Twistor string theory

Is this the new black of physics?

Study matter and its motion through spacetime...

Moderators: Calilasseia, ADParker

Twistor string theory

#1  Postby tnjrp » Jan 11, 2011 8:53 am

Based on Edward Witten's work on marrying Roger Penrose's old twistor theory (Hi twistor! :cheers:) with newer string theories, the twistor string theory appears to have made quite a few headlines in the Physics news over the late last year. An example here:
http://www.scientificamerican.com/artic ... st-of-fate

In the light of this thread and what with even the Finnish alt-physicist Matti Pitkänen going on about how faboulous it is (and how his own TGd theory is going to be found at the back of TST, obviously...) I thought it's about time I actually learned something about the subject beyond the name.

Can anyone provide something a little deeper than the above SciAm article but not too deep for a physics layman to understand on TST for starters? A personally written breakdown or a link will be good. Relationship discussion with other "extreme theories" such as the M-theory a plus.

Note that despite Witten and Arkadi-Hamed being Pitkänen's (current) heroes, this appears to be a completely legit (if not apparently yet predominant) mainstream science theory. This subject to opinion, I'm certain, given how certain forumites view the aforementioned M-theory :mrgreen:
The dog, the dog, he's at it again!
tnjrp
THREAD STARTER
 
Posts: 3587
Age: 54
Male

Finland (fi)
Print view this post

Ads by Google


Re: Twistor string theory

#2  Postby twistor59 » Jan 11, 2011 9:20 am

I was afraid someone would ask about this :lol:

Good old fashioned twistor theory I can explain, but I left the twistor programme in 1984, so all this new stuff happened WAY after my time. Having said that, as part of my current catch-up-with-physics hobby I had in mind to take a look at twistor-string (fuck, I'm interested in any application of twistor theory, no matter how tenuous !). To do that, though I needed to learn a bit about what string theory was about first - I've just started to look at that. So I'm afraid I can't answer your question in much detail yet, but I will hopefully be able to eventually.

Nima Arkani Hamed is a rising star, here's an interesting lecture from him on particle physics.

This article from Lubos's blog talks a bit about the latest twistor string work:

http://motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html

He talks about a paper which Andy Hodges contributed too. He was one of our original twistor group - nice to see some of the "old guard" still active !!
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#3  Postby tnjrp » Jan 11, 2011 9:25 am

twistor59 wrote:I was afraid someone would ask about this
I aim to displease :mrgreen:

Good old fashioned twistor theory I can explain
That might be a good start, by the way of "paving the ground" so to speak
:popcorn:

Nima Arkani Hamed is a rising star, here's an interesting lecture from him on particle physics.

This article from Lubos's blog talks a bit about the latest twistor string work:

http://motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html
Thx, have to try and take a look at those.
The dog, the dog, he's at it again!
tnjrp
THREAD STARTER
 
Posts: 3587
Age: 54
Male

Finland (fi)
Print view this post

Re: Twistor string theory

#4  Postby Someone » Jan 11, 2011 9:32 am

:popcorn:
Proper name: Toon Pine M Brown ---- AM I A WOMAN or working intimately on medical ethics?! No Period, No Say About Certain Things. Is my social philosophy. Everyone has a Hell here, so why add one to the mix if you don't need?
User avatar
Someone
Banned User
 
Name: James
Posts: 1516
Age: 55

Country: USA, mostly
Morocco (ma)
Print view this post

Re: Twistor string theory

#5  Postby twistor59 » Jan 11, 2011 9:36 am

tnjrp wrote:
Good old fashioned twistor theory I can explain
That might be a good start, by the way of "paving the ground" so to speak
:popcorn:


OK I'll write something down over the next week or so. Kick me if I forget.
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#6  Postby newolder » Jan 11, 2011 10:25 pm

ooooh, likes origami lessons... :thumbup:
Geometric forgetting gives me loops. - Nima A-H
User avatar
newolder
 
Name: Albert Ross
Posts: 6486
Age: 8
Male

Country: Feudal Estate number 9
Print view this post

Re: Twistor string theory

#7  Postby tnjrp » Jan 12, 2011 7:26 am

They kick you in origami lessons? :scratch:
The dog, the dog, he's at it again!
tnjrp
THREAD STARTER
 
Posts: 3587
Age: 54
Male

Finland (fi)
Print view this post

Ads by Google


Re: Twistor string theory

#8  Postby newolder » Jan 12, 2011 4:39 pm

:lol: it reads like we get to kick teach. iff he should forget to deliver teh goodies...
Geometric forgetting gives me loops. - Nima A-H
User avatar
newolder
 
Name: Albert Ross
Posts: 6486
Age: 8
Male

Country: Feudal Estate number 9
Print view this post

Re: Twistor string theory

#9  Postby twistor59 » Jan 19, 2011 8:52 am

I promised to say something about twistors. This will take a few posts and be spread out over some time – sorry ! Also I want to make it clear that my comments relate to the twistor theory I vaguely understood 25 years ago. I’m not familiar with what may have been done on the subject in the intervening time, including twistor string theory.

