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41 views

Action principles and covariant equations [duplicate]

Can every physically sound differential equation, that is covariant, deterministic etc. be derived by extremising a suitable action using a suitable lagrangian, that may be arbitary. Is this a ...
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0answers
71 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
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3answers
710 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
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2answers
91 views

Good Fiber Bundles reference for Physicists

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of ...
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3answers
924 views

Why does Law of Large Numbers work?

Often I see books and professors reasoning that, in order to make a good experiment, many measurements are necessary because then the average value of a quantity is closer to the expected value ...
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77 views

Several Complex Variables in QFT

After reading the very interesting quote about several complex variables in QFT: "The axiomatization of quantum field theory consists in a number of general principles, the most important of ...
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2answers
90 views

Domain of simple quantum harmonic oscillator

When discussing the spectral theory of unbounded operators, one often starts with an operator defined on a densely defined subspace of your Hilbert space, and then proves that the operator is ...
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0answers
37 views

Hopf Algebras in Quantum Groups

In the theory of quantum groups Hopf algebras arise via the Fourier transform: A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier ...
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1answer
48 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
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51 views

Learning Roadmap to Mathematical Physics [duplicate]

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...
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1answer
80 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
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6answers
639 views

Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e. $$m \frac{\mathrm{d}^2 ...
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286 views

A Theorem Due to Hodge: Hawking/Ellis

This is probably quite an obscure question but hopefully somebody has a simple answer. I'm studying the proof of the topology theorem on black holes due to Hawking and Ellis (Proposition 9.3.2, p. 335 ...
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1answer
93 views

How to choose the Correct Green's Function?

In order to solve the Green’s function of the Helmholtz operator $$(\nabla^2+k^2)G(\vec r-\vec r’)=\delta^{(3)} (\vec r-\vec r’)$$ one can obtain four different Green’s functions corresponding to four ...
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1answer
94 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 ...
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1answer
48 views

Finding expectation value of $p^2$ without integrals

So the expectation value of momentum, if you know the expectation value of position is $$\langle p \rangle = m \frac{d\langle x \rangle}{dt}$$ Is there a nice formula like this for $\langle p^2 ...
7
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0answers
88 views

Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
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1answer
72 views

Proof oriented subjects, similar to computational complexity [closed]

I'm starting my second year as an undergrad math major. I quite like the kind of thought involved in my pure math classes (analysis, abstract algebra), but I also like my physics and (theoretical) ...
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1answer
102 views

Which textbooks contain info on Bessel functions & their use as basis functions?

As an exercise my research mentor assigned me to solve the following set of equations for the constants $a$, $b$, and $c$ at the bottom. The function $f(r)$ should be a basis function for a ...
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2answers
65 views

Could a trial wavefunction providing exact eigenenergy differ from the exact eigenfunction by a zero measure function?

Given the eigenequation of a Hamiltonian $$ H |n \rangle = E_n |n \rangle \tag{1} $$ We write it in the position representation $$ \langle x | H | n \rangle = E_n \langle x | n \rangle \tag{2} $$ ...
5
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1answer
57 views

What's the definition of incompleteness of a coordinate system and a spacetime?

I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This ...
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0answers
41 views

Examples of application of detour matrices in physics?

Are there any good examples of application of detour matrices in physics?
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0answers
58 views

Greens function/resolvent of hydrogen Hamiltonian

Let $H$ be the Hamiltonian for the nonrelativistic hydrogen atom, i.e. $$H=-\frac{1}{2}\Delta-\frac{1}{r}$$ I am searching for an asymptototic expansion of the Greens function or respectively the ...
7
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1answer
121 views

Relation between cohomology and the BRST operator

Given a manifold $M$, we may define the $p$th de Rham cohomology group $H^p(M)$ as the quotient, $$C^p(M) \, / \, Z^p(M)$$ where $C^p$ and $Z^p$ are the groups of closed and exact $p$-forms ...
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272 views

TQFTs and Feynman motives

Questions Is a topological quantum field theory metrizable? or else a tqft coming from a subfactor? For a given metric, are there always renormalization and Feynman diagrams? Is there always a Feynman ...
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62 views

Relation of Betti numbers to Veneziano's scattering amplitude?

