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Homogeneous Spaces of Hilb(n)
for the action of U_g(n);
Complex Lorentzian Hypercones.

               The Isotropic Space Hilb_0(n)

   The algebraic variety Hilb_0(n) defined by those |psi> in Hilb(n) such

     <psi|G(n)|psi> = 0

   is a cone of zero curvature (cones are developable) with real dimension
   2n-1.  An odd real dimensional space cannot possess a complex structure.

        |psi>  ->  exp( i theta ) |psi>

   as in QM, the value of <psi|A|psi> is unchanged.  So as far as these
   quantities are concerned there is the usual phase indeterminacy or gauge
   freedom.  The space ~Hilb_0(n) of ray equivalence classes
   {exp( i theta ) |psi>} then has real dimension 2n-2 or complex
   dimension n-1.

   The image of the isotropic space
   in the operator algebra is the set of X in Alg(n) such that

        X  not=  0
        Tr( X G(n) )  =  0
        X^2  =  a_X X

   for a_X some constant.  The second condition expresses "purity of state"
   and with state normalization imposed, we have a_X = 1.

   Consider the isotropic space for the case n=2.
   The isotropic set is determined by the condition

     |z_0|^2 - |z_1|^2  =  0

for two complex numbers z_0 and z_1. As a real surface in R^4, the previous condition describes a one parameter family of tori T^2 = (T^1 X T^1), where each torus has its major and minor radius equal. See the subsection "spin 1/2 revisited" in [Section XVI]. The complex isotropic set then has clearly three real dimensions. Factoring out the phase, we have a two dimensional cone in a space of one complex dimension and one real dimension to describe ~Hilb_0(2). The real subset of Hilb_0(2) is then a pair of intersecting lines each of one real dimension while the real subset of ~Hilb_0(2) is one real line.

   With n=3

        |z_0|^2 + |z_1|^2 - |z_2|^2  =  0

   which describes a two parameter family of tori

        T^3  = T^2 X T^1,

   where the radii satisfy

     r_0^2 + r_1^2  =  r_2^2

   and is therefore a five real dimensional manifold.

For general n > 2, all the eigenvectors |q(n, k)> and |p(n, k)>, of Q(n) and P(n) are elements of Hilb_0(n). [Lemma 8.1]. As expected from experience with the light cone of Minkowski space, Hilb_0(n) is not a linear space, i. e., not closed under arbitrary linear combinations. We know that the Q(n) eigenbasis |q(n, k)> are isotropic; the P(n) eigenbasis |p(n, k)> are isotropic, and the eigenvectors of t_(+|-)(n) or F_(+|-)(n) defined in [Section XI], also are isotropic.

        |t^(+|-)_k>  =  SIGMA (1/sqrt(n)) exp( (+|-)i(2 pi/n) km ) |n, m>

   For a state that is a linear combination of two of the |q(n, k)>,

        |psi>  =  alpha_k |q(n, k)> + beta_j |q(n, j)>

   we have

             =  2 Re( alpha_k* beta_j  <q(n, k)| G(n) |q(n, j)> )
             =  2 |alpha_k| |alpha_j|  cos( theta_j - phi_k ) (-1)^(k-j)

   with polar representations

        alpha_k  :=  |alpha_k| exp( i theta_k )
        beta_k   :=  |beta_k|  exp( i phi_k )

   and using  Lemma 8.3.  So, <psi|G(n)|psi> is not zero generally but
   only when (theta_j - phi_k) is an half-odd integral multiple of  pi/2.

        |q_kj, (+|-), a_k, b_j>
             :=  |alpha_k| |q(n, k)> (+|-) i |alpha_j| |q(n, j)>

   is in Hilb_0(n), as is any such |psi> transformed
   by an overall phase factor

        |psi> -> exp( i delta ) |psi> 

   (The (+|-)i cannot be replaced with (+|-)1.)
   These states have the symmetry

        (-|+)i |q_kj, (+|-), a_k, b_j>  =  |q_jk, (-|+), a_j, b_k>

   and are of course not all linearly independent.

