Wick algebra



A Wick algebra is an algebra of quantum observables of free quantum fields.

In the quantum mechanics of harmonic oscillators or in quantum field theory of free fields on Minkowski spacetime one encounters linear operators {a k,a k *} kK\{a_k, a^\ast_k\}_{k \in K} that satisfy the canonical commutation relation [a i,a j *]=diag((c k)) ij[a_i, a^\ast_j] = diag((c_k))_{i j}. Then by a normal ordered polynomial or Wick polynomial (Wick 50) one means a polynomial, denoted :P::P:, which is obtained from a polynomial P((a k,a k *))P((a_k, a^\ast_k)) in these operators by ordering all “creation operators” a k *a_k^\ast to the left of all “annihiliation operators”. For example focusing on a single mode kk we have:

:a *:=a * :a:=a :a *a:=a *a :aa *:=a *a :aa *a:=a *aa etc..S \array{ :a^\ast: = a^\ast \\ :a: = a \\ :a^\ast a: = a^\ast a \\ :a a^\ast: = a^\ast a \\ :a a^\ast a: = a^\ast a a \\ etc. } \,.S

The intuitive idea is that these operators span a Hilbert space \mathcal{H} of quantum states from a vacuum state |vac\vert vac \rangle \in \mathcal{H} characterized by the condition

k(a k|vac=0) \underset{k}{\forall} \left( a_k \vert vac \rangle = 0 \right)

hence (if we think of a ka_k as acting by “removing a quantum in mode kk”) by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the quantum observables with vanishing vacuum expectation value. In quantum field theory they model scattering processes where quanta enter a reaction process (the modes corresponding to the “annihilation” operators a ka_k) and other particles come out of the reaction (the modes corresppnding to the “creation” operators a k *a^\ast_k).

The product of two Wick polynomials, computed in the ambient operator algebra and then re-expressed as a Wick polynomial, is given by computing the relevant sequence of commutators by Wick's lemma, for example

:a *a::a *a:=:a *a *aa:+c:a *a:, {:a^\ast a:} \, {:a^\ast a:} = :a^\ast a^\ast a a: + c \, :a^\ast a: \,,

where c[a,a *]c \coloneqq [a, a^\ast] is the value of the canonical commutator.

The associative algebra thus obtained is hence called the algebra of normal ordered operators or Wick polynomial algebra or just Wick algebra.

This plays a central role in perturbative quantum field theory, where the quantization of quantum observables of free fields is traditionally defined as the corresponding Wick algebra.

But the Wick algebra in quantum field theory may also be understood more systematically from first principles of quantization. It turns out that it is Moyal deformation quantization of the canonical Poisson bracket on the covariant phase space of the free field, which is the Peierls bracket modified to an almost Kähler structure by the 2-point function of a quasi-free Hadamard state (Dito 90, Dütsch-Fredenhagen 01). See example 2 and def. 7 below.

Understood in this form the construction directly generalized to quantum field theory on curved spacetimes (Brunetti-Fredenhagen 95, Brunetti-Fredenhagen 00, Hollands-Wald 01).

Finally, the shift by the quasi-free Hadamard state, which is the very source of the “normal ordering”, was understood as an example of the almost-Kähler version of the quantization recipe of Fedosov deformation quantization (Collini 16). For more on this see at locally covariant perturbative quantum field theory.


For finite-dimensional linear phase spaces


(almost Kähler vector space)

An almost Kähler vector space is a complex vector space VV equipped with two bilinear forms σ,h:V V\sigma, h \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R} such that with VV regarded as a smooth manifold and with σ,g\sigma, g regarded as constant tensors, then (V,σ,h)(V,\sigma,h) is an almost Kähler manifold.


(standard almost Kähler vector spaces)

Let V 2V \coloneqq \mathbb{R}^2 equipped with the complex structure given by the canonical identification 2\mathbb{R}^2 \simeq \mathbb{C}, let σ(0 1 1 0)\sigma \coloneqq \left( \array{0 & -1 \\ 1 & 0} \right) and h(1 0 0 1)h \coloneqq \left( \array{ 1 & 0 \\ 0 & 1} \right). Then (V,σ,g)(V,\sigma,g) is an almost Kähler vector space (def. 1).


