Dirac equation

The Dirac equation is a fundamental equation in relativistic quantum mechanics . It describes the properties and behavior of a fundamental fermion with spin 1/2 ( e.g. electron , quark ). It was developed by Paul Dirac in 1928 and, in contrast to the Schrödinger equation, fulfills the requirements of the special theory of relativity .

The Dirac equation is a partial differential equation of the first order both in the three spatial coordinates and in time, in accordance with the invariance under Lorentz transformations required by the special theory of relativity . In the non-relativistic borderline case ( ), it changes into the Pauli equation , which, in contrast to the Schrödinger equation, still contains the spin-orbit coupling and other terms. Every solution of the Dirac equation corresponds to a possible state of the particle in question, with the peculiarity that four spatial wave functions are required to represent this state (see Dirac spinor ), instead of two in the non-relativistic theory with spin or one in the case of spinless particles. For the particles described by the Dirac equation, the following applies: ${\displaystyle {\tfrac {v}{c}}\to 0}$

• For a free particle the relativistic energy-momentum relation is fulfilled. ${\displaystyle E={\sqrt {m^{2}c^{4}+p^{2}c^{2}}}}$

• For a particle in the electrostatic field of a point charge, the hydrogen spectrum results with its fine structure .

• The particle has its own angular momentum ( spin ), which has the quantum number 1/2 and - because this does not occur in classical physics - cannot go back to the rotation of a mass distribution as in a top.

• If the particle carries an electrical charge, a magnetic dipole moment is always associated with the spin (→ spin magnetism ). Compared to the magnetic dipole, which the particle would cause by a rotational movement with the same angular momentum (→ orbital magnetism ), the moment associated with the spin has twice the strength (see anomalous magnetic moment of the electron ).

• An antiparticle exists for the particle (a so-called positron to the electron ) with the same mass and the same spin, but with the opposite charge and magnetic moment.

All the properties mentioned correspond to the experimental results. At the time the Dirac equation was discovered in 1928, the first four were already known, but not their common basis. The latter property was predicted by the Dirac equation, and the first evidence of an antiparticle was made in 1932 by Carl David Anderson (see Positron ).

The differential operator occurring in the Dirac equation also plays a major role in mathematics (differential geometry) ( Dirac operator ).

Dirac equation of an uncharged particle

The Dirac equation is a system of four coupled partial differential equations for the four component functions of the Dirac spinor . The variable here stands for where the upper index 0 denotes the time and the indices 1 to 3 denote the location coordinates . ${\displaystyle \psi (x)}$ ${\displaystyle x}$ ${\displaystyle x=(x^{0},x^{1},x^{2},x^{3})\,,}$ ${\displaystyle t=x^{0}}$ ${\displaystyle (x^{1},x^{2},x^{3})}$

In natural units of measurement with is the Dirac equation for an uncharged particle of mass ${\displaystyle c=1=\hbar }$ ${\displaystyle m}$

${\displaystyle \left[\mathrm {i} \sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}-m\right]\,\psi (x)=0\,.}$

The expression in square brackets is the standard form of a Dirac operator .

The constant gamma or Dirac matrices and act in the space of the four components of the spinor and couple them to one another. The products of two gamma matrices have the following properties: ${\displaystyle \gamma ^{0},\gamma ^{1},\gamma ^{2}}$ ${\displaystyle \gamma ^{3}}$

${\displaystyle \gamma ^{\mu }\,\gamma ^{\nu }=-\gamma ^{\nu }\,\gamma ^{\mu }\quad (\,\mu ,\nu \in \{0,1,2,3\}\,,\mu \neq \nu ),}$

${\displaystyle \gamma ^{0}\gamma ^{0}=-\gamma ^{1}\gamma ^{1}=-\gamma ^{2}\gamma ^{2}=-\gamma ^{3}\gamma ^{3}=1}$

They thus form a Clifford or Dirac algebra. Becomes the Dirac operator

${\displaystyle \mathrm {i} \sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}-m}$

Applied to both sides of the Dirac equation, the four differential equations decouple and one obtains for each component of the Klein-Gordon equation : ${\displaystyle \psi }$

