Philosophizing Twistors via Fibration

The basic issue, is a question of so called time arrow. This issue is an important subject of examination in mathematical physics as well as ontology of spacetime and philosophical anthropology. It reveals crucial contradiction between the knowledge about time, provided by mathematical models of spacetime in physics and psychology of time and its ontology. The essence of the contradiction lies in the invariance of the majority of fundamental equations in physics with regard to the reversal of the direction of the time arrow (i. e. the change of a variable t to -t in equations). Neither metric continuum, constituted by the spaces of concurrency in the spacetime of the classical mechanics before the formulation of the Particular Theory of Relativity, the spacetime not having metric but only affine structure, nor Minkowski’s spacetime nor the GTR spacetime (pseudo-Riemannian), both of which have metric structure, distinguish the categories of past, present and future as the ones that are meaningful in physics. Every event may be located with the use of four coordinates with regard to any curvilinear coordinate system. That is what clashes remarkably with the human perception of time and space. Penrose realizes and understands the necessity to formulate such theory of spacetime that would remove this discrepancy. He remarked that although we feel the passage of time, we do not perceive the “passage” of any of the space dimensions. Theories of spacetime in mathematical physics, while considering continua and metric manifolds, cannot explain the difference between time dimension and space dimensions, they are also unable to explain by means of geometry the unidirection of the passage of time, which can be comprehended only by means of thermodynamics. The theory of spaces of twistors is aimed at better and crucial for the ontology of nature understanding of the problem of the uniqueness of time dimension and the question of time arrow. There are some hypotheses that the question of time arrow would be easier to solve thanks to the examination of so called spacetime singularities and the formulation of the asymmetric in time quantum theory of gravitation — or the theory of spacetime in microscale.

Although Lorentzian geometry is the mathematical framework of classical general relativity and can be seen as a good model of the world we live in, the theoretical-physics community has developed instead many models based on a complex space-time picture.

(1) When one tries to make sense of quantum field theory in flat space-time, one finds it very convenient to study the Wick-rotated version of Green functions, since this leads to well defined mathematical calculations and elliptic boundary-value problems. At the end, quantities of physical interest are evaluated by analytic continuation back to real time in Minkowski space-time.

(2) The singularity at r = 0 of the Lorentzian Schwarzschild solution disappears on the real Riemannian section of the corresponding complexified space-time, since r = 0 no longer belongs to this manifold. Hence there are real Riemannian four-manifolds which are singularity-free, and it remains to be seen whether they are the most fundamental in modern theoretical physics.

(3) Gravitational instantons shed some light on possible boundary conditions relevant for path-integral quantum gravity and quantum cosmology.  Unprimed and primed spin-spaces are not (anti-)isomorphic if Lorentzian space-time is replaced by a complex or real Riemannian manifold. Thus, for example, the Maxwell field strength is represented by two independent symmetric spinor fields, and the Weyl curvature is also represented by two independent symmetric spinor fields and since such spinor fields are no longer related by complex conjugation (i.e. the (anti-)isomorphism between the two spin-spaces), one of them may vanish without the other one having to vanish as well. This property gives rise to the so-called self-dual or anti-self-dual gauge fields, as well as to self-dual or anti-self-dual space-times.

(5) The geometric study of this special class of space-time models has made substantial progress by using twistor-theory techniques. The underlying idea is that conformally invariant concepts such as null lines and null surfaces are the basic building blocks of the world we live in, whereas space-time points should only appear as a derived concept. By using complex-manifold theory, twistor theory provides an appropriate mathematical description of this key idea.

A possible mathematical motivation for twistors can be described as follows.  In two real dimensions, many interesting problems are best tackled by using complex-variable methods. In four real dimensions, however, the introduction of two complex coordinates is not, by itself, sufficient, since no preferred choice exists. In other words, if we define the complex variables

z₁ ≡ x₁ + ix₂ —– (1)

z₂ ≡ x₃ + ix₄ —– (2)

we rely too much on this particular coordinate system, and a permutation of the four real coordinates x₁, x₂, x₃, x₄ would lead to new complex variables not well related to the first choice. One is thus led to introduce three complex variables u, z₁^u, z₂^u : the first variable u tells us which complex structure to use, and the next two are the

complex coordinates themselves. In geometric language, we start with the complex projective three-space P₃(C) with complex homogeneous coordinates (x, y, u, v), and we remove the complex projective line given by u = v = 0. Any line in P₃(C) − P₁(C) is thus given by a pair of equations

x = au + bv —– (3)

y = cu + dv —– (4)

In particular, we are interested in those lines for which c = −b^–, d = a^–. The determinant ∆ of (3) and (4) is thus given by

∆ = aa^– +bb^– + |a|² + |b|² —– (5)

which implies that the line given above never intersects the line x = y = 0, with the obvious exception of the case when they coincide. Moreover, no two lines intersect, and they fill out the whole of P₃(C) − P₁(C). This leads to the fibration P₃(C) − P₁(C) → R⁴ by assigning to each point of P₃(C) − P₁(C) the four coordinates Re(a), Im(a), Re(b), Im(b). Restriction of this fibration to a plane of the form

αu + βv = 0 —— (6)

yields an isomorphism C² ≅ R⁴, which depends on the ratio (α,β) ∈ P₁(C). This is why the picture embodies the idea of introducing complex coordinates.

∆=a

Such a fibration depends on the conformal structure of R⁴. Hence, it can be extended to the one-point compactification S⁴ of R⁴, so that we get a fibration P₃(C) → S⁴ where the line u = v = 0, previously excluded, sits over the point at ∞ of S⁴ = R⁴ ∪ ∞ . This fibration is naturally obtained if we use the quaternions H to identify C⁴ with H² and the four-sphere S⁴ with P₁(H), the quaternion projective line. We should now recall that the quaternions H are obtained from the vector space R of real numbers by adjoining three symbols i, j, k such that

i² = j² = k² =−1 —– (7)

ij = −ji = k,  jk = −kj =i,  ki = −ik = j —– (8)

Thus, a general quaternion ∈ H is defined by

x^– ≡ x₁ + x₂i + x₃j + x₄k —– (9)

where x₁, x₂, x₃, x₄ ∈ R⁴, whereas the conjugate quaternion x^– is given by

x^– ≡ x₁ – x₂i – x₃j – x₄k —– (10)

Note that conjugation obeys the identities

(xy)^– = y^– x^– —– (11)

xx^– = x^–x = ∑_μ=1⁴ x²_μ ≡ |x|² —– (12)

If a quaternion does not vanish, it has a unique inverse given by

x^-1 ≡ x^–/|x|² —– (13)

Interestingly, if we identify i with √−1, we may view the complex numbers C as contained in H taking x₃ = x₄ = 0. Moreover, every quaternion x as in (9) has a unique decomposition

x = z₁ + z₂j —– (14)

where z₁ ≡ x₁ + x₂i, z₂ ≡ x₃ + x₄i, by virtue of (8). This property enables one to identify H with C², and finally H² with C⁴, as we said following (6)

The map σ : P₃(C) → P₃(C) defined by

σ(x, y, u, v) = (−y^–, x^–, −v^–, u^–) —– (15)

preserves the fibration because c = −b^–, d = a^–, and induces the antipodal map on each fibre.

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