This my last normal post. Welcome to those who read me. TSOR is ending and from its ashes will arise another project. That is inevitable. I want to use new TeX packages, and that is not easy here, to simplify things, and to write better things I wish to tell the world.
Roger Penrose formulated twistor theory in the hope of making complex geometry and not the real geometry the fundamental arena of geometric theoretical physics, and a better way to understand quantum mechanics. Quantum Mechanics(QM) is baed on complex structure of Hilbert space of physical states (both of finite or infinite dimensions!). The probability amplitudes are the complex numbers. That is, complex numbers describe oscillations interpreted as probability waves! By the other hand, relativity implies that spacetime points is a real four dimensional (in general D-dimensional) vectors. But this is again a restricted option determined by experiments. That coordinates of spacetime are real numbers is just an hypothesis of our mathematical models, despite the fact it is well supported by experiments! The main difficulty is a consistent formulation of special relativistic quantum theory. Even when possible, that it is in the form of quantum field theory, many questions arise: entanglement, the measurement problem, the collapse of wave functions/state vector reduction and the quantum gravity issue.
Twistor theory was created with the idea of treating real coordinates of spacetime points as composed quantities of more general complex objects called twistors. Therefore, in twistor theory, the most fundamental object are twistors instead of spacetime points. Twistor theory is pointless (in the real sense) geometry.
Mathematically, a first approximation to what a twistor is comes from the conformal spinors. Complex 4-vectors in the fundamental representation of a covering conformal group called is isomorphic to . A correspondence between twistors and the spacetime points is given by the so-called incidence equation or Penrose relation.
The twistor formalism originally introduced by Penrose for 4D spacetime can be extended in two or three ways:
- Extending the Penrose-relation in a supersymmetric way one obtains a correspondence between the supertwistors and the points of superspaces.
- Replacing the complex numbers by quaternions (or octonions, Clifford-numbers) in the Penrose relation, one can bring the quaternionic (octonionic, cliffordonic) twistors into connection with the points of the (D=10, ) spacetime (superspace, C-(super)space). Even more, one can extend this quaternionic twistor formalism in a SUSY fashion introducting quaternionic fermionic degrees of freedom.
- Introducing hypertwistors, and likely hypersuperspace or C-hypersuperspace.
2. 4D spacetime as twistor composite
There is a very pedagogical introduction to 4D twistor theory and the fundamental incidence equation and the Penrose relation. It is well known that ANY spacetime point can be described by the 4D (ND) vector , or . This can be connected and linked to a hermitean matrix, using the Pauli matrics as follows:
This map is one-to-one. We can also consider the complex 4d vector
instead of real 4d vectors. The complex 4-vector describes a point of the complexified Minkovski spacetime . A similar relation to the previous equation gives us the correspondence between the points in this complexified Minkovski spacetime and the 2d complex matrices via . You can get the real Minkovski spacetime by putting the reality condition onto the complex matrix as . A point in the twistor construction or model is the use of the isomorphism between complex 2d matrices and the -plane in 4d complex vector : this is the twistor space . This isomorphism is given by the following correspondence, called Penrose relation:
More explicitly, the matrix are identified with a bitwistor, a couple of twistors :
From a mathematical viewpoint, this gives us an affine system of coordinates for the -plane in the twistor space . This subspace is a complex Grassmann manifold . In other words, the -plane is given by the two linearly independent twistors , the bitwistor in twistor space! This is also a correspondence between the complexified spacetime point and a complex -plane in the twistor space. By the other hand, there is NOT a unique relation between the pair of twistors and the -plane generated by this pair. It is clear, that every pair of twistors or bitwistor is related to nonsingular matrices by , and that that gives the same -plane in the twistor space .
Let the pair has the form of previous matrix, then any equivalent pair of twistors satisfy
and where the complex matrices are constructed of the coordinates of the twistors . Thus, we have
This is the Penrose relation in matrix form! If we write
now we get
and where . In short hand notation, we get
This is the celebrated incidence equation postulated firstly by R. Penrose, also named Penrose relation in his honor. It has a simple physical (geometrical) meaning: the pint corresponds to the twistor
It is evident that all the twistors lying on the -plane given in the same Penrose relation corresponds to a given point and for a given twistor satisfying the incidence equation, only one complex spacetime point is assigned! If anyone need to describe the real spacetime point , one should require the matrix to be hermitian, i.e., to satisfy
Using the notation we introduced above, equivalently
denotes the complex conjugation. In the twistor framework, these equations say that the twistors , the bitwistor, are null-twistors with respect to the norm
Therefore, the reality condition is equivalent to the zero condition for twistors, i.e., to the vanishing of the norm of the bitwistor and couple of twistors. The -planes generated by the null twistor (or congruence) are called totally null planes or congruence relation. In this way, we obtain a set of correspondences:
- Complex planes in twistor space are related one-to-one to points in complexified Minkovski spacetime.
- Complex planes in twistor space are related to totally null planes in twistor space, not one-to-one.
- Totally null planes in twistor space are related one-to-one to points in real Minkovski spacetime.
- Points of complexified spacetime are related to real spacetime, not one to one, to points of real spacetime.
Remark: from the viewpoint of twistor theory, it is more natural to use twistors (couple of twistors indeed, via a bitwistor), for the description of the complex Minkovski spacetime or the null twistor for the description of the real spacetime!
