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ellipticallychallenged · 8 years ago

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Twisting round space-time - follow-up

Here, I follow up on the questions raised in the previous post and make some vague notes...

...He then comments briefly on one of the conflicts between standard quantum field theory (QFT) and general relativity (GR). That is that QFT suggests that empty space is seething with activity and that the energy density of empty space is infinite. However, if such is the case, then this infinite energy density would lead in GR to an absurd picture of an infinitely small radius of curvature for the Universe....

Note 1: How is this shown? Through the Friedmann equations and the density parameter $\Omega$?

I believe that this answer is appropriate, though another way of phrasing this would be that an infinitely small radius of curvature means that curvature of the Universe is infinitely large. The picture is absurd since we observe the Universe to be (almost) flat. Baez outlines the vacuum energy conflict between QFT and GR in further detail here.

...Penrose then notes that historically, Landau, Pauli, Klein and others all suggested that a quantum theory of gravity might remove the infinities and yield a finite theory that does not require renormalisation. However, standard techniques of QFT applied to GR yield a non-finite and non-renormalisable theory....

Note 2: If possible, I should source and read the Landau, Pauli, and Klein articles.

Worst part of popular articles is the incomplete referencing. Still not quite sure what Penrose is referencing here. Maybe written in Russian/German? If I recall correctly, Kaluza–Klein was a classical theory, so imagine it’s a different approach... If I recall correctly, I think Klein pointed out that the Planck length might provide a natural cut-off that would make things finite, and I think Jordan had some early papers on quantum gravity that might reference the other papers?

Moral of story: cite articles properly and learn Russian/German...

...The first point he highlights is that complex numbers play an essential role in QM, and that physical states can be represented by rays in a complex vector space. Any two elements, $\mathbf{A}, \mathbf{B} $, of such a vector space satisfy the law of linear superposition $\mathbf{C}=\lambda\mathbf{A}+\mu\mathbf{B}$, where $\lambda$ and $\mu$ are complex numbers...

Note 3: How does the complex vector space differ to Hilbert space?

This seems like the barest possible description of a quantum state. Complex coefficients $\to$ quantum probabilities, etc. But to answer the question...Hilbert space is complete and equipped with an inner product, which leads to the concept of norms and metrics. These are not necessarily required for a vector space.

Question for the future: does twistor space (also, do complex projective spaces) have an inner product?

...Here, Penrose states that each “vector” $\mathbf{A,B,C}$ (assuming they are not the zero vector) describes the quantum state. However, the same state is obtained whenever the vector is multiplied by any non-zero complex number. (This is what is meant by the state being represented by a “ray” in the vector space). From this it follows that only distinct ratios of the complex numbers, $\lambda\colon\mu$, provide distinct quantum states...

Note 4: There appears to be some ambiguity here. The phrasing appears to suggest that each vector describes the same state?

Penrose does actually clarify this in his following comments. I believe it should read...

Penrose states that each “vector” $\mathbf{A,B,C}$ (assuming they are not the zero vector) describes a quantum state. However, the same state (corresponding to $\mathbf{C}$) is obtained whenever the vectors $\mathbf{A}$ and $\mathbf{B}$ are multiplied by any non-zero complex number. (This is what is meant by the state being represented by a “ray” in the vector space). From this it follows that only distinct ratios of the complex numbers, $\lambda\colon\mu$, provide distinct quantum states.

“...If in a state $\mathbf{A}$ the particle spins right-handedly about the upward vertical, and in state $\mathbf{B}$ about the downward vertical, then in state $\mathbf{C}$ the particle will spin right-handedly about the direction represented by $\kappa=\lambda/\mu$. In other words, the spin-directions in space correspond exactly to the points of the Riemann sphere through the linear superposition of the states of the simplest type of spinning system.”

Penrose states it thus: “the three-dimensionality of space is related intimately to the fact that complex numbers are used in the linear superposition law....”

Note 5: I don’t follow this connection between the Riemann sphere and spatial geometry particularly well, or what the right-handed spinning has to do with the Riemann sphere. Adding two right-handed states and getting a right-handed state seems like a trivial statement. Try find a more detailed discussion.

