decomposition is merely telling us that the hyperboloid K\G can be parametrized by our horospherical coordinates s and t. This suggests that a similar treatment might be possible in imaginary Lobachevsky space, giving us a new decomposition more relevant to the circumstances in which we findourselves . Unfortunately not every point of an imaginary Lobachevsky space is of the form 2 2 (sh (t) + \ \r | e*, re*, ch (t) — \ \r\ e*), for it is clear that such points must have xo + xf > 0. However, since we are identifying antipodes, it is only when x0 + xf actually vanishes that neither of the representative points x and — x satisfies this condition, and this set of recalcitrant points is of measure zero in the space. This suggests the following proposition. Almost every element g e G(f) can be written uniquely in the form of a product han, with aeA, neN, and heH', the normalizer in G(f) of (18 yT 0\ *' > * ° : (fi \ l\0 0 1/ H (If/is even we can write H' explicitly as H u HQ.) We have, in effect, already sketched a geometric proof of this, but it can also be proved by straightforward algebra. We know already that g certainly has an Iwasawa decomposition, g = lcxa^nv This will also be of the form han provided that
nx Since A normalizes N, we see that it will suffice to prove that almost every element of K has a decomposition of the required form. Rearranging this requirement slightly we need only show that for almost all b\ J the equation (8 yT 0\ /I 0 0\ /ch(r) 0 sh(r)\ /l + i|p|2 pT -\\p\^ \fi a 0j= 0 a b 0 1 0 I P 1 ~P 0 ±1/ \0 cT dj \sh(T) 0 ch(T)/ \ J|p|2 pT l-^|p|2/ can be solved for the Greek letters which occur in it (and some choice of the plus or minus sign). Carrying out the matrix multiplication and simplifying we obtain the following set of equations ± l = ed,
p = eb, 0 = c+peTd, 7 = peT, Particles in de Sitter space 289 a. = a + eTbpT, ±1 = - \\
Clearly if d = 0 there can be no solution. Otherwise we take + 1 according as d % 0. Let us suppose first that d > 0. Then we read off
y = -6rH, P = d-*b, a = a — d~1bcT. We can check by direct substitution that these do satisfy all the above equations. If d < 0 we similarly compute
p = c, y = - d^c, a = a-d-xbcT. So that provided we are not in the forbidden null set where d = 0, then we can carry out the decomposition using the above equations. Actually, once we have spotted the geometrical significance of these decomposition theorems, then we can obtain a third type of decomposition of G(f) based on its action on the cone x.x = 0. The subgroup stabilizing the point (1,0,1) is
which is isomorphic to E(f— 1), the Euclidean group in (/— 1) dimensions. If we let
F = «(£)= g then this stability subgroup is the semi-direct product of V with M. Emboldened by our previous success we conjecture that Almost every element of G(f) can be expressed uniquely as a product vman, withve V, meM, aeA, andneN. This is in fact true and there is an algebraic proof following essentially the same lines 290 K. C. HANNABUSS as that of the previous result. It is sufficient to be able to solve for almost every I T I in SO(f) the equation fl 0 0\ /ch(-r) 0 sh(r)\ a i | = |0a&)|0 1 0 ^0 c dj \sh(T) 0 ch(T)y
Our equations this time are (after a little simplification),
l-|£|2 = e^,
a = a + bpTeT,
I =e-T+\p\2er) 1 = -2cTp + (e-T\p\2eT)d, 0 = -2ap + (e-T-\p\2eT)b, whence 2 = (1 + d) eT, which gives, unless d = — 1, the solution eT = 2/(1 + d). (The orthogonality constraints ensure that \d\ ^ 1, so that'there is no possibility of this expression becoming negative.) By taking suitable combinations of the equations we read off successively
