examples of Poisson-Bracket notation:
*[(d)]
*[(f)]
* note how version (f) opts for supercripted indices for one set of canonical
coords. and subscripted for the other, while the (d) equivalent chooses
the simpler notation with all indices subscripted.
* In [Siegel] [Srednicki] [Aitchison] and others there are discussions on the
importance of the choice of indexing scheme used in the QFT commutator formalism
- which is already so complicated (;;) .
* Notation in another version (r) uses square brackets for the PB on the LHS,
eg. [f,g] , thus making it indistinguishable from the notation used to designate
the related quantum commutator, the Lie Bracket [a,b]. (Cf. introduction of the
commutator in [Shankar])
* Indeed in __Fields__, Siegel devotes §§A,B in the Symmetry chapter to
"Coordinates" and "Indices" resp. [Siegel]
-- 24.xi.2008
09 December 2008
note on notation and choice of indices
05 December 2008
linearity ≅ whole=∑parts?
In general, the main condition or step in proving a set to be a linear space is to show that for any two members f, g of the set under consideration, and for any two reals α,β
As an illusration, consider an operator qcq , L: D → ℝℝ , where D is some set of functions , and ℝℝ is the set of all real valued functions of a real variable.
This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.
This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.
The ubiquity of linearity in the abstract algebra by which we represent natural systems** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).
** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described.
a. the linear combination αf + βg is in the set
b. this satisfies closure, from whence follow the remaining axioms
if applicable.
As an illusration, consider an operator qcq , L: D → ℝℝ , where D is some set of functions , and ℝℝ is the set of all real valued functions of a real variable.
L is linear, ie, is a linear space if , ∀y∈D, ∀z∈D, ∀α∈ℝ , ∀β∈ℝ
L(αy+βz)= αL(y)+βL(z)
This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.
This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.
The ubiquity of linearity in the abstract algebra by which we represent natural systems** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).
** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described.