The main sentence that baffles me here is that complex linear superposition leads to the three-dimensionality of space? How? Why? Is this something about how states combine on the Riemann sphere?

Lots more to figure out here before this becomes clear...

...Penrose notes some subtleties here, in that there is a question of how the Riemann sphere is oriented against the sky, and the potential of relativistic abberration (where the apparent position of the stars depends on the observer’s instantaneous velocity). However, remarkably, one can transform from one choice of orientation and velocity $(\kappa)$ to another $(\kappa′)$ with the transformation: \[\kappa’=\frac{\alpha\kappa+\beta}{\gamma\kappa+\delta},\]where $\alpha, \beta, \gamma, \delta$ are fixed complex numbers defined by the relative orientations and velocities between the two choices... Note 6: Similar (i.e., Möbius or linear fractional) transformations came up regularly in my work with quartic equations and elliptic functions. Review the context there and see if there are any potential new insights or connections to be made. If the context here is similar, I am assuming that $\alpha\delta-\beta\gamma\neq 0$ needs to be satisfied.

The main context this transformation was used for was to convert general quartics to biquadratic form (cf. Tschirnhaus transformation), however it is also closely related to the modular group when $\alpha\delta-\beta\gamma=1$. Doubly periodic functions (i.e., elliptic functions) possess modular group symmetry, and so the transformation arose quite naturally. However, nothing immediately jumps out...

Somewhat appeased by Penrose noting there are some subtleties with this celestial sphere argument, as it’s not something I am wholly familiar with and I need to think about it some more. However, as a vague (and likely incorrect) method of connecting light rays with the celestial (Riemann) sphere in a slightly more explicit manner..

Light rays (i.e., null lines) satisfy \[t^2-x^2-y^2-z^2=0.\] At $t=1$, the set of all light rays defines the celestial (Riemann sphere)\[S^2=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=1\}.\]

However, it is not immediately clear how this is generalised to $t\neq 1$ cases. Rescale the units so that $t=1$? Setting $c=1$ is already implicit here, maybe this isn’t an issue? Is defining the celestial sphere for $t=1$ equivalent to defining it for all times?

...This transformation is holomorphic, i.e., the complex conjugate does not appear, where holomorphicity is essentially ‘complex smoothness’ is a key concept of twistor theory, as indeed it is in much of QFT and particle physics... Note 7: Is this aspect somehow related to conformal symmetry?

Yep. A conformal map is an orientation preserving holomorphic map.

...Thus, in order to form a complex space, Penrose replaced these light rays by spinning photons, which also possess energy. As the photons can only spin by a fixed amount in a right-, or left-handed manner, the extra variable quantity is their energy, and this then yields an abstract 6-dimensional space whose points represent spinning photons. Moreover, this space can be regarded as a complex 3-dimensional space in terms of which the symmetry transformations of space-time are purely holomorphic... Note 8: Why is the spin direction not used as the sixth variable? Why is it the energy, a much harder quantity to define uniquely?

I think answering this is going to have to wait until I have a much better grasp of twistors. However, a possible naive argument might be that the spin can’t be used as the sixth variable since it can only assume a finite number of fixed values? I expect that this argument doesn’t hold up though...

...Penrose elaborates on the motivations for twistor theory - “A more detailed aim of twistor theory is to transcribe conventional QFT into twistor terms with the hope of obtaining a finite theory, free of the infinities that plague the conventional approach. There are many mathematical difficulties that still stand in the way of this programme, though it is possible to see ways of maintaining finiteness, if certain rules can be followed...” Note 9: What are the rules needed to maintain finiteness? Would guess locality and unitarity in the space-time picture?

Again, this is something that will have to wait until I have a much better grasp of twistors. Though since twistors are non-local, not expecting locality to feature... How does Lorentz invariance appear in twistor theory?

...He also comments that twistor theory is being used to describe elementary particles, and that massive particles such as electrons or protons, as well as massless particles (e.g., photons) can be handled. However, while massless particles may be described with a single-twistor, massive particles require two or more twistors... Note 10: I believe this is the twistor diagram programme for calculating amplitudes?

