We solve the time evolution of the density matrix both for fermions and bosons in the presence of a homogeneous but time dependent external electric field. The number of particles produced by the external field, as well as their distribution in momentum space is found for finite times. Furthermore, we calculate the probability of finding a given number of particles in the ensemble. In all cases, there is a nonvanishing thermal contribution. The bosonic and the fermionic density matrices are expressed in a "functional field basis". This constitutes an extension of the "field basis" concept to fermions.

The kinematics of SL(2,R) Yang-Mills theory on a circle is considered, for reasons that are spelled out. The gauge transformations exhibit hyperbolic fixed points, and this results in a physical configuration space with a non-Hausdorff "network" topology. The ambiguity encountered in canonical quantization is then much more pronounced than in the compact case, and can not be resolved through the kind of appeal made to group theory in that case.1 Email address: ingemar@vana.physto.se 2 Email address: tfejh@fy.chalmers.seWe have studied Yang-Mills theory on a cylindrical space-time, choosing the non-compact group SL(2,R) for our structure group. Since this undertaking may appear peculiar, we will begin by spelling out our motivation. First of all the Yang-Mills Hamiltonian is not positive definite whenever the structure group is non-compact. However, this is of no concern to us, since this operator will have nothing to do with the time-development of our model. Actually, we will not be concerned with time-development at all, so that we are really interested only in setting up the model on a space which has the topology of the circle -"Yang-Mills theory", here, refers only to the phase space of the model. Our interest has to do with gravity. We know that there are four choices of structure group for Yang-Mills theory that are of physical interest: U(1), SU(2) and SU(3) -all of which are compact -and SL(2,C), which is non-compact. The latter case can be used to formulate Einstein's theory [1]. We believe that it is important to gain a broad experience of noncompact gauge theories, and this is one of two reasons for studying the toy model that we will describe. (It is true that the group theory of SL(2,R) differs in important ways from the group theory of SL(2,C), but the real case has certain simplifying features, and it seemed worthwhile to do a separate study of this case.) This is our first motivation. The second motivation is more vague, but at least as important. So far, almost all our intuition about gauge theories comes from Yang-Mills theory with compact structure groups. If we step back a bit from the problem, and view the gauge transformations in the same way as we might view the Hamiltonian flow of a dynamical system, we observe that we are then dealing with gauge transformations of a very simple, "integrable" kind. Presumably, this is not a generic case, and presumably the "gauge flow" in any theory of gravitation -where time development itself may be viewed as a gauge transformation -is a very different kettle of fish. We believe that there is a risk that our experience from compact Yang-Mills theory may be qualitatively misleading when it comes to defining a quantum theory of gravity. Our toy model is of some interest here -although the gauge flow remains integrable, it exhibits hyperbolic fixed points, which is at least a step in the "chaotic" direction.We hope that we have convinced the reader that we have chosen an interesting subject, and we will now briefly review the properties of Yang-Mi...

We consider thermal QCD in the large N C limit, mainly in 1+1 dimensions. The gauge coupling is only taken into account to minimal order, by projection onto colour singlets. An expression for the free energy, exact as N C → ∞, is then obtained. A third order phase transition will occur. The critical temperature depends on the ratio N C /L, where L is the (infinite) spatial length. In the high temperature limit, the free energy will approach the same value as in the free theory, whereas we have a mesonic like phase at low temperature. Expressions for the quark condensate, Ψ Ψ , are also obtained.

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