Minimization of the Energy of the Nonrelativistic

One-Electron Pauli-Fierz Model over Quasifree States

Volker Bach, Sébastien Breteaux, Tim Tzaneteas

Received: February 14, 2013 Revised: April 30, 2013

Communicated by Heinz Siedentop

Abstract. In this article the existence of a minimizer for the energy for the nonrelativistic one-electron Pauli-Fierz model within the class of quasifree states is established. To this end it is shown that the minimum of the energy on quasifree states coincides with the minimum of the energy on pure quasifree states, where existence and uniqueness of a minimizer holds. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is given. We also express the Lagrange equation for the minimizer in terms of the generalized one particle density matrix of the pure quasifree state.

2010 Mathematics Subject Classification: 81V10, 81L05, 46T99 Keywords and Phrases: Pauli-Fierz model, Quasifree states, Mass renormalization, Minimization techniques

I Introduction and Main Results

We begin by introducing the mathematical model studied in this paper and mention some well-known results before we describe the main results of the paper.

I.1 The Hamiltonian

According to the Standard Model of Nonrelativistic Quantum Electrodynamics [4] the unitary time evolution of a free nonrelativistic particle coupled to the quantized radiation field is generated by the Hamiltonian

H̃_{g} := \frac{1}{2}(\frac{1}{i} ∇_{x} − 𝔸⃗(x⃗))^{2} + H_{f}  (I.1)

acting on the Hilbert space L^{2}(ℝ_{x}^{3}; 𝔉) of square-integrable functions with values in the photon Fock space

𝔉 := 𝔉_{+}(𝔥) := ⨁_{n=0}^{∞} 𝔉_{+}^{(n)}(𝔥).  (I.2)

Here 𝔉_{+}^{(0)}(𝔥) = ℂ · Ω is the vacuum sector and the n-photon sector 𝔉_{+}^{(n)}(𝔥) = S_{n}(𝔥^{⊗n}) is the subspace of totally symmetric vectors on the n-fold tensor product of the one-photon Hilbert space

𝔥 = {f⃗ ∈ L^{2}(S_{σ,Λ}; ℂ ⊗ ℝ^{3}) | ∀k⃗ ∈ S_{σ,Λ}, a.e. : k⃗ · f⃗(k⃗) = 0} (I.3)

of square-integrable, transversal vector fields which are supported in the momentum shell

S_{σ,Λ} := {k⃗ ∈ ℝ^{3} | σ ≤ |k⃗| ≤ Λ},  (I.4)

where 0 ≤ σ < Λ < ∞ are infrared and ultraviolet cutoffs, respectively. The condition k⃗·f⃗(k⃗) = 0 reflects our choice of gauge, namely, the Coulomb gauge. It is convenient to fix real polarization vectors ε⃗_{±}(k⃗) ∈ ℝ^{3} such that \{ε⃗_{+}(k⃗), ε⃗_{−}(k⃗), \frac{k⃗}{|k⃗|}\} ⊆ ℝ^{3} form a right-handed orthonormal basis (Dreibein) and replace (I.3) by

𝔥 = L^{2}(S_{σ,Λ} × ℤ_{2}),  (I.5)

with the understanding that f⃗(k⃗) = ε⃗_{+} f(k⃗, +) + ε⃗_{−} f(k⃗, −). In (I.1) the energy of the photon field is represented by

H_{f} = ∫ |k| a^{∗}(k) a(k) dk,  (I.6)

where ∫ f(k)dk := ∑_{τ=±} ∫_{S_{σ,Λ}} f(k⃗, τ) d^{3}k and {a(k), a^{∗}(k)}_{k∈S_{σ,Λ}×ℤ_{2}} are the usual boson creation and annihilation operators constituting a Fock representation of the CCR on 𝔉, i.e.,

[a(k) , a(k′)] = [a^{∗}(k) , a^{∗}(k′)] = 0,  (I.7)
[a(k) , a^{∗}(k′)] = δ(k − k′) 1, a(k)Ω = 0,  (I.8)

Minimization of the Energy of the Nonrelativistic … 1483

for all k, k′ ∈ S_{σ,Λ} × ℤ_{2}. The magnetic vector potential 𝔸⃗(x⃗) is given by

𝔸⃗(x⃗) = ∫ G⃗(k)(e^{−ik⃗·x⃗} a^{∗}(k) + e^{ik⃗·x⃗} a(k)) dk,  (I.9)

with k = (k⃗, τ) ∈ ℝ^{3} × ℤ_{2},

G⃗(k⃗, τ) := g ε⃗_{τ} (k⃗) |k⃗|^{−1/2},  (I.10)

and g ∈ ℝ being the coupling constant. In our units, the mass of the particle and the speed of light equal one, so the coupling constant is given as g = \frac{1}{4π} \sqrt{α}, with α ≈ 1/137 being Sommerfeld’s fine structure constant. The Hamiltonian H̃_{g} preserves (i.e., commutes with) the total momentum operator p⃗ = \frac{1}{i} ∇_{x} + P⃗_{f} of the system, where

P⃗_{f} = ∫ k⃗ a^{∗}(k) a(k) dk  (I.11)

is the photon field momentum. This fact allows us to eliminate the particle degree of freedom. More specifically, introducing the unitary operator

𝕌 : L^{2}(ℝ_{x}^{3}; 𝔉) → L^{2}(ℝ_{p}^{3}; 𝔉) , (𝕌Ψ)(p⃗) := ∫ e^{−ix⃗·(p⃗−P⃗_{f})}Ψ(x⃗) \frac{d^{3}x}{(2π)^{3/2}} , (I.12)

one finds that

𝕌 H̃_{g} 𝕌^{∗} = ∫ ^{⊕} H_{g,p⃗} d^{3}p,  (I.13)

where

H_{g,p⃗} = \frac{1}{2}(P⃗_{f} + 𝔸⃗(0) − p⃗)^{2} + H_{f}  (I.14)

is a self-adjoint operator on dom(H_{0,0}), the natural domain of H_{0,0} = \frac{1}{2} P⃗_{f}^{2} + H_{f}.

I.2 Ground State Energy and Bogolubov-Hartree-Fock Energy

Due to (I.13), all spectral properties of H̃_{g} are obtained from those of {H_{g,p⃗}}_{p⃗∈ℝ^{3}}. Of particular physical interest is the mass shell for fixed total momentum p⃗ ∈ ℝ^{3}, coupling constant g ≥ 0, and infrared and ultraviolet cutoffs 0 ≤ σ < Λ < ∞, i.e., the value of the ground state energy

E_{gs}(g, p⃗, σ, Λ) := inf σ[H_{g,p⃗}] ≥ 0  (I.15)

and the corresponding ground states (or approximate ground states).

We express the ground state energy in terms of density matrices with finite energy expectation value and accordingly introduce

𝔇̃𝔐 := {ρ ∈ L^{1}(𝔉) | ρ ≥ 0, Tr_{𝔉}[ρ] = 1, ρ H_{0,0}, H_{0,0} ρ ∈ L^{1}(𝔉)}, (I.16)

so that the Rayleigh-Ritz principle appears in the form

E_{gs}(g, p⃗) = inf {Tr_{𝔉}[ρ H_{g,p⃗}] | ρ ∈ 𝔇̃𝔐}.  (I.17)

Note that Tr_{𝔉}[ρ H_{g,p⃗}] = Tr_{𝔉}[ρ^{1−β} H_{g,p⃗} ρ^{β}], for all 0 ≤ β ≤ 1, due to our assumption ρH_{0,0}, H_{0,0}ρ ∈ L^{1}(𝔉).

The determination of E_{gs}(g, p⃗) and the corresponding ground state ρ_{gs}(g, p⃗) ∈ 𝔇̃𝔐 (provided the infimum is attained) is a difficult task. In this paper we rather study approximations to E_{gs}(g, p⃗) and ρ_{gs}(g, p⃗) that we borrow from the quantum mechanics of atoms and molecules, namely, the Bogolubov-Hartree-Fock (BHF) approximation. We define the BHF energy as

E_{BHF} (g, p⃗, σ, Λ) = inf {Tr_{𝔉}[ρ H_{g,p⃗}(σ, Λ)] | ρ ∈ 𝔔𝔉},  (I.18)

with corresponding BHF ground state(s) ρ_{BHF} (g, p⃗, σ, Λ) ∈ 𝔔𝔉, determined by

Tr_{𝔉}[ρ_{BHF} (g, p⃗, σ, Λ) H_{g,p⃗}(σ, Λ)] = E_{BHF} (g, p⃗, σ, Λ),  (I.19)

where

𝔔𝔉 := {ρ ∈ 𝔇𝔐 | ρ is quasifree} ⊆ 𝔇𝔐  (I.20)

denotes the subset of quasifree density matrices (of finite particle number; see Sect. II.1).

I.3 Results

Our first result in Theorem IV.5 is that the minimal energy expectation value for all quasifree density matrices 𝔔𝔉 is already obtained if the variation is restricted to pure quasifree density matrices 𝔭𝔔𝔉, i.e.,

E_{BHF} (g, p⃗, σ, Λ) := inf_{ρ∈𝔔𝔉} Tr[H_{g,p⃗} ρ] = inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] ,  (I.21)

see (I.20)and (II.40). The physical relevance of the minimization over (pure) quasifree density matrices is seen by the fact that it includes density matrices of the form

ρ_{sq} = |e^{i[(a^{∗})^{2}+a^{2}]} Ω⟩⟨e^{i[(a^{∗})^{2}+a^{2}]} Ω|,  (I.22)

where a ≡ a(f) for some one-photon state f. These are important states in quantum optics known as squeezed light.

The restriction to pure quasifree states has the great advantage that the latter have a very convenient parametrization of their reduced one-particle density matrix given in Proposition IV.12. This enables us to prove the existence and uniqueness of a (pure) quasifree minimizer (f_{g,p⃗}, γ_{g,p⃗}, α̃_{g,p⃗}) which minimizes the energy (f, γ, α̃) ↦ E_{g,p⃗}(f, γ, α̃) in Theorem VIII.3. The minimizer is characterized in Theorem VIII.8 by the Euler-Lagrange equations corresponding to E_{g,p⃗}. We obtain expansions of the minimizer and the corresponding minimal energy for small g and p⃗ in Theorem VIII.6. Our proof uses a convexity and coercivity argument and the assumption that |g|, |p⃗| ≤ C are smaller than a certain constant C ≡ C(σ, Λ) which, however, is not uniformly bounded as Λ → ∞ or σ → 0.

We also determine the minimizer in the case where the variation over pure quasifree density matrices is further restricted to coherent states introduced in (II.41). More precisely, we minimize the energy functional L^{2}(S_{σ,Λ} × ℤ_{2}) ∋ f ↦ E_{g,p⃗}(f, 0, 0) and obtain the existence of a minimizer f_{g,p⃗} and its uniqueness in Theorem VI.2 again by a fixed-point argument. Compared to the general case studied in Theorem VIII.3, our assumption for the minimizing coherent state is much milder, namely, that |p⃗| ≤ 1/3 and that g^{2} is small compared to 1/ ln[Λ + 2]. As our equations are fairly explicit in the coherent state case, we determine the leading orders in the expansions of the minimizer f_{g,p⃗} and of the minimal energy E_{g,p⃗}(f_{g,p⃗}, 0, 0) in powers of g and |p⃗|. In particular, the coefficient for the term proportional to |p⃗|^{2}, which gives the “renormalized electron mass for coherent states”, is computed in Proposition VII.1 and is found to agree with the first order expansion in α of the renormalized mass of the electron, as computed for example in [12]. Our result holds uniformly in σ → 0 but not in Λ → ∞.

Outline of the article In Section II we discuss density matrices, density matrices of finite particle number, pure density matrices and quasifree density matrices in greater detail. We introduce our notation to describe the second quantization framework in Section III. Section IV introduces two parametrizations of pure quasifree states and contains the proof of Theorem IV.5. The energy functional for a fixed value of the momentum p⃗ of the dressed electron is computed in Section V, and some positivity properties of the different parts of the energy are established. From Section VI on we tacitly assume that the coupling constant |g| > 0 is small. The energy is then minimized in the particular case of coherent states in Section VI, providing a first upper bound to the energy of the ground state and a proof of Theorem VI.2. We then turn in Section VIII to the problem of minimizing the energy over all pure quasifree states. The existence and uniqueness of a minimizer among the class of pure quasifree state is then proven in Section VIII.1 provided |p⃗| is small enough. The first terms of a perturbative expansion for small g and p⃗ of the energy at the minimizer is computed in Section VIII.2. Finally the Lagrange equations associated with the problem of minimization in the generalized one particle density matrix variables are presented in Section VIII.3.

II Density Matrices and Quasifree Density Matrices

We now further discuss density matrices on Fock space and in particular give more details about quasifree density matrices.

