Dirac identity

The Dirac identity (named after Paul Dirac ) is

${\displaystyle \lim _{\eta \to 0^{+}}{\frac {1}{x\pm \mathrm {i} \eta }}={\mathcal {P}}{\bigg (}{\frac {1}{x}}{\bigg )}\mp \mathrm {i} \pi \delta (x)}$.

Therein denotes the Cauchy principal value and the Dirac delta distribution . It is to be understood as an integral operator identity, i. H. although it is noted as above, strictly speaking only applies ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \delta (x)}$

${\displaystyle {\begin{aligned}\lim _{\eta \to 0^{+}}\int \limits _{-\infty }^{\infty }{\frac {\phi (x)}{x-x_{0}\pm \mathrm {i} \eta }}\mathrm {d} x&={\mathcal {P}}\int \limits _{-\infty }^{\infty }{\frac {\phi (x)}{x-x_{0}}}\mathrm {d} x\mp \mathrm {i} \pi \int \limits _{-\infty }^{\infty }\delta (x-x_{0})\phi (x)\mathrm {d} x\\&={\mathcal {P}}\int \limits _{-\infty }^{\infty }{\frac {\phi (x)}{x-x_{0}}}\mathrm {d} x\mp \mathrm {i} \pi \phi (x_{0})\end{aligned}}}$

for a suitable test function . It can be proven within the framework of distribution theory. It is a special case of the Sokhotski – Plemelj theorem and finds z. B. in physics application. ${\displaystyle \phi (x)}$

More generally it can even be shown that it is true

${\displaystyle \lim _{\eta \to 0^{+}}{\frac {1}{(x\pm \mathrm {i} \eta )^{n}}}={\mathcal {P}}{\bigg (}{\frac {1}{x^{n}}}{\bigg )}\mp {\frac {\mathrm {i} \pi (-1)^{n-1}}{(n-1)!}}\delta ^{(n-1)}(x),}$

where denotes the -th derivative of the Dirac delta distribution. ${\displaystyle \delta ^{(n-1)}(x)}$ ${\displaystyle n}$

Derivation

In distribution theory one introduces distributions over functionals, e.g. B. for the Dirac delta distribution

${\displaystyle F_{\delta }[\rho ]:=\int \limits _{-\infty }^{\infty }\delta (x-x_{0})\rho (x)\mathrm {d} x=\rho (x_{0}),}$

d. H. the distribution is only defined in the integral with a suitable test function, which should only differ from zero in a finite range and which can be differentiated as often as desired. Part of the definition is that this creates value . A functional is identified with each singular function , which identifies a number with each matching test function . ${\displaystyle \rho (x_{0})}$ ${\displaystyle f}$ ${\displaystyle F_{f}[\rho ]}$ ${\displaystyle \rho }$

With partial integration and the fact that the test function is not equal to zero only on a finite range (" bounded support " ), one obtains ${\displaystyle \rho (\pm \infty )=0}$

${\displaystyle \int \limits _{-\infty }^{\infty }f'(x)\rho (x)\mathrm {d} x=-\int \limits _{-\infty }^{\infty }f(x)\rho '(x)\mathrm {d} x}$

and thus, for example, also for the Heaviside step function , as well (this identity is often used in solid-state physics, since the Fermi function as a function of energy at zero temperature is a step function). Note that because of the definition that is only valid in the integral, the behavior of the Heaviside distribution does not necessarily have to be specified. ${\displaystyle \theta '(x)=\delta (x)}$ ${\displaystyle \theta '(-x)=-\delta (x)}$ ${\displaystyle x=0}$

So far we've only given a brief introduction to distributions. For the Dirac identity one considers the functional , whereby the borderline case is meant here again . ${\displaystyle f(x)=\ln(x+\mathrm {i} 0^{+})}$ ${\displaystyle \lim _{\eta \to 0^{+}}\ln(x+\mathrm {i} \eta )}$

On the one hand is the derivation . ${\displaystyle f'(x)={\frac {1}{x+\mathrm {i} 0^{+}}}}$

With the branching cut of the natural logarithm along the negative real axis is for and for . Therefore, on the other hand, it follows for the derivation: ${\displaystyle f(x)=\ln |x|+\mathrm {i} \pi }$ ${\displaystyle x<0}$ ${\displaystyle f(x)=\ln |x|}$ ${\displaystyle x>0}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}\ln |x|+\mathrm {i} \pi \theta (-x){\bigg )}={\frac {\mathrm {d} }{\mathrm {d} x}}\ln |x|-\mathrm {i} \pi \delta (x)}$.

