Note on the Retarded van der Waals Potential within the Dipole Approximation

We examine the dipole approximated Pauli-Fierz Hamiltonians of the nonrelativistic QED. We assume that the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons can be approximated by harmonic potentials. By an exact diagonalization method, we prove that the binding energy of the two hydrogen atoms behaves as $R^{-7}$, provided that the distance between atoms $R$ is sufficiently large. We employ the Feynman's representation of the quantized radiation fields which enables us to diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture by Casimir and Polder.


Introduction
London was the first to explain attractive interactions between neutral atoms or molecules by applying quantum mechanics [16]. Nowadays, the attractive forces are called the van der Waals-London forces, and are described by the potential energy decaying as R −6 for R sufficiently large. 1 Here, R denotes the distance between two atoms or molecules. It is recognized that these forces come from the quantum fluctuations of the charges inside the atoms. Because even a simple hydrogen atom displays a fluctuating dipole, the van der Waals-London forces are ubiquitous and therefore very fundamental.
Casimir and Polder took the interactions between electrons and the quantized radiation fields into consideration and perfomed the fourth order perturbative computations [6]. They found that the finiteness of the speed of light weakens the correlation between nearby dipoles and causes the attractive potential between atoms to behave as where α A and α B are the static polarizability of the atoms. The potential V CP is called the Casimir-Polder potential or the retarded van der Waals potential. For reviews, see, e.g., [5,11,13,18,19]. Although this result is plausible, Casimir-Polder's arguments are heuristic, and lack mathematical rigor.

T. Miyao
There are few rigorous results concerning the Casimir-Polder potential; In [20,21], Miyao and Spohn gave a path integral formula for V CP and applied it to computing the second cumulant. Under the assumption that all of higher order cumulants behave as O R −9 and their coefficients are small enough to control, they rigorously refound that V CP behaves as R −7 as R → ∞.
Although this assumption appears to be plausible, to prove it is extremely hard. Therefore, to give a mathematical foundation of the Casimir-Polder potential is an open problem even today.
The dipole approximation (C.1) is widely accepted as a convenient procedure in the community of the nonrelativistic QED [24]. The assumption (C.2) is often useful when we study the low energy behavior of the system. Under the assumptions, we prove that the binding energy for two hydrogen atoms actually behaves as R −7 . In the context of the Born-Oppenheimer approximation, this indicates that the effective potential between two hydrogen atoms behaves as R −7 too. This result supports our assumptions for the model without dipole approximation, and is expected to become a starting point for study of the non-approximated model. Our proof relies on the fact that the dipole approximated Hamiltoninas can be diagonalized by applying Feynman's representation of the quantized radiation fields [8]. It has been believed that the dipole approximated model also exhibits R −7 behavior by the forth order perturbation theory. However, the arguments concering the error terms are completely missing. Indeed, this part is tacitly assumed to be trivial in literatures. In this paper, we actually perform systematic error estimates which are far from trivial.
