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16
votes
3answers
1k views

When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
10
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 ...
3
votes
1answer
194 views

Rigorous mathematical formalism of particle physics

Can anyone provide me with a rigorous mathematical definition of the fundamental particles (all fundamental bosons and fermions), reflecting the analogy of action of groups with interaction of ...
0
votes
0answers
65 views

Is assuming spacetime to have |2^N| points overkill?

The physical continuum is commonly assumed to have a mathematical continuum of points, that is, with a cardinality equinumerous with the Power Set of the set of natural numbers. However, since [ZFC + ...
5
votes
0answers
97 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
12
votes
6answers
1k 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 ...
7
votes
2answers
384 views

Lie algebra and Lie group about quantum harmonic oscillator

We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the ...
6
votes
2answers
344 views

How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ ...
4
votes
1answer
167 views

A question about canonical transformation

I have posted this question in math.stackexchange before with no answer till now. It may be more suitable to post here. There is a problem in Arnold's Mathematical Methods of Classical Mechanics ...
3
votes
1answer
130 views

References on $C^{*}$-algerbas, $W^{*}$-algebras and Quantum Theories

I would like to know some references regarding $C^{*}$ and $W^{*}$-algebras and quantum theories. I'm interested in concrete physical applications, models and problems. Here it is the list of ...
14
votes
5answers
1k views

What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
1
vote
2answers
273 views

Trace in non-orthogonal basis?

Physicists define the trace of an operator $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal basis, and this quantity is ...
2
votes
0answers
110 views

Solving the Schrodinger equation with appropriate symmetry

In the paper Markov Fields by Edward Nelson the introduction section claims that analytically continuing a Markov process with appropriate symmetry properties yields the solution of the Schrodinger ...
8
votes
3answers
132 views

“tmf$(n)$ is the space of supersymmetric conformal field theories of central charge $-n$”

I read this intriguing statement in John Baez' week 197 the other day, and I've been giving it some thought. The post in question is from 2003, so I was wondering if there has been any progress in ...
10
votes
0answers
295 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring ...
0
votes
2answers
254 views

Why is physics consistent? [closed]

It seems like it's a given that the laws of physics that govern our universe are consistent, and that inconsistency somehow is a reason to doubt an explanatory theory (such as famously Godel's ...
2
votes
0answers
35 views

Lie algebra and BPS stats

i would know what is a charges Lattice and the relationship between it and roots lattice ? and if there is relationship between mutation of quiver and weyl group? and how the PBS stats correspond ...
4
votes
1answer
72 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
2
votes
1answer
32 views

Consequences of tensor form due to various symmetries

I'm reading a paper on 2D hydrodynamics, specifically on the drag of a rod in a 2D fluid. It is a low-Reynolds number regime, therefore linear hydrodynamics and the velocity of the rod can be ...
36
votes
2answers
959 views

How trustworthy are numerically-obtained periodic solutions to the three body problem?

I recently found out about the discovery of 13 beautiful periodic solutions to the three-body problem, described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits. Milovan ...
2
votes
1answer
81 views

Deriving Cartan formula

I have trouble deriving Cartan formula of the form: $$ \mathrm{d} \omega (X,Y) = X[\omega(Y)] - Y[\omega(X)] - \omega([X,Y]) \tag{1} $$ where $\mathrm{d}$ is the exterior derivative, $\omega$ is a ...
54
votes
14answers
6k views

Number theory in Physics [closed]

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 ...
4
votes
2answers
217 views

Why do we require manifolds to be a topological space?

Roughly speaking, we define a manifold $M$ to be covered by a set of charts $\{(U_i , \varphi_i)\}$ such that locally the $n$-dimensional manifolds looks like $\mathbb{R}^n$. One of the conditions is ...
2
votes
0answers
53 views

Is rigorous functional analysis useful for theoretical physics? [duplicate]

I'm an undergraduate physics without much quantum mechanics at all under my belt. I'm studying functional analysis, and I want to know whether or not this will be useful for me in theoretical physics ...
0
votes
1answer
46 views

A question about Hamiltonian phase flow

Show that if a one-parameter group of difeomorphisms of a symplectic manifold preserves the symplectic structure then it is a locally hamiltonian phase flow. Note that A locally hamiltonian ...
4
votes
1answer
284 views

Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook

When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value ...
7
votes
2answers
257 views

Physical consequences of non-abelian non-trivial holonomy

The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the ...
4
votes
1answer
1k views

General procedure for Clebsch-Gordan expansions

I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
4
votes
1answer
190 views

What are type system examples of local gauge transformation- and field strength-like objects?

