Lecture 8 Global Geometry of Upper Half-Planes

So far, we have been interested in the individual properties of cycles and (relatively) localised properties of the point space. We now describe some global properties which are related to the set of cycles as a whole.

8.1 Compactification of the Point Space

In giving Definitions 10 and 11 of the maps Q and M on the cycle space, we did not properly consider their domains and ranges. For example, the point (0,0,0,1)∈ℙ³ is transformed by Q to the equation 1=0, which is not a valid equation of a conic section in any point space ℝ_σ. We have also not yet accurately investigated singular points of the Möbius map (4). It turns out that both questions are connected.

One of the standard approaches [269]*§ 1 for dealing with singularities of Möbius maps is to consider projective coordinates on the real line. More specifically, we assign, cf. Section 4.4.1, a point x∈ℝ to a vector (x,1), then linear-fractional transformations of the real line correspond to linear transformations of two-dimensional vectors, cf. (1) and (9). All vectors with a non-zero second component can be mapped back to the real line. However, vectors (x,0) do not correspond to real numbers and represent the ideal element, see [296]*Ch. 10 for a pedagogical introduction. The union of the real line with the ideal element produces the compactified real line. A similar construction is known for Möbius transformations of the complex plain and its compactification.

Since we already have a projective space of cycles, we may use it as a model for compactification of point spaces as well, as it turns out to be even more appropriate and uniform in all EPH cases. The identification of points with zero-radius cycles, cf. Exercise 18, plays an important rôle here.

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(a)   (b)
(c)

Figure 8.1: Compactification of ℝσ and stereographic projections in (a) elliptic (b) parabolic and (c) hyperbolic point spaces. The stereographic projection from the point S defines the one-to-one map P↔ Q between points of the plane (point space) and the model—surfaces of constant curvature. The red point and lines correspond to the light cone at infinity—the ideal elements of the model.

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In the elliptic case, the compactification is done by adding to ℝ_e a single point ∞ (infinity), which is, of course, the elliptic zero-radius cycle. However, in the parabolic and hyperbolic cases, singularities of the inversion z↦ 1/z are not localised in a single point. Indeed, the denominator is a zero divisor for the whole zero-radius cycle. Thus, in each EPH case, the correct compactification is made by the union ℝ_σ∪Z_∞.

It is common to identify the compactification ℂ of the space ℝ_e with a Riemann sphere. This model can be visualised by the stereographic projection (or polar projection) as follows—see Fig. 8.1(a) and [36]*§ 18.1.4 for further details. Consider a unit sphere with a centre at the origin of ℝ³ and the horizontal plane passing the centre. Any non-tangential line passing the north pole S will intersect the sphere at another point P and meet the plane at a point Q. This defines a one-to-one correspondence of the plane and the sphere within point S. If point Q moves far from the origin the point P shall approach S. Thus it is natural to associate S with infinity.

A similar model can also be provided for the parabolic and hyperbolic spaces—see Fig. 8.1(b),(c) and further discussion in [339]*§ 10 [130]. Indeed, the space ℝ_σ can be identified with a corresponding surface of constant curvature: the sphere (σ=−1), the cylinder (Fig. 8.1(b), σ=0), or the one-sheet hyperboloid (Fig. 8.1(c), σ=1). The map of a surface to ℝ_σ is given by the polar projection—see Fig. 8.1(a–c) as well as [339]*Figs 129, 135, 179  [130]*Fig. 1. The ideal elements which do not correspond to any point of the plane are shown in red in Fig. 8.1. As we may observe, these correspond exactly to the zero-radius cycles in each case: the point (elliptic), the line (parabolic) and two lines, that is, the light cone (hyperbolic) at infinity. These surfaces provide “compact” models of the corresponding ℝ_σ in the sense that Möbius transformations which are lifted from ℝ_σ to the constant curvature surface by the polar projection are not singular on these surfaces. A modern presentation of the hyperbolic case and its quantum field interpretation can be found in [298].

A more accurate realisation of compact models can be achieved through zero-radius cycles. Exercise 18 tells that a point u+ι v of ℝ_e (ℝ_p, ℝ_h) can be identified with the elliptic (elliptic, hyperbolic) centre of the zero-radius cycle (1,u,v,u²−σ v²). Then, ideal elements at infinity can be associated with zero-radius cycles which do not have finite centres, that is a cycle C=(k,l,m,n) such that: 0

Note that those cycles admit neither k-normalisation nor det_σ-normalisation.