In mathematical physics, it is known that rigorous studies of the Pauli-Fierz Hamiltonian require an extra care due to the infamous infrared problem [4,10,24]. Fortunetely, within the assumptions (C.1) and (C.2), we can control the problem relatively easily.
Before we proceed, we have additional remarks. In his Ph.D. Thesis [12], Koppen studied the retarded van der Waals potential; he examined the Pauli-Fierz model with the dipole approximation (C.1), but the condition (C.2) is not assumed in [12]. In contrast to the present study, he imposed the infrared cutoff σ on the Hamiltonian in order to apply the naive perturbation theory and obtained an expansion formula for the binding energy: E σ (R) = ∞ i=0 e i V σ i (R). Then he removed the infrared cutoff from each term: V i (R) := lim σ→+0 V σ i (R). Finally, he proved that some V i (R) satisfies (1.1). His observation could be regareded as a nice starting point of mathematical analysis of the retarded van der Waals potential, however, there are still some problems to be considered. For example, the magnetic contributions to the −R −7 decay are completely overlooked. In addition, in the mathematical study of the Pauli-Fierz model, it is well-known that to prove that lim is very hard problem, the aforementioned infrared problem.
Our contributions are • to provide a minimal QED model which can rigorously explain the Casimir-Polder potential by a relatively simple and easy way; • to perform systematic error estimates without the infrared cutoff.
In this way, the present paper and the thesis [12] are complementary to each other. Since the electrons obey Fermi-Dirac statistics, the wave functions of the two-electron system belong to (H ∧ H) ⊗ F L 2 R 3 × {1, 2} , where H = L 2 R 3 ⊗ C 2 , the Hilbert space with spin 1/2, the symbol ∧ indicates the anti-symmetric tensor product and F L 2 R 3 × {1, 2} is the Fock space over L 2 R 3 × {1, 2} . Usually, the ground state of this system is a spin singlet. Thus, the spatial part of the ground state is symmetric and we can end up with minimizing the energy in an unrestricted manner on L 2 R 3 ⊗L 2 R 3 ⊗F L 2 R 3 ×{1, 2} . For this reason, we perform our analysis on L 2 R 3 ⊗ L 2 R 3 ⊗ F L 2 R 3 × {1, 2} . 2 However, it should be mentioned that our observation here can not be extended to general N -electron systems, directly.
In fairness, we mention the following two difficulties of the assumptions (C.1) and (C.2). For details, see discussions in Section 9.
• The condition (C.2) breaks the indistinguishability of the electrons.
• Under the conditions (C.1) and (C.2), we cannot reproduce the exact cancellation of the term with R −6 decay (the van der Waals-London potential) by the contribution from the quantized Maxwell field. Note that this cancellation is known to be fundamental to explain the retarded van der Waals potential [20,21].
The present paper is organized as follows. In Section 2, we introduce the dipole approximated Pauli-Fierz Hamiltonian and state the main result. In Section 3, we switch to the Feynman representation of the quantized radiation fields. This representation enables us to diagonalize the Hamiltonians as we will see in the following sections. Further, we introduce a canonical transformation which induces the quantized displacement fields in the Hamiltonians in Section 4. Section 5 is devoted to the finite volume approximation, which is a standard method in the study of the quantum field theory [3,9]. Then we diagonalize the Hamiltonians in Sections 6 and 7. In Section 8, we give a proof of the main theorem. Section 9 is devoted to the discussions of the approximations (C.1) and (C.2). In Appendices A, B and C, we collect various auxiliary results which are needed in the main sections.