Hmm, it seems that the twistor diagram programme is actually different to the twistor particle programme. In 1987, in Gravitation and Geometry (the Ivor Robinson volume) Penrose appears to suggest that the particle programme has stalled but that the diagram theory has taken off. Not quite sure what the current state of things is, but from the amplituhedron hype, I would imagine that the diagram theory is still continuing.

...Twistor theory leads to a striking description of the “internal symmetry groups” of particles that is quite natural, and which can be used to classify the different kinds of particles...

Note 11: Which internal groups, and where does this classification break down?

Again, this is something that will have to wait until I have learned a lot more. However, if the comment above about the particle programme stalling has held true, there may be something to learn from identifying what caused it to break down.

So clearly, lots to learn before I have any real grasp of twistors. Some of the most immediate aspects appear to be:

Review of basic complex analysis

Review of special relativity and the celestial sphere.

Holomorphicity and conformal maps

Projective spaces and group representations

Much love!

#twistors #penrose #newscientist #physics #maths #spacetime

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ellipticallychallenged · 8 years ago

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Twistors 101

(Robinson congruence, from New Scientist, 1979)

As a quick heads-up, never having had any contact with twistors before, I’ll be starting from the absolute basics... I should also say that I have only had undergraduate courses on general relativity and high-energy physics, so I am by no means an expert in either. As such, while I will likely make many mistakes along the way, I am hoping that my knowledge and understanding of both will be improved by writing this blog. If others come across this blog and can elucidate my understanding further, I’d be delighted to hear from you...

So despite twistor theory being first proposed 50 years ago, I think twistors are a complete unknown for most students of physics. Naturally, over the course of my studies I had come across Penrose and seen some of his influence on mathematics and physics, but twistors had only been mentioned in throwaway remarks as an obscure formalism used in general relativity that ignored quantum mechanics, or described in a chapter of ‘The Road to Reality’ that was beyond my understanding at the time. However, they do appear to have had something of a renaissance, and indeed my current interest in them was aroused through popular articles describing their suitability for calculating amplitudes in $N=4$ Yang–Mills theory and the wider context of the ‘Amplituhedron.’

With that out of the way, I’ll begin this Twistors 101 course by reviewing a basic account of twistors presented by Roger Penrose in the New Scientist magazine, dated 31 May 1979: ‘Twisting round space-time’ (the issue is publicly available through Google Books, so hopefully I am not violating any copyright issues with the quotes and images!).

Twisting round space-time, New Scientist, 31 May 1979

The article begins with Penrose citing Sir Arthur Eddington, who in his 1927 lectures, `The Nature of the Physical World,’ commented on the revolution in our understanding of “solid” matter, i.e., that objects almost entirely consist of empty space.

Penrose highlights the fact that in the 50 years since Heisenberg, Schrōdinger, Dirac et al. clarified the fundamentals of quantum theory [technically, Penrose considers the 50 years since Eddington’s lectures], most of the attention has been on the minute particles of matter, and that the empty space which almost entirely comprises the volume of everyday objects has received only scant attention. He attributes this neglect to two almost paradoxical causes:

the feeling that we completely understand the nature of empty space; it is given to great accuracy by the Minkowski geometry underlying Einstein’s special relativity, and

the almost total lack of any really deep understanding of the nature of empty space.

He then comments briefly on one of the conflicts between standard quantum field theory (QFT) and general relativity (GR). That is that QFT suggests that empty space is seething with activity and that the energy density of empty space is infinite. However, if such is the case, then this infinite energy density would lead in GR to an absurd picture of an infinitely small radius of curvature for the Universe. [Note 1: How is this shown? Through the Friedmann equations and the density parameter $\Omega$?]

The QFT remedy to this problem is “renormalisation,” which essentially subtracts an infinite quantity from the infinite energy density such that the remainder is finite and corresponds to observable quantities. However, this procedure is considered to reflect our ignorance of the ‘true’ physics, and as such is only a stop-gap.

Penrose then notes that historically, Landau, Pauli, Klein and others all suggested that a quantum theory of gravity might remove the infinities and yield a finite theory that does not require renormalisation. However, standard techniques of QFT applied to GR yield a non-finite and non-renormalisable theory. [Note 2: If possible, I should source and read the Landau, Pauli, and Klein articles.]

