Nuclear Physics B (Proc. Suppl.) 34 (1994) 307-310 North-Holland liLgI(ll I W,_N [i "J -- k'd~'] [el,'l fl PROCEEDINGS SUPPLEMENTS QCD and QED at Finite Temperature and Chemical Potential I. Bender, H.J. Rothe, M. Plewnia, W. Wetzel ~, T. Hashimoto b, A. Nakamura ¢ and I.O. Stamatescu d aInst. Theor. Physik, Universit~t Heidelberg, 69120 Heidelberg, Germany bDepartment of Applied Physics, Faculty of Engineering, Fukui University, Fukui 910, Japan CFaculty of Education, Yamagata University, Yamagata, Japan dFEST , Schmeilweg 5, 69118 Heidelberg, Germany, and Inst. Theor. Physik, Universit~it Heidelberg, 69120 Heidelberg, Germany We report results from 3 analyses of finite temperature and density QFT on the lattice, presented in the following as section 1 [1], 2 [2] and 3 [3]. They should be understood as extended abstracts. 1. LATTICE FERMIONS AT FINITE TEMPERATURE AND DENSITY In this note we show how fermion doubling manifests itself in the thermodynamical observ- ables for a gas of free Dirac particles at finite temperature L and chemical potential/~ [1] As is well known, the use of naive lattice fermions leads to the fermion doubling problem. The chemical potential is introduced in the expo- nential form. [4] It is found that the thermody- namical observables do not receive /z-dependent divergent contributions, as has been found before at zero temperature.[4] The main message of this note will be that the positive as well as the neg- ative energy contributions to the partition func- tion each receive contributions characteristic for free particles and antiparticles. This unpleasant feature is however eliminated by the use of Wilson fermions. Consider the following lattice action for free fermions[4], "x F ^ S = E CnKnmCm, Kn,~ = (Th-4- 4r)hnm- n 1 -~ • [(r - 74)e~5o+~,,m + (r + 74)e-~5~-~,,m] /% E [(r - 7,)hn+e,,,~ + (r + 7i)hn-ei,m] (1) 2 rt,i where ¢, rh and/5 are the fermion field, mass and chemical potential measured in lattice units. The partition function is given by the product of the eigenvalues of the matrix K. The matrix can be diagonalized by going to momentum space. We first consider the case of naive fermions, i.e. r = 0 in eq.(1). The eigenvalues are found to be )~±(~, ~) = isin(~e - i/~) + E(~5) E(~5) = i~. sin2iSj + r h 2 (2) Here -4-E are the positive an negative energy eigenvalues of the lattice Hamiltonian, and dJt = (2t + 1)r/]~ are the Matsubara frequencies, with the inverse temperature in lattice units. In the large volume limit the logarithm of the partition function is given by 1 /; dZi5 E In (A+(15' ~hl)A-(i5' oht)) 1 (3) where gs is the spin degeneracy factor (gs = 2). From (3) one calculates the mean energy and charge. It is instructive to look separately at the contributions coming from the positive and neg- ative energy eigenvalues of the Hamiltonian. The sum over Matsubara frequencies can be carried out 3, and one finds for the mean charge and en- ergy that (Q)+ = (Q)_ and (/~)+ = (/~)_. Wak- 0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00273-X