a = a-bcTl(l + d) and check that these do satisfy the complete set of equations. This result was actually discovered in a disguised form for the special ease of SO0(4:, 1) by Takahashi (see Lemma 1-5, Chapter II, of Takahashi(H)). 2. Projection and inversion. The three decompositions of G as KAN, H'AN, and VMAN, give rise to three different realizations of the same coset space GjMAN. (The form of these decompositions is such as to make it convenient to use here left co sets rather than the right cosets which we used earlier when describing the Lobachevsky space itself. Of course, by the inversion or transposition of group elements the decomposition theorems above can be made to yield new decompositions like NAK, NAH', and NMAV, but this then partially obscures the simplicity of the geometrical interpretation in the Lobachevsky space.) It is interesting to enquire as Particles in de Sitter space 291 to the interrelationship between the three models of G/MAN which we obtain in this way. Using G = KAN we realise the coset space as an (/— l)-sphere, K/M. Let us suppose that the element g decomposes into the product Ica'n', where
Then the particular coset gMAN can conveniently be represented by the vector (d, b). The constraints on k force d2 + |6|2 = 1, so that this is the usual kind of representation of a sphere. Now most g (excluding only a null set) will also have both other types of decomposi- tion. From the decomposition g = han, supposing that
we can conveniently represent the coset gMAN by the vector (+ 8, ±fl) with S2— |/?|2 = 1. This is again the conventional model of the hyperboloid H'jM. Finally, to represent the coset gMAN in the Euclidean space, VMjM, which arises from the decomposition VMAN, one simply takes the parametrizing vector, £, of v (g = vman). From our algebraic relationships given above, we can immediately read off the relationships between these three realizations of a coset gMAN when g has all three types of decomposition: To (d,b) on the sphere corresponds (d"1,^"1^) on the hyperboloid, and (l + d)-^ in the Euclidean space. But these are just the equations describing stereographic projection between the unit (/— l)-sphere, the unit (/— l)-hyperboloid, and the equatorial hyperplane of the sphere (see Fig. 1). This actually gives us another interpretation of the missing null sets in the decompositions of G, as those whose MAN cosets project off to infinity. It is interesting that just such projections were used by Bander and Itzykson in their study of the de Sitter group as a dynamical group for the hydrogen atom (2). Actually the stereographic projection can be understood in another more abstract way. As subgroups of G, VM and MN are conjugate. We can easily appreciate this fact by recalling that VM was introduced as the subgroup stabilizing the point (1,0,1) on the cone. MN is the subgroup stabilizing the point (1,0, — 1). When/is even and Q is consequently in G, it is easy to see that this is just the element required to conjugate one subgroup into the other, for (1,0, \)Q = (1,0, —1) clearly. Indeed direct calcula- tion reveals that Qma{t) n(r) Q = ma( — t)v( — r). Anyway this fact means that NM\G, like VM\G, is a cone, and so too, taking the corresponding left (?-space, is GjMN. This last can actually be thought of concretely as the null-cone of horospheres. (There is a natural left action on the horospheres given by (xg). q = x. (gq), which is transitive,
19 PSP 70 292 K. C. HANNABTJSS and because the points of the horosphere q(0) were on an iV-orbit, g(0) itself is stabilized by MN.) On this cone A acts naturally by a(gMN) — (ga) MN, this being well denned because A normalizes MN. It is not difficult to check that o(i)~1gr(0) = q(t). In general the orbits of A are the rays generating the cone. But group-theoretically the orbit space of A is the coset space, GjAMN = GjMAN. We therefore have a fourth model for GjMAN: the set of rays on the cone. This is often useful in its own right.