The following observations justify our above classification in the parabolic and hyperbolic cases:

1. Cycles listed in clauses (b) and (c) represent ideal elements, but have non-empty footprint at the finite part of the point space. In contrast, no finite points belong to cycles from clauses (a).
2. Consider the inversion in the unit cycle (1,0,0,−1) which maps a cycle (k,l,n,m) to (m,l,n,k). Cycles listed in clauses (a) and (b) are mapped by the inversion to cycles representing finite points, that is cycles having finite centres. In contrast, the cycles from the clauses (c) are invariant under the inversion, thus represent the same ideal objects after the inversion. As a consequence the later ideal elements are often missed, see the discussion of this in [147] for the hyperbolic case.

Furthermore, the hyperbolic case has its own caveats which may be easily overlooked as in [130, 298], for example. A compactification of the hyperbolic space ℝ_h by a light cone—the hyperbolic zero-radius cycle—at infinity will, indeed, produce a closed Möbius-invariant object or a model of two-dimensional conformal space-time. However, it will not be satisfactory for reasons explained in the next section.

8.2 (Non)-Invariance of The Upper Half-Plane

There is an important difference between the hyperbolic case and the others.

This is illustrated in Fig. 1.3. Any cone from the family (9) intersect both planes EE′ and PP′ over a connected curve (K-orbit—a circle and parabola, respectively) belonging to a half-plane. However, the intersection of a two-sided cone with the plane HH′ is two branches of a hyperbola in different half-planes (only one of them is shown in Fig. 1.3). Thus, a rotation of the cone produces a transition of the intersection point from one half-plane to another and back again.

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t=0

  

#X2192;

  

t=0.25

  

#X2192;

  

t=0.5

t=1

  

#X2192;

  

t=2

  

#X2192;

  

t=4

Figure 8.2: Six frames from a continuous transformation from the future to the past parts of the light cone. Animations as GIF and PDF (requires Acroreader) are provided on the accompanying DVD.

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The lack of invariance of the half-planes in the hyperbolic case has many important consequences in seemingly different areas, for example:

Geometry:

ℝ_h is not split by the real axis into two disjoint pieces: there are continuous paths (through the light cone at infinity) from the upper half-plane to the lower one which do not cross the real axis, cf. a sine-li­ke curve consisting of two branches of a hyperbola in Fig. 8.3(a).

Physics:

There is no Möbius-invariant way to separate the “past” and “future” parts of the light cone [302]*Ch. II, i.e. there is a continuous family of Möbius transformations reversing the arrow of time and breaking causal orientation. For example, the family of matrices (

1	−te1
te1	1

), t∈ [0,∞) provides such a transformation. Figure 8.2 illustrates this by the corresponding images for six subsequent values of t.

Analysis:

There is no possibility of splitting the L₂(ℝ) space of functions into a direct sum of the Hardy-type space of functions having an analytic extension into the upper half-plane and its non-trivial complement, i.e. any function from L₂(ℝ) has an “analytic extension” into the upper half-plane in the sense of hyperbolic function theory—see [170].

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(a)   (b)

Figure 8.3: Hyperbolic objects in the double cover of ℝh. If we cross the light cone at infinity from one sheet, then we will appear on the other. The shaded region is the two-fold cover of the upper half-plane on (a) and the unit disk on (b). These regions are Möbius-invariant.

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All of the above problems can be resolved in the following way—see [302]*§ III.4 [170]*§ A.3. We take two copies ℝ_h⁺ and ℝ_h⁻ of the hyperbolic point space ℝ_h, depicted by the squares ACA′C″ and A′C′A″C″ in Fig. 8.3, respectively. The boundaries of these squares are light cones at infinity and we glue ℝ_h⁺ and ℝ_h⁻ in such a way that the construction is invariant under the natural action of the Möbius transformation. This is achieved if the letters A, B, C, D, E in Fig. 8.3 are identified regardless of the number of primes attached to them.

This aggregate, denoted by ℂ′ , is a two-fold cover of ℝ_h. The hyperbolic “upper” half-plane ℂ′⁺ in ℂ′ consists of the upper half-plane in ℝ_h⁺ and the lower one in ℝ_h⁻, shown as a shaded region in Fig. 8.3(a). It is Möbius-invariant and has a matching complement in ℂ′ . More formally,

The hyperbolic “upper” half-plane is bounded by two disjoint “real” axes denoted by AA′ and C′C″ in Fig. 8.3(a).

