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Basics of Quantum Field Theory at Finite Temperature and Chemical Potential Appendix A Basics of Quantum Field Theory at Finite Temperature and Chemical Potential Many of the discussions in the main part of these lectures rely on field-theoretical methods, in particular on quantum field theory at finite temperature and chemical potential. One purpose of the following basic discussion is therefore to explain how a chemical potential is introduced in quantum field theory. We shall also discuss how finite temperature enters the formalism, although for most quantities we discuss in these lecture notes we consider the zero-temperature limit, which is a good approx- imation for our purposes. For instance in the discussion of the Walecka model, Sect. 3.1, we give the finite-temperature expressions, based on Appendix A.2, before we set T = 0 in the physical discussion. In other parts, we do keep T = 0 in our results, for instance when we are interested in the cooling behavior of dense matter, see Chap. 5. We shall start with the Lagrangian for a complex bosonic field and derive the partition function in the path integral formalism, taking into account Bose–Einstein condensation. This part is particularly useful for our treatment of kaon condensation in CFL quark matter, see Sect. 4.2.1. We shall in particular see how bosonic Mat- subara frequencies are introduced and how the summation over these is performed with the help of contour integration in the complex frequency plane. In the second part of this appendix we shall then discuss the analogous derivation for fermions. A.1 Bosonic Field We start from the Lagrangian ∗ μ 2 2 4 L0 = ∂μϕ ∂ ϕ − m |ϕ| − λ|ϕ| , (A.1) with a complex scalar field ϕ with mass m and coupling constant λ.Weshallfirst show how a chemical potential μ is introduced. This will lead to a new Lagrangian L, wherefore we have denoted the Lagrangian without chemical potential by L0. The chemical potential μ must be associated with a conserved charge. We thus need to identify the conserved current. From Noether’s theorem we know that the Schmitt, A.: Basics of Quantum Field Theory at Finite Temperature and Chemical Potential. Lect. Notes Phys. 811, 123–136 (2010) DOI 10.1007/978-3-642-12866-0 c Springer-Verlag Berlin Heidelberg 2010 124 A Basics of Quantum Field Theory at Finite Temperature and Chemical Potential conserved current is related to the symmetry of the Lagrangian. We see that L0 is invariant under U(1) rotations of the field, − α ϕ → e i ϕ. (A.2) This yields the Noether current ∂L δϕ ∂L δϕ∗ μ = 0 + 0 = (ϕ∗∂μϕ − ϕ∂μϕ∗), j ∗ i (A.3) ∂(∂μϕ) δα ∂(∂μϕ ) δα μ with ∂μ j = 0, and the conserved charge (density) is ∗ ∗ j0 = i(ϕ ∂0ϕ − ϕ∂0ϕ ). (A.4) In the following we want to see how the chemical potential associated to j0 enters the Lagrangian. The partition function for a scalar field is = −β(Hˆ −μNˆ ) Z Tr e = Dπ Dϕ exp − (H − μN − iπ∂τ ϕ) . (A.5) periodic X This equation should remind you that the partition function can be written in the operator formalism in terms of the Hamiltonian Hˆ and the charge operator Nˆ ,or, as we shall use here, in terms of a functional integral over ϕ and the conjugate momentum π, with the Hamiltonian H and the charge density N = j0.Wehave abbreviated the space-time integration by β ≡ dτ d3x , (A.6) X 0 where the integration over “imaginary time” τ = it goes from 0 to the inverse temperature β = 1/T . In the following, the four-vector in position space is denoted by X ≡ (t, x) = (−iτ,x). (A.7) The term “periodic” for the ϕ integral in Eq. (A.5) means that all fields ϕ over which we integrate have to be periodic in the imaginary time direction, ϕ(0, x) = ϕ(β,x). This is essentially a consequence of the trace operation in the first line of Eq. (A.5): the partition function is formally reminiscent of a sum over transition amplitudes which have the same initial and final states at “times” 0 and β. Let us, for convenience, introduce the two real fields ϕ1, ϕ2, 1 ϕ = √ (ϕ1 + iϕ2). (A.8) 2 A.1 Bosonic Field 125 Then, the Lagrangian becomes λ 1 μ μ 2 2 2 2 2 2 L = ∂μϕ ∂ ϕ + ∂μϕ ∂ ϕ − m (ϕ + ϕ ) − (ϕ + ϕ ) . (A.9) 0 2 1 1 2 2 1 2 2 1 2 The conjugate momenta are ∂L 0 0 πi = = ∂ ϕi , i = 1, 2 . (A.10) ∂(∂0ϕi ) 0 