(Wick algebra of an almost Kähler vector space)

Let ( 2n,σ,g)(\mathbb{R}^{2n},\sigma, g) be an almost Kähler vector space (def. 1). Then its Wick algebra is the formal power series vector space [[a 1,a 1 *,,a n,a n *]][[]]\mathbb{C}[ [ a_1, a^\ast_1, \cdots, a_n, a^\ast_n ] ] [ [ \hbar ] ] equipped with the star product

P 1 ωP 2 prodexp(k 1,k 2=12nω ab a b)(P 1P 2) =P 1P 2+k 1,k 2=12nω k 1k 2( k 1P 1)( k 2P 2)+ \begin{aligned} P_1 \star_\omega P_2 & \coloneqq prod \circ \exp \left( \hbar\underoverset{k_1, k_2 = 1}{2 n}{\sum}\omega^{a b} \partial_a \otimes \partial_b \right) (P_1 \otimes P_2) \\ & = P_1 \cdot P_2 + \hbar \underoverset{k_1, k_2 = 1}{2n}{\sum}\omega^{k_1 k_2}(\partial_{k_1} P_1) \cdot (\partial_{k_2} P_2) + \cdots \end{aligned}

given by the bilinear form

ωi2σ+12g. \omega \coloneqq \tfrac{i}{2} \sigma + \tfrac{1}{2} g \,.


prod:[[a 1,a 1 *,,a n,a n *]][[]] [[a 1,a 1 *,,a n,a n *]][[]][[a 1,a 1 *,,a n,a n *]][[]] prod \;\colon\; \mathbb{C}[ [ a_1, a^\ast_1, \cdots, a_n, a^\ast_n ] ] [ [ \hbar ] ] \otimes_{\mathbb{R}} \mathbb{C}[ [ a_1, a^\ast_1, \cdots, a_n, a^\ast_n ] ] [ [ \hbar ] ] \longrightarrow \mathbb{C}[ [ a_1, a^\ast_1, \cdots, a_n, a^\ast_n ] ] [ [ \hbar ] ]

is the ordinary (commutative) product in the formal power series algebra.

To make contact with the traditional notation we decorate the elements PP in the formal power series algebra with colons and declare the notation

:P 1::P 2::P 1 ωP 2: : P_1 : \, :P_2: \;\coloneqq\; : P_1 \star_\omega P_2 :

(Wick algebra of a single mode)

Let V 2Span({x,y})V \coloneqq \mathbb{R}^2 \simeq Span(\{x,y\}) be a standard almost Kähler vector space according to example 1, with canonical coordinates denoted xx and yy. We discuss its Wick algebra according to def. 2 and show that this reproduces the traditional definition of products of “normal ordered” operators as above.

To that end, consider the complex linear combination of the coordinates to the canonical complex coordinates zx+iyz \coloneqq x + i y and z¯xiy\overline{z} \coloneqq x - i y, which we suggestively write instead as

a12(x+iy)AAAAa *12(xiy) a \coloneqq \tfrac{1}{\sqrt{2}}(x + i y) \phantom{AAAA} a^\ast \coloneqq \tfrac{1}{\sqrt{2}}(x - i y)

(with “aa” the traditional symbol for the amplitude of a field mode).

We find the value of the almost-Kähler forms on these elements to be

σ(a,a *) =12σ((x+iy),(xiy)) =i2(σ(x,y)σ(y,x)) =i \begin{aligned} \sigma(a,a^\ast) & = \tfrac{1}{2} \sigma( (x + i y), (x - i y) ) \\ & = \tfrac{-i}{2}( \sigma(x,y) - \sigma(y,x) ) \\ & = - i \end{aligned}
h(a,a *) =12h((x+iy),(xiy)) =12(h(x,x)+h(y,y)) =1 \begin{aligned} h(a, a^\ast) & = \tfrac{1}{2} h( (x + i y), (x - i y) ) \\ & = \tfrac{1}{2}(h(x,x) + h(y,y)) \\ & = 1 \end{aligned}
σ(a,a) =σ(a *,a *) =0AAAAAAby anti-symmetry \begin{aligned} \sigma(a,a) & = \sigma(a^\ast, a^\ast) \\ & = 0 \phantom{AAAAAA} \text{by anti-symmetry} \end{aligned}
h(a,a) =12(h((x+iy),(x+iy))) =12(h(x,x)h(y,y)) =0 \begin{aligned} h(a,a) & = \tfrac{1}{2}( h( (x + i y), (x + i y) ) ) \\ & = \tfrac{1}{2}( h(x,x) - h(y,y)) \\ & = 0 \end{aligned}
h(a *,a *) =12(h(x,x)h(y,y)) =0 \begin{aligned} h(a^\ast, a^\ast) & = \tfrac{1}{2}( h(x,x) - h(y,y) ) \\ & = 0 \end{aligned}

