# Straightforward twistor theory/spinor calculus question

• Bballer152
In summary: Finally, we can use the fact that \alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2 to obtain the desired expression. In summary, by supposing that the dual twistors A_\alpha and B_\alpha meet on the line L_y, we can obtain the expression \alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2. I hope this helps to clarify your confusion. Please let me know if you have any further questions. Best of luck!
Bballer152
Not sure where to post this, but this question has been frustrating me for hours now because I'm pretty sure it has a very straightforward answer. I'll reproduce the problem in a moment, but for those with the Huggett and Tod's Intro to Twistor Theory, I don't understand how we get the expression immediately before eq. 8.5 from the given info.

We're given the twistor function $$f(Z^\alpha)=\frac{1}{(A_\alpha Z^\alpha)(B_\beta Z^\beta)}$$ and we restrict the function to the line $$L_x$$ so that

$$A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}\equiv \alpha^{A'} \pi_{A'}$$ and
$$B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}\equiv \beta^{A'} \pi_{A'}$$.

Now comes the part I don't understand. Apparently, by supposing that the dual twistors $$A_\alpha$$ and $$B_\alpha$$ meet in the line $$L_y,$$ then we should get $$\alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2.$$ When I do this I get a similar answer but my contractions don't work out this way. In particular, my spinors indices contract in a different order and I have no idea how to "split them up", if that makes sense. Any help would be GREATLY appreciated!

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Thank you for your question regarding twistor theory. I understand that you are struggling with understanding how to obtain the expression before equation 8.5 in Huggett and Tod's Intro to Twistor Theory. I will try my best to explain this to you.

Firstly, let's review the given information. We have the twistor function f(Z^\alpha)=\frac{1}{(A_\alpha Z^\alpha)(B_\beta Z^\beta)}, which we are restricting to the line L_x. This means that A_\alpha Z^\alpha and B_\alpha Z^\alpha are both on the line L_x. We can rewrite these expressions as A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'} and B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}.

Next, we are supposing that the dual twistors A_\alpha and B_\alpha meet on the line L_y. This means that we can set \alpha^{A'}=\beta_{A'} and obtain the expression \alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2.

Now, let's look at how we can get this expression. We can start by expanding the expression for A_\alpha Z^\alpha and B_\alpha Z^\alpha on the line L_x. This gives us A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}=(iA_A x^{AA'}+A^{A'})\pi_{A'} and B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}=(iB_A x^{AA'}+B^{A'})\pi_{A'}.

Next, we can substitute in the expression \alpha^{A'}=\beta_{A'} and obtain A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}=(iA_A x^{AA'}+A^{A'})\pi_{A'} and B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}=(iB_A x^{AA'}+B^{A'})\pi_{A'}

## 1. What is twistor theory?

Twistor theory is a mathematical framework developed by Roger Penrose for describing the geometry of spacetime in terms of twistors, which are mathematical objects that encode both space and time information. It provides a new perspective on the nature of spacetime and has applications in various fields such as physics, mathematics, and engineering.

## 2. How does twistor theory relate to spinor calculus?

Spinor calculus is a mathematical tool used in twistor theory to manipulate twistor objects. It allows for the calculation of various quantities such as spinors, tensors, and spinor fields, which are essential in studying the structure of spacetime. In twistor theory, spinor calculus is used to describe the geometric properties of twistors and their interactions with other physical systems.

## 3. What are the advantages of using twistor theory over traditional approaches?

One advantage of twistor theory is that it provides a more elegant and intuitive way of understanding the geometry of spacetime compared to traditional approaches such as the use of tensors. It also allows for the description of physical phenomena that are difficult to explain using conventional methods. Additionally, twistor theory has applications in both classical and quantum physics, making it a versatile tool for studying the fundamental laws of nature.

## 4. Is twistor theory supported by experimental evidence?

While twistor theory has not been directly tested through experiments, it has made significant contributions to our understanding of physical phenomena. For example, it has been used to derive the scattering amplitudes of particles in quantum field theory, which have been confirmed through experiments. Additionally, twistor theory has been applied to various areas of physics, such as general relativity and quantum gravity, where it has led to new insights and predictions.

## 5. What are some potential future developments in twistor theory?

One potential development in twistor theory is its application in the study of quantum gravity, as it provides a promising approach for unifying general relativity and quantum mechanics. It may also have applications in other areas of physics, such as cosmology and high-energy physics, where it could lead to new insights and advancements. Another potential direction is the development of new mathematical tools and techniques within twistor theory, which could further enhance our understanding of the underlying structure of spacetime.

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