Quantum Field Theory And Quantum Gravity In Twistor Space

Peter Woit (author of the "Not Even Wrong" blog and a math professor at Columbia University) is working on developing a Twistor space based unification of the Standard Model and gravity. He feels he has made some breakthroughs and has provided to his blog readers a 48 page pre-pre-print working draft of his latest paper on the topic. I'll lay out what he's doing as simply as possible below, and the below the fold provide some additional quoted details about it.

In the real world of physics we directly measure only real number valued quantities (the real numbers are all of the integers, i.e. ... -3, -2, -1, 0, 1, 2, 3 . . . , and all of the numbers in between the integers).

But, mathematically, complex number valued quantities, in which each number has a real number part and an imaginary number part, have a lot of attractive properties (among other things, it is more "complete" in that it is harder to do mathematically operations that produce undefined quantities like taking the square root of negative numbers in the real numbers, and can store the same amount of information in half as many numbers).

Twistor physics is basically a way to describe physics by transforming real number valued point-like physical properties and objects (scalars) and real number valued directional physical properties (vectors), into a closely related parallel description in terms of complex valued scalars and vectors.

Euclidian Twistor space transforms the space-time of classical physics into a complex valued twistor space. Minkowski Twistor space transforms the space-time of quantum field theory, in which special relativity is observed, into a complex valued twistor space. 

There is a way to generalize concepts in Euclidian Twistor space into Minkowski Twistor space, although for a variety of technical reasons, this is much trickier than it seems because the underlying mathematical structure of Euclidian Twistor space is not parallel in a crude way to Minkowski Twistor space that is formulated in a quite different way. 

To make this transformation from Euclidian Twistor space to Minkowski Twistor space, you have to identify an imaginary time direction in Twistor space, and the key to doing that in physics applications involves the Higgs field.

It turns out that Standard Model physics can be described in a very natural and comparatively easy to mathematically manipulate form by describing it in a non-physical Minkowski Twistor space. Spinors (which are complex valued objects sometimes crudely described as the square root of vectors) in Twistor space are a very compact and useful way to describe fundamental particles in Standard Model physics. It may even be that the non-physical Twistor space-spinor description of the world is more fundamental than the world we actually observe and that the world we actually observe is emergent from this more fundamental description of it.

In particular, Peter Woit has observed that, in a Twistor space formulation:

Exactly the internal symmetries of the Standard Model occur.

The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.

Conformal symmetry is built into the picture in a fundamental way.

A Twistor space formulation, among other things, tames some of the mathematical wildness and intractability of the general case of quantum gravity by imposing a conformal symmetry (which does not a priori hold in four dimensional general relativity, although a conformal symmetry version of general relativity has been actively explored in the classical case, and some authors have argued that this may even address some dark matter and dark energy issues). 

For a given spacetime and metric, a conformal transformation is a transformation that preserves angles, and a conformal symmetry is closely related to properties of a physical law that apply without respect to scale (i.e. scale invariance), but in a more mathematically sublime manner that is relevant to preserving renormalizability in quantum field theories. 

So, by imposing a very subtle additional symmetry upon general relativity, it can be made to be much better behaved mathematically, which is a big deal because one of the main problems with quantum gravity has been its non-renormalizability and its general mathematical intractability.

Peter Woit has also concluded that in a Twistor space approach that "One gets a new chiral formulation of gravity, unified with the Standard Model."

The group theory formulation of the Standard Model is SU(3) x SU(2) x U(1), where SU(3) is the color (strong force) part, the SU(2) is the weak (left handed) part, and the U(1) is the electromagnetic part.

The approach Peter Woit is taking to unify the Standard Model and Quantum Gravity is basically to observe that you can describe gravity with the right handed SU(2) group that is an exact counterpart to the left handed SU(2) group that describes the Standard Model weak force, into a combined real valued SU(4) group, which corresponds to, in the necessary ways, the complex valued SL(2,C) group that can be found in the appropriate Twistor space. The right handed SU(2) group has the same number of degrees of freedom and the proper interrelationships among its parts necessary to describe gravity.

This is an approach that I've previously described as a gravito-weak unification.

There are details to be worked out, but it is one of the more promising approaches to quantum gravity, along with the QCD squared approach, emergent gravity approaches (with a thermodynamic/entropy/holographic orientation), the string theory concept of a graviton (and Twister theory and string theory are not necessarily inherently incompatible), supersymmetric supergravity theories, and  loop quantum gravity oriented approaches.