Using this, we find the star product as follows (where we write ()()(-)\cdot (-) for the plain commutative product in the formal power series algebra):

a ωa * =aa *+2(iσ(a,a *)=1+h(a,a *)=1) =a *a+ a * ωa =a *a+2(iσ(a *,a)=1+h(a *,a)=1) =a *a a ωa =aa a * ωa * =a *a * \begin{aligned} a \star_\omega a^\ast & = a \cdot a^\ast + \tfrac{\hbar}{2} \left( \underset{= 1}{\underbrace{i \sigma( a, a^\ast )}} + \underset{= 1}{\underbrace{h(a, a^\ast)}} \right) \\ & = a^\ast \cdot a + \hbar \\ \\ a^\ast \star_\omega a & = a^\ast \cdot a + \tfrac{\hbar}{2} \left( \underset{= -1}{\underbrace{i \sigma( a^\ast , a )}} + \underset{ = 1}{\underbrace{h( a^\ast, a )}} \right) \\ & = a^\ast \cdot a \\ \\ a \star_\omega a & = a \cdot a \\ \\ a^\ast \star_\omega a^\ast & = a^\ast \cdot a^\ast \end{aligned}

These four cases are sufficient to see that in the star-product P 1 ωP 2P_1 \star_\omega P_2 of general elements, we obtain correction term \hbar to the ordinary commutative product precisely for every pair consisting of a factor of aa in P 1P_1 and a factor a *a^\ast in P 2P_2. This is exactly the “normal ordering” prescription.

For free fields on curved spacetimes

The genuine Wick algebras in quantum field theory are the analogues to those induced by finite-dimensional almost Kähler vector spaces as above, where now the underlying vector space is a space of (“microcausal”) functionals on the space of smooth functions on a spacetime, and where the symplectic form is given by the causal propagator of the wave equation/Klein-Gordon equation.

We give the definition below. First we briefly recall the relevant ingredients:

Covariant phase space of the free scalar field


(point evaluation observable)

Let XX be a smooth manifold. Then the mapping space C (X,)C^\infty(X, \mathbb{R}) is naturally a smooth space.

For xXx \in X there is the smooth function

C (X) Φ(x) ϕ ϕ(x) \array{ C^\infty(X) &\overset{\Phi(x)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \phi(x) }

evaluating at xXx \in X.

More generally:


(smooth functionals on smooth functions)

For fC c (X)f \in C_c^\infty(X) a bump function, it induces the function given by integration

C (X) Φ(f) ϕ Xfϕdvol g, \array{ C^\infty(X) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_X f \cdot \phi dvol_g } \,,

where dvol gdvol_g denotes the volume form of the given pseudo-Riemannian metric, and similarly if fC c (X n)f \in C_c^\infty(X^n) is a bump function on the nn-fold Cartesian product manifold X nX^n, then there is

C (X n) Φ(f) ϕ X nfϕ ndvol g n, \array{ C^\infty(X^n) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_{X^n} f \cdot \phi^n dvol_g^n } \,,

Write reg\mathcal{F}_{reg} for the sub-algebra of smooth functions on the smooth space C (X)C^\infty(X) which is generated from the functions Φ(f)\Phi(f) for nn \in \mathbb{N} and fC c (X n)f \in C^\infty_c(X^n).

A further evident generalization of this takes f(X n)f \in \mathcal{E}'(X^n) to be a compactly supported distribution and induces the function

C (X n) Φ(f) ϕ f,ϕ n g. \array{ C^\infty(X^n) &\overset{\Phi(f)}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \langle f, \phi^{\otimes^n} \rangle_g } \,.

Write dist\mathcal{F}_{dist} for the subalgebra generated by these functionals. (After deformation quantization below, this is the origin of “operator-valued distributions” in perturbative quantum field theory).

