We’re rewarding the question askers & reputations are being recalculated! Read more.

Questions tagged [mathematical-physics]

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 mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

387 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
2
votes
0answers
37 views

The role of $\dim H_n$ in the definition of asymptotically continous functions on vectors

When considering the asymptotic continuity of quantum states, one works with asymptoticly continuous functions. In the definition one has the following, a funtion f is asymptotically cts if for a ...
2
votes
0answers
65 views

Relation between representations of the Poincaré group and linear PDEs

What is the relation between finite dimensional representations of the Poincaré group and Poincaré-invariant linear homogenous partial differential equations? (It that simplifies the task or provides ...
2
votes
0answers
104 views

't Hooft twisted torus construction and its relation to characteristic (e.g. Stiefel-Whitney) class

It is known that the $PSU(2) = SO(3)$ and there is an associated global anomaly labeled by the second Stiefel-Whitney class $w_2$. This second Stiefel-Whitney class $w_2$ can detect the 1+1 ...
2
votes
0answers
149 views

Diffeomorphic manifolds of inequivalent smooth structures

Given two smooth structures $\mathcal A_1$ and $\mathcal A_2$ of a spacetime manifold $\mathcal M$, we say that they are equivalent if $\mathcal A_1 \cup \mathcal A_2$ is itself a smooth structure (or ...
2
votes
0answers
186 views

A question about Gel'fand-Yaglom method of calculating functional determinants

I know that the Gel'fand-Yaglom method is a way to calculate determinants of 1D differential operators. For instance, let us consider an operator $-\partial_r^2+W(r)$. Let us define $\psi_{0}(r)$ as ...
2
votes
0answers
79 views

How to make sense of this definition of a reference frame?

Reference frames in General Relativity seem simple to understand and hard to come out with one agreed definition. In the intuition it represents a certain "point of view" of several observers. In ...
2
votes
0answers
185 views

Theorem of QFT as an operator-valued function theory

From Axioms of relativistic quantum field theory, I find the following theorem : We conclude this section with the following result of Wightman which demonstrates that in QFT it is necessary to ...
2
votes
0answers
108 views

QED in Minkowski space VS a finite volume and their mathematical formulations

I am currently trying to figure out the relationship between two different presentations of non-interacting quantum electrodynamics. The first is the usual Fock space formulation in infinite Minkowski ...
2
votes
0answers
53 views

Equivalence of definitions of harmonic (or wave) coordinates

In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves: $$ \Box_g x^{\alpha} = 0 $$ where $ \Box_g = g_{\mu \nu}\nabla^{\mu}...
2
votes
0answers
421 views

Can d'Alembert's Formula for the Wave Equation in one dimension (1+1D) be used in three dimensions (3+1D)?

The 3+1D wave equation for spherically symmetric waves is $$\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) $$...
2
votes
0answers
392 views

Jet bundles for physicists

In order to make Classical Field Theory rigorous we need the idea of jet bundles. I've seem some books on the subject, but most of them are aimed at mathematicians and tend to go quite deep in the ...
2
votes
0answers
78 views

Rigorous way of box normalisation

This is follow up from an answer to my previous question about unitarity in rigged Hilbert space. As it turns out, that there is no idea of unitarity in rigged Hilbert space (hence no meaningful QM ...
2
votes
0answers
110 views

The super Grassmannian $G_{2|2}(4|4)$

In the paper, the super Grassmannian $G_{2|2}(4|4)$ is defined by (12)--(18). An element of $G_{2|2}(4|4)$ can be written as a $(2 | 2) \times (4 | 4)$ matrix of full rank modulo the left action by ...
2
votes
1answer
115 views

which is correct way to find error in degree of linear polarization?

I have a set of "degree of polarization (DOP)" values for a star. Assume there are 10 DOP values in the set. DOP is defined as square root of sum of squares of fractional Stokes parameters, namely q ...
2
votes
0answers
71 views

Have Witten-type TQFT's nonconservation of energy and momentum in interactions?

Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be $G$ a geometric field. Then every observable expectation value ...
2
votes
0answers
108 views

Difference between vacuum and pseudovacuum vector?

What exactly is the difference between the vacuum and pseudovacuum vector? In my case the ground state of a system is the vacuum vector and by letting operators act on that vacuum vector magnons are ...
2
votes
0answers
119 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) \...
2
votes
0answers
181 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?
2
votes
0answers
378 views

What insights does category theory offer in terms of grand unified theories?

What insights does category theory offer in terms of grand unified theories? Any references to books or papers that give categorical descriptions of any of the common grand unified theories would be ...
2
votes
0answers
163 views

What physical effects cause materialization of a system of particles for a short time?

It is well-known from physics that a photon with enough energy creates a pair of particles: one electron and one positron. This pair of particles can only exist for a short time. This process is ...
2
votes
0answers
83 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol x,\...
2
votes
0answers
28 views

Main differences between elastodynamic and light scattering when using S-matrix to find bound states

What are the main differences (top 5 if question is too broad), for using the S-matrix to find bound states, between elastodynamic and light scattering? (if it facilitates a higher quality question/...
2
votes
0answers
485 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 ...
2
votes
0answers
293 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 ...
2
votes
0answers
33 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 ...
2
votes
0answers
2k views

How to calculate the dispersion relation for a wave equation with non-constant speed of wave propagation?

