Infinite-Dimensional Clifford Algebras and CAR with Inner Conjugation

Introduction

The Clifford algebra of a finite-dimensional quadratic space is generated by finitely many anticommuting elements and is a finite-dimensional algebra. Passing to an infinite-dimensional space changes two things at once: the algebra becomes infinite-dimensional, and the completion needed to obtain a well-behaved normed algebra introduces the theory of operator algebras. The object that results is the canonical anticommutation relation algebra, or CAR algebra, and it is the infinite-dimensional Clifford algebra in the same sense that the finite-dimensional one is the quotient of a tensor algebra: it is generated by a Hilbert space subject to $c(f)^2=\|f\|^2$, completed in the operator norm of its Fock representation. The Fock representation on the exterior algebra is the analogue of the irreducible Clifford module, and its vacuum representation is unique up to unitary equivalence, which is the infinite-dimensional substitute for the uniqueness of the irreducible module over a simple finite-dimensional algebra.

The finite-dimensional theory used here is that of the previous articles: the Clifford algebra $\mathrm{Cl}(V,q)$ with its parity grading, the spinor module, the complex classification $\mathbb{C}\mathrm{l}_{2m}\cong M_{2^m}(\mathbb{C})$, $\mathbb{C}\mathrm{l}_{2m+1}\cong M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C})$, and the eightfold periodicity. Only the finite-dimensional facts are taken as given; the passage to the infinite dimension is the work of this article.

The anticommutation relations are read over a complex Hilbert space $\mathcal{H}$ with inner product $\langle\cdot,\cdot\rangle$, conjugate-linear in the first variable and linear in the second. Writing $\mathcal{H}_{\mathbb{R}}$ for the underlying real Hilbert space with the quadratic form $q(f)=\langle f,f\rangle=\|f\|^2$, the algebraic Clifford algebra of $\mathcal{H}_{\mathbb{R}}$ is generated by $\mathcal{H}_{\mathbb{R}}$ subject to $c(f)^2=q(f)1$; the CAR algebra is its appropriate completion.

The Clifford Algebra of a Hilbert Space

Definition. Let $\mathcal{H}$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $q(f)=\langle f,f\rangle$. The algebraic Clifford algebra $\mathrm{Cl}(\mathcal{H})$ is the quotient of the tensor algebra $T(\mathcal{H})$ by the two-sided ideal generated by the elements $f\otimes f-q(f)1$.

Proposition. $\mathrm{Cl}(\mathcal{H})$ is the union, over the finite-dimensional subspaces $V\subseteq\mathcal{H}$, of the finite-dimensional Clifford algebras $\mathrm{Cl}(V,q)$:

$$ \mathrm{Cl}(\mathcal{H})=\varinjlim_{V}\mathrm{Cl}(V,q). $$

It is a unital associative algebra, generated by $\mathcal{H}$, with the relations $fg+gf=2\operatorname{Re}\langle f,g\rangle1$ in the real case and, in the complex case, the complex-linear relations obtained by complexification.

Proof. A finite-dimensional subspace $V$ determines a Clifford algebra $\mathrm{Cl}(V,q)$, and an inclusion $V\subseteq W$ induces an injective algebra homomorphism $\mathrm{Cl}(V,q)\to\mathrm{Cl}(W,q)$, because the larger algebra is free over the smaller with a basis of products of vectors outside $V$; the transition maps are therefore injective and the colimit is an algebra. The relations are those of each finite stage.

Remark. The algebraic Clifford algebra carries a $\mathbb{Z}/2$-grading and admits all the finite-dimensional structure of the previous articles — the parity grading, the reversion and conjugation anti-involutions, the volume elements of finite-dimensional subspaces — but it is not complete in any natural norm, and it has no finite-dimensional irreducible module: the irreducible modules appear only after completion, as the representation spaces of the completed algebra. The completion is the subject of the next sections.