Penrose thus states - “It seems that we are as far away as ever from understanding empty space. So what prospect is there of understanding the detailed nature of the particles which inhabit that space if we do not understand the space itself?”

Moreover, Heisenberg’s uncertainty principle appears to imply that on the scale of atoms (and below), our geometrical picture becomes especially inadequate and must be relinquished.

It is at this point that Penrose begins to motivate an alternate geometric formulation to our conventional space-time picture, noting however, that on account of the successes of the existing theory, this new picture must be essentially equivalent to the standard theory.

Penrose highlights the fact that although quantum electrodynamics (QED) appears to show that our conventional space-time picture is valid to $10^{-17}$ m (at least), this need not be the only acceptable picture. In order to elucidate this point, he uses the analogy of the Newtonian and GR pictures of celestial motion. While Newtonian gravity is remarkably accurate for celestial objects, it suggests that gravitational forces are real, and that space at the dimensions relevant to planetary motions is Euclidian. On the other hand, GR denies both of these viewpoints, clarifies some anomalous results, and leads to a ‘deeper’ view of the nature of space.

Penrose thus states his view as - “So I am suggesting that some geometric reformulation may represent a key to understanding the geometry that governs behaviour at the submicroscopic level. And in order to be able to reproduce the successful physics of our day, this reformulation must incorporate both quantum mechanics and the flat Minkowski geometry of special relativity.”

Note that, here, Penrose is not quite proposing a theory of quantum gravity. Rather, he is proposing an equivalent formulation of existing physics, and indeed, he comments that the curved geometry of GR is something that must be accommodated at a later stage. Thus, this proposed reformulation is akin to the Lagrangian or Hamiltonian formulations of Newtonian mechanics; the results must be equivalent but the insight gained may be deeper, e.g., symmetries and conserved quantities.

Penrose now turns his attention to certain aspects of quantum mechanics (QM) and wave–particle duality.

The first point he highlights is that complex numbers play an essential role in QM, and that physical states can be represented by rays in a complex vector space. Any two elements, $\mathbf{A}, \mathbf{B}$, of such a vector space satisfy the law of linear superposition \[\mathbf{C}=\lambda\mathbf{A}+\mu\mathbf{B},\]where $\lambda$ and $\mu$ are complex numbers. [Note 3: How does the complex vector space differ to Hilbert space?]

Here, Penrose states that each “vector” $\mathbf{A}, \mathbf{B}, \mathbf{C}$ (assuming they are not the zero vector) describes the quantum state. However, the same state is obtained whenever the vector is multiplied by any non-zero complex number. (This is what is meant by the state being represented by a “ray” in the vector space). From this it follows that only distinct ratios of the complex numbers, $\lambda\colon\mu$, provide distinct quantum states. [Note 4: There appears to be some ambiguity here. The phrasing appears to suggest that each vector describes the same state?]

In other words: “for any two distinct quantum states, a whole array of other allowed quantum states exists, formed from the original two by linear superposition; precisely one state of this array corresponds to each distinct ratio $\lambda\colon\mu$,” and this is a fundamental axiom of quantum mechanics.

Penrose now introduces a way of visualising this array. He folds the Argand plane (representing all possible choices of the two complex numbers) into the Riemann sphere, including a point at $\infty$. The relation between the Riemann sphere and the Argand plane is then given by stereographic projection from the north pole of the sphere to its equatorial plane.

Every point on the Riemann sphere then represents a particular complex ratio, $\kappa=\lambda/\mu$, where the north pole represents an infinite value of $\kappa$ corresponding to $\mu=0$, and every other point of the sphere projects down to a unique point $\kappa=a+ib$ of the Argand plane. Thus, if the north pole represents the quantum state $\mathbf{A}$, the south pole would represent the quantum state $\mathbf{B}$, while the point $\kappa$ represents the state $\mathbf{C}=\lambda\mathbf{A}+\mu\mathbf{B}$.

As a result, the linear superposition of any two quantum states gives rise to an array of states which has the topological structure of a sphere.

