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46

Something I posted on reddit answers this question quite well, I think: "Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number. By choosing the unit you use to express a ...


39

The answer is no, because although a "Theory of Everything" means a computational method of describing any situation, it does not allow you to predict the eventual outcome of the evolution an infinite time into the future, but only to plod along, predicting the outcome little by little as you go on. Gödel's theorem is a statement that it is impossible to ...


34

I'm not sure i'll be able to post all the links i'd like to (not enough 'reputation points' yet), but i'll try to point to the major refs i know. Matilde Marcolli has a nice paper entitled "Number Theory in Physics" explaining the several places in Physics where Number Theory shows up. [Tangentially, there's a paper by Christopher Deninger entitled "Some ...


31

Rigor is clarity of concepts and precision of arguments. Therefore in the end there is no question that we want rigor. To get there we need freedom for speculation, first, but for good speculation we need... ...solid ground, which is the only ground that serves as a good jumping-off point for further speculation. in the words of our review, which is ...


30

I think Conway's Game Of Life is a great example here. We have the "Theory of Everything" for Conway's Game Of Life--the laws that determine the behavior of every system. They're extremely simple, just a few sentences! These simple "rules of the game" are analogous to a "theory of everything" that would satisfy a physicist living in the Game Of Life ...


27

Rigorous arguments are very similar to computer programming--- you need to write a proof which can (in principle) ultimately be carried out in a formal system. This is not easy, and requires defining many data-structures (definitions), and writing many subroutines (lemmas), which you use again and again. Then you prove many results along the way, only some ...


27

I still think that it's not the right place for this type of questions. Nevertheless, the topic itself is interesting, and I'll also have a shot at it. Since I'm neither a philosopher of science, nor an historian (and there are probably very few such people on this site, one of the reasons this question might not be suitable), I'll focus on my own restricted ...


26

Part of it is that since Newtonian mechanics is described in terms of calculus. When we consider vibrational motions, we're talking about some particle that tends to not be displaced from some equilibrium position. That is, the force on the particle, at displacement $x$, $F(x)$, is equal to some function of displacement $x$, $g(x)$. There are two ways ...


23

There are a number of imprecisions in your question, mostly having to do with confusing the Lie group and its Lie algebra. I suppose this will make it hard to read the mathematical literature. Having said that, the first volume of Kobayashi and Nomizu is probably the canonical reference. Let me try to summarise. Let me assume that $H$ is connected. The ...


20

This is a so called Feynman diagram you see on the board. It is a suggestive way to write the formula written below the diagram. Each aspect of the diagram directly translates to part of the formula via the so called "Feynman Rules" With Feynman diagrams you can calculate the "amplitude" (that is related to the quantum mechanical probability for a process ...


20

Yes, logarithms always give dimensionless numbers, but no, it's not physical to take the logarithm of anything with units. Instead, there is always some standard unit. For your example, the standard is the kilometer. Then 20 km, under the log transformation, becomes $\ln(20\;\textrm{km}\;/\;\textrm{km}\;)$. Similarly, the log of 10 cm, with this scale is ...


19

The last book I read on "background in math for physicists" was "Mathematics for Physics" by Stone and Goldbart, and I enjoyed it quite a bit. (Since then I've tended to hit the pure math books, but that's a different story). Even better, a version of the book is available online at Paul Goldbart's webpage. Here's a list of topics: * Calculus of ...


19

I feel very strongly about this question. I believe that for an experimentalist, it's fine to not go very deeply into advanced mathematics whatsoever. Mostly experimentalists need to understand one particular experiment at a time extremely well, and there are so many skills an experimentalist needs to focus all of their time/energy on developing as ...


19

Arthur Suvorov gives a nice comment, I am just going to give a list of a few specific physical problems I can think of from the top of my head. Yang Mills existence and mass gap (Millenium Prize) and generally the problematic of rigorous definitions and constructions of quantum field theories Navier Stokes equations and smoothness (also Millenium) - it's ...


18

Vector spaces because we need superposition. Tensor product because this is how one combines smaller systems to obtain a bigger system when the systems are represented by vector space. Hermitation operator because this allows for the possibility of having discrete-valued observables. Hilbert space because we need scalar products to get probability ...