   <psi|G(n)|psi> attains its maximal absolute value when
   when (theta_j - theta_k) is an integral
   multiple of pi; then,

        (theta_j - theta_k)  =  m  pi
        <psi|G(n)|psi>  =  2 (-1)^(m+k-j) |alpha_k| |alpha_j|

                    >  0,  if (m + k - j) is even
                    <  0,  if (m + k - j) is odd

   From the discussion of the ordering of the eigenvalues of Q(n) after
   diagonalization by XI(n), it can be seen that in the lightlike basis
   |q(n, k)>, that the subspaces spanned by the pairs
   (|q(n, k)>, |q(n, n-1-k)>), span a reducible representation space
   admitting a parity operator for the group ISO(2), the Euclidean group of
   the plane.  If n is odd, the sequence of pairs terminates in a one
   dimensional trivial representation associated with the zero eigenvalue of
   Q(n).  The operators

     K_(+|-)  =  K_1 (+|-) iK_2  =  rho exp( (+|-)i alpha )
     K_3  =  -i(d/d alpha)

   acting on a Hilbert space of functions defined on a circle of radius rho
   in the complex plane, provide a realization of the algebra of the group
   ISO(2).  The Hilbert space has a basis

     f_m( alpha )  =  exp( i m alpha )

   with m = 0, (+|-)1/2, (+|-)1, (+|-)3/2, (+|-)2, ....
   The commutation relations for the Lie algebra

        [K_+, K_-]  =  0,   [K_+, K_3]  =  -K_+,  [K_-, K_3]  =  +K_-

   are easily verified.  Each (|q(n, k)>, |q(n, n-1-k)>) pair is
   associated with eigenvalues that are negatives of each other, take the
   positive value as the value of rho.

        rho_k  =  q(n, k)

   for k = 0, 1, ..., (n-3)/2 if n is odd and
       k = 0, 1, ..., n/2 if n is even.

   For each pair form the basis for the two dimensional
   representation by:

        |(rho_k, alpha; m)>

   =  exp( +i m alpha ) |q(n, k)> DIRECT-SUM exp( -i m alpha ) |q(n, n-1-k)>

   Now form the operators

        K_+-      =  K_+  DIRECT-SUM  K_-
        K_-+      =  K_-  DIRECT-SUM  K_+
        K_(+|-)3  =  K_3  DIRECT-SUM  -K_3

        K_+-|(rho_k, alpha; m)>      =  rho_k |(rho_k, alpha; m+1)>
        K_-+|(rho_k, alpha; m)>      =  rho_k |(rho_k, alpha; m-1)>
        K_(+|-)3|(rho_k, alpha; m)>  =  m |(rho_k, alpha; m)>

   For each two dimensional subspace the parity operator interchanges the
   positive and negative eigenvalues
   q(n, k) and q(n, n-1-k) and so has the form

              |0  1|
              |    |
              |1  0|

   on each of the subspaces of pairs.

When dealing with ISO(2) as an isotropy subgroup of the Poincare group in its action on Minkowski space, customarily, each value of m associated with a two dimensional subspace is associated with a massless particle of spin m. The two dimensional subspace is then associated with the two helicity states of a massless particle: neutrinos are associated with m=(+|-)1/2, photons with m=(+|-)1, presumably gravitons with m=(+|-)2.

In the |n, k> basis, any vector of Hilb_0(n) can be brought by a unitary transformation of U(n-2) into the form that is a multiple of the vector

           |n, n-2> + (1/sqrt(n-1)) |n, n-1>

   and the transformations of the form 

     |exp( i theta )  z  |
     |                   |
     | 0  exp(-i theta ) |

where z is complex and theta real, acting on the space spanned by |n, n-2> and |n, n-1> constitute a reducible but not completely reducible representation of the stability group of IU(n-2). If (z, R(theta)) represents such a transformation, the composition law is expressed by

        (z_1, R(theta_1)) (z_2, R(theta_2))  =

        (z_1 + R(theta_1)z_2, R(theta_1)R(theta_2))

                 Cones in Complex Hermitean spaces and
          Bases of Hilb(n) That Consist of All G(n)-null Vectors/U>

Bases of Hilb(n) That Consist of All G(n)-null Vectors:

