Causality and the Kramers-Kronig relations

The fact that the response functions /(Z) in Eqs (11.19) and (11.49) vanish for Z 0 is physically significant It implies that an effect cannot precede its cause, that is, the effect at time Z cannot depend on the value of the cause at a later time. This obvious physical property is also reflected in an interesting and useful mathematical property of such response functions. To see this we express the response function XBA(.t) = 9(Z)([,55/(Z), J ])o in tenns of the eigenstates (and corresponding energies) of Ho 

The appearance of ir/ in the denominators here defines the analytical properties of this function The fact that / (cy) is analytic on the upper half of the complex m plane and has simple poles (associated with the spectrum of Hq ) on the lower half is equivalent to the casual nature of its Fourier transform—the fact that it vanishes for Z 0. An interesting mathematical property follows. For any function / (cy) that is (1) analytic in the half plane Recy 0 and (2) vanishes fast enough for cy - oo we can write (see Section 1.1.6) 

The transformation defined by (11.56) is called Hilbert transform, and we have found that the real and imaginary parts of a function that is analytic in half of the complex plane and vanishes at infinity on that plane are Hilbert transforms of each other. Thus, causality, by which response functions have such analytical properties, also implies this relation. On the practical level this tells us that if we know the real (or imaginary) part of a response function we can find its imaginary (or real) part by using this transform. 

Problem 11.4. Note that if xi (u ) is symmetric under sign inversion of w, that is, Xilco ) = /I(— ), then /2( w) is antisymmetric, /2( w) = —/2(— ) Show that in this case Eqs (11.56) can be rewritten in the form 

In this form the equations are known as the Kramers-Kronig relations. 

Now consider the Fourier transform of this function, x (ft ) = dte x (0-We use the identity  

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