Gauge theory

This article deals with gauge theories in general. Non-Belian gauge theories are dealt with in the article Yang-Mills Theory .

A gauge theory or gauge field theory is understood to be a physical field theory that satisfies local gauge symmetry .

This clearly means that the interactions predicted by the theory do not change if a certain quantity is freely selected locally. This possibility of setting a size independently at each location - to calibrate it like a yardstick - prompted the German mathematician Hermann Weyl to choose the name gauge symmetry or gauge invariance in the 1920s .

A distinction is made between local and global calibration transformations , depending on whether the transformation is location-dependent (local) or not (global). Calibration fields occur in local calibration transformations and ensure the invariance of the dynamic system in local calibration transformations.

history

The vector potential was used in electrodynamic theory as early as the 19th century , e.g. B. by Franz Ernst Neumann  (1847), Gustav Kirchhoff  (1857) and Hermann von Helmholtz  (1870 to 1874). The latter was already close to the discovery of invariance under gauge transformations and introduced a Lorenz gauge , but only for quasi-static problems.

The invariance under gauge transformations was also described by James Clerk Maxwell z. B. formulated in his main work Treatise on Electricity and Magnetism , but not yet in the most general form (he preferred the Coulomb calibration ). The Lorenz calibration for full retarded potentials comes from Ludvig Lorenz  (1867) and was also presented around 25 years later by Hendrik Antoon Lorentz .

The modern conception of a gauge theory as a consequence of a locally variable phase factor of the wave function is mostly attributed to Hermann Weyl (1929), but it is also formulated by Wladimir Fock in 1926 . This happened in the context of the discussion of the relativistic wave equation for massive scalar particles, whereby the vector potential flows in via the minimal coupling (see below). At the same time as Fock, Erwin Schrödinger and Oskar Klein published corresponding works.

As early as 1919, before the development of quantum mechanics, Weyl introduced a locally variable length scale as a calibration factor as part of an attempt to expand the general theory of relativity , which also includes electrodynamics. By reformulating it to include complex phases in the context of quantum mechanics, in 1929 he gave the formulation of gauge theories in today's sense, which Fritz London had already done independently before .

Electrodynamics is the simplest case of a gauge theory with Abelian gauge group U (1) , the case of non- Abelian gauge groups ( Yang-Mills theory , non- Abelian gauge theory) was first dealt with by Chen Ning Yang and Robert L. Mills in 1954.

Gauge theories in the physics of elementary particles

Modern particle physics strives to derive the behavior of elementary particles from the simplest possible first principles. A useful aid is the requirement for a group of transformations (e.g. rotations) of the fields involved , under which the dynamics of the particles remain invariant. This symmetry or freedom from calibration limits the shape of the Lagrangian to be constructed enormously and thus helps in the construction of the theory sought.

In general, a covariant derivation can be defined in a gauge theory , from which a field strength tensor and thus a Lagrangian and an effect can be constructed, from which the equations of motion and conservation quantities result from variation .

The standard model of elementary particle physics contains two such gauge theories:

• the theory of the electroweak interaction with the symmetry group and ${\displaystyle SU(2)_{I}\times U(1)_{Y}}$
• the theory of strong interaction with the symmetry group . ${\displaystyle SU(3)\,\!_{C}}$

The Noether theorem guarantees that every particle that is subject to the interaction to be described can be clearly assigned a received charge , e.g. B. electric charge , hypercharge , weak isospin , color charge . ${\displaystyle Q}$ ${\displaystyle Y}$ ${\displaystyle I_{3}}$ ${\displaystyle C}$

There is also a gauge theory formulation of gravity, both general relativity  (GTR) and extended theories. This was first recognized by Ryoyu Utiyama in  1956, who used the Lorentz group as a calibration group. That was not yet completely correct, the correct calibration group is the Poincaré group (which also includes translations), as Dennis Sciama and TWB Kibble recognized in 1961. The Einstein-Cartan theory, as a generalization of the GTR , also fits into this context (with it, the spin of matter is connected with the torsion of space-time , analogous to the connection of energy-momentum with the Riemann curvature tensor in the GTR). ${\displaystyle SO(3,1)}$

Calibration theory using the example of electrodynamics

Gauge symmetry of the equation of motion of point particles

The energy of a particle in an external static potential can be written as

${\displaystyle E={\frac {1}{2}}m{\vec {v}}^{\,2}+V({\vec {x}})}$

with a given potential . ${\displaystyle V({\vec {x}})}$

If we now define the impulse as

${\displaystyle {\vec {p}}=m{\vec {v}}}$,

so one can also write the energy as

${\displaystyle E={\frac {{\vec {p}}^{\,2}}{2m}}+V({\vec {x}})}$.