17

No, physics is not rigorous in the sense of mathematics. There are standards of rigor for experiments, but that is a different kind of thing entirely. That is not to say that physicists just wave their hands in their arguments [only sometimes ;) ], but rather that it does not come even close to a formal axiomatized foundation like in mathematics. Here's an ...


17

It is implicit in the wording of the question that string theory is an example of math coming first, but this is false. String theory grew out of Regge theory which was and is a phenomenological theory of the strong interactions which applies to scattering processes at high energies but small momentum transfer. This in turn is connected to the Regge ...


17

Here's one "mathematical" but highly unphysical answer. Using that $km\cdot km = (km)^2$ etc, we can formally define arithmetic of numbers with units over a graded algebra $A = \oplus_{k\in \mathbb{N}} V_k$ where $V_k = \otimes^k V$ where $V$ is treated as a one-dimensional real vector space ($V_0$ is the scalar $\mathbb{R}$). The choice of unit is the ...


16

There are many attempts for a physical proof of the Riemann hypothesis. The major work in this direction was summarized in a recent review by: Schumayer and Hutchinson. One of these attempts was proposed by: Berry and Keating. Their suggestion is within the framework of the Hilbert–Pólya conjecture, according to which, the Hilbert–Pólya Hamiltonian, whose ...


15

1. I am in love with Fecko's Differential Geometry and Lie Groups for Physicists. Despite not being just about mechanics (but rather about more or less all rudimentary modern theoretical physics) it discusses both Lagrangian and Hamiltonian formalism. It also provides countless exercises (with nice hints) so that you can really get a feel for the matter. ...


15

I can by no means claim to give a full answer on this question, but perhaps a partial answer is better than no answer at all. As regards (1) perhaps the most famous example is the Navier-Stokes equation. We know it produces extremely good results for modeling fluid flow, but we can't even show that there always exists a solution. Indeed, there is a Clay ...


15

If you're a mathematician and you want to understand QFT, you're going to have to grapple with renormalization sooner or later. Your life will be easier if you understand from the beginning that the Wilson-Weinberg-etc 'effective field theory' philosophy is the essential organizing principle for the whole subject. In particular, you're going to need to ...


15

Sean Carroll's Lecture Notes on General Relativity contain a superb introduction to the mathematics of GR (differential geometry on Riemann manifolds). These also also published in modified form in his book, Spacetime and Geometry. Spivak's Calculus on Manifolds is a gem. Bishop's Tensor Analysis on Manifolds is a great introduction to the subject, and ...


15

No, nothing in physics depends on the validity of the axiom of choice because physics deals with the explanation of observable phenomena. Infinite collections of sets – and they're the issue of the axiom of choice – are obviously not observable (we only observe a finite number of objects), so experimental physics may say nothing about the validity of the ...


14

There are two aspects to this question: 1) Which sources try to communicate the usual vague and speculative physics story in a way that mathematicians are more likely to appreciate? 2) Which sources try to give an actual mathematical treatment of QFT, something that lives up to being maths? For the first, Deligne et al's Quantum Fields and Strings is ...


13

Frankly I think Yuji's deleted answer hits the nail on the head, in that concerning yourself with the sociology of various departments is rather a bad idea. "Mathematician"'s answer is essentially the opposite of this advice, making very generic claims about the nature of researchers in various departments, and I believe this is poor advice. The exact ...


13

In the simplest form the saddle point method is used to approximate integrals of the form $I \equiv \int_{-\infty}^{\infty} dx\,e^{-f(x)}$ The idea is that the negative exponential function is so rapidly decreasing -- $e^{-10}$ is $10000$ times smaller $e^{-1}$ -- that we only need to look at the contribution from where $f(x)$ is at its minimum. Lets say ...


13

I depends on the book you've chosen to read. But usually some basics in Calculus, Linear Algebra, Differential equations and Probability theory is enough. For example, if you start with Griffiths' Introduction to Quantum Mechanics, the author kindly provides you with the review of Linear Algebra in the Appendix as well as with some basic tips on probability ...


13

The set of irrational numbers densely fills the number line. Even assuming that quantum mechanics doesn't disable the preimse of your question, the probability that you will randomly pick an irrational number out of a hat of all numbers is roughly $1 - \frac{1}{\infty} \approx 1$. So the question should be "is it possible to have an object with rational ...



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