Questions tagged [mp.mathematical-physics]

Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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Product of Heavisides: calculus vs Fourier transform vs wavefront set

I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
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174 views

If NP=P, then does quantum mechanics have a classical interpretation? [closed]

Is it true that $BQP\subset NP$, if $NP=P$?
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3 votes
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Can any POVM be induced by a quantum instrument?

I suspect this is the obvious result of something in operator algebras, but that's far outside my field. Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
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2 votes
1 answer
90 views
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$2\mathrm{d}$ area maximizing short embeddings

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...
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3 votes
1 answer
188 views

Diophantine equations

It has been proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? Or any other ...
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3 votes
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Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed

I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ...
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3 votes
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Representations of minimal model primary fields in the Coulomb-gas Formalism

This question is in some sense a follow-up to [1]: is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? (This would ...
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6 votes
2 answers
96 views

Deriving Sommerfeld radiation condition from limiting absorption principle

For the Helmholtz equation $$ -(\Delta + k ^2) u = f, \label{1}\tag{1} $$ imposing the Sommerfeld radiation condition $$ \lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0 $$ on $u$ ...
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Find $\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right)$

I am doing a quantum optimization where the final problem has the following form $$\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right),$$ where $V \in \mathbb{C}^{d\times ...
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2 votes
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Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
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6 votes
2 answers
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Freeman Dyson's approach to string theory [closed]

Context: In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]: My dream is that I will live to see the day when our ...
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Most general space for the Wigner-Weyl transform

The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...
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2 votes
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Orthonormal basis of eigenvectors of Hamiltonian - Is there any theorem justifying the physicist approach?

In his book The Principles of Quantum Mechanics, Dirac states: "We call a real dynamical variable whose eigenstates form a complete set an observable." To Dirac, any observable has a ...
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2 votes
1 answer
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Kramers' escape problem: statistical physics vs. Large deviations

I'm almost not at all knowledgable in either Freidlin-Wentzel theory or Kramers' escape problem as it is known in the physics community, so please excuse some of my naivety. One can use Freidlin-...
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1 vote
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What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
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Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
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Vacuum vector and basis defined by anti-commuting operators

Let $\mathcal{H}$ be a finite-dimensional inner product space over $\mathbb{C}$. Suppose $A_{1},...,A_{N}$ are linear operators on $\mathcal{H}$ such that: $$\{A_{i},A_{j}\} = 0 \quad \mbox{and} \quad ...
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1 vote
1 answer
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Reference request: a mathematical model of classical physics

Before entering the university I studied the book "University Physics with Modern Physics" (written by Young and Freedman). In that time I used to study physical problems with methods taught ...
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Slice in momentum space?

This is probably a very basic question but I tried physics stack exchange already and I got no answers, so I'm asking the same question here. I was reading this article and the author considers the ...
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5 votes
1 answer
268 views

In search of a combinatorial proof on particular set of partitions

Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...
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12 votes
1 answer
259 views

Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?

Motivation The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...
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4 votes
1 answer
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Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...
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7 votes
1 answer
411 views

Along what curve does an ellipse roll down the fastest?

For the original Brachistochrone problem, Johan Bernoulli proved that it is the cycloid curve along which a circle* rolls down (without friction) the fastest under the influence of a uniform ...
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Trouble understanding Lax method for KDV equation for inverse scattering method

I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
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4 votes
0 answers
102 views

Axiomatic string theory?

There have been many proposal of a mathematical definition of Quantum Field Theory, for instance through Wightman or Osterwalder-Schrader axioms. Were there any efforts toward doing the same for ...
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3 votes
1 answer
71 views

Diagonalization of a specific Dirac operator

A few hours ago, a question was posed asking for the eigenvalues and eigenvectors of the Dirac operator $$ H=\begin{pmatrix} x & 0 & -i\partial_{x} & \bar{z} \\ 0 & x & z & i\...
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1 vote
0 answers
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$H^s$ norm of dispersive semigroup

The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm $$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
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2 votes
0 answers
73 views

Factorization algebras as factorizable cosheaves on the (extended) Ran Space

A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...
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2 votes
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68 views

On the "integrality condition" of the bilinear form in the Chern-Simons action

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be a principal $G$-bundle over a smooth orientable manifold $\mathcal{M}$. Furthermore, let $\langle\cdot,\cdot\...
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1 vote
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32 views

Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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1 answer
109 views

SU(2) and entangled particles [closed]

We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$ $$ \left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0 \right\rangle_A\otimes \left| ...
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2 votes
0 answers
64 views

Mathematical reason for scatter states being special?

