Learn Observables in Mathematical Quantum Field Theory

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

The previous chapter is 6. Symmetries.

The next chapter is 8. Phase space.

7. Observables

In this chapter we discuss these topics:

• General observables
• Polynomial off-shell observables and Distributions
• Polynomial on-shell observables and Distributional solutions to PDEs
• Local observables and Transgression
• Infinitesimal observables
• States

Given a Lagrangian field theory (def. 5.4), then a general observable quantity or just observable for short (def. 7.1 below), is a smooth function

$$
A \;\colon\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C}
$$

on the on-shell space of field histories (example 3.12, example 3.46) hence a smooth “functional” of field histories. We think of this as assigning to each physically realizable field history $\Phi$ the value $A(\Phi)$ of the given quantity as exhibited by that field history. For instance concepts like “average field strength in the compact spacetime region $\mathcal{O}$” should be observables. In particular the field amplitude at spacetime point $x$ should be an observable, the “field observable” denoted $\mathbf{\Phi}^a(x)$.

Beware that in much of the literature on field theory, these point-evaluation field observables $\mathbf{\Phi}^a(x)$ (example below 7.2) are eventually referred to as “fields” themselves, blurring the distinction between

1. types of fields/field bundles $E$,
2. field histories/sections $\Phi$,
3. functions on the space of field histories $\mathbf{\Phi}^a(x)$.

In particular, the process of quantization (discussed in Quantization below) affects the third of these concepts only, in that it deforms the algebra structure on observables to a non-commutative algebra of quantum observables. For this reason, the field observables $\mathbf{\Phi}^a(x)$ is often referred to as quantum fields. But to understand the conceptual nature of quantum field theory it is important that the $mathbf{Phi}^a(x)$ are really the observables or quantum observables on the space of field histories.

fields

aspect	term	type	description	def.
field component	$\phi^a$, $\phi^a_{,\mu}$	$J^\infty_\Sigma(E) \to \mathbb{R}$	coordinate function on jet bundle of field bundle	def. 3.1, def. 4.1
field history	$\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$	$\Sigma \to J^\infty_\Sigma(E)$	jet prolongation of section of field bundle	def. 3.1, def. 4.2
field observable	$\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x),$	$\Gamma_{\Sigma}(E) \to \mathbb{R}$	derivatives of delta-functional on space of sections	def. 7.1, example 7.2
averaging of field observable	$\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$	$\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$	observable-valued distribution	def. 7.30
algebra of quantum observables	$\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$	$\mathbb{C}Alg$	non-commutative algebra structure on field observables	def. link , def. link

There are various further conditions on observables which we will eventually consider, forming subspaces of gauge-invariant observables (def. 11.2), local observables (def. 7.39 below), Hamiltonian local observables (def. 8.12 below) and microcausal observables (def. link ). While in the end, it is only these special kinds of observables that matter, it is useful to first consider the unconstrained concept and then consecutively characterize smaller subspaces of well-behaved observables. In fact, it is useful to consider yet more generally the observables on the full space of field histories (not just the on-shell subspace), called the off-shell observables.

In the case that the field bundle is a vector bundle (example 3.4), the off-shell space of field histories is canonically a vector space and hence it makes sense to consider linear off-shell observables, i.e. those observables $A$ with $A(c \Phi) = c A(\Phi)$ and $A(\Phi_1 + \Phi_2) = A(\Phi_1) + A(\Phi_2)$. It turns out that these are precisely the compactly supported distributions in the sense of Laurent Schwartz (prop. 7.5 below). This fact makes powerful tools from functional analysis and microlocal analysis available for the analysis of field theory (discussed below).

More generally there are the multilinear off-shell observables, and these are analogously given by distributions of several variables (def. 7.13 below). In fully perturbative quantum field theory one considers only the infinitesimal neighbourhood (example 3.30) of single on-shell field history and in this case, all observables are in fact given by such multilinear observables (def. 7.43 below).

For a free field theory (def. 5.25) whose Euler-Lagrange equations of motion are given by a linear differential operator which behaves well in that it is “Green hyperbolic” (def. 7.19 below) it follows that the actual on-shell linear observables are equivalently those off-shell observables which are spatially compactly supported distributional solutions to the formally adjoint equation of motion (prop. 7.28 below), and this equivalence is exhibited by composition with the causal Green function (def. 7.18 below):

This is theorem 7.29 below, which is pivotal for passing from classical field theory to quantum field theory:

$$
\left\{
\,\,
\array{
\text{polynomial}
\\
\text{on-shell}
\\
\text{observables}
}
\,\,
\right\}
\underset{\simeq}{\overset{\text{restriction}}{\longleftarrow}}
\left\{
\array{
\text{polynomial}
\\
\text{off-shell}
\\
\text{observables}
\\
\text{modulo equations of motion}
}
\right\}
\underset{\simeq}{\overset{\text{causal propagator}}{\longleftarrow}}
\left\{
\array{
\text{spatially compactly supported}
\\
\text{distributions in several variables}
\\
\text{which are distributional solutions}
\\
\text{to the adjoint equations of motion}
}
\right\}
$$

This fact makes, in addition, the distributional analysis of linear differential equations available for the analysis of free field theory, notably the theory of propagators, such as Feynman propagators (def. 9.61 below), which we turn to in Propagators below.

The functional analysis and microlocal analysis (below) of linear observables re-expressed in distribution theory via theorem 7.29 solve the issues that the original formulation of perturbative quantum field theory by Schwinger-Tomonaga-Feynman-Dyson in the 1940s was notorious for suffering from (Feynman 85): The normal ordered product of quantum observables in a Wick algebra of observables follows from Hörmander’s criterion for the product of distributions to be well-defined (this we discuss in Free quantum fields below) and the renormalization freedom in the construction of the S-matrix is governed by the mechanism of extensions of distributions (this we discuss in Renormalization below).

Among the polynomial on-shell observables characterized this way, the focus is furthermore on the local observables:

In local field theory, the idea is that both the equations of motion as well as the observations are fully determined by their restriction to infinitesimal neighbourhoods of spacetime points (events). For the equations of motion, this means that they are partial differential equations as we have seen above. For the observables, it should mean that they must be averages over regions of spacetime of functions of the value of the field histories and their derivatives at any point of spacetime. Now a “smooth function of the value of the field histories and their derivatives at any point” is precisely a smooth function on the jet bundle of the field bundle (example 4.1) pulled back via jet prolongation (def. 4.2). If this is to be averaged over spacetime it needs to be the coefficient of a horizontal $p+1$-form (prop. 4.11).

In mathematical terminology these desiderata say that the local observables in a local field theory should be precisely the “transgressions” (def. 7.32 below) of horizontal variational $p+1$-forms (with compact spacetime support, def. 7.31 below) to the space of field histories (example 3.12). This is def. 7.39 below.

A key example of a local observable in Lagrangian field theory (def. 5.1) is the action functional (example 7.34 below). This is the transgression of the Lagrangian density itself, or rather of its product with an “adiabatic switching function” that localizes its support in a compact spacetime region. In typical cases, the physical quantity whose observation is represented by the action functional is the difference of the kinetic energy-momentum minus the potential energy of a field history averaged over the given region of spacetime.

The equations of motion of a Lagrangian field theory say that those field histories are physically realized which are critical points of this action functional observable. This is the principle of extreme action (prop. 7.38 below).

In summary, we find the following system of types of observables:

types of observables in perturbative quantum field theory:

$$
\array{
&&
\text{local}
\\
&&
& \searrow
\\
\text{field}
&\longrightarrow&
\text{linear}
&\longrightarrow&
\text{microcausal}
&\longrightarrow&
\text{polynomial}
&\longrightarrow&
\text{general}
\\
&&
& \nearrow
\\
&&
\text{regular}
}
$$

In the chapter Free quantum fields we will see that the space of all polynomial observables is too large to admit quantization, while the space of regular local observables is too small to contain the usual interaction terms for perturbative quantum field theory (example 7.42) below. The space of microcausal polynomial observables (def. 14.2 below) is in between these two extremes and evades both of these obstacles.

Given the concept of observables, it remains to formalize what it means for the physical system to be in some definite state so that the observable quantities take some definite value, reflecting the properties of that state.

Whatever formalization for states of a field theory one considers, at the very least the space of states $States$ should come with a pairing linear map

$$
\array{
Obs \otimes States & \longrightarrow& \mathcal{C}
\\
\left( A , \langle – \rangle \right) &\mapsto& \langle A \rangle
}
$$

which reads in an observable quantity $A$ and a state, to be denoted $\langle – \rangle$, and produces the complex number $\langle A \rangle$ which is the “value of the observable quantity $A$ in the case that the physical system is in the state $\langle -\rangle$”.

One might imagine that it is fundamentally possible to pinpoint the exact field history that the physical system is found in. From this perspective, fixing a state should simply mean to pick such a field history, namely an element $\Phi \in \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ in the on-shell space of field histories. If we write $\langle -\rangle_{\Phi}$ for this state, its pairing map with the observables would simply be an evaluation of the observable, being a function on the field history space, on that particular element in this space:

$$
\langle A \rangle_{\Phi} := A(\Phi)
\,.
$$

However, in the practice of experiment a field history can never be known precisely, without remaining uncertainty. Moreover, quantum physics (to which we finally come below), suggests that this is true not just in practice, but even in principle. Therefore we should allow states to be a kind of probability distributions on the space of field histories, and regard the pairing $\langle A \rangle$ of a state $\langle – \rangle$ with an observable $A$ as a kind of expected value of the function $A$ averaged with respect to this probability distribution. Specifically, if the observable quantity $A$ is (a smooth approximation to) a characteristic function of a subset $S \subset \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ of the space of field histories, then its value in a given state should be the probability to find the physical system in that subset of field histories.

But, moreover, the superposition principle of quantum physics says that the actually observable observables are only those of the form $A^\ast A$ (for $A^\ast$ the image under the star-operation on the star algebra of observables.

This finally leads to the definition of states in def. 7.47 below.

General observables

Definition 7.1. (observables)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) with $\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0}$ its on-shell space of field histories (def. 5.24).

Then the space of observables is the super formal smooth set (def. 3.40) which is the mapping space

$$
Obs(E,\mathbf{L})
\;:=\;
\left[
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
\,,\,
\mathbb{C}
\right]
$$

from the on-shell space of field histories to the complex numbers.

Similarly, there is the space of off-shell observables

$$
Obs(E)
\;:=\;
\left[
\Gamma_\Sigma(E)
\,,\,
\mathbb{C}
\right]
\,.
$$

Every off-shell observables induce an on-shell observable by restriction, this yields a smooth function

$$
\label{OffShellObservablesRestrictToOnShellObservables}
Obs(E)
\overset{(-)_{\delta_{EL}\mathbf{L} = 0}}{\longrightarrow}
Obs(E,\mathbf{L})
$$ (84)

similarly, we may consider the observables on the sup-spaces of field histories with restricted causal support according to def. 2.36. We write

$$
Obs(E_{scp})
\;:=\;
\left[ \Gamma_{\Sigma,scp}(E), \mathbb{C} \right]
$$

and

$$
\label{SpaceOfObservablesOnFieldHistoriesOfSpatiallyCompactSupport}
Obs(E_{scp}, \mathbf{L})
\;:=\;
\left[ \Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}, \mathbb{C} \right]
$$ (85)

for the spaces of (off-shell) observables on field histories with spatially compact support (def. 2.36).

Observables form a commutative algebra under the pointwise product:

$$
\label{ObservablesPointwiseProduct}
\array{
Obs(E) \otimes Obs(E) &\overset{(-)\cdot (-)}{\longrightarrow}& Obs(E)
\\
(A_1, A_2) &\mapsto& A_1 \cdot A_2
}
$$ (86)

given by

$$
(A_1 \cdot A_2)(\Phi_{(-)}) := A_1(\Phi_{(-)}) \cdot A_2(\Phi_{(-)})
\,,
$$

where on the right we have the product in $\mathbb{C}$.

(Suitable subspaces of observables will in addition carry other products, notably non-commutative algebra structures, this is the topic of the chapters Free quantum fields and Quantum observables below.)

