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Can you, please, help me to understand the following

  1. How is the analyticity of a complex-valued function in the upper half plane related to causality and Kramers-Kronig relations? Namely, why is it not the lower half-plane?

  2. Does it imply that the poles of the response function have to be in the lower half plane or at the real axis to fulfill causality?

The causality can be imposed by multiplying the response function by the Heaviside function with the following Fourier transform:

$$F[H(t)] = \frac{\delta(\omega)}{2}+\frac{1}{i2\pi\omega}$$

The convolutional theorem leads to:

$$\chi(\omega) = \frac{1}{\pi}\int\limits_{-\infty}^{\infty} \frac{\chi(\Omega)d\Omega}{i(\omega-\Omega)}$$

Now imagine that $\chi(\Omega)$ has a pole in the upper half-plane. How does it violate this equality?

The reference where the requirement of analyticity in the upper half-plane is established is: John S. Toll, "Causality and the Dispersion Relation: Logical Foundations", Phys. Rev. 104 (1760)

Edit:

Two expressions shown above seem to be the proof of the first consequence of Titchmarsh's theorem 95 (see E. C. Titchmarsh "Introduction to the Theory of Fourier Integrals", 1948, p. 128). What I want is to explain the proof of the second consequence.

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    $\begingroup$ "a complex-valued function" Is there any particular function you are interested in? $\endgroup$
    – hft
    Commented Jun 23, 2023 at 14:51
  • $\begingroup$ not really, any function indeed $\endgroup$
    – freude
    Commented Jun 23, 2023 at 14:52
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    $\begingroup$ If "any function" then I don't understand why there would be any connection to causality. Just some general "any function" does not necessarily have anything to do with causality. Are you sure you are not interested in some particular response function? $\endgroup$
    – hft
    Commented Jun 23, 2023 at 14:54
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    $\begingroup$ But you wrote "causality" too, right? $\endgroup$
    – hft
    Commented Jun 23, 2023 at 14:57
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    $\begingroup$ What are you quoting from? $\endgroup$
    – hft
    Commented Jun 23, 2023 at 14:58

2 Answers 2

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  1. How is the analyticity of a complex-valued function in the upper half plane related to causality and Kramers-Kronig relations? Namely, why is it not the lower half-plane?

A response function $\chi(t-t')$ provides a relationship between some input that I control $D(t')$ and some output $E(t)$: $$ E(t) = \int dt' \chi(t-t') D(t')\;. $$

Causality means that: $$ \chi(t-t') = 0 $$ when $t-t'<0$. That is, the output doesn't happen before the input.

In the frequency domain: $$ E(\omega) = \tilde \chi(\omega)D(\omega) $$

We have: $$ \chi(t) = \int d\omega e^{-i\omega t}\tilde \chi(\omega)\;, $$ so if $t<0$ and if there are no poles in the UHP then I find $\chi(t) = 0$.

Note that, there is an element of convention here, in particular the time fourier transform: $$ f(t) = \int d\omega e^{-i\omega t}\tilde f(\omega)\;, $$ where it is conventional in physics to put a minus sign in the exponent.


  1. Does it imply that the poles of the response function have to be in the lower half plane or at the real axis to fulfill causality?

The lower half plane, not on the real axis. You can't integrate through a pole. But often the poles are brought close to the real axis and we say we are just slightly avoiding them.

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – rob
    Commented Jun 24, 2023 at 5:37
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After carefully reading the cited paper, I have realized that the confusion was related to the fact that I missed an additional requirement for the response function. Specifically, the theorem holds only if it is assumed that the response function is a square-integrable function. To fulfill this requirement we need the decaying exponents.

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