CCR algebra, Weyl algebra, and quasi-free states

Intuition Beginner

The canonical commutation relation $[\hat{q},\hat{p}]=i\hslash$ is the algebraic heart of quantum mechanics. Position and momentum cannot share precise values at once, and the failure of commutativity measures exactly how much they refuse to. In ordinary quantum mechanics, you write down $\hat{q}$ as multiplication by $x$, write down $\hat{p}$ as the derivative $-i\hslash d/dx$, check the commutator on test functions, and move on. The relation is a consequence of the representation; the operators come first, the algebra is read off afterward.

Algebraic quantum field theory inverts that order. The relation $[\hat{q},\hat{p}]=i\hslash$ is taken as the starting point, and the question becomes: what is the universal object that algebraically encodes this commutator without committing to any particular Hilbert-space representation? The answer is a $\ast$-algebra called the CCR algebra. Different choices of representation — Schrödinger on $L^2(\mathbb{R})$, Bargmann-Fock on holomorphic functions, harmonic-oscillator on the number-state basis — are then different incarnations of the same algebra, related by changes of basis.

The next refinement is technical. The operators $\hat{q},\hat{p}$ are unbounded, and unbounded operators are awkward to package into a clean algebraic theory. So you exponentiate. The bounded unitaries $U(s)=e^{is\hat{q}}$ and $V(t)=e^{it\hat{p}}$ satisfy a relation that no longer involves a commutator: $U(s)V(t)=e^{-ist\hslash }V(t)U(s)$. This is the Weyl form of the canonical commutation relation, and the $C^{\ast }$-algebra these unitaries generate is the Weyl algebra. Everything in the operator-side picture lifts cleanly to the bounded Weyl side, and the algebraic theory acquires the tools of $C^{\ast }$-algebras.

A state in this framework is not a wavefunction; it is a linear functional that assigns expectation values to algebra elements. The vacuum, a thermal state, a coherent state — all are states on the algebra, distinguished by their two-point functions. A quasi-free state is a state whose entire content is fixed by its two-point function, with higher-point functions determined by a Wick-style rule. These are the algebraic versions of Gaussian states, and they include every physically reasonable free-field vacuum.

The framework lets you talk about "the algebra of observables" of a free quantum field before deciding which Fock space, which vacuum, which inequivalent representation to work in — exactly the freedom you need when the spacetime is curved and no single vacuum is preferred.

Visual Beginner

Picture a real symplectic vector space $(V,\sigma )$ — the classical phase space of the system, with a bilinear pairing $\sigma$ that tracks the conjugacy between position-like and momentum-like directions. Above each $v\in V$ sits an abstract algebra element $\varphi (v)$, the "smeared field" in that direction. The CCR algebra is the free associative algebra generated by these $\varphi (v)$, modulo the single relation $[\varphi (v),\varphi (w)]=i\sigma (v,w)1$.

The bounded Weyl side replaces each $\varphi (v)$ by a formal unitary $W(v)="e^{i\varphi (v)}"$ with the relation $W(v)W(w)=e^{-i\sigma (v,w)/2}W(v+w)$. The map $v\mapsto W(v)$ is a projective unitary representation of the additive group of $V$, with cocycle $\sigma$. The $C^{\ast }$-algebra generated by all the $W(v)$ is the Weyl algebra $\mathcal{W}(V,\sigma )$.

A state $\omega$ on the algebra is a linear functional $\omega :\mathcal{W}\to \mathbb{C}$ that is positive (it assigns non-negative numbers to $A^{\ast }A$) and normalised ($\omega (1)=1$). Different states give different physical pictures of the same abstract observables: the Minkowski vacuum, a thermal Gibbs state, a Bunch-Davies vacuum on de Sitter, a Hartle-Hawking vacuum on Schwarzschild — all are different functionals on the same Weyl algebra of the same free scalar field. The GNS construction turns any state $\omega$ into a Hilbert-space representation in which $\omega$ becomes the expectation value in a distinguished cyclic vector ${\Omega }_{\omega }$; this is the bridge from the algebraic side back to the more familiar Hilbert-space side.

Worked example Beginner

Take the smallest interesting case: one harmonic oscillator. The symplectic vector space is $V={\mathbb{R}}^2$ with coordinates $(q,p)$ and the standard symplectic form $\sigma ((q_1,p_1),(q_2,p_2))=q_1{p}_2-p_1{q}_2$. Write $v_1=(1,0)$ for the "position" direction and $v_2=(0,1)$ for the "momentum" direction.

The CCR algebra is generated by $\varphi (v_1)=\hat{q}$ and $\varphi (v_2)=\hat{p}$, modulo $[\hat{q},\hat{p}]=i\sigma (v_1,v_2)1=i\cdot 1\cdot 1$ (using natural units with $\hslash =1$). The Weyl elements are $W(v_1)=e^{i\hat{q}}$ and $W(v_2)=e^{i\hat{p}}$, satisfying $W(v_1)W(v_2)=e^{-i/2}W(v_1+v_2)$ — the formal Baker-Campbell-Hausdorff identity with the central commutator collapsing the higher-order terms.

The vacuum state is determined by its action on a general Weyl element. For $v=(q,p)\in {\mathbb{R}}^2$, the standard oscillator vacuum gives

$${\omega }_0(W(v))=e^{-(q^2+p^2)/4}.$$

You can read off the two-point function $\eta ((q_1,p_1),(q_2,p_2))=\frac{1}{2}(q_1{q}_2+p_1{p}_2)$, which encodes both the position-squared and momentum-squared zero-point fluctuations. Computing the field two-point function via ${\omega }_0(\varphi (v_1)\varphi (v_2))=\frac{1}{2}⟨v_1,v_2⟩+\frac{i}{2}\sigma (v_1,v_2)$, the real part is the symmetric covariance and the imaginary part is the antisymmetric commutator.

Plug in numbers. For $v_1=(1,0)$ and $v_2=(0,1)$: ${\omega }_0(\hat{q}\hat{p})=\frac{1}{2}(1\cdot 0+0\cdot 1)+\frac{i}{2}\cdot 1=\frac{i}{2}$. For $v_1=v_2=(1,0)$: ${\omega }_0({\hat{q}}^2)=\frac{1}{2}(1+0)+0=\frac{1}{2}$. Similarly ${\omega }_0({\hat{p}}^2)=\frac{1}{2}$. The product ${\omega }_0({\hat{q}}^2){\omega }_0({\hat{p}}^2)=1/4$ — exactly the Heisenberg minimum, as the vacuum is a minimum-uncertainty Gaussian state.