If one describes energy as a function of position and momentum according to Hamiltonian mechanics , that is

${\displaystyle E=H({\vec {x}},{\vec {p}})}$

then one obtains the equations of motion from their derivatives :

${\displaystyle {\dot {x}}_{i}={\frac {\partial H}{\partial p_{i}}}}$

${\displaystyle {\dot {p}}_{i}=-{\frac {\partial H}{\partial x_{i}}}}$

For the energy mentioned above this results in:

${\displaystyle {\dot {x}}_{i}={\frac {p_{i}}{m}}=v_{i}}$

${\displaystyle {\dot {p}}_{i}=-{\frac {\partial V}{\partial x_{i}}}=F_{i}}$

If you add a constant term to the potential and to the momentum, i.e. define:

${\displaystyle V_{1}({\vec {x}})=V({\vec {x}})+V_{0}}$

${\displaystyle {\vec {p}}_{1}=m{\vec {v}}+{\vec {p}}_{0}}$

and then describes the movement of the particle by means of the "index 1 quantities", that is the energy

${\displaystyle E={\frac {|{\vec {p}}_{1}-{\vec {p}}_{0}|^{2}}{2m}}+V_{1}({\vec {x}})-V_{0}}$

and the equations of motion are:

${\displaystyle {\dot {x}}_{i}={\frac {\partial H}{\partial (p_{1})_{i}}}={\frac {(p_{1})_{i}-(p_{0})_{i}}{m}}=v_{i}}$

${\displaystyle ({\dot {p}}_{1})_{i}=-{\frac {\partial V}{\partial x_{i}}}=F_{i}}$

Since also

${\displaystyle {\dot {\vec {p}}}={\dot {\vec {p}}}_{1}}$

holds (because constants disappear in the derivative), they are exactly the same equations of motion.

It is therefore possible to define a constant summand for both the energy and the momentum without changing the physics described thereby. This property is called global gauge symmetry.

Now the question arises whether one can also add non-constant quantities instead without changing the equations of motion, i.e. in general

${\displaystyle V_{1}(x)=V(x)+q\phi (x,t)\,}$

${\displaystyle {\vec {p}}_{1}(x)=m{\vec {v}}+q{\vec {A}}(x,t),}$

where the constant q has been extracted because it will prove useful afterwards; but this fact has no bearing on the argument.

It is immediately clear that it is not possible to use arbitrary functions for and , since e.g. B. any one acts like an additional potential. Assuming arbitrary functions for both quantities, recalculation shows that the equations of motion are given by: ${\displaystyle \phi }$ ${\displaystyle {\vec {A}}}$ ${\displaystyle \phi }$

${\displaystyle {\dot {\vec {x}}}={\vec {v}}}$

${\displaystyle {\dot {\vec {p}}}=q{\vec {v}}\times \operatorname {rot} \,\,{\vec {A}}-q\,{\frac {\partial {\vec {A}}}{\partial t}}-q\,\operatorname {grad} \,\,\phi -\operatorname {grad} \,\,V({\vec {x}})}$

But these are precisely the equations of motion that one would expect if the particle has the charge q and is not only in the potential V but also in the electric field

${\displaystyle {\vec {E}}=-{\frac {\partial {\vec {A}}}{\partial t}}-\operatorname {grad} \,\,\phi }$

and in the magnetic field

${\displaystyle {\vec {B}}=\operatorname {rot} \,\,{\vec {A}}}$

emotional.