Equatorial hyperplane Hyperboloid
Fig. 1
An H' orbit
A K orbit A VM orbit
Fig. 2
Now one can conveniently visualize this by picking some set on the cone which meets each ray just once. For example, the orbit of q(0) under the action of K is just such a set. It is naturally an (/— l)-sphere if-isomorphic to KjM by the map K: JcM->kMN. One can imbed H'jM and VMjM in the cone similarly by maps 6 and v as orbits of q(0) under H' and V respectively. (d(hM) = hMN, v(vM) = vMN.) These do not meet every ray, but they miss only null-sets of rays, so that for many purposes they are admissible. If one projects the cone down onto its tangent space at q(0) then one sees that the points in which a given ray meets these various sets are stereo- graphically related (see Fig. 2). Particles in de Sitter space 293 The fact that VAM is conjugate to MAN also gives rise to some amusing elementary geometry, for it means that there is a natural correspondence between the points of GjMAN and those of GjMA V. This can be found in a variety of ways ranging from matrix manipulation to pure projective geometry. We shall work rather from the group-theoretic point of view, and for convenience we shall take/to be even. Since MA V = QMANQ it is naturally the stabilizer of the image under Q of the base point of GjMAN. Thus the natural base point, MA V, of GjMA V corresponds to QMAN in GjMAN. Conversely MAN in GjMAN corresponds to QMA V in GjMA V. Both GjMAN and GjMA V can conveniently be identified with Euclidean spaces by using the decompositions G = VMAN and G = NMA V almost everywhere. The identifications are in fact p = v(p) MAN, and r = n(r) MA V. Now, in GjMA V, r is the image of the base-point MA V under n(r), and so it should correspond to the image of QMAN under n(r), that is to n(r) QMAN = Qv( - r) MAN = v(Q(-r))MAN. In fact the image of — r under Q is r/|r|2, as we can easily see by identifying GjMAN with the horosphere on the cone. Then 2 v{-r) = (l + |r|2)-2r,l-|r| ) and Qv(-r) = (l + |r|2,2r, -l + |r|8) which is on the same ray as (|r|-2-l-1, 2r/|r|2,1 — |r|~2) = (r/|r|2). Thus r in GjMAV corresponds to rj\r\2 in GjMAN: geometrically they are inverse points. The corre- spondence between points on the hyperboloid H'jM and in GjMAV is now easily found. For (5,/?) on the hyperboloid corresponds to /?/(l + £) in GjMAN, and this point corresponds to
-1) in GjMAV. This relationship is again one of stereographic projection. (In fact we could have proved this first, and then deduced the inversion formula.) 3. The real mass representations. Returning after this long digression to our original idea, the decomposition of almost all G as H'AN shows that in particular, almost every de Sitter transformation is the composition of something which is essentially a Lorentz transformation, h, a time translation, a, and a spatial translation, n; a decomposition which mimics that of the Poincare group. More generally the Lobachevsky group, (?(/), decomposes in a manner very similar to that of the corresponding pseudo- Euclidean group, Rf.G(f- 1) = E(f- 1, 1). Now when we come to induce representations of (?(/) the fact that null sets are excluded from this decomposition is unimportant, since one is interested only in equivalence classes of functions on a homogeneous space of the group, and not in the 19-2 294 K. C. HANNABITSS functions themselves. (Two functions differing only on a null set are regarded as being the same, so that it does not matter if we exclude a certain null set altogether.) Thus the first principal series of induced representations can be realized on functions whose domain is the hyperboloid, H'jM. This already shows an important difference between the representations of the Lobachevsky group and those of the corresponding pseudo- Euclidean group, as well as an obvious similarity. In both cases one has a representa- tion on functions over a' momentum' hyperboloid. In the case of the pseudo-Euclidean group, however, one need only take one sheet of the two-sheeted hyperboloid (the positive mass shell, to use the terminology of the Poincare group, which is paradigmatic of the general case), whereas in the case of the Lobachevsky group it is necessary to use both sheets. (Because H' is not actually a copy of G(f— 1) but contains it as a subgroup of index two.) Looking at it from the conventional point of view one is being forced to include both positive and negative mass shells together. This is a manifestation of the fact that the energy in de Sitter space is not positive definite (Philips and Wigner(9)). Specializing once more to the case of/ = 4, it is now possible to give a single pre- scription which covers the construction of the 'real mass' representations of both the Poincare and de Sitter groups. Decompose the group into the produce of Lorentz type transformations, time trans- lations and spatial motions. Pick out those Lorentz transformations which commute with the time translations. This will be a subgroup isomorphic to the rotation group. Now take a character of the time translation group {xp say, sending translation through t to e1^), and lift it to the whole of the space-time translations. Pick also an irreducible (ray) representation, Ds, of the rotation subgroup, and take its product with the lifted character to get a representation of the product