In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics. My understanding is, if $O \in B(H)$ is a self-...
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2 votes
0 answers
80 views

Why do quantum observables form an associative algebra in some contexts?

In elementary quantum mechanics, we learn that quantum observables are self-adjoint operators that act on the Hilbert space of states. However, in more advanced context, we talk of local operators, ...
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Reed-Simon Vol. IV: Question regarding convergence of eigenvalues

I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
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1 vote
0 answers
67 views

Calculation of a multi-dimensional Fourier transform

I am interested in the following multi-dimensional Fourier transform: $$ \int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
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9 votes
2 answers
675 views

Undergraduate research in Topological Quantum Field Theory

I'm really interested in Topological Quantum Field Theory (TQFT) and am currently planning to focus on it in my undergraduate thesis. My university, unfortunately, does not allow double majors in ...
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4 votes
1 answer
140 views

Motivation for the axioms in Wick product

Here is a link for the definition of Wick product https://encyclopediaofmath.org/wiki/Wick_product, which defines the Wick product recursively. My question is where do these two equations come from? I ...
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2 votes
0 answers
65 views

Covariant momenta associated to higher order Lagrangians

Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$). Suppose that $L\in\Omega^m_{...
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16 votes
1 answer
680 views

From a physicist: How do I show certain superelliptic curves are also hyperelliptic?

As the title suggests, I am a physicist and have a question about how to show certain superelliptic curves are also hyperelliptic. The superelliptic Riemann surfaces in question has the form $$w^n = \...
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4 votes
0 answers
151 views

What is the natural framework for Lagrangians in QFT?

I wonder what is the natural geometric setting for Lagrangians in QFT, in the case of a general polynomial $P(\phi_i)$ of fields which could be scalars, or spinors etc: Are there natural, geometrical ...
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10 votes
2 answers
893 views

Early successes of Schwartz distribution theory

What are the early successes of Schwartz distributions theory? What are the hard theorems that became simple and what open problems were solved with this new tool soon after Laurent Schwartz released ...
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Eigenvalues of large block matrices arising in quantum many-body problems

I have a real block matrix that looks like: $$ M = \begin{pmatrix} D_0 &C_{01} & 0 & ... & 0& 0 & 0 \\ C_{10} & D_1 & C_{12} & ......
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5 votes
2 answers
586 views

Polygamma function in mathematical physics

Are there situations in which the polygamma pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the ...
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2 votes
2 answers
436 views

About a "magical" rearrangement-type inequality (used to prove Chandrasekhar mass limit in Hawking & Ellis' book)

The following excerpt from Hawking & Ellis' "The large-scale structure of spacetime" contains a proof of the Chandrasekhar limit to the mass of white dwarfs. The highlighted inequality ...
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2 votes
0 answers
42 views

Lie group and symmetry concept for weak notions of surfaces

I am studying measure-theoretic and functional analytic notions of surfaces such as varifolds and, since my background comes from physics I am wondering whether there is a simiar concept such as Lie ...
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38 votes
4 answers
2k views

Interesting and surprising applications of the Ising Model

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in ...
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4 votes
1 answer
120 views

Resource on spectral theory for differential operators with symmetry groups

In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that "A part of the analysis of [the periodic ...
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1 vote
0 answers
44 views

Intuition behind bound of second moment of Greens function by fractional moment

Consider the Hilbert space $ \mathcal{H} = l^2(\mathbb{Z}^d)$ for some dimension $d$ with basis given by the basisvectors $\{ \vert {x} \rangle \}_{x \in \mathbb{Z}^d} $. Let $A$ be an either self-...
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4 votes
2 answers
400 views

In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?

The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit. We begin with a Hilbert space $\...
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2 votes
0 answers
101 views

How to calculate the Lagrangian subgroup of $G\oplus\hat{G}$?

Let $G$ be an finite abelian group. We have known the following things: Denote the Drinfeld center of $\operatorname{Rep}(G)$ by $\mathfrak{Z}_1(\operatorname{Rep}...
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