The movement is now not changed if a change from and to and the fields and does not change (i.e. especially leaves the fields at zero if they were previously zero). That means that and the equations and must satisfy. Since the rotation of a gradient field is always zero, it is clear that the first of these equations is fulfilled (and therefore the magnetic field remains unchanged) if any time- and location-dependent function is chosen for the gradient. In order to satisfy the second equation, one then has to set this function as the time derivative, i.e. reduce the potential accordingly. If one chooses the position and negative time derivative of one and the same function as and , the equations of motion for the particle do not change. Such a choice therefore gives a local calibration symmetry . ${\displaystyle \phi }$ ${\displaystyle {\vec {A}}}$ ${\displaystyle \phi '=\phi +\delta \phi }$ ${\displaystyle {\vec {A}}'={\vec {A}}+\delta {\vec {A}}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \delta \phi }$ ${\displaystyle \delta {\vec {A}}}$ ${\displaystyle \textstyle \operatorname {rot} \delta {\vec {A}}=0}$ ${\displaystyle \textstyle \partial _{t}\delta {\vec {A}}+\operatorname {grad} \delta \phi =0}$ ${\displaystyle \delta {\vec {A}}}$ ${\displaystyle -\delta \phi }$ ${\displaystyle \delta {\vec {A}}}$ ${\displaystyle \delta \phi }$

Gauge symmetry of the quantum mechanical wave function

In quantum mechanics, particles are no longer described by location and momentum, but by the so-called wave function . This is a field, i.e. a function of space and time, and is generally complex (e.g. it is a complex scalar in the non-relativistic Schrödinger equation and a complex spinor in the Dirac equation ). However, it is not unambiguous: The wave functions and with arbitrarily chosen real both describe the same state. Again, this is a global symmetry. Mathematically, this symmetry is described by the Lie group U (1) , because it consists exactly of the numbers . ${\displaystyle \psi ({\vec {x}},t)}$ ${\displaystyle \psi ({\vec {x}},t)}$ ${\displaystyle \mathrm {e} ^{\mathrm {i} \phi }\psi ({\vec {x}},t)}$ ${\displaystyle \phi }$ ${\displaystyle \mathrm {e} ^{\mathrm {i} \phi }}$

As before in the case of the classical equation of motion, the question arises here whether one could also introduce a location- and time-dependent phase instead of the global phase. However , partial derivatives appear in the equation of motion of the wave function ( Schrödinger equation , Dirac equation, etc.) , which lead to additional terms in the wave function changed in this way:

${\displaystyle {\frac {\partial }{\partial t}}\left(\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}\psi ({\vec {x}},t)\right)=\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}{\frac {\partial }{\partial t}}\psi ({\vec {x}},t)+\mathrm {i} {\frac {\partial \phi ({\vec {x}},t)}{\partial t}}\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}\psi ({\vec {x}},t)}$

${\displaystyle {\frac {\partial }{\partial x_{i}}}\left(\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}\psi ({\vec {x}},t)\right)=\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}{\frac {\partial }{\partial x_{i}}}\psi ({\vec {x}},t)+\mathrm {i} {\frac {\partial \phi ({\vec {x}},t)}{\partial x_{i}}}\mathrm {e} ^{\mathrm {i} \phi ({\vec {x}},t)}\psi ({\vec {x}},t)}$

These relationships can also be interpreted in such a way that the partial position and time derivatives are given by the derivative operators

${\displaystyle \operatorname {D} _{t}={\frac {\partial }{\partial t}}+\mathrm {i} {\frac {\partial \phi ({\vec {x}},t)}{\partial t}}}$

${\displaystyle \operatorname {D} _{x_{i}}={\frac {\partial }{\partial x_{i}}}+\mathrm {i} {\frac {\partial \phi ({\vec {x}},t)}{\partial x_{i}}}}$

be replaced. The connection with the electromagnetic field becomes clear when one considers the form of the Schrödinger equation:

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\psi ({\vec {x}},t)={\hat {H}}\psi ({\vec {x}},t),}$

where in the Hamilton operator the local derivatives over the components of the momentum operator ${\displaystyle {\hat {H}}}$

${\displaystyle {\hat {p}}_{i}=-\mathrm {i} \hbar {\frac {\partial }{\partial x_{i}}}}$

occur. We replace the momentum operator now by , we get: ${\displaystyle {\frac {\partial }{\partial x_{i}}}}$ ${\displaystyle \operatorname {D} _{x_{i}}}$

${\displaystyle {\hat {p}}_{i}=-\mathrm {i} \hbar \operatorname {D} _{x_{i}}=-\mathrm {i} \hbar {\frac {\partial }{\partial x_{i}}}+\hbar {\frac {\partial \phi ({\vec {x}},t)}{\partial x_{i}}}}$

