Proca equation

The Proca equation is a fundamental equation in relativistic quantum mechanics . It describes the properties and behavior of a fundamental boson with spin  1 and mass, such as the W boson and the Z boson . It was discovered by the Romanian physicist Alexandru Proca . It belongs to the standard model of elementary particle physics .

In the following the (+ −−−) signature of the metric tensor is used.

Lagrangian density

The field to be described is generally a complex wave function :

${\displaystyle B^{\mu }=\left({\frac {\phi }{c}},\mathbf {B} \right)}$

With

• a generalized electrical potential ${\displaystyle \phi }$
• the speed of light ${\displaystyle c}$
• a generalized magnetic potential . ${\displaystyle \mathbf {B} }$

With a Lorentz transformation of the coordinates, the four wave functions transform like a four-vector . If a field is to be described with a charge, the four field functions are generally four complex-valued functions. If a field is to be described without an electric charge, the four field functions are four real functions. ${\displaystyle B^{\mu }}$ ${\displaystyle B^{\mu }}$ ${\displaystyle B^{\mu }}$

As with the theory of electromagnetism , it is also useful for the Proca equation to introduce a field strength tensor according to:

${\displaystyle G^{\mu \nu }:=\partial ^{\mu }B^{\nu }-\partial ^{\nu }B^{\mu }}$

The Lagrange density of the field is thus :

${\displaystyle {\begin{alignedat}{2}{\mathcal {L}}&=-{\frac {1}{2}}G_{\mu \nu }^{*}G^{\mu \nu }&&+{\frac {m^{2}c^{2}}{\hbar ^{2}}}B_{\nu }^{*}B^{\nu }\\&=-{\frac {1}{2}}(\partial _{\mu }B_{\nu }^{*}-\partial _{\nu }B_{\mu }^{*})(\partial ^{\mu }B^{\nu }-\partial ^{\nu }B^{\mu })&&+{\frac {m^{2}c^{2}}{\hbar ^{2}}}B_{\nu }^{*}B^{\nu }.\end{alignedat}}}$

With

• the reduced Planckian quantum of action ${\displaystyle \hbar }$
• the four gradient ${\displaystyle \partial _{\mu }}$
• the complex conjugation . ${\displaystyle {}^{*}}$

The Lagrange density of the Proca equation (also Proca effect) can be understood as a special case of the Stückelberg effect with the help of the Higgs mechanism and a special choice of calibration .

Representations

With the help of the Euler-Lagrange equations , the actual Proca equation can be derived from the above Lagrange density:

${\displaystyle {\begin{alignedat}{2}\partial _{\mu }(\partial ^{\mu }B^{\nu }-\partial ^{\nu }B^{\mu })+\left({\frac {mc}{\hbar }}\right)^{2}B^{\nu }&=0\\\Leftrightarrow \partial _{\mu }G^{\mu \nu }+\left({\frac {mc}{\hbar }}\right)^{2}B^{\nu }&=0.\end{alignedat}}}$

In this case , this equation for the uncharged field is reduced to the inhomogeneous Maxwell equations without charge current. ${\displaystyle m=0}$

In this case , the Proca equation with the names from the vector analysis is also written as: ${\displaystyle m\neq 0}$

${\displaystyle \Box \phi +\left({\frac {mc}{\hbar }}\right)^{2}\phi =0}$

${\displaystyle \Box \mathbf {B} +\left({\frac {mc}{\hbar }}\right)^{2}\mathbf {B} =0}$

and

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {B} =0}$

With

• the D'Alembert operator ${\displaystyle \Box }$
• the nabla operator . ${\displaystyle \nabla }$

The Proca equation is closely related to the Klein-Gordon equation . It is also a second order equation in Minkowski spacetime .

In a slightly different and less common form, the Proca equation was introduced by Nicholas Kemmer in 1939 , which is why the terms Kemmer equation and Proca-Kremmer equation appear in the literature. They are formally similar to the Dirac equation , but use a ten-dimensional spinor and corresponding 10 × 10 matrices.

Coupling to the electromagnetic field

Since the electrical charge of the negatively charged W boson is equal to the elementary charge, as is the case with the electron , the minimum coupling can be applied to the field equations in order to derive the field equations for the . As with the Klein-Gordon equation, the minimal coupling is obtained by using the covariant derivative . The proca field determined in this way is transformed in accordance with a calibration transformation of the electromagnetic field ${\displaystyle W^{-}}$ ${\displaystyle D_{\mu }=\partial _{\mu }+{\tfrac {ie}{\hbar c}}A_{\mu }}$

${\displaystyle B^{\mu }\rightarrow B^{\mu }e^{{\frac {ie}{\hbar c}}f(x^{\mu })}}$

It is

• ${\displaystyle f}$ a freely selectable function
• ${\displaystyle e}$ equal to the elementary charge

The complex conjugation of the field equations changes the sign of the charge in the covariant derivative. The complex conjugation of the field equation gives the equations for the associated antiparticle, i.e. the . This also shows that the complex conjugation of the proca field corresponds to the charge conjugation . ${\displaystyle W^{+}}$ ${\displaystyle B^{\mu }\rightarrow B^{*\mu }}$

