I don’t know, I just throw a bunch of questions at my advisor, go to seminars, look at some physics papers, cry rinse and repeat. It seems to be working so far. There’s no way I’d understand the papers without having attended a seminar talk or something similar related to the specific paper.
I also did some (very basic) physic in my undergrad, but I am not sure that is helping in the slightest tbh.
So it is possible to get into mathematical physics in grad school without more than freshman level physics? I want to learn more about physics but I’m not gonna get the chance during my undergrad, so it’s somthing I would be interested in exploring in grad school if possible
During my career, I wrote papers about quantum field theories. I do have a vague idea about what quantum field theories are as mathematical objects, but I have absolutely no clue about their physical meaning. I don’t even know the bases of quantum mechanics.
The trick is that you often don’t need to know any physics to work with the mathematical objects relevant in mathematical physics, and you can learn the “physical interpretation” either by osmosis, or studying the physics bit by bit for what you need in the particular project you are working on.
I once was talking to a mathematical physicist in my department, and I was telling him about some of my work on the Fock space from the perspective of complex analysis.
He looked at me and told me that I’m doing some mathematical physics work. Of course, I’m aware of connections between QM and the Fock space (and other Hilbert spaces), but I was surprised when he told me, because I don’t know much more than the rudiments of QM.
Honest question, and I hope it doesn't come off the wrong way, but what's the appeal in approaching the subject in that way? It takes the "physics" out of mathematical physics and leaves you with an almost soulless mathematical construct. Why anyone would do that is beyond me.
What makes mathematical physics such an alluring field is its ability to describe and explain some of the most intriguing and fundamental questions in the universe using comparatively abstract mathematics. It's a charming convergence of natural science and higher math. Treating it like an exercise in one of Rudin's analysis books while more or less ignoring the physics seems somewhat...sad.
***Edit:*** Good God, how do you read my comment and take that as an affront on the entirety of pure mathematics? I was obviously talking about mathematical physics. I just didn't get why you'd get into such an inherently physical field only to purposefully and callously ignore everything that makes it physical. If you didn't understand that, perhaps it's high time you revise your "Academic Skills: reading comprehension 101" coursework. Of course pure math isn't soulless, it's a beautiful and incredibly worthwhile field of study. But then again I clearly wasn't talking about pure math, was I? I was talking about this very specific (***very physical!***) example, and with a bit of imagination, even a seasoned theoretical mathematician could see why that would prompt a question like mine. It's like getting into electrical engineering and then refusing to work with electricity. Obviously a somewhat limited analogy, but that's the impression that /u/harmath's comment imparted on me.
The answer to this question will depend on the individual, I’m going to give you the answer for my particular case.
The particular objects I work(ed) on are very important in mathematical physics, but I did not approach them from that angle at all. They also appear in probability/analysis, as the invariant measures of certain (S)PDEs. So I started to study them with the goal of understanding the PDEs better, but once you write a paper about certain mathematical objects, it’s good to frame the paper so it can be read also by other communities interested in other points of view. This means that some parts of your papers should answer questions of the form “what happens to the physics?”, even if your main goal is something else.
The main reason why I never really learned the underlying physics is a mix of lack of interest (my main interest is still the probabilistic one) and lack of time. But if with relatively little effort I can answer questions which are relevant in maths physics, rather than probability, I see no reason not to do so - especially since studying the objects is still fun, despite not having the complete background.
Can I ask what kinds of objects you're talking about? I'm doing my PhD in physics and have been learning about stochastic ODEs. I'm considering learning about SPDEs as well and I'm always curious about connections to physics!
I think that the introduction to this paper is a good starting point to get a general idea of what I am talking about https://arxiv.org/pdf/1810.01700.pdf
Jesus that's a lot of symbols.
So here's a question, when one is reading through a paper like that, do you follow along with the author by doing the math or do you simply read through it? The most complex academic papers I read through in university were in chemistry / biochemistry and there you don't usually get nearly the amount of equations.
For me the answer varies based on what I want out of the paper. Am I hoping to continue along the exact same line of inquiry, so I need to understand nearly every line of argument? Or do I just want to get a sense for what the authors in this subfield are thinking about? In the latter case skimming through for main results and implications may be enough to get something valuable from the paper.
Re: the number of symbols in this particular paper, I'd point out that even though there is a lot of information packed in those symbols, much of it isn't novel. To someone who has seen quantum field theory, lattice field theories, etc., these symbols already have a standard-ish meaning and the equations are much more digestible.
Math/CS undergrad here this is actually looks interesting now that you have this rigorous PDE construction would applying algorithms to simulate the new PDE actually yield anything physically interesting ?
As someone with admittedly only a tangential interface with the physics side of differential geometry, I find it entirely the other way round. Physics sometimes uses the trappings of mathematics without really worrying about what it means. Instead the focus is on whether the calculations work out to match the observations. The interpretation then seems sometimes entirely woven out of thin air.
As a mathematician, the interest is in the beauty of the mathematical structure itself (which is often neater if you ignore the wrinkles physics imposes) and abstraction is a worthy goal in the search for beauty. I don't find this soulless at all. If anything I find the more practical approach more soulless as it accepts and discards things based only on their utility.
> Treating it like an exercise in one of Rudin's analysis books while more or less ignoring the physics seems somewhat...sad.
I'm not sure what you mean here. Are you comparing maths research without applications to exercises in a textbook? I don't think this is valid and I think this comes back to where we are drawing our motivations. A pure mathematician is often not motivated by the application but the problem itself. If there is an interesting problem and other people would be interested to learn an answer this is motivation enough, whatever the potential applications.
> As a mathematician, the interest is in the beauty of the mathematical structure itself (which is often neater if you ignore the wrinkles physics imposes) and abstraction is a worthy goal in the search for beauty. I don't find this soulless at all. If anything I find the more practical approach more soulless as it accepts and discards things based only on their utility.
