Learn Quantization in Mathematical Quantum Field Theory

The following is one chapter in a series on Mathematical Quantum Field Theory.

The previous chapter is 12. Gauge fixing.

The next chapter is 14. Free quantum fields.

13. Quantization

In this chapter we discuss the following topics:

• Motivation from Lie theory
• Geometric quantization
• Moyal star products
• Moyal star product as deformation quantization
• Moyal star product via geometric quantization
• Example: Wick algebra of normal ordered product on Kähler vector space
• Star-product on regular polynomial observables in field theory

In the previous chapters, we had found the Peierls-Poisson bracket (theorem 8.7) on the covariant phase space (prop. 8.6) of a gauge fixed (def. 12.1) free Lagrangian field theory (def. 5.25).

This Poisson bracket (def. 13.7 below) is a Lie bracket and hence reflects infinitesimal symmetries acting on the covariant phase space. Just as with the infinitesimal symmetries of the Lagrangian and the BRST-reduced field bundle (example 10.28), we may hard-wire these Hamiltonian symmetries into the very geometry of the phase space by forming their homotopy quotient given by the corresponding Lie algebroid (def. 10.22): here this is called the Poisson Lie algebroid. Its Lie integration to a finite (instead of infinitesimal) structure is called the symplectic groupoid. This is the original covariant phase space, but with its Hamiltonian flows hard-wired into its higher differential geometry (Bongers 14, section 4).

Where smooth functions on the plain covariant phase space form the commutative algebra of observables under their pointwise product (def. 7.1), the smooth functions on this symplectic groupoid-refinement of the phase space are multiplied by the groupoid convolution product and as such become a non-commutative algebra of quantum observables. This passage from the commutative to the non-commutative algebra of observables is called quantization, here specifically geometric quantization of symplectic groupoids (Hawkins 04, Nuiten 13).

Instead of discussing this in generality, we here focus right away on the simple special case relevant for the quantization of gauge fixed free Lagrangian field theories in the next chapter.

After an informal motivation of geometric quantization from Lie theory below (for a self-contained introduction see Bongers 14), we first showcase geometric quantization by discussing how the archetypical example of quantum mechanics in the Schrödinger representation arises from the polarized action of the Poisson bracket Lie algebra (example 13.1 below). With the concept of polarization thus motivated, we use this to find the polarized groupoid convolution algebra of the symplectic groupoid of a free theory (prop. 13.12 below).

The result is the “Moyal-star product” (def. 13.2 below). This is the exponentiation of the integral kernel of the Poisson bracket plus possibly asymmetric shift (prop. 13.6 below); it turns out to be (example 13.9 below) a formal deformation quantization of the original commutative pointwise product (def. 13.8 below).

Below we spell out the (elementary) proofs of these statements for the case of phase spaces which are finite-dimensional vector spaces. But these proofs manifestly depend only on elementary algebraic properties of polynomials and hence go through in more general contexts as long as these basic algebraic properties are retained.

In the context of free Lagrangian field theory, the analog of the formal power series algebras on a linear phase space is, a priori, the algebra of polynomial observables (def. 7.13). These are effectively polynomials in the field observables $\mathbf{\Phi}^a(x)$ (def. 7.2) whose coefficients, however, are distributions of several variables. By microlocal analysis, such polynomial distributions do satisfy the usual algebraic properties of ordinary polynomials if potential UV-divergences (remark 9.27) encoded in their wave front set (def. 9.28) vanish, according to Hörmander’s criterion (prop. 9.34).

This criterion is always met on the subspace of regular polynomial observables and hence every propagator induces a star product on these (prop. 13.18 below). In particular thus the star product of the causal propagator of a gauge fixed free Lagrangian field theory is a formal deformation quantization of its algebra of regular polynomial observables (cor. 13.19 below). In order to extend this to local observables, one may appeal to certain quantization freedom (prop. 13.6 below) and shift the causal propagator by a symmetric contribution, such that it becomes the Wightman propagator; this is the topic of the following chapters (remark 13.20 at the end below).

In conclusion, for free gauge fixed Lagrangian field theory the product in the algebra of quantum observables is given by exponentiating propagators. It is the combinatorics of these exponentiated propagator expressions that yield the hallmark structures of perturbative quantum field theory, namely the combinatorics of Wick’s lemma for the Wick algebra of free fields, and the combinatorics of Feynman diagrams for the time-ordered products. This is the topic of the following chapters Free quantum fields and Scattering. Here we conclude just by discussing the finite-dimensional toy version of the normal-ordered product in the Wick algebra (example 13.17 below).

motivation from Lie theory

Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.

But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics?

The following spells out an argument to this effect. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.

So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold $(X, \omega)$. A symplectic manifold is, in particular, a Poisson manifold, which means that the algebra of functions on phase space $X$, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance, if $H \in C^\infty(X)$ is the function on phase space which is interpreted as assigning to each configuration of the system its energy — the Hamiltonian function — then the Poisson bracket with $H$ yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton’s equations.

Something to take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra $\mathfrak{g}$, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) $G$. One also says that $G$ is a Lie integration of $\mathfrak{g}$ and that $\mathfrak{g}$ is the Lie differentiation of $G$.

Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under the Poisson bracket, what then is the corresponding Lie group?

The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs that state the answer. But, maybe surprisingly, the answer to this question is not (at the time of this writing) a widely advertised fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads to the quantum mechanics of the system.

Before we spell this out in more detail, we need a brief technical aside: of course, Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.

The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian line Lie algebra $\mathbb{R}$. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just $\mathbb{R}$ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group

$$
U(1) = \mathbb{R}/\mathbb{Z}
\,.
$$

Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology but can be traced back to the heart of what is “quantized” about quantum mechanics.

