**Wigner Funktionen in ``Single Time'' und Explizit
Kovarianter Formulierung**

**Autor:** Stefan Ochs

**Betreuer:** Prof. Dr. Ulrich Heinz

**Abgabe:** 17. Dezember 1996

**Abstract:**

To describe the formation of a Quark Gluon Plasma, we need a transport theory for quarks and gluons. The basis for such a transport theory from first principles is the Wigner formalism [1].

The aim of this thesis was to establish the connection between the (gauge-) covariant formulation of Wigner functions [3] and the recently proposed so-called ``Single time'' formulation [2]. The definition of the covariant Wigner function is given through a four-dimensional Fourier transform of the gauge-covariant density operator , whereas the ``Single time'' Wigner function takes the density operator at equal times and performs then a three-dimensional Fourier transform over .

I established the connection between the two formulations by taking moments of the covariant Wigner function with respect to the energy variable . This gives me back for the lowest moment the ``Single time'' formulation.

To derive the equations of motion for the moments, I studied first the
covariant theory for spinor fields with U(1) (QED) and SU(*n*) (QCD with
*n*=3) symmetry, discussing mass shell and transport equations, the
classical limit, color decomposition,
the external field limit and chiral theory.
Then I derived the equations for the moments, finding to sets of equations:
the first set gave me the explicit time derivative of the
moment as
expressed through the lowerlying moments from 0 to n. The second set had no
explicit time derivative and coupled the moment n to the next higher moment.
This hierarchy of moment equations in general does not truncate. For
external fields however I could proove, that it is sufficient to
consider the equation for the time evolution of the lowest (zeroth)
moment. The
higher moments could then all be explicitely constructed from the lowest
one. This established the connection with the ``Single time'' formalism,
which only gave an equation for the time evolution of the lowest moment, but
could not tell on how to calculate the higher moments.