Kramers - Kronig relations

The Kramers – Kronig relation is an integral relationship between the real and imaginary parts of any complex function that is analytic in the upper half-plane. Often used in physics to describe the relationship between the real and imaginary parts of the response function of a physical system, since the analyticity of the response function implies that the system satisfies the causality principle , and vice versa ^[1] . In particular, the Kramers – Kronig relations express the relationship between the real and imaginary parts of the dielectric constant in classical electrodynamics and the amplitude of the transition probability ( matrix element ) between two states in quantum field theory .

Content

Definition

For complex function $\chi (\omega )=\chi _{1}(\omega )+i\chi _{2}(\omega )$ ${\ displaystyle \ chi (\ omega) = \ chi _ {1} (\ omega) + i \ chi _ {2} (\ omega)}$ complex variable $\omega ,$ ${\ displaystyle \ omega,}$ analytic in the upper half-plane $\omega$ ${\ displaystyle \ omega}$ and tending to zero at $|\omega |\rightarrow \infty ,$ ${\ displaystyle | \ omega | \ rightarrow \ infty,}$ Kramers-Kronig relations are written as follows:

\chi _{1}(\omega )={1 \over \pi }v.p.\int \limits _{-\infty }^{\infty }{\chi _{2}(\omega ') \over \omega '-\omega }\,d\omega '

{\ displaystyle \ chi _ {1} (\ omega) = {1 \ over \ pi} vp \ int \ limits _ {- \ infty} ^ {\ infty} {\ chi _ {2} (\ omega ') \ over \ omega '- \ omega} \, d \ omega'}

and

\chi _{2}(\omega )=-{1 \over \pi }v.p.\int \limits _{-\infty }^{\infty }{\chi _{1}(\omega ') \over \omega '-\omega }\,d\omega ',

{\ displaystyle \ chi _ {2} (\ omega) = - {1 \ over \ pi} vp \ int \ limits _ {- \ infty} ^ {\ infty} {\ chi _ {1} (\ omega ') \ over \ omega '- \ omega} \, d \ omega',}

where are the characters $v . p .$ ${\ displaystyle vp}$ means taking the integral in the sense of the main value (according to Cauchy) . It's clear that $\chi _{1}(\omega )$ ${\ displaystyle \ chi _ {1} (\ omega)}$ and $\chi _{2}(\omega )$ ${\ displaystyle \ chi _ {2} (\ omega)}$ are not independent, which means that the full function can be restored if only its real or imaginary part is specified.

In a more compact form:

\chi (\omega )={1 \over i\pi }v.p.\int \limits _{-\infty }^{\infty }{\chi (\omega ') \over \omega '-\omega }\,d\omega '.

{\ displaystyle \ chi (\ omega) = {1 \ over i \ pi} vp \ int \ limits _ {- \ infty} ^ {\ infty} {\ chi (\ omega ') \ over \ omega' - \ omega } \, d \ omega '.}