${\displaystyle {\Bigl [}{\frac {\partial ^{2}}{(\partial {x^{0}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{1}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{2}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{3}})^{2}}}+m^{2}{\Bigr ]}\,\psi (x)=0}$

Applying a Dirac operator twice leads to the Klein-Gordon equation, which is why the Dirac equation is also regarded as the “root” of the Klein-Gordon equation. For a particle in a momentum eigenstate, the Klein-Gordon equation gives (in the order of its terms) , i.e. the relativistic energy-momentum relationship of a particle of mass ${\displaystyle -E^{2}+{\vec {p}}^{\,2}+m^{2}=0}$ ${\displaystyle m\,.}$

Every irreducible representation of Dirac algebra consists of matrices. In the standard or Dirac representation they have the following form (vanishing matrix elements with the value zero are not written): ${\displaystyle 4\times 4}$

${\displaystyle {\begin{array}{c c}\gamma ^{0}={\begin{pmatrix}1&&&\\&1&&\\&&-1&\\&&&-1\end{pmatrix}}\,,&\gamma ^{1}={\begin{pmatrix}&&&1\\&&1&\\&-1&&\\-1&&&\end{pmatrix}}\,,\\\,&\,\\\gamma ^{2}={\begin{pmatrix}&&&-\mathrm {i} \\&&\mathrm {i} &\\&\mathrm {i} &&\\-\mathrm {i} &&&\end{pmatrix}}\,,&\gamma ^{3}={\begin{pmatrix}&&1&\\&&&-1\\-1&&&\\&1&&\end{pmatrix}}\,.\end{array}}}$

The first two components of form the two-component identity matrix, the last two components their negatives. Similarly, the top two components of the second, third and fourth matrix result in the three 2 × 2 Pauli matrices and the last two components of their negatives. In the non-relativistic limit case, the latter tend to approach zero. This representation, the standard representation, is therefore particularly suitable for treating slowly moving electrons. In the Weyl representation, which is mathematically and physically equivalent to this, the spinor transformation behavior in Lorentz transformations is particularly simple; in the Majorana representation, which is also equivalent, the Dirac equation is a real system of equations. Further representations are obtained through equivalence transformations. ${\displaystyle \gamma _{0}}$ ${\displaystyle \gamma }$ ${\displaystyle \sigma _{j}}$ ${\displaystyle \gamma _{j}}$ ${\displaystyle v/c}$

The four gamma matrices can be written symbolically to the contravariant 4-vector

${\displaystyle \gamma ^{\mu }={\begin{pmatrix}\gamma ^{0}\\\gamma ^{1}\\\gamma ^{2}\\\gamma ^{3}\end{pmatrix}}}$

sum up. Then the first term of the Dirac equation has the form of a scalar product of the vectors and . However, this is not invariant in the Lorentz transformation because it remains constant. The Lorentz invariance of the Dirac theory only results from the fact that the Dirac operator acts on a spinor whose four components are also suitably transformed. In the end result, a solution of the Dirac equation by Lorentz transformation turns into a solution of the correspondingly transformed Dirac equation. ${\displaystyle \gamma ^{\mu }}$ ${\displaystyle {\frac {\partial }{\partial x^{\mu }}}}$ ${\displaystyle \gamma ^{\mu }}$ ${\displaystyle \psi }$ ${\displaystyle \psi (x)}$

Momentum space and slash notation

In addition to the form just described in spatial space, the Dirac equation can also be written in momentum space. It then reads

${\displaystyle {\Bigl [}\gamma ^{\mu }p_{\mu }-m{\Bigr ]}\,\psi (p)=0\,,}$

Einstein's summation convention was used for abbreviation (which means that the same indices are used for summing). In the still further simplified Feynman Slash notation , the scalar product with the gamma matrices is expressed by a slash symbol. It arises in local space

