One way to "see" Prigogine's ideas, in some sense to give specific content to @ArnoldNeumaier's comment, is to add one extra dimension to a field model, something akin to a renormalization scale $\mu$, say. One could then write equations that relate the field at different scales to each other, such as, say,
$$\frac{\partial^2\phi(x,\mu)}{\partial x^\alpha\partial x_\alpha}+(\mu-\mu_0)^2\phi(x,\mu)+\phi(x,\mu_0)=0.$$
This particular equation makes the scale $\mu_0$ special, in that what happens at the scale $\mu_0$ asymmetrically affects what happens at both larger and smaller scales. Obviously such a model requires interaction strengths to be chosen and $(\mu-\mu_0)^2$ could be any function of the scaling dimension; also dependency on differentials w.r.t. $\mu$ could be introduced without affecting Lorentz invariance; some kind of singular behavior at specific scales might be introduced. It would certainly have to be asked how this might be quantized, which as an effectively 5-dimensional field theory would surely be problematic, albeit the symmetry is still Lorentzian 3+1-dimensional.
In this type of model, Prigogine's specific idea, that the higher scale DoFs are subject to thermodynamics, would come from the impossibility of measuring the field $\phi(x,\mu)$ at all scales $\mu$ (even supposing that we could measure $\phi(x,\mu_0)$ for all $x$ at some scale $\mu_0$). Thermodynamics would be a consequence of averaging over DoFs at all scales below $\mu_0$, say.
There must surely be other ways to formalize Prigogine's ideas; I would make no claim that this is the only way to do the job or that such an approach might be capable of being empirically better than standard Lorentzian QFT. Adding an extra dimension in this way can only be more general than a 4-dimensional field, however the effects of the extra dimension could be engineered to be as small as necessary. The introduction of the extra dimension makes the field at every scale independent of the field at every other scale, but the coupling to the field at larger and smaller scales can be arbitrarily weak or strong, with arbitrarily chosen functional behavior (so that the independence can be arbitrarily constrained, giving the appearance that there is no independence for different scales).
Really outlandish behavior could be introduced with this kind of model; Planck's constant and the metric could change very slowly as the scale changes, for example, so that at scale $\log_{10}(\mu/meter) = -100 \ll -34$, say, phenomena could be much different than at scales above the Planck length, but making detailed contact with experiment for such a model wouldn't be easy!
EDIT: A Poincaré-invariant free field quantization, giving a Gaussian field, seems not too difficult: we can define the 2-point VEV of the vacuum state, in momentum space, as
$$\langle 0|\tilde\phi(k,\mu)\tilde\phi(k',\mu')|0\rangle=\sum\limits_\alpha(2\pi)^4\delta^4(k-k')2\pi\delta\bigl(k^2-m^2(\alpha)\bigr)\theta(k_0)M_\alpha(\mu,\mu'),$$ where $M_\alpha(\mu,\mu')$ is an $\alpha$-indexed set of positive semi-definite matrices. $(2\pi)^4\delta^4(k-k')$ is required to ensure translation invariance and $2\pi\delta\bigl(k^2-m^2(\alpha)\bigr)\theta(k_0)$ restricts to the forward light-cone, with an $\alpha$-dependent mass, so that $\bigl[\hat\phi(x,\mu),\hat\phi(x',\mu')\bigr]=0$ when $x-x'$ is space-like, for all $\mu,\mu'$, but there can also be non-trivial vacuum state correlations between scales $\mu$ and $\mu'$.
Of course one could say that $\mu$ is just one more dimension, not necessarily "scale", but I suppose there to be interpretations that would take "collective behavior [to follow not only] from the more detailed model" (modeled on the phrasing of @ArnoldNeumaier's third comment on your question), but also from additional information that is encoded in $\hat\phi(x,\mu)$ for different values of $\mu$. I am, of course, waving my arms a little here.
Q&A (4868)