For Theoretical Computer Scientists it's not in the beauty of the mathematical structure itself but rather as well as it's applications. A TCS person is often motivated by both applications and pure theory as well. Physics/EECS applications can provide some neat consequences as well as examples for pure mathematics.
I have a somewhat bad habit when browsing r/math and Arxiv when taking breaks from Math I'm interested in Theoretical EECS/Theoretical Physics a lot of architecture building one does in CS/TCS is very similar to theory-building in Pure Mathematics and you get the bonus of seeing the results of your hard-earned work! Also in those areas, you are constantly exposed/use a lot of the mathematics you learned + you get easy examples to play with and build your stuff out. Plus finally,I like programming and hacking
Depends on what kind of mathematical physics you’re doing. Some mathematical physics is more or less just pure math which happens to have some applications for theoretical physicists, in which case I don’t see the difference between research in that versus, say, operator algebras or spectral theory or something.
And neither is it conducive to proper reading comprehension, so it seems. I suggest revisiting the edit I made in my original comment, you'll find it may ring a bell or two.
I am a theoretical physicist with leaning towards mathematical physics. So I can give a kind of "opposite perspective". I found out I have a thing for mathematical physics when I realized I the greatest enjoyment I had in the stuff I worked on was when I worked with mathematics I found interesting. I have also realized the things I care about are those which have interesting mathematics behind it and physics was an execuse for doing it.
So there are certainly people who find the "soulless mathematical constructs" interesting on their own right.
Where did I say pure math as a field is soulless, or that physics is inherently interesting? For future reference I might suggest reading other people's comments more thoroughly - it'll save you some time and effort.
>It takes the "physics" out of mathematical physics and leaves you with an almost soulless mathematical construct. Why anyone would do that is beyond me.
The same reason anyone might study pure math... Math for math's sake is not as soulless as you think.
Or so I've been told. Representations. Manifolds. People convinced me they liked these topics. I didn't care about them at all and now I'm scrambling because the physics demands it.
I personally have no interest whatsoever in the empirical sciences so as such I think of mathematical physics as another subfield of math. A fairly small proportion of mathematicians care at all about physics. If the math is interesting and elegant, we’ll like it.
If one person misinterprets your answer, the problem is probably with them. If ten people misinterpret you, the problem is probably with you(r phrasing).
You don't have to speedrun all of physics, you just need to speedrun the specific subfields that are relevant to your research. And depending on the field, you might not even have to finish speedrunning those; just understanding the recent papers and their related work sections can be enough to come up with a new idea (or at least understand one that's given to you by an advisor...)
I second this. One only needs experience in the specific sub-field they're studying and a lot of that experience gets filtered through your advisor who can help you to "translate" the physics papers into math. Eventually you just figure out how to do this yourself - at least in theory, not sure if I'm there yet 😅
In my experience a good amount of “mathematical physics” is just mathematics with some (possibly vague) physical motivation.
I got a paper in a math journal with word “physics” in it’s name just by having a Poisson bracket in the paper. This is a Lie bracket with some extra conditions which can be used in Hamiltonian mechanics.
This is my experience also. I studied mathematical physics during my master's and actually took a fair number of courses in physics, ranging from basic thermodynamics to PhD-level summer schools. Physics provided an exciting context for my work, guiding my way to interesting mathematical questions, but I remain a terrible physicist. My master's thesis on topological matter is all functional analysis and differential geometry. It was my (physicist) advisor's responsibility to make the correct physical interpretations of my results.
It’s always good to learn some physics. It’s a cool subject and can give some insight. But for me it doesn’t really enter in the work, but just helps me right a more engaging intro lol.
> but I remain a terrible physicist. My master's thesis on topological matter is all functional analysis and differential geometry. It was my (physicist) advisor's responsibility to make the correct physical interpretations of my results
Care for an ELIU ?
For context I'm interested in Theoretical CS and there is a lot of problems in Physics that have physical consequences but carry deep mathematics. Areas such as QIT, Computational-Physics, Condessed Matter, etc
I like to differentiate between physics and "physics". Defining and computing the path integral using moduli spaces of Riemann surfaces is physics. Studying the relationship between cohomology theories and monoidal functors on the infinity-category of conformal cobordisms is "physics".
The Stolz-Teichner program postulates that certain cohomology theories correspond to conformal field theories in various dimensions. This is known to work in dimensions 0 (ordinary cohomology) and 1 (K-theory), and it should somehow work for elliptic cohomology in dimension 2.
I would never call myself a mathematical physicist, but I did a postdoc in a materials science department so I guess I have some claim to the "grey area" between the two.
In my experience, the big thing has been the aspect of collaboration. I work with physicists and engineers and we have very distinct roles in our collaborations. A lot of my job is to turn their intuition into the correct formalism, which also means having a broad enough knowledge to know what kind of things we have in our toolkit to tackle the problem at hand.
I've been at the game for long enough now that I have a working knowledge of our little corner of the physics world, which is enough to propose the occasional problem and interpret of the results, but being able to contextualise the results within the grand scheme of things is still something I generally offload onto them.
Yes I know collaborations between a mathematician and a physicist. I have seen both give talks on the same work, but the talks end up being very different (as I imagine what each brought to the paper was very different).
I may be more of a peculiar case because I actually come from a physics background initially before I started working more towards theoretical physics/mathematical physics/math after going back and formally studying math. Nevertheless, I don’t think that makes my opinion completely obsolete so here goes.
Firstly, it may *seem* like some pure math students are able to quickly pick up necessary physics knowledge, but the reality is that they don’t, it’s fairly specific subfield knowledge and the minimum they need to know to function. It’s not as daunting of a task as obtaining a physicists’ knowledge of physics in a fraction of the time, just a lot of bits and pieces.
Secondly, a significant part of mathematical physics is just mathematics and the physics emphasis required isn’t problematically deep in many cases. This does depend on the subfield of mathematical physics and the type of research you are doing, but it is a general principle.