The GNS construction applied to ${\omega }_0$ recovers the familiar Bargmann-Fock Hilbert space of holomorphic functions of $z=q+ip$, with ${\Omega }_{\omega }$ the constant function 1. The takeaway: a single Gaussian functional on the abstract Weyl algebra, plus the GNS construction, regenerates the entire harmonic-oscillator Hilbert space without ever writing down $L^2(\mathbb{R})$.

Check your understanding Beginner

Formal definition Intermediate+

Symplectic vector space and the abstract CCR algebra

Let $(V,\sigma )$ be a real symplectic vector space: $V$ is a real vector space and $\sigma :V\times V\to \mathbb{R}$ is a bilinear antisymmetric non-degenerate form. The CCR $\ast$-algebra $\mathrm{CCR}(V,\sigma )$ is the free unital associative $\ast$-algebra over $\mathbb{C}$ generated by formal symbols $\varphi (v)$ for $v\in V$, modulo the relations

$$\varphi (v{)}^{\ast }=\varphi (v),\varphi (\lambda v+\mu w)=\lambda \varphi (v)+\mu \varphi (w),[\varphi (v),\varphi (w)]=i\sigma (v,w)1,$$

for all $v,w\in V$ and $\lambda ,\mu \in \mathbb{R}$. The first relation says $\varphi (v)$ is Hermitian; the second says $\varphi$ is real-linear in $v$; the third is the canonical commutation relation, with the symplectic pairing as right-hand side. This is the universal $\ast$-algebra in which a real-linear assignment $v\mapsto \varphi (v)$ of Hermitian generators satisfies the commutator relation.

Notation conventions

Throughout this unit we adopt the conventions of Bratteli-Robinson Vol. II ^{[Bratteli-Robinson Vol. II]}:

• $(V,\sigma )$ is a real symplectic vector space (possibly infinite-dimensional, in which case "non-degenerate" means $\sigma (v,\cdot )=0\Rightarrow v=0$).
• The CCR generators $\varphi (v)$ are real-linear in $v\in V$, with $\varphi (v{)}^{\ast }=\varphi (v)$.
• $[\cdot ,\cdot ]$ is the commutator $AB-BA$.
• $\hslash =1$ throughout. To restore physical units, replace $i\sigma (v,w)1$ by $i\hslash \sigma (v,w)1$.
• For a complex one-particle Hilbert space $\mathcal{H}$, the canonical symplectic structure on the underlying real vector space is $\sigma (f,g)=-2\mathrm{Im}⟨f,g⟩$, so that complex-linear creation and annihilation operators on the symmetric Fock space 12.13.01 reproduce the CCR with this sign convention.

Weyl form of the CCR

The CCR generators $\varphi (v)$ are unbounded in every faithful representation, so $\mathrm{CCR}(V,\sigma )$ is not a $C^{\ast }$-algebra in its raw algebraic form. The remedy is to work with formal exponentials. Define Weyl elements $W(v)$ for $v\in V$ as bounded unitary elements of a $C^{\ast }$-algebra subject to

$$W(v{)}^{\ast }=W(-v),W(v)W(w)=e^{-i\sigma (v,w)/2}W(v+w),$$

for all $v,w\in V$. The first relation says $W(v)$ is unitary; the second is the Weyl form of the CCR. The map $v\mapsto W(v)$ is a projective unitary representation of the additive group $(V,+)$ with cocycle $e^{-i\sigma (v,w)/2}$.

Theorem (Manuceau 1968 ^{[Manuceau 1968]}). There is a unique $C^$-algebra$\mathcal{W}(V, \sigma)$generatedbyelements$W(v), v \in V$,satisfyingtheWeylrelations,suchthatthemap$W : V \to \mathcal{W}(V, \sigma)$isaWeylrepresentationandanyothersuchWeylrepresentationfactorsuniquelythrough$\mathcal{W}(V, \sigma)$.*

The algebra $\mathcal{W}(V,\sigma )$ is the Weyl algebra (also called the CCR $C^{\ast }$-algebra) of $(V,\sigma )$. It is simple ($\mathcal{W}(V,\sigma )$ has no proper non-zero two-sided closed ideals when $\sigma$ is non-degenerate), and the linear span of the Weyl elements is norm-dense in it. In any faithful Hilbert-space representation in which the $\varphi (v)$ are essentially self-adjoint, $W(v)=e^{i\overline{\varphi (v)}}$ is the bounded incarnation of $\varphi (v)$ via Stone's theorem ^{[Reed-Simon Vol. I §VIII.5]}.

Stone-von Neumann theorem (finite degrees of freedom)

Suppose $\dim V=2n<\infty$ (finitely many degrees of freedom). A Weyl representation is regular if for each $v\in V$ the one-parameter group $t\mapsto W(tv)$ is strongly continuous; by Stone's theorem this is equivalent to the existence of self-adjoint generators $\varphi (v)$.

Theorem (Stone-von Neumann; Stone 1930, von Neumann 1931 ^{[von Neumann 1931]}). Let $\dim V=2n$ be finite-dimensional, and let $\pi :\mathcal{W}(V,\sigma )\to \mathcal{B}(K)$ be any regular irreducible Weyl representation on a separable Hilbert space $K$. Then $\pi$ is unitarily equivalent to the Schrödinger representation on $L^2({\mathbb{R}}^n)$ given by ${\pi }_{\mathrm{Sch}}(W(q,p))=e^{i{\sum }_k{q}_k{\hat{x}}_k}{e}^{i{\sum }_k{p}_k{\hat{p}}_k}{e}^{-i{\sum }_k{q}_k{p}_k/2}$, where ${\hat{x}}_k$ is multiplication by $x_k$ and ${\hat{p}}_k=-i{\partial }_{x_k}$.

The Bargmann-Fock representation of 12.13.01 is one such regular irreducible Weyl representation, equivalent to the Schrödinger representation by the Bargmann transform. Uniqueness fails in infinite dimensions: when $\dim V=\infty$ (the case of free quantum field theory), the Weyl algebra admits uncountably many unitarily inequivalent regular irreducible representations, classified by Bogoliubov transformations and the Shale-Stinespring criterion. This failure is the algebraic source of Haag's theorem (Haag 1955 ^{[Haag 1955]}) — the inequivalence of free and interacting Heisenberg-picture representations in QFT — and the reason the algebraic-QFT framework treats the Weyl algebra and its states as the primary objects rather than any single Hilbert-space representation.