II.1 Density Matrices of Finite Particle Number

Recall that the ground state energy is obtained as

E_{gs}(g, p⃗) = inf {Tr_{𝔉}[ρ H_{g,p⃗}] | ρ ∈ 𝔇̃𝔐}.  (II.23)

It is not difficult to see that E_{gs}(g, p⃗) is already obtained as an infimum over all density matrices

𝔇𝔐 := {ρ ∈ 𝔇̃𝔐 | ρ N_{f}, N_{f} ρ ∈ L^{1}(𝔉)}  (II.24)

of finite photon number expectation value, where

N_{f} = ∫ a^{∗}(k) a(k) dk  (II.25)

is the photon number operator. Indeed, if σ > 0 then

H_{g,p⃗} ≥ H_{f} ≥ σ N_{f},  (II.26)

and 𝔇𝔐 = 𝔇̃𝔐 is automatic. Furthermore, if σ = 0 then it is not hard to see [4] that E_{gs}(g, p⃗, 0, Λ) = lim_{σ↘0} E_{gs}(g, p⃗, σ, Λ), by using the standard relative bound

∥𝔸⃗_{<σ}(0) ψ∥ ≤ O(σ) ∥(H_{f,<σ} + 1)^{1/2} ψ∥,  (II.27)

where 𝔸⃗_{<σ}(0) and H_{f,<σ} are the quantized magnetic vector potential and field energy, respectively, for momenta below σ. So, for all 0 ≤ σ < Λ < ∞, we have that

E_{gs}(g, p⃗, σ, Λ) = inf {Tr_{𝔉}[ρ H_{g,p⃗}(σ, Λ)] | ρ ∈ 𝔇𝔐}.  (II.28)

Indeed, if the infimum (II.28) is attained at ρ_{gs}(g, p⃗, σ, Λ) ∈ 𝔇𝔐 then we call ρ_{gs}(g, p⃗, σ, Λ) a ground state of H_{g,p⃗}(σ, Λ). Since 𝔇𝔐 is convex, we may restrict the density matrices in (II.28) to vary only over pure density matrices,

E_{gs}(g, p⃗, σ, Λ) = inf {Tr_{𝔉}[ρ H_{g,p⃗}(σ, Λ)] | ρ ∈ 𝔭𝔇𝔐},  (II.29)

where pure density matrices are those of rank one,

{ \atop 𝔭𝔇𝔐} := \{ρ ∈ 𝔇̃𝔐 | ∃Ψ ∈ 𝔉, ∥Ψ∥ = 1 : ρ = |Ψ⟩⟨Ψ|\},  (II.30)

and

𝔭𝔇𝔐 := 𝔇𝔐 ∩ { \atop 𝔭𝔇𝔐}.  (II.31)

Another class of states that play an important role in our work is the set of centered density matrices,

𝔠𝔇𝔐 := {ρ ∈ 𝔇𝔐 | ∀f ∈ 𝔥 : Tr_{𝔉}[ρ a^{∗}(f)] = 0}.  (II.32)

II.2 Quasifree Density Matrices

A density matrix ρ ∈ 𝔇𝔐 is called quasifree, if there exist f_{ρ} ∈ 𝔥, a symplectomorphism T_{ρ} (see Definition III.6) and a positive, self-adjoint operator h_{ρ} = h_{ρ}^{∗} ≥ 0 on 𝔥 such that

⟨W (\sqrt{2}f/i)⟩_{ρ} := Tr_{𝔉}[ρ W (\sqrt{2}f/i)]
= exp [2i Im⟨f_{ρ}|f⟩ − \frac{1}{2}⟨T_{ρ}f|(1 + h_{ρ})T_{ρ}f⟩],  (II.33)

for all f ∈ 𝔥, where

W (f) := exp [iΦ(f)] := exp [\frac{i}{\sqrt{2}} (a^{∗}(f) + a(f))]  (II.34)

denotes the Weyl operator corresponding to f and we write expectation values w.r.t. the density matrix ρ as ⟨·⟩_{ρ}. There are several important facts about quasifree density matrices, which do not hold true for general density matrices in 𝔇𝔐. See, e.g., [5, 13, 7, 8]. The first such fact is that if ρ ∈ 𝔔𝔉 is a quasifree density matrix then so is W (g)^{∗}ρW (g) ∈ 𝔔𝔉, for any g ∈ 𝔥, as follows from the Weyl commutation relations

∀ f, g ∈ 𝔥 :  W (f) W (g) = e^{− \frac{i}{2} Im⟨f|g⟩} W (f + g).  (II.35)

Choosing g := −i\sqrt{2}f_{ρ}, we find that ρ̃:=W (−i\sqrt{2}f_{ρ})^{∗} ρ W (−i\sqrt{2}f_{ρ}) is a centered quasifree density matrix, i.e.,

ρ̃:=W (\sqrt{2}f_{ρ}/i)^{∗} ρ W (\sqrt{2}f_{ρ}/i) ∈ 𝔠𝔔𝔉 := 𝔔𝔉 ∩ 𝔠𝔇𝔐.  (II.36)

A characterization of centered quasifree density matrices is given in Appendix VII. A second important fact is that any quasifree state ρ ∈ 𝔔𝔉 is completely determined by its one-point function ⟨a(φ)⟩_{ρ}= ⟨φ, f_{ρ}⟩ and its two-point function (one-particle reduced density matrix)

Γ[γ_{ρ}, α̃_{ρ}] := (\begin{matrix}γ_{ρ} & α̃_{ρ} \\ α̃_{ρ}^{∗} & 1 + J γ_{ρ} J \end{matrix} ) ∈ B(𝔥 ⊕ 𝔥),  (II.37)

where the operators γ_{ρ}, α̃_{ρ} ∈ B(𝔥) are defined as

⟨φ, γ_{ρ} ψ⟩ := ⟨a^{∗}(ψ) a(φ)⟩ρ̃ and ⟨φ, α̃_{ρ} ψ⟩ := ⟨a(φ) a(J ψ)⟩ρ̃, (II.38)

and J : 𝔥 → 𝔥 is a conjugation. See Definition IV.8and Remark IV.10. The positivity of the density matrix ρ implies that Γ[γ_{ρ}, α̃_{ρ}] ≥ 0 and, in particular, γ_{ρ} ≥ 0, too. Moreover, the additional finiteness of the particle number expectation value, which distinguishes 𝔇𝔐 from 𝔇̃𝔐, ensures that γ_{ρ} ∈ L^{1}(𝔥) is trace-class, namely,

Tr_{𝔥}[γ_{ρ}] = ⟨N_{f}⟩_{ρ} < ∞,  (II.39)

and that α̃_{ρ} ∈ L^{2}(𝔥) is Hilbert-Schmidt. Similar to (II.30)-(II.31), we introduce pure quasifree density matrices,

𝔭𝔔𝔉 := 𝔔𝔉 ∩ { \atop 𝔭𝔇𝔐}.  (II.40)

A subset of 𝔭𝔔𝔉 of special interest is given by coherent states, i.e., pure quasifree states of the form |W (−i\sqrt{2}f)Ω⟩⟨W (−i\sqrt{2}f)Ω|, which we collect in

coh := \{|W (−i\sqrt{2}f)Ω⟩⟨W (−i\sqrt{2}f)Ω| | f ∈ 𝔥\}.  (II.41)

For these, γ_{ρ} = α̃_{ρ} = 0. Conversely, if γ ∈ L_{+}^{1}(𝔥) is a positive trace-class operator and α̃ ∈ L^{2}(𝔥) is a Hilbert-Schmidt operator such that Γ[γ, α̃] ≥ 0 is positive then there exists a unique centered quasifree density matrix ρ ∈ 𝔠𝔔𝔉 such that γ = γ_{ρ} and α = α_{ρ} are its one-particle reduced density matrices. Summarizing these two relations, the set 𝔔𝔉 of quasifree density matrices is in oneto-one correspondence to the convex set

1−pdm := {(f, γ, α̃) ∈ 𝔥 ⊕ L_{+}^{1}(𝔥) ⊕ L^{2}(𝔥) | Γ[γ, α̃] ≥ 0}.  (II.42)

Note that coherent states correspond to elements of 1−pdm of the form (f, 0, 0). Next, we observe in accordance with (II.42) that, if ρ ∈ 𝔔𝔉 is quasifree then its energy expectation value ⟨H_{g,p⃗}⟩_{ρ} is a functional of (f_{ρ}, γ_{ρ}, α̃_{ρ}), namely,

⟨H_{g,p⃗}⟩_{ρ} = E_{g,p⃗}(f_{ρ}, γ_{ρ}, α̃_{ρ}),  (II.43)

where, as shown in Section V,

E_{g,p⃗}(f, γ, α̃) = \frac{1}{2}\{(Tr[γk⃗] + ⟨f, k⃗f⟩ + 2Re(⟨f, G⃗⟩) − p⃗)^{·2}
+ Tr[γk⃗ · γk⃗] + Tr[α̃^{∗}k⃗ · α̃k⃗] + Tr[|k⃗|^{2}γ]

+ 2Re⟨\overline{G⃗ + k⃗f},·α̃(G⃗ + k⃗f)⟩ + ⟨G⃗ + k⃗f, ·(2γ + 1)(G⃗ + k⃗f)⟩\}

+ Tr[γ|k⃗|] + ⟨f, |k⃗|f⟩ ,  (II.44)

where |a⃗⟩ · ⟨a⃗| = ∑_{j=1}^{3} |a_{j}⟩⟨a_{j}|. Note that in this expression k⃗ denotes the triple of multiplication operators (k_{1}, k_{3}, k_{3}). We also use the same notation k⃗ for the momentum variable, the meaning being clear from the context.

III Second Quantization

In this section we describe in detail the second quantization framework we use, and in particular we explain the notation introduced below, which may be unfamiliar to some readers. In what follows 𝔥 will denote a ℂ-Hilbert space with a scalar product ℂ-linear in the right variable and ℂ-antilinear in the left variable. Let B(X; Y) be the space of bounded operators between two Banach spaces X and Y, and L^{1}(𝔥) the space of trace class operators on 𝔥. Given two ℂ-Hilbert spaces (𝔥_{j}, ⟨·, ·⟩_{j}), j = 1, 2 and a bounded linear operator A : 𝔥_{1} → 𝔥_{2}, set A^{∗} : 𝔥_{2} → 𝔥_{1} to be the operator such that

∀z_{1} ∈ 𝔥_{1}, z_{2} ∈ 𝔥_{2}, ⟨z_{2}, Az_{1}⟩_{2} = \overline{⟨z_{1}, A^{∗}z_{2}⟩_{1}} ,

and ReA := \frac{1}{2}(A ⊕ A^{∗}), ImA := \frac{1}{2i}(A ⊕ (−A^{∗})) ∈ B(𝔥_{1}, 𝔥_{2}) ⊕ B(𝔥_{2}, 𝔥_{1}).

Example III.1. For z, z′ ∈ 𝔥,

⟨z, z′⟩ = z^{∗}z′ .

The adjoint of a bounded operator A on 𝔥 is A^{∗}.

Remark III.2. This notation applies in particular to one-particle vectors f ∈ 𝔥 identified with linear applications from ℂ to 𝔥 or to two-particle vectors α ∈ 𝔥^{⊗2} identified with linear applications from ℂ to 𝔥^{⊗2}. For this purpose a slight generalization of the Dirac notation with bras and kets would have been sufficient, but we would like to emphasize that in some situations, like in Equation (IV.58), it is natural to apply this operation to more general objects. For vectors and operators in a finite dimensional space 𝔥 this notation is consistent with the usual notation on matrices.

The symmetrization operator S_{n} on 𝔥^{⊗n} is the orthogonal projection defined by

S_{n}(z_{1} ⊗ ⋯ ⊗ z_{n}) = \frac{1}{n!} ∑_{π∈𝔖_{n}} z_{π_{1}} ⊗ ⋯ ⊗ z_{π_{n}}

and extension by linearity and continuity. The symmetric tensor product for vectors is z_{1} ∨ z_{2} = S_{n_{1}+n_{2}}(z_{1} ⊗ z_{2}) and more generally for operators is A_{1} ∨ A_{2} = S_{q_{1}+q_{2}} ∘ (A_{1} ⊗ A_{2}) ∘ S_{p_{1}+p_{2}} for A_{j} ∈ B(𝔥^{⊗p_{j}}; 𝔥^{⊗q_{j}}). We set

𝔥^{∨n} := S_{n}𝔥^{⊗n}, B^{p,q} := B(𝔥^{⊗p}; 𝔥^{⊗q}).

Definition III.3. The symmetric Fock space on a Hilbert space 𝔥 is defined to be

𝔉_{+}(𝔥) := ⨁_{n=0}^{∞} 𝔥^{∨n} ,

where 𝔥^{∨0} := ℂΩ, Ω being the normalized vacuum vector.

For a linear operator C on 𝔥 such that ∥C∥_{B(𝔥)} ≤ 1, let Γ(C) defined on each 𝔥^{∨n} by C^{∨n} and extended by continuity to the symmetric Fock space on 𝔥. For an operator A on 𝔥, the second quantization dΓ(A) of A is defined on each 𝔥^{∨n} by

dΓ(A)|𝔥_{∨n} = n 1_{𝔥}^{∨n−1} ∨ A

and extended by linearity to ⨁_{n≥0}^{alg} 𝔥^{∨n}. The number operator is N_{f} = dΓ(1_{𝔥}). For a vector f in 𝔥, the creation and annihilation operators in f are the linear operators such that a(f)Ω = 0, a^{∗}(f)Ω = f, and

a(f)g^{∨n} = \sqrt{n}(f^{∗}g) g^{∨n−1} , and a^{∗}(f)g^{∨n} = \sqrt{n + 1}f ∨ g^{∨n} ,  (III.45)

for all g ∈ 𝔥. By the polarization identity

∀g_{1}, …, g_{n}, g_{1} ∨ ⋯ ∨ g_{n} = \frac{1}{2^{n}n!} ∑_{ε_{i}=±1} ε_{1} ⋯ ε_{n}( ∑_{j=1}^{n} ε_{j}g_{j})^{⊗n}

Eq. (III.45) extends to 𝔥^{∨n} and hence also to ⨁_{n≥0}^{alg} 𝔥^{∨n}. They satisfy the canonical commutation relations [a(f), a^{∗}(g)] = f^{∗}g, [a(f), a(g)] = [a^{∗}(f), a^{∗}(g)] = 0. The self-adjoint field operator associated to f is Φ(f) = \frac{1}{\sqrt{2}}(a^{∗}(f)+a(f)). For more details on the second quantization see the book of Berezin [6].

A dot “·” denotes an operation analogous to the scalar product in ℝ^{3}. For every two objects a⃗ = (a_{1}, a_{2}, a_{3}) and b⃗ = (b_{1}, b_{2}, b_{3}) with three components such that the products a_{j}b_{j} are well defined

a⃗ · b⃗ := ∑_{j=1}^{3} a_{j}b_{j} .

Example III.4. With p⃗ ∈ ℝ^{3}, G⃗ ∈ 𝔥^{3}, k⃗ ∈ (B^{1,1})^{3}

p^{·2} = ∑_{j=1}^{3} p_{j}^{2} ∈ ℝ ,  k⃗ · p⃗ = ∑_{j=1}^{3} p_{j}k_{j} ∈ B^{1,1}, p⃗ · G⃗ = ∑_{j=1}^{3} p_{j}G_{j} ∈ 𝔥 ,

k⃗^{·2} = ∑_{j=1}^{3} k_{j}^{2} ∈ B^{1,1},  k⃗ · G⃗ = ∑_{j=1}^{3} k_{j}G_{j} ∈ 𝔥 , G⃗^{∗} · k⃗ = ∑_{j=1}^{3} G_{j}^{∗}k_{j} ∈ 𝔥^{∗},

G⃗ · G⃗^{∗} = ∑_{j=1}^{3} G_{j}G_{j}^{∗} ∈ B^{1,1}, G⃗^{∗}· G⃗ = ∑_{j=1}^{3} G_{j}^{∗}G_{j} ∈ ℂ ,

where for an object with three components a⃗ = (a_{1}, a_{2}, a_{3}) such that a_{j}^{∗} is welldefined, a^{∗} := (a_{1}^{∗}, a_{2}^{∗}, a_{3}^{∗}). We sometimes use the notation p^{·2} = |p⃗|^{2}, or k⃗^{·2} = |k⃗|^{2}. And with another product, such as the symmetric tensor product ∨,

k⃗^{·∨2} = ∑_{j=1}^{3} k_{j}^{∨2} ∈ B^{2,2} ,  k⃗ ·∨G⃗ = ∑_{j=1}^{3} k_{j} ∨ G_{j} ∈ B^{2,3} .