Observables on bosonic fields

In the case that $E$ is a purely bosonic field bundle in smooth manifolds so that $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$ is a diffeological space (def. 3.12, def. 5.24) this means that a single observable $A \in Obs_{E,\mathbf{L}}$ is equivalently a smooth function (def. 3.10)

$$
A
\;\colon\;
\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0}
\longrightarrow
\mathbb{C}
\,.
$$

Explicitly, by def. 3.14 (and similarly by def. 3.40) this means that $A$ is for each Cartesian space $U$ (generally: super Cartesian space, def. 3.37) a natural function of plots

$$
A_U
\;\colon\;
\left\{
\array{
U \times \Sigma && \overset{\Phi_{(-)}}{\longrightarrow} && E
\\
& {}_{\llap{pr_2}}\searrow && \swarrow_{\rlap{fb}}
\\
&& \Sigma
}
\right\}_{\delta_{EL}\mathbf{L} = 0}
\;\overset{}{\longrightarrow}\;
\left\{
U \to \mathbb{C}
\right\}
\,.
$$

Observables on fermionic fields

In the case that $E$ has purely fermionic fibers (def. 3.45), such as for the Dirac field (example 3.50) with $E = \Sigma\times S_{odd}$ then the only points in $Obs_{E}$, namely morphisms $\mathbb{R}^0 \to Obs_E$ are observables depending on an even power of field histories; while general observables appear as possibly odd-parameterized families

$$
(\theta \mapsto \theta \Psi)
\;\colon\;
\mathbb{R}^{0\vert 1}
\longrightarrow
Obs_{E,\mathbf{L}}
$$

whose component $\mathbf{\Psi}$ is a section of the even-graded field bundle, regarded in odd degree, via prop. 3.51. See example 7.14 below.

The most basic kind of observables are the following:

Example 7.2. (point evaluation observables — field observables)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) whose field bundle (def. 3.1) over some spacetime $\Sigma$ happens to be a trivial vector bundle in even degree (i.e. bosonic) with field fiber coordinates $(\phi^a)$ (example 3.4). With respect to these coordinates a field history, hence a section of the field bundle

$$
\Phi \;\in \; \Gamma_\Sigma(E)
$$

has components $(\Phi^a)$ which are smooth functions on spacetime.

Then for every index $a$ and every point $x \in \Sigma$ in spacetime (every event) there is an observable (def. 7.1) denoted $\mathbf{\Phi}^a(x)$ which is given by

$$
\mathbf{\Phi}^a(x)
\;\colon\;
\Phi_{(-)}
\mapsto
\Phi_{(-)}^a(x)
\,,
$$

hence which on a test space $U$ (a Cartesian space or more generally super Cartesian space, def. 3.37) sends a $U$-parameterized collection of fields

$$
\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$

to their $U$-parameterized collection of values at $x$ of their $a$-th component.

Notice how the various aspects of the concept of “field” are involved here, all closely related but crucially different:

$$
\array{
\mathbf{\Phi}^a(x)
&\colon&
\Phi
&\overset{\phantom{AA}}{\mapsto}&
\Phi^a(x)
&=&
\phi^a & \circ \Phi(x)
\\
\array{
\text{field}
\\
\text{observable}
}
&&
\array{
\text{field}
\\
\text{history}
}
&&
\array{
\text{field}
\\
\text{value}
}
&&
\array{
\text{field}
\\
\text{component}
}
}
$$

Polynomial off-shell Observables and Distributions

We consider here linear observables (def. 7.3 below) and more generally quadratic observables (def. 7.12) and generally polynomial observables (def. 7.13 below) for free field theories and discuss how these are equivalently given by integration against generalized functions called distributions (prop. 7.5 and prop. 7.6 below).

This is the basis for the discussion of quantum observables for free field theories further below.

Definition 7.3. (linear off-shell observables)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) whose field bundle $E$ (def. 3.1) is a super vector bundle (as in example 3.4 and as opposed to more general non-linear fiber bundles).

This means that the off-shell space of field histories $\Gamma_\Sigma(E)$ (example 3.46) inherits the structure of a super vector space by spacetime-pointwise (i.e. event-wise) scaling and addition of field histories.

Then an off-shell observable (def. 7.1)

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

is a linear observable if it is a linear function with respect to this vector space structure, hence if

$$
A\left( c \Phi_{(-)}) = c A(\Phi_{(-)} \right)
\phantom{AAAA}
\text{and}
\phantom{AAAA}
A\left(\Phi_{(-)} + \Phi’_{(-)} \right)
=
A\left( \Phi_{(-)}) + A(\Phi’_{(-)} \right)
$$

for all plots of field histories $\Phi_{(-)}, \Phi’_{(-)}$.

If moreover $(E,\mathbf{L})$ is a free field theory (def. 5.25) then the on-shell space of field histories inherits this linear structure and we may similarly speak of linear on-shell observables.

We write

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

for the subspace of linear observables inside all observables (def. 7.1) and similarly

$$
LinObs(E) \hookrightarrow Obs(E)
$$

for the linear off-shell observables inside all off-shell observables, and similarly for the subspaces of linear observables on field histories of spatially compact supprt (85):

$$
\label{LinearObservablesOnSpatiallyCompactlySupportedOnShellFieldHistories}
LinObs(E_{scp}, \mathbf{L})
\hookrightarrow
Obs(E_{scp}, \mathbf{L})
$$ (87)

and

$$
LinObs(E_{scp})
\hookrightarrow
Obs(E_{scp})
\,.
$$

Example 7.4. (point evaluation observables are linear)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) over Minkowski spacetime (def. 2.17), whose field bundle $E$ (def. 3.1) is the trivial vector bundle with field coordinates $(\phi^a)$ (example 3.4).

Then for each field component index $a$ and point $x \in \Sigma$ of spacetime (each event) the point evaluation observable (example 7.2)

$$
\array{
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
&\overset{\mathbf{\Phi}^a(x)}{\longrightarrow}&
\mathbb{C}
\\
\phi &\mapsto& \phi^a(x)
}
$$

is a linear observable according to def. 7.3. The distribution that it corresponds to under prop. 7.5
is the Dirac delta-distribution at the point $x$ combined with the Kronecker delta on the index $a$: In the generalized function-notation of remark 7.7 this reads:

$$
\Phi^a(x) \;\colon\; \Phi \mapsto \int_\Sigma \Phi^b(y) \delta_b^a \delta(x,y) \, dvol_\Sigma(y)
\,.
$$

Proposition 7.5. (linear off-shell observables of scalar field are the compactly supported distributions)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) over Minkowski spacetime (def. 2.17), whose field bundle $E$ (def. 3.1) is the trivial real line bundle (as for the real scalar field, example 3.5). This means that the off-shell space of field histories $\Gamma_\Sigma(E) \simeq C^\infty(\Sigma)$ (19) is the real vector space of smooth functions on Minkowski spacetime and that every linear observable $A$ (def. 7.3) gives a linear function

$$
A_\ast \;\colon\; C^\infty(\Sigma)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C}
\,.
$$

This linear function $A_\ast$ is in fact a compactly supported distribution, in the sense of functional analysis, in that it satisfies the following Fréchet vector space continuity condition:

• Fréchet continuous linear functionalA linear function $A_\ast \;\colon\; C^\infty(\mathbb{R}^{p,1}) \to \mathbb{R}$ is called continuous if there exists
  1. a compact subset $K \subset \mathbb{R}^{p,1}$ of Minkowski spacetime;
  2. a natural number $k \in \mathbb{N}$;
  3. a positive real number $C \in \mathbb{R}_+$

  such that for all on-shell field histories

  $$
  \Phi \in C^\infty(\Sigma)_{\delta_{EL}\mathbf{L} = 0}
  $$

  the following inequality of absolute values ${\vert -\vert}$ of partial derivatives holds

  $$
  {\vert A_\ast(\Phi)\vert}
  \;\leq\;
  C \underset{{\vert \alpha \vert} \leq k}{\sum} \, \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x)\vert}
  \,,
  $$

  where the sum is over all multi-indices $\alpha \in \mathbb{N}^{p+1}$ (1) whose total degree ${\vert \alpha\vert} := \alpha_0 + \cdots + \alpha_{p}$ is bounded by $k$, and where

  $$
  \partial^\alpha \Phi
  \;:=\;
  \frac{\partial^{{\vert \alpha\vert}} \Phi }{ \partial^{\alpha_0} x^0 \partial^{\alpha_1} x^1 \cdots \partial^{\alpha^p} x^p }
  $$

  denotes the corresponding partial derivative (1).

This identification constitutes a linear isomorphism

$$
\array{
LinObs(\Sigma \times \mathbb{R}) &\overset{\simeq}{\longrightarrow}& \mathcal{E}'(\Sigma)
\\
\array{
\text{linear off-shell}
\\
\text{observables}
\\
\text{of the scalar field}
}
&&
\array{
\text{compactly supported}
\\
\text{distributions}
\\
\text{on spacetime}
}
}
\,,
$$

saying that all compactly supported distributions arise from linear off-shell observables of the scalar field this way, and uniquely so.

For proof see at distributions are the smooth linear functionals, this prop.

The identification from prop. 7.5 of linear off-shell observables with compactly supported distributions makes available powerful tools from functional analysis. The key fact is the following:

Proposition 7.6. (distributions are generalized functions)

For $n \in \mathbb{N}$, every compactly supported smooth function $b \in C^\infty_{cp}(\mathbb{R}^n)$ on the Cartesian space $\mathbb{R}^n$ induces a distribution (prop. 7.5), hence a continuous linear functional, by integration against $b$ times the volume form.

$$
\array{
C^\infty(\mathbb{R}^n) &\longrightarrow& \mathbb{R}
\\
f &\mapsto& \int_{\mathbb{R}^n} f(x) b(x) \, dvol(x)
}
$$

The distributions arising this way are called the non-singular distributions.

This construction is clearly a linear inclusion

$$
C^\infty_{cp}(\mathbb{R}^n) \overset{\phantom{AAA}}{\hookrightarrow} \mathcal{E}'(\mathbb{R}^n)
$$

and in fact, this is a dense subspace inclusion for the space of compactly supported distributions $\mathcal{E}'(\mathbb{R}^n)$ equipped with the dual space topology (this def.) to the Fréchet space structure on $C^\infty(\mathbb{R}^n)$ from prop. 7.5.

Hence every compactly supported distribution $u$ is the limit of a sequence $\{b_n\}_{n \in \mathbb{N}}$ of compactly supported smooth functions in that for every smooth function $f \in C^\infty(\mathbb{R}^n)$ we have that the value $u(f) \in \mathbb{R}$ is the limit of integrals against $b_n dvol$:

$$
u(f)
\;=\;
\underset{n \to \infty}{\lim}\, \int_{\mathbb{R}^n} f(x) b_n(x) dvol(x)
\,.
$$

(e. g. Hörmander 90, theorem 4.1.5)

Proposition 7.6 with prop. 7.5 implies that with due care we may think of all linear off-shell observables as arising from the integration of field histories against some “generalized smooth functions” (namely a limit of actual smooth functions):

Remark 7.7. (linear off-shell observables of real scalar field as integration against generalized functions)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) over Minkowski spacetime (def. 2.17), whose field bundle $E$ (def. 3.1) is a trivial vector bundle with field coordinates $(\phi^a)$.

Prop. 7.5 implies immediately that in this situation linear off-shell observables $A$ (def. 7.3) correspond to tuples $(A_a)$ of compactly supported distributions via

$$
A(\Phi) = \underset{a}{\sum} A_a(\Phi^a)
\,.
$$

With prop. 7.6 it follows furthermore that there is a sequence of tuples of smooth functions $\{(\alpha_n)_{a}\}_{n \in \mathbb{N}}$ such that $A_a$ is the limit of the integrations against these:

$$
A(\Phi)
\;=\;
\underset{n \to \infty}{\lim}
\,
\int_\Sigma \Phi^a(x) (\alpha_n)_a(x) \, dvol(x)
\,,
$$

where now the sum over the index $a$ is again left notationally implicit.

For handling distributions/linear off-shell observables it is, therefore, useful to adopt, with due care, shorthand notation as if the limits of the sequences of smooth functions $(\alpha_n)_a$ actually existed, as “generalized functions” $\alpha_a$, and to set

$$
\int_\Sigma \Phi^a(x) \alpha_a(x) \, dvol(x)
\;:=\;
A(\Phi)
\,,
$$

This suggests that basic operations on functions, such as their pointwise product, should be extended to distributions, e.g. to a product of distributions. This turns out to exist, as long as the high-frequency modes in the Fourier transform of the distributions being multiplied cancel out — the mathematical reflection of “UV-divergences” in quantum field theory. This we turn to in Free quantum fields below.

