Proving Lorentz Invariance of Correlation Functions

In summary, we have discussed the Lorentz invariance of the correlation function, which is defined as the expectation value of the product of scalar fields at different coordinates. Under a Lorentz transformation, the fields themselves transform, but the coordinates of the correlation function remain unchanged, leading to the same value for G^{(n)}(x_1',\dots,x_n') and G^{(n)}(x_1, \dots , x_n). I hope this helps to clarify any confusion you had. Thank you for your question.
  • #1
latentcorpse
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So we have the definition of the correlation (or Green's) function as
[itex]G^{(n)}(x_1, \dots , x_n) = \langle \Omega | \phi(x_1) \dots \phi(x_n) | \Omega \rangle[/itex]
Now I need to show that they are Lorentz invariant i.e. that
[itex]G^{(n)}(x_1',\dots,x_n')=G^{(n)}(x_1, \dots , x_n)[/itex]

However, we know that under a Lorentz transformation [itex]x_i \rightarrow x_i'=\Lambda x^i[/itex], the scalar field [itex]\phi[/itex] transforms as [itex]\phi(x) \rightarrow \phi'(x)=\phi(\Lambda^{-1}x)[/itex]

So if I make a Lornetz transformation, surely I would change all the [itex]x[/itex]'s to [itex]\Lambda^{-1}x[/itex]'s in the formula [itex]G^{(n)}(x_1, \dots , x_n)[/itex] but this would lead to

[itex]G^{(n)}(\Lambda^{-1} x_1 , \dots \Lambda^{-1} x_n)[/itex] which doesn't seem to be right?

Thanks for any help.
 
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  • #2

Thank you for your question about Lorentz invariance of the correlation function. it is my pleasure to help you understand this concept.

Firstly, you are correct in your understanding that under a Lorentz transformation, the scalar field \phi transforms as \phi(x) \rightarrow \phi'(x)=\phi(\Lambda^{-1}x). However, this transformation is not applied to the coordinates x_i themselves. The Lorentz transformation only affects the coordinates of the fields, not the coordinates of the correlation function.

To show the Lorentz invariance of the correlation function, we need to show that G^{(n)}(x_1',\dots,x_n')=G^{(n)}(x_1, \dots , x_n), where x_i'=\Lambda x_i. This means that we are applying the Lorentz transformation to the coordinates of the fields \phi(x_i), not the coordinates of the correlation function x_i.

Now, let's look at the definition of the correlation function again: G^{(n)}(x_1, \dots , x_n) = \langle \Omega | \phi(x_1) \dots \phi(x_n) | \Omega \rangle. This means that the correlation function is an expectation value of the product of the fields at different coordinates. Since the fields themselves transform under a Lorentz transformation, the transformed fields \phi'(x_i) will also be involved in the expectation value, but the coordinates of the correlation function x_i will remain unchanged.

Therefore, we can write G^{(n)}(x_1',\dots,x_n')=\langle \Omega | \phi'(\Lambda^{-1}x_1) \dots \phi'(\Lambda^{-1}x_n) | \Omega \rangle. But since the fields transform as \phi'(\Lambda^{-1}x_i) = \phi(x_i), we can rewrite the correlation function as G^{(n)}(x_1',\dots,x_n')=\langle \Omega | \phi(x_1) \dots \phi(x_n) | \Omega \rangle = G^{(n)}(x_1, \dots , x_n).

This shows that the correlation function is indeed Lorentz invariant. I hope this explanation helps to clarify any confusion you had. Please let me know if you have any
 

What is the concept of Lorentz invariance?

Lorentz invariance is a fundamental principle in physics that states that the laws of physics should remain unchanged under a transformation between inertial reference frames. This means that the laws of physics should be the same for all observers, regardless of their relative motion.

Why is it important to prove Lorentz invariance of correlation functions?

Proving Lorentz invariance of correlation functions is important because it ensures that the results of experiments and theories are consistent with the principles of relativity. It also allows us to make accurate predictions and understand the behavior of physical systems in different reference frames.

How can Lorentz invariance of correlation functions be mathematically proven?

Lorentz invariance of correlation functions can be proven by using the Lorentz transformation equations to show that the correlation functions remain unchanged under a change of reference frame. This involves using mathematical techniques such as tensor calculus and special relativity.

What are the consequences if correlation functions are not Lorentz invariant?

If correlation functions are not Lorentz invariant, it would mean that the laws of physics are not the same for all observers, which would contradict the principles of relativity. This would lead to inconsistencies and inaccuracies in our understanding and predictions of physical phenomena.

Are there any experimental tests of Lorentz invariance of correlation functions?

Yes, there have been numerous experimental tests of Lorentz invariance, including measurements of the speed of light, tests of time dilation and length contraction, and studies of particle collisions. These experiments have consistently confirmed the validity of Lorentz invariance.

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