States and the GNS construction

A state on the Weyl algebra is a linear functional $\omega :\mathcal{W}(V,\sigma )\to \mathbb{C}$ that is positive ($\omega (A^{\ast }A)\ge 0$ for all $A\in \mathcal{W}$) and normalised ($\omega (1)=1$). The set of states is convex; extreme points are pure states, and convex combinations are mixed states.

The Gelfand-Naimark-Segal (GNS) theorem ^{[Bratteli-Robinson Vol. II]} asserts that every state $\omega$ on a $C^{\ast }$-algebra $\mathcal{A}$ gives rise to a representation $({\mathcal{H}}_{\omega },{\pi }_{\omega },{\Omega }_{\omega })$ where ${\mathcal{H}}_{\omega }$ is a Hilbert space, ${\pi }_{\omega }:\mathcal{A}\to \mathcal{B}({\mathcal{H}}_{\omega })$ is a $\ast$-representation, and ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$ is a cyclic unit vector with

$$\omega (A)=⟨{\Omega }_{\omega },{\pi }_{\omega }(A){\Omega }_{\omega }{⟩}_{{\mathcal{H}}_{\omega }}$$

for all $A\in \mathcal{A}$. The triple $({\mathcal{H}}_{\omega },{\pi }_{\omega },{\Omega }_{\omega })$ is unique up to unitary equivalence intertwining the cyclic vectors. The state $\omega$ is pure iff the representation ${\pi }_{\omega }$ is irreducible. In algebraic QFT, the GNS construction is the bridge from the algebra-and-state side back to the more familiar Hilbert-space side.

Quasi-free (Gaussian) states

A quasi-free state on $\mathcal{W}(V,\sigma )$ is a state ${\omega }_{\eta }$ whose generating functional has the Gaussian form

$${\omega }_{\eta }(W(v))=e^{-\eta (v,v)/2},v\in V,$$

where $\eta :V\times V\to \mathbb{R}$ is a real symmetric bilinear form. The state ${\omega }_{\eta }$ is genuinely a state on $\mathcal{W}(V,\sigma )$ iff $\eta$ satisfies the CCR positivity inequality

$$\eta (v,v)\cdot \eta (w,w)\ge \frac{1}{4}\sigma (v,w{)}^2,v,w\in V,$$

equivalently iff the complex bilinear form $\eta +\frac{i}{2}\sigma$ is positive semidefinite on the complexification $V{\otimes }_{\mathbb{R}}\mathbb{C}$. When $\eta$ saturates the inequality on a Lagrangian subspace (equivalently, when there is a compatible complex structure $J$ on $V$ with $\sigma (v,Jw)$ a positive-definite inner product and $\eta =\frac{1}{2}\sigma (\cdot ,J\cdot )$), the state ${\omega }_{\eta }$ is pure; otherwise it is mixed.

The two-point function of a quasi-free state is, in field-operator form, ${\omega }_{\eta }(\varphi (v)\varphi (w))=\eta (v,w)+\frac{i}{2}\sigma (v,w)$. All higher $n$-point functions of ${\omega }_{\eta }$ are determined by the two-point function via the bosonic Wick rule: ${\omega }_{\eta }(\varphi (v_1)\cdots \varphi (v_{2n}))={\sum }_{\text{pairings}}\prod {\omega }_{\eta }(\varphi (v_{i_k})\varphi (v_{j_k}))$, summed over all pairings of $\{1,\dots ,2n\}$, and odd $n$-point functions vanish. This is the algebraic content of "Gaussian": the state is fully characterised by its covariance.

Counterexamples to common slips

• The CCR relation $[\varphi (v),\varphi (w)]=i\sigma (v,w)1$ uses the symplectic pairing on the right, not the Euclidean inner product. The pairing is antisymmetric — $\sigma (v,w)=-\sigma (w,v)$ — which is necessary for the commutator (also antisymmetric in $v,w$) to be consistent. A symmetric form there would force $[\varphi (v),\varphi (w)]=[\varphi (w),\varphi (v)]$, hence both equal zero, collapsing the algebra to the commutative polynomial algebra in the $\varphi (v)$.

• The Weyl relation $W(v)W(w)=e^{-i\sigma (v,w)/2}W(v+w)$ has a $1/2$ in the cocycle, not 1. The factor comes from the formal Baker-Campbell-Hausdorff identity $e^A{e}^B=e^{A+B+[A,B]/2}$ for elements whose commutator is central. Dropping the $1/2$ produces a different algebra (it is still associative but the resulting projective representation has a different cocycle class), not a notational variant.

• Stone-von Neumann uniqueness applies to regular (strongly continuous in each direction) irreducible representations on a separable Hilbert space, in finite dimensions. Removing any of the four hypotheses breaks the theorem: non-regular representations include the position-eigenstate representation (Schrödinger as a measure on ${\mathbb{R}}^n$); reducible representations include direct sums of Schrödinger; non-separable Hilbert spaces include uncountable tensor products; and infinite dimensions break it most violently, as in Haag's theorem.

• A "state" in algebraic QFT is a functional, not a vector. Many physically important states — the thermal Gibbs state at non-zero temperature, the Hadamard states on a generic globally hyperbolic spacetime — are not vector states for any single Fock representation. They live entirely on the algebra side, and the GNS construction produces a different Hilbert space representation for each one. Conflating "state" with "wavefunction" smuggles in a representation-dependence that algebraic QFT was designed to avoid.

Key theorem with proof Intermediate+

Theorem (GNS construction for a quasi-free state). Let $(V,\sigma )$ be a real symplectic vector space and $\eta :V\times V\to \mathbb{R}$ a real symmetric bilinear form satisfying the CCR positivity inequality $\eta (v,v)\eta (w,w)\ge \frac{1}{4}\sigma (v,w{)}^2$. Let ${\omega }_{\eta }$ be the quasi-free state on $\mathcal{W}(V,\sigma )$ defined by ${\omega }_{\eta }(W(v))=e^{-\eta (v,v)/2}$. There exist a complex Hilbert space ${\mathcal{H}}_{\omega }$, a representation ${\pi }_{\omega }:\mathcal{W}(V,\sigma )\to \mathcal{B}({\mathcal{H}}_{\omega })$, and a cyclic unit vector ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$ such that ${\omega }_{\eta }(A)=⟨{\Omega }_{\omega },{\pi }_{\omega }(A){\Omega }_{\omega }⟩$ for all $A\in \mathcal{W}(V,\sigma )$. The triple is unique up to unitary equivalence intertwining the cyclic vectors. When $\eta$ comes from a compatible complex structure $J$ on $V$ via $\eta (v,w)=\frac{1}{2}\sigma (v,Jw)$, the GNS representation is unitarily equivalent to the Fock representation on the bosonic Fock space ${\mathcal{F}}_s(\mathcal{H})$ over the one-particle Hilbert space $(V,J)$ with inner product $⟨v,w{⟩}_{\mathcal{H}}=\eta (v,w)+\frac{i}{2}\sigma (v,w)$.