These considerations generalize from the field bundle of the real scalar field to general field bundles (def. 3.1) as long as they are smooth vector bundles (def. 1.10):

Definition 7.8. (Fréchet topological vector space on spaces of smooth sections of a smooth vector bundle)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle (def. 3.1) which is a smooth vector bundle (def. 1.10) over Minkowski spacetime (def. 2.17); hence, up to isomorphism, a trivial vector bundle as in example 3.4.

On its real vector space $\Gamma_\Sigma(E)$ of smooth sections consider the seminorms indexed by a compact subset $K \subset \Sigma$ and a natural number $k \in \mathbb{N}$ and given by

$$
\array{
\Gamma_\Sigma(E) &\overset{p_K^k}{\longrightarrow}&
[0,\infty)
\\
\Phi &\mapsto& \underset{ {\vert \alpha\vert} \leq k}{max} \left( \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x)\vert}\right) \,,
}
$$

where on the right we have the absolute values of the partial derivatives of $\Phi$ index by $\alpha$ (1) with respect to any choice of norm on the fibers.

This makes $\Gamma_\Sigma(E)$ a Fréchet topological vector space.

For $K \subset \Sigma$ any closed subset then the sub-space of sections

$$
\Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)
$$

of sections whose support is inside $K$ becomes a Fréchet topological vector spaces with the induced subspace topology, which makes these be closed subspaces.

Finally, the vector spaces of smooth sections with prescribed causal support (def. 2.36) are inductive limits of vector spaces $\Gamma_{\Sigma,K}(E)$ as above, and hence they inherit topological vector space structure by forming the corresponding inductive limit in the category of topological vector spaces. For instance

$$
\Gamma_{\Sigma,cp}(E)
\;:=\;
\underset{\underset{ {K \subset \Sigma} \atop {K\, \text{compact}} }{\longrightarrow}}{\lim}
\Gamma_{\Sigma,K}(E)
$$

etc.

(Bär 14, 2.1)

Definition 7.9. (distributional sections)

Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle (def. 1.10) over Minkowski spacetime (def. 2.17).

The vector spaces of smooth sections with restricted support from def. 2.36 structures of topological vector spaces via def. 7.8. We denote the dual topological vector spaces by

$$
\Gamma’_{\Sigma}( E ^*)
\;:=\;
(\Gamma_{\Sigma,cp}(E))^*
\,.
$$

This is called the space of distributional sections of the dual vector bundle ${E}^*$.

The support of a distributional section $supp(u)$ is the set of points in $\Sigma$ such that for every neighbourhood of that point $u$ does not vanish on all sections with support in that neighbourhood.

Imposing the same restrictions to the supports of distributional sections as in def. 2.36, we have the following subspaces of distributional sections:

$$
\Gamma’_{\Sigma,cp}(E^\ast) ,
\Gamma’_{\Sigma,\pm cp}(E^\ast) ,
\Gamma’_{\Sigma,scp}(E^\ast) ,
\Gamma’_{\Sigma,fcp}(E^\ast) ,
\Gamma’_{\Sigma,pcp}(E^\ast) ,
\Gamma’_{\Sigma,tcp}(E^\ast)
\;\subset\;
\Gamma’_{\Sigma}(E^\ast) .
$$

(Sanders 13, Bär 14)

As before in prop. 7.6 the actual smooth sections yield examples of distributional sections, and all distributional sections arise as limits of integrations against smooth sections:

Proposition 7.10. (non-singular distributional sections)

Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle over Minkowski spacetime and let $s \in \{cp, \pm cp, scp, tcp\}$ be any of the support conditions from def. 2.36.

Then the operation of regarding a compactly supported smooth section of the dual vector bundle as a functional on sections with this support property is a dense subspace inclusion into the topological vector space of distributional sections from def. 7.9:

$$
\array{
\Gamma_{\Sigma,cp}(E^\ast)
&\overset{\phantom{A}u_{(-)}\phantom{A} }{\hookrightarrow}&
\Gamma’_{\Sigma,s}(E)
\\
b &\mapsto& \left( \Phi \mapsto \underset{\Sigma}{\int} b(x) \cdot \Phi(x) \, dvol_\Sigma(x) \right)
}
$$

(Bär 14, lemma 2.15)

Proposition 7.11. (distribution dualities with causally restricted supports)

Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle (def. 1.10) over Minkowski spacetime (def. 2.17).

Then there are the following isomorphisms of topological vector spaces between a) dual spaces of spaces of sections with restricted causal support (def. 2.36) and equipped with the topology from def. 7.8 and b) spaces of distributional sections with restricted supports, according to def. 7.9:

$$
\begin{aligned}
\Gamma_{\Sigma,cp}(E)^* &\simeq \Gamma’_{\Sigma}(E^\ast) ,
\\
\Gamma_{\Sigma,+cp}(E)^* &\simeq \Gamma’_{\Sigma,fcp}(E^\ast) ,
\\
\Gamma_{\Sigma,-cp}(E)^* &\simeq \Gamma’_{\Sigma,pcp}(E^\ast) ,
\\
\Gamma_{\Sigma,scp}(E)^* &\simeq \Gamma’_{\Sigma,tcp}(E^\ast) ,
\\
\Gamma_{\Sigma,fcp}(E)^* &\simeq \Gamma’_{\Sigma,+cp}(E^\ast) ,
\\
\Gamma_{\Sigma,pcp}(E)^* &\simeq \Gamma’_{\Sigma,-cp}(E^\ast) ,
\\
\Gamma_{\Sigma,tcp}(E)^* &\simeq \Gamma’_{\Sigma,scp}(E^\ast) ,
\\
\Gamma_{\Sigma}(E)^* &\simeq \Gamma’_{\Sigma,cp}(E^\ast) .
\end{aligned}
$$

(Sanders 13, thm. 4.3, Bär 14, lem. 2.14)

The concept of linear observables naturally generalizes to that of multilinear observables:

Definition 7.12. (quadratic off-shell observables)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) over a spacetime $\Sigma$ whose field bundle $E$ (def. 3.1) is a super vector bundle.

The external tensor product of vector bundles of the field bundle $E \overset{fb}{\to} \Sigma$ with itself, denoted

$$
E \boxtimes E \overset{}{\to} \Sigma \times \Sigma
$$

is the vector bundle over the Cartesian product $\Sigma \times \Sigma$, of spacetime with itself, whose fiber over a pair of points $(x_1,x_2)$ is the tensor product $E_{x_1} \otimes E_{x_2}$ of the corresponding field fibers.

Given a field history, hence a section $\phi \in \Gamma_\Sigma(E)$ of the field bundle, there is then the induced section $\phi \boxtimes \phi \in \Gamma_{\Sigma \times \Sigma}(E \boxtimes E)$.

We say that an off-shell observable

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

is quadratic if it comes from a “graded-symmetric bilinear observable”, namely a smooth function on the space of sections of the external tensor product of the field bundle with itself

$$
B
\;\colon\;
\Gamma_{\Sigma \times \Sigma}(E \boxtimes E)_{\delta_{EL}\mathbf{L} = 0}
\longrightarrow
\mathbb{C}
\,,
$$

as

$$
A(\Phi) = B(\Phi,\Phi)
\,.
$$

More explicitly: By prop. 7.5 the quadratic observable $A$ is given by a compactly supported distribution of two variables which in the notation of remark 7.7 comes from a graded-symmetric matrix of generalized functions $\beta_{a_1 a_2} \in \mathcal{E}'(\Sigma \times \Sigma, E \boxtimes E)$ as

$$
A(\Phi)
\;=\;
\int_{\Sigma \times \Sigma}
\beta_{a_1 a_2}(x_1,x_2)
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2)\, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
\,.
$$

This notation makes manifest how the concept of quadratic observables is a generalization of that of quadratic forms coming from bilinear formsS.

Definition 7.13. (off-shell polynomial observables)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) over a spacetime $\Sigma$ whose field bundle $E$ (def. 3.1) is a super vector bundle.

An off-shell observable (def. 7.1)

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

is a polynomial observable if it is the sum of a constant, and a linear observable (def. 7.3), and a quadratic observable (def. 7.12) and so on:

$$
\label{ExpansionOfPolynomialObservables}
\begin{aligned}
A(\Phi)
& = \phantom{+}
\alpha^{(0)}
\\
&
\phantom{=}
+
\int_{\Sigma} \Phi^a(x) \alpha^{(1)}_a(x) \, dvol_\Sigma(x)
\\
&
\phantom{=}
+
\int_{\Sigma^2}
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2)
\alpha^{(2)}_{a_1 a_2}(x_1, x_2) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
\\
&
\phantom{=}
+
\int_{\Sigma^3}
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2) \cdot \Phi^{a_3}(x_3)
\alpha^{(3)}_{a_1 a_2 a_3}(x_1,x_2,x_3)
\, dvol_\Sigma(x_1) dvol_\Sigma(x_2) dvol_\Sigma(x^3)
\\
&
\phantom{=}
+
\cdots
\,.
\end{aligned}
$$ (88)

If all the coefficient distributions $\alpha^{(k)}$ are non-singular distributions, then we say that $A$ is a regular polynomial observable.

We write

$$
PolyObs(E)_{reg} \hookrightarrow PolyObs(E) \hookrightarrow Obs(E)
$$

for the subspace of (regular) polynomial off-shell observables.

Example 7.14. (polynomial observables of the Dirac field)

Let $E = \Sigma \times S_{odd}$ be the field bundle of the Dirac field (example 3.50).

Then, by prop. 3.51, an $\mathbb{R}^{0\vert 1}$-parameterized plot of the space of off-shell polynomial observables (def. 7.13)

$$
A_{(-)}
\;\colon\;
\mathbb{R}^{0 \vert 1}
\longrightarrow
PolyObs(\Sigma \times S_{odd})
$$

is of the form

$$
\begin{aligned}
A_{(-)}
& =
a^{(0)}
\\
& \phantom{=}
+
\theta
\underset{\Sigma}{\int}
a^{(1)}_{\alpha}(x)
\mathbf{\Psi}^\alpha(x)
dvol_\Sigma(x)
\\
&
\phantom{=}
+
\underset{\Sigma^2}{\int}
a^{(2)}_{\alpha_1 \alpha_2}(x,y)
\mathbf{\Psi}^{\alpha_1}(x_1)
\cdot
\mathbf{\Psi}^{\alpha_2}(x_2)
\,
dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
\\
& \phantom{=}
+
\theta
\underset{\Sigma}{\int}
a^{(3)}_{\alpha_1 \alpha_2 \alpha_3}(x_1, x_2, x_3)
\mathbf{\Psi}^{\alpha_1}(x_1)
\cdot
\mathbf{\Psi}^{\alpha_2}(x_2)
\cdot
\mathbf{\Psi}^{\alpha_3}(x_3)
\, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \, dvol_\Sigma(x_3)
\\
& \phantom{=}
+ \cdots
\end{aligned}
$$

for any distributions of several variables $a^{(k)}_{\alpha_1, \cdots , \alpha_k}$. Here

$$
\mathbf{\Psi}^\alpha(x)
\;\colon\;
\Gamma_\Sigma(\Sigma \times S_{even})
\longrightarrow
\mathbb{C}
$$

are the point-evaluation field observables (example 7.2) on the spinor bundle, and

$$
\theta \in C^\infty(\mathbb{R}^{0\vert 1})_{odd}
$$

is the canonical odd-graded coordinate function on the superpoint $\mathbb{R}^{0 \vert 1}$ (def. 3.37).

Hence all the odd powers of the Dirac-field observables are proportional to $\theta$. In particular if one considers just a point in the space of polynomial observables

$$
A \;\colon\; \mathbb{R}^{0} \longrightarrow PolyObs(E \times S_{odd})
$$

then all the odd monomials in the field observables of the Dirac field disappear.