Main result
Let us consider a single hydrogen atom with an infinitely heavy nucleus located at the origin 0. The nonrelativistic QED Hamiltonian for this system is given by The nucleus has charge e > 0, and the electron has charge −e. We assume that the charge distribution satisfies the following properties: . Thus the Fourier transformationˆ is real.
, of rapid decrease and smooth.
The smeared Coulomb potential V is given by The photon annihilation operator is denoted by a(k, λ). As usual, this operator satisfies the standard commutation relation: [a(k, λ), a(k , λ ) * ] = δ λλ δ(k − k ). 2 Or we could simply say that one considers the "distinguishable particles", see Section 9 for detail.

Electron 1
Electron 2 x 1 x 2 Nucleus 1 Nucleus 2 r The quantized vector potential A(x) is defined by where ε(k, λ) = (ε 1 (k, λ), ε 2 (k, λ), ε 3 (k, λ)), λ = 1, 2 are polarization vectors. For concreteness, we choose as Note that A(x) is essentially self-adjoint. We will denote its closure by the same symbol. The field energy H f is given by The operator H 1e acts in the Hilbert space Here, ⊗ s indicates the symmetric tensor product. To examine the Casimir-Polder potential, we consider two hydrogen atoms, one located at the origin and the other at r = (0, 0, R) with R > 0. For computational convenience, we define the position of the second electron relative to r, see Figure 1. Then the two-electron Hamiltonian reads The operator H 2e acts in L 2 (R 3 ). The dipole approximation (C. 1) means the following replacement: (2.10) The quantized vector potential A(x) is defined by where ε(k, λ) = (ε 1 (k, λ), ε 2 (k, λ), ε 3 (k, λ)), λ = 1, 2 are polarization vectors. For concreteness, we choose as Note that A(x) is essentially self-adjoint. We will denote its closure by the same symbol. The field energy H f is given by The operator H 1e acts in the Hilbert space h ⊗sn . Here, ⊗ s indicates the symmetric tensor product.
To examine the Casimir-Polder potential, we consider two hydrogen atoms, one located at the origin and the other at r = (0, 0, R) with R > 0. For computational convenience, we define the position of the second electron relative to r, see Fig. 1. Then the two-electron Hamiltonian reads The operator H 2e acts in L 2 R 3 . The dipole approximation (C.1) means the following replacement: By the assumption (C.2), we can take x 1 and x 2 sufficiently small. Therefore, we assume that the Coulomb potential of the nuclei together with the Coulomb interaction between the electrons can be approximated by harmonic potentials. Then one has Note that H D1e and H D2e are self-adjoint and bounded from below [14], because the cross-term dkˆ (k) 2 e ik·r x 1 ·k x 2 ·k becomes very small provided that R is large enough. As for physical discussions of the approximation above, see Section 9 in detail.
In what follows, we assume an additional condition: (A.4) We regard ν 0 as a parameter. Thus, ν 0 is independent of .
Hence, there are three parameters e, R and ν 0 in our models.
1. The constant α E,at is the dipole moment of a decoupled atom, i.e., where h at = − 1 2 ∆ + e 2 ν 2 0 2 x 2 and ψ at is the ground state of h at . Note that x j ψ at is orthogonal to ψ at : ψ at |x j ψ at = 0. Thus, the vectors (h at − 3eν 0 /2) −1 x j ψ at in (2.5) are mathematically meaningful.
2. The restrictions of the parameters in Theorem 2.1 come from technical reasons: As we will see in the later sections, these are needed in order to control the perturbative expansions for E and E(R).

Preliminaries
To prove our main result, let us introduce Feynman Hamiltonians of the nonrelativistic QED [8]. These Hamiltonians can be diagonalized readily as we will see in Sections 6 and 7. First, remark the following identification: For notational convenience, we denote by ε j (·, λ) the multiplication operator by the function ε j (·, λ). We begin with the following lemma.
where A X indicates the restriction of A to X. Then H 1 , H 2 , H 3 and H 4 are subspaces of L 2 R 3 .

Second quantized operators in
Let a(f 1 ⊕ f 2 ) be the annihilation operator acting in As usual, we express this operator as Let F be a real-valued function on R 3 which is finite almost everywhere. The multiplication operator by F is also written as F . The second quantization of F ⊕ F is then given by where indicates the algebraic tensor product. We will denote the closure of dΓ(F ⊕ F ) by the same symbol. Symbolically, we express dΓ(F ⊕ F ) as

Second quantized operators in
Let a λ (f λ ) be the annihilation operator on F(H λ ). We employ the following identifications: F(H λ ). Let F be a real-valued function on R 3 .
Suppose that F is even: F (−k) = F (k) a.e.. dΓ λ (F ) denotes the second quantization of F which acts in F(H λ ). As before, we can also regard dΓ λ (F ) as a linear operator acting in The Fock vacuum in F(H λ ) is denoted by Ψ λ . We will freely use the following notations:

Identifications between
Lemma 3.3. We define a linear operator V : Then V can be extended to the unitary operator. In what follows, we denote the extension by the same symbol. Then we have where the bar indicates the closure of the operator.
To see this, note that µ (1) ii (k) and µ (2) ii (k) are even functions. Thus, Accordingly, we have To check other commutation relations are easy.
Using the above fact, we readily confirm that , it follows that the subspace spanned by the set of vectors and the subspace spanned by the set of vectors Hence, V can be extended to the unitary operator. To check (3.5) is easy.
Lemma 3.4. Let F be a real-valued even function on R 3 . Assume that F is continuous. Then we obtain Proof . For readers' convenience, we will provide a sketch of the proof. We will continue to use the notations in the proof of Lemma 3.3. Set We define dense subspaces of F L 2 R 3 ⊕ L 2 R 3 and H λ by where Lin(S) indicates the linear span of S. As is well-known, dΓ(F ⊕ F ) and 4 λ=1 dΓ λ (F ) are essentially self-adjoint on V 1 and V 2 , respectively. We readily confirm that Therefore, by Lemma 3.3, we obtain This concludes the proof of Lemma 3.4.