Consequently, with j = ϕ2π1 − ϕ1π2, which follows from Eqs. (A.4), (A.8), and (A.10), we have H − μN = π ∂ ϕ + π ∂ ϕ − L − μN 1 0 1 2 0 2 0 1 = π 2 + π 2 + (∇ϕ )2 + (∇ϕ )2 + m2(ϕ2 + ϕ2) 2 1 2 1 2 1 2 −μ(ϕ2π1 − ϕ1π2). (A.11) The integration over the conjugate momenta π1, π2 can be separated from the inte- gration over the fields ϕ1, ϕ2 after introducing the shifted momenta π˜1 ≡ π1 − ∂0ϕ1 − μϕ2 , π˜2 ≡ π2 − ∂0ϕ2 + μϕ1 . (A.12) This yields 1 π ∂ ϕ + π ∂ ϕ − H + μN =− (π˜ 2 +˜π 2) + L , (A.13) 1 0 1 2 0 2 2 1 2 where the new Lagrangian L now includes the chemical potential, 1 μ μ L = ∂μϕ ∂ ϕ + ∂μϕ ∂ ϕ + 2μ(ϕ ∂ ϕ − ϕ ∂ ϕ ) 2 1 1 2 2 2 0 1 1 0 2 λ +(μ2 − m2)(ϕ2 + ϕ2) − (ϕ2 + ϕ2)2 . (A.14) 1 2 2 1 2 Thus we see that the chemical potential produces, besides the expected term μj0, μ2 ϕ2 +ϕ2 / 0 the additional term 1 2 2. This is due to the momentum-dependence of j . In terms of the complex field ϕ, the Lagrangian reads 2 2 2 2 4 L =|(∂0 − iμ)ϕ| −|∇ϕ| − m |ϕ| − λ|ϕ| , (A.15) which shows that the chemical potential looks like the temporal component of a gauge field. We can now insert Eq. (A.13) into the partition function (A.5). The 126 A Basics of Quantum Field Theory at Finite Temperature and Chemical Potential integration over conjugate momenta and over fields factorize, and the momentum integral yields an irrelevant constant N, such that we can write Z = N Dϕ1Dϕ2 exp L . (A.16) periodic X In order to take into account Bose–Einstein condensation, we divide the field into a constant background field and fluctuations around this background, ϕi → φi + ϕi . A nonzero condensate φ1 + iφ2 picks a direction in the U(1) degeneracy space and thus breaks the symmetry spontaneously. We can choose φ2 = 0 and thus may denote φ ≡ φ1. Then, the Lagrangian (A.14) becomes ( ) ( ) ( ) L =−U(φ2) + L 2 + L 3 + L 4 , (A.17) with the tree-level potential m2 − μ2 λ U(φ2) = φ2 + φ4 , (A.18) 2 4 and terms of second, third, and fourth order in the fluctuations, (2) 1 μ μ L =− −∂μϕ1∂ ϕ1 − ∂μϕ2∂ ϕ2 − 2μ(ϕ2∂0ϕ1 − ϕ1∂0ϕ2) 2 + 2 − μ2 ϕ2 + ϕ2 + λφ2ϕ2 + λφ2ϕ2 , m 1 2 3 1 2 (A.19a) L(3) =−λφϕ ϕ2 + ϕ2 , 1 1 2 (A.19b) ( ) λ 2 L 4 =− ϕ2 + ϕ2 . (A.19c) 4 1 2 We have omitted the linear terms since they do not contribute to the functional inte- gral. Note that the cubic interactions are induced by the condensate. In this appendix we are only interested in the tree-level contributions U φ2 and L(2) in order to explain the basic calculation of the partition function for the simplest case. We therefore shall ignore the cubic and quartic contributions L(3) and L(4).We introduce the Fourier transforms of the fluctuation fields via 1 − · 1 (ω τ+ · ) ϕ(X) = √ e iK X ϕ(K ) = √ ei n k x ϕ(K ), (A.20) TV K TV K with the four-momentum K ≡ (k0, k) = (−iωn, k), (A.21) and with the Minkowski scalar product K · X = k0x0 − k · x =−(τωn + k · x). (Although for convenience we have defined the time components with a factor i and A.1 Bosonic Field 127 thus can use Minkowski notation, the scalar product is essentially Euclidean.) The normalization is chosen such that the Fourier-transformed fields ϕ(K ) are dimen- sionless. The 0-component of the four-momentum is given by the Matsubara fre- quency ωn. To fulfill the periodicity requirement ϕ(0, x) = ϕ(β,x) we need iω β e n = 1, i.e., ωnβ has to be an integer multiple of 2π,or ωn = 2πnT , n ∈ Z . (A.22) With the Fourier transform (A.20), and iK·X V e = δK,0 , (A.23) X T we have −1( ) (2) 1 D0 K ϕ1(K ) L =− (ϕ1(−K ), ϕ2(−K )) , (A.24) 2 T 2 ϕ2(K ) X K with the free inverse propagator in momentum space − 2 + 2 + λφ2 − μ2 − μ −1( ) = K m 3 2i k0 . D0 K 2 2 2 2 (A.25) 2iμk0 −K + m + λφ − μ With Eqs. (A.16), (A.24) and using that ϕ(K ) = ϕ∗(−K ) (because ϕ(X) is real) we can write the tree-level thermodynamic potential as Ω T =− ln Z V V −1( ) 2 T 1 D0 K ϕ1(K ) = U(φ ) − ln Dϕ1Dϕ2 exp − (ϕ1(−K ), ϕ2(−K )) V 2 T 2 ϕ2(K ) K − T D 1(K ) = U(φ2) + ln det 0 , (A.26) 2V T 2 where the determinant is taken over 2 × 2 space and momentum space. Here we have used the general formula D − 1 x·Aˆx D/2 ˆ −1/2 d xe 2 = (2π) (det A) , (A.27) for a Hermitian, positive definite matrix Aˆ, which is a generalization of the one- dimensional Gaussian integral % ∞ π − 1 αx2 2 dx e 2 = .
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