Recall that the Weyl operators are the unitary operators W (f) = exp(iΦ(f)) satisfying the relations

∀z_{1}, z_{2} ∈ 𝔥 :  W (z_{1})W (z_{2}) = e^{− \frac{i}{2} Im(z_{1}^{∗}z_{2})}W (z_{1} + z_{1}) ,  (III.46)
∀z ∈ 𝔥 :  W (−i\sqrt{2}z)Ω = e^{− \frac{|z|^{2}}{2}} ∑_{n=0}^{∞} \frac{z^{∨n}}{\sqrt{n!}} .  (III.47)

We now introduce the usual parametrization of coherent states by vectors in 𝔥 and of Bogolubov transformations by symplectomorphisms.

Definition III.5. The coherent vectors are the vectors of the form

E_{z} = W (−i\sqrt{2}z)Ω

for some z ∈ 𝔥 and the coherent states are the states of the form

|E_{z}⟩⟨E_{z}| .

Definition III.6. A symplectomorphism T for the symplectic form Im⟨·, ·⟩ on a ℂ-Hilbert space 𝔥 is a continuous ℝ-linear automorphism on 𝔥 which preserves this symplectic form, i.e.,

∀z_{1}, z_{2} ∈ 𝔥 : Im⟨T z_{1}, T z_{2}⟩ = Im⟨z_{1}, z_{2}⟩ .

A symplectomorphism T is implementable if there is a unitary operator 𝕌_{T} on 𝔉_{+}(𝔥) such that

∀z ∈ 𝔥 , 𝕌_{T} W (z)𝕌_{T}^{∗} = W (T z) .

In this case 𝕌_{T} is a Bogolubov transformation corresponding to T.

We recall a well-known parametrization, in the spirit of the polar decomposition, of implementable symplectomorphisms.

Proposition III.7. The set of implementable symplectomorphisms is the set of operators

T = u exp[r̂] = u ∑_{n=0}^{∞} \frac{1}{n!}r̂^{n} ,

where u is an isometry and r̂ is an antilinear operator, self-adjoint in the sense that ∀z, z′ ∈ 𝔥, ⟨z, r̂z′⟩ = ⟨z′, r̂z⟩, and Hilbert-Schmidt in the sense that the positive operator r̂^{2} is trace-class. Equivalently, there exist a Hilbert basis (φ_{j})_{j∈ℕ} of 𝔥 and (r̂_{i,j})_{i,j} ∈ ℓ^{2}(ℕ^{2}; ℂ) such that

r̂ = ∑_{i,j=1}^{∞} r̂_{i,j} ⟨·, φ_{j}⟩ φ_{i} , ∀i, j ∈ ℕ^{2} : r̂_{i,j} = r̂_{j,i} , and ∑_{i,j=1}^{∞} |r̂_{i,j}|^{2} < ∞.

Proof. On the one hand, every operator of the form T = u exp[r̂] with u a unitary operator and r̂ a self-adjoint antilinear operator is a symplectomorphism. Since a unitary operator is a symplectomorphism, and the set of symplectomorphisms is a group for the composition, it is enough to prove that exp[r̂] is a symplectomorphism. It is indeed the case since, for all z, z′ in 𝔥,

Im⟨er̂z, er̂z′⟩ = Im⟨er̂z, cosh(r̂)z′⟩ + Im⟨er̂z, sinh(r̂)z′⟩
= Im⟨cosh(r̂)er̂z, z′⟩ + Im⟨z′, sinh(r̂)er̂z⟩
= Im⟨cosh(r̂)er̂z, z′⟩ − Im⟨sinh(r̂)er̂z, z′⟩
= Im⟨e^{−r̂}er̂z, z′⟩ .

The implementability condition is then satisfied if we suppose r̂ to be Hilbert-Schmidt. On the other hand, to get exactly this formulation we give the step to go from the result given in Appendix A in [9] to the decomposition in Proposition III.7. In [9] an implementable symplectomorphism is decomposed as

T = ue^{cr̃} ,  (III.48)

where u is a unitary operator, c is a conjugation and r̃ is a Hilbert-Schmidt, selfadjoint, non-negative operator commuting with c. It is then enough to set r̂ = cr̃ to get the expected decomposition. To check the self-adjointness of r̂, observe that, for all z, z′ in 𝔥,

⟨z′, r̂z⟩ = ⟨z′, r̃cz⟩ = ⟨r̃z′, cz⟩ = ⟨z, cr̃z′⟩ = ⟨z, r̂z′⟩ .

For the convenience of the reader we recall the main steps to obtain the decomposition in Eq. (III.48). First decompose T in its ℂ-linear and antilinear parts, T = L+A, then write the polar decomposition L = u|L|. It is then enough to prove that |L|+u^{∗}A is of the form e^{cr̃}. >From certain properties of symplectomorphisms (also recalled in [9]) it follows that the antilinear operator u^{∗}A is self-adjoint and |L|^{2} + 1_{𝔥} = (u^{∗}A)^{2}. A decomposition of the positive trace class operator (u^{∗}A)^{2} = ∑_{j} λ_{j}^{2}e_{j}e_{j}^{∗} with e_{j} an orthonormal basis of 𝔥 yields |L| = ∑_{j}(1 + λ_{j}^{2})^{1/2}e_{j}e_{j}^{∗}. Using that λ_{j} → 0 one can study the operator |L| and u^{∗}A on the finite dimensional subspaces ker(|L| − μ1_{𝔥}) which are invariant under u^{∗}A. It is then enough to prove that for a ℂ-antilinear self-adjoint operator f such that ff^{∗} = λ^{2} on a finite dimensional space, there is an orthonormal basis {φ_{k}}_{k} such that f(φ_{k}) = λφ_{k}. The conjugation is then defined such that c(∑ β_{k}φ_{k}) = ∑ β̄_{k}φ_{k} and r̃ = sinh^{−1}(λ_{j})1 on that subspace.

IV Pure Quasifree States

In this section we give a characterization of quasifree states and use this to show that the infimum of the energy over quasifree states is equal to the infimum of the energy over pure quasifree states. This result has been generalized to a wider class of Hamiltonians and also to the case of fermion Fock space in [2].

IV.1 From Quasifree States to Pure Quasifree States

Let 𝔥 be the ℂ-Hilbert space L^{2}(S_{σ,Λ} × ℤ_{2}). We make use of the following characterization of quasifree density matrices.

Lemma IV.1. The set of quasifree density matrices and pure quasifree density matrices, respectively, of finite photon number expectation value can be characterized by

𝔔𝔉 = 𝔇𝔐 ⋂ \{W (−i\sqrt{2}f)𝕌^{∗} \frac{Γ(C)}{Tr[Γ(C)]}𝕌W (−i\sqrt{2}f)^{∗}
| f ∈ 𝔥, 𝕌 a Bogolubov transformation,
C ∈ L^{1}(𝔥), C ≥ 0, ∥C∥_{B(𝔥)} < 1}

𝔭𝔔𝔉 = 𝔇𝔐 ⋂ \{W (−i\sqrt{2}f)𝕌^{∗}|Ω⟩⟨Ω|𝕌W (−i\sqrt{2}f)^{∗}
| f ∈ 𝔥, 𝕌 a Bogolubov transformation}

Proof. We only sketch the argument, details can be found in [6, 13]. It is not difficult to see that any density matrix of the form W (−i\sqrt{2}f)𝕌^{∗} \frac{Γ(C)}{Tr[Γ(C)]}𝕌W (−i\sqrt{2}f)^{∗} is indeed quasifree. Conversely, if ρ ∈ 𝔔𝔉 is a quasifree density matrix then it is fully characterized by its one-point function f_{ρ} ∈ 𝔥 and two-point functions (γ_{ρ}, α̃_{ρ}). Moreover, W (−i\sqrt{2}f_{ρ})^{∗} ρ W (−i\sqrt{2}f_{ρ}) ∈ 𝔠𝔔𝔉 is a centered quasifree density matrix with the same one-particle density matrix, that is, the density matrix W (−i\sqrt{2}f_{ρ})^{∗} ρ W (−i\sqrt{2}f_{ρ}) corresponds to (0, γ_{ρ} − f_{ρ}f_{ρ}^{∗}, α̃_{ρ} − f_{ρ}f̄_{ρ}^{∗}). Obviously, γ_{ρ} − f_{ρ}f_{ρ}^{∗} is again trace-class and α̃_{ρ} − f_{ρ}f̄_{ρ}^{∗} is Hilbert-Schmidt. Now, we use that there exists a Bogolubov transformation 𝕌 which eliminates α̃_{ρ}, i.e., 𝕌^{∗}W (\sqrt{2}f_{ρ}/i)^{∗} ρ W (\sqrt{2}f_{ρ}/i)𝕌 corresponds to (0, γ̃_{ρ}, 0). While this is the only nontrivial step of the proof, we note that if 𝕌 is characterized by u and v as in Lemma IV.2 then there is an involved, but explicit formula that determines u and v. Again γ̃_{ρ} is trace-class because the photon number operator N_{f} transforms under 𝕌^{∗} to itself plus lower order corrections, 𝕌^{∗} N_{f} 𝕌 = N_{f} + O(N_{f}^{1/2} + 1). Finally, it is easy to see that (0, γ̃_{ρ}, 0) corresponds to the quasifree density matrix Γ(C_{ρ})/Tr[Γ(C_{ρ})] with C_{ρ} := γ̃_{ρ}(1 + γ̃_{ρ})^{−1}. Following these steps we finally obtain

ρ = W (f_{ρ})𝕌\frac{Γ(C_{ρ})}{Tr[Γ(C_{ρ})]}𝕌^{∗}W (f_{ρ})^{∗} ,

as asserted. The additional characterization of pure quasifree density matrices is obvious.

Lemma IV.2. Let 𝕌 ∈ B(𝔉) be a unitary operator. The following statements are equivalent:

𝕌 ∈ B(𝔉) is a Bogolubov transformation;  (IV.49)
⇔ ∃T implementable symplectomorphism,  (IV.50)

𝕌 = Ũ_{T} , Ũ_{T} W (f)Ũ^{∗} = W (T f).

⇔ ∃u ∈ B(𝔥), v ∈ L^{2}(𝔥) ∀f ∈ 𝔥 :  (IV.51)

𝕌a^{∗}(f)𝕌^{∗} = a^{∗}(uf) + a(J vJ f);

⇔ 𝕌 = exp(iH), where H = H^{∗} is a semibounded operator,  (IV.52)

quadratic in a^{∗} and a and without linear term.

Proof. Again, we only sketch the argument. First note that (IV.49)⇔(IV.50) is the definition of a Bogolubov transformation. Secondly, Ũ_{T} W (f)Ũ_{T}^{∗} = W (T f) is equivalent to Ũ_{T} Φ(f)Ũ_{T}^{∗} = Φ(T f). Hence, using that a^{∗}(f) = \frac{1}{\sqrt{2}}[Φ(f) − iΦ(if)] and a(f) = \frac{1}{\sqrt{2}}[Φ(f) + iΦ(if)] we obtain the equivalence (IV.50)⇔(IV.51). Thirdly, setting U_{λ} = exp(iλH) and a_{λ}^{∗}(f) := U_{λ}a^{∗}(f)U_{λ}^{∗}, we observe that ∂_{λ}a_{λ}^{∗}(f) = i[H, a_{λ}^{∗}(f)]. Furthermore, [H, a_{λ}^{∗}(f)] is linear in a^{∗} and a if, and only if, H is quadratic in a^{∗} and a. Solving this linear differential equation, we finally obtain (IV.51)⇔(IV.52).

As a consequence, the class of quasifree states (resp. centered quasifree states) is invariant under conjugation by Weyl transformations and Bogolubov transformations (resp. Bogolubov transformations):

Lemma IV.3. For all Bogolubov transformations 𝕌 and all g ∈ 𝔥:

W (g)𝕌 𝔔𝔉 𝕌^{∗}W (g)^{∗} = 𝔔𝔉,  (IV.53)
𝕌 𝔠𝔔𝔉 𝕌^{∗} = 𝔠𝔔𝔉.  (IV.54)

Remark IV.4. A pure quasifree state is a particular case of quasifree state with C = 0, that is Γ(C) = |Ω⟩⟨Ω|.

We come to the main result of this section.

Theorem IV.5. Let 0 ≤ σ < Λ < ∞, g ∈ ℝ and p⃗ ∈ ℝ^{3}. Minimizing the energy over quasifree states is the same as minimizing the energy over pure quasifree states, i.e.,

E_{BHF} (g, p⃗, σ, Λ) := inf_{ρ∈𝔔𝔉} Tr[H_{g,p⃗} ρ] = inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] .

For the proof of Theorem IV.5 we derive a couple of preparatory lemmata.

Proposition IV.6. Let C be a non-negative operator on 𝔥, then

{Tr[Γ(C)] < ∞}  ⇔  {C ∈ L^{1}(𝔥) and ∥C∥_{B(𝔥)} < 1} .

In this case Tr[Γ(C)] = det(1 − C)^{−1}. (We refrain from defining the determinant.) For the direction ⇐ the non-negativity assumption is not necessary.

Proof. Let us decompose 𝔥 = ⨁_{j≥0} ℂe_{j} where C = ∑ c_{j}e_{j}e_{j}^{∗} with (e_{j})_{j≥0} an orthonormal basis of 𝔥. Then 𝔉_{+}(𝔥) = ⨂_{j≥0} 𝔉_{+}(ℂe_{j}) and

Tr[Γ(C)] = Tr[⨂_{j≥0} Γ(c_{j})] = ∏_{j≥0} Tr[Γ(c_{j})] = ∏_{j≥0} \frac{1}{1 − c_{j}}

and the infinite product converges exactly when C ∈ L^{1}(𝔥) and ∥C∥_{B(𝔥)} < 1.

Lemma IV.7. Suppose 𝔥_{d} is of dimension d < ∞. Then, for any non-negative operator C_{d} ≠ 0 such that C_{d} ∈ L^{1}(𝔥_{d}) and ∥C_{d}∥_{B(𝔥_{d})} < 1, there exist a non-negative measure μ_{d} (depending on C) of mass one on 𝔥_{d} and a family {ρ_{d}(z_{d})}_{z_{d}∈𝔥_{d}} of pure quasifree states such that

\frac{Γ(C)}{Tr[Γ(C)]} = ∫𝔥_{d} ρ_{d}(z_{d}) dμ_{d}(z_{d}) .

Proof. In finite dimension d we can use a resolution of the identity with coherent states (see, e.g., [6])

1_{Γ(𝔥_{d})} = ∫𝔥_{d} |E_{z_{d}}⟩⟨E_{z_{d}}| \frac{dz_{d}}{π^{d}} ,

where 𝔥_{d} is identified with ℂ^{d} and dz_{d} = dx_{d} dy_{d}, z_{d} = x_{d} + iy_{d}. Using Equation (III.47) we get

Γ(C) = ∫𝔥_{d} Γ(C^{1/2})|E_{z_{d}}⟩⟨E_{z_{d}}|Γ(C^{1/2}) \frac{dz_{d}}{π^{d}}

= ∫𝔥_{d} |E_{C^{1/2}z_{d}}⟩⟨E_{C^{1/2}z_{d}}| \frac{exp(|C^{1/2}z_{d}|^{2} − |z_{d}|^{2})dz_{d}}{π^{d}} .

