DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com The mathematics tag covers non-applied pure mathematical disciplines that are traditionally not part of the mathematical physics ...

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46
votes
13answers
10k views

Best books for mathematical background?

What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory? Some subjects off the top of my head that probably need covering: ...
22
votes
9answers
5k views

How should a physics student study mathematics? [closed]

Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house). This question is broad, involved, and ...
32
votes
8answers
3k views

Does Gödel preclude a workable ToE?

Gödel's incompleteness theorem prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a theory of everything for physics? Edit: I haven't before seen ...
54
votes
6answers
2k views

The Role of Rigor

The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into ...
7
votes
4answers
10k views

What is the math knowledge necessary for starting Quantum Mechanics?

Could someone experienced in the field tell me what the minimal math knowledge one must obtain in order to grasp the introductory Quantum Mechanics book/course? I do have math knowledge but I must ...
25
votes
8answers
3k views

Classical mechanics without coordinates book

I am a math grad student who would like to learn some classical mechanics. The caveat is I am not to interested in the standard coordinate approach. I can't help but think of the fields that arise in ...
52
votes
14answers
5k views

Number theory in Physics

As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications ...
26
votes
6answers
2k views

Formalizing Quantum Field Theory

I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or ...
10
votes
4answers
1k views

What is the physical meaning of a “complete” Hilbert space in QM?

What does the word "complete" means from the physical point of view? I do not understand what it physically means to say that a Hilbert space is a complete vector space.
10
votes
3answers
511 views

infinite grid of planets with newtonian gravity

Assuming only Newtonian gravity, suppose that the universe consists of an infinite number of uniform planets, uniformly distributed in a two-dimensional grid infinite in both directions and not moving ...
14
votes
8answers
3k views

Crash course on algebraic geometry with view to applications in physics

Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ...
43
votes
11answers
5k views

Quantum Field Theory from a mathematical point of view

I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view. Are there any good books or other reference ...
20
votes
10answers
4k views

Applications of Algebraic Topology to physics

I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ...
18
votes
6answers
877 views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
11
votes
7answers
1k views

Is physics rigorous in the mathematical sense?

I am a student studying Mathematics with no prior knowledge of Physics whatsoever except for very simple equations. I would like to ask, due to my experience with Mathematics: Is there a set of ...
8
votes
3answers
2k views

A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
30
votes
10answers
1k views

Examples of number theory showing up in physics

My question is very simple: Are there any interesting examples of number theory showing up unexpectedly in physics? This probably sounds like rather strange question, or rather like one of the ...
20
votes
11answers
6k views

What is the logarithm of a kilometer? Is it a dimensionless number?

In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is. Suppose I have something with length $L = 1 km$. $\lg L = \lg km$ It ...
4
votes
10answers
1k views

Is it possible for a physical object to have a irrational length?

Suppose I have a caliper that is infinitely precise. Also suppose that this caliper returns not a number, but rather whether the precise length is rational or irrational. If I were to use this ...
2
votes
3answers
556 views

Mathematics for Quantum Mechanics [duplicate]

What math should I study if I want to get a basic understanding of quantum mechanics and especially to be able to use the Schrodinger's equation.
2
votes
1answer
231 views

Why is physical space equivalent to $\mathbb{R}^3$?

Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$? I am trying to understand what would be the logical reasons behind our assumption that our physical space is ...
0
votes
2answers
882 views

Level of calculus required for physics [closed]

First time for me here so kindly let me know if I violate the rules - especially if this is a duplicate. After reading the page how to become a good theoretical phycist, I started a serious revision ...
6
votes
2answers
885 views

Book covering Topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
11
votes
3answers
549 views

What happened with Hilbert's sixth problem (the axiomatization of physics) after Gödel's work?

I'll write the question but I'm not fully confident of the premises I'm making here. I'm sorry if my proposal is too silly. Hilbert's sixth problem consisted roughly about finding axioms for physics ...
7
votes
4answers
1k views

Number of dimensions in string theory and possible link with number theory

This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For k = 0 one gets ...
4
votes
2answers
473 views

Normalization of the path integral

When one defines the path integral propagator, there is the need to normalize the propagator (since it would give you a probability density). There are two formulas which are used. 1) Original ...
3
votes
2answers
3k views

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”? [duplicate]

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics? I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis ...
14
votes
4answers
1k views

Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
9
votes
3answers
851 views

What is a dual / cotangent space?

Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
7
votes
3answers
2k views

What math do I need for mathematical physics? In what manner should I learn math? [closed]

I'm a freshman undergraduate. I've got my sight on mathematical physics. I love math but I don't have the talent nor the inclination for purely abstract mathematics. I also love physics. The only ...
12
votes
17answers
3k views

Can pure maths create new theories in physics or does the “idea” ALWAYS come before the math?

I am in a debate with a friend about the value of string theory in physics. He is concerned that we are wasting valuable intellectual and financial resources on a path that is fanciful and can't ever ...
11
votes
3answers
715 views

Mathematical Physics Book Recommendation [duplicate]

Possible Duplicate: Best books for mathematical background? I want to learn contemporary mathematical physics, so that, for example, I can read Witten's latest paper without checking other ...
9
votes
6answers
597 views

In coordinate-free relativity, how do we define a vector?

Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change ...
6
votes
4answers
3k views

How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
13
votes
4answers
657 views

Does the axiom of choice appear to be “true” in the context of physics?

I have been wondering about the axiom of choice and how it relates to physics. In particular, I was wondering how many (if any) experimentally-verified physical theories require axiom of choice (or ...
6
votes
5answers
712 views

What is the meaning of following expresion $C=\frac{\delta Q}{dT}$ mathematicly

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even wikipedia) writes wrong expressions (from ...
6
votes
5answers
633 views

Are the solutions in radicals of cubic and quartic of any use in physics?

We all know that there are analytic formulae to solve quadratic, cubic and quartic polynomial equations. But it seems to me that the only solution that widely used is physics is the solution of ...
4
votes
2answers
369 views

Sum total distance of electrons on a spherical surface

What is the sum total distance between every possible pair of point charges when there are n point charges on a spherical surface? All point charges can only and are located on the infinitesimal ...
3
votes
2answers
407 views

What are some interesting calculus of variation problems? [closed]

That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)
9
votes
2answers
162 views

If a theory of everything exists, is it necessarily unique?

There is a lot of interesting debate over whether a "theory of everything" (ToE) is allowed to exist in the mathematical sense, see Does Gödel preclude a workable ToE?, Final Theory in Physics: a ...
6
votes
7answers
1k views

Why are radians more natural than any other angle unit?

I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for ...
0
votes
0answers
92 views

Literature request for books / review papers on gravitation, gauge theories and related mathematics [duplicate]

Similar to this reference, are there more such references / works [including textbooks] available in the literature? (A list would be greatly welcomed and appreciated.) With great appreciation.
-1
votes
2answers
241 views

Is there any phenomenon in physics which is sensitive to irrational numbers?

We can measure only rational numbers by our scale. Here is an example where irrational numbers does makes sense. If so then this question may have some theoretical importance. Is irrational numbers ...
13
votes
7answers
1k views

The philosophy behind the mathematics of quantum mechanics

My field of study is computer science, and I recently had some readings on quantum physics and computation. This is surely a basic question for the physics researcher, but the answer helps me a lot ...
4
votes
3answers
809 views

Direct Sum of Hilbert spaces

I am a physicist who is not that well-versed in mathematical rigour (a shame, I know! But I'm working on it.) In Wald's book on QFT in Curved spacetimes, I found the following definitions of the ...
4
votes
1answer
1k views

Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
14
votes
5answers
10k views

How can area be a vector?

My professor told me recently that Area is a vector. A Google search gave me the following definition for a vector: Noun: A quantity having direction as well as magnitude, esp. as determining ...
4
votes
1answer
92 views

Metric of a manifold foliated by maximally symmetric submanifold

I am reading the last chapter (Schwarzchild solution and Black Holes) of Sean Caroll's GR notes (http://arxiv.org/abs/gr-qc/9712019). While talking about spherical symmetry, he says how the ...
3
votes
4answers
3k views

How many digits of Pi are required in physics?

In other words: which physics experiment requires to know Pi with the highest precision?
2
votes
4answers
2k views

Necessary condition for square integrable functions?

I'm studying Quantum Mechanics and I came across this which I don't quite understand: For a vector space of functions $f(x)$ to be square integrable (i.e $\int{|f(x)|^2dx < \infty)}$, the necessary ...