Bogoliubov Transformations Beyond Shale–Stinespring: Generic $v^{*}v$ for bosons

Bogoliubov transformations are a versatile tool for elucidating the physical properties of many–body and quantum field theory (QFT) models in mathematical physics. Applications include the simplification of Hamiltonians in interacting bosonic or fermionic $N$–body systems [1, 2, 3, 4, 5], QFT toy models [6, 7, 8] and relativistic quantum dynamics [9, 10].
In simple words, a Bogoliubov transformation $\mathcal{V}$ is a linear replacement of creation– and annihilation operators $a^{\dagger}\mapsto b^{\dagger},a\mapsto b$ such that $b,b^{\dagger}$ satisfy the same commutation relations as $a,a^{\dagger}$. There are cases where $\mathcal{V}$ allows for finding a unitary operator on Fock space $\mathbb{U}_{\mathcal{V}}:\mathscr{F}\to\mathscr{F}$ such that $\mathbb{U}_{\mathcal{V}}a^{\dagger}\mathbb{U}_{\mathcal{V}}^{-1}=b^{\dagger}$ and $\mathbb{U}_{\mathcal{V}}a\mathbb{U}_{\mathcal{V}}^{-1}=b$. In this case, $\mathbb{U}_{\mathcal{V}}$ is called an implementer of $\mathcal{V}$ and $\mathcal{V}$ is called implementable. The case of implementability is interesting for the following reason: Suppose, we are given a Hamiltonian $H$, which is an inconvenient sum of products of $a,a^{\dagger}$, but there exists a Bogoliubov transformation $\mathcal{V}$ and a constant $c\in\mathbb{C}$, such that $(H+c)$ takes a much more convenient form in the $b,b^{\dagger}$–operators. This situation arises frequently for quadratic Hamiltonians, see Remark 5. Then, $\mathcal{V}^{-1}$, which is implemented by $\mathbb{U}_{\mathcal{V}}^{-1}=\mathbb{U}_{\mathcal{V}}^{*}$, replaces $b,b^{\dagger}$ by $a,a^{\dagger}$. So, under the replacement, $(H+c)$ becomes

$$\widetilde{H}:=\mathbb{U}_{\mathcal{V}}^{-1}(H+c)\mathbb{U}_{\mathcal{V}},$$

(1)

which has a convenient form in the $a,a^{\dagger}$–operators. This convenient form allows for a simple description of the dynamics generated by $\widetilde{H}$, which is by (1) unitarily equivalent to the dynamics generated by $H$.

The important question when $\mathcal{V}$ is implementable has long been settled by Shale and Stinespring [11, 12]: This is the case if and only if for some operator $v$ (16), characterizing $\mathcal{V}$, we have

$$\mathrm{tr}(v^{*}v)<\infty.$$

This so–called Shale–Stinespring condition is rather restrictive. Therefore, the author recently [13] proposed the construction of an implementer in an extended sense, $\mathbb{U}_{\mathcal{V}}$, which also achieves the replacement $a^{\sharp}\mapsto b^{\sharp}$ induced by $\mathcal{V}$, but can be constructed for a much greater class of Bogoliubov transformations. In [13], two Fock space extensions $\widehat{\mathscr{H}},\overline{\mathscr{F}}$ were constructed, each together with an operator $\mathbb{U}_{\mathcal{V}}:\mathcal{D}_{\mathscr{F}}\to\widehat{\mathscr{H}}$ or $\mathbb{U}_{\mathcal{V}}:\mathcal{D}_{\mathscr{F}}\to\overline{\mathscr{F}}$, with $\mathcal{D}_{\mathscr{F}}\subset\mathscr{F}$ being a dense subspace, such that (1) holds as a strong operator identity on $\mathcal{D}_{\mathscr{F}}$. Here, $\widehat{\mathscr{H}}$ is an infinite tensor product space of the form introduced by von Neumann [14] and $\overline{\mathscr{F}}$ is an “extended state space” as recently introduced in [15]. The main result in [13] was that $\mathcal{V}$ can be implemented in the extended sense, whenever