The measure dμ_{d}(z_{d}) = π^{−d} exp(|C^{1/2}z_{d}|^{2} − |z_{d}|^{2})dz_{d}/Tr[Γ(C)] has mass one. Indeed

∫𝔥_{d} exp(−z_{d}^{∗}(1𝔥_{d} − C)z_{d})\frac{dz_{d}}{π^{d}} = ∏_{j=1}^{d} ∫_{ℝ^{2}} exp(−(1 − c_{j})(x^{2} + y^{2}))\frac{dx dy}{π}

= ∏_{j=1}^{d} \frac{1}{1 − c_{j}} = Tr[Γ(C)],

where C = ∑_{j=1}^{d} c_{j}e_{j}e_{j}^{∗} with (e_{j})_{j=1}^{d} an orthonormal basis of 𝔥_{d}.

Proof of Theorem IV.5. The inclusion 𝔭𝔔𝔉 ⊂ 𝔔𝔉 implies that

inf_{ρ∈𝔔𝔉} Tr[H_{g,p⃗} ρ] ≤ inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] ,

and it is hence enough to prove for any quasifree state

ρ_{qf} = W (−i\sqrt{2}f) 𝕌_{T}^{∗} \frac{Γ(C)}{Tr[Γ(C)]} 𝕌_{T} W (−i\sqrt{2}f)^{∗} ,

that the inequality

Tr[H_{g,p⃗} ρ_{qf}] ≥ inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ]

holds true. The operator C is decomposed as C = ∑_{j≥0} c_{j}e_{j}e_{j}^{∗} where (e_{j}) is an orthonormal basis of the Hilbert space 𝔥 and c_{j} ≥ 0. Let C_{d} = ∑_{j≤d} c_{j}e_{j}e_{j}^{∗}. Let

ρ_{qf,d} = W (−i\sqrt{2}f) 𝕌_{T}^{∗} \frac{Γ(C_{d})}{Tr[Γ(C_{d})]} 𝕌_{T} W (−i\sqrt{2}f)^{∗} ,

then using Lemma IV.7with 𝔥_{d} = ⨁_{j≤d} ℂe_{j}, 𝔉_{+}𝔥 = 𝔉_{+}(𝔥_{d} ⊕𝔥_{d}^{⊥}) ≅ 𝔉_{+}𝔥_{d} ⊗𝔉_{+}𝔥_{d}^{⊥} and the extension of the operator Γ(C_{d}) on 𝔉_{+}𝔥_{d} to 𝔉_{+}𝔥_{d} ⊗ 𝔉_{+}𝔥_{d}^{⊥} by Γ(C_{d}) ⊗ (|Ω𝔥_{⊥_{d}}⟩⟨Ω𝔥_{⊥_{d}}|) (which we still denote by Γ(C_{d})), we obtain

ρ_{qf,d} = ∫𝔥_{d} ρ_{d}(z_{d}) dμ_{d}(z_{d}) ,

where ρ_{d}(z_{d}) are pure quasifree states and the μ_{d} are non-negative measures with mass one. Note that

ν_{d} := \frac{Tr[Γ(C_{d})]}{Tr[Γ(C)]} = ∏_{j>d}(1 − c_{j}) ↗ 1 ,

as d → ∞. Further note that ρ_{qf} ≥ ν_{d} ρ_{qf,d}, for any d ∈ ℕ, since Γ(C) ≥ Γ(C_{d}). Thus

Tr[H_{g,p⃗} ρ_{qf}] ≥ Tr[H_{g,p⃗} ν_{d}ρ_{qf,d}]

= ν_{d} ∫𝔥_{d} Tr[H_{g,p⃗} ρ_{d}(z_{d})] dμ_{d}(z_{d})

≥ ν_{d} inf_{z_{d}∈𝔥_{d}} Tr[H_{g,p⃗} ρ_{d}(z_{d})]

≥ ν_{d} inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] ,

for all d ∈ ℕ, and in the limit d → ∞, we obtain

Tr[H_{g,p⃗} ρ_{qf}] ≥ lim_{d→∞}{ν_{d}} inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] = inf_{ρ∈𝔭𝔔𝔉} Tr[H_{g,p⃗} ρ] .

IV.2 Pure Quasifree States and their One-Particle Density Matrices

We now define reduced density matrices ρ^{p,q} resulting as marginals from a given density matrix ρ on Fock space and derive a convenient parametrization for them in case ρ is pure and quasifree. We also recall a characterization of the admissible one-particle density matrices for pure quasifree states. Let 𝔥 be a ℂ-Hilbert space.

Definition IV.8. Let ρ ∈ 𝔇𝔐 be a density matrix on the bosonic Fock space 𝔉_{+}(𝔥)

over 𝔥. If Tr[ρNf^{\frac{p+q}{2}} ] < ∞, we define ρ^{p,q} ∈ B^{p,q}(𝔥) through

∀φ, ψ ∈ 𝔥 , ψ^{∗∨p}ρ^{p,q}φ^{∨q} = Tr[a^{∗}(φ)^{q}a(ψ)^{p}ρ] .

We single out

f = ρ^{0,1} ∈ B^{0,1} ≅ 𝔥 ,

i.e., f_{ρ} ∈ 𝔥 is the unique vector such that Tr[a(ψ) ρ] = ψ^{∗}f_{ρ}, for all ψ ∈ 𝔥. Furthermore, with ρ̃ = W (\sqrt{2}f_{ρ}/i)^{∗}ρW (\sqrt{2}f_{ρ}/i), the matrix elements of the (generalized) one-particle density matrix are defined by

γ_{ρ} = ρ̃^{1,1} ∈ B^{1,1} and α_{ρ} = ρ̃^{0,2} ∈ B^{0,2} ≅ 𝔥^{∨2} ,

in other words

∀φ, ψ ∈ 𝔥 :  ⟨ψ , γ_{ρ} φ⟩ = Tr[ρ̃a^{∗}(φ)a(ψ)] ,
⟨ψ ⊗ φ , α_{ρ}⟩ = Tr[ρ̃a(ψ)a(φ)] .

Note that f_{ρ}, γ_{ρ}, and α_{ρ} exist for any ρ ∈ 𝔇𝔐 since N_{f}ρ, ρN_{f} ∈ L^{1}(𝔉_{+}).

Remark IV.9. For a centered pure quasifree state ρ̃, ρ̃^{p,q} vanishes when p+q is odd.

Remark IV.10. Another definition of the one-particle density matrix γ_{ρ} would be through the relation ⟨ψ, γ_{ρ}φ⟩ = Tr[a^{∗}(φ)a(ψ)ρ]. We prefer here a definition with a “centered” version ρ̃of the state ρ, because this centered quasifree state ρ̃then satisfies the usual Wick theorem. The same considerations hold for α_{ρ}. Hence, any quasifree density matrix is characterized by (f_{ρ}, γ_{ρ}, α_{ρ}), since ρ^{p,q} can be expressed in terms of (f_{ρ}, γ_{ρ}, α_{ρ}). When f_{ρ} = 0, the definition of γ_{ρ} is consistent with the usual one, for z_{1}, z_{2} ∈ 𝔥, ⟨z_{1}, γ_{ρ}z_{2}⟩ = Tr[a^{∗}(z_{2})a(z_{1})ρ]. The definition of α_{ρ} is related with the definition of the operator α̂_{ρ} (here denoted with a hat for clarity) used in the article of Bach, Lieb and Solovej [5], through the relation ⟨z_{1} ⊗ z_{2}, α_{ρ}⟩𝔥_{⊗2} = ⟨z_{1}, α̃_{ρ}cz_{2}⟩_{𝔥} with c a conjugation on 𝔥.

Example IV.11. A centered pure quasifree state satisfies the relation,

ρ̃^{2,2} = γ ⊗ γ + γ ⊗ γ Ex +αα^{∗} ∈ B^{2,2} ,  (IV.55)

where the exchange operator is the linear operator on 𝔥^{⊗2} such that

∀z_{1}, z_{2} ∈ 𝔥, Ex(z_{1} ⊗ z_{2}) = z_{2} ⊗ z_{1}

and where for any b ∈ 𝔥^{⊗2}, αα^{∗}b = ⟨α, b⟩𝔥_{⊗2} α.

We now turn to another parametrization of quasifree states, by vectors in a real Hilbert space. This parametrization enables us to use convexity arguments.

Proposition IV.12. Let T = uer̂ be an implementable symplectomorphism and ρ a quasifree state of the form ρ = 𝕌_{T}^{∗} |Ω⟩⟨Ω|𝕌_{T}. Then

γ_{ρ} = \frac{1}{2}(cosh(2r̂) − 1) ,  (IV.56)
∀z_{1}, z_{2} ∈ 𝔥 : ⟨z_{1} ⊗ z_{2}, α_{ρ}⟩𝔥_{⊗2} = ⟨z_{1}, \frac{1}{2} sinh(2r̂)z_{2}⟩ .  (IV.57)

Proof of Proposition IV.12. We have T i = uer̂i = uie^{−r̂} = iue^{−r̂} and for all z ∈ 𝔥

Tr[ρW (−i\sqrt{2}z)] = Tr[𝕌_{T}^{∗} |Ω⟩⟨Ω|𝕌_{T} W (−i\sqrt{2}z)]
= ⟨Ω|W (uer̂(−i\sqrt{2}z))|Ω⟩
= ⟨Ω|W (−i\sqrt{2}ue^{−r̂}z)|Ω⟩
= exp ( − \frac{1}{2}|ue^{−r̂}z|^{2})
= exp ( − \frac{1}{2}|e^{−r̂}z|^{2}).

>From this formula we can easily compute the function

h(t, s) := Tr[ρW (−ti\sqrt{2}z)W (−si\sqrt{2}z)] = exp ( − \frac{1}{2}|e^{−r̂}(t + s)z|^{2}),

whose derivative ∂_{t}∂_{s} at (t, s) = (0, 0) involves α and γ:

∂_{t}∂_{s}h(0, 0) = Tr[ρ(a^{∗}(z) − a(z))^{2}]
= −2z^{∗}γz + 2Re(α^{∗}z^{∨2}) − z^{∗}z .

But we also have

∂_{t}∂_{s} exp(−\frac{1}{2}|e^{−r̂}(t + s)z|^{2})|_{t=s=0}
= −(e^{−r̂}z)^{∗}(e^{−r̂}z)
= −(cosh(r̂)z − sinh(r̂)z)^{∗}(cosh(r̂)z − sinh(r̂)z)
= −(cosh(r̂)z)^{∗}(cosh(r̂)z)
+ 2Re(sinh(r̂)z)^{∗}(cosh(r̂)z) − (sinh(r̂)z)^{∗}(sinh(r̂)z)
= −z^{∗}(cosh^{2} r̂ + sinh^{2} r̂)z + 2Re(z^{∗}(sinh r̂cosh r̂)z)
= −z^{∗} cosh(2r̂)z + 2Re(z^{∗} \frac{1}{2} sinh(2r̂)z)

and hence, using the polarization identity

4z ∨ z′ = (z + z′)^{⊗2} − (z − z′)^{⊗2}

to recover every vector from 𝔥^{∨2} from linear combinations of vectors of the form z^{∨2}, we arrive at (IV.56)-(IV.57).

Proposition IV.13. The admissible γ, α for a pure quasifree state are exactly those satisfying the relation

γ + γ^{2} = (α ⊗ 1)^{∗}(1 ⊗ α) ,  (IV.58)

with γ ≥ 0.

This is the constraint when we minimize the energy as a function of (f, γ, α) with the method of Lagrange multipliers in Section VIII.3.

Proof. If γ, α are associated with a quasifree state, then there is an r̂ such that γ, α and r̂ satisfy Equations (IV.56) and (IV.57), then

⟨z_{1}, (α^{∗} ⊗ 1)(1 ⊗ α)z_{2}⟩ = (α^{∗} ⊗ z_{1}^{∗})(z_{2} ⊗ α)

= ([α^{∗}(z_{2} ⊗ 1)] ⊗ z_{1}^{∗})α

= ⟨α^{∗}(z_{2} ⊗ 1), \frac{1}{2} sinh(2r̂)z_{1}⟩_{𝔥}

= ⟨α^{∗}, z_{2} ⊗ \frac{1}{2} sinh(2r̂)z_{1}⟩𝔥_{⊗2}

= ⟨\frac{1}{4} sinh^{2}(2r̂)z_{1}, z_{2}⟩_{𝔥}

= ⟨(\frac{1}{2}(cosh(2r̂) − 1) + \frac{1}{4}(cosh(2r̂) − 1)^{2})z_{1}, z_{2}⟩_{𝔥} .

Conversely, if γ and α satisfy Eq. (IV.58) then we define the ℂ-antilinear operator α̂ such that ⟨z_{1}, α̂z_{2}⟩ = (z_{1} ⊗ z_{2})^{∗}α, and set r̂ = \frac{1}{2} sinh^{−1}(2α̂), then

∀z_{1}, z_{2} ∈ 𝔥 : ⟨z_{1} ⊗ z_{2}, α_{ρ}⟩𝔥_{2} = ⟨z_{1}, α̂z_{2}⟩ = ⟨z_{1}, \frac{1}{2} sinh(2r̂)z_{2}⟩ ,

which, in turn, implies that (α^{∗} ⊗ 1)(1 ⊗ α) = \frac{1}{4} sinh^{2}(2r̂). Hence, we have

γ + γ^{2} = \frac{1}{4} sinh^{2}(2r̂)

and as γ ≥ 0, it follows that γ = \frac{1}{2}(cosh(2r̂) − 1). Then γ, α is associated with the centered pure quasifree state whose symplectic transformation is exp[r̂].

V Energy Functional

In this section we calculate the energy of a quasifree state in terms of its characterizing parameters, i.e., in terms of (f, γ, α) and (f, r). Notation: We first recall that, as before, we denote by k⃗, and |k⃗| the multiplication operators k⃗ ⊗ 1_{ℂ^{2}} and |k⃗| ⊗ 1_{ℂ^{2}} on 𝔥 = L^{2}(S_{σ,Λ} × ℤ_{2}), with three components in the case of k⃗. We now work at fixed values of total momentum p⃗ ∈ ℝ^{3}. The operator H_{g,p⃗} is given by

H_{g,p⃗} = \frac{1}{2}(dΓ(k⃗) + 2Re a^{∗}(G⃗) − p⃗)^{·2} + dΓ(|k⃗|) ,

where G⃗(k) = G⃗(k⃗, ±) := g|k⃗|^{−1/2}ε⃗_{±}(k⃗). The energy of a quasifree state ρ associated with f ∈ 𝔥, γ ∈ L^{1}(𝔥), α ∈ 𝔥^{∨2} is

E_{g,p⃗}(f, γ, α) := Tr[H_{g,p⃗} ρ] ,  (V.59)

where 𝔥 is the ℂ-Hilbert space 𝔥 = L^{2}(S_{σ,Λ} × ℤ_{2}) and L^{1}(𝔥) is the space of trace class operators on 𝔥.