Consider the canonical eigenbasis of N(n) in which G(n) is diagonal, and the real subspace R^n of Hilb(n) that it spans. The group of transformations on R^n that preserves G(n) is conjugate in GL(n, R) to O(n-1, 1). Geometrically we can construct a class of "nice" real G(n)-null bases this R^n as follows:

We know that G(n)-null space is a cone in R^n with dimension (n-1). If an (n-1)-dimensional hyperplane parallel to the positive (n-1)-dimensional subspace spanned by the |n, k> for k = 0, 1, 2, ..., (n-2), and orthogonal to |n, n-1> cuts the cone, it intersects the cone in an (n-2)-dimensional hypersurface of positive constant curvature. We know that the vertex angle of the cone is given by

          tan( (1/2)theta_v )  =  sqrt(n-1)

so as n increases, the vertex angle opens up approaching + pi in the limit of infinite n. The angle that a cone vector makes with |n, n-1> is (1/2)theta_v and for any |s> in the positive subspace

             |n, n-1>/sqrt(n-1) +|- |s>

will be G(n)-null, because it has been constructed to lie in the G(n)-null cone. In the positive subspace of R^n construct and (n-1)-dimensional solid sphere with radius 1. Construct a regular (n-1)-simplex (of n points) with centroid at the center of the solid hypersphere. The cross section of the cone at |n, n-1>/sqrt(n-1) is the (n-2) dimensional surface of this solid hypersphere. Construct the vectors from the common center of simplex and solid hypersphere to the n verticies of the simplex, all of which will lie in the (n-2)-dimensional hypersuface, and call these |s_k>. All the |s_k> are G(n)-orthogonal to |n, n-1>, i.e.,

             <n, n-1| G(n) |s_k>  =  0

for all k; but in the positive subspace of R^n, the |s_k> cannot constitute a linearly independent set, since the space is (n-1)-dimensional and there are n vectors; and of course, they cannot be mutually orthogonal with respect to G(n). The |s_k> are not fixed since the orientation of the simplex has not been fixed. All the possible sets of |s_k> using a regular simplex rather, constitute an orbit for the action of the Lie group O(n-1) on the positive subspace of R^n. (More freedom and irregularity can be intrroduced by relaxing the requirement that the simplex be regular, and allowing distortions of the side lengths. The essential requirement of the simplex is that its dimension be (n-1), and that it not be allowed to degenerate so its verticies lie in a subspace of lesser dimension.)

Now construct the set of 2n vectors

          |n_k>(+|-)  =  |n, n-1>/sqrt(n-1) +|- |s_k>

These are all G(n)-null, as are all individual scalings of them. With the positive sign, there are n G(n)-null vectors; they lie in the forward cone of R^n and span R^n, With the negative sign there are again n G(n)-null vectors spanning R^n but these all lie in the backward cone.

Positive real linear combinations of |n_k>+ will fill the interior of the forward cone while positive linear combinations of the |n_k>- will fill the interior of the backward cone.

If the positive real linear combinations are restricted to be also convex, solid n-dimensional cones truncated by an (n-1)-dimensional spherical surface section result.

Now complexify the real space by allowing complex linear combinations of the |n_k>(+|-).

In R^n, the forward and backward cones are disconnected; not so when our real subspace is complexified to Hilb(n). With complex coefficients, the |n_k>+ or the |n_k>- can be used equally. The Invariance group U(G, n) of G(n) in Hilb(n) is conjugate in GL(n, C) to U(n-1, 1) and it is fully connected unlike O(n-1, 1), and so U(G, n) connects the forward and backward cones continuously. Unlike SO(n) and SO(n-1, 1) and SO(G, n) which are doubly connected manifolds, U(n), U(n-1, 1), SU(n), SU(n-1, 1), hence U(G, n) and SU(G, n) are all simply connected.

   Two examples of easily constructed G(n)-null basis.

   G(n)-null basis #1

        For k = 0, 1, 2, ..., n-2

             |n_k>+|-  =  |n, n-1>/sqrt(n-1) +|- |n, k>

        For k = n-1

             |n_(n-1)>+|-   =  (1/sqrt(n-1))(|n, n-1> +|-  SIGMA |n, j>)

     Geometry: The simplex formed has a set of verticies formed
               by the tips of the n-1 unit vectors in the positive
               subspace of R^n, which is the base of the simplex;
               together with the point that is the reflection of
               the origin orthogonal to the simplex-base hyperplane.