The Canonical Anticommutation Relations

The generators of the complex Clifford algebra are reorganised into creation and annihilation operators, and the resulting relations are the canonical anticommutation relations.

Definition. Let $\mathcal{H}$ be a complex Hilbert space. The canonical anticommutation relation algebra (CAR algebra) is the universal unital $\mathbb{C}$-algebra generated by elements $a(f)$, $f\in\mathcal{H}$, subject to

$$ a(f)a(g)+a(g)a(f)=0, \qquad a(f)a(g)^{*}+a(g)^{*}a(f)=\langle f,g\rangle1, $$

for all $f,g\in\mathcal{H}$, with $f\mapsto a(f)$ antilinear. The creation operators are the adjoints $a(f)^{*}$, and the field operators are

$$ c(f)=a(f)+a(f)^{*}. $$

Proposition. The field operators satisfy the Clifford relations in the real form

$$ c(f)c(g)+c(g)c(f)=2\operatorname{Re}\langle f,g\rangle1, \qquad c(f)^{2}=\|f\|^{2}1, $$

and for $\lambda\in\mathbb{C}$ one has $c(\lambda f)=\operatorname{Re}\lambda\,c(f)+\operatorname{Im}\lambda\,c(if)$, the complex structure entering through $c(if)=i\bigl(a(f)^{*}-a(f)\bigr)$, as follows from the antilinearity of $f\mapsto a(f)$ and the linearity of $f\mapsto a(f)^{*}$. The map $f\mapsto c(f)$ extends to an algebra isomorphism of the complexified Clifford algebra of $\mathcal{H}_{\mathbb{R}}$ onto the algebra generated by the $c(f)$.

Proof. Expanding $c(f)c(g)+c(g)c(f)$ gives $a(f)a(g)+a(f)a(g)^{*}+a(f)^{*}a(g)+a(f)^{*}a(g)^{*}$, together with the same four terms for $g,f$; the terms $aa$ and $a^*a^*$ cancel in pairs, and the mixed terms combine by the defining relation to $\langle f,g\rangle+\langle g,f\rangle=2\operatorname{Re}\langle f,g\rangle$. The square follows from $g=f$. The last statement is the complexification of the real Clifford algebra, since $c(f)$ is real-linear in $f$ only through the real structure and the operators $c(f),c(if)$ together generate the complexified algebra.

Remark. The CAR relations are therefore the Clifford relations in operator form. The two descriptions differ in bookkeeping: the Clifford description uses the real Hilbert space and its quadratic form, and the CAR description splits the complexification into the $\pm i$-eigenspaces of the complex structure, so that the creation operators correspond to one maximal isotropic subspace and the annihilation operators to the other. This is the infinite-dimensional form of the splitting $V=W\oplus W'$ of the spinor construction.

The Fock Representation

The exterior algebra of $\mathcal{H}$ carries a canonical representation of the CAR, and it is the infinite-dimensional spinor module.

Definition. The Fock space is the Hilbert space completion of the exterior algebra

$$ \mathcal{F}=\overline{\bigoplus_{k\ge0}\Lambda^{k}\mathcal{H}}, $$

with the inner product in which the exterior powers are mutually orthogonal. The unit vector $\Omega=1\in\Lambda^0\mathcal{H}$, of norm one, is the vacuum.

The representation is defined on the exterior algebra by creation and annihilation: for $f\in\mathcal{H}$, let $a(f)^{*}$ be exterior multiplication by $f$ and let $a(f)$ be its adjoint, the contraction. Explicitly, on decomposable elements,

$$ a(f)^{*}(g_1\wedge\cdots\wedge g_k)=f\wedge g_1\wedge\cdots\wedge g_k, $$

$$ a(f)(g_1\wedge\cdots\wedge g_k)=\sum_{j=1}^{k}(-1)^{j-1}\langle f,g_j\rangle\,g_1\wedge\cdots\widehat{g_j}\cdots\wedge g_k. $$

Theorem. The operators $a(f),a(f)^{*}$ extend to bounded operators on $\mathcal{F}$ and satisfy the canonical anticommutation relations. The representation is irreducible: the only closed subspaces invariant under all $a(f),a(f)^{*}$ are $\{0\}$ and $\mathcal{F}$. The vacuum $\Omega$ is annihilated by every $a(f)$ and is cyclic.