$$v^{*}v\text{ has {discrete spectrum}}.$$

In this general case, $c$ becomes an “infinite renormalization constant”, which was rigorously interpreted in [13] as an element of some vector space $\mathrm{Ren}_{1}$.
The purpose of this article is to propose a new Fock space extension $\overline{\mathcal{E}_{\mathscr{F}}}$, related to $\overline{\mathscr{F}}$, which allows for an extended implementer $\mathbb{U}_{\mathcal{V}}:\mathcal{D}_{\mathscr{F}}\to\overline{\mathcal{E}_{\mathscr{F}}}$

$$\text{in the {bosonic} case for {arbitrary} }v^{*}v,$$

as long as $v$ is defined on a suitable domain. We establish the notation in Section 2, construct $\overline{\mathcal{E}_{\mathscr{F}}}$ in Section 3, state our main result, Theorem 4.1, in Section 4 and prove it in Section 5.
An analogous result can be expected to hold for fermions, as long as 1 is not an eigenvalue of infinite multiplicity of $v^{*}v$, i.e., $\mathcal{V}$ only performs a particle–hole transformation on finitely many modes. The main technical complication in the fermionic case comes from the fact that the operator $\mathcal{O}$ (47) is unbounded, whereas it is bounded in the bosonic case.
An extended implementer $\mathbb{U}_{\mathcal{V}}$ is particularly useful in cases where $H$ is an algebraic expression that does not define an operator on a dense subspace of $\mathscr{F}$, while $\widetilde{H}=\mathbb{U}_{\mathcal{V}}^{-1}(H+c)\mathbb{U}_{\mathcal{V}}$ maps $\mathcal{D}_{\mathscr{F}}$ into itself and allows for a self–adjoint extension. In that case, we can mathematically make sense of $(H+c)$ as an operator mapping $\mathbb{U}_{\mathcal{V}}(\mathcal{D}_{\mathscr{F}})$ into itself (see Figure 1) and use the dynamics generated by $\widetilde{H}$ on $\mathscr{F}$ for a physical interpretation.
The construction of $\overline{\mathcal{E}_{\mathscr{F}}}$ is significantly shorter than that of $\overline{\mathscr{F}}$ in [13]. We intend a future use of $\overline{\mathcal{E}_{\mathscr{F}}}$ as a bookkeeping tool for more general operator transformations that go beyond Bogoliubov transformations.

The notation is adapted from [13] in great parts. We consider a system with an indeterminate number of particles $N$, whose one–particle sector is given by the sequence space $\ell^{2}$.³³3Note that any separable Hilbert space can be identified with $\ell^{2}$ by fixing an orthonormal basis, so the description with $\ell^{2}$ is quite general. The configuration of the system is given by a vector of mode numbers $(j_{1},\ldots,j_{N})$, which is an element of configuration space⁴⁴4The symbol $\sqcup$ denotes the disjoint union of two sets. This notation is chosen to emphasize that, e.g., $\mathbb{N}\sqcup\mathbb{N}^{2}\sqcup\ldots\sqcup\mathbb{N}^{N}$ is not just $\mathbb{N}^{N}$.

$$\mathcal{Q}:=\bigsqcup_{N\in\mathbb{N}_{0}}\mathcal{Q}^{(N)}:=\bigsqcup_{N\in\mathbb{N}_{0}}\mathbb{N}^{N}.$$

(2)

Here, $\mathcal{Q}^{(N)}=\mathbb{N}^{N}$ is also called the $N$–particle sector of configuration space. The state of the system is described by a normed vector in the (mode–) Fock space

$$\mathscr{F}:=L^{2}(\mathcal{Q}).$$

(3)

We impose bosonic symmetry using the symmetrization operator $S_{+}:\mathscr{F}\to\mathscr{F}$ with

$$(S_{+}\Psi)(j_{1},\ldots,j_{N}):=\frac{1}{N!}\sum_{\sigma\in S_{N}}\Psi(j_{\sigma(1)},\ldots,j_{\sigma(N)}),$$