Proof. By definition of supergeometric mapping spaces (def. 3.47), there is a natural bijection between $\mathbb{R}^{0 \vert 1}$-plots $A_{(-)}$ of the space of observables and smooth functions out of the Cartesian product of $\mathbb{R}^{0 \vert 1}$ with the space of field histories to the complex numbers:

$$
\frac{
\mathbb{R}^{0\vert 1}
\overset{ A_{(-)} }{\longrightarrow}
[ \Gamma_\Sigma(\Sigma \times S_{odd}), \mathbb{C} ]
}
{
\mathbb{R}^{0 \vert 1} \times \Gamma_\Sigma(\Sigma \times S_{odd})
\longrightarrow
\mathbb{C}
}
$$

Moreover, by prop. 3.51 we have that the coordinate functions on the space of field histories of the Dirac bundle are given by the field observables $\mathbf{\Psi}^\alpha(x)$ regarded in odd degree. Now a homomorphism as above has to pull back the even coordinate function on $\mathbb{C}$ to even coordinate functions on this Cartesian product, hence to joint even powers of $\theta$ and $\mathbf{\Psi}^\alpha(x)$.

Next we discuss the restriction of these off-shell polynomial observables to the shell to yield on-shell polynomial observables, characterized by theorem 7.29 below.

Polynomial on-shell Observables and Distributional solutions to PDEs

The evident on-shell version of def. 7.13 is this:

Definition 7.15. (on-shell polynomial observables)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory (def. 5.25) with on-shell space of field histories $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Then an on-shell observable (def. 7.1)

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

is an on-shell polynomial observable if it is the restriction of an off-shell polynomial observable $A_{off}$ according to def. 7.13:

$$
\array{
\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} &\overset{\phantom{A}A\phantom{A}}{\longrightarrow}&
\\
\downarrow & \nearrow_{\rlap{A_{off}}}
\\
\Gamma_\Sigma(E)
}
\,.
$$

Similarly $A$ is an on-shell linear observable or on-shell regular polynomial observable etc. if it is the restriction of a linear observable or regular polynomial observable, respectively, according to def. 7.13. We write

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

for the subspace of polynomial on-shell observables inside all on-shell observables, and similarly

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

and

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

etc.

$$

While by def. 7.15 every off-shell observable induces an on-shell observable simply by restriction (84), different off-shell observables may restrict to the same on-shell observale. It is therefore useful to find a condition on off-shell observables that makes them equivalent to on-shell observables under restriction. We now discuss such precise characterizations of the off-shell polynomial observables for the case of sufficiently well behaved free field equations of motion — namely Green hyperbolic differential equations, def. 7.19 below. The main result is theorem 7.29 below.

While in general the equations of motion are not Green hyperbolic — namely not in the presence of implicit infinitesimal gauge symmetries discussed in Gauge symmetries below — it turns out that up to a suitable notion of equivalence they are equivalent to those that are; this we discuss in the chapter Gauge fixing below.

Definition 7.16. (derivatives of distributions and distributional solutions of PDEs)

Given a pair of formally adjoint differential operators $P, P^\ast \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$ (def. 4.9) then the distributional derivative of a distributional section $u \in \Gamma’_\Sigma(E)$ (def. 7.9) by $P$ is the distributional section $P u \in \Gamma’_\Sigma(E^\ast)$

$$
P u
\;:=\;
u(P^\ast(-)) \;\colon\; \Gamma_{\Sigma,cp}(E^\ast)
\,.
$$

If

$$
P u = 0 \;\in\; \Gamma’_\Sigma(E^\ast)
$$

then we say that $u$ is a distributional solution (or generalized solution) of the homogeneous differential equation defined by $P$.

Example 7.17. (ordinary PDE solutions are generalized solutions)

Let $E \overset{fb}{\to} \Sigma$ be a smooth vector bundle over Minkowski spacetime and let $P, P^\ast \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$ be a pair of formally adjoint differential operators.

Then for every non-singular distributional section $u_{\Phi} \in \Gamma’_{\Sigma}(E^\ast)$ coming from an actual smooth section $\Phi \in \Gamma_\Sigma(E)$ via prop. 7.10
the derivative of distributions (def. 7.16) is the distributional section induced from the ordinary derivative of smooth functions:

$$
P u_\Phi \;=\; u_{P \Phi}
\,.
$$

In particular $u_\Phi$ is a distributional solution to the PDE precisely if $\Phi$ is an ordinary solution:

$$
P u_\Phi \;=\; 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
P \Phi = 0
\,.
$$

Proof. For all $b \in \Gamma_{\Sigma,cp}(E)$ we have

$$
\begin{aligned}
(P u_\Phi)(b)
& =
u_\Phi(P^\ast b)
\\
& =
\int u \cdot P^\ast b \, dvol
\\
& =
\int (P u) \cdot b \, dvol
\\
& =
u_{P \Phi}(b)
\end{aligned}
$$

where all steps are by the definitions except the third, which is by the definition of formally adjoint differential operator (def. 4.9), using that by the compact support of $b$ and the Stokes theorem (prop. 1.25) the term $K(\Phi,b)$ in def. 4.9 does not contribute to the integral.

Definition 7.18. (advanced and retarded Green functions and causal Green function)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle (def. 3.1) which is a vector bundle (def. 1.10) over Minkowski spacetime (def. 2.17). Let $P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$ be a differential operator (def. 4.7) on its space of smooth sections.

Then a linear map

$$
\mathrm{G}_{P,\pm}
\;\colon\;
\Gamma_{\Sigma, cp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, \pm cp}(E)
$$

from spaces of smooth sections of compact support to spaces of sections of causally sourced future/past support (def. 2.36) is called an advanced or retarded Green function for $P$, respectively, if

1. for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have
   

   $$
   \label{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator}
   G_{P,\pm} \circ P(\Phi) = \Phi
   $$ (89)

   and

   $$
   \label{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator}
   P \circ G_{P,\pm}(\Phi) = \Phi
   $$ (90)

2. the support of $G_{P,\pm}(\Phi)$ is in the closed future cone or closed past cone of the support of $\Phi$, respectively.

If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference

$$
\label{CausalGreenFunction}
\mathrm{G}_P := \mathrm{G}_{P,+} – \mathrm{G}_{P,-}
$$ (91)

is called the causal Green function.

(e.g. Bär 14, def. 3.2, cor. 3.10)

Definition 7.19. (Green hyperbolic differential equation)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle (def. 3.1) which is a vector bundle (def. 1.10) over Minkowski spacetime (def. 2.17).

A differential operator (def. 4.8)

$$
P
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\Gamma_{\Sigma}(E^\ast)
$$

is called a Green hyperbolic differential operator if $P$ as well as its formal adjoint differential operator $P^\ast$ (def. 4.9) admit advanced and retarded Green functions (def. 7.18).

(Bär 14, def. 3.2, Khavkine 14, def. 2.2)

The two archtypical examples of Green hyperbolic differential equations are the Klein-Gordon equation and the Dirac equation on Minkowski spacetime. For the moment we just cite the existence of the advanced and retarded Green functions for these, we will work these out in detail below in Propagators.

Example 7.20. (Klein-Gordon equation is a Green hyperbolic differential equation)

The Klein-Gordon equation, hence the Euler-Lagrange equation of motion of the free scalar field (example 5.27) is a Green hyperbolic differential equation (def. 7.19) and formally self-adjoint (example 5.28).

(e. g. Bär-Ginoux-Pfaeffle 07, Bär 14, example 3.3)

Example 7.21. (Dirac operator is Green hyperbolic)

The Dirac equation, hence the Euler-Lagrange equation of motion of the massive free Dirac field (example 5.30) is a Green hyperbolic differential equation (def. 7.19) and formally anti self-adjoint (example 5.32).

(Bär 14, corollary 3.15, example 3.16)

Example 7.22. (causal Green functions of formally adjoint Green hyperbolic differential operators are formally adjoint)

Let

$$
P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(E^\ast)
$$

be a pair of Green hyperbolic differential operators (def. 7.19) which are formally adjoint (def. 4.9). Then also their causal Green functions $\mathrm{G}_P$ and $G_{P^\ast}$ (def. 7.18) are formally adjoint differential operators, up to a sign:

$$
\left(
\mathrm{G}_P
\right)^\ast
\;=\;
– \mathrm{G}_{P^\ast}
\,.
$$

(Khavkine 14, (24), (25))

We did not require that the advanced and retarded Green functions of a Green hyperbolic differential operator are unique; in fact this is automatic:

Proposition 7.23. (advanced and retarded Green functions of Green hyperbolic differential operator are unique)

The advanced and retarded Green functions (def. 7.18) of a Green hyperbolic differential operator (def. 7.19) are unique.

(Bär 14, cor. 3.12

Moreover we did not require that the advanced and retarded Green functions of a Green hyperbolic differential operator come from integral kernels (“propagators”). This, too, is automatic:

Proposition 7.24. (causal Green functions of Green hyperbolic differential operators are continuous linear maps)

Given a Green hyperbolic differential operator $P$ (def. 7.19), the advanced, retarded and causal Green functions of $P$ (def. 7.18) are continuous linear maps with respect to the topological vector space structure from def. 7.8 and also have a unique continuous extension to the spaces of sections with larger support (def. 2.36) as follows:

$$
\begin{aligned}
\mathrm{G}_{P,+}
&\;\colon\;
\Gamma_{\Sigma, pcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, pcp}(E) ,
\\
\mathrm{G}_{P,-}
&\;\colon\;
\Gamma_{\Sigma, fcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, fcp}(E) ,
\\
\mathrm{G}_{P}
&\;\colon\;
\Gamma_{\Sigma, tcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma}(E) ,
\end{aligned}
$$

such that we still have the relation

$$
\mathrm{G}_P = \mathrm{G}_{P,+} – \mathrm{G}_{P,-}
$$

and

$$
P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id
$$

and

$$
supp \mathrm{G}_{P,\pm}({\alpha}^*) \subseteq J^\pm(supp {\alpha}^*)
\,.
$$

By the Schwartz kernel theorem the continuity of $\mathrm{G}_{\pm}, \mathrm{G}$ implies that there are integral kernels

$$
\Delta_{\pm} \;\in\; \Gamma’_{\Sigma \times \Sigma}( E \boxtimes_\Sigma E )
$$

such that, in the notation of generalized functions,

$$
(G_{\pm} \alpha^\ast)(x)
\;=\;
\underset{\Sigma}{\int}
\Delta_\pm(x,y) \cdot \alpha^\ast(y) \, dvol_\Sigma(y)
\,.
$$

These integral kernels are called the advanced and retarded propagators. Similarly the combination

$$
\label{CausalPropagator}
\Delta \;:=\; \Delta_+ – \Delta_-
$$ (92)

is called the causal propagator.

(Bär 14, thm. 3.8, cor. 3.11)

We now come to the main theorem on polynomial observables:

Lemma 7.25. (exact sequence of Green hyperbolic differential operator)

Let $\Gamma_\Sigma(E) \overset{P}{\longrightarrow} \Gamma_\Sigma(E^\ast)$ be a Green hyperbolic differential operator (def. 7.19) with causal Green function $\mathrm{G}$ (def. 7.19). Then the sequences

$$
\label{GreenOperatorExactSequenceFirst}
\array{
0
&\to&
\Gamma_{\Sigma,cp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,cp}(E^\ast)
&\overset{\mathrm{G}_P}{\longrightarrow}&
\Gamma_{\Sigma,scp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,scp}(E^\ast)
&\to&
0
\\
\\
0
&\to&
\Gamma_{\Sigma,tcp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,tcp}(E^\ast)
&\overset{\mathrm{G}_P}{\longrightarrow}&
\Gamma_{\Sigma}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma}(E^\ast)
&\to&
0
}
$$ (93)

of these operators restricted to functions with causally restricted supports as indicated (def. 2.36) are exact sequences of topological vector spaces and continuous linear maps between them.

Under passing to dual spaces and using the isomorphisms of spaces of distributional sections (def. 7.9) from prop. 7.11 this yields the following dual exact sequence of topological vector spaces and continuous linear maps between them:

$$
\label{GreenHyperbolicOperatorDualExactSequence}
\array{
0
&\to&
\Gamma’_{\Sigma,tcp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma’_{\Sigma,tcp}(E^\ast)
&\overset{-\mathrm{G}_{P^*}}{\longrightarrow}&
\Gamma’_{\Sigma}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma’_{\Sigma}(E^\ast)
&\to&
0
\\
\\
0
&\to&
\Gamma’_{\Sigma,cp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma’_{\Sigma,cp}(E^\ast)
&\overset{-\mathrm{G}_{P^*}}{\longrightarrow}&
\Gamma’_{\Sigma,scp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma’_{\Sigma,scp}(E^\ast)
&\to&
0
}
$$ (94)

This is due to Igor Khavkine, based on (Khavkine 14, prop. 2.1); for proof see at Green hyperbolic differential operator this lemma.