Conclusion

Let be $f(x)$ ${\ displaystyle f (x)}$ - continuous function of a complex variable $x$ ${\ displaystyle x}$ . Let us estimate the sum of the integrals along the contours slightly higher and slightly lower than the real axis: $\lim _{\epsilon \to 0}\left[\int _{-\infty }^{\infty }{\frac {f(x)dx}{x-i\epsilon }}+\int _{-\infty }^{\infty }{\frac {f(x)dx}{x+i\epsilon }}\right]=\lim _{\epsilon \to 0}\left[\int _{-\infty -i\epsilon }^{\infty -i\epsilon }{\frac {f(x'+i\epsilon )dx'}{x'}}+\int _{-\infty +i\epsilon }^{\infty +i\epsilon }{\frac {f(x'-i\epsilon )dx'}{x'}}\right]=2v.p.\int _{-\infty }^{\infty }{\frac {f(x)}{x}}dx$ ${\ displaystyle \ lim _ {\ epsilon \ to 0} \ left [\ int _ {- \ infty} ^ {\ infty} {\ frac {f (x) dx} {xi \ epsilon}} + \ int _ { - \ infty} ^ {\ infty} {\ frac {f (x) dx} {x + i \ epsilon}} \ right] = \ lim _ {\ epsilon \ to 0} \ left [\ int _ {- \ infty -i \ epsilon} ^ {\ infty -i \ epsilon} {\ frac {f (x '+ i \ epsilon) dx'} {x '}} + \ int _ {- \ infty + i \ epsilon} ^ {\ infty + i \ epsilon} {\ frac {f (x'-i \ epsilon) dx '} {x'}} \ right] = 2v.p. \ int _ {- \ infty} ^ {\ infty} {\ frac {f (x)} {x}} dx}$ . Let us estimate the difference of the integrals along the contours slightly higher and slightly lower than the real axis: $\lim _{\epsilon \to 0}\left[\int _{-\infty }^{\infty }{\frac {f(x)dx}{x-i\epsilon }}-\int _{-\infty }^{\infty }{\frac {f(x)dx}{x+i\epsilon }}\right]=\oint _{-\infty }^{\infty }{\frac {f(x)}{x}}dx=2\pi if(0)$ ${\ displaystyle \ lim _ {\ epsilon \ to 0} \ left [\ int _ {- \ infty} ^ {\ infty} {\ frac {f (x) dx} {xi \ epsilon}} - \ int _ { - \ infty} ^ {\ infty} {\ frac {f (x) dx} {x + i \ epsilon}} \ right] = \ oint _ {- \ infty} ^ {\ infty} {\ frac {f ( x)} {x}} dx = 2 \ pi if (0)}$ ( Cauchy integral formula ). Combining these two equalities, we find $\int _{-\infty }^{\infty }{\frac {f(x)dx}{x\pm i\epsilon }}=v.p.\int _{-\infty }^{\infty }{\frac {f(x)}{x}}dx\mp i\pi f(0)$ ${\ displaystyle \ int _ {- \ infty} ^ {\ infty} {\ frac {f (x) dx} {x \ pm i \ epsilon}} = vp \ int _ {- \ infty} ^ {\ infty} {\ frac {f (x)} {x}} dx \ mp i \ pi f (0)}$ . Consider the function $f(z)$ ${\ displaystyle f (z)}$ integrated variable $z$ ${\ displaystyle z}$ analytic in the upper half-plane and consider the contour integral $\oint {\frac {f(z)dz}{z-a}}$ ${\ displaystyle \ oint {\ frac {f (z) dz} {za}}}$ along a contour on a complex plane formed by a semicircle including a point inside $a$ ${\ displaystyle a}$ and a segment of the real axis as the diameter of the semicircle. We will unlimitedly increase the diameter of this semicircle. By virtue of the Cauchy integral formula, this integral will be equal to $2\pi if(a)$ ${\ displaystyle 2 \ pi if (a)}$ . If the function $f(z)$ ${\ displaystyle f (z)}$ quickly decreases at infinity, then the contribution from integration over the semicircle of unlimited radius disappears and only the contribution from integration along the real axis remains: $2\pi if(a)=\int _{-\infty }^{\infty }{\frac {f(z)dz}{z-a}}$ ${\ displaystyle 2 \ pi if (a) = \ int _ {- \ infty} ^ {\ infty} {\ frac {f (z) dz} {za}}}$ . If put $Ima=\epsilon$ ${\ displaystyle Ima = \ epsilon}$ where $\epsilon$ ${\ displaystyle \ epsilon}$ is an infinitesimal quantity, then $\lim _{\epsilon \to 0}f(a+i\epsilon )={\frac {1}{2\pi i}}\lim _{\epsilon \to 0}\int _{-\infty }^{\infty }{\frac {f(z)dz}{z-a-i\epsilon }}={\frac {1}{2\pi i}}\left[v.p.\int _{-\infty }^{\infty }{\frac {f(z)dz}{z-a}}+i\pi f(a)\right]$ ${\ displaystyle \ lim _ {\ epsilon \ to 0} f (a + i \ epsilon) = {\ frac {1} {2 \ pi i}} \ lim _ {\ epsilon \ to 0} \ int _ {- \ infty} ^ {\ infty} {\ frac {f (z) dz} {zai \ epsilon}} = {\ frac {1} {2 \ pi i}} \ left [vp \ int _ {- \ infty} ^ {\ infty} {\ frac {f (z) dz} {za}} + i \ pi f (a) \ right]}$ . From here we get $f(x)={\frac {1}{\pi i}}v.p.\int _{-\infty }^{\infty }{\frac {f(z)dz}{z-x}}$ ${\ displaystyle f (x) = {\ frac {1} {\ pi i}} vp \ int _ {- \ infty} ^ {\ infty} {\ frac {f (z) dz} {zx}}}$ Assuming $f(x)=Ref(x)+iImf(x)$ ${\ displaystyle f (x) = Ref (x) + iImf (x)}$ , we arrive at the dispersion relations: $Ref(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {Imf(z)dz}{z-x}}$ ${\ displaystyle Ref (x) = {\ frac {1} {\ pi}} \ int _ {- \ infty} ^ {\ infty} {\ frac {Imf (z) dz} {zx}}}$ , $Imf(x)=-{\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {Ref(z)dz}{z-x}}$ ${\ displaystyle Imf (x) = - {\ frac {1} {\ pi}} \ int _ {- \ infty} ^ {\ infty} {\ frac {Ref (z) dz} {zx}}}$ ^[2] .

Kramers - Kronig relations in physics

Classical electrodynamics ^[3] ^[4]

An important example of the application of the Kramers - Kronig relations in physics is the expression of dispersion relations in classical electrodynamics . In this case $\varepsilon (\omega )=\varepsilon ^{\prime }(\omega )+i\varepsilon ^{\prime \prime }(\omega )$ ${\ displaystyle \ varepsilon (\ omega) = \ varepsilon ^ {\ prime} (\ omega) + i \ varepsilon ^ {\ prime \ prime} (\ omega)}$ where $\varepsilon$ ${\ displaystyle \ varepsilon}$ Is the dielectric constant , ω is the frequency .