Post # 1: Spinors

Twistor theory was developed in the hope that it would provide a new way to formulate physics and hopefully lead to some new predictions. I have to say right up front that, as far as I’m aware, the latter hope was never realized. However, the former was realized and there were advantages to be gained from this. My own view of the twistor approach is that, rightly or wrongly, I think of it a bit like people think of Fourier transformations. In communications theory, people transform between the time domain and the frequency domain. Some operations which are complicated in one domain are simple in the other domain and vice versa.

The important transformation in twistor theory is the Penrose transform which maps Spacetime < - - > Twistor Space

Twistor theory uses quite a few fairly abstract mathematical constructs, much of which are taken from the field of algebraic geometry, so unavoidably, I’m going to have to write down a teensy bit of mathematics.

The definition of twistors relies heavily on much better-known objects called spinors, and it’s virtually impossible to talk twistor without talking spinor language, so I’ll briefly describe spinors in this post. I make no pretence that any of this is rigorous or complete – I merely intend to give a flavour of the subject. OK caveats done, let’s start with spin and spinors.

We’ll restrict ourselves for now to flat spacetime, Minkowski space. This is the set of points with Cartesian coordinates (x0, x1, x2, x3). x0 is the time coordinate and x1, x2, x3 are the three space coordinates (x, y, z). These coordinates are considered to define events in an inertial frame, and the whole big deal about special relativity is that you can transform to a new inertial frame with coordinates (x’0, x’1, x’2, x’3) and the transformation will preserve the Minkowski metric i.e.

(dx’0)2-(dx’1)2- (dx’2)2- (dx’3)2 = (dx0)2-(dx1)2- (dx2)2- (dx3)2........(1)

The mapping (x0, x1, x2, x3) -> (x’0, x’1, x’2, x’3), known as a Lorentz transformation, is linear and hence can be represented by a 4x4 matrix.

For example for relative motion in a straight line in the x direction (people call this type of transformation a “boost”) would be represented by the matrix


(ɣ ...-βɣ...0...0)
(-βɣ...ɣ....0...0)...............(2)
(0 ....0.....1...0)
(0.....0 ....0...1)

where ɣ = 1/sqrt(1-v2/c2), and β = v/c. I apologize for the shitty looking depiction of matrices, this is the best I could manage with Unicode. Also, I had to put the dots in to separate the entries – I was having trouble with the spaces (but maybe that’s because I was drafting in MS Word, which has a mind of its own).

In addition to transformations like this, which describe relative motion, by mixing up the time and space coordinates (x0 and x1 in the example) the matrix can also describe rotations, which define new space coordinates which are tilted with respect to the old ones. Not any old 4x4 matrix is an allowable Lorentz transformation – they have certain restrictions (such as ensuring (1) is respected, for example). The set of allowed Lorentz transformations is a group – you can do stuff like compose two transformations to get a new one, and define inverse transformations etc. Mathematicians and physicists like groups. They like them very much indeed.

The Lorentz group is an abstract mathematical structure, and a large part of particle physics/field theory is concerned with representations of this group (strictly the Poincare group, but I’ll ignore the difference for now). What’s a representation you may ask.


Spin 1

Well you’ve already met the first example of a representation of the Lorentz group, namely 4x4 matrices of which (2) is an example. We can think of a representation as a mapping of the abstract group to a bunch of matrices. These 4x4 matrices, together with the vectors they act upon constitute the vector representation, which is descibed as “having spin equal to 1”. What this means intuitively is that, restricting attention to the spatial (3x3) part of the matrix and forgetting about the boosts for the moment, the matrix performs rotations on the vectors on which it acts. For example a rotation through angle θ anticlockwise about the z axis would be the matrix

(cosθ...sinθ....0)
(-sinθ...cosθ...0) ...................... (3)
(0.........0......1)


It’s easy to see that as θ goes from 0 to 2π, the vector on which the matrix acts rotates and goes back to where it started from. We’ll say that this sort of behaviour, i.e. rotation by 2π bringing you back to where you started from is “spin 1” behaviour. We can think of the rotation as acting on an arrow, anchored at its tail, the arrow representing the vector.