Corollary 7.26. (on-shell space of field histories for Green hyperbolic free field theories)

Let $(E,\mathbf{L})$ be a free field theory Lagrangian field theory (def. 5.9) whose Euler-Lagrange equation of motion $P \Phi = 0$ is Green hyperbolic (def. 7.19).

Then the on-shell space of field histories (or of field histories with spatially compact support, def. 2.36) is, as a vector space, linearly isomorphic to the quotient space of compactly supported sections (or of temporally compactly supported sections, def. 2.36) by the image of the differential operator $P$, and this isomorphism is given by the causal Green function $\mathrm{G}_P$ (91)

$$
\label{SolutionSpaceIsomorphicToQuotientByImP}
\array{
\Gamma_{\Sigma,tcp}(E^\ast)/im(P)
&\underset{\simeq}{\overset{\phantom{A}\mathrm{G}_P \phantom{A}}{\longrightarrow}}&
ker(P) \;=\; \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}
\\
\Gamma_{\Sigma,cp}(E^\ast)/im(P)
&\underset{\simeq}{\overset{\phantom{A}\mathrm{G}_P\phantom{A}}{\longrightarrow}}&
ker_{scp}(P) \;=\; \Gamma_{\Sigma,scp}(E)_{\delta_{EL}\mathbf{L} = 0}
\,.
}
$$ (95)

Proof. This is a direct consequence of the exactness of the sequence (93) in lemma 7.25.

We spell this out for the statement for $\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}$, which follows from the first line in (93), the first statement similarly follows from the second line of (93):

First the on-shell space of field histories is the kernel of $P$, by definition of free field theory (def. 5.9)

$$
\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}
\;=\;
ker_{scp}(P)
\,.
$$

Second, exactness of the sequence (93) at $\Gamma_{\Sigma,scp}(E)$ means that the kernel $ker_{scp}(P)$ of $P$ equals the image $im(\mathrm{G}_{P})$. But by exactness of the sequence at $\Gamma_{\Sigma,cp}(E^\ast)$ it follows that $\mathrm{G}_P$ becomes injective on the quotient space $\Gamma_{\Sigma,cp}(E)^\ast/im(P)$. Therefore on this quotient space it becomes an isomorphism onto its image.

Remark 7.27.

Under passing to dual vector spaces, the linear isomorphism in corollary 7.26
in turn yields linear isomorphisms of the form

$$
\label{DualSolutionSpaceIsomorphicToQuotientByImP}
\array{
\left(\Gamma_{\Sigma,cp}(E^\ast)/im(P)\right)^\ast
&\underset{\simeq}{\overset{(-)\circ \mathrm{G}_P}{\longleftarrow}}&
\left(ker_{scp}(P)\right)^\ast
\\
\left(\Gamma_\Sigma(E^\ast)/im(P)\right)^\ast
&\underset{\simeq}{\overset{(-)\circ \mathrm{G}_P }{\longleftarrow}}&
\left(ker(P)\right)^\ast
}
\,.
$$ (96)

Except possibly for the issue of continuity this says that the linear on-shell observables (def. 7.3) of a Green hyperbolic free field theory are equivalently those linear off-shell observables which are generalized solutions of the formally dual equation of motion according to def. 7.16.

That this remains true also for topological vector space structure follows with the dual exact sequence (94). This is the statement of prop. 7.28 below.

Proposition 7.28. (distributional sections on a Green hyperbolic solution space are the generalized PDE solutions)

Let $P, P \ast \;\colon\; \Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(E^\ast)$ be a pair of Green hyperbolic differential operators (def. 7.19) which are formally adjoint (def. 4.9).

Then

1. the canonical pairing (from prop. 7.11)$$
   \array{
   \Gamma’_{\Sigma,cp}(E^\ast) &\otimes& \Gamma_\Sigma(E)
   &\overset{}{\longrightarrow}&
   \mathbb{C}
   \\
   \alpha^\ast &,& \Phi &\mapsto& \int \alpha^\ast_a(x) \Phi^a(x)\, dvol_\Sigma(x)
   }
   $$induces a continuous linear isomorphism

   $$
   \label{LinearDualOfSolutionSpaceIsLinearDualOfFullSpaceModuloImageOfDifferentialOperator}
   (ker(P))^\ast
   \;\simeq\;
   \Gamma’_{\Sigma,cp}(E^\ast)/im_{cp}(P^\ast)
   $$ (97)

2. a continuous linear functional on the solution space$$
   u_{sol} \in \left(ker(P)\right)^\ast
   $$is equivalently a distributional section (def. 7.9) whose support is spacelike compact (def. 2.36, prop. 7.11)$$
   u \in \Gamma’_{\Sigma,scp}(E)
   $$and which is a distributional solution (def. 7.16) to the differential equation$$
   P^\ast u = 0
   \,.
   $$Similarly, a continuous linear functional on the subspace of solutions that have spatially compact support (def. 2.36)$$
   u_{sol} \in \left(ker(P)_{scp}\right)^\ast
   $$is equivalently a distributional section (def. 7.9) without constraint on its distributional support

   $$
   u \in \Gamma’_{\Sigma}(E)
   $$

   and which is a distributional solution (def. 7.16) to the differential equation

   $$
   P^\ast u = 0
   \,.
   $$

   Moreover, these linear isomorphisms are both given by composition with the causal Green function $\mathrm{G}$ (def. 7.18):

   $$
   \array{
   \left(ker(P)\right)^\ast
   &\underset{\simeq}{\overset{(-)\circ \mathrm{G}}{\longrightarrow}}&
   \left\{
   u \in \Gamma’_{\Sigma,scp}(E)
   \,\vert\,
   P^\ast u = 0
   \right\}
   \\
   \left(ker_{scp}(P)\right)^\ast
   &\underset{\simeq}{\overset{(-)\circ \mathrm{G}}{\longrightarrow}}&
   \left\{
   u \in \Gamma’_{\Sigma}(E)
   \,\vert\,
   P^\ast u = 0
   \right\}
   }
   \,.
   $$

This follows from the exact sequence in lemma 7.25. For details of the proof see at Green hyperbolic differential operator this prop., due to Igor Khavkine.

In conclusion we have found the following:

Theorem 7.29. (linear observables of Green free field theory are the distributional solutions to the formally adjoint equations of motion)

Let $(E,\mathbf{L})$ be a Lagrangian free field theory (def. 5.25) which is a free field theory (def. 5.25) whose Euler-Lagrange differential equation of motion $P \Phi = 0$ (def. 5.24) is Green hyperbolic (def. 7.19), such as the Klein-Gordon equation (example 7.20) or the Dirac equation (example 7.21). Then:

1. The linear off-shell observables (def. 7.3) are equivalently the compactly supported distributional sections (def. 7.9) of the field bundle:$$
   LinObs(E)
   \;\simeq\;
   \Gamma’_{\Sigma,cp}(E)
   $$
2. The linear on-shell observables (def. 7.3) are equivalently the linear off-shell observables modulo the image of the differential operator $P$:
   

   $$
   \label{LinearOnShellObservablesAreLinearOffShellobservableModuloTheEquationsOfMotion}
   LinObs(E,\mathbf{L})
   \simeq
   LinObs(E)/im(P)
   \,.
   $$ (98)

   More generally the on-shell polynomial observables are identified with the off-shell polynomial observables (def. 7.13) modulo the image of $P$:

   $$
   \label{PolynomialOnShellObservablesArePolynomialOffShellobservableModuloTheEquationsOfMotion}
   PolyObObs(E,\mathbf{L})
   \simeq
   PolyObs(E)/im(P)
   \,.
   $$ (99)

3. The linear on-shell observables (def. 7.3) are also equivalently those spacelike compactly supported compactly distributional sections (def. 7.9) which are distributional solutions of the formally adjoint equations of motion (def. 4.9), and this isomorphism is exhibited by precomposition with the causal propagator $\mathrm{G}$:$$
   LinObs(E,\mathbf{L})
   \;\underset{\simeq}{\overset{\phantom{A}(-)\circ\mathrm{G}_P \phantom{A}}{\longrightarrow}}\;
   \left\{
   A \in \Gamma’_{\Sigma,scp}(E)
   \;\vert\;
   P^\ast A = 0
   \right\}
   $$Similarly the linear on-shell observables on spacelike compactly supported on-shell field histories (85) are equivalently the distributional solutions without constraint on their support:$$
   LinObs(E_{scp},\mathbf{L})
   \;\underset{\simeq}{\overset{\phantom{A}(-) \circ \mathrm{G}_P \phantom{A}}{\longrightarrow}}\;
   \left\{
   A \in \Gamma’_{\Sigma}(E)
   \;\vert\;
   P^\ast A = 0
   \right\}
   $$

Proof. The first statement follows with prop. 7.5
applied componentwise. The same proof applies verbatim to the subspace of solutions, showing that $LinObs(E,\mathbf{L}) \simeq \left( ker(P)\right)^\ast$, with the dual topological vector space on the right. With this the second and third statement follows by prop. 7.28.

We will be interested in those linear observables which under the identification from theorem 7.29 correspond to the non-singular distributions (because on these the Poisson-Peierls bracket of the theory is defined, theorem 8.7 below):

Definition 7.30. (linear regular observables and observable-valued distributions)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory (def. 5.25) whose Euler-Lagrange equations of motion (prop. 11.19) is Green hyperbolic (def. 7.19).

Define the regular linear observables among the linear on-shell observables (def. 7.3) to be the non-singular distributions on the on-shell space of field histories, hence the image

$$
LinObs(E_{scp},\mathbf{L})_{reg}
\hookrightarrow
LinObs(E_{scp},\mathbf{L})
$$

of the map

$$
\label{RegularLinearObservables}
\array{
\mathbf{\Phi}
&\colon&
\Gamma_{\Sigma,cp}(E^\ast)
&\longrightarrow&
LinObs(E_{scp},\mathbf{L})
&\hookrightarrow&
Obs(E_{scp},\mathbf{L})
\\
&& \alpha^\ast
&\mapsto&
\left(
\Phi
\mapsto
\underset{\Sigma}{\int} \alpha^\ast_a(x) \Phi^a(x)
\, dvol_\Sigma(x)
\right)
}
$$ (100)

By theorem 7.29 we have the identification (97) (98)

$$
\label{RegularLinearObservablesAreCompactlySupportedSectionsModuloImageOfP}
LinObs(E_{scp},\mathbf{L})_{reg}
\;\simeq\;
\Gamma_{\Sigma,scp}(E^\ast)/im(P)
\,.
$$ (101)

The point-evaluation field observables $\mathbf{\Phi}^a(x)$ (example 7.2) are linear observables (example 7.4) but far from being regular (100) (except in spacetime dimension $p +1 = 0+1$). But the regular observables are precisely the averages (“smearings”) of these point evaluation observables against compactly supported weights.

Viewed this way, the defining inclusion of the regular linear observables (100) is itself an observable valued distribution

$$
\label{AverageOfFieldObservableIsRegularLinearObservables}
\array{
\mathbf{\Phi} &\colon& \Gamma_{\Sigma,cp}(E^\ast) &\hookrightarrow& LinObs(E,\mathbf{L})
\\
&& \alpha^\ast &\mapsto& \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)
}
$$ (102)

which to a “smearing function” $\alpha^\ast$ assigns the observable which is the field observable smeared by (i.e. averaged against) that smearing function.

Below in Free quantum fields we discuss how the polynomial Poisson algebra of regular polynomial observables of a free field theory may be deformed to a non-commutative algebra of quantum observables. Often this may be represented by linear operators acting on some Hilbert space. In this case then $\mathbf{\Phi}$ above becomes a continuous linear functional from $\Gamma_{\Sigma,cp}(E)$ to a space of linear operators on some Hilbert space. As such it is then called an operator-valued distribution.

Local observables

We now discuss the sub-class of those observables which are “local”.

Definition 7.31. (spacetime support)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 3.1), with induced jet bundle $J^\infty_\Sigma(E)$

For every subset $S \subset \Sigma$ let

$$
\array{
J^\infty_\Sigma(E)\vert_S
&\overset{\iota_S}{\hookrightarrow}&
J^\infty_\Sigma(E)
\\
\downarrow &(pb)& \downarrow
\\
S &\hookrightarrow& \Sigma
}
$$

be the corresponding restriction of the jet bundle of $E$.

