1) Equation (2·8) should read к×vк=0. 2) The summand in the first term ori the right-hand side of Eq. (2·9) should be supplemented by a factor δ0,p+q. 3) Every summation symbol Σ in Eqs. (3·7) and (3·14) should be replaced by Σ′. 4) The inner product (p·s) in the expression for FII(3, 11) (p, q, r; s) in Eq. (3·25) should be followed by a factor √λqλrλs/λp in place of √λpλrλs/λp. 6) On the right-hand side of (4·22), the factor 2 should be dropped from the second term in the square bracket [⋯]. 7) The denominator (Zλ0)2 on the right-hand side of (4·23) should be replaced by (4λ0)2). 8) The first term in the square bracket [⋯] in Eq. (4·24) should read (1-λ02) in place of (1-λ0)2. 10) The last term on the right-hand side of Eq. (A·1) should be multiplied by 2.
Equation (2·8) should read к×vк=0. 2) The summand in the first term ori the right-hand side of Eq. (2·9) should be supplemented by a factor δ0,p+q. 3) Every summation symbol Σ in Eqs. (3·7) and (3·14) should be replaced by Σ′. 4) The inner product (p·s) in the expression for FII(3, 11) (p, q, r; s) in Eq. (3·25) should be followed by a factor √λqλrλs/λp in place of √λpλrλs/λp. 6) On the right-hand side of (4·22), the factor 2 should be dropped from the second term in the square bracket [⋯]. 7) The denominator (Zλ0)2 on the right-hand side of (4·23) should be replaced by (4λ0)2). 8) The first term in the square bracket [⋯] in Eq. (4·24) should read (1-λ02) in place of (1-λ0)2. 10) The last term on the right-hand side of Eq. (A·1) should be multiplied by 2.
The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.
In Ref. [1] we have found errata. They are composed of two parts: one part is for the body, which is also explained in our recently published book [2], while the other part is for the appendix, which is mainly a result of the corrections to Ref. [3]. They do not influence the result of the manuscript. Rather, the latter part provides us a new additional result: the Schrödinger equation described with the Hamiltonian |H^B| has been derived for the future state |∣B(t)⟩| via the Feynman path integral in the complex action theory. In the fifth line below Eq. (5.8), where |f(DD†)−1| should have been replaced with |(f(D)f(D)†)−1|, we have chosen |f(D)| such that |(P†)−1(f(D)f(D)†)−1P†=F(H^†)|, which is rewritten as |(f(D)f(D)†)−1=F(D†)|. However, this relation does not stand, because the left-hand side is Hermitian, while the right-hand side is not Hermitian. Accordingly, the expression |Q′=F(H^†)Q| below Eq. (5.8), which was introduced based on the above relation, has to be corrected. In addition, the next statement, “|F(H^†)Q≃F(H^eff†)Q| for the restricted subspace,” is not right. This is because, for any reasonable function |h| and any state |∣A(t)⟩=∑iai(t)∣λi⟩| that obeys the Schrödinger equation |iℏdtd∣A(t)⟩=H^∣A(t)⟩|, the following relation holds for large |t−TA|: |h(H^)∣A(t)⟩≃h(H^eff+iBΛA)∣A~(t)⟩≡h~(H^eff)∣A~(t)⟩|, where we have used the automatic Hermiticity mechanism and introduced |∣A~(t)⟩≡∑i∈Aai(t)∣λi⟩|, |ΛA≡∑i∈A∣λi⟩⟨λi∣Q|, and another function |h~| such that |h~(Reλi)=h(Reλi+iB)|. Similarly, the statement “|Q2=F(H^eff†)Q| for the restricted subspace” given in Eq. (5.6) has to be corrected. A |dt|-dependent normalization factor, say |α(dt)1|, should be inserted on the right-hand sides of Eq. (A.2) and of the first line of Eq. (A.4). The following sentence should be inserted after the sentence “|C| is an arbitrary |…| complex plane” below Eq. (A.2): “In addition, |α(dt)| is a |dt|-dependent normalization factor, which is properly fixed later.” The factor |m2πiℏdt| in the second line of Eq. (A.4) should be deleted. The following sentences should be inserted after the phrase “where |…| Eq. (3.7)” below Eq. (A.4): “Here we have taken |α(dt)=m2πiℏdt| so that both sides of Eq. (A.4) correspond to each other in the vanishing limit of |dt|. Then Eq. (A.4) is reduced to |∣ψ(t+dt)⟩=e−ℏiH^dt∣ψ(t)⟩|.” The next sentence, “Thus we have found that |…| Eq. (A.2),” below Eq. (A.4) should be replaced with “Thus we have derived the Schrödinger equation and found that |…| Eq.(A.2).” The following sentence should be added after the above replaced sentence: “Such a derivation of the Schrödinger equation is well known in the real action theory [4].” Factors |α(dt)∗1|, |α(−dt′)∗1|, and |α(−dt)1| should be inserted on the right-hand side of the equation in the second sentence of the last paragraph of the appendix, on the right-hand sides of Eqs (A.5) and (A.6), respectively. The second sentence below Eq. (A.6), “Indeed, |H^B| is given |…||H^†|,” should be replaced with “Indeed, we obtain the Schrödinger equation |∣B(t−dt)⟩=eℏiH^Bdt∣B(t)⟩|, where |H^B| is given |…||H^†|.”
In Ref. [1] we have found errata, which are mainly a result of the corrections to Ref. [2]. They do not influence the result of the manuscript. A |dt|-dependent normalization factor, say |α(dt)1|, should be inserted on the right-hand sides of Eq. (34) and of the first line of Eq. (37). The following sentence should be inserted after the sentence “|C| is an arbitrary ... complex plane” below Eq. (34): “In addition, |α(dt)| is a |dt|-dependent normalization factor, which is properly fixed later.” On the right-hand side of the first line of Eq. (39), |∫C′dξ| should be replaced with |α(dt)1∫C′dξ|. The factor |m2πiℏdt| in the second line of Eq. (39) should be deleted. The phrase “where |H^| is given by” below Eq. (39) should be replaced with the following phrase: “where we have taken |α(dt)=m2πiℏdt| so that both sides of Eq. (39) correspond to each other in the vanishing limit of |dt|, and |H^| is given by”. The following sentence should be inserted below Eq. (40): “Then Eq. (39) is reduced to |∣ψ(t+dt)⟩=e−ℏiH^dt∣ψ(t)⟩|.” The following sentence should be added after the last sentence of Sect. 3.1: “In addition, we have derived the Schrödinger equation. Such a derivation of the Schrödinger equation is well known in the RAT [3].”