Proof. The GNS construction proceeds in three stages: build a pre-Hilbert space from ${\omega }_{\eta }$ on the algebra, complete to a Hilbert space, and identify the representation.

Stage 1 (pre-Hilbert structure). Define a sesquilinear pairing on $\mathcal{W}(V,\sigma )$ by

$$⟨A,B{⟩}_{\omega }:={\omega }_{\eta }(A^{\ast }B).$$

Bilinearity is immediate. Conjugate symmetry follows from ${\omega }_{\eta }(B^{\ast }A)=\overline{{\omega }_{\eta }(A^{\ast }B)}$, which holds because ${\omega }_{\eta }$ is real on Hermitian elements and the involution swaps the order. Positivity is the defining property of states: $⟨A,A{⟩}_{\omega }={\omega }_{\eta }(A^{\ast }A)\ge 0$. The set $\mathcal{N}=\{A\in \mathcal{W}:{\omega }_{\eta }(A^{\ast }A)=0\}$ is a left ideal (by the Cauchy-Schwarz inequality $|{\omega }_{\eta }(A^{\ast }B){|}^2\le {\omega }_{\eta }(A^{\ast }A){\omega }_{\eta }(B^{\ast }B)$), and the quotient $\mathcal{W}/\mathcal{N}$ is a pre-Hilbert space under the inherited inner product. Write $[A]\in \mathcal{W}/\mathcal{N}$ for the equivalence class of $A$.

Stage 2 (completion). Let ${\mathcal{H}}_{\omega }$ be the Hilbert-space completion of $\mathcal{W}/\mathcal{N}$. The cyclic vector is ${\Omega }_{\omega }:=[1]$, the equivalence class of the algebra identity. The representation ${\pi }_{\omega }$ is defined by left-multiplication: for $A\in \mathcal{W}$, the operator ${\pi }_{\omega }(A)$ acts on the dense subspace $\mathcal{W}/\mathcal{N}$ as ${\pi }_{\omega }(A)[B]:=[AB]$. This is well-defined because $\mathcal{N}$ is a left ideal. The operator ${\pi }_{\omega }(A)$ is bounded with $\Vert {\pi }_{\omega }(A)\Vert \le \Vert A{\Vert }_{\mathcal{W}}$ in the $C^{\ast }$-norm of $\mathcal{W}$, by

$$\Vert {\pi }_{\omega }(A)[B]{\Vert }^2={\omega }_{\eta }(B^{\ast }{A}^{\ast }AB)\le \Vert A^{\ast }A\Vert {\omega }_{\eta }(B^{\ast }B)=\Vert A{\Vert }^2\Vert [B]{\Vert }^2,$$

where the middle inequality uses positivity of the state on the element $\Vert A^{\ast }A\Vert 1-A^{\ast }A\ge 0$. The bounded operator extends to all of ${\mathcal{H}}_{\omega }$ by density. The map ${\pi }_{\omega }:\mathcal{W}\to \mathcal{B}({\mathcal{H}}_{\omega })$ is a $\ast$-representation: linearity and multiplicativity are inherited from left-multiplication, and the $\ast$-property ${\pi }_{\omega }(A^{\ast })={\pi }_{\omega }(A{)}^{\ast }$ follows from $⟨[B],{\pi }_{\omega }(A)[C]{⟩}_{\omega }={\omega }_{\eta }(B^{\ast }AC)=⟨[A^{\ast }B],[C]{⟩}_{\omega }=⟨{\pi }_{\omega }(A^{\ast })[B],[C]{⟩}_{\omega }$.

Cyclicity. The vector ${\Omega }_{\omega }=[1]$ generates a dense subspace: ${\pi }_{\omega }(\mathcal{W}){\Omega }_{\omega }=\{[A]:A\in \mathcal{W}\}=\mathcal{W}/\mathcal{N}$, which is dense in ${\mathcal{H}}_{\omega }$ by construction. The expectation-value identity ${\omega }_{\eta }(A)=⟨{\Omega }_{\omega },{\pi }_{\omega }(A){\Omega }_{\omega }⟩={\omega }_{\eta }(1^{\ast }A1)={\omega }_{\eta }(A)$ is tautological.

Uniqueness. Suppose $({\mathcal{H}}^{\prime },{\pi }^{\prime },{\Omega }^{\prime })$ is another GNS triple for the same state. The map $U:{\mathcal{H}}_{\omega }\to {\mathcal{H}}^{\prime }$ defined on the dense subspace by $U[A]:={\pi }^{\prime }(A){\Omega }^{\prime }$ is an isometry because $⟨U[A],U[B]{⟩}^{\prime }=⟨{\pi }^{\prime }(A){\Omega }^{\prime },{\pi }^{\prime }(B){\Omega }^{\prime }{⟩}^{\prime }=⟨{\Omega }^{\prime },{\pi }^{\prime }(A^{\ast }B){\Omega }^{\prime }{⟩}^{\prime }={\omega }_{\eta }(A^{\ast }B)=⟨[A],[B]{⟩}_{\omega }$. It extends to a unitary $U:{\mathcal{H}}_{\omega }\to {\mathcal{H}}^{\prime }$ by density and cyclicity, intertwining ${\pi }_{\omega }$ with ${\pi }^{\prime }$ and mapping ${\Omega }_{\omega }$ to ${\Omega }^{\prime }$.