Spin1.jpg
Spin1.jpg (12.65 KiB) Viewed 4750 times



Spin 2

All very easy to imagine. Now there is another type of behaviour, called spin 2. This sort of behaviour is exhibited, not by a vector, but by an object which rejoices in the name of “symmetric traceless second rank tensor”. Instead of the single index possessed by a vector (which labels the vector’s components), take an object with a pair of indices Tij. The pair gives it the “second rank” designation. “Symmetric” means Tij = Tji. “Traceless” means = Tii = 0. Note here we adopt the Einstein summation convention, whereby if an index is repeated in a term, that index is assumed to be summed over. Thus

Tii = Sum over i ( Tii )

With this convention, the rule for transforming a vector by multiplying it by a matrix is written (where Lij are the components of the matrix, and the primed values V' are the new components of V after the rotation)

V’i = LijVj

And the rule for transforming a second rank tensor is just

T’ij = LikLjlTkl...........(4)

If you perform the rotation (3) about the z azis, and just look at the x and y directions, the relevant part of the matrix (3) is just

(cosθ... sinθ)
(-sinθ...cosθ)........... (5)

Using this in (4), we find that, for θ = π, T’ij = Tij, i.e. T has rotated back to itself. In other words, you only need to rotate through π to get back where you started. So, you can visualise a spin 2 "object" as a double headed arrow - a 180deg rotation gets you back to where you started.


Spin2.jpg
Spin2.jpg (11.11 KiB) Viewed 4750 times



Gravity is modelled as a spin 2 field.


Spin 1/2

So far we have spin 1 and spin 2. The key spin for our twistorial purposes, however, will be spin ½. Given that you have to rotate through 2π to restore originality for spin 1 and through π for spin 2, you might guess that you’d have to rotate through 4π to restore originality for spin ½. Indeed you do, and the object in question is called a spinor.

To define spinors, again it’s the rotation part of the Lorentz group we’re concentrating on. A generic rotation in the x y z space is given by a matrix parameterised by 3 angles (the Euler angles ϕ1, θ, ϕ2)



(cosϕ2cosϕ1-cosθsinϕ1sinϕ2....-cosϕ2sinϕ1-cosθcosϕ1sinϕ2......sinϕ2sinθ)
(sinϕ2cosϕ1+cosθsinϕ1cosϕ2....-sinϕ2sinϕ1+cosθcosϕ1cosϕ2....-cosϕ2sinθ)
(sinθsinϕ1...........................sinθcosϕ1...............................-cosθ)



Rotation matrices can be mapped to complex 2x2 matrices, by pasting the Euler angles into the matrix


(cos(θ/2)ei(ϕ21)/2... isin(θ/2)ei(ϕ21)/2)
(isin(θ/2)ei(ϕ21)/2... cos(θ/2)ei(ϕ21)/2).............(6)


As a 2x2 matrix, this one operates on column vectors which have 2 complex components. The group of such matrices is called SU(2) and the complex “vectors” on which it operates are called spinors. The mapping from the 3 real-dimensional rotation matrices to the two complex-dimensional SU(2) matrices is double valued i.e. there are two SU(2) matrices for each rotation matrix.

Now the thing to note about the SU(2) matrix is that the angles appearing in there are divided by 2. This is why rotating a spinor (by multiplying by the SU(2) matrix (6) )requires a 4π rotation angle to restore it to its original state.

We had the pictures of the arrow and the double headed arrow to visualise spin 1 and spin 2 “objects”, but we’re going to have to work a bit harder to visualise spin ½. Given any physical object, 2π rotation will put it back where it started, so we need something more than just an object ! In fact what we use is an object together with its relationship with the environment. This is quite nicely illustrated by this video. The object is the glass, and the “relationship with the environment” is dude’s arm. Half way through, the glass has rotated through 2π, but its relationship with the environment is different (dude’s arm is now twisted). A further rotation of 2π again puts it back where it started and dude’s arm is untwisted. This is sometimes referred to as the “orientation-entanglement” view of the spinor. There are other illustrations of it – for example there’s a nice picture in The Road to Reality, but it would be naughty to copy it here.

Now a little aside from the main story – we all know the spin-statistics theorem – in particular that spin ½ particles are fermions. We now know that spin ½ particles are described by spinors which have this orientation-entanglement interpretation. We can use this to heuristically motivate the fact that these must be fermions. (Note I said "motivate", not "prove". This is just a bit of handwaving !)


SpinStats.jpg
SpinStats.jpg (44.31 KiB) Viewed 4750 times



Being a fermion means that the system's wavefunction changes sign when you swap places with another fermion. Look at the diagram to see what happens when we interchange two spin ½ particles. For each particle, part of its “environment” is the other particle and we show its relation to the other particle by attaching a pair of strings. Label the particles black and white and exchange them. We end up with the situation in step 3. The particles are exchanged, but the strings are twisted. To untwist I can flip the black one twice, i.e. rotate it through 2π.