Namely, one finds that the Poisson bracket Lie algebra $\mathfrak{poiss}(X,\omega)$ of the classical observables on phase space is (for $X$ a connected manifold) a Lie algebra extension of the Lie algebra $\mathfrak{ham}(X)$ of Hamiltonian vector fields on $X$ by the line Lie algebra:

$$
\mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X)
\,.
$$

This means that under Lie integration the Poisson bracket turns into a central extension of the group of Hamiltonian symplectomorphisms of $(X,\omega)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting central extension by the circle group $U(1)$. For this non-trivial Lie integration to existing, $(X,\omega)$ needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this $U(1)$-central extension of the group $Ham(X,\omega)$ of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group $QuantMorph(X,\omega)$:

$$
U(1) \longrightarrow QuantMorph(X,\omega) \longrightarrow Ham(X,\omega)
\,.
$$

While important, for some reason this group is not very well known, which is striking because it contains a small subgroup that is famous in quantum mechanics: the Heisenberg group.

More precisely, whenever $(X,\omega)$ itself has a compatible group structure, notably if $(X,\omega)$ is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space $(X,\omega)$ on itself. This is the corresponding Heisenberg group $Heis(X,\omega)$, which in turn is a $U(1)$-central extension of the group $X$ itself:

$$
U(1) \longrightarrow Heis(X,\omega) \longrightarrow X
\,.
$$

At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics:

if we think of Lie integrating $\mathbb{R}$ to the interesting circle group $U(1)$ instead of to the uninteresting translation group $\mathbb{R}$, then the name of its canonical basis element $1 \in \mathbb{R}$ is canonically “$i$”, the imaginary unit. Therefore one often writes the above central extension instead as follows:

$$
i \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X,\omega)
$$

in order to amplify this. But now consider the simple special case where $(X,\omega) = (\mathbb{R}^2, d p \wedge d q)$ is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions $p$ and $q$ of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i \mathbb{R}$, hence purely Lie theoretically it is to be called “$i$”.

With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads

$$
[q,p] = i
\,.
$$

Since the choice of basis element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing real number without changing this statement. If we write “$\hbar$” for this element, then the Poisson bracket instead reads

$$
[q,p] = i \hbar
\,.
$$

This is of course the hallmark equation for quantum physics if we interpret $\hbar$ here indeed as Planck’s constant. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.

This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here with a brief outlook.

The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form $\omega$, it is natural to ask if it is the curvature 2-form of a $U(1)$-principal connection $\nabla$ on complex line bundle $L$ over $X$ (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the field strength 2-form of electromagnetism on spacetime). If so, such a connection $(L, \nabla)$ is called a prequantum line bundle of the phase space $(X,\omega)$. The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).

As such, the quantomorphism group naturally acts on the space of sections of $L$. Such a section is like a wavefunction, except that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend only on the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance, in the simple case mentioned before where $(X,\omega)$ is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.

geometric quantization

We had seen that every Lagrangian field theory induces a presymplectic current $\Omega_{BFV}$ (def. 5.12) on the jet bundle of its field bundle in terms of which there is a concept of Hamiltonian differential forms and Hamiltonian vector fields on the jet bundle (def. 6.19). The concept of quantization is induced by this local phase space structure.

In order to disentangle the core concept of quantization from the technicalities involved in fully-fledged field theory, we now first discuss the finite-dimensional situation.

Example 13.1. (Schrödinger representation via geometric quantization)

Consider the Cartesian space $\mathbb{R}^2$ (def. 1.1) with canonical coordinate functions denoted $\{q,p\}$ and to be called the canonical coordinate $q$ and its canonical momentum $p$ (as in example 5.15) and equipped with the constant differential 2-form given in in (58) by

$$
\label{R2SymplecticForm}
\omega = d p \wedge d q
\,.
$$ (207)

This is closed in that $d \omega = 0$, and invertible in that the contraction of tangent vector fields into it (def. 1.20) is an isomorphism to differential 1-forms, and as such it is a symplectic form.

A choice of presymplectic potential for this symplectic form is

$$
\label{CanonicalSymplecticPotentialOnR2}
\theta \;:=\; – q \, d p
$$ (208)

in that $d \theta = \omega$. (Other choices are possible, notably $\theta = p \, d q$).

For

$$
A
\;\colon\;
\mathbb{R}^2
\longrightarrow
\mathbb{C}
$$

a smooth function (an observable), we say that a Hamiltonian vector field for it (as in def. 6.19) is a tangent vector field $v_A$ (example 1.12) whose contraction (def. 1.20) into the symplectic form (207) is the de Rham differential of $A$:

$$
\label{HamiltonianVectorFieldOnR2}
\iota_{v_A} \omega = d A
\,.
$$ (209)

Consider the foliation of this phase space by constant-$q$-slices

$$
\label{ConstantqSlicesOnR2}
\Lambda_q = \subset \mathbb{R}^2
\,.
$$ (210)

These are also called the leaves of a real polarization of the phase space.

(Other choices of polarization are possible, notably the constant $p$-slices.)

We says that a smooth function

$$
\psi \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{C}
$$

is polarized if its covariant derivative with connection on a bundle $i \theta$ along the leaves vanishes; which for the choice of polarization in (210) means that

$$
\nabla_{\partial_p} \psi = 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\iota_{\partial_p} \left( d \psi + i \theta \psi \right)
=
0
\,,
$$

which in turn, for the choice of presymplectic potential in (208), means that

$$
\frac{\partial}{\partial p} \psi – i q \psi = 0
\,.
$$

The solutions to this differential equation are of the form

$$
\label{PolarizedFunctionsForBasicExampleOnR2}
\Psi(q,p) = \psi(q) \, \exp(+ i p q)
$$ (211)

for $\psi \colon \mathbb{R} \to \mathbb{C}$ any smooth function, now called a wave function.

This establishes a linear isomorphism between polarized smooth functions and wave functions.