of the space-time translations with the rotations. This induced to the whole group gives an irreducible representation which seems to describe a particle of mass /t and spin s. These repre- sentations can be taken to act on functions defined on a 'momentum' hyperboloid. This same construction can be applied almost immediately to the general case of the groups G(f) and E(f— 1,1). The only minor change is that Ds will no longer be a representation of the rotation group, but will rather represent the group SO(f— 1), so that the parameter, s, will no longer be an half-integer, but something more complicated. One knows the results of this kind of construction in the case of a pseudo-Euclidean group. There a group element g, representing translation through an /-vector, x, together with a transformation XeG(f- 1), can be represented by U^^ig) where this acts on a function
Here (, ) is the G{f— l)-invariant inner product of/-vectors, (jp,p) = /i2, and h(p) picks a coset representative for the point p, that is h(p) is an element of the group G(f-l) such that h{p)SO(f-l) = peG(f-l)/SO{f- 1). This embodies hi a geo- metrical notation the results of Wigner's investigation of the Poincare' group,(13). For a Lobachevsky group, since we have a two-sheeted hyperboloid, we have to allow our coset representative, h(p), to be in the group H'. Also, because the measure Particles in de Sitter space 295 on this hyperboloid is not invariant under the full group, G, but only quasi-invariant, we must introduce a Radon-Nikodym derivative in compensation when inducing. Then, by general theory (see Appendix), if we call the induced representation Hf>s, we have
(The term ^i(f-l) which has been added to the parameter ft of the character is the promised Radon-Nikodym derivative.) There are many ways of simplifying this expression and making it more explicit, but perhaps the most attractive is to proceed along the following lines. We introduce a coset representative, £(x), for those points, x, of Lobachevsky space not sitting in the wayward null-set. That is H'£,(x) = xeH'\O, or if we write x(0) for the base-point (the coset H' itself) then x(0) E,(x) = x. Next we apply the decomposition theorem, 0 = H'AN, to the element h(p)~1g~1h(p), and deduce that, unless this is in the forbidden null-set, g can be written in the form h(p) anhJi(p)~l. The introduction of A = h(p) hh(p)-x e H', and the point x = x(0) anhlp)-1 of Lobachevsky space enables us to write this as g = h(p)^(xh(p))h(p)~1A. This can be rearranged using the fact that hip)-1 AA(A~1p) e M, which normalizes AN. (Indeed mr^B^x) m = £,{xm) for meM.) So for almost every g we may suppose that g = hWiizMpfiMp)-1*. = A/KA-^zA^A-^^A-V)-1.
Using either of these expressions, together with the definition of the coset representa- tive, h(p), and the fact that AN stabilizes the base point of the momentum hyper- boloid H'/M = GjMAN, we deduce that g~*p = A-1p. Thus, as far as its action on p goes, g looks like the transformation A. The action on x(0), which is stabilized by H' is slightly more adventurous and x(0)g = xA. We can summarize both these facets of its behaviour in the single statement that g acts like a transformation through space- time to the point x, together with an #'-transformation A. (Though care needs to be exercised when interpreting this, since both x and A actually depend on p.) Now using the second expression for g as well as its computed action on p,
This effects the decomposition of h(p)-x gMg^p) into its parts in M and in AN, so that
(H»'{g) $) (p) = Because x is really a character of A lifted to AN its value depends only on that part of ^{xAhlA^p)) which is in A. But if this is a(t) we have seen that
= ki\xAh(A-1p).q{0)\. The action of the group can be thrown across onto the horosphere to give t = ]n\x.Ah(A-1p)q(O)\, 296 K. C. HANNABUSS and then using the fact that q(0) is the image of the base point of the hyperboloid under 6, which is an -ff'-isomorphism, t = ln\x.6{p)\. Incorporating this expression into our formula for the group action
The similarity between this and the corresponding representation of E(f— 1,1), which is already obvious, becomes even more striking when one notices that the inner product (x, p) which occurs in the exponential factor in E/>>s is just fi times the distance of the pseudo-Euclidean point x from the hyperplane whose normal is p. This means that really the sole formal difference between the Lobachevsky group representation and the pseudo-Euclidean representation is the absence of the term £(/— 1) in the latter. The effect of this minor difference should only be noticeable when T(X, 6(p)) becomes large, that is over appreciable portions of the universe. (For known physical particles the mass fi is sufficiently large that even over small distances the exponential cannot be ignored altogether.) If the group element, g, is actually inside the subgroup H, then we have in our decomposition x = x(0), so that everything simplifies down to g = A, independently of p. Because of the way in which the cross-section 6 was chosen x(0).9(p) = 1 for any p, so that the exponential term vanishes leaving us with (-ff<"-8(A) $) (p) = So when restricted to their subgroups O(f— 1), the real mass representations of the Lobachevsky and pseudo-Euclidean groups are formally identical. In the