So, for spinor particles, one exchange is equivalent to a 2π rotation. Now a 2π rotation using the matrix (6) simply multiplies the spinor by -1, so there's something distinctly "fermionic" going on !

Note that I've talked about spatial rotations to simplify things, so I end up with the spinors being representations of SU(2). However, there are also boosts in the Lorentz group, and if you include those in the picture, the 2x2 matrices that act on the spinors are now elements of the group SL(2,ℂ) rather than SU(2), but as far as the heuristics above are concerned, this is just a technicality.

Next time - a bit more about spinors and Minkowski space (we'll get to twistors eventually).

Edit: Got the signature of the metric wrong !
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#10  Postby tnjrp » Jan 21, 2011 8:30 am

Formulae -- they burns us! :ahrr:

Seriously tho, looks informative so far: you've offically avoided a kicking :cheers:
The dog, the dog, he's at it again!
tnjrp
THREAD STARTER
 
Posts: 3587
Age: 54
Male

Finland (fi)
Print view this post

Re: Twistor string theory

#11  Postby twistor59 » Jan 27, 2011 4:18 pm

Post# 2: A Few more Spinors and some more Minkowski Space


Last post I introduced spinors as the two-component complex vectors that are acted on by the 2x2 complex matrices that represent elements of the group SU(2). The SU(2) matrices represent rotations in ordinary 3 dimensional space. The spinors have this weird property that if you multiply one by a matrix representing a rotation of 2π you don’t get back the spinor you started with – in fact you need to rotate through 4π to do this.

We can think of the rotations in 3 space as a subset of the Lorentz transformations (with Lorentz transformations, we don’t just have rotations, but also “boosts” where one frame is moving with respect to another. The boosts mix up the time and space coordinates). In the same way, we can extend the group SU(2) to include matrices representing these boosts as well as the rotations. The “extended” group is SL(2, ℂ) and it contains SU(2) as a subgroup.

Elements of SL(2, ℂ) are still 2x2 complex matrices and they still act on the 2 component spinors.

Conventionally spinors are represented by notation such as ξA. A = 0, 1 labels the two spinor components, each a complex number. Now for a little bit of tedious, but essential, abstract algebra:

Our spinors ξA are elements in a two complex-dimensional vector space V. (Just means they each have two elements, each of which is a complex number). Now, given a vector space, what does a mathematician immediately do with it ? – well of course they define the “dual space”. This is just the space of linear maps from V to whatever field V was defined over. In this case, it’s the set of linear maps from V -> ℂ. Elements of this dual space are denoted with lower indices, for example ηA.

The complex number obtained by ηA acting on the spinor ξA is just given by ηAξA (remember repeated indices are summed over).

If you like, you can think of ξA as a column vector and ηA as a row vector, so ηAξA is just what you get when you multiply a row vector by a column vector.

Unfortunately, to define twistors, I just need one more spinor thing. If V is the vector space of spinors, ξA, then we can consider another vector space Vbar, the space of complex conjugate spinors (V should have a bar over the top, but I can’t see a Unicode way to do that). The coordinates of a conjugate spinor are denoted ξA’. Of course, there are also dual conjugate spinors ηA’.

Compactified Minkowski Space

Minkowski space M is the set of quadruples (x0, x1, x2, x3) with metric
(dx0)2-(dx1)2- (dx2)2- (dx3)2..........(1)

Of course, that’s far too simple to understand, so we’ll have to make it a bit more complicated. One of the original motivations behind twistor theory was the idea that spacetime points may not be the most fundamental objects in our models of the universe, but that the conformal structure might be more fundamental. What do I mean by the conformal structure ?

In the diagram I show a picture of a null cone in (2+1) dimensional Minkowski space. (Ideally it would be 3+1 dimensional, but not possible to draw). This is the null cone of the origin and the surface of the cone defines the points for which the expression (1) is zero. The straight lines passing through the origin which make up the surface of the cone are potential world lines of light rays passing through the origin. Anything passing through the origin at a speed slower than the speed of light would ride on a world line in the interior of the cone.

There are two halves to the cone – the blue half and the red half. The blue half defines the spacetime points which could possibly influence the origin, and the red half defines the spacetime points which could possibly be influenced by the origin. Points outside the cone cannot possibly influence, or be influenced by the origin since a signal would have to travel faster than light for that to happen.

NullCone.jpg
NullCone.jpg (29.8 KiB) Viewed 4713 times


There is a conceptual null cone such as this one at every spacetime point. From the argument above, these null cones define the causal structure of the spacetime.

