**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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*.

**types of observables in perturbative quantum field theory**:

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

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:

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

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

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.

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. ).

**(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*.

Last revised on February 8, 2020 at 11:06:55. See the history of this page for a list of all contributions to it.