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comment Free Particle Propagator Using Path Integrals
Since the limits are $-\infty \to \infty$, each factor is an (analytically continued) Gaussian, which is explicitly computable with no special functions needed. This is exactly what happens in the piecewise-linear regularization, so it's a good sign that the right answer is sure to pop out (once the $N$-dependent normalization is determined).
comment What criteria distinguishes causality from retrocausality?
But your comments are elaboration, and I agree with much of what you have said. I have more to say, which I might put into an answer. Since you have engaged in a healthy discussion and have defended your answer, I would be more than happy to remove my downvote. However, it seems that the SE software won't allow me to change my vote unless you edit your answer.
comment What criteria distinguishes causality from retrocausality?
I don't mean to sound contrarian, and am absolutely sincere--if there is nothing to say, then why say it?
comment Why do some anomalies (only) lead to inconsistent quantum field theories
I'm sure this has been said many times before, but "gauge symmetry" is absolutely a misnomer as what we really have is a gauge redundancy. Mathematically, we have a space of connections $\mathcal{A}$, and gauge theories are all about quantizing the Poisson space $T^\ast \mathcal{A} / \mathcal{G}$, rather than the symplectic space $T^\ast \mathcal{A}$ (which does have a gauge symmetry). From this point of view, of course an anomalous redundancy plays a different role, both conceptually and mathematically, than an anomalous symmetry.
comment What criteria distinguishes causality from retrocausality?
-1. "However, on the larger scale these many interactions form a system where time moves forward (i.e. positive) not backward (negative)." As far as I can tell, this is exactly what the questioner is asking and this response is a mindless regurgitation without any physical or mathematical content. If you could please elaborate and convince me otherwise I would be more than happy to remove my downvote (and possibly upvote instead!).
comment What does symplecticity imply?
Even on a Poisson manifold, the dynamics given any particular initial condition are always constrained to a symplectic leaf, so you will have trouble finding examples that are "genuinely Poisson". You might find more interesting examples by looking at systems with constraints, but then even Poisson geometry is not entirely adequate for this--see, e.g., Dirac geometry.
comment Why/How is this Wick's theorem?
I like this answer just the way it is, as it makes it very clear that Wick's theorem is not a theorem about QFT, but is rather a completely general theorem about algebras obeying a few simple axioms. The standard textbook treatment with fields and VEV's obfuscates what is really just basic algebra.
comment Why is it important that Hamilton's equations have the four symplectic properties and what do they mean?
The course would depend on the department and who's teaching it, but there are lots of good books. Arnold's Mathematical Methods of Classical Mechanics is a good companion to Goldstein (and is in some ways better), but places much more emphasis on the geometry. There is Foundations of Mechanics by Abraham and Marsden, and Symplectic Techniques in Physics by Guillemin and Sternberg. The are lecture notes by Cannas da Silva which are quite good and available online for free. But this is just the tip of the iceberg, as symplectic geometry is a huge field in modern mathematics.