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 Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced ...
25
votes
10answers
4k 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:
...
15
votes
5answers
2k views
Why does calculus of variations work?
How does it make sense to vary the position and the velocity independently?
Edit:
Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
16
votes
7answers
2k 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 ...
9
votes
5answers
1k views
Mathematically-oriented Treatment of General Relativity
Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would
Prove all theorems used
Use modern "mathematical notation" as ...
22
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 ...
11
votes
4answers
939 views
If all conserved quantities of a system are known, can they be explained by symmetries?
If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1 conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver ...
37
votes
14answers
4k views
Number theory in Physics
As a Graduate Mathematics student, my interests 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 ...
20
votes
4answers
756 views
In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?
A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential.
Is there a one to one correspondence between the potential and its spectrum?
If the ...
8
votes
2answers
952 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
21
votes
9answers
2k views
Rigor in quantum field theory
Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
12
votes
8answers
2k 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 ...
13
votes
6answers
1k views
Quantum mechanics on a manifold
In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
11
votes
5answers
505 views
Reading list in topological QFT
I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm ...
12
votes
4answers
566 views
Reason for the discreteness arising in quantum mechanics?
What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
9
votes
2answers
570 views
Are there books on Regularization and Renormalization in QFT at an Introductory level?
Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one?
Added: I posted at math.SE the question Reference ...
12
votes
8answers
1k views
Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?
In the time dependent Schrodinger equation $\displaystyle, H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ , the Hamiltonian operator is given by
$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V$
...
10
votes
3answers
322 views
Representing forces as one-forms
First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics.
This question ...
2
votes
5answers
3k views
Mathematical background for Quantum Mechanics
What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
21
votes
13answers
2k views
Suggested reading for renormalization (not only in QFT)
What papers/books/reviews can you suggest to learn what renormalization "really" is? Standard QFT textbooks are usually computation-heavy and provide little physical insight in this respect - after my ...
26
votes
7answers
2k views
Is there something similar to Noether's theorem for discrete symmetries?
Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
6
votes
10answers
2k views
Physics for mathematicians
How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages ...
9
votes
3answers
884 views
Why can't General Relativity be written in terms of physical variables?
I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically ...
9
votes
5answers
784 views
Hubble's law and conservation of energy
If all distances are constantly increasing, as Hubble's law say, then lots of potential energies of form ~$\frac{1}{r}$ changes, so how is the total energy of the Universe conserved with Hubble's ...
6
votes
2answers
390 views
Introduction to string theory
I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I ...
4
votes
2answers
344 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 ...
23
votes
9answers
551 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 ...
3
votes
1answer
650 views
Are There Strings that aren't Chew-ish?
String theory is made from Chew-ish strings, strings which follow Geoffrey Chew's S-matrix principle. These strings have the property that all their scattering is via string exchange, so that the ...
4
votes
1answer
924 views
Gabriele Veneziano, strong nuclear force and beta-function
Background to the question:
From The History of String Theory:
Gabriele Veneziano, a research fellow at CERN (a European particle accelerator lab) in 1968, observed a strange coincidence - many ...
8
votes
2answers
489 views
Introductory texts for functionals and calculus of variation
I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good ...
5
votes
4answers
726 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
12
votes
4answers
2k views
A pedestrian explanation of conformal blocks
I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding ...
25
votes
4answers
1k views
Trace of a commutator is zero - but what about the commutator of $x$ and $p$?
Operators can be cyclically interchanged inside a trace:
$${\rm Tr} (AB)~=~{\rm Tr} (BA).$$
This means the trace of a commutator of any two operators is zero:
$${\rm Tr} ([A,B])~=~0.$$
But what about ...
13
votes
7answers
2k 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 ...
12
votes
1answer
370 views
Intuitive meaning of Hilbert Space formalism
I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:
The observables are given by self-adjoint operators on the ...
12
votes
3answers
1k views
What is the relation between renormalization in physics and divergent series in mathematics?
The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from ...
33
votes
18answers
2k 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 ...
12
votes
4answers
1k views
Haag's theorem and practical QFT computations
There exists this famous Haag's theorem which basically states that the interaction picture in QFT cannot exist. Yet, everyone uses it to calculate almost everything in QFT and it works beautifully.
...
8
votes
2answers
345 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
7
votes
2answers
700 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 ...
9
votes
2answers
682 views
What does John Conway and Simon Kochen's “Free Will” Theorem mean?
The way it is sometimes stated is that
if we have a certain amount of "free will", then, subject to certain assumptions, so must some elementary particles."(Wikipedia)
That is confusing to me, ...
8
votes
5answers
604 views
Does the Banach-Tarski paradox contradict our understanding of nature?
Since the Banach-Tarski paradox makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My ...
3
votes
2answers
275 views
Fractals in physics. Suggest book titles
I'm looking for some good books on Fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis ...
5
votes
6answers
761 views
Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)
We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
...
4
votes
3answers
284 views
Guessing what a simple partial differential equation is describing physically
Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this:
...
4
votes
5answers
475 views
If the size of universe doubled
My question is silly formulated, but I want to know if there is some sensible physical question buried in it:
Suppose an exact copy of our universe is made, but where spatial distances and sizes are ...
12
votes
2answers
181 views
What is a good introduction to integrable models in physics?
I would be interested in a good mathematician-friendly introduction to integrable models in physics, either a book or expository article.
Related MathOverflow question: what-is-an-integrable-system.
7
votes
3answers
452 views
Boundary layer theory in fluids learning resources
I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. ...
6
votes
1answer
329 views
Why Zeta regularization is not valid for multiple-loops?
Why zeta regularization only valid at one-loop?
I mean there are zeta regularizations for multiple zeta sums.
Also we could use the zeta regularization iteratively on each variable
to obtain finite ...
4
votes
2answers
394 views
Positive Mass Theorem and Geodesic Deviation
This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem.
The positive mass theorem was proven by Schoen and Yau using ...
10
votes
4answers
1k views
Where is the Atiyah-Singer index theorem used in physics?
I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my ...
