I am extremely passionate about entropy/physics, but I don't think I can get into grad school. What can I do to pursue these topics sustainably? I am a recent graduate in Chemical Engineering from the University of Maryland with a 2.94 GPA and a 3.01 major GPA and I am completely obsessed with entropy and the foundations of quantum physics. I've spent hours and hours pouring over entropy, reading all the way from its foundations in classical thermodynamics to modern bayesian entropy (Information Theory and Statistical Mechanics all the way to Entropic Inference and The Foundations of Physics). The only part of entropy I would (personally) say I don't understand is black hole entropy, but I'm working on that. I've read through much of Sakurai's Modern Quantum Mechanics and even audited a graduate level physics course at my university (the only non-engineering physics course I've taken). I just started reading Schwartz' Quantum Field Theory and the Standard Model last night in fact. 

I don't know what to do and I'm a bit terrified. In undergrad I had major mental health issues. I was barely functional to tell the truth, and I don't really understand how I got through my major. In my spring semester of senior year I failed all my non-trivial classes. Only within the past six months I've been able to get my life together in some fashion and been treated properly. My life is significantly improved over what it used to be and I don't have the problems that messed my GPA up so significantly.

Getting healthier has helped me realize understand how meaningful I find these ideas. I really, really want to do research on entropy and quantum physics, you might even call them my passion (yes I know, cliche). But I have a major problem: I don't think I get into research the standard way, grad school (at least right now). My transcript is terrible, I have no research experience, and what I'm interested in is barely related to my undergrad major. I do not know any professors who could give me a decent letter of recommendation. I haven't even taken the GRE. Although I believe that I am suited for research, I doubt I could convince a university of that.

I know you can publish without being with a university and that seems like a decent option since it would personally meaningful and it may make it easier to be admitted to a grad school. I even have two and a half partial manuscripts I'd like publish at some point. But I'm unsure of this pathway for three reasons:


I need an income. 
I have no experience publishing
I need a long-term life trajectory. 


For 1, I've thought about entering into a job, making some savings and publishing on my free time but I suspect that publishing and working would seriously conflict each other. I have minimal savings at this point, so I doubt I could work on exclusively publishing for more than a month. 2 has prevented me from even submitting my work for publication. I've never felt that what I've written is up to par with what journals want (and I only have a foggy notion of what journals I'd like to publish in). The professors I've contacted about this have either not been helpful, or have not been interested in my ideas so far. Should I just keep blasting out emails to people? 3 scares me because if I fail to get into grad school, or I turn out to be completely wrong, I might permanently impact my life trajectory in a negative way. 

What are the best ways to overcome these obstacles? I'm extremely driven, so even if you think a path is difficult I would love to hear it.
Passion will not make you a successful PhD student.  Health problems will prevent you from being a successful PhD student.  

I suggest you find a job that is relevant to physics.  Considering your interests and the current job market, maybe a job in data science?  Demonstrate over a period of at least a year that you can be successful at that job.  Study for the Physics GRE and get a GRE score.  Then you might be able to get into a decent PhD program.

Enjoying studying entropy does not mean you will enjoy publishing about it; these are quite different activities.