By (209) we have the Hamiltonian vector fields

$$
v_q = \partial_p
\phantom{AAAA}
v_p = -\partial_q
\,.
$$

The corresponding Poisson bracket is

$$
\label{R2PoissonBracket}
\begin{aligned}
\{q,p\}
& :=
\iota_{v_p} \iota_{v_q} \omega
\\
& =
-\iota_{\partial_q} \iota_{\partial_p} d p \wedge d q
=
\\
& =
– 1
\end{aligned}
$$ (212)

The action of the corresponding quantum operators $\hat q$ and $\hat p$ on the polarized functions (211) is as follows

$$
\begin{aligned}
\hat q \Psi(q,p)
& =
– i \nabla_{\partial_p}\Psi(q,p) + q \Psi(q,p)
\\
& = -i
\underset{ = 0 }{
\underbrace{
\left(
\underset{
= i q \Psi(q,p)
}{
\underbrace{
\frac{\partial}{\partial p}
\left(
\psi(q) e^{i q p}
\right)
}
}
– i q \Psi(q,p)
\right)
}
}
+
q \Psi(q,p)
\\
& = \left( q \psi(q) \right) e^{i q p}
\end{aligned}
$$

and

$$
\begin{aligned}
\hat p \Psi(q,p)
& =
i \nabla_{\partial_q} \Psi(q,p) + p \Psi(q,p)
\\
& = i \frac{\partial}{\partial q} (\psi(q)e^{i q p}) + p \Psi(q,p)
\\
& =
\left(
i \frac{\partial}{\partial q}\psi(q)
\right) e^{i q p}
+
\underset{ = 0}{
\underbrace{
\underset{
= – p \Psi(q,p)
}{
\underbrace{
\psi(q)
\left(
i \frac{\partial}{\partial q} e^{i q p}
\right)
}
}
+
p \Psi(q,p)
}
}
\\
& =
\left(
i \frac{\partial}{\partial q}\psi(q)
\right) e^{i p q}
\end{aligned}
\,.
$$

Hence under the identification (211) we have

$$
\hat q \psi = q \psi
\phantom{AAAA}
\hat p \psi = i \frac{\partial}{\partial q} \psi
\,.
$$

This is called the Schrödinger representation of the canonical commutation relation (212).

Moyal star products

Let $V$ be a finite-dimensional vector space and let $\pi \in V \otimes V$ be an element of the tensor product (not necessarily skew-symmetric at the moment).

We may canonically regard $V$ as a smooth manifold, in which case $\pi$ is canonically regarded as a constant rank-2 tensor. As such it has a canonical action by forming derivatives on the tensor product of the space of smooth functions:

$$
\pi
\;\colon\;
C^\infty(V) \otimes C^\infty(V)
\longrightarrow
C^\infty(V) \otimes C^\infty(V)
\,.
$$

If $\{\partial_i\}$ is a linear basis for $V$, identified, as before, with a basis for $\Gamma(T V)$, then in this basis, this operation reads

$$
\pi(f \otimes g)
\;=\;
\pi^{i j}
(\partial_i f) \otimes (\partial_j g)
\,,
$$

where $\partial_i f := \frac{\partial f}{\partial x^i}$ denotes the partial derivative of the smooth function $f$ along the $i$th coordinate, and where we use the Einstein summation convention.

For emphasis we write

$$
\array{
C^\infty(V) \otimes C^\infty(V)
&\overset{prod}{\longrightarrow}&
C^\infty(V)
\\
f \otimes g &\mapsto& f \cdot g
}
$$

for the pointwise product of smooth functions.

Definition 13.2. (star product induced by constant rank-2 tensor)

Given $(V,\pi)$ as above, then the star product induced by $\pi$ on the formal power series algebra $C^\infty(V) [ [\hbar] ]$ in a formal variable $\hbar$ (“Planck’s constant”) with coefficients in the smooth functions on $V$ is the linear map

$$
(-) \star_\pi (-)
\;\colon\;
C^\infty(V)[ [ \hbar ] ] \otimes C^\infty(V)[ [ \hbar ] ]
\longrightarrow
C^\infty(V)[ [\hbar] ]
$$

given by

$$
(-) \star_\pi (-)
\;:=\;
prod
\circ \exp\left(
\hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
\right)
$$

Hence

$$
f \star_\pi g
\;:=\;
1
+
\hbar \pi^{i j}
\frac{\partial f}{\partial x^i} \cdot \frac{\partial g}{\partial x^j}
+
\hbar^2
\tfrac{ 1 }{2}
\pi^{i j}
\pi^{k l}
\frac{\partial^2 f}{\partial x^{i} \partial x^{k}}
\cdot
\frac{\partial^2 g}{\partial x^{j} \partial x^{l}}
+
\cdots
\,.
$$

Example 13.3. (star product degenerating to pointwise product)

If $\pi = 0$ in def. 13.2, then the star product $\star_0 = \cdot$ is the plain pointwise product of functions.

Example 13.4. (Moyal star product)

If the tensor $\pi$ in def. 13.2 is skew-symmetric, it may be regarded as a constant Poisson tensor on the smooth manifold $V$. In this case $\star_{\tfrac{1}{2}\pi}$ is called a Moyal star product and the star-product algebra $C^\infty(V)[ [\hbar] ], \star_\pi)$ is called the Moyal deformation quantization of the Poisson manifold $(V,\pi)$.

Proposition 13.5. (star product is associative and unital)

Given $(V,\pi)$ as above, then the star product $(-) \star_\pi (-)$ from def. 13.2
is associative and unital with unit the constant function $1 \in C^\infty(V) \hookrightarrow C^\infty(V)[ [ \hbar ] ]$.

Hence the vector space $C^\infty(V)$ equipped with the star product $\pi$ is a unital associative algebra.