Proof. Exterior multiplication and contraction are bounded on the finite exterior powers and preserve the direct sum decomposition by degree up to one, so they extend to bounded operators of norm $\|f\|$. A direct computation on decomposables verifies the relations. For irreducibility, the span of the vectors $a(f_1)^{*}\cdots a(f_k)^{*}\Omega$ is the full exterior algebra, and any invariant closed subspace containing a nonzero vector contains, by applying an appropriate sequence of annihilators and creators, a multiple of $\Omega$; since $\Omega$ is annihilated by all $a(f)$ and the creators generate from $\Omega$, invariance forces the whole space. The annihilation of the vacuum is the definition of the contraction on $\Lambda^0$, and cyclicity is the same spanning statement.

Uniqueness of the Fock Representation

The vacuum determines the representation, and this is the infinite-dimensional form of the uniqueness of the irreducible module.

Definition. A vacuum representation of the CAR algebra is a representation on a Hilbert space together with a unit vector $\Omega$ such that $a(f)\Omega=0$ for all $f$ and such that $\Omega$ is cyclic for the algebra generated by the creation operators.

Theorem. The Fock representation is the unique irreducible vacuum representation, up to unitary equivalence. Equivalently, the vacuum state $\omega$, the state with $a(f)\Omega=0$ for all $f$, is unique, and its GNS representation is the Fock representation on $\Lambda^{\bullet}\mathcal{H}$.

Proof. Let $\omega$ be a state with $a(f)\Omega=0$ for all $f$ in its GNS representation. Then the value of $\omega$ on any word in the $a$ and $a^{*}$ is computed by moving all annihilators to the right with the anticommutation relations; because $a(f)\Omega=0$, every word with more annihilators than creators vanishes, and every word with equal numbers reduces to a product of inner products. Hence $\omega$ is determined by the two-point function $\omega(a(g)a(f)^{*})=\langle g,f\rangle$, every word being evaluated by Wick's theorem: $\omega$ of a product of annihilation and creation operators is the sum over all complete pairings of the operators of the products of the two-point functions, with the sign of the permutation that brings each pair together. In particular $\omega$ vanishes on every word in which the numbers of annihilators and creators differ. This is the vacuum state of the Fock representation, and two representations with the same state are unitarily equivalent by the GNS construction.

Corollary. The Fock representation is faithful, and the CAR algebra is a simple algebra with a unique vacuum state. For finite-dimensional $\mathcal{H}$ of dimension $m$ the Fock space has dimension $2^m$ and the representation is the irreducible spinor module of the finite-dimensional classification; the vacuum is the line $\Lambda^0\mathcal{H}$.

Remark. Faithfulness is not the same as uniqueness of all representations. The vacuum state is unique, but there are many other states — for instance the tracial state, and the states defined by quasifree assignments other than the vacuum — and their GNS representations are generally inequivalent to the Fock representation. The uniqueness statement is therefore about the vacuum representation, which is the one that generalises the irreducible Clifford module, and not about the representation theory of the algebra as a whole, which is much richer. What is genuinely unique is the Fock space $\Lambda^{\bullet}\mathcal{H}$ together with its vacuum line; everything else is the ordinary multiplicity theory of modules.

The CAR Algebra as a C*-Algebra

The bounded operators of the Fock representation generate a C*-algebra, and it is a familiar one.

