Wick algebra in nLab

Context

Algebraic Quantum Field Theory

• Idea
• Definition
• Abstract Wick algebra
• Abstract time-ordered product
• Operator product notation
• Hadamard vacuum states
• Related concepts
• References

Definition

(microcausal polynomial observables) Let $E \overset{fb}{\to}$ be field bundle which is a vector bundle. An off-shell polynomial observable is a smooth function

$$A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}$$

on the on-shell space of sections of the field bundle $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

$$\begin{aligned} A(\Phi) & = \phantom{+} \alpha^{(0)} \\ & \phantom{=} + \int_\Sigma \alpha^{(1)}_a(x) \Phi^a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \int_\Sigma \int_\Sigma \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2) \,dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \phantom{=} + \cdots \,, \end{aligned}$$

where

$$\alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)$$

is a compactly supported distribution of k variables on the $k$-fold graded-symmetric external tensor product of vector bundles of the field bundle with itself.

Write

$$PolyObs(E) \hookrightarrow Obs(E)$$

for the subspace of off-shell polynomial observables onside all off-shell observables.

Let moreover $(E,\mathbf{L})$ be a free Lagrangian field theory whose equations of motion are Green hyperbolic differential equations. Then an on-shell polynomial observable is the restriction of an off-shell polynomial observable along the inclusion of the on-shell space of field histories $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

$$PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})$$

for the subspace of all on-shell polynomial observables inside all on-shell observables.

By this prop. restriction yields an isomorphism between polynomial on-shell observables and polynomial off-shell observables modulo the image of the differential operator $P$:

$$PolyObs(E,\mathbf{L}) \underoverset{\simeq}{\text{restriction}}{\longleftarrow} PolyObs(E)/im(P) \,.$$

Finally a polynomial observable is a microcausal observable if each coefficient $\alpha^{(k)}$ as above has wave front set away from those points where the $k$ wave vectors are all in the future cone or all in the past cone. We write

$$\array{ PolyObs(E)_{mc} &\hookrightarrow& PolyObs(E) \\ PolyObs(E,\mathbf{L})_{mc} \simeq PolyObs(E)_{mc}/im(P) &\hookrightarrow& PolyObs(E,\mathbf{L}) }$$

for the subspace of off-shell/on-shell microcausal observables inside all off-shell/on-shell polynomial observables.

Proposition

(Hadamard-Moyal star product on microcausal observables – abstract Wick algebra)

Let $(E,\mathbf{L})$ a free Lagrangian field theory with Green hyperbolic equations of motion $P \Phi = 0$. Write $\Delta$ for the causal propagator and let

$$\Delta_H \;=\; \tfrac{i}{2}\Delta + H$$

be a corresponding Wightman propagator (Hadamard 2-point function).

Then the star product induced by $\Delta_H$

$$A \star_H A \;\coloneqq\; prod \circ \exp\left( \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g \right) (P_1 \otimes P_2)$$

on off-shell microcausal observables $A_1, A_2 \in \mathcal{F}_{mc}$ (def. ) is well defined in that the wave front sets involved in the products of distributions that appear in expanding out the exponential satisfy Hörmander's criterion.

Hence by the general properties of star products (this prop.) this yields a unital associative algebra structure on the space of formal power series in $\hbar$ of off-shell microcausal observables

$$\left( PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H \right) \,.$$

This is the off-shell Wick algebra corresponding to the choice of Wightman propagator $H$.

Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the on-shell microcausal observables to yield the on-shell Wick algebra

$$\left( PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H \right) \,.$$

Finally, under complex conjugation $(-)^\ast$ these are star algebras in that

$$\left( A_1 \star_H A_2 \right)^\ast = A_2^\ast \star_H A_1^\ast \,.$$

Proof

By prop. this is immediate from the general properties of the star product (this example).

Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then

$$\begin{aligned} A_1 \star_H A_2 & = A_1 A_2 + \hbar \int \left( \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2) \right) \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)} \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)} \;mod\; \hbar^2 \\ & = A_1 A_2 + \hbar \left( \int (\alpha_1)_{a_1}(x_1) \left( \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1, x_2) \right) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \end{aligned}$$

Now since $\Delta$ is skew-symmetric while $H$ is symmetric is follows that

$$\begin{aligned} A_1 \star_H A_2 - A_2 \star_H A_1 & = i \hbar \left( \int (\alpha_1)_{a_1}(x_1) \Delta^{a_1 a_2}(x_1, x_2) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \\ & = i \hbar \, \left\{ A_1, A_2\right\} \end{aligned} \,.$$

The right hand side is the integral kernel-expression for the Poisson-Peierls bracket, as shown in the second line.

