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.)

  • 1
    $\begingroup$ 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

1 Answer 1


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 $\epsilon\to 0^+$.

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.