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32

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 ...

29

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 ...

29

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 ...

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 ...

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 ...

19

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 ...

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

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 ...

16

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 ...

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

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 ...

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

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

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 ...

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

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

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 ...

12

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 ...

12

Here is a toy example; I don't know how interesting this will be to physicists. The eigenvalues of the Laplacian acting on, say, smooth functions $\mathbb{R}^k/(2\pi \mathbb{Z})^k \to \mathbb{C}$ are given by $$\{ m_1^2 + ... + m_k^2 : m_i \in \mathbb{Z} \}.$$ as a multiset (that is, with multiplicities). These are the energy eigenvalues of $n$ free ...

11

This question is the analogous of : Is Romeo and Juliet the result of grammar and syntax or is there an external input necessary? In my opinion mathematics as far as physics goes, is a tool. A beautiful tool, tools are very important for creating stuff, but still physics is a meta level to mathematics. Mathematics is necessary for rigorous physics but not ...

11

This might be more of a math question. This is a peculiar thing about three-dimensional space. Note that in three dimensions, an area such as a plane is a two dimensional subspace. On a sheet of paper you only need two numbers to unambiguously denote a point. Now imagine standing on the sheet of paper, the direction your head points to will always be a way ...

11

People tend to take Gödel's theorem and bend it, stretch it, misstate it, misapply it, and generally do things to it that, if you did them to a cockroach in Texas, would get you arrested for animal cruelty. But there is a book, Franzén (2005), that should be enough to inoculate any responsible adult against such naughty behavior. Some points made by Franzén: ...

11

To give an example where convergence of Cauchy sequences is important: time-evolution is typically calculated as $$|\psi(t)\rangle = e^{i\hbar^{-1} \hat{H}\cdot t}|\psi_0\rangle$$ now, the exponential of an operator is defined1 by $$e^{\hat{A}} = \sum_{i=0}^\infty\frac{\hat{A}^i}{i!}$$ where the sum in turn is defined by $$... 11 From a pure mathematical point of view the answer is negative. As you probably know, wavefunctions are all of the functions \psi from, say, R to C such that |\psi(x)|^2 has finite (Lebesgue) integral, namely \psi belongs to the Hilbert space L^2(R). One can simply construct functions that belong to L^2(R) and that oscillate with larger and ... 10 With suitable boundary conditions, the decomposition is unique. Without them, it's not. Suppose that (\phi,{\bf G}) and (\phi',{\bf G}') are two different decompositions for the same function. Then$$ \nabla(\phi-\phi')+\nabla\times({\bf G}-{\bf G}')=0. $$Take the divergence of both sides to find that$$ \nabla^2(\phi-\phi')=0. ...

10

Your proof is right (and I voted it up accordingly). But this is a result that's worth proving a few different ways, because the different ways lead to different insights. So I'll give some alternative proofs. Proof 2 (yours is proof 1): Taking linear combinations of the equations we're trying to solve, we get an equivalent pair of equations. $$... 10 This is a fun question. I have a hard time getting a good grip on the transformation that is ln so I'll write things in terms of exponents.$$value = \ln(10\ \mathrm{ km})e^{value} = 10\ \mathrm{ km} The number $e$ is, of course, unit-less. If I raise a number to a power, what are the permissible units of the power? If I write $x^2$, I have an ...

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