Proof

Recall the following facts:

1. the advanced and retarded propagators $\Delta_{\pm}$ by definition are supported in the future cone/past cone, respectively

   $$supp(\Delta_{\pm}) \subset \overline{V}^{\pm}$$

2. they turn into each other under exchange of their arguments (this cor.):

   $$\Delta_\pm(y,x) = \Delta_{\mp}(x,y) \,.$$

3. the real part $H$ of the Feynman propagator, which by definition is the real part of the Wightman propagator is symmetric (by definition or else by this prop.):

   $$H(x,y) = H(y,x)$$

Using this we compute as follows:

(1)$$\begin{aligned} A_1 \underset{\Delta_{F}}{\star} A_2 \; & = A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2 \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \end{aligned}$$

Proposition

(time-ordered product on regular polynomial observables isomorphic to pointwise product)

The time-ordered product on regular polynomial observables (def. ) is isomorphic to the pointwise product of observables (this def.) via the linear isomorphism

$$\mathcal{T} \;\colon\; PolyObs(E)_{reg}[ [\hbar] ] \longrightarrow PolyObs(E)_{reg}[ [\hbar] ]$$

given by

(2)$$\mathcal{T}A \;\coloneqq\; \exp\left( \tfrac{1}{2} \hbar \underset{\Sigma}{\int} \Delta_F(x,y)^{a b} \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right) A$$

in that

$$\begin{aligned} T(A_1 A_2) & \coloneqq A_1 \star_{F} A_2 \\ & = \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) ) \end{aligned}$$

hence

$$\array{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-)\cdot (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq \\ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-) \star_F (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] }$$

Definition

(notation for operator product and normal-ordered product)

It is traditional to use the following alternative notation for the product structures on microcausal polynomial observables:

1. The Wick algebra-product, hence the star product $\star_H$ for the Wightman propagator (def. ), is rewritten as plain juxtaposition:

   $$\text{"operator product"} \phantom{AAA} A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1 \phantom{AAAA} \array{ \text{star product of} \\ \text{Wightman propagator} } \,.$$

2. The pointwise product of observables (this def.) $A_1 \cdot A_2$ is equivalently written as plain juxtaposition enclosed by colons:

   $$\array{ \text{"normal-ordered} \\ \text{product"} } \phantom{AAAA} :A_1 A_2: \phantom{AA}\coloneqq\phantom{AA} A_1 \cdot A_2 \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}$$

3. The time-ordered product, hence the star product for the Feynman propagator $\star_F$ (def. ) is equivalently written as plain juxtaposition prefixed by a “$T$”

   $$\array{ \text{"time-ordered} \\ \text{product"} } \phantom{AAAA} T(A_1 A_2) \phantom{AA}\coloneqq\phantom{AA} A_1 \star_F A_2 \phantom{AAAA} \array{ \text{star product of} \\ \text{Feynman propagator} }$$

Under representation of the Wick algebra on a Fock Hilbert space by linear operators the first product become the operator product, while the second becomes the operator poduct applied after suitable re-ordering, called “normal odering” of the factors.

Disregarding the Fock space-representation, which is faithful, we may still refer to these “abstract” products as the “operator product” and the “normal-ordered product”, respectively.

Proposition

(canonical vacuum states on abstract Wick algebra)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory with Green-hyperbolic Euler-Lagrange equations of motion; and let $\Delta_H$ be a compatible Wightman propagator.

For

$$\Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$$

any on-shell field history (i.e. solving the equations of motion), consider the function from the Wick algebra to formal power series in $\hbar$ with coefficients in the complex numbers which evaluates any microcausal polynomial observable on $\Phi_0$

$$\array{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ A &\mapsto& A(\Phi_0) }$$

Specifically for $\Phi_0 = 0$ (which is a solution of the equations of motion by the assumption that $(E,\mathbf{L})$ defines a free field theory) this is the function

$$\array{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_0}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ \left. \begin{aligned} A & = \alpha^{(0)} \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \cdots \end{aligned} \right\} &\mapsto& A(0) = \alpha^{(0)} }$$

which sends each microcausal polynomial observable to its value $A(\Phi = 0)$ on the zero field history, hence to the constant contribution $\alpha^{(0)}$ in its polynomial expansion.