The null cone is given by the expression
(dx0)2-(dx1)2- (dx2)2- (dx3)2 = 0

i.e. ηabdxadxb where

ηab = diag(1, -1, -1, -1) is the Minkowski metric.

Now here’s the useful thing – it doesn’t matter a toss if you multiply the metric by a scalar function Ω2(x). In other words if I define a new metric η^ab = Ω2(x)ηab, then η^ab defines precisely the same null cones as ηab, so it defines precisely the same causal structure. The metric defines the geometry in the sense that it allows you to compute lengths and angles. Multiplying the metric by a scalar changes the length measurements, but not the angles. Now the one case where it doesn’t screw up the lengths is if the length is zero, as it is with the worldline of a light ray.

We can use this freedom to rescale the metrics in order to embed Minkowski space inside a “slightly” larger manifold. This manifold, which rejoices in the name of “compactified Minkowski space” can be thought of as taking the bottom half of a null cone (just the cone surface, not the interior) and “sticking” it onto the top half of Minkowski space, and taking the top half of a cone and sticking it on the bottom half of Minkowski space. So you end up with a spinning top-shaped picture:

CompMinkSpace.jpg
CompMinkSpace.jpg (20.88 KiB) Viewed 4713 times


I’ve drawn an arbitrary point P in Minkowski space and a couple of photons being fired off from it in different directions. They eventually hit the top half null cone at two different points. This half cone is called “future null infinity” and is denoted by the squiggly “I” which I don’t have in Unicode, but is called “scri plus”. Correspondingly there is a past null cone at infinity, denoted by scri-minus.

There are also points representing spacelike and timelike infinity but I won’t discuss those. Although I haven’t described it here in all its gory detail, this procedure of appending the null cone at infinity is mathematically rigorous and is an enormously useful technique, and a generalisation of it played a vital part in the proof of the Hawking-Penrose singularity theorems.

Diagrams such as these, where you explicitly attach some points to represent “infinity” in the spacetime in question are generally referred to as “Penrose diagrams” in the literature, although Roger Penrose always maintained that they should be called “Carter diagrams” after Brandon Carter since the original inspiration came from him apparently.

Minkowski space with its "null-cone-at-infinity" is called compactified Minkowski space. In the next post, I'll describe twistor space and its correspondance with compactified Minkowski space. All that work on defining spinors will then have been worth while !!
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#12  Postby Nautilidae » Jan 27, 2011 4:47 pm

:popcorn:
User avatar
Nautilidae
RS Donator
 
Posts: 4230
Age: 25
Male

Country: United States
United States (us)
Print view this post

Re: Twistor string theory

#13  Postby iamthereforeithink » Jan 27, 2011 6:54 pm

:popcorn: + :coffee:
“The supreme art of war is to subdue the enemy without fighting.” ― Sun Tzu, The Art of War
User avatar
iamthereforeithink
 
Posts: 3332
Age: 9
Male

Country: USA/ EU
European Union (eur)
Print view this post

Re: Twistor string theory

#14  Postby newolder » Jan 28, 2011 6:06 pm

poetry
:thumbup:
Geometric forgetting gives me loops. - Nima A-H
User avatar
newolder
 
Name: Albert Ross
Posts: 6486
Age: 8
Male

Country: Feudal Estate number 9
Print view this post

Re: Twistor string theory

#15  Postby twistor59 » Feb 03, 2011 5:54 pm

Post # 3


We have spinors ξA and we have their conjugate versions 𝜒A’. There’s nothing to stop us taking tensor products of spinor spaces and forming spinors with several indices, like αABA’CD (the position and order of the indices is significant).
There is one very important spinor with two indices

εAB = (0.....1)
.......(-1....0)

This is used to raise spinor indices (a bit like you use the metric tensor to raise and lower vector indices):
ξA = εAB ξB

Similarly we have εAB, which has the same components as εAB, which is used to lower spinor indices. Also we have the primed versions (same components again) which raise and lower primed spinor indices.

Now heres the thing – suppose we have a Lorentz spacetime vector (x0, x1, x2, x3). I can build out of this a two-spinor

xAA’ whose components are

(x00’....x01’) = (x0+x1..... x2+ix3)
(x10’....x11’) ..(x2-ix3..... x0-x1)

(There’s probably a 1/sqrt(2) for normalisation, but I won’t bother). Twistor people use this spinor representation, so whenever anything has a Lorentz index, you can replace it with an unprimed+primed spinor index pair. Thus an arbitrary Lorentz vector is written as vAA’ instead of va.

A Lorentz tensor with two lower indices would be FAA’BB’ instead of Fab

Etc.....