Woit's approach, if it worked, would eliminate the inherent theoretical inconsistencies between classically formulated general relativity, and the quantum mechanically formulated Standard Model of Particle Physics, in a mathematically tractable way, which would be a big deal. 

But, it would not, on its own, do anything that existing classical general relativity (possibly refined to include a conformal symmetry) does not do, to address issues like dark matter and dark energy. It would solve those problems only if general relativity, or general relativity with conformal symmetry, can already do that in the classical case (as some investigators, who are minority in the scientific community, have claimed).

Woit's approach also does not include a "grand unified theory" that describes all of the forces of nature in terms of a single unified group of which the four known forces of nature are merely parts, or with a unification of all of the fundamental forces as aspects of the same underlying force at high enough energies.

More Technical Background

Roger Penrose is commonly credited with the developing the concept of a "twistor" in physics applications (in 1967).

Mathematically, projective twistor space ${\displaystyle \mathbb {PT} }$ is a 3-dimensional complex manifold, complex projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$. It has the physical interpretation of the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space ${\displaystyle \mathbb {T} }$ with a Hermitian form of signature (2,2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weyl) spinors for the conformal group ${\displaystyle SO(4,2)/\mathbb {Z} _{2}}$ of Minkowski space; it is the fundamental representation of the spin group ${\displaystyle SU(2,2)}$ of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group.

In its original form, twistor theory encodes physical fields on Minkowski space into complex analytic objects on twistor space via the Penrose transform.

These are in turn related to spinors.

In geometry and physics, spinors /spɪnər/ are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).

It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.

Spinors are characterized by the specific way in which they behave under rotations. 

. . .

To obtain the spinors of physics, such as the Dirac spinor, one extends the construction to obtain a spin structure on 4-dimensional space-time (Minkowski space). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fibre bundle, the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation, or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves, having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called fermions; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another.

It appears that all fundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino. There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ_2,2(${\displaystyle \mathbb {R} }$), the Majorana spinor. There also does not seem to be any particular prohibition to having Weyl spinors appear in nature as fundamental particles.

The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

Weyl spinors are insufficient to describe massive particles, such as electrons, since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed. The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor. However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors. It is not known whether Weyl spinor fundamental particles exist in nature. 

Peter Woit has been active in further developing it. In a recent post, he sums up what he finds promising about this path to unifying the quantum field theory of the Standard Model and quantum gravity in Twistor space:

There are several (to me at least…) attractive aspects:

• Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
• Analytic continuation between Minkowski and Euclidean space-time can be naturally performed, since twistor geometry provides their joint complexification.
• Exactly the internal symmetries of the Standard Model occur.
• The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
• One gets a new chiral formulation of gravity, unified with the Standard Model.
• Conformal symmetry is built into the picture in a fundamental way.

There’s more in this version about how quantum gravity fits into this, when formulated in terms of chiral variables (i.e. Ashtekar variables). This gives a new context for old questions about quantizing in these variables (this is in Eucldean signature, the other chirality is not space-time geometry but internal Yang-Mills geometry, and the imaginary time component of the vierbein is distinguished and given the dynamics of a Higgs field). I haven’t spent much time on this yet, but suspect this new context may help overcome problems that people trying to pursue quantum gravity in this chiral connection framework have run into in the past.

One common reaction I’ve gotten to these ideas is the one I myself had in the past: analytic continuation relates expectation values of field operators in Euclidean and Minkowski signature, so my left-handed SU(2) after analytic continuation gives part of Lorentz symmetry, not an internal symmetry. What took me a long time to realize is just how different Euclidean and Minkowski signature QFT is. Yes, Schwinger functions and Wightman functions can be related by analytic continuation (in a rather subtle way, the Wightman functions aren’t functions, but boundary values of holomorphic functions). But at the level of states and operators things are very different. It’s just not true that there is some holomorphic formulation of QFT states and operators, with Euclidean and Minkowski space restrictions related by analytic continuation. There’s a lot of explanation about this in the paper.

One objection I’ve run into is that by distinguishing a direction in Euclidean space I’m breaking Lorentz symmetry. What’s true is quite the opposite: having such a distinguished direction is needed to get Lorentz symmetry after analytic continuation. If you want to start in Euclidean space and get Lorentz symmetry, you have to do something like distinguish a direction and get an Osterwalder-Schrader reflection in that direction, which you need to get from SO(4) to SL(2,C). From the other direction, if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. This is most easily seen in the twistor formalism, where the Minkowski space-time geometry is determined by a quadratic form that picks out a 5-dimensional hypersurface in PT. This will project down to an imaginary time = 0 subspace of Euclidean space-time, which picks out the imaginary time direction.