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 ...

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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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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
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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1answer
91 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
245 views

Paths in the path integral

In the path integral approach one defines in some heuristic way the functional path integral \begin{equation} Z=\int{\cal{D}}\phi e^{iS(\phi)} \end{equation} and the one claims that one must ...
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1answer
211 views

Research problems in application of Lie groups to differential equations [closed]

Are there any open problems in physics involving Lie groups and differential equations for a phd theses. Some applications are say, Noether's theorem in classical or quantum field theory. But I am ...
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760 views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow I have already difficulties in penetrating the literature... I'd highly appreciate any ...
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4answers
168 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
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 ...
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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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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
71 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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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 $$ \langle x | H | n \rangle = E_n \langle x | n \rangle \tag{2} $$ ...
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2k 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 ...
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2answers
153 views

What makes an abstract physical system describable by a “fluid” equations of motion?

We can describe (some of) the dynamics of many systems using fluid mechanics. Of course these include classical fluids like water, more exotic fluids like photon gases and the universe as a whole and ...
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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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2answers
185 views

Can we describe Quantum Mechanics using filters and matrices? [closed]

Can mathematical filters or ultrafilters be used to predict quantum physics 'events' as accurately as using matrices like Schrodinger did? Is there a way to explain some of the predictive power of ...
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1answer
146 views

Self-adjointness

I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by ...
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0answers
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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41 views
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1answer
159 views

What is the physical application of Navier-Stokes existence and smoothness?

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: Existence ...
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1answer
116 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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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 ...
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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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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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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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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 ...
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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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1answer
61 views

Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns. In arbitrary dimensions, what is the relation, if any, between solutions to ...
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3answers
521 views

What is the Hilbert space of a single electron?

Is it the same as the space of all possible descriptions of a single electron? If not, how do they differ? Please give the mathematical name or specification of this space or these spaces.
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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) ...
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565 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 ...
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3answers
602 views

Intuition for Path Integrals and How to Evaluate Them

I'm just starting to come across path integrals in quantum field theory, and want to get the right intuition for the them from the start. The amplitude for propagation from $x_a$ to $x_b$ is typically ...
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3answers
455 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
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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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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 ...
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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 ...
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3answers
291 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( ...
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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 ...
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1answer
125 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) = ...
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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 ...
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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 $$|n\rangle\langle n| =\prod_{m\ne n} \frac{A-a_m}{a_n-a_m}$$ ...
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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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.
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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 ...
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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 ...
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3answers
676 views

Intuitive meaning of the exponential form of an unitary operator in Quantum Mechanics

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator ...
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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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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. ...