Definition of the Feynman Hamiltonians
In this subsection, we introduce the Feynman Hamiltonians. To this end, let Here, h.c. denotes the hermite conjugates of the preceeding terms. Note that A (x) is essentially self-adjoint on V 2 defined by (3.6). We denote its closure by the same symbol. By (3.4) and (3.5), we have the following: Now we define the two-electron Feynman Hamiltonian H F2e by F(H λ ) and is bounded from below, provided that R is sufficiently large.
The following proposition plays an important role in the present paper.
As for the one-electron Feynman Hamiltonian, we obtain the following.
In Remark 7.2, we will explain why the Feynman Hamiltonians are useful.

Canonical transformations
Let U be a unitary operator on L 2 R 3 Then one readily confirms that Here, we used the following fact: where ν 2 = 2ν 2 0 and .
Since due to the assumption (A.3) the last term in (4.3) gives a rapidly decreasing contribution as a function of R to the ground state energy, we ignore this term from now on. Finally, we define By an argument similar to the construction of U , we can construct a unitary operator u on The reason why the last term in the right-hand side of (4.2) appears is as follows. After performing the unitary transformation, we see that U * HF2eU contains the term concerning x1 ·k x2 ·k and (ε(k, λ) · x1)(ε(k, λ) · x2), which is given by Here, we used the fact that dkˆ (k) 2 sin(k · r) x1 ·k x2 ·k = 0. By applying the basic property . We can confirm that [π(k, λ), φ(k , λ )] = −iδ λλ δ(k − k ). Recalling the fact [−id/dx, x] = −i, π(k, λ) and φ(k, λ) can be regarded as a multiplication operator and a differential operator, respectively. Now, we readily check that e iπN f /2 π(k, λ)e −iπN f /2 = φ(k, λ) holds, which corresponds to the relation FxF −1 = −id/dx, where F is the Fourier transformation on L 2 (R). This similarity is a reason why we refer to the unitary operator e iπN f /2 as the Fourier transformation.

Lattice approximated Hamiltonians
In order to exactly compute the ground state energies of H and K, we will first introduce the lattice approximation of Hamiltonians. As we will see in later sections, the approximated Hamiltonians can be regarded as Hamiltonians of finite dimensional harmonic oscillator, which are exactly solvable.
Let M be the (momentum) lattice with a cutoff Λ, namely, For later use, we label the elements of M as Then the lattice approximated Hamiltonians are defined by The lattice approximated operators act in the Hilbert , and we regard 2 * (M ) as a closed subspace of L 2 R 3 . Note that p(k, λ) and q(k, λ) are essentially self-adjoint on the finite particle subspace of 4 λ=1 F(H L,Λ,λ ). In what follows, we denote their closures by same symbols, respectively. q(k, λ) and p(k, λ) is a canonical pair of the photonic displacement coordinate and its conjugate momentum satisfying the standard commutation relations: Recall the identification F(C) = L 2 (R). Using this, we can naturally embed In addition, p(k, λ) and q(k, λ) can be regarded as the differential and multiplication operators, respectively.
The following proposition is a basis for our computation.
in the operator norm topology.
provided that 1 ≤ e 2 ν 2 . This concludes the proof of Lemma 6.1.
Therefore, the ground state energy of K L,Λ is given by the following formula.
Applying the elementary fact we have that Since Q is off-diagonal, (6.4) becomes where Q(s) = s 2 + ω 0 −1/2 Q s 2 + ω 0 −1/2 . In what follows, we will examine the convergence of the right-hand side of (6.5). As we will see, this series absolutely converges and (6.5) is rigorously justified if ν 0 is large enough. We begin with the following basic lemma. Lemma 6.3. We have the following Proof . For each f ∈ 2 (M × {1, . . . , 4}), we have, by (A.3), Then we have the following: (i) For all s ∈ R, holds for all Λ > 0 by the assumption in Theorem 2.1. Thus, the condition a < 1 is satisfied provided that L is sufficiently large.
Applying the result for n = 1, we get the desired result for n ≥ 2. (ii) immediately follows from (i).