Recall the following general facts about the wave equation/Klein-Gordon equation:


(causal propagator)

Let (X,g)(X,g) be a time-oriented globally hyperbolic spacetime and let m 0m \in \mathbb{R}_{\geq 0} (the “mass”). Then the Klein-Gordon equation

( g+m 2)ϕ=0 (\Box_g + m^2) \phi = 0

(a partial differential equation on smooth functions fC (X,)f \in C^\infty(X,\mathbb{R}) ) has unique advanced and retarded Green functions E R/AE^{R/A}, namely continuous linear functionals

E A/R:C c (X)C (X) E^{A/R} \;\colon\; C^\infty_c(X) \longrightarrow C^\infty(X)

(from bump functions to general smooth functions) which are fundamental solutions in that

( g+m 2)E A/R=δAAAAE A/R( g+m 2)=δ (\Box_g + m^2) \circ E^{A/R} = \delta \phantom{AAAA} E^{A/R} \circ (\Box_g + m^2) = \delta

and which have advanced/retarded support of a distribution when viewed (via the Schwartz kernel theorem) as distributions on the Cartesian product manifold X×XX \times X

supp(E A/R){(x 1,x 2)X×X|x 1J (x 2)}. supp( E^{A/R}) \subset \{ (x_1, x_2) \in X \times X \;\vert\; x_1 \in J^{\mp} (x_2) \} \,.

In fact these two fundamental solutions are related by switching their arguments

E A/R(x 1,x 2)=E R/A(x 2,x 1). E^{A/R}(x_1, x_2) = E^{R/A}(x_2, x_1) \,.

Finally their wave front set is

WF(E A/R)={((x 1,x 2),(k 1,k 2))|x 1J (x 2)and((x 1,k 1)(x 2,k 2))or((x 1=x 2)andk 1=k 2)}. WF(E^{A/R}) \;=\; \left\{ ((x_1, x_2), (k_1, -k_2)) \;\vert\; x_1 \in J^{\mp}(x_2) \;\text{and}\; \left( (x_1, k_1) \sim (x_2, k_2) \right) \;\text{or}\; \left( (x_1 = x_2) \,\text{and}\, k_1 = -k_2 \right) \right\} \,.

Here the relation (x 1,k 1)(x 2,k 2)(x_1, k_1) \sim (x_2, k_2) means that there exists a lightlike geodesic from x 1x_1 to x 2x_2 with cotangent vector k 1k_1 at x 1x_1 and k 2k_2 at x 2x_2.

It follows that the wave front set of their difference (the causal propagator)

EE AE R E \;\coloneqq\; E^A - E^R


WF(E)={((x 1x 2),(k 1,k 2))|(x 1,k 1)(x 2,k 2)}. WF(E) \;=\; \left\{ ((x_1 x_2), (k_1, -k_2)) \;\vert\; (x_1, k_1) \sim (x_2, k_2) \right\} \,.

This causal propagator gives the Poisson bracket on the covariant phase space of the free scalar field in that, as distributions

{Φ(x 1),Φ(x 2)}=E(x 1,x 2), \left\{ \Phi(x_1) , \Phi(x_2) \right\} \;=\; E(x_1, x_2) \,,

where on the left {,}\{-,-\} is the Poisson bracket and Φ(x)\Phi(x) is the point evaluation function from def. 3. This means that on the algebra reg\mathcal{F}_{reg} of regular functionals from def. 4 the bracket is given by

{Φ(f 1),Φ(f 2)}=E,f 1f 2. \{\Phi(f_1), \Phi(f_2)\} \;=\; \langle E, f_1 \cdot f_2 \rangle \,.

The point now is that:


The algebra reg\mathcal{F}_{reg} of regular functionals (def. 4) is too small a subalgebra of all smooth functionals on C (X)C^\infty(X) to be of interest (for instance the point interaction terms such as gΦ(x) n:ϕ Xg(x)(ϕ(x)) xdvolg\Phi(x)^n \colon \phi \mapsto \int_X g(x) (\phi(x))^x dvol (for gg an adiabatic switching coupling constant) needed for phi^4 theory etc. are not contained). On the other hand dist\mathcal{F}_{dist} is too large: extending the above Poisson bracket to this algebra would mean to form a product of distributions with EE, and in general this is not defined.

But the obstruction to this product being defined is well characterized: One may multiply with EE all those distributions ff such that the sum of the wave front sets WF(E)+WF(f)WF(E) + WF(f) does not intersect the zero-section.

Therefore if EE could be modified such that its wave front set shrinks, then this variant may be applied to larger subalgebras of smooth functionals. Now def. 2 suggests that sensible modifications of the causal propagator are those by adding distributions HH on X 2X^2 that are symmetric in their two arguments.