Proposition V.1. The energy functional (V.59) is

E_{g,p⃗}(f, γ, α) = \frac{1}{2}\{(Tr[γk⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2}
+ Tr[γk⃗ · γk⃗] + α^{∗}(k⃗ · ⊗k⃗)α + Tr[|k⃗|^{2}γ]

+ 2Re{α^{∗}[(G⃗ + k⃗f)^{·∨2}]} + Tr[(2γ + 1)(G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗}]}

+ Tr[γ|k⃗|] + f^{∗}|k⃗|f .  (V.60)

where the following positivity properties hold

(Tr[γk⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2} ≥ 0 ,
Tr[γk⃗ · γk⃗] + Tr[γk⃗]^{·2} + α^{∗}(k⃗ · ⊗k⃗)α + Tr[|k⃗|^{2}γ] ≥ 0 ,
(Tr[γk⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2}
+Tr[γk⃗ · γk⃗] + α^{∗}(k⃗ · ⊗k⃗)α + Tr[|k⃗|^{2}γ] ≥ 0 ,
2Re(α^{∗}((G⃗ + k⃗f)^{·∨2})) + Tr[(2γ + 1)(G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗}] ≥ 0 .

The energy of a pure quasifree state in the variables f and r̂ is

Ê_{g,p⃗}(f, r̂) = \frac{1}{2}\{(Tr[\frac{1}{2}(cosh(2r̂) − 1)k⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2}
+ Tr[\frac{1}{2}(cosh(2r̂) − 1)k⃗ · \frac{1}{2}(cosh(2r̂) − 1)k⃗]
+ Tr[\frac{1}{2} sinh(2r̂)k⃗ · \frac{1}{2} sinh(2r)k⃗] + Tr[|k⃗|^{2} \frac{1}{2}(cosh(2r̂) − 1)]
+ 2Re⟨\frac{1}{2} sinh(2r̂)(G⃗ + k⃗f); (G⃗ + k⃗f)⟩
+ Tr[(2\frac{1}{2}(cosh(2r̂) − 1) + 1)(G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗}]\}
+ Tr[\frac{1}{2}(cosh(2r̂) − 1)|k⃗|] + f^{∗}|k⃗|f .  (V.61)

Proof. Using the Weyl operators,

E_{g,p⃗}(f, γ, α) := Tr[H_{g,p⃗}ρ] = Tr[H_{g,p⃗}(f)ρ̃]

where H_{g,p⃗}(f) = W (\sqrt{2}f/i)^{∗}H_{g,p⃗}W (\sqrt{2}f/i) and ρ̃ = W (\sqrt{2}f/i)^{∗}ρW (\sqrt{2}f/i), so that ρ̃ is centered. Modulo terms of odd order, which vanish when we take the trace against a centered quasifree state, H_{g,p⃗}(f) equals

H_{g,p⃗}(f) = \frac{1}{2}(dΓ(k⃗) + f^{∗}k⃗f + 2Re(a^{∗}(k⃗f + G⃗)) + 2Re(f^{∗}G⃗) − p⃗)^{·2}
+ dΓ(|k⃗|) + f^{∗}|k⃗|f + odd

= \frac{1}{2}(dΓ(k⃗) + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2}

+ \frac{1}{2}(2Re(a^{∗}(k⃗f + G⃗)))^{·2} + dΓ(|k⃗|) + f^{∗}|k⃗|f + odd .

Minimization of the Energy of the Nonrelativistic … 1501 To compute E(f, γ, α) we are thus lead to compute, for φ⃗ ∈ 𝔥^{3} and u⃗ ∈ ℝ^{3},

Tr[ρ̃(dΓ(k⃗) + u⃗)^{·2}] and Tr[ρ̃(2Re{a(φ⃗)})^{·2}] .

The expression of the energy as a function of (f, γ, α) then follows from Propositions V.2 and V.4. The expression of the energy as a function of (f, r) follows from Proposition IV.12. Proposition V.2. Let u⃗ ∈ ℝ^{3}, then

0 ≤ Tr[ρ̃(dΓ(k⃗) + u⃗)^{·2}] = (Tr[γk⃗] + u⃗)^{·2} − Tr[γk⃗]^{·2}

+ Tr[γk⃗ · γk⃗] + Tr[γk⃗]^{·2} + α^{∗}(k⃗ ·⊗k⃗)α + Tr[|k⃗|^{2}γ] .

This condition is used with u⃗ = p⃗ − f^{∗}k⃗f − 2Re(f^{∗}G⃗). Proof. Indeed,

(dΓ(k⃗) + u⃗)^{·2} = dΓ(k⃗)^{·2} + 2dΓ(k⃗) · u⃗ + u⃗^{·2} .

Then we use that Tr[ρ̃dΓ(k⃗)] = Tr[γk⃗], add and substract Tr[γk⃗]^{·2} to complete the square and compute Tr[ρ̃dΓ(k⃗)^{·2}] using Lemma V.3. Lemma V.3. Let X ∈ B^{1,1}, then

0 ≤ Tr[ρ̃dΓ(X)dΓ(X)^{∗}] =

= Tr[γXγX^{∗}] + |Tr[γX]|^{2} + α^{∗}(X ⊗ X^{∗})α + Tr[XX^{∗}γ] .

Proof. Indeed, using Equation (IV.55), Tr[ρ̃dΓ(X)dΓ(X)^{∗}] = Tr[ρ̃(∫ X(k_{1}, k′_{1})X(k_{2}, k′_{2})a^{∗}(k_{1})a^{∗}(k_{2})a(k′_{2})a(k′_{1})dk_{1}dk_{2}dk′_{1}dk′_{2} + dΓ(XX^{∗})] = Tr[(γ ⊗ γ + γ ⊗ γ Ex + αα^{∗})(X ⊗ X^{∗})] + Tr[γ XX^{∗}] = Tr[γX]Tr[γX^{∗}] + Tr[γXγX^{∗}] + α^{∗}(X ⊗ X^{∗})α + Tr[γ XX^{∗}] . Proposition V.4. Let φ ∈ 𝔥, then

0 ≤ Tr[ρ̃(a^{∗}(φ) + a(φ))^{2}] = 2Re(α^{∗}(φ^{∨2})) + Tr[(2γ + 1)φφ^{∗}]  (V.62)

and |2Re(α^{∗}(φ^{·∨2}))| ≤ Tr[(2γ + 1)φφ^{∗}]. This condition is used with the three components of φ⃗ = G⃗ + k⃗f. Proof. A computation using the canonical commutation relations yields

T r[ρ̃(a^{∗}(φ) + a(φ))^{2}]
= Tr[ρ̃(a^{∗}(φ))^{2} + ρ̃(a(φ))^{2} + ρ̃(a^{∗}(φ)a(φ) + a(φ)a^{∗}(φ))]
= α^{∗}φ^{∨2} + φ^{∨2∗}α + Tr[γ φφ^{∗} + (γ + 1)ψψ^{∗}].

VI Minimization over Coherent States

In this section we consider the problem of minimizing the energy over coherent states and show that there is a unique minimizer. We also calculate the lower orders of the energy at the minimizer seen as a function of the total momentum p⃗. For this section we can take σ = 0 if we consider the parameter f in the energy to be in 𝔥̃ := L^{2}(S_{σ,Λ} × ℤ_{2}, (\frac{1}{2}|k⃗|^{2} + |k⃗|)dk). Recall that S_{σ,Λ} = \{k⃗ ∈ ℝ^{3} | σ ≤ |k⃗| ≤ Λ\}. We also recall that for z and z′ in some Hilbert space, z^{∗}z′ = ⟨z, z′⟩ (see Section III).

Remark VI.1. For a coherent state (see Definition III.5) the energy reduces to

E_{g,p⃗}(f) = \frac{1}{2}∥G⃗∥^{2} + \frac{1}{2}(f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2} + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗|)f . (VI.63)

Note that, for σ > 0, 𝔥 = L^{2}(S_{σ,Λ} × ℤ_{2}, dk) = 𝔥̃, while for σ = 0, 𝔥 ⊂ 𝔥̃, and E_{g,p⃗}(f) extends to 𝔥̃ by using Equation (VI.63).

Theorem VI.2. There exists a universal constant C < ∞ such that, for 0 ≤ σ < Λ < ∞, g^{2} ln(Λ + 2) ≤ C and |p⃗| ≤ 1/3, there exists a unique fp⃗ which minimizes E_{g,p⃗} in 𝔥̃. The minimizer fp⃗ solves the system of equations

fp⃗ = \frac{u⃗p⃗ · G⃗}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗p⃗} ,  (VI.64)
u⃗p⃗ = p⃗ − 2Re(f ∗ ; p⃗G⃗) − f ∗ ; p⃗k⃗fp⃗ ,  (VI.65)

with |u⃗p⃗| ≤ |p⃗|.

Remark VI.3. Our hypotheses are similar those of Chen, Fröhlich, and Pizzo [10], where their vector ∇Eσ ; p⃗ is analogous to u⃗p⃗ in our notations. The construction of u⃗p⃗ as the solution of a fixed point problem and the dependency in the parameter p⃗ imply that the map p⃗ ↦ u⃗p⃗ is of class C^{∞}.

Remark VI.4. We note that we also expect to have u⃗p⃗ in the neighborhood of p⃗.

Remark VI.5. The minimizer is constructed as the solution of a fixed point problem. As a result the application

(σ, Λ, g, p⃗) ↦ inf_{ρ∈coh}Tr[H_{g,p⃗} ρ]

is continuous on the domain defined by Theorem VI.2, and at σ, Λ fixed,

(g, p⃗) ↦ inf_{ρ∈coh}Tr[H_{g,p⃗} ρ]

is analytic for g^{2} < C/ ln(Λ + 2) and |p⃗| < 1/3.

Remark VI.6. The assumption on |p⃗| ≤ 1/3 is much weaker than in the quasifree state case where we need to have p⃗ smaller than a constant C which may be small. In fact the 1/3 is arbitrary and one may suppose only |p⃗| ≤ R for some constant R < 1, but since this would result in a heavier exposition without providing additional relevant information, we restrict to |p⃗| ≤ 1/3.

To prove Theorem VI.2 we first show that the Equations (VI.64) and (VI.65) are necessarily verified by a minimizer. We then show the existence and uniqueness of a solution to these equations by a fixed point argument.

Proof of Theorem VI.2. Assume there is a point fp⃗ where the minimum is attained. The partial derivative of the energy at the point fp⃗

∂_{f^{∗}}E(fp⃗)

= ((f ∗ ; p⃗k⃗fp⃗ − p⃗ + 2Re(f ∗ ; p⃗G⃗)) · k⃗ + \frac{1}{2}|k⃗|^{2} + |k⃗|)fp⃗ − (p⃗ − f ∗ ; p⃗k⃗fp⃗ − 2Re(f ∗ ; p⃗G⃗)) · G⃗

then vanishes, where the derivative ∂_{f^{∗}}E(f) at a point f is the unique vector in 𝔥̃^{∗} ≅ L^{2}(S_{σ,Λ}, (\frac{1}{2}|k⃗|^{2} + |k⃗|)^{−1}dk) defined by

E(f + δf) − E(f) = 2Re(δf^{∗} ∂_{f^{∗}}E(f)) + o(∥δf∥𝔥̃)

with f, δf ∈ 𝔥̃. Observe that

0 ≤ E_{g,p⃗}(0) − E_{g,p⃗}(fp⃗)

= \frac{1}{2}|p⃗|^{·2} − \frac{1}{2}(f ∗ ; p⃗k⃗fp⃗ + 2Re(f ∗ ; p⃗G⃗) − p⃗)^{·2} − f ∗ ; p⃗(\frac{1}{2}|k⃗|^{2} + |k⃗|)fp⃗

and hence |p⃗| ≥ |u⃗p⃗| with u⃗p⃗ := p⃗ − f ∗ ; p⃗k⃗fp⃗ − 2Re(f ∗ ; p⃗G⃗). Since |u⃗p⃗| ≤ |p⃗| < 1, it

makes sense to write

fp⃗ = \frac{u⃗p⃗ · G⃗}{\frac{1}{2}|k⃗|^{2} + |k⃗| − u⃗p⃗ · k⃗} .

Hence the minimum point fp⃗ satisfies Equations (VI.64) and (VI.65). It is in particular sufficient to prove that there exist a unique u⃗p⃗ in a ball B̄(0, r) with r ≥ |p⃗| such that the function in Equation (VI.64) satisfies also Equation (VI.65) to prove the existence and uniqueness of a minimizer. Proof of the existence and uniqueness of a solution. Let \frac{1}{3} < r < 1, u⃗ ∈ ℝ^{3}, |u⃗| ≤ r < 1 and

Φu⃗(k⃗) = \frac{u⃗ · G⃗(k⃗)}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗} .

Observe that Φu⃗ ∈ Z̃, indeed, if |u⃗| < 1 then \frac{1}{2}|k⃗|^{2} +|k⃗|−k⃗·u⃗ ≥ (1−r)(\frac{1}{2}|k⃗|^{2} +|k⃗|), and with ε⃗(k⃗) = ε⃗(k⃗, +) + ε⃗(k⃗, −),

Documenta Mathematica 18 (2013) 1481–1519

∫_{|k⃗|∈[σ,Λ]}(\frac{1}{2}|k⃗|^{2} + |k⃗|)|Φu⃗(k⃗)|^{2}dk ≤ g^{2} ∫_{|k⃗|∈[σ,Λ]} \frac{1}{|k⃗|} \frac{1}{(1 − r)^{2}} \frac{|u⃗ · ε⃗(k⃗)|^{2}}{\frac{1}{2}|k⃗|^{2} + |k⃗|}dk < +∞ .

≤ C_{0}g^{2} ln(Λ + 2)\frac{|u⃗|^{2}}{(1 − r)^{2}}

for some universal constant C_{0} > 0. Observe then that

∫_{|k⃗|∈[σ,Λ]} \frac{|G⃗(k)|^{2}}{\frac{1}{2}|k⃗|^{2} + |k⃗|}dk ≤ C_{0}g^{2} ln(Λ + 2)

for some universal constant C_{0} > 0. It follows that Φ∗ ; u⃗G⃗ ∈ L^{1}(S_{σ,Λ} × ℤ_{2}). Note that if σ = 0 then Φu⃗ ∉ L^{2}(S_{σ,Λ} × ℤ_{2}) (for u⃗ ≠ 0). We can thus define the application

B̄(0, r) ∋ u⃗ ↦ Ψ⃗(u⃗) := p⃗ − Φ∗ ; u⃗k⃗Φu⃗ − 2Re(Φ∗ ; u⃗G⃗) ∈ ℝ^{3} .

