This is an announcement, in the Physics Forums context, of my "Lie fields revisited", http://arxiv.org/abs/0704.3420" [Broken], which has recently been accepted by J. Math. Phys., and an attempt to elaborate on why you might read it and to elicit critiques. For a little over ten years, I have been trying to understand and characterize in detail what differences there are between quantum fields and their closest classical equivalent, random fields. Random fields at finite temperature have nonlocal correlations, just as do quantum fields, so what is the difference? I showed in "A succinct presentation of the quantized Klein–Gordon field, and a similar quantum presentation of the classical Klein–Gordon random field", http://dx.doi.org/10.1016/j.physleta.2005.02.019" [Broken] and papers. If you're wondering at this point what a classical random field IS, it's almost trivial, it's an indexed set of random variables (that's to say, if you know about random variables, a random field is easy, http://en.wikipedia.org/wiki/Random_variables" [Broken], but all we need here is that a Physical state and a random variable together give us an expected value associated with the random variable in that state). Two random variables, [tex]V_1[/tex] and [tex]V_2[/tex], say, are enough. A continuous random field effectively makes the index set be the points of space-time, so that we could talk about random variables [tex]\phi(x)[/tex] at every point of a (Minkowski) space-time, very similarly to a quantum field, but there are technical (and also notational) matters that make it mathematically much better to make the index set be a linear space of functions, which are called "test functions". It is possible to talk of a continuous random field [tex]\phi(x)[/tex] as a "random variable valued distribution", which is fairly directly comparable to the "operator valued distribution" way of talking about a quantum field. That continuous random fields are conceptually quite close to quantum fields, and can be presented in a Hilbert space and operator algebra formalism that is very close to quantum field theory, is a large part of why there is a small hope of casting some light on quantum mechanics. From here on, all the random fields are continuous. With honorable exceptions, almost all attempts to understand quantum mechanics have been through the nonrelativistic quantum theory and classical particles, quantum field theory has been felt to be too complicated for any understanding of it to be possible until quantum mechanics has already been understood. Big mistake, I believe. Of course, as classical models, random fields prima facie allow Bell inequalities to be derived, contradicting experiment, but in "Bell inequalities for random fields", http://dx.doi.org/10.1088/0305-4470/39/23/018" [Broken], I showed that Bell inequalities cannot be derived for random fields that have thermal or quantum fluctuations. Essentially, equilibrium states have non-local correlations at all times, and an absence of any pre-existing nonlocal correlations is required for Bell inequalities to be derived. Technically, this is usually called the conspiracy loophole. A lot of detail is required to get a paper on Bell inequalities into a good Physics journal; "Bell inequalities for random fields" takes several different approaches and gives a number of supporting arguments and references to papers that have previously argued that classical field theories allow the derivation of Bell inequalities. John Bell wrote the seminal paper, as so often, "The theory of local beables", which is in "Speakable and unspeakable in quantum mechanics", but it has a flaw, that it effectively works with particle properties and with random fields even-handedly, when the assumptions that are largely reasonable for one are not as reasonable for the other. Note that I'm claiming that the voluminous literature that proves that classical particle property models cannot be used as physical models is just taking on straw man theories, and is irrelevant to classical random fields that have non-zero fluctuations. Personally, I think that a classical particle property model that I like is not possible, despite the claims of the detection loophole, de Broglie-Bohm models, Nelson models, etc. My earlier papers do some mathematics, but they are largely non-constructive, so they have not so far provoked comment or rebuttal. "Lie fields revisited", in contrast, constructs a new class of interacting random field models that are not available to quantum field theory. "Lie fields" were introduced in the 1960s, but it became apparent after a few years that they are incompatible with the Wightman axioms, so they became moribund. Once we admit classical random fields as a possibility, and adopt an algebraic approach to constructing them that exploits the mathematics of quantum field theory, it turns out that we can sidestep renormalization completely, and we discover a mathematically well-defined concept of interacting particles that is quite closely analogous to the free quantum field's Fock space structure (without having to work with asymptotic states). The approach I take attempts to be as empirical as possible, to the extent that it takes a view that is common in Physics, that correlations of "the field" are the observables for the purpose of theoretical physics. Renormalization, for example, takes as its empirical starting point that correlation functions must be finite. The class of models that I construct, however, generalizes the generalized free quantum field, with an arbitrary (complexification of a) Källén-Lehmann weight function required to specify a particular dynamics, so there is a "landscape" of possible models, as they say in string theory. From the point of view of empirical usefulness, this is good, but such a large class of models is essentially not very explanatory. Still, it seems good to have an intermediate explanation of the relationship between quantum (field) theory and classical modeling that I like better than anything I have seen elsewhere in the literature. The most discursive account of my approach (but which pre-dates the mathematical developments in "Lie fields revisited") is probably in a Växjö Conference Proceedings, http://dx.doi.org/10.1063/1.2713457" [Broken]. I will post separately a couple of responses that I have received from main-line physicists, and my responses to them, which hopefully will give some indication of the sorts of interpretational and mathematical objections that can be raised about my approach.