Stage 3 (Fock identification when $\eta$ is Kähler). Suppose $J:V\to V$ is a real-linear map with $J^2=-1$, $\sigma (Jv,Jw)=\sigma (v,w)$ (symplectic), and $\sigma (v,Jv)>0$ for $v\ne 0$ (positivity). Then $(V,J)$ becomes a complex vector space with complex multiplication $(\alpha +i\beta )\cdot v=\alpha v+\beta Jv$, and the formula

$$⟨v,w{⟩}_{\mathcal{H}}:=\frac{1}{2}\sigma (v,Jw)+\frac{i}{2}\sigma (v,w)=\eta (v,w)+\frac{i}{2}\sigma (v,w)$$

is a Hermitian inner product making $\mathcal{H}:=(V,J,⟨\cdot ,\cdot {⟩}_{\mathcal{H}})$ a complex pre-Hilbert space; complete to a Hilbert space if $V$ is infinite-dimensional.

On the bosonic Fock space ${\mathcal{F}}_s(\mathcal{H})$ of 12.13.01, the field operators ${\varphi }_F(v):=\frac{1}{\sqrt{2}}(a(v)+a^{\ast }(v))$ are essentially self-adjoint on the finite-particle subspace and satisfy $[{\varphi }_F(v),{\varphi }_F(w)]=i\sigma (v,w)1$, so they define a representation of the CCR relations. The exponentiated Weyl operators $W_F(v):=e^{i\overline{{\varphi }_F(v)}}$ generate a Weyl representation of $\mathcal{W}(V,\sigma )$. The Fock vacuum ${\Omega }_F$ satisfies $a(v){\Omega }_F=0$, and a direct computation using Baker-Campbell-Hausdorff (the commutator $[a(v),a^{\ast }(w)]=⟨v,w{⟩}_{\mathcal{H}}$ is central) gives

$$⟨{\Omega }_F,W_F(v){\Omega }_F⟩=e^{-⟨v,v{⟩}_{\mathcal{H}}/2}=e^{-\eta (v,v)/2}={\omega }_{\eta }(W(v)).$$

So the Fock vacuum is a cyclic vector implementing the state ${\omega }_{\eta }$ in the Fock representation. By the uniqueness clause of the GNS theorem, the Fock triple $({\mathcal{F}}_s(\mathcal{H}),{\pi }_F,{\Omega }_F)$ is unitarily equivalent to the GNS triple $({\mathcal{H}}_{\omega },{\pi }_{\omega },{\Omega }_{\omega })$. $\square$

Bridge. The CCR-and-Weyl framework and the quasi-free-state construction package the algebraic side of free quantum field theory into a representation-independent algebra plus a parametrised family of states, which builds toward the dedicated successor unit on Hadamard states via the wave-front-set criterion 13.09.03 and appears again in every algebraic-QFT-on-curved-spacetime construction, the algebraic-statistical-mechanics treatment of thermal-equilibrium states via KMS, and the modular-theory derivation of the Unruh effect. Two patterns recur. The first pattern is algebra first, representation second: the relations $[\varphi (v),\varphi (w)]=i\sigma (v,w)1$ (or the Weyl form $W(v)W(w)=e^{-i\sigma (v,w)/2}W(v+w)$) define the abstract algebra, and the GNS construction turns any state on it into a Hilbert-space representation; the second-quantisation functor $\Gamma$ of 12.13.01 becomes the special case where the state is the Kähler vacuum and the GNS Hilbert space is the symmetric Fock space. The second pattern is the state encodes the physics: the Minkowski vacuum, a thermal Gibbs state at inverse temperature $\beta$, a Bunch-Davies vacuum on de Sitter, and a Hadamard state on a curved background are all different quasi-free states on the same CCR algebra of the free scalar field, distinguished only by their two-point covariances $\eta$. Together these patterns are exactly what algebraic QFT on curved spacetime needs: in the absence of a global timelike Killing field, no Hilbert space representation is canonically preferred, so the entire physics has to be carried by the algebra and the choice of state.

Exercises Intermediate+

A graded set covering the CCR/Weyl algebra, quasi-free states, the Stone-von Neumann theorem, the GNS construction, and Bogoliubov transformations.

Lean formalization Intermediate+

lean_status: none — Mathlib has the building blocks (symplectic forms on finite-dimensional vector spaces via Mathlib.LinearAlgebra.SymplecticGroup, $C^{\ast }$-algebra abstractions via Mathlib.Analysis.CStarAlgebra.Basic, bounded operators on Hilbert spaces, the matrix exponential for finite-dimensional Weyl-relation identities, and the symmetric tensor algebra for the Fock-space substrate). What is not in Mathlib at this date: the abstract CCR $\ast$-algebra over a real symplectic vector space; the Weyl algebra as the unique $C^{\ast }$-completion of the abstract Weyl relations $W(v)W(w)=e^{-i\sigma (v,w)/2}W(v+w)$ in the sense of Manuceau 1968; the Stone-von Neumann theorem for Weyl-form representations on finite-dimensional symplectic vector spaces; the GNS construction from a state on a $C^{\ast }$-algebra to a cyclic representation; the classification of quasi-free states by their covariance bilinear forms; the Araki-Woods bosonic Fock representation attached to a quasi-free state with Kähler one-particle structure; the Shale criterion for Bogoliubov-transformation implementability. Each item is a substantial Mathlib contribution. This unit is reviewer-attested; see the Mathlib gap analysis for the full enumeration.

Inequivalent representations and Bogoliubov transformations Master

The structural failure of Stone-von Neumann uniqueness in infinite dimensions is the source of much of the interesting physics in algebraic QFT. A Bogoliubov transformation $T:V\to V$ is a real-linear symplectic isomorphism. Relative to a Kähler structure $J$ on $V$ giving a complex Hilbert space $\mathcal{H}=(V,J)$, the transformation $T$ decomposes uniquely as $T=T_{+}+T_{-}$ with $T_{+}$ complex-linear and $T_{-}$ complex-antilinear. In creation/annihilation language, $T$ acts by

$$a(v)⟼a(T_{+}v)+a^{\ast }(\overline{T_{-}v}),$$

mixing $a$ and $a^{\ast }$ — the "Bogoliubov rotation". The transformation lifts to a unitary $U_T$ on the Fock representation ${\mathcal{F}}_s(\mathcal{H})$ if and only if $T_{-}$ is Hilbert-Schmidt as a complex-antilinear map $\mathcal{H}\to \mathcal{H}$ (Shale 1962). When this Shale criterion fails, the transformed Weyl operators define a Weyl representation on a different Fock space attached to the new Kähler structure $J^{\prime }=TJ{T}^{-1}$, and the two representations are unitarily inequivalent.