Diagonalization II: Two-electron Hamiltonian
Next we will diagonalize H L,Λ . This is actually possible because we employ the Feynman Hamiltonian, see Remark 7.2 for details. By an argument similar to that of the proof of (6.1), H L,Λ can be expressed as By setting Φ = (x 1 , x 2 , q) and Π = (−i∇ 1 , −i∇ 2 , p), we have that where By an argument similar to that in the proof of Proposition 6.2, we get the following useful formula.
Proposition 7.1. Let E L,Λ (R) = inf spec(H L,Λ ). If 1 ≤ √ 2eν 0 and √ 2e ˆ * < 1, then Ω ≥ 0 and Remark 7.2 (Why are the Feynman Hamiltonians helpful?). From the expression (5.1), we see thatÊ L,Λ (x) can be written as a sum of multiplication operators q(k, λ). As we already knew, this fact is a key to the diagonalization of H L,Λ . In contrast to the Feynman Hamiltonians, in the standard representation,Ê L,Λ (x) corresponds to the following operator: In (7.1), both multiplication and differential operators appear, provided that x = 0. At first glance, it appears that diagonalizing the Hamiltonians in this representation requires extra efforts.
Proof . Note that s 2 + Ω 0 −1 , Q 1 (s) 2 and Q 2 (s) 2 are diagonal operators, while Q 1 (s)Q 2 (s) and Q 2 (s)Q 1 (s) are off-diagonal operators, see Appendix A. Hence, if |I| is an odd number, then and where Q * I\{i 1 } (s) = Q I\{i 1 } (s) * . For all R > 0, we have the following: (i) For each s ∈ R and I ∈ I (e) 2j , where D(s) is given by (6.6). Thus, Note that as we mentioned in Lemma 6.4, the condition a < 1/4 is satisfied provided that L is large enough.
which implies that we conclude (i). From Lemma 6.3 and (6.8), we obtain that Hence, In the second inequality, we have used the fact that #I by (6.7).

Proof of Theorem 2.1
For each I ∈ I (e) 2j , #I indicates the cardinality of I. Notice that #I is different from |I| = i 1 + · · · + i 2j .

Analysis of Q I with #I = 2
We claim that In this subsection, we will examine the following terms: In Appendix B, we will prove the following lemmas.
where g is a constant independent of e, ν 0 and R. Moreover,

Case 1
In Appendix B, we will prove the following lemma.
where C is a positive number independent of e, I, R and ν 0 .

Case 2
The purpose here is to prove Lemma 8.6 below. To this end, we begin with the following lemma. By (A.3) and (A.4), we readily show that This concludes the proof of Lemma 8.5.
Proof . By the assumption in the condition Case 2, there exist at least two numbers m, n ∈ {1, 2, . . . , 2j − 1} such that i m + i m+1 = i n + i n+1 = 3. Hence, I can be decomposed as Without loss of generality, we may assume that {i m , i m+1 } = {i n , i n+1 } = {1, 2}. Thus, Let Q I (s) = Q i 1 (s)Q i 2 (s) · · · Q i 2j (s). By the Schwarz inequality, we have First, we estimate Φ 1 . By the cyclic property of the trace, we have where A = {a 1 , a 2 , . . . , a #A }, we have, by Lemma 8.5, Thus, by (8.5) and the cyclic property of the trace, As for Φ 2 , we have By an argument similar to the one in the proof of (8.6), one obtains that By using the fact Q B (s) ≤ √ 2 ν ˆ * 2#B and (8.7), we have Combining (8.4), (8.6) and (8.8), we arrive at (8.9) We will estimate the three terms in the right-hand side of (8.9 (8.10) Note that because ˆ 2 L 2 /3ν 2 < 1, the right-hand side of (8.10) converges. On the other hand, using Lemma 8.6, one obtains that j≥3 I∈I (e) 2j,2 Note that because lim L→∞ c L = c ∞ < 1/2, the right-hand side of (8.11) converges, provided that L is sufficiently large. Combining (8.9), (8.10) and (8.11), and using Lemma 8.2, we finally arrive at This concludes the proof of Theorem 2.1.