neighbourhood of x(0) we can make some approximations within our general formula. We first notice that if we introduce the ' commutator' a(x, A) = ^(a;)-1 Ag(zA) A-1 then g = £(x) a{xh(p)) A. This commutator is always in ^(x)~x H'E,{x), but when x is close to x(0) and we evaluate it to first order in the distance between x(0) and x then it is found to lie within H' itself. In this case the formula g = £(x) a(x, h(p)) A gives a straightforward decomposition of g into parts in AN and in H'. This shows that in this approximation x is determined by g alone, and does not depend onp. A does still depend onp, but tx(x, h(p)) A does not, and for x close to x(0), a(x, h(p)) is close to the identity, so that the dependence of A on p is a weak one. If x is parametrized by t and r (that is if £(x) = a(0TC(r))> an(i P is taken to be (ji^Po, fi^p) (corresponding to 6(p) = {pr^p^^p, 1), and with the null condition 2 2 forcing pi — \p\ —/t = 0), then, to first order in r and t, T(X, 8{p)) = n~\tp0 — r.p). So to this order of approximation
(#*%) ft (p) = exp ((»-1(/-1)1 fi) (tpo-r.p))D°(h(p)-iAh(A-ip))
The induced representation acts on functions over GjMA V, and it is useful to realise this as the Euclidean space NMjM by utilizing the decomposition G = NMA V (obtained by inverting or transposing G = VMAN). We do this by picking n(r) as the coset representative of the point r in the Euchdean space. If we call the representation induced in this way N* s then and, by the calculations contained in the Appendix, N*8 is equivalent to HP-8 by the map which sends ijr to 0, where (using the precise nature of the correspondence between GjMAN and GjMA V, which we showed to be one of stereographic projection)
The main reason for the interest in iV>>s lies in the fact that the space over which it is realized, NMjM, is a model of the purely spatial part of Lobachevsky space. For space was to be identified with the intrinsically Euchdean horosphere which is the orbit of x(0) under N, and this is preserved under the action of NM. The group of spatial symmetries in the Lobachevsky space is therefore NM, and the space itself can be identified with NMjM. (This is the left coset space equivalent to M\NM, which is how we should write the horosphere as a subspace of the Lobachevsky space which is a right (?-space.) It is interesting to know what the real mass representations look like when restricted to this spatial symmetry group, and N* " readily provides the answer. To simplify the formula for the representation we use several times in various guises the fact that n(r')n(r) = n(r'+r), or equivalently that n(r')~hr = n(r — r'). We need also the fact that mn{r)mrx = n(mr). Then when g = n(r') m,
= n(r' — r) mn(m~1(r — r')) = m Q centralizes M so ) = Ds{m)T/r(mr1(r-r')). 298 K. C. HANNABUSS This is visibly an ordinary position representation for the Euclidean group NM. It is a representation of NM induced from the representation Ds of M, and is therefore localizable in the sense of Mackey and Wightman(6,12). We could actually have deduced this in a more abstract way, from Mackey's theorem on restrictions of induced representations (Mackey(7) p. 128, Theorem E), but the advantage of being so explicit is that we can actually read off the position operators. In the representation N*1'8 they are naturally multiplication by the components of r. Going across to the equivalent representation H?-s, at a point p = (JI^PQ, /i~xp) on the hyperboloid they 1 1 1 will therefore be multiplication by the components o£/i- pHji~ j>0 — 1) = pl(po~l )- In all this we have worked with the fashionable interpretation of how a Lobachevsky space should be split into time and space, but we could do a very similar, if somewhat simpler calculation using de Sitter's interpretation of the model. According to that the observer at the origin of time sees space as being the (/— 1)-sphere which is the orbit of x(0) under K. Thus the spatial symmetry group is K, and space itself is identified with KjM. But this last is precisely the model which one obtains for G/MAN if one uses the Iwasawa decomposition of G. If we make use of this version of G then we can induce a representation, Kf-s, of G equivalent to the previous two. Restriction of this to the sub-group K is entirely analogous to the restriction ofH?'s to H. Taking k{u) E K as coset representative of the point u on the sphere, and ty to be a function defined on the sphere (K» %) f) («) = D'ikiu)-1 gk(g-ht)) f[g-ht) for g e K. This representation is, of course, one induced from M, and so is also localizable in the sense of Mackey and Wightman. From this we can, as before, read off position operators. It is interesting to compare this with the work of Philips and Wigner who introduce a rather different concept of localizability for the group G(2) (Philips and Wigner (9)). In this case K/M is a circle, and functions on it can conveniently be taken to be functions of an angular variable, a. A state rjr can be localized in the interval (— e, e) in the Mackey-Wightman sense by multiplying by the characteristic function of that interval. In particular if xjr is itself that characteristic function then it is already localized in the interval. To compare this with the results of Philips and Wigner ((9), equation 73) we Fourier analyse the characteristic function.