Unruh effect (Bisognano-Wichmann 1975/1976 ^{[Kay-Wald 1991]}). The free scalar field on Minkowski space, restricted to the Rindler wedge $\{x^1>|x^0|\}$, has a CCR algebra of test functions supported in the wedge. The Minkowski vacuum restricted to this algebra is a thermal KMS state at the Unruh temperature $T=a/(2\pi )$ relative to the Lorentz-boost flow (which acts as time translation for a uniformly accelerated observer with acceleration $a$). The relation between the Minkowski-vacuum Fock representation and the Rindler-vacuum Fock representation is a Bogoliubov transformation that fails the Shale criterion; the two vacua lie in unitarily inequivalent representations of the wedge algebra. The modular automorphism of the wedge von Neumann algebra coincides with the boost subgroup (Bisognano-Wichmann theorem), which is the modular-theoretic statement of the Unruh effect.

Cosmological particle production. On de Sitter spacetime, the Bunch-Davies state is the unique de-Sitter-invariant Hadamard quasi-free state for the massive Klein-Gordon field. Restricted to a static patch, it appears as a thermal state at the Gibbons-Hawking temperature $T=H/(2\pi )$ where $H$ is the Hubble parameter — exactly the Unruh phenomenon with the Hubble rate playing the role of the acceleration. The inequivalence between the Bunch-Davies representation and the static-patch vacuum representation is again a non-Shale Bogoliubov transformation.

Spontaneous symmetry breaking. In a quantum field theory with a global continuous symmetry, distinct vacua corresponding to different orientations of the order parameter (e.g., distinct directions of magnetisation in a Heisenberg ferromagnet at $T=0$) live in unitarily inequivalent representations of the field algebra. The Bogoliubov transformation rotating one vacuum to another fails the Shale condition in infinite volume, which is exactly the algebraic content of "different superselection sectors". Goldstone's theorem (Goldstone 1961; Goldstone-Salam-Weinberg 1962) is the corresponding statement about massless excitations in each broken-symmetry sector.

KMS condition and thermal-equilibrium states. A state $\omega$ on a $C^{\ast }$-algebra $\mathcal{A}$ equipped with a one-parameter automorphism group ${\alpha }_t$ (modelling time evolution) is a KMS state at inverse temperature $\beta$ (Kubo 1957; Martin-Schwinger 1959; Haag-Hugenholtz-Winnink 1967) if for every pair $A,B\in \mathcal{A}$ the function $F_{A,B}(t):=\omega (A{\alpha }_t(B))$ extends to a holomorphic function on the strip $\{0<\mathrm{Im}z<\beta \}$ with boundary value $F_{A,B}(t+i\beta )=\omega ({\alpha }_t(B)A)$. This is the algebraic formulation of thermal equilibrium, replacing the finite-volume Gibbs density matrix $\rho =e^{-\beta H}/Z$ (which makes no sense in infinite volume, where $e^{-\beta H}$ is not trace-class). Every KMS state has a distinguished GNS representation, and Tomita-Takesaki modular theory (Tomita 1967; Takesaki 1970) provides the structural framework — see the dedicated successor unit 12.14.02 (proposed).

Connections Master

• Bosonic Fock space and second quantisation 12.13.01. Direct sibling and the canonical example of the GNS construction. The bosonic Fock space ${\mathcal{F}}_s(\mathcal{H})$ over a complex Hilbert space $\mathcal{H}$ is the GNS Hilbert space of the unique pure quasi-free state on $\mathcal{W}(V,\sigma )$ corresponding to the Kähler structure on $V={\mathcal{H}}_{\mathbb{R}}$. The field operators ${\varphi }_F(v)=\frac{1}{\sqrt{2}}(a(v)+a^{\ast }(v))$ on Fock space realise the CCR algebra concretely, and the Weyl operators $W_F(v)=e^{i\overline{{\varphi }_F(v)}}$ realise the Weyl algebra; the second-quantisation functor $\Gamma$ of the prior unit is the Fock-side incarnation of the natural functoriality of the Weyl algebra in symplectic isomorphisms. The two units are dual sides of the same construction: the prior unit emphasises the operator-on-Hilbert-space side; this unit emphasises the algebra-and-state side.

• Canonical quantum field theory 12.12.01. Chapter prerequisite. The free scalar field of canonical QFT — operator-valued tempered distribution $\hat{\varphi }(x)$ on Minkowski space with $[\hat{\varphi }(x),\hat{\pi }(y)]=i\hslash {\delta }^3(\vec{x}-\vec{y})$ — is the operator-side incarnation of the smeared CCR. The smeared fields $\hat{\varphi }(f)=\int \hat{\varphi }(x)f(x)d^{4}x$ for Schwartz test functions $f$ generate a CCR algebra over the symplectic vector space of test-function Cauchy data, and the Minkowski vacuum is the distinguished quasi-free state. Algebraic QFT inverts the conventional development: the algebra-and-state framework of this unit comes first; canonical quantisation on Minkowski is the special case where the Poincaré group acts and selects a preferred vacuum.

• Hilbert space 02.11.08 and bounded/unbounded operators [02.11.01, 02.11.03]. Substrate. The GNS Hilbert space is the analytic completion of a state-defined pre-Hilbert quotient; bounded Weyl operators and unbounded CCR generators each play their role, related by Stone's theorem on one-parameter unitary groups ^{[Reed-Simon Vol. I §VIII.5]}. The CCR generators are necessarily unbounded (no pair of bounded operators on a Hilbert space can satisfy $[A,B]=i1$, by an operator-norm argument detailed in 12.13.01), forcing the exponentiated Weyl form as the input to any $C^{\ast }$-algebraic treatment.

• Inner product space 02.11.07. The complex Hilbert space $\mathcal{H}$ attached to a Kähler structure on $(V,\sigma )$ has inner product $⟨v,w{⟩}_{\mathcal{H}}=\eta (v,w)+\frac{i}{2}\sigma (v,w)$ — the real part is the quasi-free covariance, the imaginary part is half the symplectic pairing. The Fock representation built from this $\mathcal{H}$ is the GNS representation of the corresponding quasi-free state.