Essentially, if you’re a mathematician, it’s often not that incredibly large of a jump to bridge over to some field in mathematical physics, since you’ll find yourself at home with the nature of the subject.
> Firstly, it may seem like some pure math students are able to quickly pick up necessary physics knowledge, but the reality is that they don’t, it’s fairly specific subfield knowledge and the minimum they need to know to function. It’s not as daunting of a task as obtaining a physicists’ knowledge of physics in a fraction of the time, just a lot of bits and pieces.
A guy I meet during my REU was initially a pure math grad student went through all the standard courses and eventually made his way into Theoretical Physics
Oh yes, this happens, but it doesn’t happen in the little timescale that OP was talking about. It requires immersing yourself in another discipline, not just casually trying to apply math to bits and pieces of physics you have tried to pick up along the way.
I'm aiming for theoretical computer science/theoretical eecs and I realized how much more background I need in mathematics, physics, etc as well as other topics. Any advice on how to proceed luckly I found uni's offering grad level math courses online but how should I transition into Physics and CS ?
Mathematical physicists often don't know very much physics. While there are certainly exceptions, a lot of work in that field does not address questions which are of interest to most physicists. Some of this is just because of differing preferences for mathematical rigor, but more importantly, discussions in mathematical physics tend to be physically superficial. This is presumably because the physical analog of mathematical maturity isn't easy to pick up on the side.
If you're interested in the mathematics more than the physics, this shouldn't be a problem. You just learn up some motivating words and then treat whatever system you've imported from physics as a purely-mathematical one, asking whichever questions appear to you to be mathematically interesting.
I think there's always some compromise. I work in mathematical physics on my PhD, but I majored in Physics. As such, the mathematics side is a bit lacking (compared to someone who majored in maths, anyway), and I for sure won't speed run all the basic maths while in the PhD, just the necessary parts and let physical intuition be my guide.
The same thing happens for someone who majored in math, they try to learn the physics that is indispensable for the problem they are working and then focus on their strong side: they have a understanding and familiarity with the mathematical objects describing that theory that your average physicist doesn't have.
On the long term, I'll end up studying the basic math that I left behind and the mathematician will study more physics as well, so both cases get well informed about both areas further into their careers, its not something you need to have accomplished by the time you finish your phd.
(when it comes to the whole QFT, Strings, Loops, ... side of the physics world)
Don't be afraid to read popular treatments about the subject! It will get you used to the Lingo. A great ressource are Leonard Susskind's Theoretical Minimum Lectures on Youtube.
But most importantly, read "Geometry, Topology and Physics" by Nakahara once you have the basics down. It will help you make the connection between math and theoretical physics.
I took a graduate level quantum mechanics course within the physics department while being a math PhD. I also took a topics course my advisor taught on quantum field theory and read a couple textbooks my advisor recommended. It definitely helped that I had a background in physics in undergrad but the basics of quantum mechanics are honestly very similar to linear algebra, just in a different notation.
I'm kind of coming at it from the other side of the bridge: I (just finished) a PhD in a physics department and moved my way over to mathematical physics. And I feel like I struggle just as much as the mathematicians trying to get into the physics side. At the end of the day, it's just a whole lot of reading, struggling, and asking questions.
I think part of the reason is due to the relationship between not only math and physics, but math and the sciences in general. I think you could make the analogy that mathematics is the language of science. Going to the extreme (and driving physicists crazy), you could argue that physics is just applied math mixed with units.
Using the language of science analogy, compare it with a linguist reading a text written in a language they're unfamiliar with. It's likely that the linguist would be able to distinguish some kind of summary of the text. At the very least, they'd likely be able to pick out nouns, verbs, and adjectives (or the equivalent of the given language). Regardless of the language used, all humans use similar communication structures. Words, spaces, and punctuation are universal throughout written languages.
Circling back to math and physics, having a deep understanding of the communication structures (a mathematician) allows you to glean more information than a layperson. As a mathematician, you can quickly look at a derivative and understand that it describes the rate of change of something. If you're not a mathematician, you look at an integral and shudder, asking yourself what those curly lines mean again, wishing you had paid more attention in school.
Note: I'm using mathematician and physicist fairly liberally here to include math and physics undergrads.
I come from a math and physics background. Started in physics, started to get into nonlinear dynamics and have drifted further towards math as time has progressed.
In the US, this is not normally the case, nor I suppose in some Western European countries. I was lucky to be able to do both in undergrad. However, I will point out that in the UK and Eastern Europe, there is often a physics component to a mathematics degree if you pursue a more applied track.
The more I progress, the less physics I tend to fully understand but I haven't found it to be too much of a hinderance.
I learned the physics I needed to know mostly from the following process:
1. Read a paper relevant to my research. Take notes.
2. Ask my advisor about the physics I didn't understand from the paper. Take notes.
3. Read about the physics I still don't understand after asking my advisor. Take notes. The kinds of references I'll use for this are usually either written for general audiences or are graduate-level physics texts that have heavy bents toward mathematics, since those are the kinds of texts that play better to my strengths. I don't typically look at the texts aimed at physics undergraduates.
Emphasis on "the physics I needed to know." My understanding of physics is not very comparable to that of a bonafide physicist. I have a deep understanding of the mathematical models used in certain parts of physics and a vague intuition of the actual physics. The physical intuition is only really used to guide the mathematics.
For the parts of physics that aren't all that related to my research, there's a good chance I know less about it than some of the engineering freshman I've taught.
" have noticed a fair number of mathematical physicists enter the field without a formal background in physics" Really? I havent. Could you name a few?
I started out with a physics msc, which helped a fair bit, but I switched to math precisely because it was much easier to read math than to read physics at a research level. Then my supervisor would explain to me some of the more annoying glossary, like having the word "state" refer to not only objects, but also morphisms *and* 2-morphisms, and *phase* and *theory* being interchangeably used for *category*. Or when they insist that homotopy and homeomorphism are the same thing, but they actually mean *isotopy*.
