I know this is still a current topic of study, so my question is less about mathematical underpinnings, and more about the transition from the "sum-over-all-possible-intermediate-points" measure Dx=limN→∞N∏i=0dxi
to the "sum-over-all-possible-field-configurations" measure found in the non-abelian partition function for Yang-Mills, which I've been told has the form:
D[A]=limN→∞N∏i=03∏μ=0Nc∏c=0dAcμ(xi)
where Nc is the is the dimension of gauge group. The relevant passages I found in Peskin and Schroeder glossed over the topic. (p.294 Eq. (9.50) begins the discussion, but it is for Abelian groups, where I imagine we can treat each component of Aμ as a scalar field, so the product makes sense to me (and the color product isn't present of course). I've yet to find a proper expansion/derivation of the same measure for non-abelian gauge configurations. There seems to be a brief mention of the functional integral measure around p. 665 Eq. (19.67) as well, but not quite what I'm looking to understand.)
Answer
Forget about gauge fields. Consider a scalar field ϕ:V⊆R4→R for simplicity. The relevant equation in Peskin & Schroeder is instead Dϕ = ∏idϕ(xi).
No comments:
Post a Comment