The spacetime support $supp_\Sigma(A)$ of a differential form $A \in \Omega^\bullet(J^\infty_\Sigma(E))$ on the jet bundle of $E$ is the topological closure of the maximal subset $S \subset \Sigma$ such that the restriction of $A$ to the jet bundle restrited to this subset vanishes:

$$
supp_\Sigma(A) := Cl( \{ x \in \Sigma | \iota_{\{x\}^\ast A = 0} \} )
$$

We write

$$
\Omega^{r,s}_{\Sigma,cp}(E)
:=
\left\{
A \in \Omega^{r,s}_\Sigma(E)
\;\vert\;
supp_\Sigma(A) \, \text{is compact}
\right\}
\;\hookrightarrow\;
\Omega^{r,s}_\Sigma(E)
$$

for the subspace of differential forms on the jet bundle whose spacetime support is a compact subspace.

Definition 7.32. (transgression of variational differential forms to space of field histories)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 3.1). and let $$
\Sigma_r \hookrightarrow \Sigma
$$

be a submanifold of spacetime of dimension $r \in \mathbb{N}$. Recall the space of field histories restricted to its infinitesimal neighbourhood, denoted $\Gamma_{\Sigma_r}(E)$ (def. 3.1).

Then the operation of transgression of variational differential forms to $\Sigma_r$ is the linear map

$$
\tau_{\Sigma_r}
\;\colon\;
\Omega^{\bullet,\bullet}_{\Sigma,cp}(E)
\overset{ }{\longrightarrow}
\Omega^\bullet\left(
\Gamma_{\Sigma_r}(E)
\right)
$$

that sends a variational differential form $A \in \Omega^{\bullet,\bullet}_{\Sigma,cp}(E)$ to the differential form $\tau_{\Sigma_r} \in \Omega^\bullet(\Gamma_{\Sigma_r}(E))$ (def. 3.18, example 3.44) which to a smooth family on field histories

$$
\Phi_{(-)}(-) \;\colon\; U \times N_\Sigma \Sigma_r \longrightarrow E
$$

assigns the differential form given by first forming the pullback of differential forms along the family of jet prolongation $j^\infty_\Sigma(\Phi_{(-)})$ followed by the integration of differential forms over $\Sigma_r$:

$$
(\tau_{\Sigma}A)_\Phi
\;:=\;
\int_{\Sigma_r} (j^\infty_\Sigma(\Phi_{(-)}))^\ast A
\;\in\;
\Omega^\bullet(U)
\,.
$$

Remark 7.33. (transgression to dimension $r$ picks out horizontal $r$-forms)

In def. 7.32
we regard integration of differential forms over $\Sigma_r$ as an operation defined on differential forms of all degrees, which vanishes except on forms of degree $r$, and hence transgression of variational differential forms to $\Sigma_r$ vanishes except on the subspace

$$
\Omega^{r,\bullet}_\Sigma(E)
\;\subset\;
\Omega^{\bullet,\bullet}_\Sigma(E)
$$

of forms of horizontal degree $r$.

Example 7.34. (adiabatically switched action functional)

Given a field bundle $E \overset{fb}{\longrightarrow} \Sigma$, consider a local Lagrangian density (def. 5.1)

$$
\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)
\,.
$$

For any bump function $b \in C^\infty_{cp}(\Sigma)$, the transgression of $b \mathbf{L}$ (def. 7.32) is called the action functional

$$
\mathcal{S}_b \mathbf{L}
:=
\tau_{\Sigma}
\left(
b \mathbf{L}
\right)
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\mathbb{R}
$$

induced by $\mathbf{L}$, “adiabatically switched” by $b$.

Specifically if the field bundle is a trivial vector bundle as in example 3.4, such that the Lagrangian density may be written in the form

$$
\mathbf{L}
\;=\;
L
\left(
(x^\mu), (\phi^a), (\phi^a_{,\mu}),
\cdots
\right)
\,
b dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_{\Sigma,cp}( E )
\,.
$$

then its action functional takes a field history $\Phi$ to the value

$$
\mathcal{S}_{b \mathbf{L}}(\Phi)
\:\colon\;
\int_\Sigma
L
\left(
x,
\left( \Phi^a(x) \right),
\left(\frac{\partial \Phi^a}{\partial x^\mu}(x)\right),
\cdots
\right)
\,
b(x)
dvol_\Sigma(x)
$$

Proposition 7.35. (transgression compatible with variational derivative)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. 3.1) and let $\Sigma_r \hookrightarrow \Sigma$ be a submanifold possibly with boundary $\partial \Sigma_r \hookrightarrow \Sigma_r$. Write

$$
\Gamma_{\Sigma_r}(E) \overset{(-)\vert_{\partial \Sigma_r}}{\longrightarrow} \Gamma_{\partial \Sigma_r}(E)
$$

for the boundary restriction map.

Then the operation of transgression of variational differential forms (def. 7.32)

$$
\tau_{\Sigma} \;\colon\; \Omega^{\bullet,\bullet}_{\Sigma,cp}(E) \longrightarrow \Omega^\bullet\left(\Gamma_{\Sigma_r}(E)\right)
$$

is compatible with the variational derivative $\delta$ and with the total spacetime derivative $d$ in the following way:

1. On variational forms that are in the image of the total spacetime derivative a transgressive variant of the Stokes’ theorem (prop. 1.25) holds:$$
   \tau_{\Sigma_r}(d \alpha) \;=\; ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma_r}( \alpha)
   $$
2. Transgression intertwines, up to a sign, the variational derivative $\delta$ on variational differential forms with the plain de Rham differential on the space of field histories:$$
   \tau_{\Sigma}\left(
   \delta \alpha
   \right)
   \;=\;
   (-1)^{p+1}\, d \,\tau_{\Sigma}(\alpha)
   \,.
   $$

Proof. Regarding the first statement, consider a horizontally exact variational form

$$
d \alpha \in \Omega^{r,s}_{\Sigma,cp}(E)
\,.
$$

By prop. 4.13 the pullback of this form along the jet prolongation of fields is exact in the $\Sigma$-direction:

$$
(j^\infty_\Sigma\Phi_{(-)})^\ast(d \alpha )
\;=\;
d_\Sigma (j^\infty_\Sigma\Phi_{(-)})^\ast \alpha
\,,
$$

(where we write $d = d_U + d_\Sigma$ for the de Rham differential on $U \times \Sigma$). Hence by the ordinary Stokes’ theorem (prop. 1.25) restricted to any $\Phi_{(-)} \colon U \to \Gamma_{\Sigma_r}(E)$ with restriction $(-)\vert_{\partial \Sigma_r} \circ \Phi_{(-)} \colon U \to \Gamma_{\Sigma_r}(E)$ the relation

$$
\begin{aligned}
(\Phi_{(-)})^\ast \tau_{\Sigma_r}(d \alpha)
& =
\int_{\Sigma_r}
d_{\Sigma_r} (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha
\\
& = \int_{\partial \Sigma_r} (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha
\\
& = \int_{\partial \Sigma_r} (j^\infty_\Sigma ( (-)\vert_\Sigma \circ \Phi_{(-)}) )^\ast\alpha
\\
& =
( (-)\vert_\Sigma \circ \Phi_{(-)} )^\ast \tau_{\partial \Sigma_r}(\alpha)
\\
& =
(\Phi_{(-)})^\ast ((-)\vert_{\Sigma_r})^\ast \tau_{\partial \Sigma_r}(\alpha)
\,.
\end{aligned}
\,.
$$

Regarding the second statement: by the Leibniz rule for de Rham differential (product law of differentiation>) it is sufficient to check the claim on variational derivatives of local coordinate functions

$$
\delta \phi^a_{\mu_1 \cdots \mu_k} b
\in
\Omega^{0,1}_\Sigma(E)
\,.
$$

The pullback of differential forms (prop. 1.21) along the jet prolongation $j^\infty_\Sigma(\Phi_{(-)}) \colon U \times \Sigma \to J^\infty_\Sigma(E)$ has two contributions: one from the variation along $\Sigma$, the other from variation along $U$:

1. By prop. 4.13, for fixed $u \in U$ the pullback of $\delta \phi^a_{\mu_1 \cdots \mu_k}$ along the jet prolongation vanishes.
2. For fixed $x \in \Sigma$, the pullback of the full de Rham differential $\mathbf{d} \phi^a_{\mu_1\cdots \mu_k}$ is$$
   \begin{aligned}
   (\Phi_{(-)}(x))^\ast( \mathbf{d} \phi^a_{\mu_1\cdots \mu_k} )
   & =
   d_U (\Phi_{(-)}(x))^\ast(\phi^a_{\mu_1\cdots \mu_k})
   \\
   & =
   d_U \frac{ \partial^k \Phi_{(-)}(x)}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}
   \end{aligned}
   $$(since the full de Rham differentials always commute with pullback of differential forms by prop. 1.21), while the pullback of the horizontal derivative $d \phi^a_{\mu_1\cdots \mu_k} = \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \mathbf{d}x^{\mu_{k+1}}$ vanishes at fixed $x \in \Sigma$.

This implies over the given smooth family $\Phi_{(-)}$ that

$$
\begin{aligned}
\tau_\Sigma\left(
\delta \phi^a_{,\mu_1 \cdots \mu_k} b
\right)\vert_{\Phi_{(-)}}
& =
\tau_\Sigma\left(
\mathbf{d} ( \phi^a_{,\mu_1 \cdots \mu_k} b)
\right)
\vert_{\Phi_{(-)}}
–
\underset{ = 0 }{
\underbrace{
\tau_\Sigma
\left(
d (\phi^a_{,\mu_1 \cdots \mu_k} b)
\right)\vert_{\Phi_{(-)}}
}}
\\
& =
\int_\Sigma d_U (\Phi_{(-)})^\ast ( \phi^a_{\mu_1\cdots \mu_k} b )
\\
& = (-1)^{p+1} d_U \int_\Sigma (\Phi_{(-)})^\ast ( \phi^a_{\mu_1\cdots \mu_k} b )
\\
& = (-1)^{p+1} d_U \tau_{\Sigma}( \Phi_{(-)} )^\ast ( \phi^a_{\mu_1 \cdots \mu_k} )
\,.
\end{aligned}
$$

and since this holds covariantly for all smooth families $\Phi_{(-)}$, this implies the claim.

Example 7.36. (cohomological integration by parts on the jet bundle)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle (def. 3.1).

Prop. 7.35 says in particular that the operation of integration by parts in an integral is “localized” to a cohomological statement on horizontal differential forms: Let

$$
\alpha_1, \alpha_2 \;\in\; \Omega^{\bullet,\bullet}_\Sigma(E)
$$

be two variational differential forms (def. 4.11), of total horizontal degree $p$ (hence one less than the dimension of spacetime $\Sigma$).

Then the derivation-property of the total spacetime derivative says that

$$
\label{IntegrationByPartsCohomologicallyOnJetBundle}
(d \alpha_1) \wedge \alpha_2
\;=\;
– (-1)^{deg(\alpha_1)} \alpha_1 \wedge ( d \alpha_2 )
\;\;
d( \alpha_1 \wedge \alpha_2 )
\;\in\;
\Omega^{p+1,\bullet}_\Sigma(E)
\,,
$$ (103)

hence that we may “throw over” the spacetime derivative from the factor $\alpha_1$ to the factor $\alpha_2$, up to a sign, and up to a total spacetime derivative $d (\alpha_1 \wedge \alpha_2)$. By prop. 7.35 this last term vanishes under transgression $\tau_\sigma$ to a spacetime without manifold with boundary, so that the above equation becomes

$$
\tau_\Sigma( d \alpha_1) \wedge \alpha_2 )
\;=\;
– (-1)^{deg(\alpha_1)}
\tau_\Sigma( \alpha_1 \wedge d \alpha_2 )
\,,
$$

hence

$$
\underset{\Sigma}{\int}
(d j^\infty_\sigma(\alpha_1)) \wedge j^\infty_\Sigma(\alpha_2)
\;=\;
– (-1)^{deg(\alpha_1)}
\underset{\Sigma}{\int}
j^\infty_\Sigma(\alpha_1) \wedge d j^\infty_\Sigma(\alpha_2)
\,.
$$

This last statement is the statement of integration by parts under an integral.

Notice that these integrals (and hence the actual integration by parts-rule) only exist if $\alpha_1 \wedge \alpha_2$ has compact spacetime support, while the “cohomological” avatar (103) of this relation on the jet bundle holds without such a restriction.