${\displaystyle {\Bigl [}i\partial \!\!\!/\ -m{\Bigr ]}\,\psi (x)=0\,,}$

and holds in momentum space

${\displaystyle {\Bigl [}p\!\!\!/\ -m{\Bigr ]}\,\psi (p)=0\,.}$

Gauge invariance and electromagnetic interaction

If the Dirac equation solves, then the spinor multiplied by a phase also solves the Dirac equation. Since all physically measurable quantities with each factor also contain the conjugate complex factor , they and the Dirac equation are invariant under this phase transformation of the Dirac spinor . ${\displaystyle \psi (x)}$ ${\displaystyle \alpha }$ ${\displaystyle \mathrm {e} ^{\mathrm {i} q\alpha }\psi }$ ${\displaystyle \psi }$ ${\displaystyle \psi ^{*}}$ ${\displaystyle \psi }$

In the case of non-constant, this results in an additional U (1) gauge invariance, and the partial derivatives must be replaced by so-called covariant derivatives: From the requirement of invariance under all phase transformations, which depend continuously and differentially on time and place, ${\displaystyle \alpha }$

${\displaystyle \psi ^{\prime }(x)=\mathrm {e} ^{\mathrm {i} q\alpha (x)}\psi (x)\,,}$ the need arises that

replace partial derivatives in the Dirac equation with the covariant derivative:

${\displaystyle D_{\mu }={\frac {\partial }{\partial x^{\mu }}}+\mathrm {i} qA_{\mu }}$

The four functions occurring here form the so-called four-potential or calibration field in physics . Mathematically, it is a connection or a relationship . Define the transformed calibration field through ${\displaystyle A_{\mu }}$

${\displaystyle A_{\mu }^{\prime }=A_{\mu }-{\frac {\partial \alpha (x)}{\partial x^{\mu }}}\,,}$

then solves the Dirac equation with the calibration field ${\displaystyle \psi }$ ${\displaystyle A_{\mu }}$

${\displaystyle \left(\mathrm {i} \sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}-q\sum _{\mu =0}^{3}\gamma ^{\mu }A_{\mu }-m\right)\psi (x)=0}$

or in slash notation

${\displaystyle \left(\mathrm {i} \partial \!\!\!/\,-qA\!\!\!/\,-m\right)\psi (x)=0}$

if and only if the transformed Dirac spinor satisfies the Dirac equation with the transformed gauge field. Transformations, the parameters of which, like the phase here,  may depend on time and place, are called local calibration transformations in physics . ${\displaystyle \alpha (x)}$

The calibration field is the scalar potential and the vector potential of electrodynamics, ${\displaystyle \phi }$ ${\displaystyle {\vec {A}}}$

${\displaystyle (A_{0},A_{1},A_{2},A_{3})=(\phi ,-A_{x},-A_{y},-A_{z})\,.}$

If you transform them as indicated, the electric and magnetic field strengths remain

${\displaystyle {\vec {E}}=-{\vec {\nabla }}\phi -{\frac {\partial {\vec {A}}}{\partial t}}\quad {\mbox{sowie}}\quad {\vec {B}}={\vec {\nabla }}\times {\vec {A}}}$

and all other measurable quantities unchanged.

The Dirac equation with covariant derivative and the electrodynamics are invariant under any time- and position-dependent transformations of the phase of the Dirac spinor. The parameter in the covariant derivative determines the strength of the coupling of the electromagnetic potentials to the Dirac spinor. It corresponds exactly to the electrical charge of the particle. ${\displaystyle q}$

The replacement of the partial derivatives in the Dirac equation by a covariant derivative couples the electromagnetic potentials to the Dirac spinor. One speaks of so-called minimal coupling in contrast to a coupling term such as “magnetic field strength times Dirac spinor”, ​​which would also be gauge invariant, but is not required to supplement a derivative with a covariant derivative.

