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46
votes
2answers
2k views

Intuitively, why are bundles so important in Physics?

I've seem the notion of bundles, fiber bundles, connections on bundles and so on being used in many different places on Physics. Now, in mathematics a bundle is introduced to generalize the ...
0
votes
2answers
52 views

Examples of measure zero sets in physics [on hold]

The problem with Riemann integration of functions like ${\chi }_{ℚ}(x)$ (a function which takes the value of 1 if $x\in ℚ$, and 0 otherwise) is usually used to show the need for defining the Lebesgue ...
1
vote
1answer
27 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
4
votes
0answers
68 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
4
votes
1answer
73 views

State space of interacting theories

Haag's theorem states that in general, an interacting quantum field and the corresponding free field have unitary-inequivalent state space representations. I would like to have an example of a state ...
0
votes
1answer
59 views

Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
0
votes
0answers
47 views

How does the functional Bethe Ansatz work?

This question is a bit general and less specific. I've understood the algebraic Bethe Ansatz so far, but since we have a completeness problem Sklyanin suggested a different method, the so called ...
2
votes
1answer
54 views

Show that the boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by transformation techniques. Any help would be ...
2
votes
0answers
51 views

Flow between two infinite plates [closed]

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
3
votes
1answer
118 views

Path dependent phase in quantum mechanics

In elementary treatments of quantum mechanics, we are taught that the wavefunction of a single particle is complex valued ($\Psi : \mathbb{R}^3 \to \mathbb{C}$). In particular, the wavefunction has a ...
0
votes
1answer
60 views

How to deal with eigenvectors which are not square integrable?

In Quantum Mechanics there is one type of situation I'm still unsure on how to deal with. First of all, I want to make clear I'm trying to understand how to deal with this rigorously. What I'm talking ...
4
votes
1answer
304 views

Resources for theory of distributions (generalized functions) for physicists

I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.
5
votes
0answers
127 views
+50

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as ...
29
votes
5answers
1k 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 ...
5
votes
1answer
72 views

Can we reconstruct 1D potentials in QM from the spectrum? [duplicate]

Knowing the potential, we can find the spectrum of the Schrödinger operator. The converse question is: Knowing the spectrum, can we reconstruct the potential? As an example, a harmonic potential has ...
1
vote
0answers
32 views

What does complexification mean for our particles in physics

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) ...
8
votes
1answer
397 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
5
votes
2answers
67 views

Conservation Laws and Symmtery

The toughest of topics in physics, like Quantum Mechanics, Relativity, String theory, can be explained in layman words and many have done so. Though there is no substitute to the understanding a ...
-6
votes
2answers
44 views
1
vote
1answer
27 views

How does height of a parachute affect air resistance compared to circumference or diameter?

I'm trying to find out how much a double in height (making it more ovular or oblong in shape) of the parachute affects air resistance compared to a double in circumference or diameter. Can someone ...
2
votes
2answers
128 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
1
vote
0answers
39 views

Wightman axioms always imply triviality in 4D?

Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial ...
9
votes
1answer
648 views

String theory from a mathematical point of view

I have a great interest in the area of string theory, but since I am more focused on mathematics, I was wondering if there is any book out there that covers mathematical aspects of string theory. I ...
2
votes
1answer
122 views

Divergence Theorem, mathematical approach to Gauss's Law?

Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is ...
1
vote
1answer
79 views

How does one make sense of a delta function of a scalar field?

Disclaimer: Originally posted on math SE, but thought that it was better in physics SE, so deleted my post on math SE and posted here. In the classic review summary of stochastic quantization here, ...
6
votes
1answer
278 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group? [duplicate]

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
2
votes
1answer
208 views

Floquet quasienergy spectrum, continuous or discrete?