We check that the hypotheses of the Banach-Picard fixed point theorem are verified on the ball B̄(0, r), which will prove the result. Stability: If g^{2} ln(Λ + 2) is sufficiently small, we get from

|Ψ⃗(u⃗)| ≤ |Φ∗ ; u⃗k⃗Φu⃗| + |2Re(Φ∗ ; u⃗G⃗)| + |p⃗|

and the estimates above that the sum of the two first terms is smaller than r − 1/3 and since |p⃗| ≤ 1/3 the map Ψ⃗ sends B̄(0, r) into itself,

Ψ⃗(B̄(0, r)) ⊆ B̄(0, r) .

Contraction: For u⃗ and v⃗ in B̄(0, r), we have that

|Φu⃗(k⃗) − Φv⃗(k⃗)|(\frac{1}{2}|k⃗|^{2} + |k⃗|)

= | \frac{u⃗.G⃗(k⃗)}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗} − \frac{v⃗.G⃗(k⃗)}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗.v⃗} |(\frac{1}{2}|k⃗|^{2} + |k⃗|)

≤ ( \frac{|u⃗ − v⃗||G⃗(k⃗)|}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗}

+ |v⃗||G⃗(k⃗)| | \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · v⃗} − \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗}|)(\frac{1}{2}|k⃗|^{2} + |k⃗|)

≤ |u⃗ − v⃗||G⃗(k⃗)|\frac{1}{(1 − r)}(1 + \frac{r|k⃗|}{\frac{1}{2}|k⃗|^{2} + |k⃗|(1 − r)})

≤ |u⃗ − v⃗||G⃗(k⃗)|\frac{1}{(1 − r)^{2}} .

Documenta Mathematica 18 (2013) 1481–1519

Minimization of the Energy of the Nonrelativistic … 1505 For the term 2Re(Φ∗ ; u⃗G⃗), we observe that

|2Re(Φ∗ ; u⃗G⃗) − 2Re(Φ∗ ; v⃗G⃗)|

≤ g^{2}2|u⃗ − v⃗|\frac{1}{(1 − r)^{2}} ∫_{|k⃗|∈[σ,Λ]} \frac{1}{|k⃗|} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}d^{3}k

≤ C_{1}g^{2} ln(2 + Λ)2|u⃗ − v⃗|\frac{1}{(1 − r)^{2}} .

Note that, for g^{2} ln(2 + Λ) < (1 − r)^{2}/(3C_{1}),

|2Re(Φ∗ ; u⃗G⃗) − 2Re(Φ∗ ; v⃗G⃗)| < \frac{1}{3}|u⃗ − v⃗| .

Finally, for the term Φ∗ ; u⃗k⃗Φu⃗, we obtain the estimate

|Φ∗ ; u⃗k⃗Φu⃗ − Φ∗ ; v⃗k⃗Φv⃗|

≤ ∫_{|k⃗|∈[σ,Λ]}(\frac{1}{2}|k⃗|^{2} + |k⃗|)|Φu⃗(k⃗) − Φv⃗(k⃗)|(|Φu⃗(k⃗)| + |Φv⃗(k⃗)|)d^{3}k

≤ \frac{|u⃗ − v⃗|}{(1 − r)^{2}} ∫_{|k⃗|∈[σ,Λ]} |G⃗(k⃗)|(|Φu⃗(k⃗)| + |Φv⃗(k⃗)|)dk⃗

≤ \frac{|u⃗ − v⃗|}{(1 − r)^{2}} ∥(\frac{1}{2}|k⃗|^{2} + |k⃗|)^{−1/2}G∥(∥\sqrt{\frac{1}{2}|k⃗|^{2} + |k⃗|}Φu⃗∥ + ∥\sqrt{\frac{1}{2}|k⃗|^{2} + |k⃗|}Φv⃗∥)

≤ C_{2}|u⃗ − v⃗| (|u⃗| + |v⃗|)g^{2} ln(Λ + 2) , and thus this term can be controlled for |g ln(Λ + 2)|^{2} sufficiently small by \frac{1}{3}|u⃗ − v⃗|. We thus get a contraction

|Ψ⃗(u⃗) − Ψ⃗(u′)| ≤ \frac{2}{3}|u⃗ − u′|

and with fp⃗ = Φu⃗p⃗ Equation (VI.64) is solved. We then obtain several explicit results from the expression of this minimizer. Corollary VI.7. With the same hypotheses as in Theorem VI.2:

1. For 0 ≤ σ < Λ < ∞,

inf_{f∈𝔥} E_{g,p⃗}(f) = inf_{f∈𝔥̃} E_{g,p⃗}(f) = E_{g,p⃗}(fp⃗) ,

and for 0 < σ < Λ < ∞, we have that fp⃗ ∈ 𝔥.

2. For fixed g, σ, Λ, as a function of p⃗,

E_{g,p⃗}(fp⃗) = E_{g,p⃗}(0) − p⃗ · G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗}} G⃗ · p⃗ + O(|p⃗|^{3}) .

Documenta Mathematica 18 (2013) 1481–1519

3. For all f in 𝔥̃,

E_{g,p⃗}(fp⃗ + f) = E_{g,p⃗}(fp⃗) + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗| − u⃗p⃗ · k⃗)f

+ \frac{1}{2}(f^{∗}k⃗f + 2Re(f ∗ ; p⃗k⃗f) + 2Re(f^{∗}G⃗))^{·2} . (VI.66)

4. The energy E_{g,p⃗}(fp⃗) of the minimizer compared to the energy of the vacuum state E_{g,p⃗}(0) is

E_{g,p⃗}(fp⃗) = E_{g,p⃗}(0) − \frac{1}{2}2Re(f ∗ ; p⃗u⃗p⃗ · G⃗) − \frac{1}{2}|u⃗p⃗ − p⃗|^{2} .

Note that the term 2Re(f ∗ ; p⃗u⃗p⃗ · G⃗) is non-negative.

Proof of 1. is straightforward for σ > 0. For σ = 0 one can approximate the minimizer in 𝔥̃ by functions in 𝔥. Proof of 2. The expression of the energy E_{g,p⃗}(f) given in Equation (VI.63) implies that E_{g,p⃗}(f) ≥ \frac{1}{2}∥G⃗∥^{2}, and for p⃗ = 0 this minimum is only attained at the point f0 = 0. It follows that fp⃗ = ∂p⃗f0 · p⃗ + O(|p⃗|^{2}) . >From Equation (VI.65) we deduce

u⃗p⃗ = p⃗ − 2Re((∂p⃗f0 · p⃗)^{∗}G⃗) + O(|p⃗|^{2})

and thus

fp⃗ = \frac{(p⃗ − 2Re((∂p⃗f0 · p⃗)^{∗}G⃗)).G⃗}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗p⃗} + O(|p⃗|^{2})

= (\frac{1}{2}|k⃗|^{2} + |k⃗|)^{−1}(p⃗ − 2Re((∂p⃗f0 · p⃗)^{∗}G⃗)) · G⃗ + O(|p⃗|^{2}) .

Expanding the left hand side of this equality in 0 brings

∂p⃗f0 · p⃗ = (\frac{1}{2}|k⃗|^{2} + |k⃗|)^{−1}(p⃗ − 2Re((∂p⃗f0 · p⃗)^{∗}G⃗)) · G⃗

and hence ∂p⃗f0 = (\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗})^{−1}G⃗. The expansion of fp⃗ to the second order is then

fp⃗ = (\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗})^{−1}G⃗ · p⃗ + O(|p⃗|^{2}) .

We can compute the energy modulo error terms in O(|p⃗|^{3}). To have less heavy com-

Minimization of the Energy of the Nonrelativistic … 1507 putations we set A = \frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗} and get

E_{g,p⃗}(fp⃗) − \frac{1}{2}∥G⃗∥^{2} − \frac{1}{2}|p⃗|^{2}

≡ −\frac{1}{2}|p⃗|^{2} + \frac{1}{2}(2Re(p⃗ · ∂p⃗f ∗ ; 0G⃗) − p⃗)^{·2} + p⃗ · ∂p⃗f ∗ ; 0(\frac{1}{2}|k⃗|^{2} + |k⃗|)∂p⃗f ∗ ; 0 · p⃗

≡ \frac{1}{2}(2Re(p⃗ · G⃗^{∗}A^{−1}G⃗))^{·2} − 2p⃗ · G⃗^{∗}A^{−1}G⃗ · p⃗

+ p⃗ · G⃗^{∗}A^{−1}(\frac{1}{2}|k⃗|^{2} + |k⃗|)A^{−1}G⃗ · p⃗

≡ 2(p⃗ · G⃗^{∗}A^{−1}G⃗)^{·2} + p⃗ · G⃗^{∗}A^{−1}((\frac{1}{2}|k⃗|^{2} + |k⃗|) − 2A)A^{−1}G⃗ · p⃗

≡ p⃗ · G⃗^{∗}A^{−1}2G⃗ · G⃗^{∗}A^{−1}G⃗ · p⃗ − p⃗ · G⃗^{∗}A^{−1}(\frac{1}{2}|k⃗|^{2} + |k⃗| + 4G⃗ · G⃗^{∗}))A^{−1}G⃗ · p⃗

≡ −p⃗ · G⃗^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗})^{−1}G⃗ · p⃗

which yields the result. Proof of 3. The Taylor expansion of the energy around fp⃗ is

E_{g,p⃗}(fp⃗ + f) = E_{g,p⃗}(fp⃗) + f^{∗} ∂_{f^{∗}}E(fp⃗) + ∂_{f}E(fp⃗) f

+ \frac{1}{2}\{(f^{∗}k⃗f + 2Re(f^{∗}G⃗) + 2Re(f ∗ ; p⃗k⃗f))^{.2}

+ 2(f ∗ ; p⃗k⃗fp⃗ + 2Re(f ∗ ; p⃗G⃗) − p⃗) · f^{∗}k⃗f + f^{∗}|k⃗|^{2}f} + f^{∗}|k⃗|f .

Since ∂_{f^{∗}}E(fp⃗) vanishes this gives Equation (VI.66). Proof of 4. It is sufficient to replace f by −fp⃗ in Equation (VI.66). The observation

f ∗ ; p⃗u⃗p⃗ · G⃗ = ∫ \frac{(u⃗p⃗ · G⃗(k⃗))^{2}dk}{\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗p⃗}

shows that 2Re(f ∗ ; p⃗u⃗p⃗ · G⃗) is non-negative since |u⃗p⃗| < 1. VII Renormalized Electron Mass for Coherent states In this section we use the coherent state minimizer found in Section VI in order to determine the renormalized electron mass. Proposition VII.1. The renormalized electron mass for coherent states m_{coh}(g, Λ) defined by

E_{g,p⃗}(f_{g,p⃗}) − E_{g,0}(f_{g,0}) = \frac{1}{2m_{coh}(g, Λ)}|p⃗|^{2} + O(|p⃗|^{3}) .

is

m_{coh}(g, Λ) = 1 + \frac{32π}{3} g^{2} ln(1 + \frac{Λ}{2}) .

Remark VII.2. This result agrees with m_{eff}/m obtained in [12] to second order in g, taking into account that g in [12] equals \frac{\sqrt{α}}{2π} in the present paper, that ω(k⃗) = |k⃗|, and that the mass m of the electron is one in our units. See also (among others) [11] or [3].

Proof. From

E_{g,p⃗}(f_{g,p⃗}) − E_{g,p⃗}(0) = \frac{1}{2}|p⃗|^{2} − p⃗·G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗}} G⃗·p⃗ + O(|p⃗|^{3})

and the fact that for p⃗ = 0 the minimizer is f_{g,0} = 0, it follows that

E_{g,p⃗}(f_{g,p⃗}) − E_{g,0}(f_{g,0}) = E_{g,p⃗}(f_{g,p⃗}) − E_{g,0}(0)

= E_{g,p⃗}(f_{g,p⃗}) − E_{g,p⃗}(0) + \frac{1}{2}|p⃗|^{2}

= \frac{1}{2}|p⃗|^{2} − p⃗·G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗}} G⃗·p⃗ + O(|p⃗|^{3})

where we used point 2 of Theorem VI.2 for the last equality. The power expansion of (1 − t)^{−1} yields

\frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗| + 2G⃗ · G⃗^{∗}} = ∑_{j=0}^{∞} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}( − 2G⃗ · G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|})^{j}

and hence

E_{g,p⃗}(f_{g,p⃗}) − E_{g,0}(f_{g,0}) = \frac{1}{2}|p⃗|^{2} − \frac{1}{2} ∑_{j=1}^{∞} p⃗·( − 2G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗)^{j}·p⃗ + O(|p⃗|^{3})

= \frac{1}{2}p⃗ · (Id_{ℝ^{3}} + 2G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗)^{−1}·p⃗ + O(|p⃗|^{3}) .

The coherent renormalized mass is thus

m_{coh} = Id_{ℝ^{3}} + 2G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗ .

The proof is achieved by the computation of an integral that we give in a separate lemma since it will be useful again later.

Lemma VII.3. With σ = 0,

2G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗ = g^{2} 8π \frac{4}{3} ln(1 + \frac{Λ}{2}) Id_{ℝ^{3}} .

Proof. Using spherical coordinates

2p⃗ · G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗ · p⃗ = 2g^{2} ∫_{|k⃗|≤Λ} \frac{|p⃗·ε⃗_{+}(k⃗)|^{2} + |p⃗·ε⃗_{−}(k⃗)|^{2}}{\frac{1}{2}|k⃗|^{3} + |k⃗|^{2}} dk⃗

= 2g^{2}|p⃗|^{2} ∫_{|k⃗|≤Λ} \frac{1 − |\frac{p⃗}{|p⃗|}·\frac{k⃗}{|k⃗|}|^{2}}{\frac{1}{2}|k⃗|^{3} + |k⃗|^{2}} dk⃗

= 2g^{2}|p⃗|^{2} ∫_{0}^{Λ} ∫_{0}^{2π} ∫_{0}^{π} \frac{1 − cos^{2} φ}{\frac{1}{2}ρ^{3} + ρ^{2}} sin φ dφ dθ ρ^{2}dρ

= g^{2}|p⃗|^{2} 8π \frac{4}{3} ln(1 + \frac{Λ}{2}) ,

which yields the result.

VIII Minimization over Quasifree States

In this section we minimize the energy functional over the set of pure quasifree states. We first obtain the existence and uniqueness of a pure quasifree state in Theorem VIII.3. As shown in Section IV this proves the existence (but not the uniqueness) of a minimizer in the class of quasifree states. We then compute asymptotics for small coupling g and momentum p of the parameters f_{g,p⃗} and r_{g,p⃗} describing the pure quasifree state minimizing the energy in Theorem VIII.6. From these asymptotics we deduce an expansion of the energy at the minimizer. Finally we give in Theorem VIII.8 the Lagrange equations in terms of the parameters f, α and γ.