Definition. The CAR algebra $\mathcal{A}(\mathcal{H})$ is the norm closure in the Fock representation of the algebra generated by the $a(f)$ and $a(f)^{*}$:

$$ \mathcal{A}(\mathcal{H})=\overline{\operatorname{span}\bigl\{\,a(f_1)^{*}\cdots a(f_k)^{*}a(g_l)\cdots a(g_1)\ :\ k,l\geq0,\ f_i,g_j\in\mathcal{H}\,\bigr\}}^{\ \|\cdot\|}\subseteq \mathcal{B}(\mathcal{F}). $$

Theorem. $\mathcal{A}(\mathcal{H})$ is a unital simple C*-algebra. If $\mathcal{H}$ is separable and infinite-dimensional, then

$$ \mathcal{A}(\mathcal{H})\cong M_{2^{\infty}}(\mathbb{C})=\bigotimes_{k=1}^{\infty}M_2(\mathbb{C}), $$

the infinite tensor product of the $2\times2$ complex matrix algebras, called the uniformly hyperfinite (UHF) algebra of type $2^{\infty}$.

Proof. Choose an orthonormal basis $e_1,e_2,\dots$ of $\mathcal{H}$. The subspace $\mathcal{H}_m=\operatorname{span}\{e_1,\dots,e_m\}$ gives a finite-dimensional Clifford algebra with Fock space $\Lambda^{\bullet}\mathcal{H}_m$ of dimension $2^m$ and operator algebra $M_{2^m}(\mathbb{C})$; the inclusion $\mathcal{H}_m\subseteq\mathcal{H}_{m+1}$ gives an injective unital embedding $M_{2^m}(\mathbb{C})\hookrightarrow M_{2^{m+1}}(\mathbb{C})$ whose image is $M_{2^m}(\mathbb{C})\otimes1$, so the norm closure is the infinite tensor product of copies of $M_2(\mathbb{C})$. Simplicity is inherited from the simple finite-dimensional stages and the fact that the union is dense; the details are the standard theory of UHF algebras.

Remark. The isomorphism exhibits the infinite-dimensional Clifford algebra as a direct limit of the finite-dimensional ones, and it is the precise sense in which the CAR algebra is the "completed Clifford algebra". The finite-dimensional algebras of the classification are the truncations, and the eightfold structure of the finite case is replaced in the infinite case by the classification of the states and of the automorphism group.

The Exterior Algebra and the Complex Structure

The Fock space is the exterior algebra, and the complex structure on $\mathcal{H}$ is the data that separates creation from annihilation.

Theorem. The Fock space $\mathcal{F}$ is isomorphic to the exterior algebra $\Lambda^{\bullet}\mathcal{H}$ as a graded $\mathcal{A}(\mathcal{H})$-module, with the grading by exterior degree. The number operator $N=\sum_j a(e_j)^{*}a(e_j)$, defined as the generator of the one-parameter group that multiplies $\Lambda^{k}\mathcal{H}$ by $e^{itk}$, is an unbounded self-adjoint operator with spectrum the non-negative integers, and $a(f)$ and $a(f)^{*}$ lower and raise the degree by one.

Proof. This is the reading of the definitions: the creation operator is exterior multiplication and the annihilator is contraction, so the degree is raised and lowered. The series defining $N$ converges as a quadratic form on the finite exterior powers and extends to a self-adjoint operator on its domain; the spectrum is the set of degrees, each with the finite multiplicity $\binom{m}{k}$ when the dimension is finite.

The Hodge structure. On the exterior algebra of a finite-dimensional $\mathcal{H}$ there is the Hodge star operator, which sends $\Lambda^{k}\mathcal{H}$ to $\Lambda^{m-k}\mathcal{H}$ and is a conjugation up to a sign; it is the finite-dimensional incarnation of the reality condition on the spinor module. In the infinite dimension the star operator does not extend to a bounded operator, and the reality conditions of Real Spinors and Reality Conditions with Inner Conjugation are formulated instead in terms of the charge-conjugation operator on the Fock space, which is an antilinear unitary commuting or anticommuting with the field operators according to the parity of the spin.

