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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2017
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   Author: Urs
   Format: MarkdownItexAm starting _[[Green hyperbolic differential equation]]_ from * [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv:1402.1282](https://arxiv.org/abs/1402.1282)) So far I have the definition and then the statement of the first remarkable proposition from this article: [here](https://ncatlab.org/nlab/show/Green+hyperbolic+partial+differential+equation#Properties).

   Am starting Green hyperbolic differential equation from

   • Igor Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv:1402.1282)

   So far I have the definition and then the statement of the first remarkable proposition from this article: here.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2017
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   Author: Urs
   Format: MarkdownItexadded now also the statement of Igor's lemma 2.5, together with some required infrastructure: [here](https://ncatlab.org/nlab/show/Green+hyperbolic+partial+differential+equation#PoissonToPPeierls)

   added now also the statement of Igor’s lemma 2.5, together with some required infrastructure: here

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2017
   • (edited Dec 27th 2017)
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   Author: Urs
   Format: MarkdownItexI have expanded the statement of the main propositon ([here](https://ncatlab.org/nlab/show/Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions)) making explicit the other isomorphism which previously only appeared in an intermediate step in the proof: $$ (ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast) $$ (For Green hyperbolic differential equations $P \Phi = 0$.) This is the one more frequently used, and I should have highlighted it earlier. This implies for instance that for free field theories with Green-hyperbolic equations of motion $P \Phi = 0$ , the on-shell polynomial observables are equivalently the off-shell polynomial observables modulo the image of $P$, which is used in the construction of the abstract Wick algebra. I have added this to _[[polynomial observables]]_ and to _[[microlocal observables]]_. Am working on bringing _[[Wick algebra]]_ into shape.

   I have expanded the statement of the main propositon (here) making explicit the other isomorphism which previously only appeared in an intermediate step in the proof:

   $$(ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast)$$

   (For Green hyperbolic differential equations $P \Phi = 0$.)

   This is the one more frequently used, and I should have highlighted it earlier. This implies for instance that for free field theories with Green-hyperbolic equations of motion $P \Phi = 0$ , the on-shell polynomial observables are equivalently the off-shell polynomial observables modulo the image of $P$, which is used in the construction of the abstract Wick algebra. I have added this to polynomial observables and to microlocal observables. Am working on bringing Wick algebra into shape.