The function $\langle -\rangle_0$ is

1. linear over $\mathbb{C}[ [\hbar] ]$;

2. real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$

   $$\langle A^\ast \rangle = \langle A \rangle^\ast$$

3. positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that

   $$\langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A \,,$$

4. normalized, in that

   $$\langle 1\rangle_H = 1$$

where $(-)^\ast$ denotes componet-wise complex conjugation.

This means that $\langle -\rangle_{0}$ is a states on the Wick star-algebra $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. ). One says that

• $\langle - \rangle_0$ is a Hadamard vacuum state;

and generally

• $\langle - \rangle_{\Phi_0}$ is called a coherent state.

Proof

The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a representation of the Wick algebra on a Fock Hilbert space (this algebra homomorphism is Wick's lemma), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the inner product of the Hilbert space.

Example

(operator product of two linear observables)

Let

$$A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}$$

for $i \in \{1,2\}$ be two linear microcausal observables represented by distributions which in generalized function-notation are given by

$$A_i \;=\; \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i) \,.$$

Then their Hadamard-Moyal star product (prop. ) is the sum of their pointwise product with their value

(3)$$\langle A_1 \star_H A_2 \rangle_0 \;\coloneqq\; i \hbar \int \int (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2) \,dvol_\Sigma(x_1) \,dvol_\Sigma(x_2)$$

in the Wightman propagator, which is the value of the Hadamard vacuum state from prop.

$$A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;+\; \langle A_1 \star_H A_2 \rangle_0$$

In the operator product/normal-ordered product-notation of def. this reads

$$A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle \,.$$

Example

(Wightman propagator is 2-point function in the Hadamard vacuum state)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory with Green-hyperbolic Euler-Lagrange equations of motion; and let $\Delta_H$ be a compatible Wightman propagator.

With respect to the induced Hadamard vacuum state $\langle - \rangle_0$ from prop. , the Wightman propagator $\Delta_H(x,y)$ itself is the 2-point function, namely the distributional vacuum expectation value of the operator product of two field observables:

$$\left\langle \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y') \right\rangle }}$$

by example .

Equivalently in the operator product-notation of def. this reads:

$$\left\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \hbar \Delta_H(x,y) \,.$$

Example

(Feynman propagator is time-ordered 2-point function in the Hadamard vacuum state)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory with Green-hyperbolic Euler-Lagrange equations of motion; and let $\Delta_H$ be a compatible Wightman propagator with induced Feynman propagator $\Delta_F$.

With respect to the induced Hadamard vacuum state $\langle - \rangle_0$ from prop. , the Feynman propagator $\Delta_F(x,y)$ itself is the time-ordered 2-point function, namely the distributional vacuum expectation value of the time-ordered product (def. ) of two field observables:

$$\left\langle T\left( \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y') \right\rangle }}$$

analogous to example .

Equivalently in the operator product-notation of def. this reads:

$$\left\langle T\left( \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \hbar \Delta_F(x,y) \,.$$

propagators (i.e. integral kernels of Green functions)
for the wave operator and Klein-Gordon operator
on a globally hyperbolic spacetime such as Minkowski spacetime:

name	symbol	wave front set	as vacuum exp. value of field operators	as a product of field operators
causal propagator	$\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}$	$\phantom{A}\,\,\,-$	$\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned}$	Peierls-Poisson bracket
advanced propagator	$\Delta_+$		$\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned}$	future part of Peierls-Poisson bracket
retarded propagator	$\Delta_-$		$\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}$	past part of Peierls-Poisson bracket
Wightman propagator	$\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned}$	positive frequency of Peierls-Poisson bracket, Wick algebra-product, 2-point function $\phantom{=}$ of vacuum state $\phantom{=}$ or generally of $\phantom{=}$ Hadamard state
Feynman propagator	$\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}$	time-ordered product

(see also Kocic‘s overview: pdf)

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