Example 7.37. (variation of the action functional)

Given a Lagrangian field theory $(E,\mathbf{L})$ (def. 5.1) then the derivative of its adiabatically switched action functional (def. 7.34) equals the transgression of the Euler-Lagrange variational derivative $\delta_{EL} \mathbf{L}$ (def. 5.12):

$$
d \mathcal{S}_{b \mathbf{L}}
\;=\;
\tau_\Sigma( b \delta_{EL}\mathbf{L} )
\,.
$$

Proof. By the second statement of prop. 7.35 we have

$$
\begin{aligned}
d \mathcal{S}_{b \mathbf{L}}
& =
\tau_\Sigma( \delta ( b \mathbf{L} ) )
\end{aligned}
\,,
$$

Moreover, by prop. 5.12 this is

$$
\begin{aligned}
\cdots
& =
\tau_\Sigma( \delta_{EL} b \mathbf{L} + d \Theta_{BFV,b} )
\\
& =
\tau_\Sigma( \delta_{EL} b \mathbf{L} ) + \underset{= 0}{\underbrace{\tau_\Sigma( d \Theta_{BFV,b} )}}
\end{aligned}
\,,
$$

where the second term vanishes by the first statement of prop. 7.35.

Proposition 7.38. (principle of extremal action)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1).

The de Rham differential $d \mathcal{S}_{b\mathbf{L}}$ of the action functional (example 7.37) vanishes at a field history

$$
\Phi \in \Gamma_\Sigma(E)
$$

for all adiabatic switchings $b \in C^\infty_{cp}(\Sigma)$ constant on some subset $\mathcal{O} \subset \Sigma$ (def. 2.39) on those smooth collections of field histories

$$
\Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E)
$$

around $\Phi$ which, as functions on $U$, are constant outside $\mathcal{O}$ (example 3.12, example 3.46) precisely if $\Phi$ solves the Euler-Lagrange equations of motion (def. 5.24):

$$
\left(
\underset{ { {\mathcal{O} \subset \Sigma} \atop { b\vert_{\mathcal{O}} = const } } \atop { \Phi_{(-)}\vert_{\Sigma \setminus \mathcal{O}} = const } }{\forall}
\left(
(\Phi_{(-)})^\ast d \mathcal{S}_{b \mathbf{L}}(\Phi) = 0
\right)
\right)
\;\Leftrightarrow\;
\left(
j^\infty_\Sigma(\Phi)^\ast \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) = 0
\right)
\,.
$$

Proof. By prop. 7.35 we have

$$
(\Phi_{(-)})^\ast d \mathcal{S}_{b \mathbf{L}}
\;=\;
\int_\Sigma
j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} b \mathbf{L} )
\,.
$$

By the assumption on $\Phi_{(-)}$ it follows that after pullback to $U$ the switching function $b$ is constant, so that it commutes with the differentials:

$$
(\Phi_{(-)})^\ast
d \mathcal{S}_{b \mathbf{L}}
\;=\;
\int_\Sigma
b
j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} \mathbf{L} )
\,.
$$

This vanishes at $\Phi$ for all $\Phi_{(-)}$ precisely if all components of $j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} \mathbf{L} )$ vanish, which is the statement of the Euler-Lagrange equations of motion.

Definition 7.39. (local observables)

Given a Lagrangian field theory $(E,\mathbf{L})$ (def. 5.1) the local observables are the horizontal p+1-forms

1. of compact spacetime support (def. 7.31)
2. modulo total spacetime derivatives

$$
LocObs(E)
\;:=\;
\left(\Omega^{p+1,0}_{\Sigma,cp}(E)/(im(d))\right)\vert_{\mathcal{E}^\infty}
$$

which we may identify with the subspace of all observables link on those that arise as the image under transgression of variational differential forms $\tau_\Sigma$ (def. 7.32) of local observables to functionals on the on-shell space of field histories (65):

$$
\array{
LocObs(E)
&\overset{\tau_\Sigma}{\hookrightarrow}&
Obs(E)
\\
\alpha &\maptos& \tau_\Sigma \alpha
}
\,.
$$

This is a sub-vector space inside all observables which is however not closed under the pointwise product of observables (86) (unless $E =0). We write

$$
MultiLocObs(E)
\hookrightarrow
Obs(E)
$$

for the smallest subalgebra of observables, under the pointwise product (86), that contains all the local observables. This is called the algebra of multilocal observables.

The intersection of the (multi-)local observables with the off-shell polynomial observables (def. 7.13) are the (multi-)local polynomial observables

$$
\label{InclusionOfPolynomialLocalObservablesIntoPolynomialObservables}
PolyLocObs(E)
\hookrightarrow
PolyMultiLocObs(E)
\overset{\text{dense}}{\hookrightarrow}
PolyObs(E)
\hookrightarrow
Obs(E)
$$ (104)

Example 7.40. (local observables of the real scalar field)

Consider the field bundle of the real scalar field (example 3.5).

A typical example of local observables (def. 7.39) in this case is the “field amplitude averaged over a given spacetime region” determined by a bump function $b \in C^\infty_{cp}(\Sigma)$. On an on-shell field history $\Phi$ this observable takes as value the integral

$$
\tau_\Sigma(b \phi)(\Phi) \;=\; \int_\Sigma \Phi(x) b(x) dvol_\Sigma(x)
\,.
$$

Example 7.41. (local observables of the electromagnetic field)

Consider the field bundle for free electromagnetism on Minkowski spacetime $\Sigma$.

Then for $b \in C^\infty(\Sigma)$ a bump function on spacetime, the transgression of the universal Faraday tensor (def. 4.4) against $b$ times the volume form is a local observable (def. 7.39), namely the field strength (20) of the electromagnetic field averaged over spacetime.

For the construction of the algebra of quantum observables it will be important to notice that the intersection between local observables and regular polynomial observables is very small:

Example 7.42. (local regular polynomial observables are linear observables)

An observable (def. 7.1) which is

1. a regular polynomial observable (def. 7.13);
2. a local observable (def. 7.39)

is necessarily

• a linear observable (def. 7.3).

This is because non-linear local expressions are polynomials in the sense of def. 7.13
with delta distribution-coefficients, for instance for the real scalar field the $\Phi^2$ interaction term is

$$
\int (\Phi(x))^2 \, dvol_\Sigma(x)
\;=\;
\int \int \Phi(x) \Phi(y) \underset{ = \alpha^{(2)}(x,y) }{\underbrace{\delta(x-y)}} \, dvol_\Sigma(y)
$$

and so its coefficient $\alpha^{(2)}$ is manifestly not a non-singular distribution.

Infinitesimal observables

The definition of observables in def. 7.1 and specifically of local observables in def. 7.39 uses explicit restriction to the shell, hence, by the principle of extremal action (prop. 7.38) to the “critical locus” of the action functional. Such critical loci are often hard to handle explicitly. It helps to consider a “homological resolution” that is given, in good circumstances, by the corresponding “derived critical locus”. These we consider in detail below in Reduced phase space. In order to have good control over these resolutions, we here consider the first perturbative aspect of field theory, namely we consider the restriction of local observables to just an infinitesimal neighbourhood of a background on-shell field history:

Definition 7.43. (local observables around infinitesimal neighbourhood of background on-shell field history)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) whose field bundle $E$ is a trivial vector bundle (example 3.4) and whose Lagrangian density $\mathbf{L}$ is spacetime-independent (example 5.14). Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (57) as in example 5.14.

Then we write

$$
LocObs_\Sigma(E,\varphi)
$$

for the restriction of the local observables (def. 7.39) to the fiberwise infinitesimal neighbourhood (example 3.30) of $\Sigma \times \{\varphi\}$.

Explicitly, this means the following:

First of all, by prop. 4.6 the dependence of the Lagrangian density $\mathbf{L}$ on the order of field derivatives is bounded by some $k \in \mathbb{N}$ on some neighbourhood of $\varphi$ and hence, by the spacetime independence of $\mathbf{L}$, on some neighbourhood of $\Sigma \times \{\varphi\}$.

Therefore we may restrict without loss to the order-$k$ jets. By slight abuse of notation we still write

$$
\mathcal{E} \hookrightarrow J^k_\Sigma(E)
$$

for the corresponding shell. It follows then that the restriction of the ring $\Omega^{0,0}_{\Sigma,cp}(E)$ of smooth functions on the jet bundle to the infinitesimal neighbourhood (example 3.30) is equivalently the formal power series ring over $C^\infty_{cp}(\Sigma)$ in the variables

$$
((\phi^a- \varphi^a),
(\phi^a_{,\mu}- \varphi^a_{,\mu}),
\cdots,
(\phi^a_{,\mu_1 \cdots \mu_k} – \varphi^a_{,\mu_1 \cdots \mu_k})
)
$$

We denote this by

$$
\label{FunctionsOnInfNbh}
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\;:=\;
C^\infty_{cp}(\Sigma)\left[ \left[ (\phi^a – \varphi^a ), (\phi^a_{,\mu} -\varphi^a_{,\mu}), \cdots, (\phi^a_{,\mu_1 \cdots \mu_k}- \varphi^a_{,\mu_1 \cdots \mu_k}) \right] \right]
\,.
$$ (105)

A key consequence is that the further restriction of this ring to the shell $\mathcal{E}^\infty$ (50) is now simply the further quotient ring by the ideal generated by the total spacetime derivatives of the components $\frac{\partial_{EL}L}{\delta \phi^a}$ of the Euler-Lagrange form (prop. 5.12).

$$
\label{ObservablesOnInfinitesimalNeighbourhoodOfZeroInShellInFieldFiber}
\begin{aligned}
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}
& :=
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
/
\left(
\frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\right)_{ { a \in \{1, \cdots, s\} } \atop { { l \in \{1, \cdots, k\} } \atop { \mu_r \in \{0, \cdots, p\} } } }
\\
& =
C^\infty_{cp}(\Sigma)\left[ \left[ (\phi^a – \varphi^a ), (\phi^a_{,\mu} -\varphi^a_{,\mu}), \cdots, (\phi^a_{,\mu_1 \cdots \mu_k}- \varphi^a_{,\mu_1 \cdots \mu_k}) \right] \right]
/
\left(
\frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\right)_{ { a \in \{1, \cdots, s\} } \atop { { l \in \{1, \cdots, k\} } \atop { \mu_r \in \{0, \cdots, p\} } } }
\end{aligned}
\,.
$$ (106)

Finally the local observables restricted to the infinitesimal neighbourhood is the module

$$
\label{LocalObservablesRestrictedToInfinitesimalNeighbourhood}
LocObs_\Sigma(E,\varphi)
\;\simeq\;
\left(
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}} \langle dvol_\Sigma \rangle
\right)/(im(d))
\,.
$$ (107)

The space of local observables in def. 7.43 is the quotient of a formal power series algebra by the components of the Euler-Lagrange form and by the image of the horizontal spacetime de Rham differential. It is convenient to also conceive of the components of the Euler-Lagrange form as the image of a differential, for then the algebra of local observables obtaines a cohomological interpretation, which will lend itself to computation. This differential, whose image is the components of the Euler-Lagrange form, is called the _BV-differential. We introduce this now first (def. 7.44 below) in a direct ad-hoc way. Further below we discuss the conceptual nature of this differential as part of the construction of the reduced phase space as a derived critical locus (example 11.22 below).

Definition 7.44. (local BV-complex of ordinary Lagrangian density)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. 5.1) whose field bundle $E$ is a trivial vector bundle (example 3.4) and whose Lagrangian density $\mathbf{L}$ is spacetime-independent (example 7.43). Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}^\infty$ be a constant section of the shell (57).

In correspondence with def. 7.43, write

$$
\Gamma_{\Sigma,cp}(T_\Sigma J^\infty_\Sigma E,\varphi)
\simeq
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)
\;\in\;
\Omega^{0,0}_{\Sigma,cp}(E) Mod
$$

for the restriction of vertical vector fields on the jet bundle to the fiberwise infinitesimal neighbourhood (example 3.30) of $\Sigma \times {\varphi}$.

Now we regard this as a graded module over $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ (105) concentrated in degree $-1$:

$$
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\;\in\;
\Omega^{0,0}_{\Sigma,cp}(E) Mod^{\mathbb{Z}}
\,.
$$

This is called the module of antifields corresponding the given type of fields encoded by $E$.