• Hadamard states via the wave-front-set criterion 13.09.03 (proposed successor). On a globally hyperbolic Lorentzian manifold $(M,g)$, the CCR algebra of Klein-Gordon Cauchy data carries a parametrised family of quasi-free states. The Hadamard condition (Radzikowski 1996) selects the physically reasonable subset by a wave-front-set criterion on the two-point distribution: $\mathrm{WF}({\omega }_2)=\{((x_1,{\xi }_1),(x_2,-{\xi }_2)):(x_1,{\xi }_1){\sim }_g(x_2,{\xi }_2),{\xi }_1\in \overline{V_{x_1}^{+}}\}$. The Hadamard property is preserved under causal propagation and is the input for the perturbative renormalisation programme of Brunetti-Fredenhagen-Köhler 1996 and Hollands-Wald 2001/2002. The dedicated curved-spacetime unit cites the present unit for the algebraic-state framework and develops the WF criterion in full.

• GNS construction and Tomita-Takesaki modular theory 12.14.02 (proposed successor). The GNS construction sketched here is the special case of the general theorem (Gelfand-Naimark-Segal 1943/1947) that every state on a $C^{\ast }$-algebra gives a cyclic representation; the successor unit develops the full theorem, introduces the KMS condition for thermal equilibrium, and unpacks Tomita-Takesaki modular theory (the modular operator $\Delta$, modular conjugation $J$, and modular automorphism group ${\sigma }_t$ attached to a cyclic and separating vector). The Bisognano-Wichmann theorem identifying the modular automorphism of a Rindler-wedge algebra with the boost subgroup is the entry point to the rigorous Unruh effect.

• CAR algebra and quasi-free fermionic states 12.14.03 (proposed successor). The fermionic analogue of this unit. The CAR algebra over a complex Hilbert space $\mathcal{H}$ is the $C^{\ast }$-algebra generated by creation/annihilation operators $b^{\ast }(f),b(f)$ satisfying $\{b(f),b^{\ast }(g)\}=⟨f,g⟩1$ — anticommutators in place of commutators. Quasi-free fermionic states are determined by a covariance $S:\mathcal{H}\to \mathcal{H}$ with $0\le S\le 1$ (a "one-particle density matrix"). The construction runs parallel to the bosonic story of this unit, with the substitutions (symplectic ↔ Hermitian, commutator ↔ anticommutator, Shale criterion ↔ Shale-Stinespring criterion, unbounded CCR ↔ bounded CAR). Both algebras together form the input to the algebraic-QFT treatment of free fields of any spin.

• Symplectic vector space (chapter 05-symplectic). Substrate. The CCR algebra is built directly on a real symplectic vector space $(V,\sigma )$. Classical Hamiltonian mechanics on $V$ has Poisson brackets $\{f,g{\}}_{\sigma }$ that the canonical commutation relation $[\varphi (v),\varphi (w)]=i\sigma (v,w)1$ "quantises". The Kähler-structure choice that selects a vacuum is a choice of compatible complex structure on the symplectic phase space — the same data that geometric quantisation uses to define a polarisation. Bogoliubov transformations are precisely the linear symplectomorphisms of $(V,\sigma )$, and their failure to be Hilbert-Schmidt is the algebraic obstruction to lifting them to unitaries on a single Fock representation.

• Globally hyperbolic Lorentzian manifolds 13.09.01 (proposed prereq for 13.09.03). The symplectic vector space $(V,\sigma )$ supplying the input to the CCR construction in the curved-spacetime setting is the space of compactly supported smooth Cauchy data on a Cauchy surface of a globally hyperbolic Lorentzian manifold, with $\sigma$ the symplectic form induced by the Klein-Gordon Wronskian. The dedicated curved-spacetime geometry unit supplies this input; the present unit supplies the algebraic-quantisation framework that consumes it.

Historical & philosophical context Master

The algebraic formulation of canonical commutation relations originated in Hermann Weyl's 1927 paper Quantenmechanik und Gruppentheorie ^{[Weyl 1927]}, appearing in Zeitschrift für Physik 46, 1. Weyl proposed the exponentiated relation $U(s)V(t)=e^{ist}V(t)U(s)$ for one-parameter unitary groups generated by position and momentum, sidestepping the domain issues of the unbounded operators $\hat{q},\hat{p}$ themselves. The mathematical content of Weyl's proposal was made precise by von Neumann 1931 ^{[von Neumann 1931]} in Mathematische Annalen 104, 570, which proved the uniqueness up to unitary equivalence of the Schrödinger representation on $L^2({\mathbb{R}}^n)$ as the only regular irreducible representation of the Weyl-form CCR on finitely many degrees of freedom. The Stone-von Neumann theorem combining von Neumann's uniqueness with Stone's 1930 theorem on one-parameter unitary groups settled the foundational kinematic question of finite-degree quantum mechanics: there is one quantum mechanics, not many.

The $C^{\ast }$-algebraic formulation was developed by Irving Segal in the 1940s-50s, beginning with his 1947 Annals of Mathematics paper Postulates for general quantum mechanics ^{[Segal 1947]} which proposed taking the algebra of observables — rather than the Hilbert space — as the primary object of quantum theory. Segal's framework absorbed both quantum mechanics and quantum field theory under a single algebraic umbrella, with states as positive normalised functionals and the Gelfand-Naimark-Segal construction (Gelfand-Naimark 1943; Segal 1947) recovering a Hilbert-space representation from any state. Manuceau 1968 ^{[Manuceau 1968]} in Annales de l'Institut Henri Poincaré 8, 139 supplied the proof that the Weyl algebra over a symplectic vector space is the unique $C^{\ast }$-completion of the abstract Weyl relations — the technical foundation for treating $\mathcal{W}(V,\sigma )$ as a well-defined object independent of any representation.

The infinite-degree case turned out to be radically different from the finite-degree one. Rudolf Haag's 1955 paper On quantum field theories ^{[Haag 1955]} in Det. Kgl. Danske Vidensk. Selsk. Mat.-fys. Medd. 29, no. 12 proved that the interaction picture of perturbative QFT is incompatible with the existence of a single Fock representation in which both free and interacting Heisenberg-picture fields act — Haag's theorem, now a basic obstacle in any rigorous treatment of interacting QFT. The structural reason is the failure of Stone-von Neumann uniqueness in infinite dimensions: the Weyl algebra over the infinite-dimensional symplectic vector space of QFT test-function Cauchy data admits uncountably many unitarily inequivalent regular irreducible representations, classified by Bogoliubov transformations and the Shale 1962 / Shale-Stinespring 1965 Hilbert-Schmidt criteria. Wightman's late-1950s axiomatic framework (Streater-Wightman 1964) and the Haag-Kastler 1964 algebraic-QFT framework codified the resulting paradigm shift: in QFT, the algebra of observables is primary, the choice of physically relevant state is the dynamical input, and the GNS representation is a derived construction.