Indistinguishability of the electrons
The original Hamiltonian H 2e has the indistinguishability of the electrons, i.e., the Hamiltonian is unchanged under the exchange of x 1 ↔ x 2 + r. In contrast to this, the approximated Hamiltonian H D2e breaks the indistinguishability. Nevertheless, the Hamiltonian H D2e does explain the Casimi-Polder potential as we show in Theorem 2.1. The distinguishability comes from the assumptions (C.1) and (C.2). However, to justify the assumptions is still open. One way to avoid the unjustified derivation of H D2e is to directly start with the Hamiltonian H given by (4.3) without the last term, which can for instance be directly taken from [24, equation (13.127)] and then extended to the two-particle case. Alternatively and equivalently, the many-particle case is presented, e.g., in [17,Section 4]. If we start from this form, the necessary assumptions are stated as follows: • We assume distinguishability of the two electrons by localizing electron 1 at 0, such that electron 1 experiences the fieldÊ(0), while electron 2 is localized at r and hence experiences the fieldÊ(r).
• We discard all self-interaction terms and approximate the atomic Coulomb potential by a harmonic potential.
In this manner, we can construct a minimal QED model which describes the Casimir-Polder potential. Note that, since the particle 1 and 2 only communicate via the photon field, and due to distinguishability, the actual choice of coordinate systems is insubstantial such that we can choose for particle 2 a coordinate system that is centered at r.

Cancellation mechanism of the van der Waals-London force
As we performed in [20], the attractive R −7 decay (the retarded van der Waals potential) appears due to the exact cancellation of the terms with R −6 decay (the van der Waals-London potential) originating from V R by the contribution from the quantized radiation field. Note that the conditions (A.1)-(A.3) are assumed in [20] as well, but (C.1) and (C.2) are not. As we saw in the present paper, this kind of the cancellation mechanism cannot be reproduced under the conditions (C.1), (C.2) and (A.1)-(A.4). In this sense, our assumptions, especially (C.1) and (C.2) would be unphysical. In many literatures, the retardation on the van der Waals potential is examined under the condition (C.1) alone. In these studies, the cancellation of the terms with R −6 decay is presupposed and only the 4-th order perturbation theory is performed without estimating higher order terms. 6 As far as we know, to examine the exact cancellation mechanism under only the condition (C.1) is still unsolved. This problem could be a key to achieving mathematically complete understanding of the retarded van der Waals potential.
For readers' convenience, we will explain how to compute the integral (B.6). Let ϕ(x) be the Fourier transformation ofˆ 2 rad (r): ϕ(x) = (2π) −1/2 R e −irxˆ 2 rad (r)dr. Here, we extendˆ 2 rad to a function on R byˆ 2 rad (−r) :=ˆ 2 rad (r) for r > 0. Note that ϕ(x) decays rapidly by the assumption (A.3). By the convolution theorem in the Fourier analysis, we have and ∞ 0 dr [Here, we explain how we derive (B.7). First, we observe that we have where ϕ R (x) = Rϕ(Rx) and (g * h)(x) = R g(y)h(x − y)dy. Because we get (B.7).] Hence, by the dominated convergence theorem, we obtain Similarly, we obtain that Summarizing the above results, we arrive at where we used the following formula in [20]: As for the contribution from I ir , we have, by an argument similar to that of the computation concerning with I re , dX 2 S(X 1 , X 2 ) r 4 1 r 4 2 e ir 1 X 1 e ir 2 X 2 × I ir e 2 ν 2 ; r 2 1 /R 2 ; r 2 2 /R 2 ˆ 2 rad (r 1 /R)ˆ 2 rad (r 2 /R) = const · R −9 + o R −9 .
To summarize, we obtain that This concludes the proof of Lemma 8.1.
To justify this rough argument, we carefully have to treat the oscillatory integral as we did in the proof of Lemma 8.1. Similarly, we see that lim Λ→∞ lim L→∞ Q 1 Q 2 Q 2 Q 1 = g e 2 ν 6 R −9 + o R −9 .
Since ϕ(x) decays rapidly, we readily see that the integral in (B.14) is uniformly bounded provided that R is sufficiently large.