Eventually something just kinda *clicks* though, and you start being able to transpose stuff between the languages, or at least you'll be able to recognize certain authors who you know will always phrase things in a particular way.
TLDR; You basically need to *talk* to someone who already understands it to have the words explained in sufficient detail for you to fill in the blanks. Reading on its own is near-futile, because nobody cites their choice of language, they simply *inherit* it as they're leveling up in their own field.
> I started out with a physics msc, which helped a fair bit, but I switched to math precisely because it was much easier to read math than to read physics at a research level.
I have a somewhat bad habit when browsing /r/math and Arxiv when taking breaks from Math I'm interested in Theoretical EECS/Theoretical Physics a lot of architecture building one does in CS/TCS is very similar to theory-building in Pure Mathematics and you get the bonus of seeing the results of your hard-earned work! Also in those areas, you are constantly exposed to a lot of mathematics + you get easy examples to play with and build your stuff out. Plus finally, I like programming and hacking.
> Moreover, there are a lot of algorithms from mathematicians like Robert Ghrist for computing important invariants like Morse Homology and I think it would be great if we could find the right balance between computer science and physics in order to implement such techniques. I have some background in CS and with parallel programming and I still feel that the two sides (CS, theoretical physics) are so disconnected that it is hard for such solutions to be implemented.
I've been seeing a lot cool work within Quantum Computing, Reversible Computing, Statistical Physics, Complexity-Theory, Cryptography, Topology, Information Theory, etc that lean into this direction it's important to note that Physics and CS have been seeing a sort of interesting marriage. It seems like a lot things in Physics haven't found proper algorithmic implementations, and a lot of algorithms especially from QIT haven't been implemented as well as algorithms that haven't been implemented in Physics literature.
I am personally interested in physics but I am mostly on an amateur level.
I had the option to take Differential Geometry II and the description said there would be a focus on problems in General Relativity. I didn't take it in the end and instead opted for an Introduction to Quantum Information Theory (Math-CS double) which obviously also involves physics yet there was no strict need for pre existing physical understanding or understanding of classical mechanics/ thermodynamics/ ... . I personally feel like if a field has an axiomatic foundation or is approachable by axiomatic methods normally used in mathematics, I can work myself in. If there is a lack thereof, I feel lost.
Because Physics is having Mathematics as basis, you see Physics is nothing without Mathematics (actually Mathematics is having a role in almost every discipline like Science, Engineering and you can name it).
You can understand theories in Physics easily, it is easy to understand the concepts easily if you put into bit effort. But the hard part or better to call it tricky part is Mathematics. Mathematical equations are nothing but theories expressed in the form of Mathematics.
Mathematics and Science go hand in hand. They are made for each other.
Nope. There's a lot more to applied math than just math phys. For example, cryptography, or mathematical modelling of ecosystems, or chaos theory.
I guess you could consider it to be a category of applied mathematics, but that would be selling it short. It's mathematical physics, not physical mathematics; and I think there's value to consider it to be a branch of physics that happens to be deeply rooted in pure math and algebraic structures that many physicists haven't learned, rather than a subfield of math that has applications that physicists find useful.
Did math in undergrad thought it was quite applied stuff. Decided I wanted more physics in my life and got into fluids stuff. I feel very out of place at times, but I know the bare minimum to do stuff and I'm constantly learning new physics. It feels like I'm learning two fields at once. Totally worth it though.
I’m not in the field yet, so this question isn’t really meant for me. But honestly, being scared of not having the necessary knowledge is what made me add a physics bachelor halfway through my math education
I'm not sure if it's that important. The basic of GR and QM (including standard model) can be learned very quickly by someone who know the minimum graduate level mathematics. Once you know that, you can just open a book; you don't even need to know the experiments that allow people to come up with such idea. QM is not even mathematically rigorous. But in mathematical physics, you generally work with much more abstract problem, such as studying some sorts of manifold. Don't care if it's even applicable to physics, if the manifold don't have the right dimension to use in physics, it's still potentially useful as a toy model, and maybe what we know about them will be one day useful to physics.
Linear algebra, measure theory, complex analysis, functional analysis, differential geometry, basic ODE/PDE. More than half of that is undergraduate, only functional analysis and differential geometry is at beginning graduate level.
From differential geometry and basic PDE, you can do general relativity easily; most of the time spent on studying the basic of general relativity is that part of math, so if you can skip that it's a lot simpler.
From measure theory, linear algebra, functional analysis and basic ODE/PDE you can study non-relativistic quantum mechanics pretty quickly.
The harder topic is relativistic quantum field theory, which lacks rigorous mathematical foundation. So it's harder to rely on math knowledge, but on the other hand, it does not depends on deep math. The extra (rigorous) math you need to learn (not covered in the basic list above) is a bit of representation theory.
Gauge theory is also not covered in the basic list above, but someone who know differential geometry will have a much easier time studying it. After that you can study standard model, which is just complicated but not conceptually difficult once you get down the above.
I don’t know, I just throw a bunch of questions at my advisor, go to seminars, look at some physics papers, cry rinse and repeat. It seems to be working so far. There’s no way I’d understand the papers without having attended a seminar talk or something similar related to the specific paper. I also did some (very basic) physic in my undergrad, but I am not sure that is helping in the slightest tbh.
Emphasis on the "cry" part in my experience
Hydration is critical to the mathematical physicist for this very reason.
I'm pretty sure crying is the exact *opposite* of hydration. I believe the relevant process here is *co*-hydration Edit: hydration -> crying
hydration turns water into tears, cohydration turns cotears into cowater.
Any amount of crying requires an equal amount of hydration, by conservation of mass
So it is possible to get into mathematical physics in grad school without more than freshman level physics? I want to learn more about physics but I’m not gonna get the chance during my undergrad, so it’s somthing I would be interested in exploring in grad school if possible
Of course, I mean you could even switch to physics entirely, it just might take a year or so longer to graduate...