(4)

where $S_{N}$ is the permutation group. The bosonic Fock space is then given by

$$\mathscr{F}_{+}:=S_{+}(\mathscr{F}).$$

(5)

Equivalently, using the symmetric tensor product

$$\phi\otimes_{S}\phi:=S_{+}(\phi\otimes\phi),$$

(6)

we can write

$$\mathscr{F}_{+}=\bigoplus_{N\in\mathbb{N}_{0}}(\ell^{2})^{\otimes_{S}N}.$$

(7)

Denote by $(\boldsymbol{e}_{j})_{j\in\mathbb{N}}$ the canonical basis of $\ell^{2}$ and by $a^{\dagger}_{j},a_{j}$ the creation and annihilation operators corresponding to $\boldsymbol{e}_{j}$, which are a priori just symbolic expressions. Products of these expressions or countable sums thereof are not necessarily defined as Fock space operators. Instead, we interpret them as elements of the ^∗–algebra $\overline{\mathcal{A}}$, defined as follows: Denote by

$$\Pi_{\boldsymbol{e}}:=\big{\{}P_{\boldsymbol{e}}=a_{j_{1}}^{\sharp_{1}}\ldots a_{j_{m}}^{\sharp_{m}}\;\big{|}\;m\in\mathbb{N}_{0},j_{\ell}\in\mathbb{N},\sharp_{\ell}\in\{\cdot,\dagger\}\big{\}}$$

(8)

the set of all finite operator products with respect to the basis $(\boldsymbol{e}_{j})_{j\in\mathbb{N}}$. Then, $\overline{\mathcal{A}}$ is defined as the set of all countable sums

$$\overline{\mathcal{A}}:=\left\{H=\sum_{P_{\boldsymbol{e}}\in\Pi_{\boldsymbol{e}}}H_{j_{1},\sharp_{1},\ldots,j_{m},\sharp_{m}}P_{\boldsymbol{e}}\;\middle|\;H_{j_{1},\sharp_{1},\ldots,j_{m},\sharp_{m}}\in\mathbb{C}\right\},$$

(9)

with $m$ depending on $P_{\boldsymbol{e}}$. Let us denote the space of all complex–valued sequences by

$$\mathcal{E}=\{\mathbb{N}\to\mathbb{C}\}.$$

(10)

For $\boldsymbol{\phi}\in\mathcal{E}$, we then introduce the creation/annihilation operators

$$a^{\dagger}(\boldsymbol{\phi}):=\sum_{j\in\mathbb{N}}\phi_{j}a^{\dagger}_{j},\qquad a(\boldsymbol{\phi}):=\sum_{j\in\mathbb{N}}\overline{\phi_{j}}a_{j},\qquad a^{\dagger}(\boldsymbol{\phi}),a(\boldsymbol{\phi})\in\overline{\mathcal{A}},$$

(11)

where the overline denotes complex conjugation. Alternatively, complex conjugation is described by the complex conjugation operator

$$J:\mathcal{E}\to\mathcal{E},\qquad(J\boldsymbol{\phi})_{j}=\overline{\phi_{j}},$$

(12)

with $J^{2}=1$ and whose adjoint satisfies $(J^{*})^{2}=1$. We will interchangeably use the notations $\phi_{j},(\boldsymbol{\phi})_{j}$ and $(\boldsymbol{\phi})(j)$ for elements of a series $\boldsymbol{\phi}\in\mathcal{E}$. Further, for $\boldsymbol{\phi},\boldsymbol{\psi}\in\ell^{2}$ we impose the canonical commutation relations (CCR) on $\overline{\mathcal{A}}$

$$[a(\boldsymbol{\phi}),a(\boldsymbol{\psi})]=[a^{\dagger}(\boldsymbol{\phi}),a^{\dagger}(\boldsymbol{\psi})]=0,\qquad[a(\boldsymbol{\phi}),a^{\dagger}(\boldsymbol{\psi})]=\langle\boldsymbol{\phi},\boldsymbol{\psi}\rangle.$$

