Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = vawa = 0. By definition, their null tangent vectors have the two-component spinor form λAπA, where λA is varying and πA is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λAπA μAπA = 0 = η(v, v) = λAπA λAπA . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field πA is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP3. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.


4 thoughts on “Revisiting Twistors

  1. Hello again.

    This is it particularly about this post here.

    I am wondering: classical and quantum physics.

    For my lame in mind, my understanding is that classical physics breaks down at certain levels. Quantum physics bus explains a level of reality that defies classical.

    Is there an intermediary? I mean is there some sort of math or can you explain to me the intersection between these two types of physics?

    Is there a Segway, the continuity of some sort whereby what is classical becomes quantum and what is quantum moves into classical?

    Does that make any sense?

  2. In the quantum world, pairs of photons are “entangled” – connected so that measurements performed on one affect the other, even when separated by great distances. This concept – which Albert Einstein called “spooky action at a distance” – leads to another counter-intuitive claim: that, when unobserved, the photons exist in all possible states simultaneously.

    Using a well-established technique called broadband four-wave mixing (FWM), when a laser is through an optical fiber, it generates entangled photon pairs, or bi-photons. FWM is an important source of single bi-photons for quantum communication schemes, especially for in-fiber applications.

    Rather than generating the bi-photons with high- or low-power laser pulses – which would cause the system to demonstrate either pure classical or quantum behavior – the focus is on the intermediate power regime. At this intermediate power level, one is able to observe the transition point where the cross-over between ‘spooky’ quantum behavior and ‘classical’ wave physics takes place.

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