The plan of SUSY is to give a unified mathematical description ob bosonic and fermionic fields. Therefore, one can consider bosons and fermions using the same theoretical background. SUSY or supersymmetry allows us to transorm the descriptions of bosonic fields into fermionic fields and vice versa. In order to have a possible description of bosonic and fermionic fields using the twistor theory, we has to extend it using SUSY. What is SUSY? Surprise…
SUSY replaces the notation of any space-time point by an appropiate superpoint
Here, the superspace point extends spacetime with a new class of numbers, , , , for all . These numbers are called Grassmann numbers. These numbers allow us to handle fermions, since they anticommute themselves. We can define a supervector representing the -extended superspace as follows:
in such a way
Commuting coordinates of any supervector are called bosonic coordinates, anticommuting coordinates (c-numbers) are called fermionic coordinatese. In the same spirit, we could generalize twistor theory and the twistor approach introducing N-extended bosonic supertwistors
and the fermionic N-extended supertwistors
where the quantities are fermionic coordinates ant the quantities are the bosonic degrees of freedom. We will discuss the case, simple supersymmetry, for simplicity. Firstly, two linearly independent supertwistors span (2,0)-superplane int he superspace . In analogy with the no superspace case, we define and get
Here, are complex matrices made up of bosonic elements. This can also be expressed using equations
Then, the supersymmetric extension of Penrose relation reads off
These equations mean that every supertwistor corresponds to a superspace point or superpoint. However, note that it is not the only option to generalize the Penrose relation!
Apply three linearly independent supertwistors , where the latter is a fermionic twistor, such as is our superspace.
where the fermionic supertwistor includes the four fermionic components
and also the bosonic . The (2;1)-superplane is parametrized by a matrix of type with elements satisfying the following generalized incidence relations:
and where the first equation is the bosonic incidence relation, and the second one is the fermionic incidence relation. Thus, this is a different generalization of Penrose relation. For N=1 suspersymmetry, there are 2 possible extensions of Penrose’s relation/incidence equations. In case of the N-extended SUSY, one can generalize those equations in different ways!
4. Quaternionic extension of Penrose incidence in D=6 spacetime
Taking into account the previous section, there are 2 possible approaches to 6D twistors:
- Extend Penrose relation from D=4 to D=6 as it has been done in bibligraphy or following these lines.
- Replace the complex matrices Z by quaternionic ones. Use quaternionic matrices describing naturally a 6d real Minkovski spacetime. The previous approach is equivalent to this one if the description of 6d spacetime is careful.
Consider the first case at the moment. 6d twistors are objects
whose structure is determined by the norm of the spinors for 8d complex orthogonal group given by:
Points in 6d complex Minkovski spacetime are represented by a antisymmetric matrix . The Penrose relation becomes
with . This equation has a nontrivial solution if the twistors are pure, i.e., if
that is, they have vanishing norm. The points of the real 6d spacetime are representeed by complex, antisymmetric matrices satisfying a reality condition in the form of
and denotes the hermitian conjugated matrix. This reality condition for is equivalent to the following equations
The overline means complex conjugation. Indeed, this equation is in fact the condition of vanishing the norm. Thus, 6d twistors describe the points of the real Minkovski spacetime if the following two norms are zero:
The first equation above is the norm, and the second equation is the norm. It means that 6d twistors describing points in are, indeed, invariant under the quaternionic orthogonal group , covering the six dimensional group . The chain:
is true as group isomorphism. Therefore, one can look for the quaternionic extension of 4D twistor formalism which can describe Minkovski spacetime. Quaternions are algebraic objects
and . Real numbers are naturally embedded in quaternions. We can also define quaternionic conjugation
and the norm
The quaternion algebra has a natural structure of euclidean 4d spacetime. Complex numbers can be seen too as certain subset of quaternion algebras. Identifying complex numbers is easy from quaternions, if you take the couple
In analogy to previous arguments, we can associate a -plane in 4D quaternionic space , in quaternionic twistor space as follows: take and associate to it the subspace by columns of quaternionic matrices
By a similar procedure, we get quaternionic Penrose relations
and . The quaternionic twistor will be now . A real 6d Minkovski spacetime point is desecribed by a 6d vector
which can be mapped on a quaternionic hermitian matrix
and , with the imaginary quaternion units. The reality condition , with the plus meaning quaternionic conjugation and transposition, is equivalent to the following condition for quaternionic twistors :
Thus, twistors describe a poitn of if their norm, on , vanishes. Using the decomposition of quaternionic coordinates of twistor in quaternions one can show that Penrose relations are equivalent to the incidence relations, so descriptions of by 6d complex twistors and D=6 quaternions are equivalent.
We can summarize some simple possible definitions of twistors related to the written stuff:
- A twistor is a solution of the twistor equation , that is called twistor space.
- A spinor of the conformal group (with two elements!) is a twistor.
- Point in twistor space=null-line in Minkovski spacetimes.
- Point in Minkovski spacetime=line in twistor space.
Twistors are a powerful tool for spacetime geometry in complex manifolds or even real manifolds. We note that 2 approaches seen are equivalent only for real spacetime, though. 6D spacetime is special. 6D spacetime can be xtended in two nonequivalent ways: by complexification or quaternionization methods! The quaternionic formulation of twistor theory leads to serious issues in general, and that is why it is not popular. The main issue is the quantization of twistors because of non-commutative of quaternions. However, the description of 6d spacetime with quaternionic procedures allow us to use the same geometry as in case of the complex description of 4d spacetime! In fact, it is natural to extend this to octonions and 10d spacetime. The problem there is that octonions are generally non-associative and matrix multiplications become nasty due to that: octonionic matrices are hardly associative!
There can only be one!!!!!!