🚀 go-pugleaf

: The simplest example arises in the theory of free boson fields.
: Here one treats ``Gaussian measures'' of the form 
:                 e^{Q(x,x)} dx  
: where Q is a complex-valued symmetric bilinear form on the real vector
: space V.  When V is finite-dimensional and dx denotes Lebesgue
: measure, e^{Q(x,x)} dx is indeed a perfectly nice measure
: on V, and it's a finite measure when Q is negative definite, for example
:                 Q(x,x) = -(x_1^2 + ... + x_n^2)
: on R^n.

Simplest, eh? O Shit [tm].

: The point
: is that a nice measure (i.e., a finite regular Borel measure) on a nice
: space (i.e., a locally compact Hausdorff space - for example, a finite-
: dimensional vector space) allows one to integrate all bounded continuous  
: functions on the space, but in practice one is willing to settle for
: much less.  

Hit the nail on the head there. MUCH less.

: Given a compact Lie group G (the "gauge group") and a trivial
: principal G-bundle P over a 3-dimensional manifold M, the
: Chern-Simons action of a connection A on P is given by
:    	S(A) = (k/4pi) \int_M tr(A ^ dA + (2/3)A ^ A ^ A) 
: where the ``level'' k is a nonnegative integer.   The quantity S(A)
: changes by an integer multiple of 2pi under gauge transformations of A, so we
: may formally define a ``measure''
:              e^{iS(A)} DA
: on the space of smooth connections modulo gauge transformations, where
: DA is the (purely heuristic) ``Lebesgue measure'' on this space. 

: The problem is in making this sort of stuff make
: sense, because the space of connections modulo gauge transformations is
: a fairly nasty infinite-dimensional space, not even a manifold, so there
: really is no Lebesgue measure on it.

Well, I agree with the first part of that last sentence.

lordSnooty@theGiantPoisonedElectricHead <suitably humbled>