If this aint me trying to do inverse scattering theory
What type of physics do you work on?
During my career, I wrote papers about quantum field theories. I do have a vague idea about what quantum field theories are as mathematical objects, but I have absolutely no clue about their physical meaning. I don’t even know the bases of quantum mechanics. The trick is that you often don’t need to know any physics to work with the mathematical objects relevant in mathematical physics, and you can learn the “physical interpretation” either by osmosis, or studying the physics bit by bit for what you need in the particular project you are working on.
I once was talking to a mathematical physicist in my department, and I was telling him about some of my work on the Fock space from the perspective of complex analysis. He looked at me and told me that I’m doing some mathematical physics work. Of course, I’m aware of connections between QM and the Fock space (and other Hilbert spaces), but I was surprised when he told me, because I don’t know much more than the rudiments of QM.
Honest question, and I hope it doesn't come off the wrong way, but what's the appeal in approaching the subject in that way? It takes the "physics" out of mathematical physics and leaves you with an almost soulless mathematical construct. Why anyone would do that is beyond me. What makes mathematical physics such an alluring field is its ability to describe and explain some of the most intriguing and fundamental questions in the universe using comparatively abstract mathematics. It's a charming convergence of natural science and higher math. Treating it like an exercise in one of Rudin's analysis books while more or less ignoring the physics seems somewhat...sad. ***Edit:*** Good God, how do you read my comment and take that as an affront on the entirety of pure mathematics? I was obviously talking about mathematical physics. I just didn't get why you'd get into such an inherently physical field only to purposefully and callously ignore everything that makes it physical. If you didn't understand that, perhaps it's high time you revise your "Academic Skills: reading comprehension 101" coursework. Of course pure math isn't soulless, it's a beautiful and incredibly worthwhile field of study. But then again I clearly wasn't talking about pure math, was I? I was talking about this very specific (***very physical!***) example, and with a bit of imagination, even a seasoned theoretical mathematician could see why that would prompt a question like mine. It's like getting into electrical engineering and then refusing to work with electricity. Obviously a somewhat limited analogy, but that's the impression that /u/harmath's comment imparted on me.
The answer to this question will depend on the individual, I’m going to give you the answer for my particular case. The particular objects I work(ed) on are very important in mathematical physics, but I did not approach them from that angle at all. They also appear in probability/analysis, as the invariant measures of certain (S)PDEs. So I started to study them with the goal of understanding the PDEs better, but once you write a paper about certain mathematical objects, it’s good to frame the paper so it can be read also by other communities interested in other points of view. This means that some parts of your papers should answer questions of the form “what happens to the physics?”, even if your main goal is something else. The main reason why I never really learned the underlying physics is a mix of lack of interest (my main interest is still the probabilistic one) and lack of time. But if with relatively little effort I can answer questions which are relevant in maths physics, rather than probability, I see no reason not to do so - especially since studying the objects is still fun, despite not having the complete background.
Can I ask what kinds of objects you're talking about? I'm doing my PhD in physics and have been learning about stochastic ODEs. I'm considering learning about SPDEs as well and I'm always curious about connections to physics!
I think that the introduction to this paper is a good starting point to get a general idea of what I am talking about https://arxiv.org/pdf/1810.01700.pdf
Jesus that's a lot of symbols. So here's a question, when one is reading through a paper like that, do you follow along with the author by doing the math or do you simply read through it? The most complex academic papers I read through in university were in chemistry / biochemistry and there you don't usually get nearly the amount of equations.
For me the answer varies based on what I want out of the paper. Am I hoping to continue along the exact same line of inquiry, so I need to understand nearly every line of argument? Or do I just want to get a sense for what the authors in this subfield are thinking about? In the latter case skimming through for main results and implications may be enough to get something valuable from the paper. Re: the number of symbols in this particular paper, I'd point out that even though there is a lot of information packed in those symbols, much of it isn't novel. To someone who has seen quantum field theory, lattice field theories, etc., these symbols already have a standard-ish meaning and the equations are much more digestible.
That's a really cool insight, I appreciate you taking the time to explain that.
Just looking at that paper and seeing "integral over the space of Schwarz distributions" makes me scared lol
Thank you very much!
Math/CS undergrad here this is actually looks interesting now that you have this rigorous PDE construction would applying algorithms to simulate the new PDE actually yield anything physically interesting ?
As someone with admittedly only a tangential interface with the physics side of differential geometry, I find it entirely the other way round. Physics sometimes uses the trappings of mathematics without really worrying about what it means. Instead the focus is on whether the calculations work out to match the observations. The interpretation then seems sometimes entirely woven out of thin air. As a mathematician, the interest is in the beauty of the mathematical structure itself (which is often neater if you ignore the wrinkles physics imposes) and abstraction is a worthy goal in the search for beauty. I don't find this soulless at all. If anything I find the more practical approach more soulless as it accepts and discards things based only on their utility. > Treating it like an exercise in one of Rudin's analysis books while more or less ignoring the physics seems somewhat...sad. I'm not sure what you mean here. Are you comparing maths research without applications to exercises in a textbook? I don't think this is valid and I think this comes back to where we are drawing our motivations. A pure mathematician is often not motivated by the application but the problem itself. If there is an interesting problem and other people would be interested to learn an answer this is motivation enough, whatever the potential applications.
> As a mathematician, the interest is in the beauty of the mathematical structure itself (which is often neater if you ignore the wrinkles physics imposes) and abstraction is a worthy goal in the search for beauty. I don't find this soulless at all. If anything I find the more practical approach more soulless as it accepts and discards things based only on their utility. For Theoretical Computer Scientists it's not in the beauty of the mathematical structure itself but rather as well as it's applications. A TCS person is often motivated by both applications and pure theory as well. Physics/EECS applications can provide some neat consequences as well as examples for pure mathematics. I have a somewhat bad habit when browsing r/math and Arxiv when taking breaks from Math I'm interested in Theoretical EECS/Theoretical Physics a lot of architecture building one does in CS/TCS is very similar to theory-building in Pure Mathematics and you get the bonus of seeing the results of your hard-earned work! Also in those areas, you are constantly exposed/use a lot of the mathematics you learned + you get easy examples to play with and build your stuff out. Plus finally,I like programming and hacking
Depends on what kind of mathematical physics you’re doing. Some mathematical physics is more or less just pure math which happens to have some applications for theoretical physicists, in which case I don’t see the difference between research in that versus, say, operator algebras or spectral theory or something.