I came across Veneziano's famous formula for the scattering amplitude for four tachyons written as $$A(s,t)= \sum_{n \geq0} \frac{(-1)^n}{n-1 + ...
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81 views

Disclinations, dislocations, lattices, Displacement fields and scaling

I am looking up Frank, and Burger vectors and associated material on dislocation/disclination. It seems straightforward describing a lattice and what dislocation means. It is even possible to restrict ...
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2answers
86 views

Conductors and Uniqueness Theorem

I'm working with Griffiths Electrodynamics, and he introduces a uniqueness theorem: First Uniqueness Theorem: The potential $V$ in a volume $\Omega$ is uniquely determined if (a) the charge ...
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2answers
91 views

System without ground state is not real in nature?

We know that Coulomb force is real phenomena in nature and with Coulomb potential $V(x) \thicksim -\frac{1}{|x|}$ lowest energy is bounded in hydrogen atom. But it's mathematically clear that if ...
3
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0answers
76 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
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0answers
48 views

How to carry out the perturbation expansion of an anharmonic oscillator to high orders?

I think this is a standard problem in quantum mechanics. Consider the anharmonic oscillator $E \psi = \left(- \frac{1}{2} \frac{\partial^2}{\partial^2 x } + \frac{1}{2}x^2 + \epsilon x^4 \right) ...
4
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0answers
128 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
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22 views

Can isotropy (or anisotropy) be expressed in terms of intervals ($s^2$) between pairs of events?

Considering a set $\mathcal S$ of events and given the values of intervals $s^2[~P, Q~] \in \mathbb R$ for all pairs of events $P, Q \in \mathcal S$ (up to a common non-zero scale factor): how can ...
5
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4answers
169 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 ...
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1answer
34 views

What are absolutely continuous spectrum and singularly continuous spectrum?

I am now reading some mathematical note on Anderson localization. It mentioned two types of continuous spectrum. What are absolutely continuous spectrum and singularly continuous spectrum? I only had ...
1
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1answer
48 views

Mapping Issues with Unbounded Operators

Consider the operator-valued generalized function $\phi^{(k)}_{l}:=\phi^{(k)}_{l}$ on space-time $\mathcal{M}$. Now, smooth the operator-valued generalized function with test function $f(x)$ ...
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3answers
292 views

Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?

I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( ...
1
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3answers
76 views

What is the relationship between $V(t)$ and $V(x,y,z)$

I was recently asked this by a friend. He told me that coming from a physics background, he does not understand $V(t)$ and he believes it is purely theoretical construct made up by circuit ...
7
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2answers
184 views

Mathematical physics text with plenty of applications

I'm looking for texts on mathematical physics. I've seen various other threads, but the texts recommended in those threads were mathematical methods of theoretical physics texts, that is to say those ...
2
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1answer
66 views

Approach to expressing $|n\rangle\langle n| $ as a polynomial when eigenvalues are degenerate?

If ${|n\rangle}$ are eigenvectors of an operator $A$ then $|n\rangle\langle n| $ can be expressed in terms of a finite order polynomial $$|n\rangle\langle n| =\prod_{m\ne n} \frac{A-a_m}{a_n-a_m}$$ ...
2
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1answer
82 views

Is the fine structure constant a rational number?

Since the fine structure constant (denoted alpha) is a pure real number, it just occured to me to ask if it is a rational number or not.
1
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1answer
89 views

Moduli spaces in string theory vs. soliton theory

In both string theory and soliton theory, moduli spaces are frequently used. As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for ...
6
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1answer
180 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
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2answers
94 views

A new way of looking and formulation of Observables for a new quantum theory

I often think of basic of QM (although it wasnt discovered that way) is that we have a physical parameter/observable and the favourite one is the displacement $x$ of a particle $P$. Its conjugate ...
2
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0answers
64 views

What is elliptic genera?

What is elliptic genera in physics? Reading many relevant papers, they just defined elliptic genus as sort of partition function. I try to find useful materials to explain it, but I couldn't find ...
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0answers
41 views

Mathematical books to become a successful mathematical physicists [duplicate]

My understanding of algebraic topology and Riemannian geometry come from Nakahara's Geometry, Topology, and Physics, which I do not think is sufficient. I am first year PhD student, and I want to do ...
1
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1answer
75 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
5
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2answers
143 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
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1answer
37 views

Charge and current density fields

The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this ...
3
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1answer
127 views

Graph Theory and Feynman Integrals

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals, Section 2.3, the alpha representation of general Feynman integral takes the form $$ F_{\Gamma}(q_1,\ldots,q_n;d) = ...