I want to get a better understanding of relativity and quantum physics. Not just what we know, but how we know what we know. In other words being able to understand the math behind things like why don’t have a theory of quantum gravity. The physicist on a podcast I listen to puts it something like this. “The math doesn’t work when you try to unify the two. You get infinities or negative values where those kinds of results don’t make sense.” If I wanted to do that math for myself to gain a better understanding of the current field of knowledge, what math classes would I have to take?
I suppose I would start with typical university-level courses like (multi)linear algebra, multivariable calculus, differential geometry, functional analysis, etc. in addition to physics curricula. Sounds like a lot but, fortunately, all that stuff is available at the MIT Open Courseware site.
That renormalization stuff might start to require more advanced concepts like Lie theory, abstract algebra, complex analysis, maybe algebraic geometry, and so on.
My wife, who was a double major undergrad in Mathematics and Physics, and who just finished her Doctorate in Physics, says “All of it.”.
But after jokingly answering the above she says complex algebra, calculus, and linear algebra as a minimum, differential equations will be very helpful.
And that’s leaving out the physics, studies first of the limitations of Newtonian physics that lead to the theories behind quantum.
“you have chosen wisely, but it will be very confusing for a long time” were her exact words in closing.
“Both crying and alcoholism is allowed in physics” being her wry enjoinder.
One thing to understand is that quantum mechanics (QM), and especially quantum field theory (QFT, the special relativistic theory of quantum mechanics) is essentially all math, and a lot of it beyond classical wave theory is pretty abstract, so there is little reason to go study Lie algebra or group theory separately unless you are interested in them for their own sake; you’ll pick that up as you study the physics, and it will make more sense in the application. What you do need, beyond multivariate calculus, differential equations, and an essential understanding of probability and descriptive frequentist statistics, is a good grasp of classical wave mechanics, linear algebra, and at least an introduction into statistical mechanics which is both the historical and conceptual precursor into quantum mechanics. (I’m sure someone will come along and say that you don’t actually need statistical mechanics to understand quantum theory, and while that is technically true it will give you a lot of tools to understand how quantum mechanics actually relates to real world phenomena instead of just being this weird abstraction of rules that only apply at the level of individual particle interactions.)
If you want to get into general relativity (a.k.a. Einstein gravity)…be prepared for a lot of math, much of it tensors. Get really familiar with tensors, differential geometry, and maybe learn up on quaternions (also of use in some areas of QM but you can get by without them, and most physics students don’t even learn about quaternions until they come across someone else using them). Honestly, you can self-study the basics of QM and QED in about the equivalent of four classes; grasping all of Einstein gravity is a far greater challenge. And in the end, you are just going to understand that while general relativity is a remarkable theory, it is also really incomplete and very difficult to apply in all but almost trivial conditions without a serious amount of computing power and approximation methods.
There are a number of books on the topic suitable for self-study for a suitably motivated autodidact but I recommend the In A Nutshell series, and particularly Tony Zee’s Quantum Field Theory in a Nutshell and Einstein Gravity in a Nutshell. These are graduate-level texts but they do start with a relatively gentle introduction and don’t through you into the deep end MTW does. I have recommended to me A Most Incomprehensible Thing: Notes toward a very gentle introduction to the mathematics of relativity, which unfortunately has been sitting on my shelf unread for several years, as has Quantum Field Theory for the Gifted Amateur; they both look like pretty good introductions but I haven’t spent enough time to really read through them. A lot of people try to start with Feynman’s Lectures but they were intended as lecture notes for undergraduates and I think they leave something to be desired in terms of really getting a student up to a working level, and they are ‘Sixties era understanding that really predates the current understanding of the Standard Model and current understanding of the effects of general relativity in quantized conditions (such as it is), so they’re more for familiarization and enjoyment than a comprehensive understanding.
Stranger
Wife sends her Newtonian/Magnetic Love to Stranger’s detailed review (she’s Thin Films Magnetic Specialist) and agrees and amplifies his much more detailed recommendations than hers.
She has and loves the Feynman’s Lectures (and reads them in the tub) but thinks it’s more fun to watch some of the videos of the lectures online, but doesn’t dispute that to enjoy the lectures, you should already have your basic math under your belt already.
There is very little in mathematics that the combination of QFT and GM doesn’t touch in some way, and even the things that seem totally abstract or not immediately applicable like information theory or Noether’s theorem crop up unexpectedly. So, you can’t really just go in with the intent of learning all of the math first, and it probably makes a lot more sense to learn them in parallel so the abstract concepts in mathematics have some correspondence to physical systems.