(13)

In case $\boldsymbol{\phi},\boldsymbol{\psi}\in\ell^{2}$, we may also densely define $a^{\sharp}(\boldsymbol{\phi})$ as Fock space operators

	$\displaystyle(a^{\dagger}(\boldsymbol{\phi})\Psi)(j_{1},\ldots,j_{N})$	$\displaystyle=\frac{1}{\sqrt{N}}\sum_{\ell=1}^{N}\phi_{j_{\ell}}\Psi(j_{1},\ldots,j_{\ell-1},j_{\ell+1},\ldots,j_{N})$		(14)
	$\displaystyle(a(\boldsymbol{\phi})\Psi)(j_{1},\ldots,j_{N})$	$\displaystyle=\sqrt{(N+1)}\sum_{j\in\mathbb{N}}\overline{\phi_{j}}\Psi(j_{1},\ldots,j_{N},j),$

which preserve symmetry and satisfy (13) as a strong operator identity on a dense domain in Fock space.
Creation and annihilation operators $a^{\sharp}(\boldsymbol{\phi})$ are particularly easy to handle, if the sum over $j$ is finite, or equivalently, if the form factor is an element of the following space:

$$\mathcal{D}:=\{\boldsymbol{\phi}\in\ell^{2}\;\mid\;\phi_{j}=0\;\text{for all but finitely many}\;j\in\mathbb{N}\}.$$

(15)

Note that $\mathcal{D}\subset\ell^{2}\subset\mathcal{E}$ and $\mathcal{E}=\mathcal{D}^{\prime}$, i.e., $\mathcal{E}$ is the dual space of $\mathcal{D}$ with respect to a suitable seminorm–induced topology, see [16, Chap. 1, Example 1.6].
By an algebraic Bogoliubov transformation, we mean a map $\mathcal{V}_{\mathcal{A}}:\overline{\mathcal{A}}\to\overline{\mathcal{A}}$, that sends

	$\displaystyle a^{\dagger}(\boldsymbol{\phi})$	$\displaystyle\mapsto b^{\dagger}(\boldsymbol{\phi}):=a^{\dagger}(u\boldsymbol{\phi})+a(v\overline{\boldsymbol{\phi}}),\qquad$		(16)
	$\displaystyle a(\boldsymbol{\phi})$	$\displaystyle\mapsto b(\boldsymbol{\phi}):=a^{\dagger}(v\overline{\boldsymbol{\phi}})+a(u\boldsymbol{\phi})$

for all $\boldsymbol{\phi}\in\mathcal{D}$, where $u,v:\mathcal{D}\to\ell^{2}$ satisfy the bosonic Bogoliubov relations as a weak operator identity:

	$\displaystyle u^{*}u-v^{T}\overline{v}$	$\displaystyle=1\qquad$	$\displaystyle u^{*}v-v^{T}\overline{u}=0$		(17)
	$\displaystyle uu^{*}-vv^{*}$	$\displaystyle=1\qquad$	$\displaystyle uv^{T}-vu^{T}=0.$

Here, $u^{*}$ is the adjoint of $u$ and the conjugate and transposed operators are given by $\overline{u}=JuJ$ and $u^{T}=Ju^{*}J$ (and the same for $v$). Writing $u,v$ as matrices of infinite size, $u=(u_{jk})_{j,k\in\mathbb{N}},v=(v_{jk})_{j,k\in\mathbb{N}}$, we equivalently have $(u^{T})_{jk}=(u)_{kj}$, $(u^{*})_{jk}=\overline{(u)_{kj}}$ and $(\overline{u})_{jk}=\overline{(u)_{jk}}$. Further, we may keep track of $\mathcal{V}_{\mathcal{A}}$ by a block matrix $\mathcal{V}=\big{(}\begin{smallmatrix}u&v\\ \overline{v}&\overline{u}\end{smallmatrix}\big{)}$. In [13, Lemma 4.1] it was proven that (17) holds, whenever $\mathcal{V}$ and $\mathcal{V}^{*}$ both preserve the CCR, that is, (13) also holds for $a^{\sharp}$ replaced by $b^{\sharp}$.

