The problem is that a space with this metric signature does not conform with the standard view about causality. The challenge is to find a physical interpretation consistent with the metric signature of Minkowski space: somehow M4 or at least light-cone boundary should be mapped to twistor space. The (2,2) resp. (4,4) signature of the metric of the target space is a problem of also N=2 resp. N=4 super-conformal string theories, and N=4 super-conformal string theory could be relevant for quantum TGD. The identification of the target space of N=4 theory as twistor space T looks natural since it has metric with the required real signature(4,4).
Number theoretical compactification implies dual slicings of the space-time surface to string world sheets and partonic 2-surfaces. Finite measurement resolution reduces light-like 3-surfaces to braids defining boundaries of string world sheets. String model in T is obtained if one can lift the string world sheets from CD×CP2 to T (CD denotes causal diamond defined as intersection of future and past directed light-cones). It turns out that this is possible and one can also find an interpretation for the phases associated with the spinors defining the twistor.
1. General remarks
Some remarks are in order before considering detailed proposal for how to achieve this goal.
- Penrose ends up with the notion of twistor by expressing Pauli-Lubanski vector and four-momentum vector of massless particle in terms of two spinors and their conjugates. Twistor Z consists of a pair (μa, λa') of spinors in representations (1/2,0) and (0,1/2) of Lorentz group. The hermitian matrix defined by the tensor product of λa and its conjugate characterizes the four-momentum of massless particle in the representation paσa using Pauli's sigma matrices. μa characterizes the angular angular momentum of the particle: spin is given by s=ZαZbarα. The representation is not unique since λa is fixed only apart from a phase factor, which might be called "twist". The phases of two spinors are completely correlated.
- This interpretation is not equivalent with that discussed mostly in Witten's paper. Two-component spinors replace light-like momentum also in this approach as a kinematic variable and a phase factor emerges as an additional kinematic variable. Scattering amplitudes are therefore not functions of momenta and polarizations but of a spinor, its conjugate defining light-like momentum, and helicity having values ±1. Fourier transform with respect to spinor or its conjugate gives scattering amplitude as a function of a twistor variable.In Minkowski space with Lorentz signature the momentum as kinematic variable is replaced with spinor and its conjugate and spinor is defined apart from a phase factor. In the article of both Witten and Nima and collaborators the signature of Minkowski space is taken to be (2,2) so that the situation changes dramatically. Light rays assignable to twistors are 2-D light-like light-like surfaces and the spinor associated with light-like point decomposes to two independent real spinors replacing light-like momentum as a kinematic variable. The phase factor as an additional kinematic variable is replaced by a real scaling factors t and 1/t for the two spinors. Fourier transform with respect to the real spinor or its conjugate is possible and gives scattering amplitude as a function of a twistor variable. In Lorentz signature the twistor Fourier transform in this sense is not possible so one cannot replace spinor and its conjugate by a twistor.
- Twistor space -call it T- has Kähler metric with complex signature (2,2) and real signature (4,4) and could correspond to the target space of N=4 super-conformally symmetric string theory with strings identified as T lifts of the string world sheets. The minimum requirement is that one can assign to each point of string world sheet a twistor.
2. What twistor Fourier transform could mean in TGD framework?
The twistor transform described in Wittens' article deserves some remarks.
- From Witten's paper one learns that twistor-space scattering amplitudes obtained as Fourier-transforms with respect to the conjugate spinor correspond in Minkowski space correspond to incoming and outgoing states for which the wave functions are not plane waves but are located to sub-spaces of Minkowski space defined by the equation
In a more familiar notation one has xμσμλ = μ. The solution is unique apart from the shift xμ→ xμ+ kpμ, where pμ is the light-like momentum associated with λ identified as a solution of massless Dirac equation. Clearly, twistor transform corresponds to a wave function located at light-like ray of δM4+/- and momentum eigen state is represented as a superposition of this kind of wave functions localized at parallel light rays in the direction of momentum and labeled by μ.
- If the equivalent of twistor Fourier transform exists in some sense in Lorentz signature, the geometric interpretation would be as a decomposition of massless plane wave to a superposition of wave functions localized to light-like rays in the direction of momentum. Uncertainty Principle does not deny the existence of this kind of wave functions. These highly singular wave functions would be labeled by momentum and one point at the light ray or equivalently (apart from the phase factor) by λa and μa defining the twistor. The wave functions would be constant at the rays and thus wave functions in a 3-dimensional sub-manifold of M4 labeling the light rays. This sub-manifold could be taken light-cone boundary as is easy to see so that the overlap of different wave functions would take place only at the tip of the light-cone. Fields in twistor space would be fields in the space of light-rays characterized by a wave vector. Since twistor Fourier transform does not work, one must invent some other manner to introduce these wave functions. Here the lifting of space-time surface to twistor space suggests itself.
- One can assign to space-time sheet conserved four-momentum identifiable by quantum classical correspondence as its quantal variant. This option would fix λ to be same at each point of the space-time surface about from a possible phase factor depending on space-time point. The resulting surfaces in twistor space would be rather boring.