Proof. Observe that the product rule of differentiation> says that

$$
\partial_i \circ prod
=
prod \circ ( \partial_i \otimes id \;+\; id \otimes \partial_i )
\,.
$$

Using this we compute as follows:

$$
\begin{aligned}
& (f \star_\pi g) \star_\pi h
\\
& =
prod \circ
\exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
\left(
\left( prod \circ \exp( \pi^{k l} \partial_k \otimes \partial_l ) \right) \otimes id
\right)
(f \otimes g \otimes g)
\\
& =
prod \circ
\exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
(prod \otimes id)
\circ
\left(
\exp( \pi^{k l} \partial_k \otimes \partial_l ) \otimes id
\right)
(f \otimes g \otimes g)
\\
& =
prod \circ
(prod \otimes id) \circ
\exp( \pi^{i j} ( \partial_i \otimes id \otimes \partial_j +id \otimes \partial_i \otimes \partial_j )
\circ
\exp( \pi^{k l} \partial_k \otimes \partial_l ) \otimes id
(f \otimes g \otimes g)
\\
& =
prod \circ
(prod \otimes id) \circ
\exp( \pi^{i j} \partial_i \otimes id \otimes \partial_j )
\circ
\exp( \pi^{i j} id \otimes \partial_i \otimes \partial_j )
\circ
\exp( \pi^{k l} \partial_k \otimes \partial_l \otimes id )
(f \otimes g \otimes g)
\\
& =
prod_3
\circ
\exp( \pi^{i j} ( \partial_i \otimes \partial_j \otimes id + \partial_i \otimes id \otimes \partial_j + id \otimes \partial_i \otimes \partial_j) )
\end{aligned}
$$

In the last line we used that the ordinary pointwise product of functions is associative, and wrote $prod_3 \colon C^\infty(V) \otimes C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ for the unique pointwise product of three functions.

The last expression above is manifestly independent of the choice of order of the arguments in the triple star product, and hence it is clear that an analogous computation yields

$$
\cdots = f \star_\pi (g \star_\pi h)
\,.
$$

Proposition 13.6. (shift by symmetric contribution is an isomorphism of star products)

Let $V$ be a vector space, $\pi \in V \otimes V$ a rank-2 tensor and $\alpha \in Sym(V \otimes V)$ a symmetric rank-2 tensor.

Then the linear map

$$
\array{
C^\infty(V)
&\overset{\exp\left(\tfrac{1}{2}\alpha \right)}{\longrightarrow}&
C^\infty(V)
\\
f
&\mapsto&
\exp\left( \tfrac{1}{2}\hbar \alpha^{i j} \partial_i \partial_j \right) f
}
$$

constitutes an isomorphism of star product algebras (prop. 13.5) of the form

$$
\exp\left(\hbar\tfrac{1}{2}\hbar\alpha \right)
\;\colon\;
(C^\infty(V)[ [\hbar] ], \star_{\pi})
\overset{\simeq}{\longrightarrow}
(C^\infty(V))[ [\hbar] ], \star_{\pi + \alpha})
\,,
$$

hence identifying the star product induced from $\pi$ with that induced from $\pi + \alpha$.

In particular every star product algebra $(C^\infty(V)[ [\hbar] ],\star_\pi)$ is isomorphic to a Moyal star product algebra $\star_{\tfrac{1}{2}\pi}$ (example 13.4) with $\tfrac{1}{2}\pi_{skew}^{i j} = \tfrac{1}{2}(\pi^{i j} – \pi^{j i})$ the skew-symmetric part of $\pi$, this isomorphism being exhibited by the symmetric part $2\alpha^{i j} = \tfrac{1}{2}(\pi^{i j} + \pi^{j i})$.

Proof. We need to show that

$$
\array{
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
&
\overset{
\exp\left( \tfrac{1}{2}\hbar \alpha \right)
\otimes
\exp\left( \tfrac{1}{2}\hbar \alpha \right)
}{\longrightarrow}&
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
\\
{}^{\llap{\star_{\pi}}}\downarrow
&&
\downarrow^{\rlap{\star_{\pi + \alpha}}}
\\
C^\infty(V)[ [\hbar] ]
&\underset{\exp\left( \tfrac{1}{2} \alpha \right) }{\longrightarrow}&
C^\infty(V)[ [\hbar] ]
}
$$

hence that

$$
prod \circ
\exp( \hbar(\pi + \alpha) )
\circ
\left(
\exp\left( \tfrac{1}{2}\alpha\right)
\otimes
\exp\left( \tfrac{1}{2}\alpha \right)
\right)
\;=\;
\exp\left( \tfrac{1}{2}\alpha \right)
\circ
prod \circ \exp( \pi )
\,.
$$

To this end, observe that the product rule of differentiation> applied twice in a row implies that

$$
\partial_i \partial_j \circ prod
\;=\;
prod \circ
\left(
(\partial_i \partial_j) \otimes id
+
id \otimes (\partial_i \partial_j)
+
\partial_i \otimes \partial_j
+
\partial_j \otimes \partial_i
\right)
\,.
$$

Using this we compute

$$
\begin{aligned}
&
\exp\left( \hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right)
\circ
prod
\circ
\exp( \hbar \pi^{i j} \partial_{i} \otimes \partial_j )
\\
& =
prod
\circ
\exp\left(
\hbar \tfrac{1}{2}\alpha^{i j}
\left(
(\partial_i \partial_j) \otimes id
+
id \otimes (\partial_i \partial_j)
+
\partial_i \otimes \partial_j
+
\partial_j \otimes \partial_i
\right)
\right)
\circ
\exp( \hbar \pi^{i j} \partial_{k} \otimes \partial_l )
\\
&
=
prod
\circ
\exp\left(
\hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
\right)
\circ
\exp\left(
\hbar \tfrac{1}{2} \alpha^{i j} (\partial_i \partial_j) \otimes id
\hbar \tfrac{1}{2} \alpha^{i j} id \otimes (\partial_i \partial_j)
\right)
\\
& =
prod
\circ
\exp\left(
\hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
\right)
\circ
\left(
\exp\left( \tfrac{1}{2} \hbar \alpha \right)
\otimes
\exp\left( \tfrac{1}{2} \hbar \alpha \right)
\right)
\end{aligned}
$$

Moyal star product as deformation quantization

Definition 13.7. (super-Poisson algebra)

A super-Poisson algebra is

1. a supercommutative algebra $\mathcal{A}$ (here: over the real numbers)
2. a bilinear function$$
   \{-,-\}
   \;\colon\;
   \mathcal{A} \otimes \mathcal{A}
   \longrightarrow
   A
   $$to be called the Poisson bracket

such that

1. $\{-,-\}$ is a super Lie bracket on $\mathcal{A}$, hence it
   1. is graded skew-symmetric;
   2. satisfies the super-Jacobi identity;
2. for each $A \in \mathcal{A}$ of homogeneous degree, the operation$$
   \left\{ A, -\right\}
   \;\colon\;
   \mathcal{A}
   \longrightarrow
   \mathcal{A}
   $$is a graded derivation on $\mathcal{A}$ of the same degree as $A$.

