Is an analogy between karma and modern day physics possible?

No I don't think it's possible. At best it's a weak or stretched analogy; at worst it's a misleading one (implying too much equivalence between the system-of-kamma and the system-of-physics); or it's an unfalsifiable theory with cherry-picked results (e.g. "we observe phenomenon X in one system ... what phenomenon in the other system is analogous?"). The problem there is that the analogy exists because you created it, not because they're genuinely (or useful, predictively) analogous.

It is (or was, has been) especially common to confuse things using Quantum Physics (see Quantum woo and Quantum mysticism).

A cleverer teacher than me (e.g. the Buddha) might use what you know already to give analogies which explain his Dhamma; so if you knew Physics you might appreciate any apt Physical analogies; but I can't think of any. The only analogy I've ever made is that kamma is like the weather: things happen when the conditions are right but the details are complicated to predict.

More on the subject of Quantum Mechanics, see also Observer effect (physics):

An especially unusual version of the observer effect occurs in quantum mechanics, as best demonstrated by the double-slit experiment. Physicists have found that even passive observation of quantum phenomena (by changing the test apparatus and passively 'ruling out' all but one possibility), can actually change the measured result; the 1998 Weizmann experiment is a particularly famous example.[1] These findings have led to a popular misconception that observation by a conscious mind can directly affect reality,[2] though this has been rejected by mainstream science. This misconception is rooted in a poor understanding of the quantum wave function ψ and the quantum measurement process.[3][4][5]

If you read the references [3] through [5] it's clear (if you understand e.g. what the famous "double slit experiment" is) that the "observation" it's talking about is the act of installing the physical test equipment (which I'd also know as "probing the system"), or the action (effect) of the equipment itself ... and it's not to do with whether any "conscious mind" observes the results of the experiment.

I kind of like jcarpenter2's answer, because I thought this was an unanswerable question but that answer (perhaps successfully) talks about the limitations of logic and not just about physics.

I could try to add another analogy along those lines -- inspired by Heisenberg's Uncertainty and the start of the Tao Te Ching -- that making a verbal observation (i.e. a comment or description) about something will affect (disturb, interfere with) the thing described (or the perception of the idea described) ... but I don't think that's a useful analogy. The Buddhist analogies are purposeful, have a goal (liberation or cessation). Inventing further useless analogies is just proliferating views, leading to a "thicket of views" of the sort that we're warned doesn't lead to liberation. Even Occam's razor (also Einstein) suggests preferring simpler theories.

I think it's better to try to understand Buddhism, to approach Buddhism, on its own terms: understand what it's saying, understand what the Buddha would have been saying to the people he said it to ... without preconceptions such as that it's talking about Rutherford's theory of atoms. Perhaps the only really useful preconceptions are ones you might have learned in pre-school, i.e. morality, of the sort that allows the Buddha to tell the Kalamas they should judge for themselves.

I need to do some more reading about quantum physics, I really do. It seems very interesting.

What I can say, is there is a clear analogy (to me at least) between Buddhism (or how I understand it) and modern mathematics.

It is my opinion that the single most important theorem in mathematics is Godel's 1st Incompleteness Theorem. Basically, this theorem shows that no system of constructing proofs of arithmetic is expressive enough to prove all true theorems and disprove all false ones. No system of logic can tell all tales and solve all problems.

This hydra has many heads. Another well-known one is the Halting Problem in computer science. This one states that there is no single algorithm that can determine whether any algorithm halts. This is an extremely deep result assuming the Church-Turing thesis, which states that any computable function at all is computable by a Turing machine.

It's very similar to the beginning of the Tao Te Ching. While this is not a Buddhist text, nonetheless it opens with "The Tao that can be spoken is not the eternal Tao. The name that can be named is not the eternal name." We are extremely fortunate that it can be proven logically that all logic is faulty in some way. We are privy to the knowledge that all knowledge is false, and that that is the only true knowledge.

It is my belief that there is no truth. When somebody says "This statement is false", a good and proper response is, "Yes, every statement is false." Newton's third law is not true, nor is any other physical theory true. Edge cases will continue to be discovered ad infinitum. If any physical theory were true, and the universe operated deterministically according to its laws, there would be no experience.

The only things that are real are the things that transcend this world, such as our Karma. Of course, to us there is no difference between the things that transcend this world, since we cannot access or imagine them.

Do not rely on Karma to bring good results in this life, since the distinction between good Karma and bad Karma itself is just a dualistic notion. Just know that you are here for a reason, and have faith.

https://www.youtube.com/watch?v=uIFgZD4errQ