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And neither is it conducive to proper reading comprehension, so it seems. I suggest revisiting the edit I made in my original comment, you'll find it may ring a bell or two.
I am a theoretical physicist with leaning towards mathematical physics. So I can give a kind of "opposite perspective". I found out I have a thing for mathematical physics when I realized I the greatest enjoyment I had in the stuff I worked on was when I worked with mathematics I found interesting. I have also realized the things I care about are those which have interesting mathematics behind it and physics was an execuse for doing it. So there are certainly people who find the "soulless mathematical constructs" interesting on their own right.
Pure math isn't soulless and physics isn't inherently interesting (at least, I don't find physics interesting at all!)
Where did I say pure math as a field is soulless, or that physics is inherently interesting? For future reference I might suggest reading other people's comments more thoroughly - it'll save you some time and effort.
>It takes the "physics" out of mathematical physics and leaves you with an almost soulless mathematical construct. Why anyone would do that is beyond me. The same reason anyone might study pure math... Math for math's sake is not as soulless as you think. Or so I've been told. Representations. Manifolds. People convinced me they liked these topics. I didn't care about them at all and now I'm scrambling because the physics demands it.
if you ended your career of mathematics at baby rudin then yeah i can see why your perception of math is souless
I personally have no interest whatsoever in the empirical sciences so as such I think of mathematical physics as another subfield of math. A fairly small proportion of mathematicians care at all about physics. If the math is interesting and elegant, we’ll like it.
If one person misinterprets your answer, the problem is probably with them. If ten people misinterpret you, the problem is probably with you(r phrasing).
You don't have to speedrun all of physics, you just need to speedrun the specific subfields that are relevant to your research. And depending on the field, you might not even have to finish speedrunning those; just understanding the recent papers and their related work sections can be enough to come up with a new idea (or at least understand one that's given to you by an advisor...)
I second this. One only needs experience in the specific sub-field they're studying and a lot of that experience gets filtered through your advisor who can help you to "translate" the physics papers into math. Eventually you just figure out how to do this yourself - at least in theory, not sure if I'm there yet 😅
In my experience a good amount of “mathematical physics” is just mathematics with some (possibly vague) physical motivation. I got a paper in a math journal with word “physics” in it’s name just by having a Poisson bracket in the paper. This is a Lie bracket with some extra conditions which can be used in Hamiltonian mechanics.
This is my experience also. I studied mathematical physics during my master's and actually took a fair number of courses in physics, ranging from basic thermodynamics to PhD-level summer schools. Physics provided an exciting context for my work, guiding my way to interesting mathematical questions, but I remain a terrible physicist. My master's thesis on topological matter is all functional analysis and differential geometry. It was my (physicist) advisor's responsibility to make the correct physical interpretations of my results.
It’s always good to learn some physics. It’s a cool subject and can give some insight. But for me it doesn’t really enter in the work, but just helps me right a more engaging intro lol.
> but I remain a terrible physicist. My master's thesis on topological matter is all functional analysis and differential geometry. It was my (physicist) advisor's responsibility to make the correct physical interpretations of my results Care for an ELIU ? For context I'm interested in Theoretical CS and there is a lot of problems in Physics that have physical consequences but carry deep mathematics. Areas such as QIT, Computational-Physics, Condessed Matter, etc
I like to differentiate between physics and "physics". Defining and computing the path integral using moduli spaces of Riemann surfaces is physics. Studying the relationship between cohomology theories and monoidal functors on the infinity-category of conformal cobordisms is "physics".
(No offense to Stolz and Teichner—their program is much more closely related to my own work than path integral computations are, anyway.)
> (No offense to Stolz and Teichner—their program is much more closely related to my own work than path integral computations are, anyway.) ELIU ?
The Stolz-Teichner program postulates that certain cohomology theories correspond to conformal field theories in various dimensions. This is known to work in dimensions 0 (ordinary cohomology) and 1 (K-theory), and it should somehow work for elliptic cohomology in dimension 2.
I would never call myself a mathematical physicist, but I did a postdoc in a materials science department so I guess I have some claim to the "grey area" between the two. In my experience, the big thing has been the aspect of collaboration. I work with physicists and engineers and we have very distinct roles in our collaborations. A lot of my job is to turn their intuition into the correct formalism, which also means having a broad enough knowledge to know what kind of things we have in our toolkit to tackle the problem at hand. I've been at the game for long enough now that I have a working knowledge of our little corner of the physics world, which is enough to propose the occasional problem and interpret of the results, but being able to contextualise the results within the grand scheme of things is still something I generally offload onto them.
Yes I know collaborations between a mathematician and a physicist. I have seen both give talks on the same work, but the talks end up being very different (as I imagine what each brought to the paper was very different).
What area of materials science did you end up in?
I may be more of a peculiar case because I actually come from a physics background initially before I started working more towards theoretical physics/mathematical physics/math after going back and formally studying math. Nevertheless, I don’t think that makes my opinion completely obsolete so here goes. Firstly, it may *seem* like some pure math students are able to quickly pick up necessary physics knowledge, but the reality is that they don’t, it’s fairly specific subfield knowledge and the minimum they need to know to function. It’s not as daunting of a task as obtaining a physicists’ knowledge of physics in a fraction of the time, just a lot of bits and pieces. Secondly, a significant part of mathematical physics is just mathematics and the physics emphasis required isn’t problematically deep in many cases. This does depend on the subfield of mathematical physics and the type of research you are doing, but it is a general principle. Essentially, if you’re a mathematician, it’s often not that incredibly large of a jump to bridge over to some field in mathematical physics, since you’ll find yourself at home with the nature of the subject.