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Figure 8.4: Double cover of the hyperbolic space, cf. Fig. 8.1(c). The second hyperboloid is shown as a blue skeleton. It is attached to the first one along the light cone at infinity, which is represented by two red lines. A crossing of the light cone implies a transition from one hyperboloid to another.

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The corresponding model through a stereographic projection is presented in Fig. 8.4. In comparison with the single hyperboloid in Fig. 8.1(c), we add the second hyperboloid intersecting the first one over the light cone at infinity. A crossing of the light cone in any direction implies a swap of hyperboloids, cf. the flat map in Fig. 8.3. A similar conformally-invariant two-fold cover of the Minkowski space-time was constructed in [302]*§ III.4 in connection with the red shift problem in extragalactic astronomy—see Section 8.4 for further information.

8.3 Optics and Mechanics

We have already used many physical terms (light cone, space-time, etc.) to describe the hyperbolic point space. It will be useful to outline more physical connections for all EPH cases. Our list may not be exhaustive, but it illustrates that SL₂(ℝ) not only presents some distinct areas but also links them in a useful way.

8.3.1 Optics

Consider an optical system consisting of centred lenses. The propagation of rays close to the symmetry axis through such a device is the subject of paraxial optics. See [110]*Ch. 2 for a pedagogical presentation of matrix methods in this area—we give only a briefly outline here. A ray at a certain point can be described by a pair of numbers P=(y, V) in respect to the symmetry axis A—see Fig. 8.5. Here, y is the height (positive or negative) of the ray above the axis A and V=ncosv, where v is the angle of the ray with the axis and n is the refractive index of the medium.

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System	Transfer matrices
<img src="epal093.png">	Propagation in a homogeneous and isotropic medium with refractive index n: (       y2 V2 ) = (       1  t/n 0 1 ) (       y1 V1 )
<img src="epal094.png">	A circular boundary between two regions with refractive indices n1 and n2: (       y2 V2 ) = (       1 0 n1−n2 r 1 ) (       y1 V1 )
<img src="epal095.png">	A ray emitted from the focal plane. The output direction v2 depends only on y1: (       y2 V2 ) = (       a b c 0 ) (       y1 V1 )

Figure 8.5: Some elementary optical systems and their transfer matrices

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The paraxial approximation to geometric optics provides a straightforward recipe for evaluating the output components from the given data:

If two paraxial systems are aggregated one after another, then the composite is described by the product of the respective transfer matrices of the subsystems. In other words, we obtained an action of the group SL₂(ℝ) on the space of rays. More complicated optical systems can be approximated locally by paraxial models.

There is a covariance of the theory generated by the conjugation automorphism g: g′↦ gg′g⁻¹ of SL₂(ℝ). Indeed, we can simultaneously replace rays by g P and a system’s matrices by gAg⁻¹ for any fixed g∈SL₂(ℝ). Another important invariant can be constructed as follows. For matrices from SL₂(ℝ), we note the remarkable relation

Subsequently, we define a symplectic form on ℝ² using the matrix J:

Then, this form is invariant under the SL₂(ℝ)-action (2), due to (3):

where P₁=AP and P′₁=AP′. In other words, the symplectic form is an invariant of the covariant action of SL₂(ℝ) on the optical system, cf. Exercise 3.

8.3.2 Classical Mechanics

A Hamiltonian formalism in classical mechanics was motivated by an analogy between optics and mechanics—see [11]*§ 46. For a one-dimensional system, it replaces the description of rays through (y,V) by a point (q,p) in the phase space ℝ². The component q gives the coordinate of a particle and p is its momentum.

Paraxial optics corresponds to transformations of the phase space over a fixed period of time t under quadratic Hamiltonians. They are also represented by linear transformations of ℝ² with matrices from SL₂(ℝ), preserve the symplectic form (3) and are covariant under the linear changes of coordinates in the phase space with matrices from SL₂(ℝ).

For a generic Hamiltonian, we can approximate it by a quadratic one at the infinitesimal scale of phase space and time interval t. Thus, the symplectic form becomes an invariant object in the tangent space of the phase space. There is a wide and important class of non-linear transformations of the phase space whose derived form preserves the symplectic form on the tangent space to every point. They are called canonical transformations. In particular, Hamiltonian dynamics is a one-parameter group of canonical transformations.

8.3.3 Quantum Mechanics

Having a transformation φ of a set X we can always extend it to a linear transformation φ^* in a function space defined on X through the “change of variables”: [φ^* f](x)=f(φ(x)). Using this for transformations of the phase space, we obtain a language for working with statistical ensembles: functions on X can describe the probability distribution on the set.