The complex structure. The complex structure $J_0$ of $\mathcal{H}$ — multiplication by $i$ — defines the splitting of the complexified Clifford algebra into the $+i$ and $-i$ eigenspaces, and the creation and annihilation operators are the projections onto the two eigenspaces. A different complex structure on the same real Hilbert space produces a different splitting and a different Fock representation; the two vacua are related by a Bogoliubov transformation, and the transformation is implementable by a unitary exactly when the two complex structures differ by a Hilbert–Schmidt operator. This is the content of the theorem of the next section.

Bogoliubov Transformations

An orthogonal transformation of the underlying real Hilbert space induces an automorphism of the Clifford algebra, and the question of which automorphisms preserve the Fock representation is the infinite-dimensional replacement for the action of the finite orthogonal group.

Definition. Let $T\colon\mathcal{H}_{\mathbb{R}}\to\mathcal{H}_{\mathbb{R}}$ be a real orthogonal transformation, $\langle Tf,Tg\rangle=\langle f,g\rangle$. By the universal property it induces an algebra automorphism $\alpha_T$ of $\mathrm{Cl}(\mathcal{H})$ with $\alpha_T(c(f))=c(Tf)$. The induced map on the CAR algebra is a Bogoliubov transformation.

Theorem (Shale–Stinespring). Let $T$ be a real orthogonal transformation of $\mathcal{H}$. The automorphism $\alpha_T$ of the CAR algebra is implementable in the Fock representation — that is, there is a unitary $U_T$ on $\mathcal{F}$ with $\alpha_T(x)=U_TxU_T^{*}$ for all $x$ — if and only if

$$ T-\mathrm{id} \ \text{ is Hilbert–Schmidt}. $$

When this holds, $U_T$ is determined up to scalar of modulus one, and the assignment $T\mapsto U_T$ is a projective representation of the group of implementable transformations.

Proof sketch. The criterion is the standard one. Writing $T$ in a basis that puts it in block form with respect to a polarisation, the unitary is formally an exponential of a quadratic expression in the creation and annihilation operators, and its convergence is equivalent to the Hilbert–Schmidt condition on the off-diagonal blocks, which for an orthogonal $T$ is equivalent to the condition on $T-\mathrm{id}$. The scalar ambiguity comes from the phase of the vacuum.

Corollary. The restricted orthogonal group $O_2(\mathcal{H})$, the group of real orthogonal transformations $T$ with $T-\mathrm{id}$ Hilbert–Schmidt, acts on the Fock representation by unitaries, and the resulting map is a strongly continuous projective representation. Its finite-dimensional truncations are the orthogonal groups $O(V)$ of the finite-dimensional subspaces $V\subseteq\mathcal{H}$ of the classification, whose identity components are the special orthogonal groups. The transformations with $T-\mathrm{id}$ not Hilbert–Schmidt induce automorphisms of the algebra that are not spatially implemented; this is a genuinely infinite-dimensional phenomenon, since in the finite-dimensional case every isometry is a product of reflections and every automorphism of the Clifford algebra is realised on its module.

The Infinite-Dimensional Pin and Spin Groups

The Clifford group and the Pin and Spin groups have infinite-dimensional analogues, obtained by the same construction as in the finite case from the units of the algebra.

Definition. Let $\mathrm{Cl}(\mathcal{H})=\bigcup_V\mathrm{Cl}(V,q)$ be the algebraic Clifford algebra, the union of the Clifford algebras of the finite-dimensional subspaces $V\subseteq\mathcal{H}_{\mathbb{R}}$. Let $\Gamma(\mathcal{H})$ be the group of its units $x$ for which the signed inner conjugation $\mathrm{Ad}^{\alpha}_x$ maps the space of field operators, that is $\mathcal{H}_{\mathbb{R}}$, into itself, and let $\mathrm{Pin}(\mathcal{H})$ be the subgroup on which the Clifford norm is $\pm1$; the group $\mathrm{Spin}(\mathcal{H})$ is its even part. Since every element of $\mathrm{Cl}(\mathcal{H})$ lies in $\mathrm{Cl}(V)$ for some finite-dimensional $V$, one has $\Gamma(\mathcal{H})=\bigcup_V\Gamma(V)$ and $\mathrm{Pin}(\mathcal{H})=\bigcup_V\mathrm{Pin}(V)$.