Penrose now states that there is a physical relation between this abstract mathematical sphere and spatial geometry that can be understood in terms of the physical states of a spinning particle.

“If in a state $\mathbf{A}$ the particle spins right-handedly about the upward vertical, and in state $\mathbf{B}$ about the downward vertical, then in state $\mathbf{C}$ the particle will spin right-handedly about the direction represented by $\kappa=\lambda/\mu$. In other words, the spin-directions in space correspond exactly to the points of the Riemann sphere through the linear superposition of the states of the simplest type of spinning system.”

Penrose states it thus: “the three-dimensionality of space is related intimately to the fact that complex numbers are used in the linear superposition law.” [Note 5: I don’t follow this connection between the Riemann sphere and spatial geometry particularly well, or what the right-handed spinning has to do with the Riemann sphere. Adding two right-handed states and getting a right-handed state seems like a trivial statement. Try find a more detailed discussion.]

In order to further clarify how space-time geometry is related to complex structure, Penrose then outlines another visualisation, this time of the Riemann sphere. Now, we imagine an observer at the centre of the sphere looking out at the ‘celestial sphere’ or ‘sky,’ and this observer labels every star that he can see with a complex ratio $\kappa$ corresponding to the direction that the light from that star comes from.

Penrose notes some subtleties here, in that there is a question of how the Riemann sphere is oriented against the sky, and the potential of relativistic abberration (where the apparent position of the stars depends on the observer’s instantaneous velocity). However, remarkably, one can transform from one choice of orientation and velocity $(\kappa)$ to another $(\kappa’)$ with the transformation: $$\kappa ’=\frac{\alpha\kappa+\beta}{\gamma\kappa+\delta},$$where $\alpha, \beta, \gamma$ and $\delta$ are fixed complex numbers defined by the relative orientations and velocities between the two choices. [Note 6: Similar (i.e., Möbius or linear fractional) transformations came up regularly in my work with quartic equations and elliptic functions. Review the context there and see if there are any potential new insights or connections to be made. If the context here is similar, I am assuming that $\alpha\delta-\beta\gamma\neq 0$ needs to be satisfied.]

This transformation is holomorphic, i.e., the complex conjugate does not appear, where holomorphicity is essentially ‘complex smoothness’ is a key concept of twistor theory, as indeed it is in much of QFT and particle physics. [Note 7: Is this aspect somehow related to conformal symmetry?]

Here, Penrose states one of the aims of twistor theory - “In fact, one aim of twistor theory is actually to reduce all the equations of physics simply to ‘pure holomorphicity.’”

The holomorphic transformation above is such an example, and is the most general purely holomorphic transformation that sends the Riemann sphere to itself. This group of transformations is known as the “restricted Lorentz group” and is the most fundamental symmetry group in physics, describing the special relativistic symmetry of space-time about a point.

Penrose now continues to extend the proposed visualisation to further relate the Riemann sphere with space-time geometry as follows:

At a particular instant, the observer at the centre of the Riemann sphere can be represented by a point $O$ in four-dimensional Minkowski space-time, and the light that he detects from the stars in his celestial sphere are represented by the “past light cone.” Thus, at any point in space-time (such as $O$), the system of light rays reaching an observer generates a cone stretching back into the past. Moreover, each one of these light rays corresponds to one point of the celestial sphere (i.e., a particular $\kappa$).

However, one can also imagine a new space $\mathbf{PT}$, called projective twistor space, where each point corresponds to an entire light ray in Minkowski space-time. The light rays through $O$ will then be represented by a subset of points in this new space.

However, while the Riemann sphere defines a complex space of one complex dimension (this is essentially trivial since it was constructed from the Argand plane), the new entire light ray space does not define a complex space since light rays only form a 5-dimensional system (space, time, and direction), and complex spaces need to have an even number of dimensions.

Thus, in order to form a complex space, Penrose replaced these light rays by spinning photons, which also possess energy. As the photons can only spin by a fixed amount in a right-, or left-handed manner, the extra variable quantity is their energy, and this then yields an abstract 6-dimensional space whose points represent spinning photons. Moreover, this space can be regarded as a complex 3-dimensional space in terms of which the symmetry transformations of space-time are purely holomorphic. [Note 8: Why is the spin direction not used as the sixth variable? Why is it the energy, a much harder quantity to define uniquely?]