Remark VIII.1. We believe that the assumption σ > 0 is unnecessary, but we have no proof for this assertion and do not follow it here, because the infrared singular limit σ → 0 is not our concern in this paper.

VIII.1 Existence and Uniqueness of a Minimizer of the Energy over Pure Quasifree States

In this section we minimize the energy over all pure quasifree states and show that there is a unique minimizer.

Definition VIII.2. Let 𝔥 be a ℂ-Hilbert space. Let Y be the ℝ-Hilbert space of antilinear operators r̂on 𝔥, self-adjoint in the sense that ∀z, z′ ∈ 𝔥, ⟨z, r̂z′⟩ = ⟨z′, r̂z⟩, and Hilbert-Schmidt in the sense that the positive operator r̂^{2} is trace class. The space X = 𝔥 × Y with the scalar product

⟨(f, r̂), (f′, r̂′)⟩_{X} = f^{∗}f′ + Tr[r̂r̂′]

is an ℝ-Hilbert space. Keeping σ > 0, we only need to use 𝔥 = L^{2}(S_{σ,Λ} × ℤ_{2}) in this section.

Theorem VIII.3. Let 0 < σ < Λ < ∞. There exists C > 0 such that for g, |p⃗| ≤ C there exists a unique minimizer for Ê_{g,p⃗}(f, r̂).

Proof. This result follows from convexity and coercivity arguments. By Proposition VIII.4, Ê_{g,p⃗}(f, r̂) is strictly θ-convex (i.e., uniformly strictly convex) on B̄_{X}(0, R) for some R > 0 and θ > 0. Since Ê_{g,p⃗}(f, r̂) is strongly continuous on the closed and convex set B̄_{X}(0, R) of the Hilbert space X we get the existence and uniqueness of a minimizer in B̄_{X}(0, R). (See for example [1]. The uniform strict convexity allows to prove directly that a minimizing sequence is a Cauchy sequence.) Proposition VIII.5 then proves that it is the only minimum of Ê_{g,p⃗}(f, r̂) on the whole space. Note that to use Propositions VIII.4 and VIII.5 we need to restrict to values of g and |p⃗| smaller than some constant C > 0.

Proposition VIII.4 (Convexity). There exist 0 < C, R < ∞ such that for g ≤ C and |p⃗| ≤ \frac{1}{2}, the Hessian of the energy satisfies HÊ_{g,p⃗}(f, r̂) ≥ \frac{σ}{4} 1_{X} on the ball B_{X}(0, R).

Proof. We use that strict positivity of the Hessian implies strict convexity and thus first compute the Hessian in (0, 0). The Hessian HÊ_{g,p⃗}(f, r̂) ∈ B(X) is defined using the Fréchet derivative

Ê_{g,p⃗}(f + δf, r̂ + δr̂) − Ê_{g,p⃗}(f, r̂)

= DÊ_{g,p⃗}(f, r̂)(δf, δr̂) + \frac{1}{2}⟨(δf, δr̂) , HÊ_{g,p⃗}(f, r̂) (δf, r̂)⟩_{X} + o(∥(δf, δr̂)∥_{X}^{2})

with DÊ_{g,p⃗}(0, 0) ∈ B(X, ℝ). (Note that differentiability is granted in this case because |k⃗| ≤ Λ < ∞.) For any μ > 0, ∀(f, r̂) ∈ X,

⟨(f,r̂) , \frac{1}{2}HÊ_{g,p⃗}(0, 0) (f, r̂)⟩_{X}

= 2Re⟨r̂k⃗f; G⃗⟩ + \frac{1}{2}(2Re(f^{∗}G⃗))^{·2} + Tr[r̂^{2}G⃗ · G⃗^{∗}]

+ \frac{1}{2}\{Tr[r̂k⃗ · r̂k⃗] + Tr[|k⃗|^{2}r̂^{2}]\}

+ Tr[r̂^{2}(|k⃗| − k⃗ · p⃗)] + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · p⃗)f

≥ Tr[r̂^{2}G⃗ · G⃗^{∗}] − μ∥r̂G⃗∥^{2} − \frac{1}{μ}∥k⃗f∥^{2}

+ \frac{1}{2}\{(2Re(δf^{∗}G⃗))^{·2} + Tr[r̂k⃗ · r̂k⃗] + Tr[|k⃗|^{2}r̂^{2}]\}

+ Tr[r̂^{2}(|k⃗| − k⃗ · p⃗)] + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · p⃗)f

≥ Tr[r̂^{2}(|k⃗| − k⃗ · p⃗ + (1 − μ)G⃗ · G⃗^{∗})] + f^{∗}((\frac{1}{2} − \frac{1}{μ})|k⃗|^{2} + |k⃗| − k⃗ · p⃗)f ,

since

|2Re⟨r̂k⃗f; G⃗⟩| ≤ 2∥r̂G⃗∥∥k⃗f∥ = 2\sqrt{μ}∥r̂G⃗∥\frac{1}{\sqrt{μ}}∥k⃗f∥ ≤ μ∥r̂G⃗∥^{2} + \frac{1}{μ}∥k⃗f∥^{2} .

With μ = 2 we obtain (with |p⃗| ≤ \frac{1}{2})

\frac{1}{2}HÊ_{g,p⃗}(0, 0)(f, r̂) ≥ Tr[r̂^{2}(|k⃗| − k⃗ · p⃗ − G⃗ · G⃗^{∗})] + f^{∗}(|k⃗| − k⃗ · p⃗)f

≥ Tr[r̂^{2}(|k⃗|(1 − ∥|k⃗|^{−1/2}G⃗∥^{2}) − k⃗ · p⃗)] + f^{∗}(|k⃗| − k⃗ · p⃗)f

≥ Tr[r̂^{2}σ(\frac{1}{2} − ∥|k⃗|^{−1/2}G⃗∥^{2})] + f^{∗} \frac{σ}{2} f

and for g small enough

\frac{1}{2}HÊ_{g,p⃗}(0, 0) ≥ \frac{σ}{4} .

We then compare it with the Hessian in points near zero. Observing that the Hessian is continuous with respect to (f, r̂, p⃗, g), we deduce that there exist R < ∞ and C > 0, as asserted.

Proposition VIII.5 (Coercivity). Suppose p⃗ and C > 0 are fixed such that \frac{1}{2}|p⃗|^{2} + \frac{1}{2}∥G⃗∥^{2} < σR^{2}, with the value of R given by Proposition VIII.4, for any 0 < g < C. For every (f, r̂) ∈ X,

Ê_{g,p⃗}(f, r̂) ≥ Tr[r̂^{2}|k⃗|] + f^{∗}|k⃗|f ≥ σ ∥(f, r̂)∥_{X}^{2} .

Since Ê_{g,p⃗}(0, 0) = \frac{1}{2}|p⃗|^{2} + \frac{1}{2}∥G⃗∥^{2} < σR^{2}, any minimizing sequence can be assumed to take its values in B̄_{X}(0, R).

VIII.2 Asymptotics for small Coupling and Momentum

In this section we calculate the ground state energy for small orders of the coupling constant g and total momentum p⃗. We use below an identification between self-adjoint ℂ-antilinear Hilbert-Schmidt operator r̂ and symmetric two vector r given by the relation ⟨φ, r̂ψ⟩_{𝔥} = ⟨φ ⊗ ψ, r⟩𝔥_{⊗2}. Note that the self-adjointness condition for r̂ is equivalent to the symmetry condition r ∈ 𝔥^{∨2}.

Theorem VIII.6. Let 0 < σ < Λ < ∞. There exists C > 0 such that for |g|, |p⃗| < C, there exist two functions f_{g,p⃗} and r̂_{g,p⃗} which are smooth in (g, p⃗) such that the minimum of the energy Ê_{g,p⃗}(f, r̂) is attained at (f_{g,p⃗}, r̂_{g,p⃗}). These functions satisfy

f_{g,p⃗} = \frac{p⃗·G⃗}{\frac{1}{2}|k⃗|^{2} + |k⃗|} + O(∥(g, p⃗)∥^{3})
r_{g,p⃗} = −S^{−1}G⃗^{·∨2} + O(∥(g, p⃗)∥^{3}) ,

with S = k⃗ · ⊗k⃗ + 2(\frac{1}{2}|k⃗|^{2} + |k⃗|) ∨ 1_{𝔥} and where G⃗^{·∨2} = ∑_{j=1}^{3} G⃗_{j} ∨ G⃗_{j} ∈ 𝔥 ∨ 𝔥 (recall that ∨ denotes the symmetric tensor product). As a consequence

E_{BHF} (g, p⃗, σ, Λ)

= Ê_{g,p⃗}(0_{X}) − p⃗ · G⃗^{∗} \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}G⃗ · p⃗ − \frac{1}{2}G⃗^{·∨2∗}S^{−1}G⃗^{·∨2} + O(∥(g, p⃗)∥^{5}) .

Recall that for z and z′ in a Hilbert space, z^{∗}z′ = ⟨z, z′⟩.

Remark VIII.7. The energy in 0_{X} is the energy of the vacuum state and is Ê_{g,p⃗}(0_{X}) = \frac{1}{2}p^{·2} + \frac{1}{2}∥G⃗∥^{2}. Further note that by Lemma VII.3

2(p⃗ · G⃗^{∗}) \frac{1}{\frac{1}{2}|k⃗|^{2} + |k⃗|}(G⃗ · p⃗) = g^{2}|p⃗|^{2} \frac{32π}{3} ln (\frac{Λ + 2}{σ + 2})

and in particular does not depend on the choice of the polarization vectors ε⃗. The quantity G⃗^{·∨2∗}S^{−1}G⃗^{·∨2} does not depend on the choice of the vectors ε⃗ either since

G⃗^{·∨2∗}S^{−1}G⃗^{·∨2} = ∑_{μ,ν=±} ∫ \frac{|ε⃗(k⃗_{1}, μ) · ε⃗(k⃗_{2}, ν)|^{2}}{\sqrt{|k⃗_{1}||k⃗_{2}|}S(k⃗_{1}, k⃗_{2})} d^{3}k_{1}d^{3}k_{2}

and with Pu⃗ is the orthogonal projection on u⃗ in ℝ^{3},

∑_{μ,ν=±} |ε⃗(k⃗_{1}, μ) · ε⃗(k⃗_{2}, ν)|^{2} = ∑_{μ,ν=±} Tr_{ℝ^{3}}[P_{ε⃗(k⃗_{1},μ)}P_{ε⃗(k⃗_{2},ν)}]

= 1+Tr_{ℝ^{3}}[P ⊥ ; k⃗_{1}P ⊥ ; k⃗_{2}]

= 1 + (\frac{k⃗_{1}}{|k⃗_{1}|} · \frac{k⃗_{2}}{|k⃗_{2}|})^{2} .

Proof of Theorem VIII.6. Let

F : (g, p⃗, f, r̂) ↦ ∂_{f,r̂}Ê_{g,p⃗}(f, r̂)

and ( f ; r̂ ) (g, p⃗) := ( \begin{matrix}f(g,p⃗) \\ r̂(g,p⃗) \end{matrix} ) such that

F (g, p⃗, ( f ; r̂ ) (g, p⃗)) = 0 ,  (VIII.67)

then a derivation of Equation (VIII.67) with respect to (f, r̂) brings

∂_{g,p⃗} ( f ; r̂ ) (0_{g,p⃗}) = −[∂_{f,r̂}F (0_{g,p⃗}, 0_{f,r̂})]^{−1}∂_{g,p⃗}F (0_{g,p⃗}, 0_{f,r̂}) .

The term which is independent of (g, p⃗) and quadratic in ( f ; r̂ ) in the energy is

\frac{1}{2}\{Tr[r̂Sr̂] + f^{∗}(|k⃗|^{2} + 2|k⃗|)f\}

thus, in (0_{g,p⃗}, 0_{f,r̂}),

∂_{f,r̂}F = ( \begin{matrix}|k⃗|^{2} + 2|k⃗| & 0 \\ 0 & S \end{matrix} ) .

To compute ∂_{g,p⃗}F in 0, observe that no part in the energy is linear in (g, p⃗) and linear in (f, r̂). Thus ∂_{g,p⃗}F (0_{g,p⃗}, 0_{f,r̂}) = 0 and we get

∂_{g,p⃗}f(0_{g,p⃗}) = 0 .

Differentiating a second time Equation (VIII.67) brings

0 = ∂_{g,p⃗}^{2}F +2∂_{f,r̂}∂_{g,p⃗}F ∘∂_{g,p⃗} ( f ; r̂ )+∂_{f,r̂}F ∘∂_{g,p⃗}^{2} ( f ; r̂ )+∂_{f,r̂}^{2}F (∂_{g,p⃗} ( f ; r̂ ) , ∂_{g,p⃗} ( f ; r̂ )) .

Since ∂_{g,p⃗} ( f ; r̂ ) (0_{g,p⃗}) = 0, it follows that

∂_{g,p⃗}^{2} ( f ; r̂ ) (0_{g,p⃗}) = −[∂_{f,r̂}F (0_{g,p⃗}, 0_{f,r̂})]^{−1}∂_{g,p⃗}^{2}F (0_{g,p⃗}, 0_{f,r̂}) .

The part of the energy which is quadratic in (g, p⃗) and linear in (f, r̂) is −2Re(f^{∗}G⃗) · p⃗ + Re⟨r̂G⃗; G⃗⟩, it follows that, in (0_{g,p⃗}, 0_{f,r̂}),

∂_{g,p⃗}^{2}F = 2 ( \begin{matrix}( 1 0 ) ∨ ( 0 −2∂_{g}G⃗ ) \\ ( ∂_{g}G⃗ 0 ) · ∨ ( ∂_{g}G⃗ 0 ) \end{matrix} ) ,

which gives in 0_{g,p⃗}

∂_{g,p⃗}^{2} ( f ; r̂ ) = 2 ( \begin{matrix}( 1 0 ) ∨ ( 0 \frac{∂_{g} G⃗}{\frac{1}{2} |k⃗|^{2}+|k⃗|} ) \\ −S^{−1} ( ∂_{g}G⃗ 0 ) · ∨ ( ∂_{g}G⃗ 0 ) \end{matrix} ) .

