/sci/ - Science & Math

>>9061124
This is referred to as the symplectic reduction. Usually this can be done by quotienting out holonomic constraints.
The first few chapters of Woodhouse's book on geometric quantization walk you through these symplectic constructions.

>>9034286
>properly developing the functional calculus necessary to justify the technique and establishing that the appropriate properties of the propagator follow through makes the analysis tricky.
That's definitely the case. When you're dealing with quantized fields (i.e. operator-valued distributions) directly in the action instead of quantized measures in the partition function you cannot use classical means to evaluate the variation [math]\delta S[/math], since any Taylor expansion you might do will not in general give you the cancellations you want due to non-commutativity.
This is one of the reasons why many had abandoned the QFT approach a la Wightman or Schwinger and instead focused on the stat mech or Feynman path integral approach in order to do calculations.

>>8989068
Thanks anon. Posting here is a way for me to organize my thoughts on some interesting relationships and connections between things I've read. I'm glad people are enjoying them too.

Fix [math]k\in \mathbb{N}[/math] and let [math]\lambda\in P^{+}(k)[/math] be the set of initegrable highest weights of level [math]k[/math] of the representatoins [math]V_\lambda[/math] of the affine Lie algebra [math]\hat{\mathfrak{g}}[/math]. Let [math]L \subset \mathbb{R}^3[/math] be an oriented framed link and let [math]t_j, 1\leq j \leq n[/math] be straight lines that decompose [math]L[/math] into elementary tangles up to isotopy. In each elementary tangle, we assign the highest weights [math]\lambda_j ~(\lambda_j^*) \in P^+(k), 1\leq j \leq m[/math] to each intersection [math]q_j[/math] of the link [math]L[/math] with [math]t_j[/math] if the link segment is oriented downward (upward). Set [math]V(t_j) = V_{\lambda_1\dots\lambda_m}[/math] to be the space of conformal blocks at [math]t = t_j[/math] with weight [math]\lambda_j[/math].
Define a linear map [math]Z_j:V(t_j) \rightarrow V(t_{j+1})[/math] and let [math] Z(L,\lambda_1,\dots,\lambda_m) = (Z_{n}\circ \dots \circ Z_1)(1)[/math], then [math]J(L,\lambda_1,\dots,\lambda_m) = \prod_{j=1}^{m}Z(K_0;\lambda_j)^{-\mu(j)} Z(L,\lambda_1,\dots,\lambda_m)[/math] is a link invariant, where [math]K_0[/math] is the [math](0,0)[/math] link with two minimal and two maximal points and [math]\mu(j)[/math] is the number of maximal points in the component [math]L_j[/math] of the link [math]L[/math].
With [math]J[/math] and Dehn surgery on 3-manifolds [math]M[/math], we can now define the Witten invariant by [eqn]Z_k(M) = S_{00}e^{-\pi i \sigma(L)\frac{c}{4}}\sum_\lambda S_{0\lambda_1}\dots S_{0\lambda_m}J(;\lambda_1,\dots,\lambda_m)[/eqn], where [math]S_{\mu\nu}[/math] is the representation of the action of the conformal group on level [math]k[/math] characters [math]\chi_\lambda[/math] of [math]\mathfrak{g}[/math], and [math]\sigma(L) = \#\{n \in \operatorname{Spec}(L_i \cdotp L_j) \mid n > 0\} - \#\{n \in \operatorname{Spec}(L_i \cdotp L_j) \mid n < 0\}[/math] is the signature of the link [math]L[/math].