Definition 13.8. (formal deformation quantization)

Let $(\mathcal{A},\{-,-\})$ be a super-Poisson algebra (def. 13.7). Then a formal deformation quantization of $(A,\{-,-\})$ is

• the structure of an associative algebra on the formal power series algebra over $\mathcal{A}$ in a variable to be called $\hbar$, hence an associative and unital product$$
  \mathcal{A}[ [\hbar] ]
  \otimes
  \mathcal{A}[ [\hbar] ]
  \longrightarrow
  \mathcal{A}[ [\hbar] ]
  $$

such that for all $f,g \in \mathcal{A}$ of homogeneous degree we have

1. $f \star g \, mod \hbar = f g$
2. $f \star g – (-1)^{deg(f) deg(g)} g \star f \, \mod \hbar^2 = \hbar \{f,g\}$

meaning that

1. to zeroth order in $\hbar$ the star product coincides with the given commutative product on $\mathcal{A}$,
2. to first order in $\hbar$ the graded commutator of the star product coincides with the given Poisson bracket on $\mathcal{A}$.

Example 13.9. (Moyal star product is formal deformation quantization)

Let $(V,\pi)$ be a Poisson vector space, hence a vector space $V$, equipped with a skew-symmetric tensor $\pi \in V \wedge V$.

Then with $V$ regarded as a smooth manifold, the algebra of smooth functions $C^\infty(X)$ (def. 1.1) becomes a Poisson algebra (def. 13.7) with Poisson bracket given by

$$
\{f,g\}
\;:=\;
\pi^{i j} \frac{\partial f}{\partial x^i} \frac{\partial g}{\partial x^j}
\,.
$$

Moreover, for every symmetric tensor $\alpha \in V \otimes V$, the Moyal star product associated with $\tfrac{1}{2}\pi + \alpha$

$$
\array{
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
&\overset{\star_{\tfrac{1}{2}\pi + \alpha}}{\longrightarrow}&
C^\infty(V)[ [\hbar] ]
\\
(f,g)
&\mapsto&
((-)\cdot (-)) \circ \exp\left( (\tfrac{1}{2}\pi^{i j} + \alpha^{i j}) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j} \right\}
}
(f,g)
$$

is a formal deformation quantization (def. 13.8) of this Poisson algebra structure.

Moyal star product via geometric quantization of symplectic groupoid

Proposition 13.10. (integral representation of star product)

If $\pi$ skew-symmetric and invertible, in that there exists $\omega \in V^\ast \otimes V^\ast$ with $\pi^{i j}\omega_{j k} = \delta^i_k$, and if the functions $f,g$ admit Fourier analysis (are functions with rapidly decreasing partial derivatives), then their star product (def. 13.2) is equivalently given by the following integral expression:

$$
\begin{aligned}
\left(f \star_\pi g\right)(x)
&=
\frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} }
\int
e^{ \tfrac{1}{i \hbar} \omega((x – \tilde y),(x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\end{aligned}
$$

(Baker 58)

Proof. We compute as follows:

$$
\begin{aligned}
\left(f \star_\pi g\right)(x)
& :=
prod
\circ \exp\left(
\hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
\right)(f, g)
\\
& =
\frac{1}{(2 \pi)^{2n}}
\frac{1}{(2 \pi)^{2n}}
\int
\int
\underbrace{
e^{ i \hbar \pi(k,q) }
}
\underbrace{
e^{i k \cdot (x-y)}
f(y)
}
\underbrace{
e^{i q \cdot (x- \tilde y)}
g(\tilde y)
}
\,
d^{2 n} k
\,
d^{2 n} q
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{1}{(2 \pi)^{2n}}
\int
\delta\left( x – \tilde y + \hbar \pi \cdot k \right)
e^{i k \cdot (x-y)}
f(y)
g(\tilde y)
\,
d^{2 n} k
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{1}{(2 \pi)^{2n}}
\int
\delta\left( x – \tilde y + z \right)
e^{ \tfrac{i}{\hbar} \omega(z, (x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} z
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{(det(\pi)^{2n})}{(2 \pi \hbar)^{2n} }
\int
e^{\tfrac{1}{i \hbar}\omega((x – \tilde y),(x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\end{aligned}
$$

Here in the first step we expressed $f$ and $g$ both by their Fourier transform (inserting the Fourier expression of the delta distribution from this example) and used that under this transformation the partial derivative $\pi^{a b} \frac{\partial}{\partial\phi^a}{\frac{\partial}{\phi^b}}$ turns into the product with $i \pi^{i j} k_i k_j$ (this prop.). Then we identified again the Fourier-expansion of a delta distribution and finally we applied the change of integration variables $k = \tfrac{1}{\hbar}\omega \cdot z$ and then evaluated the delta distribution.

Next, we express this as the groupoid convolution product of polarized sections of the symplectic groupoid. To this end, we first need the following definition:

Definition 13.11. (symplectic groupoid of symplectic vector space)

Assume that $\pi$ is the inverse of a symplectic form $\omega$ on $\mathbb{R}^{2n}$. Then the Cartesian product

$$
\array{
&& \mathbb{R}^{2n} \times \mathbb{R}^{2n}
\\
& {}^{\llap{pr_1}}\swarrow && \searrow^{\rlap{pr_2}}
\\
\mathbb{R}^{2n} && && \mathbb{R}^{2n}
}
$$

inherits the symplectic structure

$$
\Omega
\;:=\;
\left(
pr_1^\ast \omega – pr_2^\ast \omega
\right)
$$

given by

$$
\begin{aligned}
\Omega
& =
\omega_{i j} d x^i \wedge d x^j – \omega_{i j} d y^i \wedge d y^j
\\
& =
\omega_{i j} ( d x^i – d y^i ) \wedge ( d x^j + d y^j )
\end{aligned}
\,.
$$

The pair groupoid on $\mathbb{R}^{2n}$ equipped with this symplectic form on its space of morphisms is a symplectic groupoid.

