How Do We Determine the Dependence of 2-Point Correlators in CFTs?

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In summary, the two- and three-point correlators of spinless fields in Conformal Field Theory can be derived from the invariance under rotations, translations, and scaling. The dependence on the relative coordinates of the quasi primary fields follows from the invariance statements, resulting in a scaling behavior of the type f(|x_1-x_2|)\sim \lambda^{\Delta_1+\Delta_2}f(\lambda|x_1-x_2|). By imposing invariance under special conformal transformations, the most general solution is found to be \frac{C(\Delta_{1}, \Delta_{2})}{|x|^{\Delta_{1} + \Delta_{2}}}, where C is
  • #1
earth2
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Hi guys,

i'm studying Conformal Field Theory using the big yellow book by Senechal et al. So far everything has been a smooth ride. I'm a bit stuck at the point where they derive the 2- and 3-point correlator for spinless fields.

Based on invariance under rotations and translations the correlator should depend only on the relative coords of the quasi primary fields and moreover - because of scaling invariance - this dependence should be of the type

[tex] f(|x_1-x_2|)\sim \lambda^{\Delta_1+\Delta_2}f(\lambda|x_1-x_2|)[/tex] where λ is the scaling and Δ the conformal weight.

But then those guys say that this is nothing but

[tex]\langle \phi(x_1)\phi(x_2)\rangle \sim \frac{1}{|x_1-x_2|^{\Delta_1+\Delta_2}} [/tex]

which is cannot follow. How do they know that the dependence is in the denominator and where does the exponent come from explicitely?
Any help is appreciated!
Thanks,
earth2
 
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  • #2
Hello! I don't know whether I have understood what you have written; but in the case in which [itex]\langle \phi(x_1)\phi(x_2)\rangle = f(|x_1-x_2|)[/itex] and the behavior you have written is not just a behavior but an equality, then in my opinion you can try the following mathematical trick: the equation
[itex]\lambda^{-\Delta_1-\Delta_2}f(|x_1-x_2|)=f(\lambda |x_1-x_2|)[/itex]
is valid for every λ; you can, furthermore, subtract [itex]f(|x_1-x_2|)[/itex] and then divide by [itex]\lambda -1[/itex] both sides of the equation. In the limit [itex]\lambda\rightarrow 1[/itex] you can find a differential equation: if I didn't make any mistake it has the following form
[itex]-\frac{(\Delta_1+\Delta_2)}{|x_1-x_2|}f(|x_1-x_2|)=f'(|x_1-x_2|)[/itex]
The solution of this differential equation is the solution you have written with up to the multiplication of an unknown constant coefficient which depends on the border conditions.
I hope I have been clear.
 
  • #3
Thank you for your answer! That is a nice way to understand this! From the book however I have the impression that the conclusion is much simpler to get and follows 'for free' from the invariance statements. But thanks a lot anyways!

earth2

P.s. you were right, the [tex] \sim [/tex] should be an equality in the first equation
 
  • #4
earth2 said:
Hi guys,

i'm studying Conformal Field Theory using the big yellow book by Senechal et al. So far everything has been a smooth ride. I'm a bit stuck at the point where they derive the 2- and 3-point correlator for spinless fields.

Based on invariance under rotations and translations the correlator should depend only on the relative coords of the quasi primary fields and moreover - because of scaling invariance - this dependence should be of the type

[tex] f(|x_1-x_2|)\sim \lambda^{\Delta_1+\Delta_2}f(\lambda|x_1-x_2|)[/tex] where λ is the scaling and Δ the conformal weight.

But then those guys say that this is nothing but

[tex]\langle \phi(x_1)\phi(x_2)\rangle \sim \frac{1}{|x_1-x_2|^{\Delta_1+\Delta_2}} [/tex]

which is cannot follow. How do they know that the dependence is in the denominator and where does the exponent come from explicitely?
Any help is appreciated!
Thanks,
earth2

Poincare invariance implies
[tex]
\langle \Phi_{\Delta_{1}}(x_{1})\Phi_{\Delta_{2}}(x_{2}) \rangle = F(|x_{1}- x_{2}|).
[/tex]
Scale invariance;
[tex]
\Phi_{\Delta_{i}}(x_{i}) \rightarrow \lambda^{\Delta_{i}} \ \Phi_{\Delta_{i}}(\lambda x_{i}), \ \ i = 1,2 ,
[/tex]
leads to
[tex]
F(|x|) = \lambda^{\Delta}F(\lambda |x|), \ \ (1)
[/tex]
where
[tex]|x| = |x_{1} - x_{2}| \ \ \mbox{and} \ \Delta = \Delta_{1} + \Delta_{2}.[/tex]
Eq(1) tells you that [itex]F(|x|)[/itex] does not depend on [itex]\lambda[/itex] and it admits the following (most general) solution,
[tex]F(|x|) = \frac{C(\Delta_{1}, \Delta_{2})}{|x|^{\Delta_{1} + \Delta_{2}}} \ \ (2).[/tex]
(Put [itex]F(|x|) \propto |x|^{N}[/itex] in eq(1), you find [itex]N = -\Delta[/itex])

Finally, demanding invariance under special conformal transformation, we find
[tex]
F(|x|) = \frac{C \ \delta_{\Delta_{1}, \Delta_{2}}}{|x|^{\Delta_{1} + \Delta_{2}}}
[/tex]
Where C is a constant depends on the type of the field. Thus, in order to have a non-vanishing two point function, the fields must have the same scaling dimension.
If it is not obvious to you that eq(2) is the most general solution to eq(1), then do the following; write
[tex]\lambda = 1 + \epsilon , \ \ |\epsilon| \ll 1,[/tex]
then, expanding to first order in [itex]\epsilon[/itex], eq(1) gives you
[tex]|x| \frac{dF(|x|)}{d|x|} = - \Delta F(|x|)[/tex]
This you can solve to find eq(2).

Sam
 
Last edited:
  • #5
Yupp, thank you for your answer!
 

1. What is a 2Point Correlator in CFTs?

A 2Point Correlator in CFTs (Conformal Field Theories) is a mathematical tool used in theoretical physics to study the behavior of quantum systems. It measures the correlation between two points in a system and is often used to analyze the scaling properties of the system.

2. How is a 2Point Correlator calculated?

A 2Point Correlator is calculated by taking the expectation value of two operators at different points in a system. These operators can represent physical quantities such as energy or momentum. The resulting value is a measure of the correlation between the two points.

3. What information can be obtained from a 2Point Correlator?

A 2Point Correlator can provide information about the scaling behavior of a system, such as the critical exponents and dimensions of operators. It can also reveal the underlying symmetries and properties of the system.

4. How is a 2Point Correlator used in CFTs?

In CFTs, a 2Point Correlator is used to study the behavior of conformally invariant systems. It is often used in conjunction with other mathematical tools, such as the conformal bootstrap, to make predictions about the properties of these systems.

5. Can a 2Point Correlator be measured experimentally?

Yes, a 2Point Correlator can be measured experimentally in systems that exhibit conformal invariance, such as certain condensed matter systems. However, it can also be calculated theoretically using mathematical techniques, making it a valuable tool for understanding the behavior of quantum systems.

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