The derivation of the logarithm must again be considered as an integral, whereby the bounded support is used as above (here in the first line and when stepping from the third to the fourth line):

${\displaystyle {\begin{aligned}\int \limits _{-\infty }^{\infty }{\frac {\mathrm {d} \ln |x|}{\mathrm {d} x}}\rho (x)\mathrm {d} x&=-\int \limits _{-\infty }^{\infty }\ln |x|\rho '(x)\mathrm {d} x\\&=-\lim _{\varepsilon \to 0^{+}}\int \limits _{|x|>\varepsilon }\ln |x|\rho '(x)\mathrm {d} x\\&=-\lim _{\varepsilon \to 0^{+}}{\bigg [}\ln |x|\rho (x){\bigg \vert }_{-\infty }^{-\varepsilon }+\ln |x|\rho (x){\bigg \vert }_{\varepsilon }^{\infty }-\int \limits _{|x|>\varepsilon }{\frac {\rho (x)}{x}}\mathrm {d} x{\bigg ]}\\&={\mathcal {P}}\int \limits _{-\infty }^{\infty }{\frac {\rho (x)}{x}}\mathrm {d} x.\end{aligned}}}$

The last step took advantage of the fact that the test function is "good-natured", i. This means that the front terms in the third line (boundary terms of the partial integration) vanish and that the integral over the whole range of numbers except for the area around the pole of the integrand is the main value integral. ${\displaystyle ]-\varepsilon ,\varepsilon [}$

This applies in the sense of distribution theory and the Dirac identity follows from a comparison of the two calculations of the derivative. The case with the other sign is treated analogously. ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln |x|={\mathcal {P}}{\bigg (}{\frac {1}{x}}{\bigg )}}$

Applications

With the Dirac identity, for example, the Kramers-Kronig relations for response functions can be elegantly proven, since these are analytical in the upper complex half-plane . Instead of closing a semicircle in the upper complex half-plane and excluding the area around the pole on the real axis (as found in the book by Charles Kittel , for example ), one now closes a semicircle in the upper complex half-plane and shifts the path along the real axis Axis up or down and applies the Dirac identity. In addition, both approaches use the fact that a closed curve integral in the complex plane is only determined by the poles inside the integration path ( residual theorem ). ${\displaystyle \chi (\omega )}$ ${\displaystyle \omega }$ ${\displaystyle \mathrm {i} 0^{+}}$

Another application is the calculation of the real and imaginary part of the dielectric function in the theory of the shielding of electrical charges according to Lindhard, because precisely such a structure is found in the expression for in this theory, which is based on 1st order perturbation theory appears in the denominator as it presupposes the Dirac identity. The separation into real and imaginary parts is here u. a. This is important because the imaginary part of the dielectric function is usually associated with the damping of the propagation of waves in the medium described. ${\displaystyle \epsilon ({\vec {q}},\omega )}$ ${\displaystyle \epsilon }$

literature

• Laurent Schwartz, Théorie des Distribution
• Gel'fand, Shilov, Generalized Functions , Vol. 1-5

Web links

Discussions in forums:

• http://math.stackexchange.com/questions/245211/principal-value-of-1-x-equivalence-of-two-definitions
• http://physics.stackexchange.com/questions/105729/principal-value-of-1-x-and-few-questions-about-complex-analysis-in-peskins-qft

An exercise sheet from the HU Berlin: http://people.physik.hu-berlin.de/~thklose/FQM-WS1314/QMU-13.pdf

Individual evidence

1. ^ Branch Cut Wolfram Research, accessed September 19, 2018.
2. ^ Kittel, Charles, Jochen Matthias Gress, and Anne Lessard. Introduction to solid state physics . Vol. 14. Munich: Oldenbourg, 1969.
3. ^ Giuliani, Gabriele, and Giovanni Vignale. Quantum theory of the electron liquid . Cambridge university press, 2005.