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By "main" research advisor I mean: with him, I got my most valuable research experience, greatest results and of course I invested most of my research time. Our field is Physics and I am making some achievements, but when I asked him for a recommendation letter for my PhD application, he refused and said I would not want his letter if he really writes one. I'll give his reason at the end, but I want to state my own question here first.

Will it be weird that I state my research experience with him very clearly (I think it will make a strong part) in my SoP but without his recommendation? Will the admission committee wonder where this advisor's recommendation is? If this is not a problem, does it even help that I merely mention this experience in SoP without the support of a letter? I have another piece of research experience from a very prestigious place and I have the LoR from my supervisor there, but that one doesn't have strong results as this main piece of experience and I am counting on this experience to make my application strong.

Here is his reason why he doesn't want to recommend me which might be trivial to my question but I guess you would want to know.

First, my undergrad major is an engineering field that has almost nothing to do with Physics. My advisor said if he is recruiting graduate students, he would only consider Physics Bachelors to be graduate students because it will be very hard and even impossible for students trained in other majors to shape their thinking to be suitable to do research in Physics. I am no exception, although I am doing part-time research with him and have taught myself most Physics core coursework (Damn, another equally prestigious Physics Professor in the neighbouring office just holds the opposite view). For me, his suggestion is that I should treat Physics as a hobby but forget about going to graduate school. (We encountered a nasty math problem which was important for logical completeness of our work some time ago, and he got around it with a trick which to me was "ridiculous" by then. I proposed my own alternative method, but in his view my method is rigid, and he then used my failure of understanding his trick to tell me that my thinking is already not suitable for Physics.)

Second, he is really really an "idealist" in doing research. Since I am applying for grad school, I want to speed up a little bit to revise our work to have a presentable draft (I had already derived the results by then, we were polishing the writing and dealing the above-mentioned math problem). He said I was appearing to be like a "pragmatist" because I was not acting or thinking slowly and calmly like an idealist. He even got a bit unhappy with me when I was arguing with him, wanting him to explain his trick (mentioned in the first reason) more clearly, when I was so desperate to understand it.

As a result, he said I would not want his letter since his opinion is just he won't recommend me to a graduate program although I did research with him.

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    You should stop spending your time on research with someone who doesn't consider your contribution important. – FourierFlux Jan 10 at 17:32
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    Its a bit hard to judge or suggest where to look without knowing more about the subject you are working in or the math trick that was used (though its probably better to mention in the comments than in the question itself, unless of course this revels their identity). I am a physicist who has a number of engineers as friends/family and could imagine scenarios where this comment makes sense, since I know their thought process tends to be different to my thought process (build something that functions vs deep understanding). Though part of a PhD/masters is to learn how to think. – N A McMahon Jan 10 at 20:16
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    I sympathize with the "two algorithms" story because I've been there myself. I have a math degree and have worked mostly on developing software for mech eng applications. Once, there was an issue proving that an algorithm would never fail in practical situations. Mathematically, that was equivalent to a statement for matrices with a certain mathematical property, there was always a sequence of submatrices such that factorization of each one had no zero diagonal terms. It was hard to even formulate the problem mathematically without getting lost in a mass of notation. – alephzero Jan 11 at 2:23
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    ... But from an engineering point of view, it was easy to see that the algorithm was equivalent to saying that it was always possible to physically construct the object which was modelled by the matrix, piece by piece, without the need for any temporary supports during the construction process. Professional engineers (and probably most people who had played with LEGO!) could easily understand why the algorithm always worked. In my case, luckily there was no need to provide a formal mathematical proof, so we just got on with using the algorithm in practice! – alephzero Jan 11 at 2:26
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    I have worked in a situation like yours (no respect for engineers). I got out of that lab, started over somewhere else and finished my PhD in 3.5 years and incorporated aspects of comp. engineering in my thesis. The world needs more people who are multi-disciplinary, not less. – shaunakde Jan 11 at 23:22
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Yes, it might look a bit weird, but you don't have control over that. And he is giving you good advice that he shouldn't write you a letter. Not every advisor is willing to be so honest.

But, I don't think it is an absolute block to your plans. Find others to write you strong letters and (US perspective) you should be ok.

What you need are letters from people who can confidently and honestly predict your future success based on their experiences with you. This advisor is not that person.

And FWIW, it is his problem, not a problem with your work mode. And no admissions committee should make any decisions based on guesses or suppositions. They should take your materials as they stand.

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Do not press for a recommendation letter of someone who is reluctant to give you one. Nothing good is coming out of that.

You have a different approach to things. It's very dangerous of you to label him as "idealist", maybe what happened is that he considers you an strong pragmatist and if you let through that you see him in the opposite side of things, this simply escalates to the point where he sees you both not to have common ground.

Engineering and Physics have a lot in common, but they are quite distinct in the way of thinking. Physics is about finding out how things work, Engineering is about how to use these rules to create something that you want to do specific things. The way people think in both areas is quite distinct and you can sometimes recognize whether the education background of someone in a 3rd field comes from physics, engineering, math, or something else. Thinking style is almost like a signature.

You ended up in a situation where your prof sees only the outcome-oriented aspect of your work. [As an aside: The rigidity he accuses your trick of is a bit of an accusation that he may have leveled against a mathematician rather than an engineer (assuming your trick is really rigorous), which is unwise, because usually it is better to use the more solid mathematician's "trick" than a handwaving one. The latter may be justified to get an insight, but not because it is better.]

That being said, that's not a bad thing. Different backgrounds can bring in different fresh ideas into a field. However, if someone holds prejudices against you, there's not much you can do. You could of course try to kick off a social movement against the discrimination of engineers by physicists (and I am only partially joking here, because this is a form of discrimination, a suitably named -ism).

But, as long as you are on your own, your best bet is to look for a different recommender. You will not get a good recommendation from him, and better to answer questions about that discrepancy during your interview once that happens.

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Get a job.
He is trying to tell you that you have no future in Physics at a uni.

You could try to see if someone else would write a letter but I doubt that would be productive.

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    "I doubt it would be productive" - how do you come to that conclusion? – Captain Emacs Jan 12 at 9:29

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