\varepsilon ^{\prime }(\omega )-1={\frac {1}{\pi }}v.p.\int \limits _{-\infty }^{\infty }{\frac {\varepsilon ^{\prime \prime }(x)}{x-\omega }}dx

{\ displaystyle \ varepsilon ^ {\ prime} (\ omega) -1 = {\ frac {1} {\ pi}} vp \ int \ limits _ {- \ infty} ^ {\ infty} {\ frac {\ varepsilon ^ {\ prime \ prime} (x)} {x- \ omega}} dx}

and

\varepsilon ^{\prime \prime }(\omega )=-{\frac {1}{\pi }}v.p.\int \limits _{-\infty }^{\infty }{\frac {\varepsilon ^{\prime }(x)-1}{x-\omega }}dx.

{\ displaystyle \ varepsilon ^ {\ prime \ prime} (\ omega) = - {\ frac {1} {\ pi}} vp \ int \ limits _ {- \ infty} ^ {\ infty} {\ frac {\ varepsilon ^ {\ prime} (x) -1} {x- \ omega}} dx.}

The real and imaginary parts of the dielectric constant determine the refractive index and absorption coefficient (optical constants) of a given medium. Thus, these indicators are not independent of one another and, therefore, a fundamental possibility arises from the spectrum of one of the optical constants to calculate the spectrum of the other, without resorting to direct measurements of the latter. This allows in some cases to reduce the amount of experimentally obtained information necessary to determine the optical constants, for example, in the region of intense absorption bands of condensed matter. The feasibility of the Kramers – Kronig relations has been repeatedly tested experimentally for various media in various states of aggregation and at different temperatures (crystals, liquids, solutions) ^[5] ^[6] .

Quantum Field Theory

In quantum field theory, when studying scattering processes, the transition probability amplitudes, considered as complex functions of the total energy of the system, transmitted momentum, etc., satisfy the dispersion relations ^[2] . This greatly facilitates the study of these phenomena.

History

The Kramers - Kronig relations were established in 1926-1927. Ralph Kronig ^[7] and Hendrick Kramers ^[8] and are named after them.

Notes

↑ John S. Toll, Causality and the Dispersion Relation: Logical Foundations , Physical Review, vol. 104 , pp. 1760-1770 (1956).
↑ ¹ ² Nishijima, 1965 , p. 153.
↑ Martin P. Sum rules Kramers - Kronig relations and transport coefficients in charged systems // Phys. Rev. . - 1967 .-- T. 161 . - S. 143 .
↑ Agranovich V.M., Ginzburg V.L. Crystal optics taking into account spatial dispersion and the theory of excitons. - M. , 1979.
↑ Alperovich L.I., Bakhshiev N.G., Zabiyakin Yu. E., Libov V.S. Kramers - Kronig relations for molecular spectra of liquids and solutions // Optics and Spectroscopy . - 1968 .-- T. 24 . - S. 60 - 63 .
↑ Zabiyakin Yu. E. Verification of the Kramers - Kronig dispersion relations in a wide temperature range // Optics and Spectroscopy . - 1968 .-- T. 24 . - S. 828 - 829 .
↑ R. de L. Kronig, On the theory of the dispersion of X-rays, J. Opt. Soc. Am., Vol. 12 , pp. 547-557 (1926).
↑ HA Kramers, La diffusion de la lumiere par les atomes, Atti Cong. Intern. Fisica, (Transactions of Volta Centenary Congress) Como, vol. 2 , p. 545-557 (1927).

Literature

Matthews J., Walker R. Mathematical Methods in Physics / Per. from English - M., Atomizdat , 1972.- 392 p.
Barton G. Dispersion methods in field theory / Per. from English - M., 1968.
Nishijima K. Fundamental particles. - M .: Mir, 1965 .-- 462 p.

[1] John S. Toll, Causality and the Dispersion Relation: Logical Foundations , Physical Review, vol. 104 , pp. 1760-1770 (1956).

[_f1f39ed22d343ae1-2] ¹ ² Nishijima, 1965 , p. 153.

[3] Martin P. Sum rules Kramers - Kronig relations and transport coefficients in charged systems // Phys. Rev. . - 1967 .-- T. 161 . - S. 143 .

[4] Agranovich V.M., Ginzburg V.L. Crystal optics taking into account spatial dispersion and the theory of excitons. - M. , 1979.

[5] Alperovich L.I., Bakhshiev N.G., Zabiyakin Yu. E., Libov V.S. Kramers - Kronig relations for molecular spectra of liquids and solutions // Optics and Spectroscopy . - 1968 .-- T. 24 . - S. 60 - 63 .

[6] Zabiyakin Yu. E. Verification of the Kramers - Kronig dispersion relations in a wide temperature range // Optics and Spectroscopy . - 1968 .-- T. 24 . - S. 828 - 829 .

[7] R. de L. Kronig, On the theory of the dispersion of X-rays, J. Opt. Soc. Am., Vol. 12 , pp. 547-557 (1926).

[8] HA Kramers, La diffusion de la lumiere par les atomes, Atti Cong. Intern. Fisica, (Transactions of Volta Centenary Congress) Como, vol. 2 , p. 545-557 (1927).