Hence the expansion of ( f ; r̂ ) up to order 2. We can thus express the energy around 0_{g,p⃗} modulo error terms in O(∥(g, p⃗)∥^{5})

min_{f,r̂} Ê_{g,p⃗}(f, r̂) − Ê_{g,p⃗}(0, 0)

≡ \frac{1}{2}\{(Tr[r̂^{2}k⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)^{·2} + Tr[r̂k⃗ · r̂k⃗] + Tr[|k⃗|^{2}r̂^{2}]

+ 2Re⟨r̂(G⃗ + k⃗f); (G⃗ + k⃗f)⟩ + ∥G⃗∥^{2} + f^{∗}|k⃗|^{2}f}

+ Tr[r̂^{2}|k⃗|] + f^{∗}|k⃗|f − Ê_{g,p⃗}(0, 0)

≡ −2Re(f^{∗}G⃗) · p⃗ + \frac{1}{2}Tr[r̂Sr̂] + Re⟨r̂G⃗; G⃗⟩ + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗|)f

≡ −2Re(f^{∗}p⃗ · G⃗) + \frac{1}{2}Tr[r̂Sr̂] + Re⟨r̂G⃗; G⃗⟩ + f^{∗}(\frac{1}{2}|k⃗|^{2} + |k⃗|)f

≡ −2\frac{(p⃗ · G⃗)^{∗}(p⃗ · G⃗)}{\frac{1}{2}|k⃗|^{2} + |k⃗|} + \frac{1}{2}G⃗^{·∨2∗}S^{−1}G⃗^{·∨2} − G⃗^{·∨2∗}S^{−1}G⃗^{·∨2} + \frac{(p⃗ · G⃗)^{∗}(p⃗ · G⃗)}{\frac{1}{2}|k⃗|^{2} + |k⃗|}

which completes the proof.

VIII.3 Lagrange Equations

In this section we derive a system of equations that determine critical points of the energy functional. We formulate the results of Section VIII.1 in terms of γ and α subject to the constraints γ + γ^{2} = (α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α), without reference to the parametrization of γ and α in terms of r̂. Suppose f ∈ 𝔥, α ∈ 𝔥^{∨2}, γ ∈ L^{1}(𝔥), λ ∈ B(𝔥) = B and u⃗ ∈ ℝ^{3}. Let A(λ) = \frac{1}{2}k⃗ ·∨k⃗ + λ ∨ 1 and G(γ) = γ + γ^{2}.

Theorem VIII.8. Suppose (f, γ, α) is a minimizer of the energy functional E such that ∥γ∥_{B(𝔥)} < \frac{1}{2}. Then there is a unique (λ, u⃗) such that (f, γ, α, λ, u⃗) satisfies the following equations, equivalent to Lagrange equations

M(γ, u⃗)f = −(k⃗γ − u⃗) · G⃗ − k⃗ ·∨(G⃗ + k⃗f)^{∗}α  (VIII.68)
A(λ)α = −\frac{1}{2}(G⃗ + k⃗f)^{·∨2}  (VIII.69)
γ = G^{−1}((α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α))  (VIII.70)
λ = ∫_{0}^{∞} e^{−t( \frac{1}{2} +γ)}(M(γ, u⃗) + (G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗})e^{−t( \frac{1}{2} +γ)}dt (VIII.71)

u⃗ = p⃗ − Tr[γk⃗] − f^{∗}k⃗f − 2Re(f^{∗}G⃗)  (VIII.72)

with M(γ, u⃗) = \frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗ · u⃗ + k⃗ · γk⃗. Assuming |p⃗| < \frac{1}{2}, sufficient conditions such that M(γ, u⃗) and A(λ) are invertible operators are |u⃗| < 1/2, γ ≥ 0 and ∥λ − (|k⃗|^{2}/2 + |k⃗| − p⃗ · k⃗)∥_{B} < σ/2. Equations (VIII.68) to (VIII.72) then form a system of coupled explicit equations.

Remark VIII.9. To prove that Equations (VIII.68) to (VIII.72) admit a solution we use here the result of existence of a minimizer proved in Section VIII.1. It can also be proved directly by a fixed point argument by defining the applications

Ψ_{f}(f, α, γ, u⃗) = −M(γ, u⃗)^{−1}(k⃗γ − u⃗) · G⃗ − k⃗ ·∨(G⃗ + k⃗f)^{∗}α
Ψ_{α}(f, λ) = −A(λ)^{−1} \frac{1}{2}(G⃗ + k⃗f)^{·∨2}
Ψ_{γ}(α) = G^{−1}((α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α))
Ψ_{λ}(f, γ, u⃗) = ∫_{0}^{∞} e^{−t( \frac{1}{2} +γ)}(M(γ, u⃗) + (G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗})e^{−t( \frac{1}{2} +γ)}dt
Ψu⃗(f, γ) = p⃗ − Tr[γk⃗] − f^{∗}k⃗f − 2Re(f^{∗}G⃗)

defined on balls of centers 0, 0, 0, \frac{1}{2}|k⃗|^{2} + |k⃗| − k⃗·p⃗ and p⃗ and proving that the application

Ψ_{(f,λ)}(f, λ) = (Ψ_{f} [f, Ψ_{α} {f, λ} , Ψ_{γ} {Ψ_{α}(f, λ)} , Ψu⃗ {f, Ψ_{γ}(Ψ_{α} [f, λ])}] , Ψ_{λ} [f, Ψ_{γ} {Ψ_{α}(f, λ)} , Ψu⃗ {f, Ψ_{γ}(Ψ_{α} [f, λ])}])

is a contraction for a convenient choice of the radiuses and a sufficiently small coupling constant g. Note that it is then convenient to consider the norm of L^{2}(S_{σ,Λ} × ℤ_{2}, |k⃗|^{2}) for f.

Proof of Theorem VIII.8. Indeed, set u⃗ = p⃗ − Tr[γk⃗] − f^{∗}k⃗f − 2Re(f^{∗}G⃗) and define the partial derivatives as ∂_{f^{∗}}E(f, γ, α) ∈ 𝔥, ∂_{α^{∗}}E(f, γ, α) ∈ 𝔥^{∨2} and ∂_{γ}E(f, γ, α) ∈ B(𝔥) ≅ L^{1}(𝔥)′ such that

E(f + δf, γ + δγ, α + δα) − E(f, γ, α)
= 2Re(δf^{∗} ∂_{f^{∗}}E(f, γ, α)) + 2Re(δα^{∗} ∂_{α^{∗}}E(f, γ, α))
+ Tr[δγ ∂_{γ}E(f, γ, α)] + o(∥(δf, δγ, δα)∥_{𝔥×L^{1}(𝔥)×𝔥^{∨2}}) .

Recall the energy functional is given by Equation (V.60) and this yields

∂_{f^{∗}}E(f, γ, α) = \frac{1}{2}\{2(k⃗f + G⃗) · (Tr[γk⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)
+ 2k⃗ ·∨(G⃗ + k⃗f)^{∗}α + k⃗ · (2γ + 1)(G⃗ + k⃗f)} + |k⃗|f
= −(k⃗f + G⃗) · u⃗ + k⃗ · ∨(G⃗ + k⃗f)^{∗}α + k⃗ · γ(G⃗ + k⃗f) + |k⃗|f
= M(γ, u⃗)f + (k⃗γ − u⃗) · G⃗ + k⃗ ·∨(G⃗ + k⃗f)^{∗}α ,
∂_{α^{∗}}E(f, γ, α) = \frac{1}{2}(k⃗ ·⊗k⃗)α + \frac{1}{2}(G⃗ + k⃗f)^{·∨2} ,
∂_{γ}E(f, γ, α) = \frac{1}{2}\{2k⃗ · (Tr[γk⃗] + f^{∗}k⃗f + 2Re(f^{∗}G⃗) − p⃗)
+ 2k⃗ · γk⃗ + |k⃗|^{2} + 2(G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗}} + |k⃗|
= M(γ, u⃗) + (G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗} .

The constraint given by Equation (IV.58) can be expressed as

C(f, γ, α) = 0  (VIII.73)

with

C : 𝔥 × L^{1}(𝔥) × 𝔥^{∨2} → L^{1}(𝔥)
(f, γ, α) ↦ γ + γ^{2} − (α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α) .

Equation (VIII.73) is equivalent to Equation (VIII.70). The application C has a differential DC(f, γ, α) : 𝔥 × L^{1}(𝔥) × 𝔥^{∨2} → L^{1}(𝔥) such that

DC(f, γ, α)(δf, δγ, δα)

= δγ + δγ γ + γ δγ − (δα^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α) − (α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ δα) .

For ∥γ∥_{B(𝔥)} < \frac{1}{2} the application DC(f, γ, α) is surjective. Indeed it is already surjective on {0}×L^{1}(𝔥)×{0}, since, for every γ′ ∈ L^{1}(𝔥) the equation δγ+δγ γ+γ δγ = γ′ with unknown δγ has at least one solution, see Proposition VIII.10. We can then apply the Lagrange multiplier rule (see for example the book of Zeidler [14]) which tells us that there exists a λ ∈ B(𝔥) such that

∀(δf, δα, δγ) , DE(f, α, γ)(δf, δα, δγ) + Tr[DC(f, α, γ)(δf, δα, δγ) λ] = 0 ,

that is to say

2Re(δf^{∗}∂_{f^{∗}}E(f, γ, α) + δα^{∗}∂_{α^{∗}}E(f, γ, α)) + Tr[∂_{γ}E(f, γ, α)δγ] + Tr[(δγ + δγ γ + γ δγ − (δα^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ α) − (α^{∗} ⊗ 1_{𝔥})(1_{𝔥} ⊗ δα))λ] = 0 .

This is equivalent to Equations (VIII.68), (VIII.69) and

λ(\frac{1}{2} + γ) + (\frac{1}{2} + γ)λ = M(γ, u⃗) + (G⃗ + k⃗f) · (G⃗ + k⃗f)^{∗}  (VIII.74)

Using again Proposition VIII.10 we get that Equation (VIII.74) is equivalent to Equation (VIII.71). For the invertibility of A(λ) note that

A(λ) = \frac{1}{4}(k⃗ ⊗ 1 + 1 ⊗ k⃗)^{·2} + (|k⃗| − k⃗.p⃗ + λ − \frac{1}{2}|k⃗|^{2} − |k⃗| + k⃗ · p⃗) ∨ 1

≥ (\frac{σ}{2} − λ − (|k⃗|^{2}/2 + |k⃗| − p⃗ · k⃗)∥_{B}) 1 ∨ 1 .

For M(γ, u⃗), M(γ, u⃗) = \frac{1}{2}|k⃗|^{2}+|k⃗|−k⃗·u⃗+k⃗·γk⃗ ≥ σ/2 if γ ≥ 0 and |u⃗| < 1/2.

Let us recall a well known expression for the solution of the Sylvester or Lyapunov equation.

Proposition VIII.10. Let A and B be bounded self-adjoint operators on a Hilbert space. Suppose A ≥ a 1 with a > 0. Then the equation

AX + XA = B

for X a bounded operator has a unique solution χ_{A}(B) = ∫_{0}^{∞} e^{−tA}Be^{−tA}dt. If B a trace class operator then the solution X is also trace class.

Proof. Indeed, χ_{A}(B) is a solution because

Aχ_{A}(B) + χ_{A}(B)A = ∫_{0}^{∞} e^{−tA}(AB + BA)e^{−tA}dt

= − ∫_{0}^{∞} \frac{d}{dt}(e^{−tA}Be^{−tA})dt = B .

Conversely, suppose that AX + XA = B, then

χ_{A}(B) = ∫_{0}^{∞} e^{−tA}(AX + XA)e^{−tA}dt

= − ∫_{0}^{∞} \frac{d}{dt}(e^{−tA}Xe^{−tA})dt = X ,

and thus any solution X is equal to χ_{A}(B). Hence the solution is unique.

A Equivalent Characterizations of Centered Quasifree Density Matrices

In this appendix we give various equivalent characterizations of quasifree states. In particular we remark that (ii) in Lemma A.1 below corresponds to the definition of quasifree states in terms of Wick’s Theorem.

Lemma A.1. Let ρ ∈ 𝔠𝔇𝔐 be a centered density matrix and denote ⟨A⟩_{ρ} := Tr_{𝔉}{ρA}. Then (i) ⇔ (ii) ⇔ (iii), where

(i) ρ ∈ 𝔠𝔔𝔉 is centered and quasifree;

(ii) All odd correlation functions and all even truncated correlation functions of ρ vanish, i.e., for all N ∈ ℕ and φ_{1}, …, φ_{2N} ∈ 𝔥, let either b_{n} := a^{∗}(φ_{n}) or b_{n} := a(φ_{n}), for all 1 ≤ n ≤ 2N. Then ⟨b_{1} ⋯ b_{2N−1}⟩_{ρ} = 0 and

⟨b_{1} b_{2} ⋯ b_{2N}⟩_{ρ} = ∑_{π∈𝔓_{2N}} ⟨b_{π(1)} b_{π(2)}⟩_{ρ} ⋯ ⟨b_{π(2N−1)} b_{π(2N)}⟩_{ρ}, (A.75)

where 𝔓_{2N} denotes the set of pairings, i.e., the set of all permutations π ∈ 𝔖_{2N} of 2N elements such that π(2n − 1) < π(2n + 1) and π(2n − 1) < π(2n), for all 1 ≤ n ≤ N − 1 and 1 ≤ n ≤ N, respectively.

(iii) There exist two commuting quadratic, semibounded Hamiltonians

H = ∑_{i,j} \{B_{i,j} a^{∗}(ψ_{i}) a(ψ_{j}) + C_{i,j}a^{∗}(ψ_{i}) a^{∗}(ψ_{j}) + \overline{C_{i,j}}a(ψ_{i}) a(ψ_{j})\} (A.76)

H′ = ∑_{i,j} \{B′_{i,j} a^{∗}(ψ_{i}) a(ψ_{j}) + C′_{i,j}a^{∗}(ψ_{i}) a^{∗}(ψ_{j}) + \overline{C′_{i,j}}a(ψ_{i}) a(ψ_{j})\} (A.77)

with B = B^{∗} ≥ 0, C = C^{T} ∈ L^{2}(𝔥), where {ψ_{i}}_{i∈ℕ} ⊆ 𝔥 is an orthonormal basis, such that exp(−H − βH′) is trace class, for all β < ∞, and

⟨A⟩_{ρ} = lim_{β→∞} \{\frac{Tr_{𝔉}[A exp(−H − βH′)]}{Tr_{𝔉}[exp(−H − βH′)]}\},  (A.78)

for all A ∈ B(𝔉).

Eq. (II.36) and the vanishing (ii) of the truncated correlation functions of a centered quasifree state imply that any quasifree state ρ ∈ 𝔔𝔉 is completely determined by its one-point function ⟨a(φ)⟩_{ρ} and its two-point function (one-particle reduced density matrix).

Acknowledgements

The third author was supported for part of this work by DFG Grant No. Ba-1477/5-1 and also by the European Research Council under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC grant agreement 20285.

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Volker Bach Technische Universität Braunschweig Institut für Analysis und Algebra Rebenring 31, A 14 38106 Braunschweig Germany v.bach@tu-bs.de Sébastien Breteaux Technische Universität Braunschweig Institut für Analysis und Algebra Rebenring 31, A 14 38106 Braunschweig Germany sebastien.breteaux@ens-cachan.org

Tim Tzaneteas

Technische Universität Braunschweig Institut für Analysis und Algebra Rebenring 31, A 14 38106 Braunschweig Germany timmy.tzaneteas@uni-tuebingen.de 1520

Documenta Mathematica 18 (2013)