Schrödinger form

After multiplication with , one can solve for the time derivative in the Dirac equation and bring the Dirac equation into the form of a Schrödinger equation , ${\displaystyle \gamma ^{0}}$ ${\displaystyle (\gamma ^{0})^{2}=1}$

${\displaystyle \mathrm {i} {\frac {\partial }{\partial t}}\,\psi =H_{\text{Dirac}}\psi }$

${\displaystyle H_{\text{Dirac}}=\sum _{k=1}^{3}\alpha ^{k}(-\mathrm {i} {\frac {\partial }{\partial x^{k}}}-qA_{k})+\gamma ^{0}m+q\phi .}$

The 4 × 4 matrices occurring here, which are slightly different from the corresponding matrices, can also be described in compact form with the help of the Pauli matrices using blocks of 2 × 2 matrices : ${\displaystyle \gamma }$ ${\displaystyle \alpha ^{k}=\gamma ^{0}\,\gamma ^{k},}$ ${\displaystyle \sigma _{2\times 2}^{k}}$

${\displaystyle \alpha ^{k}={\begin{pmatrix}&\sigma _{2\times 2}^{k}\\\sigma _{2\times 2}^{k}&\end{pmatrix}}\,,\,k\in \{1,2,3\}\,,\quad \gamma ^{0}={\begin{pmatrix}1_{2\times 2}&\\&-1_{2\times 2}\end{pmatrix}}\,.}$

The differential operator on the right-hand side of the Schrödinger equation is the Hamilton operator belonging to the Dirac equation. The possible energies of the particle are eigenvalues ​​of this Hamilton operator . ${\displaystyle H_{\text{Dirac}}\,.}$

The mathematical analysis shows, in the case of an uncharged particle ( ) that the spectrum of positive and negative values contains, as well as to the Klein-Gordon equation from the energy-momentum relation (in natural units with the positive and negative energy values) obtained . ${\displaystyle q=0}$ ${\displaystyle E^{2}-{\vec {p}}^{\,2}=m^{2}}$ ${\displaystyle c=1=\hbar }$ ${\displaystyle E=\pm {\sqrt {m^{2}+{\vec {p}}^{\,2}}}}$

Since particles with negative energy have never been observed and since a world with particles whose energies are unlimited upwards and downwards would be unstable, Dirac postulated that the vacuum is a Dirac sea in which every conceivable state of negative energy is already occupied so that further electrons could only take on positive energies. If one adds enough energy to this Dirac sea, at least the rest energy of two electrons, one can give one sea electron positive energy and the resulting hole would behave like a state with the remaining, also positive energy and the missing, opposite charge. Dirac predicted the existence of antiparticles and the pair creation of electron-positron pairs, which were observed a year later.

The idea of ​​a Dirac lake is now considered untenable and has been replaced by the Feynman-Stückelberg interpretation . She interprets the Dirac equation as an equation for a quantum field , which is mathematically an operator that creates or destroys particles or antiparticles in the quantum mechanical states. The creation and annihilation of particles during the interaction of the electron with the proton leads in quantum electrodynamics to a small shift in the energies of different states of the hydrogen atom, which would have the same energy without these creation and annihilation processes. The calculated size of this Lamb shift agrees with the measured value within the measuring accuracy of six places. ${\displaystyle \psi (x)}$

The creation and annihilation of particles during the interaction of the electron with a magnetic field also changes the Dirac value of the gyromagnetic factor . It causes a so-called anomalous magnetic moment, which is also referred to as a g- 2 anomaly. The value of calculated in quantum electrodynamics agrees with the measured value to ten decimal places. ${\displaystyle g=2}$ ${\displaystyle g}$

Derivation of the gyromagnetic factor

Based on the Schrödinger form of the Dirac equation for a particle in the electromagnetic field, the Dirac spinor is split into two two spinors.

${\displaystyle \mathrm {i} \partial _{t}{\begin{pmatrix}\varphi _{1}\\\varphi _{2}\end{pmatrix}}={\begin{pmatrix}\sigma ^{k}\pi _{k}\varphi _{2}\\\sigma ^{k}\pi _{k}\varphi _{1}\end{pmatrix}}+q\phi {\begin{pmatrix}\varphi _{1}\\\varphi _{2}\end{pmatrix}}+m{\begin{pmatrix}\varphi _{1}\\-\varphi _{2}\end{pmatrix}}}$ With  ${\displaystyle \pi _{k}=-\mathrm {i} \partial _{k}-qA_{k}}$