A choice of potential form $\Theta$ for $\Omega$, hence with $\Omega = d \Theta$, is given by

$$
\Theta
:=
-\omega_{i j} ( x^i + y^i ) d (x^j – y^j) )
$$

Choosing the real polarization spanned by $\partial_{x^i} – \partial_{y^i}$ a polarized section is function $F = F(x,y)$ such that

$$
\iota_{\partial_{x^j} – \partial_{y^j}}(d F – \tfrac{1}{i \hbar} \tfrac{1}{4} \Theta F) = 0
$$

hence

$$
\label{PolarizedSectionOnMorphismsOfSymplecticGroupoid}
F(x,y)
=
f\left( \tfrac{x + y}{2} \right) e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{x – y}{2} , \tfrac{x + y}{2} \right)}
\,.
$$ (213)

Proposition 13.12. (polarized symplectic groupoid convolution product of symplectic vector space is given by Moyal star product)

Given a symplectic vector space $(\mathbb{R}^{2n}, \omega)$, then the groupoid convolution product on polarized sections (213) on its symplectic groupoid (def. 13.11), given by convolution product followed by averaging (integration) over the polarization fiber, is given by the star product (def. 13.2) for the corresponding Poisson tensor $\pi := \omega^{-1}$, in that

$$
\begin{aligned}
\int \int
F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
& =
(f \star_\pi g)((x+y)/2)
\end{aligned}
\,.
$$

(Weinstein 91, p. 446, Garcia-Bondia & Varilly 94, section V, Hawkins 06, example6.2)

Proof. We compute as follows:

$$
\begin{aligned}
&
\int \int
F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
\\
& :=
\int \int f((x + t)/2) g( (t + y)/2 )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( x-t, x+t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-y, t + y) }
\, d^{2n} t
\, d^{2n} (x-y)
\\
& =
\int \int
f(t/2) g( (t – (x – y))/2 )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) + (x – y) – t, t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-(x+y), t – (x-y)) }
\, d^{2n} t
\, d^{2n} (x-y)
\\
& =
\int \int
f(t/2) g( \tilde t / 2)
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) – \tilde t, t ) – \tfrac{1}{i \hbar} \tfrac{1}{4} \omega((x+y)-t, \tilde t) }
\, d^{2n} t
\, d^{2n} \tilde t
\\
& =
\int \int
f(t) g( \tilde t )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) – 2 \tilde t, 2 t ) – \tfrac{1}{ii \hbar} \tfrac{1}{4} \omega((x+y)- 2 t, 2 \tilde t) }
\, d^{2n} t
\, d^{2n} \tilde t
\\
\\
& =
\int \int
f(t) g(\tilde t )
e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{1}{2}(x+y) – \tilde t, \tfrac{1}{2}(x + y) – t \right)}
\, d^{2n} t
\, d^{2n} \tilde t
\\
& =
(f \star_\omega g)((x+y)/2)
\end{aligned}
$$

The first line just unwinds the definition of polarized sections from def. 13.11, the following lines each implement a change of integration variables and finally, in the last line, we used prop. 13.10.

Example: Wick algebra of normal ordered products on Kähler vector space

Definition 13.13. (Kähler vector space)

An Kähler vector space is a real vector space $V$ equipped with a linear complex structure $J$ as well as two bilinear forms $\omega, g \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R}$ such that the following equivalent conditions hold:

1. $\omega(J v, J w) = \omega(v,w)$ and $g(v,w) = \omega(v,J w)$;
2. with $V$ regarded as a smooth manifold and with $\omega, g$ regarded as constant tensors, then $(V, \omega, g)$ is an almost Kähler manifold.

Example 13.14. (standard Kähler vector spaces)

Let $V := \mathbb{R}^2$ equipped with the complex structure $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical linear basis $(e_i)$ of $\mathbb{R}^2$, this is

$$
J = (J^i{}_j)
:=
\left(
\array{
0 & -1
\\
1 & 0
}
\right)
\,.
$$

Moreover let

$$
\omega = (\omega_{i j}) := \left( \array{0 & 1 \\ -1 & 0} \right)
$$

and

$$
g = (g_{i j}) := \left( \array{ 1 & 0 \\ 0 & 1} \right)
\,.
$$

Then $(V, J, \omega, g)$ is a Kähler vector space (def. 13.13).

The corresponding Kähler manifold is $\mathbb{R}^2$ regarded as a smooth manifold in the standard way and equipped with the bilinear forms $J, \omega g$ extended as constant rank-2 tensors over this manifold.

If we write

$$
x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for the standard coordinate functions on $\mathbb{R}^2$ with

$$
z := x + i y \;:=\; \mathbb{R}^2 \to \mathbb{C}
$$

and

$$
\overline{z} := x – i y \;:=\; \mathbb{R}^2 \to \mathbb{C}
$$

for the corresponding complex coordinates, then this translates to

$$
\omega
\in
\Omega^2(\mathbb{R}^2)
$$

being the differential 2-form given by

$$
\begin{aligned}
\omega
& =
d x \wedge d y
\\
& =
\tfrac{1}{2i} d z \wedge d \overline{z}
\end{aligned}
$$

and with Riemannian metric tensor given by

$$
g = d x \otimes d x + d y \otimes d y
\,.
$$

The Hermitian form is given by

$$
\begin{aligned}
h
& =
g – i \omega
\\
& =
d z \otimes d \overline{z}
\end{aligned}
$$

(for more see at Kähler vector space this example).