However, there is an important development of this scheme for the case of a homogeneous space X=G / H. We use maps p:G→ G/H, s: X → G and r: G → H defined in Subsection 2.2.2. Let χ: H → B(V) be a linear representation of H in a vector space V. Then, χ induces a linear representation of G in a space of V-valued functions on X given by the formula (cf. [159]*§ 13.2.(7)–(9))

where g∈ G, x∈ X and h∈ H. “*” denotes multiplication on G and “·” denotes the action (3) of G on X from the left.

One can build induced representations for the action SL₂(ℝ) on the classical phase space and, as a result, quantum mechanics emerges from classical mechanics [199, 196]. The main distinction between the two mechanics is encoded in the factor χ(r(g⁻¹ * s(x))) in (5). If this term takes complex values then there is self-interaction of functions in linear combinations. Such an effect is natural for wave packets rather than the classical statistical distributions. It is the common believe that non-commutativity of observables is the characteristic feature of quantum mechanics, however a closer consideration [200] reveals the key role of complex numbers instead.

8.4 Relativity of Space-Time

Relativity describes an invariance of a kinematics with respect to a group of transformations, generated by transitions from one admissible reference system to another. Obviously, it is a counterpart of the Erlangen programme in physics and can be equivalently stated: a physical theory studies invariants under a group of transformations, acting transitively on the set of admissible observers. One will admit that group invariance is much more respected in physics than in mathematics.

We saw an example of SL₂(ℝ) (symplectic) invariance in the previous section. The main distinction is that the transformations in kinematic relativity involve time components of space-time, while mechanical covariance was formulated for the phase space.

There are many good sources with a comprehensive discussion of relativity—see, for example, [42, 339]. We will briefly outline the main principles, only restricting ourselves to two-dimensional space-time with a one-dimensional spatial component. We also highlight the role of subgroups N′ and in the relativity formulation to make a closer connection to the origin of our development—cf. Section 3.3.

It is easy to see directly that parabolic cycles make an invariant family under the transformations (6). These parabolas are graphs of particles moving with a constant acceleration; the acceleration is the reciprocal of the focal length of this parabola (up to a factor). Thus, movements with a constant acceleration a form an invariant class in Galilean mechanics. A particular case is a=0, that is, a uniform motion, which is represented by non-vertical straight lines. Each such line can be mapped to another by a Galilean transformation.

The class of vertical lines, representing sets of simultaneous events, is also invariant under Galilean transformations. In other words, Galilean mechanics possesses the absolute time which is independent from spatial coordinates and their transformations. See [339]*§ 2 for a detailed examination of Galilean relativity.

A different type class of transformations, discovered by Lorentz and Poincare, is admitted in Minkowski space-time.

The admissible values v∈(−1,1) for transformations (7) are bounded by 1, which serves as the speed of light. Velocities greater than the speed of light are not considered in this theory. The fundamental object preserved by (7) is the light cone:

More generally, the following quadratic form is also preserved:

The light cone C₀ is obviously the collection of points d_h(t,x)=0. For other values, we obtain hyperbolas with asymmptotes formed by C₀. The light cone separates areas of space-time where d_h(t,x)>0 or d_h(t,x)<0. The first consists of time-like intervals and the second of space-like ones. They can be transformed by (7) to pure time (t,0) or pure space (0,x) intervals, respectively. However, no mixing between intervals of different kinds is admitted by Lorentz–Poincare transformations.

Furthermore, for time-like intervals, there is a preferred direction which assigns the meaning of the future (also known as the arrow of time) to one half of the cone consisting of time-like intervals, e.g. if the t-component is positive. This causal orientation [302]*§ II.1 is required by the real-world observation that we cannot remember the future states of a physical system but may affect them. Conversely, we may have a record of the system’s past but cannot change it in the present.

However, the causal orientation is not preserved if the group of admissible transformations is extended to conformal maps by an addition of inversions—see Fig. 8.2. Such an extension is motivated by a study of the red shift in astronomy. Namely, spectral lines of chemical elements (cf. Example 7) observed from remote stars are shifted toward the red part of the spectrum in comparison to values known from laboratory measurements. Conformal (rather than Lorentz–Poincare) invariance produces much better correlation to experimental data [302] than the school textbook explanation of a red shift based on the Doppler principle and an expanding universe.

Further discussion of relativity can be found in [339]*§ 11 [42].