^a) = (!/») TV This is remarkably similar to the localized state of Philips and Wigner, differing from it only by an unimportant multiplicative factor (because our function is not normal- ized), and by a phase factor in each term of the series (the ln of Philips and Wigner). Returning now to the horospherical model of space, we saw that even in the repre- sentation HP> S the position operators were multiplication by certain functions of p. If we were to take too seriously the analogy with the Poincare theory, and think of p as being the momentum, then this would be rather disturbing since it would imply that momentum and position commute, and therefore that there is no uncertainty principle in Lobachevsky space. This, of course, would be too naive an interpretation of the model. Although we have called p the momentum by analogy with the Poincare Particles in de Sitter space 299 case it does not in fact represent the true physical momentum of the system. That must be given by the infinitesimal generators of spatial translations in various direc- tions. For instance, the physical momentum in the direction of the unit vector, j, is the infinitesimal generator, (d/dy) HP-s(n(yj)). This is a differential operator and contains terms dependent on the spin of the particle. Paradoxically the exact expres- sion for the momentum is more easily expressed in the 'position' representation, 2V>'S, where it has to be the usual spin-independent — igrad. The energy here also has a fairly simple form, because (Ni''s(a(t)))f(r) = Differentiating, this gives for the energy operator, or free Hamiltonian,
(This shows, incidentally that our parameter, /i, is in this case minus the physical mass. This is really of no consequence, for, as Takahashi shows, (11), the representations parametrized by /* and — /i are unitarily equivalent.) This section illustrates one of the interesting features of the Lobachevsky spaces, that in them many of the simple properties of flat space become complicated or are lost altogether, but compensating for this many of the difficulties of the ordinary theory disappear. An example of the first kind is provided by the fact that since the compo- nents of physical momentum in a Lobachevsky space do not commute with one another it is impossible to find a representation in which they are all given by position operators. (In particular, as we saw, the 'momentum' hyperboloid does not provide such a representation.) On the other hand the above demonstration of the localizability of the real mass representations compares very favourably with the complicated trans- forms needed for the pseudo-Euclidean groups (see Newton and Wigner (8)). We really only used a stereographic projection of the hyperboloid onto space. It is tempting to try to derive an easier transform for the pseudo-Euclidean groups by looking at what happens to this as the radius of the Lobachevsky space tends to infinity. Of course, this does not in fact lead anywhere, because the scale of space is proportional to its radius, and as this increases without bound a single point blows up to fill the whole of space. The pleasant point transformation of stereographic projection is lost in the process, becoming instead the integral transform of the other theory.
