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47

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 ...


42

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 ...


41

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 ...


39

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 ...


34

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean? Perhaps you would find the geometric interpretations of the dot and cross products more intuitive: The dot product of A and B is the length of the projection of A onto B multiplied by the length of B (or the other way ...


31

Part of it is that 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 calculus ...


28

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 ...


28

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 ...


27

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 ...


26

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 ...


26

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 ...


25

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 ...


25

There are some problems with using quaternions to describe spacetime. Quaternions have two important properties: (1) they form a four-dimensional vector space; (2) you can multiply quaternions together.[1] The first property is obviously very suggestive, but it's no different from the usual four-vectors that we already use in special relativity. To ...


24

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 ...


24

As ACuriousMind has already noted, you can geometrically interpret the length of the cross product of two vectors as the area of the parallelogram (or as twice the area of the triangle) spanned by them, and (the absolute values of) its components as the areas of the projections of that parallelogram onto the coordinate planes. As for the dot product of two ...


23

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 ...


22

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 ...


22

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 ...


20

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 ...


19

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 ...


19

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 ...


19

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 ...


19

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 ...


18

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 ...


18

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 ...


17

A semi-silly idea that I've read about is the Primon gas, a model where the Riemann zeta function arises as the partition function of a quantum statistical mechanical system. More seriously, take a look at the papers of Yuri Manin and Matilde Marcolli on the hep-th arxiv, which attempt to connect the holographic principle to arithmetic geometry. I think ...



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