Now for some twistors:

Take a pair of spinors, primed and unprimed (ωA, πA’) and consider the set of points xAA’ in Minkowski space such that
ωA = i xAA’ πA’......................(1)

It’s not too difficult to convince yourself that, if the spinors satisfy the reality condition
ωA πbarA + ωbarA’ πA’ = 0..........(2)

then the set of xAA’ satisfying (1) is a null line in Minkowski space, i.e. the path of a light ray.

We talk of the 4-tuple of complex numbers (ωA, πA’) as a twistor, traditionally denoted by Zα. The space of twistors is called, well, “twistor space”, and denoted by 𝕋.

In order to neatly write the reality condition (2), we define a conjugation operation on 𝕋:

HermConjugate.jpg
HermConjugate.jpg (16.28 KiB) Viewed 4681 times


What this means is “given a twistor Zα we can define a ‘conjugate twistor’ Zbarα such that the zeroth component of Zbarα is given by the complex conjugate of Z2.....etc”

With these definitions, the condition (2) is just

ZαZbarα = 0

And twistors satisfying this condition are called “null twistors”, and they correspond to null rays in Minkowski space. However the correspondence isn’t one to one. It’s pretty easy to see that if Zα corresponds to a particular null ray, then 𝜆 Zα, where 𝜆 is any old non zero complex number, corresponds to the same null ray.

So, if we’re interested in null rays, then rather than being interested in straightforward twistor space 𝕋, we’re really more interested in 𝕋 but we want to consider two twistors Wα and Zα to be equivalent if there is a nonzero complex number 𝜆 such that Wα = 𝜆 Zα.

Mathematically speaking, if Zα is a twistor, we “identify” all twistors = 𝜆 Zα where 𝜆 runs over the non zero complex numbers. Mathematicians use the word identify to mean “make the same”, rather than “find out what it is”.

Mathematician “Entomology is a very easy subject”
Biologist “Why ?”
Mathematician “Because entomology is the identification of insects, and if you identify them, then there’s only one of them”

The things we end up with after this identification process are called “projective twistors”, and the space of them is called projective twistor space, P𝕋. As a complex manifold, P𝕋 is an object very familiar to mathematicians called ℂP3. (In fact ℂP3 is guaranteed to make an algebraic geometer jizz himself !).

OK, so we have 𝕋 projectivised to P𝕋. Now we had inside 𝕋 a subset, consisting of null twistors, which correspond to light rays. After we’ve projectivised, the projective versions of the null twistors form a subspace of P𝕋, which we will call PN “projective null twistor space”. So a point of PN is a light ray in CM (compactified Minkowski space).

What about going the other way ? What do points in Minkowski space correspond to in projective twistor space ?
A point in Minkowski space can be defined by specifying the set of light rays which intersect at that point. How many such light rays are there ? Well, if you were a point floating in space, rays could come to you from any direction, so there is a “sphere’s worth” of light rays which intersect at a point.

Having picked a point x in Minkowski space, there is a sphere’s worth of (projective) null twistors corresponding to it – each null twistor representing a light ray through x. Now, (and you may be entitled have a WTF? Moment here), this sphere is actually referred to by algebraic geometers as a “projective line”. Why line FFS ? Well, in ordinary geometry a line is a mapping of the real numbers into some manifold. It’s one dimensional. In complex geometry, the complex numbers replace the real numbers, and “one dimensional” means “one complex-dimensional”. Single complex number coordinates can be slapped onto the sphere, so it’s a one complex-dimensional object. (Actually you need two coordinate patches to cover the sphere. Points on the sphere are given complex coordinates by stereographic projection). The sphere with this complex structure slapped onto it tends to be called the “Riemann sphere”, but algebraic geometers tend to refer to it as a “projective line”, or just a “line”).

Anyway, it’s a fucking line alright ? So, points in Minkowski space correspond to lines in projective null twistor space.

This is shown schematically on the following diagram, where the point x in Minkowski space corresponds to the “line” Lx (yes it’s really a sphere) of projective null twistors corresponding to the light rays through x. I’ve drawn a couple of twistors Z and W on this line and the light rays in Minkowski space they correspond to.