We will establish implementability on a Fock space extension $\overline{\mathcal{E}_{\mathscr{F}}}\supset\mathscr{F}$ here, which is related, but not equal to the extensions $\overline{\mathscr{F}},\overline{\mathscr{F}}_{{\rm ex}}$ in [13] and [15]. Just as in [13], we define the generalized $N$–particle space and the generalized Fock space as

$$\mathcal{E}^{(N)}:=\big{\{}\Psi^{(N)}:\mathbb{N}^{N}\to\mathbb{C}\big{\}},\qquad\mathcal{E}_{\mathscr{F}}:=\big{\{}\Psi:\mathcal{Q}\to\mathbb{C}\big{\}}\supset\mathscr{F}.$$

(18)

The definition of $\overline{\mathcal{E}_{\mathscr{F}}}$ is motivated by formal and possibly divergent sums that appear when formally applying one or several operators $a(\boldsymbol{\phi}),\boldsymbol{\phi}\in\mathcal{E}$ as in (14) to functions in $\mathcal{E}_{\mathscr{F}}$. In its most general form, such a formal sum on the $(N)$–sector reads

$$\Psi^{(N)}(j_{1},\ldots,j_{N})=\sum_{j_{N+1},\ldots,j_{N+L}\in\mathbb{N}}\Psi_{(L)}^{(N+L)}(j_{1},\ldots,j_{N+L}),$$

(19)

where $N,L\in\mathbb{N}_{0}$. There are two ways to read (19): First, $\Psi^{(N)}(j_{1},\ldots,j_{N})$ can be viewed as a generalized function value, which is given by a formal sum at fixed $(j_{1},\ldots,j_{N})$. Second, (19) can be seen as a definition for a generalized function on $\mathbb{N}^{N}$ that is obtained by taking the “raw functions” $\Psi_{(L)}^{(N+L)}$ and formally “integrating out” the last $L$ variables.

Mathematically, the sector $\Psi^{(N)}$ is specified by a pair $\big{(}\Psi_{(L)}^{(N+L)},L\big{)}$. In order to allow for countable linear combinations and products of such expressions, we introduce the algebra of functions

$$\overline{\mathcal{E}_{\mathscr{F},0}}:=\Big{\{}\mathbb{N}_{0}\times\mathbb{N}_{0}\to\mathcal{E}_{\mathscr{F}}\;\Big{|}\;(N,L)\mapsto\Psi_{(L)}^{(N+L)}\in\mathcal{E}^{(N+L)}\Big{\}}.$$

(20)

Here, $\overline{\mathcal{E}_{\mathscr{F},0}}$ is to be understood as a $\mathbb{C}$–vector space with linear combinations defined argument–wise, i.e.,

$$(c\Psi+\Psi^{\prime})_{(L)}^{(N+L)}=c(\Psi)_{(L)}^{(N+L)}+(\Psi^{\prime})_{(L)}^{(N+L)},\quad c\in\mathbb{C}.$$

(21)

The algebra multiplication is given by

$$(\Psi\otimes\Psi^{\prime})_{(L)}^{(N+L)}=\sum_{\begin{subarray}{c}N_{1}+N_{2}=N\\ L_{1}+L_{2}=L\end{subarray}}(\Psi)_{(L_{1})}^{(N_{1}+L_{1})}\otimes(\Psi^{\prime})_{(L_{2})}^{(N_{2}+L_{2})},$$

(22)

with $\otimes$ denoting the topological tensor product in $\mathcal{E}_{\mathscr{F}}$. We now introduce the sum notation, where we write each function in $\overline{\mathcal{E}_{\mathscr{F},0}}$ as a formal countable sum

$$\Psi^{(N)}(j_{1},\ldots,j_{N})=\sum_{L\in\mathbb{N}_{0}}\;\sum_{j_{N+1},\ldots,j_{N+L}\in\mathbb{N}}\Psi_{(L)}^{(N+L)}(j_{1},\ldots,j_{N+L}).$$