If the field bundle is a trivial vector bundle (example 3.4) with field coordinates $(\phi^a)$, then we write

$$
\label{AntifieldCoordinates}
\phi^{\ddagger}_{a,\mu_1 \cdots \mu_l}
\;:=\;
\left(
\partial_{(\phi^a_{\mu_1 \cdots \mu_l})}
\right)[-1]
\;\in\;
\Gamma_{\Sigma,cp}(T_\Sigma J^\infty_\Sigma E,\varphi)[-1]
$$ (108)

for the vector field generator that takes derivatives along $\partial_{\phi^a_{,\mu_1 \cdots \mu_k}}$, but regarded now in degree -1.

Evaluation of vector fields in the total spacetime derivatives $\frac{d^l}{d x^{\mu_1} \cdots d x^{\mu_l}} \delta\mathbf{L} \in \Omega^{p,0}_\Sigma(E) \wedge \delta \Omega^{0,0}_\Sigma(E)$ of the variational derivative (prop. 5.12) yields a linear map over $\Omega^{\bullet,\bullet}_{\Sigma,cp}(E,\varphi)$ (106)

$$
\iota_{(-)}\delta_{EL} \mathbf{L}
\;\colon\;
\Gamma_{\Sigma,cp}( J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\longrightarrow
\Omega^{p+1,0}_{\Sigma,cp}(E,\varphi)
\,.
$$

If we use the volume form $dvol_\Sigma$ on spacetime $\Sigma$ to induce an identification

$$
\Omega^{p+1,0}_\Sigma(E) \;\simeq\; C^\infty(J^\infty_\Sigma(E))\langle dvol_\sigma\rangle
$$

with respect to which the Lagrangian density decomposes as

$$
\mathbf{L} = L dvol_\Sigma
$$

then this is a $\Omega^{0,0}_\sigma(E,\varphi)$-linear map of the form

$$
\iota_{(-)}{\delta L_{EL}}
\;\colon\;
\Gamma_{\Sigma,cp}^{ev}(T_\Sigma E,\varphi)[-1]
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\,.
$$

In the special case that the field bundle $E \overset{fb}{\to} \Sigma$ is a trivial vector bundle (example 3.4) with field coordinates $(\phi^a)$ so that the Euler-Lagrange form has the coordinate expansion

$$
\delta_{EL} \mathbf{L}
\;=\;
\frac{\delta_{EL}\mathbf{L}}{\delta \phi^a} \delta \phi^a
$$

then this map is given on the antifield basis elements (108) by

$$
\iota_{(-)} {\delta L_{EL}}
\;\colon\;
\phi^{\ddagger}_{a,\mu_1 \cdots \mu_l}
\;\mapsto\;
\frac{d^l}{d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\,.
$$

Consider then the graded symmetric algebra

$$
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\;:=\;
Sym_{\Omega^{0,0}_{\Sigma,cp}(E,\varphi)}\left(
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\right)
$$

which is generated over $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ from the module of vector fields in degree -1.

If we think of a single vector field as a fiber-wise linear function on the cotangent bundle, and of a multivector field similarly as a multilinear function on the cotangent bundle, then we may think of this as the algebra of functions on the infinitesimal neighbourhood (example 3.30) of $\varphi$ inside the graded manifold $(T_\Sigma E)[-1] \times_\Sigma E$.

Let now

$$
\label{BVDifferentialForOrdinaryLagrangian}
s_{BV}
\;\colon\;
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\;\longrightarrow\;
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
$$ (109)

be the unique extension of the linear map $\iota_{(-)}{\delta_{EL} L}$ to an $\mathbb{R}$-linear derivation of degree +1 on this algebra.

The resulting differential graded-commutative algebra over $\mathbb{R}$

$$
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}
\;:=\;
\left(
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\,,\,
s_{BV}
\right)
$$

is called the local BV-complex of the Lagrangian field theory at the background solution $\varphi$. This is the CE-algebra of the infintiesimal neighbourhood of $\Sigma \times \{\varphi\}$ in the derived prolonged shell (def. 11.20). In this case, in the absence of any explicit infinitesimal gauge symmetries, this is an example of a Koszul complex.

There are canonical homomorphisms of dgc-algebras, one from the algebra of functions $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ on the infinitesimal neighbourhood of the background solution $\varphi$ to the local BV-complex and from there to the local observables on the neighbourhood of the background solution $\varphi$ (106), all considered with compact spacetime support:

$$
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}
$$

such that the composite is the canonical quotient coprojection.

Similarly we obtain a factorization for the entire variational bicomplex:

$$
\label{ComparisonMorphismFromOrdinaryBVComplexToLocalObservables}
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)
\longrightarrow
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}
\longrightarrow
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}}
\,,
$$ (110)

where $\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}$ is now triply graded, with three anti-commuting differentials $d$ $\delta$ and $s_{BV}$.

By construction this is now such that the local observables (def. 7.39) are the cochain cohomology of this complex in horizontal form degree p+1, vertical degree 0 and BV-degree 0:

$$
LocObs_\Sigma(E) \simeq \Omega^{p+1,0}_{\Sigma,cp}(E)/(im(s_{BV} + d))
\,.
$$

States

We introduce the basics of quantum probability in terms of states defined as positive linear maps on star-algebras of observables.

Definition 7.45. (star algebra)

A star ring is a ring $R$ equipped with

• a linear map$(-)^\ast \;\colon\; R \longrightarrow R$

such that

• (involution) $((-)^\ast)^\ast = id$;
• (antihomomorphism)
  1. $(a b)^\ast = b^\ast a^\ast$ for all $a,b \in R$
  2. $1^\ast = 1$.

A homomorphism of star-rings

$$
f \;\colon\; (R_1, (-)^\ast) \longrightarrow (R_2, (-)^\dagger)
$$

is a homomorphism of the underlying rings

$$
f \;\colon\; R_1 \longrightarrow R_2
$$

which respects the star-involutions in that

$$
f \circ (-)^\ast \;=\; (-)^\dagger \circ f
\,.
$$

A star algebra $\mathcal{A}$ over a commutative star-ring $R$ in an associative algebra $\mathcal{A}$ over $R$ such that the inclusion

$$
R \hookrightarrow \mathcal{A}
$$

is a star-homomorphism.

Examples 7.46. (complex number-valued observables are star-algebra under pointwise product and pointwise complex conjugation)

The complex numbers $\mathbb{C}$ carry the structure of a star-ring (def. 7.45) with star-operation given by complex conjugation.

Given any space $X$, then the algebra of functions on $X$ with values in the complex numbers carries the structure of a star-algebra over the star-ring $\mathbb{C}$ (def. 7.45) with star-operation given by pointwise complex conjugation in the complex numbers.

In particular for $(E,\mathbf{L})$ a Lagrangian field theory (def. 5.1) then its on-shell observables $Obs(E,\mathbf{L})$ (def. 7.1) carry the structure of a star-algebra this way.

Definition 7.47. (state on a star-algebra)

Given a star algebra $(\mathcal{A}, (-)^\ast)$ (def. 7.45) over the star-ring of complex numbers (def. 7.46) a state is a function to the complex numbers

$$
\langle -\rangle
\;\colon\;
Obs_\Sigma \longrightarrow \mathbb{C}
$$

such that

1. (linearity) this is a complex-linear map:$$
   \left\langle
   c_1 A_1 + c_2 A_2
   \right\rangle
   \;=\;
   c_1 \langle A_1 \rangle + c_2 \langle A_2 \rangle
   $$
2. (positivity) for all $A \in Obs$ we have that$$
   \langle A^\ast A \rangle \geq 0 \;\in\; \mathbb{R}
   $$where on the left $A^\ast$ is the star-operation from
3. (normalization)$$
   \langle 1 \rangle \;=\; 1
   \,.
   $$

(e.g. Bordemann-Waldmann 96, Fredenhagen-Rejzner 12, def. 2.4, Khavkine-Moretti 15, def. 6)

Remark 7.48. (probability theoretic interpretation of state on a star-algebra)

A star algebra $\mathcal{A}$ (def. 7.45) equipped with a state $\mathcal{A} \overset{\langle -\rangle}{\longrightarrow} \mathbb{C}$ (def. 7.47) is also called a quantum probability space, at least when $\mathcal{A}$ is in fact a von Neumann algebra.

For this interpretation we think of each element $A \in \mathcal{A}$ as an observable as in example 7.46 and of the state as assigning expectation values.

Remark 7.49. (states form a convex set)

For $\mathcal{A}$ a unital star-algebra (def. 7.45), the set of states on $\mathcal{A}$ according to def. 7.47 is naturally a convex set: For $\langle (-)\rangle_1, \langle – \rangle_2 \colon \mathcal{A} \to \mathbb{C}$ two states then for every $p \in [0,1] \subset \mathbb{R}$ also the linear combination

$$
\array{
\mathcal{A}
&\overset{p \langle (-)\rangle_1 + (1-p) \langle (-)\rangle_2}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& p \langle A \rangle_1 + (1-p) \langle A \rangle_2
}
$$

is a state.

Definition 7.50. (pure state)

A state $\rho \colon \mathcal{A} \to \mathbb{C}$ on a unital star-algebra (def. 7.47) is called a pure state if it is extremal in the convex set of all states (remark 7.49) in that an identification

$$
\langle (- )\rangle = p \langle (-)\rangle_1 + (1-p) \langle (-)\rangle_2
$$

for $p \in (0,1)$ implies that $\langle (-) \rangle_1 = \langle (-)\rangle_2$ (hence $= \langle (-)\rangle$).

Proposition 7.51. (classical probability measure as state on measurable functions)

For $\Omega$ classical probability space, hence a measure space which normalized total measure $\int_\Omega d\mu = 1$, let $\mathcal{A} \cloneqq L^1(\Omega)$ be the algebra of Lebesgue measurable functions with values in the complex numbers, regarded as a star algebra (def. 7.45) by pointwise complex conjugation as in example 7.46. Then forming the expectation value with respect to $\mu$ defines a state (def. 7.47):

$$
\array{
L^1(\Omega)
&\overset{\langle (-)\rangle_\mu}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& \int_\Omega A d\mu
}
$$

Example 7.52. (elements of a Hilbert space as pure states on bounded operators)

Let $\mathcal{H}$ be a complex separable Hilbert space with inner product $\langle -,-\rangle$ and let $\mathcal{A} := \mathcal{B}(\mathcal{H})$ be the algebra of bounded operators, regarded as a star algebra (def. 7.45) under forming adjoint operators. Then for every element $\psi \in \mathcal{H}$ of unit norm $\langle \psi,\psi\rangle = 1$ there is the state (def. 7.47) given by

$$
\array{
\mathcal{B}(\mathcal{H})
&\overset{\langle (-)\rangle_\psi}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &:=& \langle \psi, A \psi \rangle
}
$$

These are pure states (def. 7.50).

More general states in this case are given by density matrices.

Theorem 7.53. (GNS construction)

Given

1. a star-algebra, $\mathcal{A}$ (def. 7.45);
2. a state, $\langle (-)\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$ (def. 7.47)

there exists

1. a star-representation$$
   \pi
   \;\colon\;
   \mathcal{A}
   \longrightarrow
   End(\mathcal{H})
   $$of $\mathcal{A}$ on some Hilbert space $\mathcal{H}$
2. a cyclic vector $\psi \in \mathcal{H}$

such that $\langle (-)\rangle$ is the state corresponding to $\psi$ via example 7.52, in that

$$
\begin{aligned}
\langle A \rangle
& = \langle \psi \vert\, A \, \vert \psi \rangle
\\
& :=
\langle \psi , \pi(A) \psi \rangle
\end{aligned}
$$

for all $A \in \mathcal{A}$.

(Khavkine-Moretti 15, theorem 1)

Definition 7.54. (classical state)

Given a Lagrangian field theory $(E,\mathbf{L})$ (def. 5.1) then a classical state is a state on the star algebra (def. 7.47) of on-shell observables (example 7.46):

$$
\langle -\rangle
\;\colon\;
Obs(E,\mathbf{L})
\longrightarrow
\mathbb{C}
\,.
$$

Below we consider quantum states. These are defined just as in def. 7.54, only that now the algebra of observables is equipped with another product, which changes the meaning of the product expression $A^\ast A$ and hence the positivity condition in def. 7.47.

This concludes our discussion of observables. In the next chapter we consider the construction of the covariant phase space and of the Poisson-Peierls bracket on observables.