Definition 13.15. (Wick algebra of a Kähler vector space)

Let $(\mathbb{R}^{2n},\sigma, g)$ be a Kähler vector space (def. 13.13). Then its Wick algebra is the formal power series vector space $\mathbb{C}[ [ \mathbb{R}^{2n} ] ] [ [ \hbar ] ]$ equipped with the star product (def. 13.2) which is given by the bilinear form

$$
\label{InStarProductTensorInvertingHermitianForm}
\pi := \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}
\,,
$$ (214)

hence:

$$
\begin{aligned}
A_1 \star_\pi A_2
& :=
((-)\cdot (-)) \circ \exp \left( \hbar\underset{k_1, k_2 = 1}{\overset{2 n}{\sum}}\pi^{a b} \partial_a \otimes \partial_b \right) (A_1 \otimes A_2)
\\
& =
A_1 \cdot A_2 + \hbar \underset{k_1, k_2 = 1}{\overset{2n}{\sum}}\pi^{k_1 k_2}(\partial_{k_1} A_1) \cdot (\partial_{k_2} A_2)
+ \cdots
\end{aligned}
$$

(e.g. Collini 16, def. 1)

Proposition 13.16. (star product algebra of Kähler vector space is star-algebra)

Under complex conjugation the star product $\star_\pi$ of a Kähler vector space structure (def. 13.15) is a star algebra in that for all $A_1, A_2 \in \mathbb{C}[ [\mathbb{R}^{2n}] ][ [\hbar] ]$ we have

$$
\left( A_1 \star_\pi A_2 \right)^\ast
\;=\;
A_2^\ast \star_\pi A_1^\ast
$$

Proof. This follows directly from that fact that in $\pi = \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}$ the imaginary part coincides with the skew-symmetric part, so that

$$
\begin{aligned}
(\pi^\ast)^{a b}
& =
-\tfrac{i}{2} (\omega^{-1})^{a b}
+ \tfrac{1}{2} (g^{-1})^{a b}
\\
& =
\tfrac{i}{2} (\omega^{-1})^{b a}
+
\tfrac{1}{2} (g^{-1})^{b a}
\\
& = \pi^{b a}
\,.
\end{aligned}
$$

Example 13.17. (Wick algebra of a single-mode)

Let $V := \mathbb{R}^2 \simeq Span(\{x,y\})$ be the standard Kähler vector space according to example 13.14, with canonical coordinates denoted $x$ and $y$. We discuss its Wick algebra according to def. 13.15 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

$$
z \;:=\; x + i y
\phantom{AAA}
\text{and}
\phantom{AAA}
\overline{z} := x – i y
$$

which we use in the form

$$
a^\ast \;:=\; \tfrac{1}{\sqrt{2}}(x + i y)
\phantom{AAA}
\text{and}
\phantom{AAA}
a \;:=\; \tfrac{1}{\sqrt{2}}(x – i y)
$$

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

Now

$$
\omega^{-1}
=
\frac{\partial}{\partial y} \otimes \frac{\partial}{\partial x}
–
\frac{\partial}{\partial x} \otimes \frac{\partial}{\partial y}
$$

$$
g^{-1}
=
\frac{\partial}{\partial x}
\otimes
\frac{\partial}{\partial x}
+
\frac{\partial}{\partial y}
\otimes
\frac{\partial}{\partial y}
$$

so that with

$$
\frac{\partial}{\partial z}
=
\tfrac{1}{2}
\left(
\frac{\partial}{\partial x}
-i
\frac{\partial}{\partial y}
\right)
\phantom{AAAA}
\frac{\partial}{\partial \overline{z}}
=
\frac{1}{2}
\left(
\frac{\partial}{\partial x}
+
i
\frac{\partial}{\partial y}
\right)
$$

we get

$$
\begin{aligned}
\tfrac{i \hbar}{2}\omega^{-1} + \tfrac{\hbar}{2} g^{-1}
& =
2 \hbar
\frac{\partial}{\partial \overline{z}}
\otimes
\frac{\partial}{\partial z}
\\
& =
\hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}
\end{aligned}
$$

Using this, we find the star product

$$
A \star_\pi B
\;=\;
prod \circ \exp\left( \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast} \right)
$$

to be as follows (where we write $(-)\cdot (-)$ for the plain commutative product in the formal power series algebra):

$$
\begin{aligned}
a \star_\pi a & = a \cdot a
\\
a^\ast \star_\pi a^\ast & = a^\ast \cdot a^\ast
\\
a^\ast \star_\pi a & = a^\ast \cdot a
\\
a \star_\pi a^\ast
& =
a \cdot a^\ast + \hbar
\end{aligned}
$$

and so forth, for instance

$$
\array{
(a \cdot a ) \star_\pi (a^\ast \cdot a^\ast)
& =
a^\ast \cdot a^\ast \cdot a \cdot a
+
4 \hbar a^\ast \cdot a
+
\hbar^2
}
$$

If we instead indicate the commutative pointwise product by colons and the star product by the plain juxtaposition

$$
:f g: \;:=\; f \cdot g
\phantom{AAAA}
f g \;:=\; f \star_\pi
$$

then this reads

$$
\array{
:a a: \, :a^\ast a^\ast:
& =
: a^\ast a^\ast a a :
+
4 \hbar \, : a^\ast a :
+
\hbar^2
}
$$

This is the way the Wick algebra with its operator product $\star_\pi$ and normal-ordered product $:-:$ is traditionally presented.

star products on regular polynomial observables in field theory

Proposition 13.18. (star products on regular polynomial observables induced from propagators)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory with field bundle $E \overset{fb}{\to} \Sigma$, and let $\Delta \in \Gamma’_\Sigma((E \boxtimes E)^\ast)$ be a distribution of two variables on field histories.

On the off-shell regular polynomial observables with a formal paramater $\hbar$ adjoined consider the bilinear map

$$
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg} [ [ \hbar ] ]
\overset{\star_{\Delta}}{\longrightarrow}
PolyObs(E)_{reg}[ [\hbar] ]
$$

given as in def. 13.2, but with partial derivatives replaced by functional derivatives

$$
A_1 \star_{\Delta} A_2
\;:=\;
((-)\cdot(-))
\circ
\exp\left(
\int_\Sigma \Delta^{a b}(x,y)
\frac{\delta}{\delta \Phi^a(x)}
\otimes
\frac{\delta}{\delta \Phi^b(y)}
\right)
(A_1 \otimes A_2)
$$

As in prop. 13.5 this defines an unital and associative algebra structure.