   G(n)-null basis #2

        For k = 0

             |m_0>  =  |n, n-1>/sqrt(n-1) +|- |n, 0>

        For k = 1, 2, ..., n-1

         |m_k>  =  |n, n-1>/sqrt(n-1)  +|- (1/sqrtk)( SIGMA |n, j> - |n, k> )

        with the understanding that |n, n> = 0.

     Geometry: The simplex formed has a set of verticies formed
               by a single vertex (say the reflected point in the
               previous example) being identified with tip of the
               unit vector |n, 0>, and the others constituting an
               (n-2)-simplex contained in an (n-2) dimensional
               hyperplane that is orthogonal to |n, 0>, and which
               hyperplane is spanned by the vectors

                    |n, k>

               for k = 1, 2, ..., (n-2)

          A basis for Hilb(n) induces a basis for Alg(n):

With a G-null basis for Hilb(n) we can construct a G-null basis for the algebra Alg(n) of linear operators on Hilb(n). If |v_k> is such a basis then

        V_kj  =  |v_k><v_j|

   spans Alg(n) as a vector space and

        Tr( V_kj G(n) )  =  0

   for all k,j = 0, 1, 2, ..., n-1.
   Then V_kj is basis for Alg(n) and is G-null with respect to the
   pseudometric induced on Alg(n) from by Hilb(n).

   The Geometry of the G(2)-null cone: In C^2

        |z_1|^2 - |z_2|^2  =  0
        r_1  =  r_2

   with 2 unconstrained phases.  As a surface in R^4

        (x_1^2 + y_1^2) - (x_2^2 + y_2^2)  =  0

        (x_1^2 + y_1^2)  =  (x_2^2 + y_2^2)

The complex cone is then a real 3-dimensional space which is a one parameter family of sufaces (T^1 X T^1), the topological product of two circles (2 dimensions), the parameter, supplying the third dimension and being the radius common to both T^1.

Artificially make one of the radii very large. On this large circle attach at every point a copy of the other circle whose plane is orthogonal to the tangent vector of the large circle at the point of attachment. We have generated a torus whose centroid is the large circle. If the large circle radius is contracted to match that of the smaller circle, the doughnut hole of the torus has just collapsed to a point. Thus generate the resulting 3-surface by drawing a circle (the first T^1) and its tangent at a single point. Revolve the circle about the tangent line. (The center of the first T^1 traces out the second T^1.) The swept out surface is the 2-dimensional surface (T^1 X T^1), while its size r is the third parameter. Either of the T^1 together with r determines a cone.

Consider the 3-dimensional solid of which (T^1 X T^1) is the boundary. Before contracting the large radius, say r1, to that of the small radius, so r_1 > r_2, (T^1(r_1) X T^1(r_2)) is a genuine doubly connected T^2. If r_1 < r_2, it looks like the T^2 has lost its double connectedness. Not so, the two complex planes (which contain the the individual T^1's) enter perfectly symmetrically in C^2. What has happened is that a particular projection into an R^3 has lost double connectedness. The single point closure of the doughnut hole is not sufficient to free paths that cross the "interior equator of torus". One has still two classes of paths not continuously deformable to each other: class 1 = paths that are deformable to the closure point, class 2 = paths that are not deformable to the closure point. If a path (that is not a point) goes through the closure point, it is a class 2 path.

        The 2-dimensional cross section of the 3-dimensional G(2)-null
        cone is doubly connected torus with a central hole contracted
        to a point.

Question: A normal 2-dim toroidal surface, unlike the 2-dim sphere, can be covered by a single coordinate patch with no singularities. The toroidal cross section of the G(2)-null cone, appears to have a singularity much like the northpole singularity for the sphere. If the closure point is split into two and pulled apart, the result is toplogically equivalent to a sphere with a latitude and longitude coordinate system with two singularities.

Can one say from this that the G(2)-null cone, is essentially the same topologically (modulo the point surgery) as the Minkowski light cone? Strictly speaking, of course, without the point surgery, there are still two homotopy classes of paths, and so the the topology is that of a toroid.