TwistorCorresp1.jpg
TwistorCorresp1.jpg (26.56 KiB) Viewed 4681 times



Next post: A bit of physics
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Ads by Google


Re: Twistor string theory

#16  Postby twistor59 » Feb 04, 2011 8:05 am

A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#17  Postby twistor59 » Feb 19, 2011 4:24 pm

Post #4 Twistor Treatment of Maxwell Equations

As an example of the twistor approach to physical problems, I'll show how Maxwell's equations are handled. Firstly, we need to recast them in a form which may not be so familiar to people. In the form which is familiar, the free equations (i.e. no sources and currents) are:

divB= 0
B/∂t + curl E = 0
div E = 0
E/∂t - curl B = 0

Firstly, we can write the equations in tensor form by constructing the Maxwell tensor thus:

Fab = (0.....-E1......-E2.....-E3)
.......(E1.....0.......B3......-B2)
.......(E2.....-B3......0......-B1)
.......(E3.....B2.......-B1......0)

Then the equations are
aFbc + ∇bFca + ∇cFab = 0 and

aFab = 0

To see that the first set of equations are satisfied, write Fab in terms of a potential, and it then becomes obvious:
Fab = ∇a𝜒b - ∇b𝜒a

The tricky part is to find fields satisfying the second set of equations. Everyone knows by now that true wisdom is only achieved by writing equations in spinor form (tensors are so last-year), so this is what we do.

In the last post I mentioned that a Lorentz vector index a is equivalent to a pair of spinor indices AA', so our Maxwell tensor Fab is really a spinor with two unprimed and two primed indices FAA'BB' .

If you do this, there is a nice way to decompose F:

FAA'BB' = ϕABεA'B' + ψA'B'εAB

where ε is the antisymmetric fixed spinor that I mentioned last post, and ϕAB and ψA'B' are SYMMETRIC spinor fields. If we want FAA'BB' to represent a real Maxwell field, then

ψA'B' = complex conj of ( ϕAB )

Once we've made this decomposition, Maxwell's equations are simply

AA' ϕAB = 0;

AA' ψA'B' = 0;

Twistor theory gives a nice way to write down and classify solutions to these equations. Remember the picture of (projective) twistor space I gave in the last post where the twistor space was divided into two halves by the space of (projective) null twistors ? Denote the top half by P𝕋+, denote the top half of ordinary (non projective) twistor space by 𝕋+. Pick yourself a function f on 𝕋+, which is holomorphic on a suitable region (will define this region in a minute) and homogenous of degree zero.

"Holomorphic" on a region means it's a function of the Z's only (not the Zbar's) and doesn't do pathological things in the region like sod off to infinity.

A function on twistor space is homogenous of degree n if f(λZα) = λnf(Zα)

Now, if you don't want to get a trivial solution of Maxwells equations, you shouldn't pick your function f to be holomorphic everywhere - it needs to have some singular regions. Once you've picked your function, to generate the Maxwell field, you do the following contour integral:

ϕAB(x) = ∮ρx(𝜕/𝜕ωA 𝜕/𝜕ωB f(Zα) πC'C' )

The integrand is a function of the twistor coordinates Zα = (ωA, πA’), and the function ρx just resticts the twistors to the Minkowski space point x, i.e. it forces

ωA = i xAA’ πA’
ContourIntegral.jpg
ContourIntegral.jpg (24.75 KiB) Viewed 4608 times


The contour is taken along any closed curve on Lx which threads past the singularities of f(Z) (the red bits).

It’s fairly easy to show that a function generated in this way satisfies the Maxwell equations:
AA' ϕAB = 0.

Slightly harder is to show that any solution arises in this way (this is indeed the case, but the proof requires a fair bit of machinery).

What about generating the other type of solution to Maxwell’s equations, namely the ones with primed indices ? To do this, recall that previously we picked a function f(Zα) which was homogenous of degree ZERO. This time we need to pick one which has similar singularity properties, but is homogeneous of degree MINUS 4. We then form

ϕA’B’(x) = ∮ρxA’ πB’ f(Zα) πC'C' )
Again this results in a solution of the primed Maxwell equations. The two “varieties” of Maxwell fields that we have generated, namely ϕAB and ψA'B' are fields representing “left handed” and “right handed” Maxwell fields respectively, i.e. fields with left and right circular polarization.

Positive and Negative Frequency Fields


I want to say a bit about how this is handled in the twistor picture. First I’ll explain a bit about what the issue is. First we consider some construction that you do in the first lecture on QFT. I’ll describe it for a Klein-Gordon (i.e. scalar) field because the notation is easier.

A Klein-Gordon field satisfies

(☐+m2)φ(x) = 0

We Fourier decompose it into plane waves:

φ(x) = ∫d4k 𝛿(k2-m[sup2[/sup])e-ikxa(k) (integral over whole of k space)

The delta function restricts the integration to the mass shell k2 = m2, which is a hyperboloid in momentum space, looking like this (I’ve just drawn 2 out of the 3 “spatial” momentum axes).
MassShell.jpg
MassShell.jpg (20.71 KiB) Viewed 4608 times


Another way of saying this is that only modes are present that have k0 = +ωk or k0 = -ωk where
ωk ≝ sqrt(k2 + m2)

Now we can split our momentum space function a(k) into two halves, one a+(k) on the red sheet and one a-(k) on the blue by doing this:

a(k) = 𝛳(k0)a+(k) + 𝛳(-k0)a-(k)

where 𝛳 is the Heaviside step function defined simply by 𝛳(s) = 1 if s>=0, and 0 otherwise.