Such distributions are quasi-free Hadamard states:


(quasi-free Hadamard state)

Let (X,g)(X,g) be a time oriented globally hyperbolic spacetime.

A Hadamard 2-point function for the free scalar field on (X,g)(X,g) is a distribution

ω𝒟(X×X) \omega \in \mathcal{D}'(X \times X)

on the Cartesian product manifold such that

  1. the anti-symmetric part of ω\omega is the causal propagator EE (from prop. 1)

    ω(x 1,x 2)ω(x 2,x 1)=iE(x 1,x 2) \omega(x_1, x_2) - \omega(x_2, x_1) \;=\; i E(x_1, x_2)
  2. the wave front set is one causal half that of the causal propagator:

    WF(ω)={((x 1,x 2),(k 1,k 2))|(x 1,k 1)(x 2,k 2)andk 1V x 1 +} WF(\omega) \;=\; \left\{ ((x_1, x_2), (k_1, -k_2)) \;\vert\; (x_1, k_1) \simeq (x_2, k_2) \;\;\text{and}\;\; k_1 \in V_{x_1}^+ \right\}
  3. (Klein-Gordon bi-solution) ( gm 2)ω(,x)=0(\Box_g - m^2) \omega(-,x) = 0 and ( gm 2)ω(x,)=0(\Box_g - m^2)\omega(x,-) = 0, for all xXx \in X;

  4. (positive semi-definiteness) For any complex-valued bump function bb we have that

    X×Xb *(x)ω(x,y)b(y)dvol(x)dvol(y)ω,b *b0 \int_{X \times X} b^\ast(x) \omega(x,y) b(y) \, dvol(x) dvol(y) \;\coloneqq\; \langle \omega, b^\ast \otimes b \rangle \;\geq\; 0

(existence of Hadamard distributions)

Let (X,g)(X,g) be a globally hyperbolic spacetime. Then a Hadamard distribution ω\omega according to def. 5 does exist.

(Radzikowski 96)

Wick algebra in field theory


(microcausal functionals)

Let (X,g)(X,g) be a globally hyperbolic spacetime.

Write mc dist\mathcal{F}_{mc} \subset \mathcal{F}_{dist} for the subalgebra of smooth functionals

C (X) C^\infty(X) \longrightarrow \mathbb{R}

on the smooth space of smooth functions on XX which is generated from those distributions on some Cartesian product X nX^n (as in def. 4) whose wave front set excludes those covectors to a point in X nX^n all whose components are in the future cone or all whose components are in the past cone.

(After deformation quantization below, the distributions appearing in def. 6 are the origin of “operator-valued distributions” in perturbative quantum field theory).


(regular functionals are microcausal)

Every regular functional (def. 4) is a microcausal functional (def. 6), since the wave front set of a distribution that is given by an ordinary function is empty:

reg mc. \mathcal{F}_{reg} \subset \mathcal{F}_{mc} \,.

(adiabtaically switched point interactions are microcausal)

Let gC c (X)g \in C^\infty_c(X) be a bump function, then for nn \in \mathbb{N} the smooth functional

C (X) ϕ Xg(x)(ϕ(x)) ndvol(x) \array{ C^\infty(X) &\overset{}{\longrightarrow}& \mathbb{R} \\ \phi &\mapsto& \int_X g(x) (\phi(x))^n dvol(x) }

is a microcausal functional (def. 6).

If here we think of ϕ(x) n\phi(x)^n as a point-interaction term (as for instance in phi^4 theory) then gg is to be thought of as an “adiabatically switchedcoupling constant. These are the relevant interaction terms to be quantized via causal perturbation theory.


For notational convenience, consider the case n=2n = 2, the other cases are directly analogous. The distribution in question is the delta distribution

Xg(x)ϕ(x) 2dvol(x)= X×Xg(x 1)ϕ(x 1)ϕ(x 2)dvol(x 1)dvol(x 2)=gδ(,),(ϕpr 1)(ϕpr 2) g. \int_X g(x) \phi(x)^2 dvol(x) \;=\; \int_{X \times X} g(x_1) \phi(x_1) \phi(x_2) dvol(x_1) dvol(x_2) = \langle g \cdot \delta(-,-) , (\phi \circ pr_1)\cdot (\phi \circ pr_2) \rangle_g \,.