Assuming that the particle moves only slowly, so that its energy is only slightly greater than its rest energy, the rapid time evolution that comes from the rest energy can be split off:

${\displaystyle {\begin{pmatrix}\varphi _{1}\\\varphi _{2}\end{pmatrix}}=\mathrm {e} ^{-\mathrm {i} mt}{\begin{pmatrix}\varphi \\\chi \end{pmatrix}}}$

From this approach it follows:

${\displaystyle \mathrm {i} \partial _{t}{\begin{pmatrix}\varphi \\\chi \end{pmatrix}}={\begin{pmatrix}\sigma ^{k}\pi _{k}\chi \\\sigma ^{k}\pi _{k}\varphi \end{pmatrix}}+q\phi {\begin{pmatrix}\varphi \\\chi \end{pmatrix}}+{\begin{pmatrix}0\\-2m\chi \end{pmatrix}}\,.}$

In the second line, according to the assumption, both the time derivative and the kinetic energies and the electrostatic energy are small compared to the rest energy . Hence, small is against and roughly equal ${\displaystyle m}$ ${\displaystyle \chi }$ ${\displaystyle \varphi }$

${\displaystyle \chi \approx {\frac {\sigma ^{k}\pi _{k}}{2m}}\varphi }$.

Inserted in the first line you get:

${\displaystyle \mathrm {i} \partial _{t}\varphi ={\frac {(\sigma ^{k}\pi _{k})^{2}}{2m}}\varphi +q\phi \varphi }$

For the product of the Pauli matrices one obtains

${\displaystyle (\sigma ^{k}\pi _{k})^{2}=(\delta ^{ij}+\mathrm {i} \varepsilon ^{ijk}\sigma _{k})\pi _{i}\pi _{j}={\vec {\pi }}^{2}-q\,\sigma ^{k}B_{k}.}$

The spinor therefore satisfies the Pauli equation with the non-classical value ${\displaystyle \varphi }$ ${\displaystyle g=2,}$

${\displaystyle \mathrm {i} \partial _{t}\varphi ={\frac {{\vec {\pi }}^{2}}{2m}}\varphi +q\phi \varphi -g{\frac {q}{2m}}S^{k}B_{k}\varphi .}$

Here are the components of the spin operator. ${\displaystyle S^{k}={\tfrac {\sigma ^{k}}{2}}}$

In a homogeneous magnetic field, and with ${\displaystyle \phi =0,{\vec {A}}={\tfrac {1}{2}}{\vec {B}}\times {\vec {x}}}$ ${\displaystyle p_{k}=-\mathrm {i} \partial _{k}}$

${\displaystyle ({\vec {p}}-q{\vec {A}})^{2}={\vec {p}}^{2}-q{\vec {L}}\cdot {\vec {B}},}$

if one neglects terms that are quadratic in . Then the Pauli equation says ${\displaystyle {\vec {B}}}$

${\displaystyle \mathrm {i} \partial _{t}\varphi ={\frac {{\vec {p}}^{2}}{2m}}\varphi -{\frac {q}{2m}}({\vec {L}}+g{\vec {S}})\cdot {\vec {B}}\varphi .}$

The magnetic field therefore not only couples to the orbital angular momentum  and not only contributes to the energy. The factor is the particle's magneton . In angular momentum eigenstates there is an integral multiple of the magnetic field strength . In contrast, a half- integer multiple results , which only becomes an integer after multiplication with . ${\displaystyle {\vec {L}}}$ ${\displaystyle -{\tfrac {q}{2m}}{\vec {L}}\cdot {\vec {B}}}$ ${\displaystyle {\tfrac {q}{2m}}}$ ${\displaystyle {\vec {L}}\cdot {\vec {B}}}$ ${\displaystyle |{\vec {B}}|}$ ${\displaystyle {\vec {S}}\cdot {\vec {B}}}$ ${\displaystyle g}$