This $\mathbf{PT}$ space is a higher-dimensional version of the Riemann sphere, only now the points are defined by four complex numbers, $Z^0\colon Z^1\colon Z^2\colon Z^3$, which are the components of a twistor, $Z^\alpha$. The projective twistor space can then be divided into three regions, $\mathbf{PT}^+, \mathbf{PT}^-$ and $\mathbf{PN}$, where the points of $\mathbf{PT}^+$ represent photons spinning in a right-handed sense about their directions of motion, while the points of $\mathbf{PT}^-$ represent photons spinning in a left-handed manner. The 5-dimensional surface $\mathbf{PN}$ represents the boundary between the two halves (the two types of photon), and can thus be considered to represent light rays.

Next, the Robinson congruence is presented as a simple consequence of the fact that when spin is present, the localised description of a photon in a light-ray is just an approximation. In fact, the spinning photon acquires a ‘spread-out’ non-local structure determined by its motion that can be described geometrically in space-time terms as a system of twisting lines (i.e., the Robinson congruence).

Penrose then acknowledges that this twistor presentation is actually somewhat misleading, since a photon is just one of many different particles, and twistors do not have any special relation to any particular kind of particle. In fact, they are something more basic than the concept of a particle and Penrose describes them as simply providing an alternative, more ‘particle-like,’ way of viewing the geometry of space-time.

Thus, studying the geometry of twistor space gives us a new position from which to consider space-time, and in twistor theory, Penrose essentially advocates abandoning the space-time picture. If desired, Minkowski geometry should instead be constructed from the twistor space base (space-time events being interpreted as ‘lines’ in $\mathbf{PN}$), but Penrose notes that the essential aim of twistor theory was to describe as much as possible in purely twistor terms, without reference to space-time.

Penrose elaborates on the motivations for twistor theory - “A more detailed aim of twistor theory is to transcribe conventional QFT into twistor terms with the hope of obtaining a finite theory, free of the infinities that plague the conventional approach. There are many mathematical difficulties that still stand in the way of this programme, though it is possible to see ways of maintaining finiteness, if certain rules can be followed...” [Note 9: What are the rules needed to maintain finiteness? Would guess locality and unitarity in the space-time picture?]

He also comments that twistor theory is being used to describe elementary particles, and that massive particles such as electrons or protons, as well as massless particles (e.g., photons) can be handled. However, while massless particles may be described with a single-twistor, massive particles require two or more twistors. [Note 10: I believe this is the twistor diagram program for calculating amplitudes?]

Now nearing the end of the article, Penrose highlights a few of the promising aspects of twistor theory and some successes while acknowledging that a great deal more work remains to be done to fully flesh out these aspects. The examples he lists include:

Twistor theory leads to a striking description of the “internal symmetry groups” of particles that is quite natural, and which can be used to classify the different kinds of particles. [Note 11: Which internal groups, and where does this classification break down?]

The twistor description of GR leads to some intriguing results, though again, this description is far from complete. He notes that curved space-time corresponds to a deformation of twistor space and that the curvature of empty space-time can be split into two pieces: the self-dual and anti-self-dual parts.

As the anti-self-dual part can be completely understood in terms of deformations and Einstein’s equations are reduced to pure holomorphicity in accordance with the general programme, there are promising signs that the self-dual part and for other Yang–Mills fields can also be understood in terms of twistor space deformation.

Penrose concludes by acknowledging that twistor theory is incomplete, but expresses his hope that it provides a new approach to the fundamental problems of physics. He reiterates the important relation between holomorphicity and its subtle geometrical consequences, and highlights that twistor theory seeks to explain both matter and space. Moreover, he points out that the ‘correct’ ($1+3$)-dimensionality of space-time is a necessary consequence of the twistor approach.

For my next post, I will try pick over this review a bit more and see if I can answer some of the notes and questions marked throughout. This will hopefully highlight what aspects I need to work on...

Much love.

#twistors #penrose #newscientist #robinsoncongruence #spacetime #physics #maths

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