Now for (x 1,x 2)X×X(x_1, x_2) \in X \times X and 2nUX×X\mathbb{R}^{2n} \simeq U \subset X \times X a chart around this point, the Fourier transform of gδ(,)g \cdot \delta(-,-) restricted to this chart is proportional to the Fourier transform g^\hat g of gg evaluated at the sum of the two covectors:

(k 1,k 2) 2ng(x 1)δ(x 1,x 2)exp(k 1x 1+k 2x 2)dvol(x 1)dvol(x 2) g^(k 1+k 2). \begin{aligned} (k_1, k_2) & \mapsto \int_{\mathbb{R}^{2n}} g(x_1) \delta(x_1, x_2) \exp( k_1 \cdot x_1 + k_2 \cdot x_2 ) dvol(x_1) dvol(x_2) \\ & \propto \hat g(k_1 + k_2) \end{aligned} \,.

Since gg is a plain bump function, its Fourier transform g^\hat g is quickly decaying (in the sense of wave front sets) with k 1+k 2k_1 + k_2 (this prop.). Thus only on the cone k 1+k 2=0k_1 + k_2 = 0 that function is in fact constant and in particular not decaying.

This means that the wave front set consists of the element of the form (x,(k,k))(x, (k, -k)) with k0k \neq 0. Since kk and k-k are both in the future cone or both in the past cone precisely if k=0k = 0, this situation is excluded in the wave front set and hence the distribution gδ(,)g \cdot \delta(-,-) is microcausal.

(graphics grabbed from Khavkine-Moretti 14, p. 45)

This shows that microcausality in this case is related to conservation of momentum in the point interaction.

More generally:


(compactly supported polynomial local observables)


Ω poly h,v(E) \Omega_{poly}^{h,v}(E)

for the space of differential forms on the jet bundle of EE which locally are polynomials in the field variables.

locC c (Σ)Ω poly 0,0(E)Ω poly d,0(E) \mathcal{F}_{loc} \; \subset \; C^\infty_c(\Sigma) \underset{\Omega_{poly}^{0,0}(E)}{\otimes} \Omega_{poly}^{d,0}(E)

for the subspace of horizontal differential forms of degree dd on the jet bundle (local Lagrangian densities) of those which are compactly supported with respect to Σ\Sigma and polynomial with respect to the field variables (local observables).

Every L locL \in \mathcal{F}_{loc} induces a functional

Γ Σ(E) \Gamma_\Sigma(E) \longrightarrow \mathbb{R}

by integration of the pullback of LL along the jet prolongation of a given section:

ϕ Σj (ϕ) *L. \phi \mapsto \int_{\Sigma} j^\infty(\phi)^\ast L \,.

These functionals happen to be microcausal, so that there is an inclusion

loc mc \mathcal{F}_{loc} \hookrightarrow \mathcal{F}_{mc}

into the space of microcausal functionals. (See this example at microcausal functional).


(Hadamard-Moyal star product on microcausal functionals)

Let (X,g)(X,g) be a globally hyperbolic spacetime, and let ω𝒟(X×X)\omega \in \mathcal{D}'(X \times X) be a Hadamard distribution (def. 5) which is guaranteed to exist by prop. 2.

Then the star product

P 1 ωP 2prodexp( X 2ω(x 1,x 2)δδϕ(x 1)δδϕ(x 2)dvol g)(P 1P 2) P_1 \star_\omega P_2 \;\coloneqq\; prod \circ \exp\left( \int_{X^2} \hbar \omega(x_1, x_2) \frac{\delta}{\delta \phi(x_1)} \otimes \frac{\delta}{\delta \phi(x_2)} dvol_g \right) (P_1 \otimes P_2)

on microcausal functionals P 1,P 2 mcP_1, P_2 \in \mathcal{F}_{mc} is well defined in that the products of distributions that appear in expanding out the exponential are such that the sum of the wave front sets of the factors does not intersect the zero section.


By definition of Hadamard distribution, the wave front set of powers of ω\omega has all cotangents on the first variables future pointing, and all those on the second variables past pointing. The first variables are integrated against those of P 1P_1 and the second against P 2P_2. By definition of microcausal functionals, the wave front sets of P 1P_1 and P 2P_2 are disjoint from the subsets where all components are future pointing or all components are past-pointing. Therefore the relevant sum of of the wave front covectors never vanishes.

