In the context of calculations of loop diagrams with Cutkosky-rules very often the concept of dispersion relation is mentioned. For instance this technique is used in Vol. 4 of Landau-Lifshitz for the computation of the polarization tensor in paragraph 110. It also exists a note of B.A.Kniehl "Dispersion Relations in Loop Calculations" hep-ph/9607255 that is devoted to the subject and uses as an example the computation of the photon vacuum polarization.

I will give here as an example the integral representation of the photon self-energy tensor(also to be found in Peskin& Schroeder):

$$ \Pi_{\mu\nu} = \Pi(k^2)\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}\right)$$

defined in Landau-Lifshitz paragraph 101 and further discussed in paragraph 108: There the function $\Pi(t)$ is represented as an integral along a branch cut:

$$\Pi(t) = \int_0^\infty \frac{\rho(t') dt}{t-t' +i\epsilon}$$

where $\rho(t)$ is called the spectral density function. More below it can be found the the spectral density function can be expressed by the imaginary part of $\Pi(t)$:

$$Im(\Pi(t)) = -\pi \rho(t)$$

so that one can write:

$$\Pi(t) = \frac{1}{\pi}\int_0^\infty \frac{Im(\Pi(t')}{t'-t-i\epsilon} dt'$$

This formula is quoted by Landau & Lifshitz as dispersion relation. Actually, similar formulas can be found in more modern books as Peskin& Schroeder in the sections on the optical theorem and Cutkosky rules, however, the concept dispersion relation is not mentioned.

Elder books like the 2nd volume of Bjorken & Drell dedicate a whole chapter of 75 pages on the subject. They start talking about impinging and scattered wave amplitudes (but not about a relation between frequency and the wave vector) and mention at the end of the chapter's first paragraph the Kramer-Kronig relations.

I even found the concept dispersion relation mentioned in the A.Chao's book on "Physics of collective beam instabilities in High-Energy Accelerators" in section 5.2 when the author discusses the damping effect (also called Landau damping) of a whole ensemble of particles undergoing oscillations driven by wake fields if the oscillation frequency of the particle ensemble has spectral density $\rho(\omega)$ (different from a delta function), i.e. particles can oscillate at different frequencies. For a consistent oscillatory motion a self-consistency condition has to be fulfilled:

$$ 1 = - \frac{N r_0 {\cal {W}}}{2\omega_\beta \gamma T_0}\left[ P.V. \int \frac{\rho(\omega)}{\omega-\Omega}d\omega + i \pi \rho(\Omega)\right]$$

which Chao also calls a dispersion relation. ${\cal{W}}$ stands for the wake field, the rest are some normalizing factors which are here of little interest.

I absolutely see the similarity in the problems mentioned, however, why would one call such a relation dispersion relation ? I would call such a relation as corollary of Cauchy's integral theorem, or Hilbert transform, or Kramer-Kronig relation, but why dispersion relation ? Actually, on Wikipedia a dispersion relation is defined as relation between frequency and wavelength respectively wave vector. So actually I my question is mainly about terminology. But I guess there is some physics behind why this type of relations got the name dispersion relation.

But it seems that nowadays this technique is a kind of old-fashioned as it was used a lot in S-matrix theory (bootstrap etc.) which seems to have only a few adepts today.


2 Answers 2


I don't think there's a particularly deep reason here, even though it's hard to prove a negative. Here's a story of how I imagine this happened:

You are correct that these relations are what we would today call Kramers-Kronig relations, but the place where physicists, especially early quantum theorists, most likely knew the concept of Kramers-Kronig relations from is from the notion of complex refractive indices, where it relates the absorption and dispersion to each other, and one form of the Kramers-Kronig relations for optics says that $$ n(\omega) = 1 + \frac{c}{\pi}\int \frac{\alpha(\omega')}{\omega'^2 - \omega^2} \mathrm{d}\omega'$$ for $n(\omega)$ the refractive index and $\alpha(\omega)$ the absorption coefficient. This is a dispersion relation since it's an equation for the refractive index and it does look a lot like the equations you've quoted if we squint, and so the name probably stuck to the HEP application of Kramers-Kronig relations that you have found even though they are not literally related to dispersion of waves.

Additionally, Kramers' and Kronig's original work was specifically about this dispersion relation and not about general properties of complex functions (which is also why the Wiki article on the Kramers-Kronig relations reminds us that mathematicians might call the more general version for complex functions the Sokhotski–Plemelj theorem instead). Since naming things after people tends to take a while to become universally accepted, it seems perfectly plausible to me that the early theorists who named the HEP versions "dispersion relations" learned about these results before the name "Kramers-Kronig relations" was well-established (Kramers and Kronig published their results independently in the late 1920s).


They are called dispersion relations because they are examples of the Kramers-Kronig Relations that link the real and imaginary parts of the refractive index of a material. And of course it the frequency dependent refractive index that causes a prism to disperse a light beam into its component colours.


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