Theorem (over $\mathbb{R}$). There are exact sequences

$$ 1\to\{\pm1\}\to\mathrm{Pin}(\mathcal{H})\xrightarrow{\ \mathrm{Ad}^{\alpha}\ } O_1(\mathcal{H}_{\mathbb{R}})\to1, \qquad 1\to\{\pm1\}\to\mathrm{Spin}(\mathcal{H})\xrightarrow{\ \mathrm{Ad}^{\alpha}\ } SO_1(\mathcal{H}_{\mathbb{R}})\to1, $$

where $O_1(\mathcal{H}_{\mathbb{R}})$ is the union of the finite-dimensional orthogonal groups $O(V)$ over the finite-dimensional subspaces $V\subseteq\mathcal{H}_{\mathbb{R}}$ — the subgroup of the isometries of $\mathcal{H}_{\mathbb{R}}$ fixing some finite-codimensional subspace pointwise — and $SO_1$ is its special part.

Proof. The finite-dimensional construction carries over to the colimit, since the signed inner conjugation is compatible with the inclusions $V\subseteq W$: an element $x$ of $\mathrm{Cl}(\mathcal{H})$ lies in $\mathrm{Cl}(V)$ for a finite-dimensional $V$, and for such an $x$ the signed inner conjugation preserves $V$ and acts as the identity on $V^{\perp}$: an element $y\in V^{\perp}$ anticommutes with every vector of $V$, so for $x$ a product of $k$ vectors of $V$ one has $\alpha(x)=(-1)^{k}x$ and $yx^{-1}=(-1)^{k}x^{-1}y$, whence $\mathrm{Ad}^{\alpha}_x(y)=\alpha(x)yx^{-1}=y$. The kernel is the scalars, cut down to $\{\pm1\}$ by the norm condition, exactly as in the finite case; each $\Gamma(V)$ maps onto $O(V)$ with the scalars as kernel, so the image of $\mathrm{Pin}(\mathcal{H})$ is $\bigcup_VO(V)=O_1(\mathcal{H}_{\mathbb{R}})$, and the even part gives the special orthogonal groups.

Remark. The full orthogonal group $O(\mathcal{H}_{\mathbb{R}})$ is not the image of the algebraic Clifford group: an isometry that moves every finite-dimensional subspace — a shift, for instance — is not a finite product of reflections and is not reached by the colimit. It is reached instead at the level of the completed algebra: every $T\in O(\mathcal{H}_{\mathbb{R}})$ induces an automorphism $\alpha_T$ of the CAR algebra, and by the Shale–Stinespring criterion of the previous section $\alpha_T$ is implemented by a unitary of the Fock space exactly when $T-\mathrm{id}$ is Hilbert–Schmidt, that is, exactly for $T$ in the restricted orthogonal group $O_2(\mathcal{H})$. The implementable part of the orthogonal action is therefore $O_2$, and it is this group, rather than the full orthogonal group, that plays the role of the structure group of the Fock representation: the unitaries implementing $O_2$ form a central extension of $O_2$ by the circle group, and the image of the algebraic group $\mathrm{Spin}(\mathcal{H})$ is the part of that extension covering $O_1$, the circle recording the phase of the vacuum. The finite-dimensional eightfold structure is not visible as a decomposition of the infinite group; what replaces it is the classification of the states and the periodicity of the finite truncations, and the infinite group is the colimit of the finite eightfold sequence.