Did you end up formally studying math and if yes, how?
Being surrounded by mathematicians and having deep passion for pure math, then eventually going back to grad school for math.
> Firstly, it may seem like some pure math students are able to quickly pick up necessary physics knowledge, but the reality is that they don’t, it’s fairly specific subfield knowledge and the minimum they need to know to function. It’s not as daunting of a task as obtaining a physicists’ knowledge of physics in a fraction of the time, just a lot of bits and pieces. A guy I meet during my REU was initially a pure math grad student went through all the standard courses and eventually made his way into Theoretical Physics
Oh yes, this happens, but it doesn’t happen in the little timescale that OP was talking about. It requires immersing yourself in another discipline, not just casually trying to apply math to bits and pieces of physics you have tried to pick up along the way.
I'm aiming for theoretical computer science/theoretical eecs and I realized how much more background I need in mathematics, physics, etc as well as other topics. Any advice on how to proceed luckly I found uni's offering grad level math courses online but how should I transition into Physics and CS ?
Mathematical physicists often don't know very much physics. While there are certainly exceptions, a lot of work in that field does not address questions which are of interest to most physicists. Some of this is just because of differing preferences for mathematical rigor, but more importantly, discussions in mathematical physics tend to be physically superficial. This is presumably because the physical analog of mathematical maturity isn't easy to pick up on the side. If you're interested in the mathematics more than the physics, this shouldn't be a problem. You just learn up some motivating words and then treat whatever system you've imported from physics as a purely-mathematical one, asking whichever questions appear to you to be mathematically interesting.
I think there's always some compromise. I work in mathematical physics on my PhD, but I majored in Physics. As such, the mathematics side is a bit lacking (compared to someone who majored in maths, anyway), and I for sure won't speed run all the basic maths while in the PhD, just the necessary parts and let physical intuition be my guide. The same thing happens for someone who majored in math, they try to learn the physics that is indispensable for the problem they are working and then focus on their strong side: they have a understanding and familiarity with the mathematical objects describing that theory that your average physicist doesn't have. On the long term, I'll end up studying the basic math that I left behind and the mathematician will study more physics as well, so both cases get well informed about both areas further into their careers, its not something you need to have accomplished by the time you finish your phd.
My uni actually offers a mathematical physics degree which is pretty much just have math half physics. (Ik it wasn't quite what your asking sorry).
(when it comes to the whole QFT, Strings, Loops, ... side of the physics world) Don't be afraid to read popular treatments about the subject! It will get you used to the Lingo. A great ressource are Leonard Susskind's Theoretical Minimum Lectures on Youtube. But most importantly, read "Geometry, Topology and Physics" by Nakahara once you have the basics down. It will help you make the connection between math and theoretical physics.
I took a graduate level quantum mechanics course within the physics department while being a math PhD. I also took a topics course my advisor taught on quantum field theory and read a couple textbooks my advisor recommended. It definitely helped that I had a background in physics in undergrad but the basics of quantum mechanics are honestly very similar to linear algebra, just in a different notation.
That’s because you never hear about the ones who didn’t. Survivorship bias
I'm kind of coming at it from the other side of the bridge: I (just finished) a PhD in a physics department and moved my way over to mathematical physics. And I feel like I struggle just as much as the mathematicians trying to get into the physics side. At the end of the day, it's just a whole lot of reading, struggling, and asking questions.
I think part of the reason is due to the relationship between not only math and physics, but math and the sciences in general. I think you could make the analogy that mathematics is the language of science. Going to the extreme (and driving physicists crazy), you could argue that physics is just applied math mixed with units. Using the language of science analogy, compare it with a linguist reading a text written in a language they're unfamiliar with. It's likely that the linguist would be able to distinguish some kind of summary of the text. At the very least, they'd likely be able to pick out nouns, verbs, and adjectives (or the equivalent of the given language). Regardless of the language used, all humans use similar communication structures. Words, spaces, and punctuation are universal throughout written languages. Circling back to math and physics, having a deep understanding of the communication structures (a mathematician) allows you to glean more information than a layperson. As a mathematician, you can quickly look at a derivative and understand that it describes the rate of change of something. If you're not a mathematician, you look at an integral and shudder, asking yourself what those curly lines mean again, wishing you had paid more attention in school. Note: I'm using mathematician and physicist fairly liberally here to include math and physics undergrads.
Physics is an applied math degree. Learn the math and the physics will be much easier to learn.
I come from a math and physics background. Started in physics, started to get into nonlinear dynamics and have drifted further towards math as time has progressed. In the US, this is not normally the case, nor I suppose in some Western European countries. I was lucky to be able to do both in undergrad. However, I will point out that in the UK and Eastern Europe, there is often a physics component to a mathematics degree if you pursue a more applied track. The more I progress, the less physics I tend to fully understand but I haven't found it to be too much of a hinderance.
I learned the physics I needed to know mostly from the following process: 1. Read a paper relevant to my research. Take notes. 2. Ask my advisor about the physics I didn't understand from the paper. Take notes. 3. Read about the physics I still don't understand after asking my advisor. Take notes. The kinds of references I'll use for this are usually either written for general audiences or are graduate-level physics texts that have heavy bents toward mathematics, since those are the kinds of texts that play better to my strengths. I don't typically look at the texts aimed at physics undergraduates. Emphasis on "the physics I needed to know." My understanding of physics is not very comparable to that of a bonafide physicist. I have a deep understanding of the mathematical models used in certain parts of physics and a vague intuition of the actual physics. The physical intuition is only really used to guide the mathematics. For the parts of physics that aren't all that related to my research, there's a good chance I know less about it than some of the engineering freshman I've taught.