5. The momentum cone. Finally, let us mention the one model of GjMAN which we have not previously used: the rays of the cone. Using this model the induced represen- tation comes out naturally as C^s acting on the functions which are defined to be homogeneous of degree (ift — ^lf— 1)) on the null-cone. The homogeneity ensures that the values of the function on the entire ray are determined by the value at one point of it. It arises because instead of using the full force of the equivariance condition in Mackey's formulation of induced representation theory to replace functions on the group by functions on G/MAN we can use part of the condition to reduce them to functions on the cone, GjMN, and retain the rest as a subsidiary condition. It is this which is the homogeneity condition given. 300 K. C. HANNABUSS has a particularly simple form where qeGjMN, y is a coset representative for the cone, and geG. From this can be deduced the forms of the other equivalent representations, B.^ 8 and Z>>s, by exploiting the homogeneity of/ to restrict its argument to an orbit of H or K, and picking the coset representative accordingly. In some ways the null-cone, GjMN, plays the role of the whole momentum space in the Poincare theory. On it the representations are particularly simple, and in it are imbedded all the' momentum' hyperboloids for different values of the mass parameter. It is interesting that cones rather like this have been investigated recently by those who have an eye to extending the Poincare theory (see, for example, Bakri(D). APPENDIX According to Mackey (7), the representation of G induced from the representation R of a closed subgroup H is defined on functions from G to the representation space of R which satisfy f(gh) = Rify-iftg) for all geG, and heH, and also an appropriate square-summability condition. The induced representation UR is denned by
(Actually our definition differs slightly from that of Mackey because we are using left-hand notation where he uses right-hand notation. There is an equivalence between the two, however, given by/(gr) =/(
Now y(g~1s)H = g-1s = g~1y(s)H, so y(s)~1g~1y(g~1s)eH. Therefore
If q is in G consider now the function fa defined by fq(g)=f(gq)- If keqHq-1 = H«, then
since defining for keH« Particles in de Sitter space 301 Since fq is as square-integrable as /, this calculation shows it to be in the domain of the representation induced from RQ. Now as in the explicit calculations of section 2, HQ is the subgroup of G stabilizing the image of the base-point in S under q. This image is qH, and therefore corresponds to the natural base point H9 of G/Ha. A general point, s of S, is the result of applying the transformation y(s)q~1 to this point, and so corresponds to y(s)q"1Ha in G/Hq. Suppose now that we pick a coset representative 8 for GjHq. Then to any s in S will a correspond a ts in GjH such that
This means that necessarily
1 1 or q~ 8(ts)- y(s)eH. Using this
1 = R(y(s)- 8(ts)q)f(8(ts)q) 1 = R(y(s)- 8(ts)q)f«(8(ts)). It is natural to define a function \jr on GfH" by ijr(t) = f9(8(t)). This is in the natural domain of the representation A721, induced from Rq using the coset representative 8 in much the same way that TR was induced from R using y. Moreover, we have now exhibited explicitly an equivalence between TR and A^ given by
REFERENCES (1) BAKBI, M. M. NUOVO Cimento A 51 (1967), 864r-869. (2) BAUDER, M. and ITZYKSON, C. Rev. Modern Phys. 38 (1966), 330-358. (3) DntAC, P. A. M. Ann. of Math. 36 (1935), 657-669. (4) DIXMTEB, J. Bull. Soc. Math. France 89 (1961), 9-41. (5) GEL'FAITD, I. M., GBAEV, M. I. and VLLENXTN, N. YA. Generalised Functions, Volume 5, Integral geometry and representation theory (Academic Press, New York, 1966). (6) MACKEY, G. W. Bull. Amer. Math. Soc. 69 (1963), 628-686. (7) MACKEY, G. W. Induced representations of groups and quantum mechanics (W. A. Benjamin Inc., New York, Amsterdam; and Editore Boringhieri, Torino, 1968). (8) NEWTON, T. D. and WIGNER, E. P. Rev. Modern Phys. 21 (1949), 400-406. 302 K. C. HANNABUSS (9) PHILIPS, T. and WIGNEB, E. P. In Group Theory and Its Applications, ed. E. M. Loeb (Academic Press, New York, 1968). (10) ROBEBTSON, H. P. and NOONAN, T. W. Relativity and Cosmology (W. B. Saunders Company, Philadelphia, 1968). (11) TAXAHASHI, R. Bull. Soc. Math. France 91 (1963), 289-433. (12) WIGHTMAN, A. S. Rev. Modern Phys. 34 (1962), 845-872. (13) WIGNEB, E. P. Ann. of Math. 40 (1939), 149-204.