Re-reading the o.p., I get the impression that they’re not so much looking to actually learn the math than to just understand what mathematics would be necessary to comprehend the reason for the disconnect between quantum theory and general relativity that prevents having a suitable quantized theory of gravity. In that context, “All of it” is a perfectly cromulent and succinct answer because you really need to understand how each of these theories work in their own domains and why field theory doesn’t quantize for gravity (at least, not for any current approach). The short answer is to study renormalization, but just focusing on that isn’t going to offer a very good intuition beyond the fact that the perturbation technique doesn’t work with gravity because of what a weak, long range force it is.
Stranger
Well, you won’t actually need all of math for those two fields of physics. But the math you won’t need is math subjects that you’ve probably never heard of before. Heck, some of the stuff you will need, you’ve probably never heard of before, either.
A lot of the math you’d learn in the physics courses themselves, but to even take those physics courses, you’ll certainly need multivariable calculus and linear algebra, and probably differential equations and complex analysis. Group theory isn’t strictly necessary, but I found it extremely helpful. A bit of non-Euclidean geometry wouldn’t hurt, either.
The physics sequence I took in college for my BS in civil engineering included units (about 5 weeks) on relativity and quantum theory. By that point in time, I had already taken differential, integral, and vector calculus along with differential equations and linear algebra.
The math involved with relativity and quantum theory felt pretty high-level to me but it was comprehensible thanks to the math courses I had taken.
At 5 weeks for the both of them, you would only have scratched the surface of both. Most likely, the relativity was almost all Special Relativity (which mostly doesn’t require any more than high school algebra), with at most a vague handwaving of General Relativity, and the quantum mechanics was probably mostly just electron orbitals (which are only like the fifth-simplest topic in QM, but the most practical).
The five weeks sounds about right when we studied Advanced Physics in my BSEE degree. But that was (mumble) 40 (/mumble) or so years ago, back when the universe was still forming and the physics was simpler. Based on the OP and some of the suggested readings here sounds like something I should get into now that I’m retired.
It was about 5 weeks apiece, or about 25-30% the entire 33-week physics sequence. I’m sure even then it was still just an introductory overview of the material.
Looking through my physics text, it took up about 400 pages to cover these topics (yeah, the book was that big). Binomial approximations, Lorentz transformations, relativistic momentum, equations describing the probability of locating quantons, Schrödinger’s Equation, etc. The text presents a great deal of Calculus in describing the underpinnings of all these things, but as I review the homework problems I see that the vast majority of them don’t require much beyond algebra. That more-or-less matches my recollection. Frankly I feel like E&M was more math-intensive.
And you’re right, Chronos, the relativity was all Special Relativity. The textbook author did kindly recommend a few texts if the reader chose to pursue the subject of General Relativity:
Exploring Black Holes by Taylor, Wheeler, and Bertschiger
Gravity by Hartle
A General Relativity Workbook by Moore
I have no idea if these texts are worthwhile.
I know it is super old, but I always enjoyed reading Gravitation by Misner, Thorne, and Wheeler. Similarly, Peskin and Schroeder is a completely bog-standard quantum field theory textbook.
That does not bring you to the current state-of-the-art research, which may require advanced mathematics, but one has to start somewhere (possibly at the risk of cluttering your brain with an understanding of topics like renormalization that is already quaint, so perhaps there are better recommendations for more modern textbooks?) John Baez (I mean Joan Baez’s cousin, not the folk singer) used to have a blog with a lot of links, but that only brings you up to 2010 [I suppose he still has a personal [blog](https://johncarlosbaez.wordpress.com/), but you would have to dig through it.], but that is around when Borcherds started publishing some of his work towards a more mathematically rigorous formulation of renormalization.
Gravitation (colloquially known as MTW for the author’s initials, and now available in a relatively inexpensive hardback with the 2017 printing) is a great reference for how comprehensive it is, but really not the place to start for self-study, and while it certainly covers a lot of ground it is the state of the field circa the mid-‘Seventies, which is adequate from a theoretical standpoint but doesn’t really go into modern modeling methods or anything regarding modern theories of cosmology. Once you get into Part III, it is assuming that you already have a solid understanding of differential geometry, and after that it is essentially presenting information with the assumption that you have already covered this in a more comprehensive context. I don’t know if anyone actually teaches from this book; it’s at least three or four solid semesters of material. Once you have covered the material, though, it is a great reference and has a lot of discussion that you don’t see presented well (or at all) in many other texts on GR.
