Contents

# Contents

## Idea

Give a field theory (for instance a Lagrangian field theory) whose field bundle is a vector bundle, then the space of sections of the field bundle, hence the space of field histories, is canonically a vector space and hence it makes sense to consider observables which are linear functions, or quadratic functions of the field histories, etc. A sum of such is a polynomial observable.

Since linear smooth observables are compactly supported distributions (see at distributions are the smooth linear functionals) polynomial observables are sums of diagonals of distributions of several variables.

If all these distributional coefficients are non-singular distributions one speaks of regular observables.

Of all the distributional coefficients at order $k$ satisfy the condition that their wave front sets exclude the subsets where all $k$ wave vectors are all in the future cone or all in the past cone, then one speaks of microcausal observables.

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

## Definition

###### Definition

(polynomial observable)

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

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

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

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

where

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

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

Write

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

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

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

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

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

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

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

Various special cases:

###### Definition

(linear observable)

A linear observable is a polynomial observable (def. ) all whose coefficients except possible for $\alpha^{(1)}$ are zero.

###### Definition

(regular observable)

A regular observable is a polynomial observable (def. ) all whose coefficients $\alpha^{(k)}$ are non-singular distributions.

###### Definition

(microcausal observable)

For $\Sigma$ a spacetime, hence a Lorentzian manifold with time orientation, then a microcausal observable is a polynomial observable (def. ) such that each coefficient $\alpha^{(k)}$ has wave front set excluding those points where all $k$ wave vectors are in the future cone or all in the past cone.

## Examples

###### Example

(field observables)

If the field bundle is a vector bundle, then the field observables

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

are linear observables (def. ).

###### Example

(polynomial local observables are polynomial observables)

A local observable which comes form a horizontal differential form which is a polynomial in the fields and their jets times the volume form on spacetime is a polynomial observable.

These happen to be also microcausal observables (this example).

polynomiallocal observables $\hookrightarrow$ microcausal observables $\hookrightarrow$ polynomial observables $\hookrightarrow$ observables

See the references at microcausal observable.