Summary

The Clifford algebra of an infinite-dimensional Hilbert space is the union of the finite-dimensional Clifford algebras of its finite-dimensional subspaces. Reorganised into creation and annihilation operators it becomes the canonical anticommutation relation algebra, generated by $a(f)$ with $a(f)a(g)^{*}+a(g)^{*}a(f)=\langle f,g\rangle$ and $a(f)a(g)+a(g)a(f)=0$; the field operators $c(f)=a(f)+a(f)^{*}$ satisfy the Clifford relations in the real form.

The Fock space $\mathcal{F}=\Lambda^{\bullet}\mathcal{H}$ carries the Fock representation, in which creation is exterior multiplication and annihilation is contraction; it is irreducible, faithful, and the vacuum $\Omega$ is annihilated by all annihilators and is cyclic. The vacuum representation is unique up to unitary equivalence, and the vacuum state is determined by the two-point function $\omega(a(g)a(f)^{*})=\langle g,f\rangle$, with every word evaluated by Wick's theorem; its GNS representation is the Fock representation.

The norm closure of the representation is a simple C*-algebra, isomorphic for separable infinite-dimensional $\mathcal{H}$ to the UHF algebra $M_{2^\infty}(\mathbb{C})=\bigotimes_kM_2(\mathbb{C})$, exhibiting it as the direct limit of the finite-dimensional Clifford algebras of the classification. The exterior algebra is the Fock space, graded by the number operator, and the complex structure of $\mathcal{H}$ is the data that separates creation from annihilation. Orthogonal transformations of $\mathcal{H}_{\mathbb{R}}$ induce Bogoliubov automorphisms of the CAR algebra; by the Shale–Stinespring criterion such an automorphism is implemented by a unitary of the Fock space exactly when $T-\mathrm{id}$ is Hilbert–Schmidt, and the implementable transformations form the restricted orthogonal group. The infinite-dimensional Pin and Spin groups are the units of the algebraic Clifford algebra with signed inner conjugation action on the field operators and Clifford norm $\pm1$; they double-cover the union of the finite-dimensional orthogonal groups, and the part of the orthogonal action implemented by unitaries of the Fock space is the restricted orthogonal group $O_2$.

Summary of Notation

Symbol	Meaning
$\mathcal{H}$	Complex Hilbert space, $\langle\cdot,\cdot\rangle$ conjugate-linear in the first variable
$\mathcal{H}_{\mathbb{R}}$	Underlying real Hilbert space, $q(f)=\\|f\\|^2$
$\mathrm{Cl}(\mathcal{H})$	Algebraic Clifford algebra, $\varinjlim_V\mathrm{Cl}(V,q)$
$a(f)$, $a(f)^{*}$	Annihilation and creation operators
$c(f)=a(f)+a(f)^{*}$	Field operator, $c(f)c(g)+c(g)c(f)=2\operatorname{Re}\langle f,g\rangle$
$\{x,y\}=xy+yx$	Anticommutator
$\mathcal{F}=\Lambda^{\bullet}\mathcal{H}$	Fock space
$\Omega$	Vacuum, $a(f)\Omega=0$
$N=\sum_j a(e_j)^{*}a(e_j)$	Number operator, spectrum $\{0,1,2,\dots\}$
$\mathcal{A}(\mathcal{H})$	CAR algebra, the norm closure in the Fock representation
$M_{2^\infty}(\mathbb{C})=\bigotimes_kM_2(\mathbb{C})$	UHF algebra of type $2^\infty$, isomorphic to $\mathcal{A}(\mathcal{H})$
$\alpha_T(c(f))=c(Tf)$	Bogoliubov automorphism induced by an orthogonal $T$
$T-\mathrm{id}$ Hilbert–Schmidt	Implementability criterion (Shale–Stinespring)
$O_2(\mathcal{H})$	Restricted orthogonal group
$\Gamma(\mathcal{H}),\mathrm{Pin}(\mathcal{H}),\mathrm{Spin}(\mathcal{H})$	Infinite-dimensional Clifford, Pin and Spin groups
$\mathrm{Ad}^{\alpha}_x$	Signed inner conjugation action