- Hamilton-Jacobi coordinates suggest the possibility of defining λ as a quantity depending on space-time point. The two light-like M4 coordinates u,v define preferred coordinates for the string world sheets Y2 appearing in the slicing of X4(X3l), and the light-like tangent vectors U and V of these curves define a pair (λ,tildeμ) of spinors defining twistor Z. The vector V defining the tangent vector of the braid strand is analogous to four-momentum. Twistor equation defines a point m of M4 apart from a shift along the light ray defined by V and the consistency implying that the construction is not mere triviality is that m corresponds to the projection of space-time point to M4 in coordinates having origin at the tip of CD. One could distinguish between negative and positive energy extremals according to whether the tip is upper or lower one. One can assign to λ and tildeμ also two polarization vectors by a standard procedure to be discussed later having identification as tangent vectors of coordinate curves of transversal Hamilton-Jacobi coordinates. This would give additional consistency conditions.
- In this manner space-time surface representable as a graph of a map from M4 to CP2 would be mapped to a 4-surface in twistor space apart from the non-uniqueness related to the phase factor of λ. Also various field quantities, in particular induced spinor fields at space-time surface, could be lifted to fields restricted to a 4-dimensional surface of the twistor space so that the classical dynamics in twistor space would be induced from that in imbedding space.
- This mapping would induce also a mapping of the string world sheets Y2 Ì PM4(X4(X3l)) to twistor space. V would determine λ and U -taking the role of light-cone point m - would determine tildeμ in terms of the twistor equation. 2-surfaces in twistor space would be defined as images of the 2-D string world sheets if the integrability of the distribution for (U,V) pairs implies the integrability of (λ,tildeμ) pairs.
- Twistor scattering amplitude would describe the scattering of a set of incoming light-rays to a set of outgoing light-rays so that the non-locality of interactions is obvious. Discretization of partonic 2-surfaces to discrete point sets would indeed suggest wave functions localized at light-like rays going through the braid points at the ends of X3l as a proper basis so that problems with Uncertainty Principle would be overcome. The incoming and outgoing twistor braid points would be determined by M4 projections of the braid points at the ends of X3l. By quantum classical correspondence the conservation law of classical four-momentum defined would apply to the total classical four-momentum although for individual braid strands classical four-momenta would not conserved. The interpretation would be in terms of interactions. The orbits of stringy curves connecting braid points wold give string like objects in T required by N=4 super-conformal field theory.
3. Could one define the phase factor of the twistor uniquely?
The proposed construction says nothing about the phase of the spinors assigned to the tangent vectors V and U. One can consider two possible interpretations.
- Since the tangent vectors U and V are determined only apart from over all scaling the phase indeterminacy could be interpreted by saying that projective twistors are in question.
- If one can fix the absolute magnitude of U and V -say by fixing the scale of Hamilton Jacobi coordinates by some physical argument- then the map is to twistors and one should be able to fix the phase.
It turns out that the twistor formulation of field equations taking into account also CP2 degrees of freedom to be discussed latter favors the first option. The reason why the following argument deserves a consideration is that it would force braid picture and thus replacement of space-time sheets by string world sheets in twistor formulation.
- The phase of the spinor λa associated with the light-like four-momentum and light-like point of δM4+/- should represent genuine physical information giving the twistor its "twist". Algebraically twist corresponds to a U(1) rotation along closed orbit with a physical significance, possibly a gauge rotation. Since the induced CP2 Kähler form plays a central role in the construction of quantum TGD, the "twist" could correspond to the non-integrable phase factor defined as the exponent of Kähler magnetic flux (to achieve symplectic invariance and thus zero mode property) through an area bounded by some closed curve assignable with the point of braid strand at X2. Both CP2 and δM4+/- Kähler forms define fluxes of this kind so that two kinds of phase factors are available. CP2 flux however looks a more natural choice.
- The symplectic triangulation defined by CP2 Kähler form allows to identify the closed curve as the triangle defined by the nearest three vertices to which the braid point is connected by edges. Since each point of X4(X3l) belongs to a unique partonic 2-surface X2, this identification can be made for the braid strands contained by any light-like 3-surface Y3l parallel to X3l so that phase factors can be assigned to all points of string world sheets having braid strands as their ends. One cannot assign phases to all points of X4(X3l). The exponent of this phase factor is proportional to the coupling of Kähler gauge potential to fermion and distinguishes between quarks and leptons.
- The phase factor associated with the light-like four-momentum defined by V could be identified as the non-integrable phase factor defined by -say- CP2 Kähler form and would give the phase of λ1. The basic condition relating μ to λ would fix the phase of μ. Note that the phases of the twistors are symplectic invariants and not subject to quantum fluctuations in the sense that they would contribute to the line element of the metric of the world of classical worlds. This conforms with the interpretation as kinematical variables.
- Rather remarkably, this construction can assign the non-integrable phase factor only to the points of the number theoretic braid for each Y3l parallel to X3l so that one obtains only a union of string world sheets in T rather than lifting of the entire X4(X3l) to T. The phases of the twistors would code for non-local information about space-time surface coded by the tangent space of X4(X3l) at the points of stringy curves.
For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".