   Project the z_2-plane to a line: z_2 -> r_2
        (x_1^2 + y_1^2)  =  r_2^2
   which is the equation of a cone in an R^3 as r_2 varies.
        sqrt(x_1^2 + y_1^2)  =  r_2

   Similarly project the z_1-plane to a line: z_1 -> r_1
        (x_2^2 + y_2^2)  =  r_1^2
   which is the equation of a cone in an R^3 as r_1 varies.
        sqrt(x_2^2 + y_2^2)  =  r_1

   If (z_1, z_2) is a cone point
   (wz_1, wz_2), w not= 0 is also a cone point.

   The Geometry of the G(3)-null cone: In C^3

        |z_1|^2 + |z_2|^2 - |z_3|^2  =  0
        r_1^2 + r_2^2  =  r_3^2

   with 3 unconstrained phases.  As a surface in R^6

        (x_1^2 + y_1^2) + (x_2^2 + y_2^2) - (x_3^2 + y_3^2)  =  0

        (x_1^2 + y_1^2) + (x_2^2 + y_2^2)  =  (x_3^2 + y_3^2)

If the LHS is contant, this describes a 3-dimensional spherical surface in an R^4. If the RHS side is constant a T^1 is described. Again as for n=2, the null cone

   is 2n-1 = 5 dimensional surface in R^2n

   Grossly characterized by

          [ 2n-3 dimensional cross section ] X [ size parameter ]

   The cross section has the topology: S^(2n-3) X T^1 (S^4 X T^1)
   while the size parameter has obvious topology R^1.

   Generalizing, the topology of a (Lorentzian) G(n)-null cone in Hilb(n)

        N_1(C, n)  <->  R X [ S^(2n-3) X T^1 ]

   where [ S^(2n-3) X T^1 ] is the cross section.

   Comparing a real Lorentzian null-cone N_1(R, n):

        N_1(R, n):
		Dimension: n-1
		Topology: (R^1 > 0) X S^(n-2)  +  (R^1 < 0) X S^(n-2)
		          forward cone            backward cone
			2 Disconnected pieces
                Cross section Topology: S^(n-2)
			Connected and simply connected.
     N_1(CK, n):
		Dimension: 2n-1
		Topology: (R^1 > 0) X [ S^(2n-3) X T^1 ]
		          Forward cone & backward cone are continuously
			  connected (through T^1).
                Cross section Topology: S^(2n-3) X T^1
			Connected and doubly connected.

   For null cones in spaces with signature 1 < s < n-s,
   other than Lorentian (s=1)
        S^1 = T^1

        N_s(R, n):
		Dimension: n-1
		Topology: (R^1 > 0) X S^(s-2) X S^(2n-s-2)
		          Forward cone & backward cone are continuously
			  connected (through S^(s-1).
                Cross section Topology: S^(s-1) X S^(n-s-2)
			Connected and doubly connected.
			2 classes of paths.
     N_s(C, n):
		Dimension: 2n-1
		Topology: (R^1 > 0) X [ S^(2n-2s-2) X S^(2s-1) ]
		          Forward cone & backward cone are continuously
			  connected (through T^(2s-1).
                Cross section Topology: S^(2n-2s-2) X S^(2s-1)
			Connected and doubly connected.
			2 classes of paths.

   Equalities are homeomorphisms

			     N_s(R, n) = N_n-s(R, n)
			     N_s(C, n) = N_n-s(C, n)

     N_1(C, 2)  =  N_2(R, 4)
     N_1(C, 3)  =  N_2(R, 6)
     N_1(C, 4)  =  N_2(R, 6)
     N_1(C, n)  =  N_2(R, 2n)

     N_2(C, 3)  =  N_1(C, 3)  =  N_2(R, 5)  =  N_3(R, 5)
     N_2(C, 4)                =  N_4(R, 8)
     N_2(C, 5)  =  N_3(C, 5)  =  N_4(R, 10) =  N_6(R, 10)


			     N_s(C, n)  =  N_2s(R, 2n)

            The Complex Projective Space P_n(C)

The complex Projective space can be defined as a homogeneous factor space:

        P_n(C)  =  SU(n+1)/U(n)

P_n(C) is a compact, connected and simply connected Hermitean symmetric space. It can be identified with the "ray space" of Euclidean Hilb(n). With the metric induced from the Euclidean metric on C^(n, it is a complete Kaehlerian manifold of constant scalar curvature, constant positive holomorphic sectional curvature that is also an Einstein space. [Kobayashi 1969], vol. II, pp. 170-171, p. 372.