See Collini 16, p. 25-26


(Wick algebra of free quantum field)

Let (X,g)(X,g) be a globally hyperbolic spacetime and let ω𝒟(X×X)\omega \in \mathcal{D}'(X \times X) be a Hadamard distribution (def. 5) which is guaranteed to exist by prop. 2.

Then the Wick algebra of quantum observables of the free scalar field on (X,g)(X,g) is the space of microcausal functionals mc\mathcal{F}_{mc} (def. 6) equipped with the Hadamard-Moyal star product from prop. 3:

𝒲(X,ω)( mc, ω). \mathcal{W}(X,\omega) \;\coloneqq\; \left( \mathcal{F}_{mc}, \star_\omega \right) \,.

need to quotient out ideal of elements in the image of gm 2\Box_g - m^2 to go on shell

In Minkowski spacetime

In Minkowski spacetime the Hadamard state is simply the usual vacuum state |vac\vert vac \rangle, hence the Hadamard distribution is, as a generalized function

ω(x,y)=vac|Φ(x)Φ(y)|vac. \omega(x,y) = \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle \,.

Green functions for the Klein-Gordon operator on a globally hyperbolic spacetime:

propagatorAA\phantom{AA}AA\phantom{AA} primed wave front seton Minkowski spacetimegenerally
causal propagatorΔΔ RΔ A\array{\Delta \coloneqq \Delta_R - \Delta_A } A a\array{- \\ \phantom{A} \\ \phantom{a}} Δ S(x,y)= vac|[Φ(x),Φ(y)]|vac\array{\Delta_S(x,y) = \\ \langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }Peierls-Poisson bracket
advanced propagatorΔ A\Delta_AΔ A(x,y)= Θ((yx) 0)vac|[Φ(x),Φ(y)]|vac\array{\Delta_A(x,y) = \\ \Theta((y-x)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }
retarded propagatorΔ R\Delta_RΔ R(x,y)= Θ((xy) 0)vac|[Φ(x),Φ(y)]|vac\array{\Delta_R(x,y) = \\ \Theta((x-y)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }
Dirac propagatorΔ D=12(Δ A+Δ R)\Delta_D = \tfrac{1}{2}(\Delta_A + \Delta_R) + A a\array{+ \\ \phantom{A} \\ \phantom{a}}
Hadamard propagatorω =i2Δ+H =ω FiΔ A\begin{aligned} \omega &= \tfrac{i}{2}\Delta + H \\ & = \omega_F - i \Delta_A \end{aligned}ω(x,y)= vac|Φ(x)Φ(y)|vac\array{\omega(x,y) = \\ \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle }normal-ordered product (2-point function of quasi-free state)
Feynman propagatorω F =iΔ D+H =ω+iΔ A\array{\omega_F & = i \Delta_D + H \\ & = \omega + i \Delta_A}E F(x,y)= vac|T(Φ(x)Φ(y))|vac\array{E_F(x,y) = \\ \langle vac \vert T(\Phi(x)\Phi(y)) \vert vac \rangle }time-ordered product

(see also Kocic’s overview: pdf)

Therefore the abstractly defined Wick algebra as in def. 7 in this case satisfies the relation

Xf(x,y):Φ(x)::Φ(y):dvol g= Xf(x,y)(:Φ(x)Φ(y):vac|Φ(x)Φ(y)|vac)dvol g. \int_{X} f(x,y) \; :\Phi(x): \, :\Phi(y): \; dvol_g \;=\; \int_X f(x,y) \left( :\Phi(x) \Phi(y): - \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle \right) \; dvol_g \,.

This is the traditional expression for the normal ordered Wick product on Minkowski spacetime (e.g. here).

product in perturbative QFT\,\, induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables


The construction goes back to

  • Gian-Carlo Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268-272 (1950)

Its realization as the Moyal deformation quantization of the Peierls bracket shifted by a quasi-free Hadamard state is due to

  • J. Dito, Star-product approach to quantum field theory: The free scalar field. Letters in Mathematical Physics, 20(2):125–134, 1990 (spire)

further amplified in

  • Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic field theory, and deformation quantization, in Roberto Longo (ed.), Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001 (arXiv:hep-th/0101079)

and the generalization to quantum field theory on curved spacetime is discussed in

The conceptual explanation of the shift by the Hadamard state as a step in the almost-Kähler version of Fedosov deformation quantization is due to

Revised on October 17, 2017 19:02:43 by Urs Schreiber (