Quantum transport: Self-consistent approximations
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uantum Transport: Self-Consistent Approximations1 ¶ D. N. Voskresensky Moscow Engineering Physics Institute, 115409 Moscow, Russia Theory Division, GSI mbH, D–64291 Darmstadt, Germany Abstract—Functional methods: Generating Φ{φ, G} functional of Baym on Schwinger-Keldysh contour; symmetries and conservation laws; thermodynamic consistency. (See [1]. Renormalization scheme can be found in [2].) Generalized kinetic approach: gradient approximation. (See [3]. Gradient expansion diagram technique see in [4].) Some helpful examples. (See [5].) PACS numbers: 02.30.Sa; 05.20.Dd DOI: 10.1134/S1063779608070393
1. UNSOLVED PROBLEMS OF THE FIELD • Problems for vector (gauge) fields: (i) Ward identities are not fulfilled since in working with Φ derivable theories one deals with bare vertices. (ii) The Goldstone theorem is not fulfilled. Thus, modifications to the standard Φ derivable approach are needed in case of gauge theories and for complex fields if condensates are present. • It is not clear how to construct the theory including initial higher order correlations. Does the Bogolyubov– Klimontovich principle of weakening of initial correlations work in the case of strongly interacting systems? • There are different expressions for the entropy. It is important to search for relations between them, and it is a challenge to find unique expression for the entropy at nonequilibrium. • We demonstrated the H theorem in some examples. The problem of whether the H theorem holds in the self-consistent kinetic approach in the general case remains unsolved? • Do positiveness of rates and the detailed balance equation hold in the general case and, if not, how does it affect driving the system to equilibrium? • It is a challenge to develop a numerical code that would allow us to apply the generalized kinetic scheme in realistic treatment of heavy ion collisions.
2. EXERCISES FOR STUDENTS Some examples of application of methods discussed in my lectures were presented in slides 53–59. In more detail solutions can be found in [3, 5]. Below I present two extra examples of problems and their solutions. Problem 1. The Lagrangian density of a spinless one-component boson field is 1 1 4 µ 2 2 L = --- ( ∂ µ φ∂ φ – m φ ) – --- λφ . 2 4
(1)
Let m2 < 0, λ 1. Find the critical temperature of the phase transition in a uniform warm medium described by this model using the approximation of noninteracting gas of excitations on the ground of the condensate: (way 1) Use the expression for the free energy of the ideal Bose gas. (way 2) Calculate and use the expression for the closed loop diagram (iG(x = 0)) (a) in the Matsubara technique and (b) in the Schwinger-Keldysh technique. Solution (see [6]). Average the Lagrangian density over the equilibrium state (〈L〉T) at finite temperature. Put φ = φc + φ' (φc is the classical field, φ' is responsible for excitations), and expand 〈L〉T in φ' keeping terms ∝〈(φ')2〉T. One has
These are only some of many serious problems that await solution.
– 〈 L〉 T = –L c + δF,
(2)
1 1 4 µ 2 2 – L c = – --- ( ∂ µ φ c ∂ φ
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