Realizations in high energy and solid state physics

The Dirac equation forms (after quantizing the associated classical field) the basis of the relativistic quantum field theories of high energy physics . It has only been known for a few years that realizations also exist with non-relativistic energies, namely with graphenes , that is, layer systems that are related to graphite. In fact, one only needs to consider the limit value of vanishing mass (so-called chiral limes ) , and the speed of light must also be replaced by the limit speed of the electron system, the so-called Fermi speed . As a consequence, energy and momentum are proportional to each other in this system ( ), while otherwise applies to non-relativistic electrons . In addition, there are numerous other special features. ${\displaystyle m=0}$ ${\displaystyle c}$ ${\displaystyle v_{F}}$ ${\displaystyle E}$ ${\displaystyle p}$ ${\displaystyle E\propto p}$ ${\displaystyle E\propto p^{2}}$

See also

• Dirac matrices
• Landé factor
• Pauli equation
• Trembling motion

literature

items

• PAM Dirac: The Quantum Theory of the Electron . In: Proceedings of the Royal Society of London. Series A . tape 117 , no. 778 , January 1, 1928, p. 610-624 , doi : 10.1098 / rspa.1928.0023 .
• PAM Dirac: The Quantum Theory of the Electron. Part II . In: Royal Society of London Series A Proceedings . tape 118 , February 1, 1928, p. 351-361 .
• PAM Dirac: A Theory of Electrons and Protons . In: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . tape 126 , no. 801 , 1930, p. 360-365 , JSTOR : 95359 .
• Carl D. Anderson : The Positive Electron . In: Physical Review . tape 43 , no. 6 , February 15, 1933, p. 491-494 , doi : 10.1103 / PhysRev.43.491 .

Books

• James Bjorken , Sidney Drell : Relativistic Quantum Mechanics. Mannheim, Bibliographisches Institut, 1990. (BI university pocket books; 98 / 98a), ISBN 3-411-00098-8 .
  English Original Edition: Relativistic Quantum Mechanics. McGraw-Hill, New York 1964, ISBN 0-07-005493-2 .
• James Bjorken , Sidney Drell : Relativistic Quantum Field Theory. (German transl .: J. Benecke, D. Maison, E. Riedel).
  Unchangeable Reprint: Mannheim, Zurich. BI-Wissenschaftsverlag, 1993. BI-University paperback; 101, ISBN 3-411-00101-1 .
  English Original Edition: Relativistic Quantum Fields. McGraw-Hill, New York 1965, ISBN 0-07-005494-0 .
• RP Feynman : Quantum Electrodynamics. 4th edition, ISBN 3-486-24337-3 .
• Walter Greiner : Relativistic Quantum Mechanics. Wave equations. Volume 6, ISBN 3-8171-1022-7 .
• Franz Schwabl : Quantum Mechanics for Advanced Students (QM II). ISBN 978-3-540-25904-6 .

References and footnotes

1. ^ PAM Dirac: The Quantum Theory of the Electron . In: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character . A, no. 778 , 1928, pp. 610-624 , doi : 10.1098 / rspa.1928.0023 ( online ).
2. ↑ CD Anderson: The Positive Electron . In: Physical Review . tape 43 , no. 6 , 1933, pp. 491–494 , doi : 10.1103 / PhysRev.43.491 ( online ).
3. ^ J. Schwinger : A Report on Quantum Electrodynamics. In: The Physicist's Conception of Nature. Reidel, Dordrecht 1973, p. 415.
4. ↑ In the case of isolated atoms or ions , the total orbital angular momentum and the total spin angular momentum of the atom or ion must be added to a total angular momentum  J (=  L + S ) and the so-called Landé factor  g (L, S; J) is obtained. This is 1 for pure total orbital angular momentum and 2 for pure total spindle angular momentum, and otherwise has values ​​different from 1 and 2. Furthermore, if the atoms concerned are built into a solid, additional contributions are obtained that can change significantly. The ferro- and paramagnetism of typical representatives of ferromagnetic or paramagnetic solid bodies or paramagnetic molecules is mostly spin magnetism, because experimental measurements are very often . ${\displaystyle g}$ ${\displaystyle g\sim 2}$
5. ↑ One often speaks of a second quantization .
6. ↑ ^{a b} See the article Graph .