For someone really wanting to work problems I’d start with Zee, or maybe Sean Carroll’s book (also sitting on my shelf having just skimmed through it but it seems like a good presentation). For someone who wants a technical introduction but doesn’t actually intend to work problems, Leonard Susskind’s General Relativity: The Theoretical Minimum is going to be a quicker read (relatively speaking) with a bare minimum of mathematical formality, but again, it isn’t really going to get into why renormalization is such a difficult problem with gravity.
Stranger
This is not an answer to the question of the o.p. but of more general interest as how mathematics, and specifically geometry underpins all of physics:
Stranger
I am reminded of this cartoon:
I’m a sophomore at a tech school. I believe the physics math curriculum is the entire calculus sequence (differential, integral, series, polar coordinates, multivariable), then differential equations, then vector tensor, with linear algebra and some amount of probability and statistics mixed in.
I’m an IE major so I don’t have to do all of it but have to do most of it. We only really use statistics and probability based math with the occasional derivative/integral/matrix.
Penrose’s The Road to Reality is a good rundown of many topics in physics, from the classical Newtonian to modern-day quantum field theory and gravity, to speculative approaches unifying the two. It doesn’t delve deeply into these topics, but it’s a good ‘road map’ of sorts: it mentions lots of topics and how they are connected, so you can pick and choose where to dive in more deeply.
Also, the Theoretical Minimum series by Susskind by now encompasses volumes on classical mechanics, quantum mechanics, special relativity and general relativity, based on the online lectures available at the linked site (for some reason, it seems that the latter two volumes aren’t yet referenced there).
@Stranger_On_A_Train already mentioned the excellent Quantum Field Theory in a Nutshell, but Zee has recently published a general-reader level introduction to QFT, Quantum Field Theory, as Simply as Possible. I haven’t spent any time with the book, but he’s a gifted expositor even of highly technical topics.
Thank you all for your replies 
.
I asked ChatGPT a question about relativity and QM. This is what I got…without a lot of math:
" Please review the theories of general relativity and quantum mechanics, and propose a method to send information faster that the speed of light"
As an AI language model, I cannot propose a method to send information faster than the speed of light as it violates the fundamental laws of physics. However, I can provide an overview of the theories of general relativity and quantum mechanics.
General relativity is a theory of gravitation that describes the structure of spacetime and how it is affected by matter and energy. It was developed by Albert Einstein and is based on the principle that the laws of physics are the same for all observers, regardless of their relative motion. In this theory, the force of gravity is not a force between two masses, but rather the curvature of spacetime caused by the presence of mass and energy.
On the other hand, quantum mechanics is the branch of physics that studies the behavior of matter and energy on a very small scale, such as atoms and subatomic particles. It is based on the principles of wave-particle duality, uncertainty, and superposition. The theory states that particles can exist in multiple states simultaneously, and their behavior cannot be predicted with certainty.
Regarding the possibility of sending information faster than the speed of light, it is important to note that this violates the principle of causality, which states that an effect cannot occur before its cause. Additionally, both general relativity and quantum mechanics have been extensively tested and have been shown to be accurate in their predictions. Therefore, any proposed method to send information faster than the speed of light would have to be able to reconcile with these fundamental theories of physics.
Research wormholes with respect to solutions to the field equations of general relativity and propose a method of sending information faster than light without violating causality.
Wormholes are a theoretical construct in general relativity that could potentially serve as a means of traveling faster than the speed of light or as a shortcut through spacetime. They are a solution to the field equations of general relativity and can be described as a tunnel or bridge between two distant points in space-time.
The idea of a wormhole was first proposed by physicist John Wheeler in the 1950s, and the concept has since been explored extensively in both science fiction and physics research. However, the existence of wormholes remains purely theoretical, and no experimental evidence of their existence has been found to date.
One proposed method of sending information faster than light without violating causality is the use of quantum entanglement. Quantum entanglement is a phenomenon in which two particles become connected in such a way that their states are dependent on each other, regardless of the distance between them. This connection occurs instantaneously, regardless of the distance between the particles.
Some researchers have suggested that it might be possible to use quantum entanglement to send information by encoding the information in the states of the entangled particles. While this would not violate the principle of causality since the information would still be transmitted at the speed of light, it would still face practical challenges such as the difficulty of maintaining and manipulating a sufficient number of entangled particles.
Another proposed method involves the use of cosmic strings. Cosmic strings are hypothetical one-dimensional objects that are thought to exist in the universe. They are believed to have formed during the early stages of the universe’s evolution, and their properties are governed by the laws of general relativity.