If the Euler-Lagrange equations of motion $P\Phi ) = 0$ induced by the Lagrangian density $\mathbf{L}$ are Green hyperbolic differential equations and if $\Delta$ is a homogeneous propagator for these differential equations in that $P \Delta = 0$, then this star product algebra descends to the on-shell regular polynomial observables

$$
PolyObs(E,\mathbf{L})_{reg}[ [\hbar] ]
\otimes
PolyObs(E, \mathbf{L})_{reg} [ [ \hbar ] ]
\overset{\star_{\Delta}}{\longrightarrow}
PolyObs(E, \mathbf{L})_{reg}[ [\hbar] ]
\,.
$$

Proof. The proof of prop. 13.5 goes through verbatim in the present case, as long as all products of distributions that appear when the propagator is multiplied with the coefficients of the polynomial observables are well-defined, in that Hörmander’s criterion (prop. 9.34) on the wave front sets (def. 9.28) of the propagator and of these coefficients is met. But the definition the coefficients of regular polynomial observables are non-singular distributions, whose wave front set is necessarily empty (example 9.30), so that their product of distributions is always well-defined.

Corollary 13.19. (quantization of regular polynomial observables of gauge fixed free Lagrangian field theory)

Consider a gauge fixed (def. 12.1) free Lagrangian field theory (def. 5.25) with BV-BRST-extended field bundle (remark 12.7)

$$
E_{\text{BV-BRST}}
\;:=\;
T^\ast_{\Sigma,inf}[-1]
\left(
E
\times_\Sigma
\mathcal{G}[1]
\times_\Sigma
A
\times_\Sigma
A[-1]
\right)
$$

and with causal propagator (92)

$$
\Delta
\;\in\;
\Gamma’_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
\,.
$$

Then the star product $\star_\Delta$ (def. 13.2) is well-defined on off-shell (as well as on-shell) regular polynomial observables (def. 7.13)

$$
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\overset{\star_{\tfrac{i}{2}\Delta}}{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

and the resulting non-commutative algebra structure

$$
\left(
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\,,\,
\star_\Delta
\right)
$$

is a formal deformation quantization (def. 13.8) of the Peierls-Poisson bracket on the covariant phase space (theorem 8.7), restricted to regular polynomial observables.

(Dito 90, Dütsch-Fredenhagen 00 Dütsch-Fredenhagen 01, Hirshfeld-Henselder 02)

Proof. As in prop. 13.18, the vanishing of the wave front set of the coefficients of the regular polynomial observables implies that all arguments go through as for star products on polynomial algebras on finite-dimensional vector spaces. By theorem 8.7 the causal propagator is the integral kernel of the Peierls-Poisson bracket so that the tensor $\pi$ from the definition of the Moyal star product (example 13.4) now is

$$
\pi = \Delta
\,.
$$

With this the statement follows by example 13.9.

Remark 13.20. (extending quantization beyond regular polynomial observables)

While cor. 13.19 provides a quantization of the regular polynomial observables of any gauge fixed free Lagrangian field theory, the regular polynomial observables are too small a subspace of that of all polynomial observables:

By example 7.42 the only local observables (def. 7.39) contained among the regular polynomial observables are the linear observables (def. 7.3). But in general, it is necessary to consider also non-linear polynomial local observables. Notably the interaction action functionals $S_{int}$ induced from interaction Lagrangian densities $\mathbf{L}_{int}$ (example 7.34) are non-linear polynomial observables.

For example:

• For quantum electrodynamics on Minkowski spacetime (example 5.11) the adiabatically switched action functional (example 7.34) which is the transgression of the electron-photon interaction is a cubic local observable$$
  S_{int}
  \;=\;
  i
  \underset{\Sigma}{\int}
  g_{sw}(x)
  \,
  (\gamma^\mu)^{\alpha}{}_\beta
  \,
  \overline{\mathbf{\Psi}}_\alpha(x)
  \cdot \mathbf{\Psi}^\beta(x)
  \cdot \mathbf{A}^a(x)
  \, dvol_\Sigma(x)
  $$
• For scalar field phi^n theory (example 5.5) the adiabatically switched action functional (example 7.34) which is the transgression of the phi^n interaction$$
  S_{int}
  \;=\;
  \underset{\Sigma}{\int}
  g_{sw}
  \,
  \underset{
  n \, \text{factors}
  }{
  \underbrace{
  \mathbf{\Phi}(x)
  \cdot
  \mathbf{\Phi}(x)
  \cdots
  \mathbf{\Phi}(x)
  \cdot
  \mathbf{\Phi}(x)
  }
  }
  \,
  dvol_\Sigma(x)
  $$is a local observable of order $n$.

Therefore one needs to extend the formal deformation quantization provided by corollary 13.19 to a larger subspace of polynomial observables that includes at least the local observables.

But prop. 13.6 characterizes the freedom in choosing a formal deformation quantization: We may shift the causal propagator by a symmetric contribution. In view of prop. 13.18 and in view of Hörmander’s criterion for the product of distributions (prop. 9.34) to be well defined, we are looking for symmetric integral kernels $H$ such that the sum

$$
\label{ShiftingCausalPropagatorBySymmetricContribution}
\Delta_H = \tfrac{i}{2}\Delta + H
$$ (215)

has a smaller wave front set (def. 9.28) than $\tfrac{i}{2}\Delta$ itself has. The smaller $WF(\tfrac{i}{2}\Delta + H)$, the larger the subspace of polynomial observables on which the corresponding formal deformation quantization exists.

Now by prop. 9.60 the Wightman propagator $\Delta_H$ is of the form (215) and by prop. 9.69 its wave front set is only “half” that of the causal propagator. It turns out that $\Delta_H$ does yield a formal deformation quantization of a subspace of polynomial observables that includes all local observables: this is the Wick algebra on microcausal polynomial observables. We discuss this in detail in the chapter Free quantum fields.

With such a formal deformation quantization of the local observables free field theory in hand, we may then finally obtain also a formal deformation quantization of interacting Lagrangian field theories by perturbation theory. This we discuss in the chapters Scattering and Quantum observables.

This concludes our discussion of some basic concepts of quantization. In the next chapter we apply this to discuss the algebra of quantum observables of free Lagrangian field theories. Further below in the chapter Quantum observables we then discuss also the quantization of the interacting Lagrangian field theories, perturbatively.