The application to quantum statistical mechanics and to curved-spacetime QFT was developed in the 1960s-90s. Araki and Woods 1963 ^{[Araki-Woods 1963]} in J. Math. Phys. 4, 637 constructed the Fock representations of quasi-free states on the CCR algebra of a non-relativistic free Bose gas, identifying thermal-equilibrium states with explicit cyclic vectors on a doubled Fock space. Haag, Hugenholtz, and Winnink 1967 identified thermal equilibrium with the KMS condition on the algebra of observables, generalising the finite-volume Gibbs ensemble to the infinite-volume setting where $e^{-\beta H}$ has no representation-theoretic meaning. Bratteli-Robinson Operator Algebras and Quantum Statistical Mechanics Vol. II 1981 ^{[Bratteli-Robinson Vol. II]} became the canonical reference for the algebraic-QFT and algebraic-statistical-mechanics framework, with §5.2 the definitive treatment of the CCR / Weyl / quasi-free apparatus. The curved-spacetime side opened with Kay and Wald 1991 ^{[Kay-Wald 1991]} in Phys. Rep. 207, 49 (uniqueness and thermal properties of stationary Hadamard quasi-free states on spacetimes with a bifurcate Killing horizon), followed by Radzikowski 1996 (microlocal definition of Hadamard states via the wave-front-set criterion) and the modern programme of Brunetti-Fredenhagen-Köhler 1996 and Hollands-Wald 2001/2002 (perturbative renormalisation in curved spacetime via Hadamard parametrix subtraction).

The CCR-and-Weyl framework also underlies the rigorous derivation of the Unruh effect (Unruh 1976 heuristic; Bisognano-Wichmann 1975/1976 modular theory; Sewell 1982 rigorous statement) and the Hawking effect in the formal limit of a stationary black hole, both of which are statements about the inequivalence of Fock representations attached to different timelike vector fields — Bogoliubov transformations that fail the Shale criterion. Gérard 2019 ^{[Gerard 2019]} Microlocal Analysis of Quantum Fields on Curved Spacetimes is the modern textbook consolidation of the algebraic-microlocal programme, with Ch. 6 the contemporary treatment of the CCR / Weyl / quasi-free apparatus on which the present unit is anchored.

Bibliography Master

Primary literature, originating papers (1927–1955):

• Weyl, H., Z. Phys. 46, 1 (1927). Quantenmechanik und Gruppentheorie — originator paper for the exponentiated CCR.
• von Neumann, J., Math. Ann. 104, 570 (1931). Uniqueness theorem for Schrödinger operators — Stone-von Neumann theorem.
• Segal, I. E., Ann. Math. 48, 930 (1947). Postulates for general quantum mechanics — algebraic / $C^{\ast }$-algebraic formulation.
• Haag, R., Det. Kgl. Danske Vidensk. Selsk. Mat.-fys. Medd. 29, no. 12 (1955). On quantum field theories — Haag's theorem on inequivalent representations.

Primary literature, structural developments (1962–1996):

• Shale, D., Trans. Amer. Math. Soc. 103, 149 (1962). Linear symmetries of free boson fields — Hilbert-Schmidt criterion for Bogoliubov implementability.
• Araki, H. & Woods, E. J., J. Math. Phys. 4, 637 (1963). Representations of the CCR for a nonrelativistic infinite free Bose gas — Araki-Woods construction.
• Manuceau, J., Ann. Inst. H. Poincaré 8, 139 (1968). $C^{\ast }$-algèbre de relations de commutation — uniqueness of the Weyl-algebra $C^{\ast }$-completion.
• Bisognano, J. & Wichmann, E., J. Math. Phys. 16, 985 (1975); 17, 303 (1976). On the duality condition for Hermitian scalar fields and for quantum fields — modular theory of Rindler wedges.
• Kay, B. S. & Wald, R. M., Phys. Rep. 207, 49 (1991). Theorems on uniqueness and thermal properties of stationary nonsingular quasi-free states on spacetimes with a bifurcate Killing horizon.
• Radzikowski, M., Comm. Math. Phys. 179, 529 (1996). Micro-local approach to the Hadamard condition in QFT on curved space-time — WF-set criterion.

Canonical monographs (operator-algebra / algebraic-QFT lineage):

• Bratteli, O. & Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics, Vol. II, 2nd ed. (Springer, 1997). §5.2 CCR-algebra, Weyl form, quasi-free states, GNS / Araki-Woods construction.
• Petz, D., An Invitation to the Algebra of Canonical Commutation Relations (Leuven University Press, 1990). Textbook on CCR, Fock representations, quasi-free states, Bogoliubov classification.
• Gérard, C., Microlocal Analysis of Quantum Fields on Curved Spacetimes, ESI Lectures (EMS, 2019). Ch. 6 CCR / Weyl / quasi-free states; Ch. 7 Hadamard states via WF criterion.
• Bär, C., Ginoux, N. & Pfäffle, F., Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures (EMS, 2007). Ch. 4 CCR over a symplectic phase space, quasi-free states. [Free PDF on arXiv:0806.1036 — need to add to local archive.]
• Reed, M. & Simon, B., Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (Academic Press, 1980); Vol. II: Fourier Analysis, Self-Adjointness (Academic Press, 1975). Vol. I §VIII.5 Stone's theorem; Vol. II §X.7 Fock spaces and CCR.

Modern textbooks (mathematics-side):

• Folland, G. B., Quantum Field Theory: A Tourist Guide for Mathematicians (AMS Math. Surveys 149, 2008). Ch. 4–5 Fock spaces, CCR, Weyl form.
• Hall, B. C., Quantum Theory for Mathematicians (Springer GTM 267, 2013). Ch. 14 Stone-von Neumann theorem and Bargmann transform.

Physics-side references:

• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 2nd ed. (Pearson/Cambridge, 2011). Ch. 2 §2.3 harmonic oscillator.
• Susskind, L. & Friedman, A., Quantum Mechanics: The Theoretical Minimum (Basic Books, 2014). Lecture 10.
• Tong, D., Quantum Field Theory, DAMTP Cambridge lecture notes (raw/pdfs/qft/qft.pdf). §2 canonical quantisation. [Have.]