I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The Floquet ...
3
votes
1answer
52 views

Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...
0
votes
0answers
62 views

Definition of Hamilton operator

The Hamilton operator is by definition a self-adjoint operator $H\text{: }D\left(H\right)\to\mathcal{H}$ with $D\left(H\right)\subset\mathcal{H}$ a dense linear subspace of the Hilbert space ...
1
vote
1answer
134 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
15
votes
1answer
1k views

Motivation for the use of Tsallis entropy

Every now and again I hear something about Tsallis entropy, $$ S_q(\{p_i\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't ...
0
votes
0answers
38 views

Configurations and configuration manifold in Lagrangian Optics

In Classical Mechanics, given a certain system of particles it is possible to consider the configuration manifold $Q$ which is a differentiable manifold whose points are possible configurations of the ...
9
votes
2answers
185 views

Does geodesic incompleteness in Penrose-Hawking theorems imply curvature blow up?

The singularity theorems in General Realtivity roughly stated say that given: A global causal condition An energy condition The existence of a closed trapped surface then spacetime must be ...
1
vote
0answers
17 views

Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
3
votes
1answer
68 views

For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the ...
3
votes
1answer
120 views

Can operators be argument of Dirac Delta function

In one part of Marc Bee's book on Quasielastic Neutron Scattering, he defines the pair correlation function $$ G(\textbf r,t) = \frac{1}{(2\pi)^3}\int I(\textbf Q,t)\text e^{-i\textbf Q.\textbf r}\ ...
5
votes
2answers
260 views

Examples of singularities in classical physics [closed]

I am a math teacher and I have to teach a topic called "Bruchterme" and "Bruchgleichungen" in german (I don't know the english word for it). For example $$ \frac{x^2 - 3}{(x - 2)x^2} + \frac{4}{x} + ...
0
votes
0answers
39 views

What is the first topic on mathematics,that an undergraduate physics student must study? [duplicate]

I aspire to become a theoretical physicist. I like to understand about infinite series and so on. Could you please recommend me any book?
3
votes
1answer
374 views

Instantons in Witten's supersymmetry and Morse theory

I'm reading Witten's paper on supersymmetry and Morse theory and am confused about the details of the instanton calculation which he uses to define a Morse complex (beginning at page 11 of the pdf) . ...
5
votes
0answers
129 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 $F = dA$ ...
8
votes
1answer
142 views

Lagrangian formalism and Contact Bundles

In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$: A line integral makes geometric sense only if ...
0
votes
0answers
30 views

Clarification of Type D space-time

I've asked this question on MathOverflow but received no feedback, so I thought I'd try better luck here. I've read the following formulation of the Goldberg-Sachs theorem in Chandrasekhar's ...
0
votes
0answers
36 views

Period for small oscillations is like simple harmonic motion

In Arnold's book on mechanics there is the following problem: Consider the period of oscillations near a minimum $E_0$ of the potential energy function $U$. Then he says to compute the limit of ...
4
votes
0answers
97 views

The classification of particles or fields in general spacetime- Is it still meaningful to say spin-0, 1/2 ,1 field in general spacetime? [closed]

In 3+1 dim Minkovski spacetime, the classification of particle or field, that is spin-0, 1/2 , 1..., depends on the representation of the universal covering group of $SO(1,3)$, that is $SL(2,C)$. When ...
3
votes
1answer
60 views

Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
2
votes
1answer
66 views

Infinite dimensional manifolds in general relativity [closed]

In GR the concept of a manifold is very useful. However, all of these manifolds are of finite dimension. Is it possible to define a manifold with infinite dimension (ie much like Hilbert space in QM) ...
1
vote
1answer
55 views

Can the inverse operator be expressed as a series?

I've seen the claim that a function of an operator can be defined as a series. For example, say $A: H_1 \mapsto H_2$ is an operator. Then $$ e^A \equiv \sum_{n=0}^\infty \frac{A^n}{n!}. \tag{1}$$ In ...
3
votes
1answer
181 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
2
votes
0answers
74 views

What new does geometric or deformation quantization give to physics? [closed]

What new does geometric quantization or deformation quantization give to physics? For example: prediction of new physical phenomena or just better tool for quantization. What can these schemes do in ...
1
vote
1answer
84 views

Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.