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The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.

There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.

As for quantum mechanics, while I have only some general knowledge, I think "inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝn" is true, but somewhat too optimistic. Yes, there are quantizations, but they are canonical to all terms only when you restrict yourself to linear change of coordinates (or when you do some additional constructions).

I risk being viewed as stepping on a slippery stone here, but my limited understanding is that you're actually hitting a fundamental question here — the quantum field theory that plays completely nice with diffeomorphisms would be called quantum gravity and is the Holy Grail of high-energy physics rather then a simple exercise.

The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.

There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.

As for quantum mechanics, while I have only some general knowledge, I think "inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝn" is true, but somewhat too optimistic. Yes, there are quantizations, but they are canonical to all terms only when you restrict yourself to linear change of coordinates (or when you do some additional constructions).

I risk being viewed as stepping on a slippery stone here, but my limited understanding is that you're actually hitting a fundamental question here — even some quantum field theory play nice with diffeomorphisms, but it's far from a simple exercise. Moreover, the theory that includes them as the degrees of freedom would be called quantum gravity and is the Holy Grail of high-energy physics rather then a theorem of Lie group geometry (though the latter is extensively used for the former).

The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.

There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.

The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.

There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.

As for quantum mechanics, while I have only some general knowledge, I think "inverse of this Symbol map would determine a "quantization" of the functions on T*X, corresponding to the QP quantization of ℝn" is true, but somewhat too optimistic. Yes, there are quantizations, but they are canonical to all terms only when you restrict yourself to linear change of coordinates (or when you do some additional constructions).

I risk being viewed as stepping on a slippery stone here, but my limited understanding is that you're actually hitting a fundamental question here — the quantum field theory that plays completely nice with diffeomorphisms would be called quantum gravity and is the Holy Grail of high-energy physics rather then a simple exercise.

The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols, hence the Ben's answer.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system given by the exp map, I think (not sure here).

The definition of symbol as presented in Wikipedia is not invariant — only the highest order terms. Some textbooks call those higher order terms symbols (Wikipedia suggests the name principal symbol), hence the Ben's answer, which refers to that definition.

The highest order terms are clearly most important for the properties of the differential equations, e.g. their positiveness allows to prove the existence of solutions (it's related to the fact that positively definite linear operators are invertible in linear algebra).

As for "Thus the map Symbol defines a canonical vector-space (and in fact coalgebra) isomorphism UL → SL.", this statement should be proved by induction order-by-order in a fixed coordinate system. It should be true in any coordinate system, but the homomorphism depends on it.

There is however, a canonical coordinate system given by the exp map, and this, I think (not sure here), is the canonical map referred to in the question.