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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 $$\mathcal{Dx} = \lim_{N\to\infty} \prod_{i=0}^{N} dx_{i}$$

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:

$$\mathcal{D}[A] = \lim_{N\to\infty} \prod_{i=0}^{N} \prod_{\mu=0}^{3} \prod_{c=0}^{N_{c}} dA_{\mu}^{c}(x_i) $$

where $N_{c}$ 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.)

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    $\begingroup$ Two good books: Montvay and Münster's Quantum Fields on a Lattice, and Creutz's Quarks, Gluons and Lattices. The components of the gauge field are associated with links (site-pairs) rather than lattice sites, and the formulation uses group elements rather than Lie-algebra elements. It is manifestly gauge-invariant. $\endgroup$ Nov 21, 2018 at 21:11

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Forget about gauge fields. Consider a scalar field $\phi:V\subseteq \mathbb{R}^4\to \mathbb{R}$ for simplicity. The relevant equation in Peskin & Schroeder is instead $$ {\cal D}\phi ~=~\prod_i d\phi(x_i). \tag{9.20}$$ Here $x_i$ is a point of a square lattice with a lattice spacing $\epsilon$, and $i$ runs over all lattice points. In other words, the discretized path integral measure is $$ {\cal D}\phi~=~\prod_{n\in \mathbb{Z}^4}^{\epsilon n\in V} d\phi(\epsilon n). \tag{9.20'}$$ Does this help? Ultimately, we are interested in the continuum limit $\epsilon\to 0^+$.

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  • $\begingroup$ I suppose I'll mark this as solved, I don't fully understand some things yet but I think that is due to a need of reading more. A final question, can we treat in general, $d \phi(x_{i})$ as a differential, and write is as $\frac{\partial \phi(x_{i})}{\partial x_{i}} dx_{i}$ ? $\endgroup$
    – Craig
    Nov 21, 2018 at 22:18
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    $\begingroup$ No. The integrations are over the target space variable $\phi$, not over the spacetime variable $x$ per se. $\endgroup$
    – Qmechanic
    Nov 21, 2018 at 22:37

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