Interaction with fermionic fields

Analogous to electromagnetism, the above field equations can be extended by field-generating, fermionic currents or axial vector currents . These currents can also result from the wave functions of two different fermionic particles, e.g. B. assemble electron and neutrino, because to generate a spin-1 particle due to the conservation of angular momentum, at least two spin-1/2 particles are required. This should be indicated by the two indices a and b. Examples of fermionic currents can be found in VA theory and weak interaction theory . ${\displaystyle J^{\mu }={\bar {\psi _{a}}}\gamma ^{\mu }\psi _{b}}$ ${\displaystyle J_{A}^{\mu }={\bar {\psi _{a}}}\gamma ^{\mu }\gamma ^{5}\psi _{b}}$

Since two spin 1/2 particles can also couple to a spin 0 state due to the conservation of angular momentum, there is an indication of another possible field, which is then called the Higgs field in the theory of electroweak interaction .

In addition, it must be taken into account whether the spin-1 particle to be described has a charge or not, because a charged particle is influenced by the electromagnetic field in contrast to an uncharged particle. This is accounted for by using the covariant derivative given above . ${\displaystyle D_{\mu }}$

If it is to be investigated how a charged proca field is generated by charged fermionic currents, the following equation can be used

${\displaystyle D_{\mu }G^{\mu \nu }+\left({\tfrac {m_{W}c}{\hbar }}\right)^{2}B^{\nu }=g_{W}J_{W}^{\nu }}$ ,

where the covariant derivative is to be used in the field strength tensor instead of the partial derivative. The constant determines the strength of the coupling between the charged current and the field of a massive W boson. ${\displaystyle g_{W}}$

The Lagrange density of this field equation is

${\displaystyle {\mathcal {L}}_{W}=-{\tfrac {1}{2}}G_{\mu \nu }^{*}G^{\mu \nu }+\left({\frac {m_{W}c}{\hbar }}\right)^{2}B_{\nu }^{*}B^{\nu }-g_{W}B_{\nu }J_{W}^{\nu *}-g_{W}B_{\nu }^{*}J_{W}^{\nu }}$

where the covariant derivative must also be used in the field strength tensor instead of the partial derivative.

Although the uncharged Z boson differs in mass and electrical charge from the W boson, a similar equation can also be set up for the Z boson:

${\displaystyle \partial _{\mu }G^{\mu \nu }+\left({\tfrac {m_{Z}c}{\hbar }}\right)^{2}B^{\nu }=g_{Z}J_{Z}^{\nu }}$.

The coupling to an external electromagnetic field is omitted in this case due to the missing electrical charge of this particle, so that the above covariant derivation does not have to be used here. The Lagrange density for describing the field of the Z boson has significantly fewer terms because of the lack of electrical charge:

${\displaystyle {\mathcal {L}}_{Z}=-{\frac {1}{2}}G_{\mu \nu }G^{\mu \nu }+\left({\frac {m_{Z}c}{\hbar }}\right)^{2}B_{\nu }B^{\nu }-2g_{Z}B_{\nu }J_{Z}^{\nu }}$

In contrast to the field of the electrically charged W boson, the field of the electrically neutral Z boson is described by a real four-vector vector .

It should also be mentioned that the spin-1 fields described in this way also influence the field-generating fermionic fields. Therefore the Dirac equation , which describes the movement of the fermions, has to be adapted. The consideration of all carried out nuclear physics experiments and fundamental theoretical work on weak nuclear force led to a field theory in which the electromagnetic and weak interaction are summarized and referred to as electroweak interaction. This theory is also known as the standard model of elementary particle physics .

Older models for describing the weak interaction are included in this standard model and agree with it for low particle energies.

literature

• Brian R. Martin, Graham Shaw: Particle Physics, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7 , p. 373
• Brian R. Martin: Nuclear and Particle Physics, Wiley 2008, p. 369

See also

• photon
• Quantum electrodynamics

Web links

• Spectrum Lexicon of Physics

credentials

1. ^ W. Greiner, "Relativistische Quantenmechanik", Springer, pp. 415 f., ISBN 3-540-67457-8
2. ↑ Hendrik van Hees, Quantenfeldtheorie script ( Memento of the original from February 7, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
3. ↑ ^{a b} C. Itzykson, J.-B. Zuber, "Quantum Field Theory", McGraw-Hill International Edition, 3rd Edition, 1987, pp. 134-23.
4. ↑ Greiner, Relativistic Quantum Mechanics, Springer, 3rd edition 2000, p. 361ff
5. ^ W. Greiner, "Relativistische Quantenmechanik", Verlag Harri Deutsch, 1987, pp. 442 f., ISBN 3-8171-1022-7
6. ^ W. Greiner, B. Müller, "Gauge theory of weak interaction", Verlag Harri Deutsch, 1994, ISBN 3-8171-1427-3