Some researchers have suggested that cosmic strings could be used to create a stable wormhole by bending spacetime around them in a way that would allow for a shortcut through space-time. However, this remains purely speculative, and no experimental evidence of cosmic strings or their ability to create stable wormholes has been found to date.
In conclusion, while the theoretical concept of wormholes holds promise for potentially allowing faster-than-light travel, no concrete method of utilizing them has been found, and current proposals to send information faster than the speed of light remain purely speculative. Any such method would have to reconcile with the fundamental principles of causality and the laws of physics.
Describe the information paradox in quantum mechanics
The information paradox in quantum mechanics is a long-standing problem that arises from the apparent conflict between the laws of quantum mechanics and the principles of general relativity. The paradox arises in the context of black holes, which are regions of spacetime where the gravitational pull is so strong that nothing, not even light, can escape.
According to the laws of quantum mechanics, information cannot be destroyed, and the evolution of a quantum system is always reversible. However, the formation and evaporation of a black hole appear to violate these principles, as it is believed that the information about the matter that falls into the black hole is lost forever.
This conflict between quantum mechanics and general relativity was first identified by physicist Stephen Hawking in the 1970s. Hawking proposed that black holes emit radiation, now known as Hawking radiation, due to the quantum fluctuations that occur near the event horizon of the black hole. This radiation is thought to carry away energy from the black hole and eventually cause it to evaporate completely.
However, the question of what happens to the information about the matter that falls into the black hole remains unresolved. According to the laws of quantum mechanics, this information cannot be destroyed, but if it is emitted in the form of Hawking radiation, it would be highly scrambled and difficult to extract. This has led to a paradox, as the information appears to be lost forever, in violation of the principles of quantum mechanics.
Various solutions to the information paradox have been proposed, including the idea that the information is somehow preserved within the black hole, or that it is emitted in a more recoverable form than previously thought. However, no clear resolution to the paradox has been reached, and it remains one of the most significant unresolved problems in modern physics.
Does a singularity at the center of a black hole violate quantum mechanics definitions of the planck constant
The singularity at the center of a black hole is a point where the curvature of spacetime becomes infinite, and the laws of physics, including quantum mechanics, break down. The Planck constant, which is a fundamental constant in quantum mechanics, defines the smallest possible length and the smallest possible unit of energy that can be measured.
The singularity at the center of a black hole does not violate the definition of the Planck constant per se, as it is not a violation of a specific numerical value. However, it does indicate a breakdown of the usual laws of physics, including quantum mechanics, and the inability of our current theoretical frameworks to accurately describe the behavior of matter and energy under such extreme conditions.
The Planck length, which is defined as the smallest possible length that can be measured, is thought to be on the order of 10^-35 meters. However, the singularity at the center of a black hole is thought to have zero size, which is well below the Planck length. This implies that our current understanding of physics, including quantum mechanics, is incomplete, and a more fundamental theory is needed to accurately describe the behavior of matter and energy under such extreme conditions.
Therefore, while the singularity at the center of a black hole does not violate the definition of the Planck constant directly, it does indicate the need for a more fundamental theory that can better describe the behavior of matter and energy in extreme environments where the laws of physics, including quantum mechanics, break down.
Do the theories of general relativity and quantum mechanics preclude the possibility that photons have rest mass
The theories of general relativity and quantum mechanics do not explicitly preclude the possibility that photons have rest mass, but they strongly suggest that photons are massless.
According to the theory of general relativity, gravity is the curvature of spacetime caused by the presence of mass and energy. If photons had rest mass, they would be affected by gravity and would not travel in straight lines at the speed of light, as observed. Therefore, the theory of general relativity implies that photons must be massless.
Similarly, the theory of quantum mechanics describes the behavior of particles at the subatomic level and predicts the existence of certain particles, such as the Higgs boson, based on their mass. If photons had rest mass, they would have been observed to interact with the Higgs field and acquire mass, which would have been detected in experiments. However, all experimental evidence to date indicates that photons are massless.
Additionally, the theory of quantum electrodynamics, which is a fundamental theory that describes the behavior of electromagnetism, is based on the assumption that photons are massless. If photons had rest mass, this would require a modification of the theory, which would have significant implications for our understanding of the behavior of light and other electromagnetic radiation.
Therefore, while the theories of general relativity and quantum mechanics do not explicitly preclude the possibility that photons have rest mass, the evidence strongly suggests that photons are, in fact, massless.
Well, that gave about the same level of information you would get from Wikipedia summaries on those topics, and with several of the same conceptual misperceptions. “Information-shaped sentences,” indeed.
Stranger