(23)

This notation is consistent with the notation in (19), and (22) coincides with the formal multiplication of two sums of the kind (23). However, permuting sum indices or executing convergent sums in (23) intuitively leaves the sum invariant, but results in a different associated element in $\overline{\mathcal{E}_{\mathscr{F},0}}$. Therefore, we mod out a (two–sided) ideal⁵⁵5By an ideal of an algebra, we mean that $\mathcal{I}$ shall be closed (in an algebraic sense) under the (commutative) algebra multiplication $\otimes$ and linearity. So for $\Psi_{1},\Psi_{2}\in\mathcal{I},\Psi\in\overline{\mathcal{E}_{\mathscr{F},0}}$ and $c\in\mathbb{C}$, we have $(\Psi\otimes\Psi_{1})\in\mathcal{I}$ and $c\Psi_{1}+\Psi_{2}\in\mathcal{I}$. Intuitively, the elements in $\mathcal{I}$ are the ones equivalent to 0. $\mathcal{I}\subset\overline{\mathcal{E}_{\mathscr{F},0}}$, which is generated by requiring that:

1. (A)

   $\Psi-\Psi^{\prime}\in\mathcal{I}$, whenever each $(\Psi^{\prime})_{(L)}^{(N+L)}$ can be obtained by permuting the last $L$ indices of $(\Psi)_{(L)}^{(N+L)}$. So we may swap indices that are integrated out.
   

2. (B)

   $\Psi-\Psi^{\prime}\in\mathcal{I}$, whenever we may choose for each pair $(N,L)$ some integer $0\leq\Delta L_{N,L}\leq L$, such that

   $$(\Psi^{\prime})_{(L^{\prime})}^{(N+L^{\prime})}(j_{1},\ldots,j_{N+L^{\prime}})=\sum_{L:L^{\prime}+\Delta L_{N,L}=L}\;\sum_{j_{N+L^{\prime}+1},\ldots,j_{N+L}}(\Psi)_{(L)}^{(N+L)}(j_{1},\ldots,j_{N+L})$$

   (24)

   is an absolutely convergent sum for all $N,L^{\prime}\in\mathbb{N}_{0}$ and all $(j_{1},\ldots,j_{N+L^{\prime}})\in\mathbb{N}^{L^{\prime}}$. In other words, we may execute absolutely convergent sums, where $\Delta L_{N,L}$ is the number of sums executed in $(\Psi)_{(L)}^{(N+L)}$.
   

The extended state space is then defined as the quotient algebra

$$\overline{\mathcal{E}_{\mathscr{F}}}:=\overline{\mathcal{E}_{\mathscr{F},0}}/_{\mathcal{I}}.$$

(25)

Its elements (which are cosets of $\mathcal{I}$) can thus be treated as if they were formal sums (23) and we adopt the sum notation also for elements of $\overline{\mathcal{E}_{\mathscr{F}}}$.

Consider any Bogoliubov transformation $\mathcal{V}=\left(\begin{smallmatrix}u&v\\ \overline{v}&\overline{u}\end{smallmatrix}\right)$ with $u,v:\mathcal{D}\to\ell^{2}$ (recall: $\mathcal{D}$ in (15) is the set of all sequences with finite support), as well as the dense domain

$$\mathcal{D}_{\mathscr{F}}:=\mathrm{span}\big{\{}a^{\dagger}(\boldsymbol{\phi}_{1})\ldots a^{\dagger}(\boldsymbol{\phi}_{N})\Omega,\;N\in\mathbb{N}_{0},\;\boldsymbol{\phi}_{\ell}\in\mathcal{D}\big{\}},$$

(31)

where $\Omega\in\mathscr{F}$, defined by $\Omega^{(0)}=1$ and $\Omega^{(N)}=0$ for $N\geq 1$, is the vacuum vector. In analogy to [13, Def. 5.1], we define extended implementability as follows.

Our main result is the following.