After a little bit of manipulation, we end up being able to write

φ(x) = φ+(x) + φ-(x)

Where

φ+(x) = ∫d3k (1/sqrt(ωk))e-ikxa(k)

is the “positive frequency part” of φ(x), and

φ-(x) = ∫d3k (1/sqrt(ωk))eikxa*(k)

is the “negative frequency part” of φ(x). This splitting is important because when the fields are quantized, φ+(x) contains the creation operators and φ-(x) contains the annihilation operators. (Incidentally, this splitting is what gets mixed up when you transform to an accelerated frame and is at the root of the Unruh effect).

Now there is another way to model this splitting. It is to consider fields on real Minkowski space as “boundary values” of holomorphic functions on complexified Minkowski space. Complexified Minkowski space ℂM is obtained by allowing the coordinates on Minkowski space M to become complex numbers. The positive frequency functions on M can be obtained as boundary values of holomorphic functions on the “Future Tube” (sometimes called the “Forward Tube”), denoted ℂM+. Correspondingly the negative frequency ones are boundary values of holomorphic functions on the past tube ℂM-.

If we give ℂM coordinates za = xa+iya where a = 0, 1, 2, 3 and xs and ya are real, then ℂM+ is defined to be the subset of ℂM where ya is timelike and future pointing.

The reason this funky approach is taken is that axiomatic quantum field theorists want a framework which is sufficiently general to deal, not just with functions on M, but also with distributions. The “positive frequency = holomorphic in the future tube” picture allows them to do just that. If anyone wants to dip their toes into rigorous quantum field theory a good place to start is http://www.amazon.com/PCT-Spin-Statistics-All-That/dp/0691070628

Now all this stuff fits incredibly well into the twistor picture of things. Recall in the last post, we had a twistor norm, and the twistors of zero norm, which correspond to real Minkowski space M divide twistor space into two halves P𝕋+ and P𝕋-. Well, it turns out that P𝕋+ corresponds to the future tube ℂM+ and P𝕋- corresponds to the past tube ℂM-. So twistor space rather neatly has an inbuilt mechanism for doing the positive/negative frequency splitting – considering the Penrose transform for the Maxwell field above, if I want a positive frequency solution, I just do my contour integration in P𝕋+ and vice versa.


(By the way, some people will talk of M as a boundary of ℂM+ or ℂM-. But ℂM+ or ℂM- is 4-complex (i.e. 8 real) dimensional, so a boundary would be 7 real dimensional. In fact, what is meant here is not a topological boundary, but rather a Shilov boundary.
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#18  Postby campermon » Feb 19, 2011 4:32 pm

Your sums are all wrong there twistor!

There are no numbers for a start, just funny greek symbols. What king of crazy shit is this?

Reported to the campaign for real maths.

:coffee:
Scarlett and Ironclad wrote:Campermon,...a middle aged, middle class, Guardian reading, dad of four, knackered hippy, woolly jumper wearing wino and science teacher.
User avatar
campermon
RS Donator
 
Posts: 17032
Age: 49
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#19  Postby twistor59 » Feb 19, 2011 5:51 pm

campermon wrote:
There are no numbers for a start, just funny greek symbols. What king of crazy shit is this?



Geek symbols ? how dare you. Fucking troll.... :lol:
A soul in tension that's learning to fly
Condition grounded but determined to try
Can't keep my eyes from the circling skies
Tongue-tied and twisted just an earthbound misfit, I
User avatar
twistor59
RS Donator
 
Posts: 4966
Male

United Kingdom (uk)
Print view this post

Re: Twistor string theory

#20  Postby campermon » Feb 19, 2011 6:35 pm

twistor59 wrote:
campermon wrote:
There are no numbers for a start, just funny greek symbols. What king of crazy shit is this?



Geek symbols ? how dare you. Fucking troll.... :lol:


:rofl:

I've never been called a troll!!

I likes it!

:lol: :lol: :lol:

:thumbup:
Scarlett and Ironclad wrote:Campermon,...a middle aged, middle class, Guardian reading, dad of four, knackered hippy, woolly jumper wearing wino and science teacher.
User avatar
campermon
RS Donator
 
Posts: 17032
Age: 49
Male

United Kingdom (uk)
Print view this post

Next

Return to Physics

Who is online

Users viewing this topic: No registered users and 1 guest