This is essentially a follow up motivated by this answer to my question about the gauge transformation interpretation of identity types. A field $$\psi:\mathcal M\to\mathbb C^n$$ is a section of the ...
7
votes
1answer
651 views

Why is the Hodge dual so essential?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric ...
9
votes
5answers
434 views

Motivation for tensor product in Physics

This question is about a mathematical object (the tensor product) but thinking about the motivation that comes from Physics. Algebraists motivate the tensor product like that: "given $k$ vector spaces ...
6
votes
3answers
286 views

Resources showing how to use differential forms in Physics

I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently ...
0
votes
0answers
39 views

Deriving diffusion coefficients from velocity field?

If I know the velocity, $\mathbf{v}(\mathbf{r},t)$, everywhere, is it possible to determine the diffusion coefficient, $D(\mathbf{r},t)$ everywhere as well? Would it be possible from using Fick's ...
11
votes
1answer
250 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
4
votes
2answers
107 views

Picture of supports

This questions stems from Axiomatic Quantum Field Theory and is mathematical in nature. However, I feel that an answer from physicists is more in line with what I will be asking. Let $\phi$ be a ...
9
votes
2answers
689 views

Schwinger representation of operators for n-particle 2-mode symmetric states

A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is $$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
2
votes
0answers
60 views

Solutions of PDEs in different coordinate systems

Suppose we have a PDE, for example the Helmholtz paraxial equation: $$ \nabla_\perp^2A+2ik\frac{\partial A}{\partial z}=0 $$ Solutions depend on the coordinate system we are using, i.e. we obtain ...
10
votes
1answer
662 views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
0
votes
0answers
57 views

To all experienced theoretical physicists out there, what is the step by step process in your math education? [duplicate]

I am not doing a physics degree but an engineering degree but i am planning using my free time to self study all the math in preparing myself to self study subjects in theoretical physics. (I've ...
4
votes
0answers
184 views

Topological Quantum Field Theories

I've asked this on Math.SE, but with no avail. So, I decided to ask it here. I was wondering about the following after reading the Wikipedia article on TQFTs. It is said that TQFTs have vanishing ...
1
vote
0answers
183 views

Divergent path integral

What does it mean to have a divergent path integral in a QFT? More specifically, if $$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$ What does this mean for the QFT of the field $\phi $? The field ...
38
votes
8answers
3k 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 ...
4
votes
1answer
448 views

CFT and the conformal group

Equations 2-7 on page 21 of these notes, http://www.math.ias.edu/QFT/fall/NewGaw.ps seems to give a fairly compact definition of what a CFT is. But I have two questions, This definition is ...
6
votes
2answers
161 views

Exponential of a differential operator

I have a differential operator $L$, $\displaystyle L = i (t\frac{\partial}{\partial z} - z\frac{\partial}{\partial t})$ I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, ...
5
votes
1answer
297 views

Couder-Fort Oil Bath Experiments and Quantum Entanglement Phenomena

The oil bath experiments of Couder and Fort have been able to reproduce various "pilot wave like" quantum behavior on a macroscopic scale. Particularly striking is the fact that the double-slit ...
6
votes
0answers
140 views

What does “mathematically well defined” quantum field theory mean? [duplicate]

Reading Wald's book (page 380, end of the first paragraph of section 14.1) while the author is giving an overall discussion of quantum field theories you can read However, for the more interesting ...
5
votes
2answers
214 views

Evaluate $1$-loop contribution to the $4$-point Green's function

I am trying to evaluate the following integral \begin{equation} I = \int \frac{d^d p_\text{E}}{(2 \pi)^d} \frac{1}{(p_\text{E}^2+m^2)((q_\text{E}-p_\text{E})^2 + m^2)} \tag{1} \end{equation} where ...
7
votes
1answer
155 views

Determining the Hodge numbers of some orbifold examples

I'm currently reading about complex geometry in order to get a feeling of how to determine the Hodge numbers, e.g. of certain orbifold constructions. Since I'm a physicist with no deeper mathematical ...
6
votes
1answer
146 views

Is basic quantum mechanics mathematically as robust a theory as special relativity?

This question is specifically about the robustness of mathematical models. Special relativity can be derived from very basic principles. Assuming that space is homogeneous and isotropic and that ...
3
votes
3answers
191 views

Discrete sum over a gaussian function

I have a sum of the form $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $α>0$ is some constant, and I don't mind if the limit $N\rightarrow\infty$ is taken. I know there is a possibility of ...