   Explicitly construct P_n(C) as follows:

   A point of C^(n+1) is an (n+1)-tuple of complex numbers

             z = (z_0, z_1, ..., z_n).

   The elements of P_n(C) are equivalence classes of (n+1)-tuples where
   z is said to be equivalent to w iff for some nonzero complex number

             z  =  lambda w

The coordinates z for C^(n+1) are with many values associated with a point of P_n(C) are not genuine coordinates (nor functions) on P_n(C). They are called homogeneous coordinates for P_n(C). Genuine coordinates, called inhomogeneous coordinates can be constructed by taking

             w_k  =  z_k/z_n

for k = 0, 1, ..., n-1. Any function z -> f(z) on C^(n+1) can be understood as a function on P_n(C) if it is homogeneous of degree 0. A function is called homogeneous of degree r if

             f(lambda z)  =  lambda^r f(z)

The inhomogeneous coordinates are not defined everywhere. No coordinate system can be everywhere defined since P_n(C) is compact. P_n(C) can further identified with the symmetrized product space

                S^2 X S^2 X ... X S^2

of n-1 real 2-spheres, where PI^(n-1) is the permutation group on n-1 objects, in this case 2-spheres. Geometrically, one can visualize an element in P_n(C) as a set of (n-1) unordered points on a 2-sphere, and conversely. [Bacry 1973], and [Bacry 1975]. The vector

          |psi_s>  =  SIGMA |q(n, k)>

is a linear combination of isotropic vectors for which, from Lemma 8.3,


        =    SIGMA  ( delta_(kj) - (-1)^(k-j) )

   where k,j = 0, 1, ..., n-1

        =   - SIGMA  (-1)^(k-j)
            k not= j

        =   2 nu

   for n = 2 nu and n = 2 nu + 1.

and is therefore in Hilb_+(n)
The Complex Hyperbolic Space H_n(C)

The complex Hyperbolic space can be defined as a homogeneous factor space [Appendix D]:

               H_n(C) = SU(n, 1)/U(n)

The interior of the cone of the Hilbert space Hilb(n) is foliated by a one parameter family of complex hyperbolic spaces. The cone has as its boundary the isotropic space, and elements of the interior of the cone are naturally represented as convex linear combinations of the elements of its boundary. In the real case, there are two disconnected hyperbolae; there is only one connected sheet of the complex hyperbolic space. The complex hyperbolic space can be naturally mapped to the interior of a unit ball. See [Appendix D] for details and a few remarks. For r not= 0, define a hypersurface in Hilb(n) by

          <psi| G(n) |psi>  =  -r^2

and the associated ray space by equivalence classes of |psi>,

          {{exp( i theta )|psi>}: <psi| G(n) |psi>  =  -r^2}

the resulting space is a complex hyperbolic space H_(n-1)(C) that is Kaehlerian and of constant negative curvature -(1/r^2). Using +r^2 also gives a space of constant holomorphic curvature +(1/r^2), that is positive and is called a pseudoriemannian complex hypersphere S_(n-1)(C).

   For the vector (Cf.  Lemma 8.6.)

        |psi_->  =  SIGMA  (-1)^k r_k |q(n, k)>

   with r_k > 0,


        =    SIGMA  ( delta_kj - (-1)^(k-j) (-1)^(k+j) r_k r_j )

        =   - SIGMA  r_k r_j  < 0
            k not= j

   For the vector

        |psi_->  =  SIGMA  (i)^k |q(n, k)>


        =  - SIGMA  exp( i (pi/2) (k-j) )
            k not= j

        =  - SIGMA  cos( i( pi/2)(k-j) )

           |  -(n-1),         n  =  2nu + 1
        =  |  -(n-2),         n  =  2nu, and nu is odd
           |    -n,           n  =  2nu, and nu is even

        <  0

   Therefore both vectors are in Hilb_-(n).

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Created: August 1997
Last Updated: August 17, 2000
Last Updated: November 25, 2001