So there is an additional summand that looks like a contribution to the electromagnetic vector potential . Similarly, when inserting into the Schrödinger equation, there is an additional potential term of the form . These additional electromagnetic potentials, however, just meet the calibration condition for electromagnetic fields, so that the physics is in fact not influenced by the local phase, but only the electromagnetic potentials have to be adapted in the description. ${\displaystyle {\vec {A}}}$ ${\displaystyle \operatorname {D} _{t}}$ ${\displaystyle {\frac {\partial \phi ({\vec {x}},t)}{\partial t}}}$

In connection with relationships of this kind , one often speaks of “ minimal coupling ”. ${\displaystyle {\vec {p}}\to {\vec {p}}+q{\vec {A}}}$

Gauge theories in mathematics

In mathematics , gauge theories also play an important role in the classification of four-dimensional manifolds . Edward Witten and Nathan Seiberg were able to define topological invariants , the Seiberg-Witten invariants, using gauge theory methods in 1994 .

See also

• Lattice theory

Web links

• Helmut Hörner: How the electromagnetic field, the canonical momentum and the quantum mechanical interaction between charged particles and the EM field arise from the requirement for local gauge invariance of the Schrödinger equation.

literature

• David Bailin, Alexander Love: Introduction to gauge field theory. Revised edition. Institute of Physics Publishing, Bristol u. a. 1994, ISBN 0-7503-0281-X .
• Peter Becher, Manfred Böhm, Hans Joos : gauge theories of strong and electroweak interaction . Teubner Study Books, 1983, ISBN 978-3-519-13045-1
• Ta-Pei Cheng, Ling-Fong Li: Gauge theory of elementary particle physics. Reprinted edition. Oxford University Press, Oxford u. a. 2006, ISBN 0-19-851961-3 .
• Dietmar Ebert: gauge theories. Basis of elementary particle physics. VCH-Verlag, Weinheim u. a. 1989, ISBN 3-527-27819-2 .
• Richard Healey: Gauging What's Real. The Conceptual Foundations of Gauge Theories. Oxford University Press, Oxford u. a. 2007, ISBN 978-0-19-928796-3 , review by Ward Struyve.
• Gerardus' t Hooft : Gauge theories of the forces between elementary particles . In: Scientific American , Volume 242, June 1980
• Stefan Pokorski: Gauge field theories . 2nd Edition. Cambridge University Press, 2000

About history:

• Lochlainn O'Raifeartaigh , Norbert Straumann : Early History of Gauge Theories and Kaluza-Klein Theories, with a Glance at Recent Developments. In: Reviews of Modern Physics. Volume 72, 2000, pp. 1-23, arxiv : hep-ph / 9810524 .
• John David Jackson , LB Okun : Historical roots of gauge invariance . In: Reviews of Modern Physics. Volume 73, 2001, pp. 663-680, arxiv : hep-ph / 0012061 .
• Lochlainn O'Raifeartaigh (Ed.): The dawning of gauge theory. Princeton University Press 1997 (reprint volume with commentary).

Individual evidence

1. ↑ L. Lorenz: About the identity of the vibrations of light with the electrical currents. In: Ann. der Physik und Chemie , Volume 131, 1867, pp. 243-263.
2. ↑ W. Fock: About the invariant form of the wave and motion equations for a charged mass point. In: Journal of Physics. Volume 39, 1926, pp. 226-232
3. ↑ H. Weyl: A new extension of the theory of relativity. In: Annalen der Physik , 59, 1919, pp. 101-133.
4. ↑ H. Weyl: electron and gravitation. In: Z. f. Physics. Vol. 56, 1929, pp. 330-352 ( Gravitation and the electron. In: Proc. Nat. Acad. Sci. , 15, 1929, pp. 323-334).
5. ↑ F. London: Weyl's theory and quantum mechanics. In: Natural Sciences. 15, 1927, 187.
6. ↑ F. London: Quantum mechanical interpretation of the theory of Weyl. In: Z. f. Physik , Volume 42, 1927, pp. 375-389.
7. ↑ CN Yang, RL Mills: Conservation of isotopic spin and isotopic gauge invariance. In: Physical Review. Volume 96, 1954, pp. 191-195.
8. ↑ Gronwald, Hehl: On the gauge aspects of gravity . Erice Lectures, 1995, arxiv : gr-qc / 9602013