Gauge theory

Under a gauge theory is defined as a physical field theory that satisfies a local gauge symmetry.

This clearly means that predicted by the theory of interactions do not change when a certain size is chosen locally free. This ability to independently set a limit at any place - how to calibrate a scale - prompted the German mathematician Hermann Weyl in the 1920s to the choice of the name gauge invariance and gauge symmetry.

Weyl discovered the gauge invariance in electrodynamics and first tried by applying the principle of Einstein's general relativity theory to unify electrodynamics and gravitation to a theory. However, the resulting theory proved to be wrong.

Gauge theories in the physics of elementary particles

The modern particle physics endeavors to derive the behavior of elementary particles from first principles as simple as possible. A useful tool is the demand for a group of transformations (eg rotations) of the fields involved, under which the dynamics of the particles remains invariant. This symmetry or gauge freedom restricts the form of the Lagrangian to be constructed an enormous, thus helping in the construction of this theory.

General it can be in a gauge theory a covariant derivative define, construct from this a field strength tensor and thus a Lagrangian and an effect from the result by varying the equations of motion and conserved quantities.

The Standard Model of particle physics contains two such gauge theories:

• The theory of the electroweak interaction with the symmetry group, and
• The theory of strong interactions with the symmetry group.

The Noether 's theorem guarantees that any particle that is subject to descriptive interaction, clearly a charge obtained can be assigned, such as electric charge, hypercharge, weak isospin, color. This charge indicates how strongly a particle coupled to the gauge bosons, and the sign of this coupling (eg, attractive / repulsive in the electromagnetic case).

Gauge theory on 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

With a given potential.

Then, imagine that the pulse as

So we can also write the energy as

Describing after the Hamiltonian mechanics, the energy as a function of position and momentum, ie

Then obtained from the derivatives of the equations of motion:

For energy above yields the

If we each still adds to the potential and to pulse a constant term, so defined:

And then the motion of the particle by means of the " index -1 sizes " describes you, then reads the energy

And the equations of motion are

In addition, since

( because constants vanish yes derivative), are exactly the same equations of motion.

It is thus possible to determine both the energy and the momentum for a constant term, without changing the physics it describes. This property is called global gauge symmetry.

Now the question arises whether one can also add non-constant variables instead without changing the equations of motion, ie generally

Where the constant q was pulled because it will turn out later to be practical; for reasoning but this fact has no meaning.

It is immediately obvious that it is not possible to use any function for, and since, for example, any effect as an additional potential. Assuming for both sizes of arbitrary functions, so recalculation shows that the equations of motion are given by

But these are just the equations of motion that one would expect if the particle has the charge q and, except for the potential V even in the electric field

And in the magnetic field

Moves.

The movement is now not changed if the change and not the fields and changes ( ie in particular the fields to null leaves when they were previously zero). Since the rotation of a gradient is always zero, it is clear that nothing is changed on the magnetic field through the addition of the gradient of an arbitrary space-and time -dependent scalar function to the vector potential. This changes the electrical field to the time derivative of this same gradient; this change can be compensated by the scalar potential is reduced by the time derivative of the same function.

Gauge symmetry of the quantum mechanical wave function

In quantum mechanics, particles no longer be described by the position and momentum, but by the so-called wave function. This is a field that is a function of space and time, and in general, complex (eg it is in the non-relativistic Schrödinger equation is a complex scalar and the Dirac equation is a complex spinor ). However, it is not clear: the wave functions and arbitrarily chosen both describe the same state. This is again a global symmetry. Mathematically, this symmetry is described by the Lie group.

As before in the case of the classical equation of motion here, the question arises whether one could also introduce a location-and time-dependent phase instead of the global phase. Now, however, enter into the equation of motion of the wave function ( Schrödinger equation, Dirac equation, etc.) partial derivatives, which have been altered in the wave function lead to additional terms:

These relationships can also be interpreted so that the partial spatial and time derivatives by the derivative operators

Be replaced. The relationship with the electromagnetic field becomes clear if one considers the form of the Schrödinger equation:

Where the Hamiltonian of the spatial derivatives of the components of the momentum operator

. occur If we replace the momentum operator by now, we get

It therefore occurs an additional summand which looks like a contribution to the electromagnetic vector potential. Analog arises when inserting into the Schrödinger equation, an additional potential term of the form. However, these additional electromagnetic potentials satisfy precisely the gauge condition for electromagnetic fields, so that the physics is in fact not affected by the local phase, but only in the description of the electromagnetic potentials must be adjusted.

In the context of relations of the type we often speak of " minimal coupling".

Gauge theories in mathematics

In mathematics gauge theories also play an important role in the classification of four-dimensional manifolds. So Edward Witten and Nathan Seiberg 1994 could define verifiable theoretical methods topological invariants.