Specifically, it is a one-dimensional wave equation for waves on a string with a non-constant cross-section, i. e. $$S(x)=S_1+S_2 \cos{2x}; \qquad c(x)=\sqrt{F/\rho\, S(x)}.$$ Separating the variables ...
2
votes
0answers
137 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
2
votes
0answers
161 views

How to prove that the ground state of the Hubbard model is not a Slater determinant?

Of course it is expected. But how to prove it analytically? Slater determinant is mentioned in almost every quantum mechanics textbook. But it is necessary to warn the undergraduate students that not ...
2
votes
0answers
211 views

Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four ...
2
votes
0answers
167 views

What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann ...
2
votes
0answers
264 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 ...
2
votes
0answers
74 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 ...
2
votes
0answers
101 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 ...
2
votes
0answers
76 views

How many unequivalent Seifert surfaces appear in a AdS/CFT extension?

When introducing the 't Hooft diagrams from Feynman diagrams on a torus has there been a classification in terms of knots and Seifert surfaces?
2
votes
0answers
272 views

Doubts about the Aharonov-Bohm effect

In F. Schwabl, Quantum Mechanics p.148 it is explained that if we have a particle in an electromagnetic field given by potentials $\varphi$ and $\mathbf{A}$ with wave function $\psi$, then a gauge ...
2
votes
0answers
63 views

Regular initial data

I have a very basic question. What exactly is meant by "regular" initial data in general relativity? Does it mean smooth? at least $C^{2}$? All literature on the subject just uses this term without ...
2
votes
0answers
81 views

About deriving the multi-trace index in terms of the single-trace index

This question is in reference to this paper Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function, $Z(x) = \prod_{n=1}^{n =\infty}\left [ \int ...
2
votes
0answers
122 views

Helicity for Zero Rest Mass Field Equations

I'm trying to reconcile the usual definition of the helicity operator, namely $$ h = \hat{p}.S$$ with the definition of a massless helicity $n$ field as a symmetric spinor field $\phi^{A\dots B}$ ...
2
votes
0answers
197 views

Turboshaft Turbine Mathematical Model

Are there any simplified mathematical models for how two gas coupled turbines (also called a free power turbine) should interact with one another as the speed of the driving turbine changes. (i.e.) ...
2
votes
0answers
84 views

Boundaries where AdS/CFT complementarity applies

Usually when I read about AdS/CFT complementarity as a particular case of the Holographic principle, it suggests that physics evolution on a boundary has a map to physics evolution on the bulk. But ...
2
votes
0answers
204 views

Functional determinant approximation

Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$. I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use $$\sum_{n} (z+E_{n})...
2
votes
0answers
160 views

Convergence and well-definedness of Lorentzian path integrals

Wick rotation of quantum field theories to Euclidean path integrals with a nonnegative measure everywhere is a wonderful tool. Not so with Lorentzian path integrals. Events far separated in ...
2
votes
0answers
448 views

An alternative, algebraic way to introduce interactions. Are there other ways out there?

An opening paragraph: The usual approach to introducing interactions in quantum field theory is to make the constraint on the amplitude of the field towards smaller values more forceful than ...
2
votes
0answers
245 views

Singularities in Bianchi models in general relativity ( physical science)

what are the conditions to check point type singularity in a bianchi type model ? bianchi type model are of Type I,II,III,IX,IV or u can say we use different Bianchi type models having some specific ...
1
vote
0answers
32 views

Partial trace of matrix product state

I have come accross a formula that puzzles me a bit in the proof of lemma 23 (page 32) of this paper. The authors start from a (translationally-invariant) matrix product state: $$\lvert\psi\rangle := ...
1
vote
0answers
24 views

Ingoing modes at $\mathcal{I}^+$ and outgoing modes at $\mathcal{I}^-$

This is a question about the definition below presented on this thesis on the Hawking effect. On page 73 the author defines as follows: Near $\mathcal{I}^\pm$, the solutions to (5.56) [the radial ...
1
vote
0answers
48 views

Regarding Rayleigh-Sommerfeld Diffraction Integral

While studying the Rayleigh-Sommerfeld diffraction formula I get the standard result for the following integration related to the said diffraction: $$ \int_{-\infty}^{\infty}\frac{\exp ik[\sqrt{s^2+...
1
vote
1answer
27 views

Expand superspace function into component form

In 2D (1,1) superconformal field theory, the invariant "distance" between two points $Z_1=(z_1,\theta_1)$ and $Z_1=(z_1,\theta_1)$ in superspace is $$Z_{12}=z_1-z_2-\theta_1\theta_2.$$ My question ...
1
vote
0answers
15 views

Projection operators in non-equiibrium statistical mechanics - non-euclidean function space

In the formalism proposed by Zwanzig and Mori [1,2] for projection operators, an inner product is defined for the variables of phase space which is given by, $$ (A,B) = \int d\Gamma f_{eq}(\Gamma)A(\...
1
vote
0answers
36 views

Vanishing Poisson bracket with non-vanishing Moyal bracket

Let $M=\mathbb{R}^{2n}$ be the phase space with standard Poisson bracket on smooth functions on $M$. Fix a classical hamiltonian $h$ (function on $M$) and function $f$ generating symmetry of $h$ i.e. $...