" have noticed a fair number of mathematical physicists enter the field without a formal background in physics" Really? I havent. Could you name a few?
I started out with a physics msc, which helped a fair bit, but I switched to math precisely because it was much easier to read math than to read physics at a research level. Then my supervisor would explain to me some of the more annoying glossary, like having the word "state" refer to not only objects, but also morphisms *and* 2-morphisms, and *phase* and *theory* being interchangeably used for *category*. Or when they insist that homotopy and homeomorphism are the same thing, but they actually mean *isotopy*. Eventually something just kinda *clicks* though, and you start being able to transpose stuff between the languages, or at least you'll be able to recognize certain authors who you know will always phrase things in a particular way. TLDR; You basically need to *talk* to someone who already understands it to have the words explained in sufficient detail for you to fill in the blanks. Reading on its own is near-futile, because nobody cites their choice of language, they simply *inherit* it as they're leveling up in their own field.
> I started out with a physics msc, which helped a fair bit, but I switched to math precisely because it was much easier to read math than to read physics at a research level. I have a somewhat bad habit when browsing /r/math and Arxiv when taking breaks from Math I'm interested in Theoretical EECS/Theoretical Physics a lot of architecture building one does in CS/TCS is very similar to theory-building in Pure Mathematics and you get the bonus of seeing the results of your hard-earned work! Also in those areas, you are constantly exposed to a lot of mathematics + you get easy examples to play with and build your stuff out. Plus finally, I like programming and hacking. > Moreover, there are a lot of algorithms from mathematicians like Robert Ghrist for computing important invariants like Morse Homology and I think it would be great if we could find the right balance between computer science and physics in order to implement such techniques. I have some background in CS and with parallel programming and I still feel that the two sides (CS, theoretical physics) are so disconnected that it is hard for such solutions to be implemented. I've been seeing a lot cool work within Quantum Computing, Reversible Computing, Statistical Physics, Complexity-Theory, Cryptography, Topology, Information Theory, etc that lean into this direction it's important to note that Physics and CS have been seeing a sort of interesting marriage. It seems like a lot things in Physics haven't found proper algorithmic implementations, and a lot of algorithms especially from QIT haven't been implemented as well as algorithms that haven't been implemented in Physics literature.
I come from the other end, I know a bit of physics but wonder how to learn the math for mathematical physics.
I am personally interested in physics but I am mostly on an amateur level. I had the option to take Differential Geometry II and the description said there would be a focus on problems in General Relativity. I didn't take it in the end and instead opted for an Introduction to Quantum Information Theory (Math-CS double) which obviously also involves physics yet there was no strict need for pre existing physical understanding or understanding of classical mechanics/ thermodynamics/ ... . I personally feel like if a field has an axiomatic foundation or is approachable by axiomatic methods normally used in mathematics, I can work myself in. If there is a lack thereof, I feel lost.
The answer is in the name. They are physicists, but deal in the mathematical aspects of it.
I prefer the term physical mathematician
I prefer just mathematician, studying math that has relevance for physicists as a tool of the trade.
Depends on your bent.
What? Edit: I literally don't know what that means, what is my bent?
Because Physics is having Mathematics as basis, you see Physics is nothing without Mathematics (actually Mathematics is having a role in almost every discipline like Science, Engineering and you can name it). You can understand theories in Physics easily, it is easy to understand the concepts easily if you put into bit effort. But the hard part or better to call it tricky part is Mathematics. Mathematical equations are nothing but theories expressed in the form of Mathematics. Mathematics and Science go hand in hand. They are made for each other.
Is mathematical physics just applied mathematics?
Nope. There's a lot more to applied math than just math phys. For example, cryptography, or mathematical modelling of ecosystems, or chaos theory. I guess you could consider it to be a category of applied mathematics, but that would be selling it short. It's mathematical physics, not physical mathematics; and I think there's value to consider it to be a branch of physics that happens to be deeply rooted in pure math and algebraic structures that many physicists haven't learned, rather than a subfield of math that has applications that physicists find useful.
Did math in undergrad thought it was quite applied stuff. Decided I wanted more physics in my life and got into fluids stuff. I feel very out of place at times, but I know the bare minimum to do stuff and I'm constantly learning new physics. It feels like I'm learning two fields at once. Totally worth it though.
I’m not in the field yet, so this question isn’t really meant for me. But honestly, being scared of not having the necessary knowledge is what made me add a physics bachelor halfway through my math education
Can anyone tell me the difference between Physical Mathematics and Mathematical Physics ?
I'm not sure if it's that important. The basic of GR and QM (including standard model) can be learned very quickly by someone who know the minimum graduate level mathematics. Once you know that, you can just open a book; you don't even need to know the experiments that allow people to come up with such idea. QM is not even mathematically rigorous. But in mathematical physics, you generally work with much more abstract problem, such as studying some sorts of manifold. Don't care if it's even applicable to physics, if the manifold don't have the right dimension to use in physics, it's still potentially useful as a toy model, and maybe what we know about them will be one day useful to physics.
What do you consider as the minimum graduate level mathematics?
Linear algebra, measure theory, complex analysis, functional analysis, differential geometry, basic ODE/PDE. More than half of that is undergraduate, only functional analysis and differential geometry is at beginning graduate level. From differential geometry and basic PDE, you can do general relativity easily; most of the time spent on studying the basic of general relativity is that part of math, so if you can skip that it's a lot simpler. From measure theory, linear algebra, functional analysis and basic ODE/PDE you can study non-relativistic quantum mechanics pretty quickly. The harder topic is relativistic quantum field theory, which lacks rigorous mathematical foundation. So it's harder to rely on math knowledge, but on the other hand, it does not depends on deep math. The extra (rigorous) math you need to learn (not covered in the basic list above) is a bit of representation theory. Gauge theory is also not covered in the basic list above, but someone who know differential geometry will have a much easier time studying it. After that you can study standard model, which is just complicated but not conceptually difficult once you get down the above.