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.

391 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
3
votes
0answers
252 views

From Newton to Kepler without infinitesimals

I've read some interesting calculus-free proofs of at least parts of the derivation of Kepler's Laws from Newton's gravitational force. One is of course Feyman's "Lost Lecture" (which was already ...
3
votes
0answers
100 views

Underlying C*Algebra operators in standard quantum mechanics?

Linearity in standard quantum mechanics (QM) is the key to making the math possible in this field, but the presence of nonlinear operators in QM is what is more generally dealt with. Working with the ...
3
votes
1answer
104 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 ...
3
votes
1answer
95 views

What is meant by the following divergent formula?

I have encountered the following formula a couple of times (in different physics contexts which I do not have a good understanding of) $$\int_{0}^\infty \frac{dt}{t}e^{-tx}=-\log x$$ Formally one ...
3
votes
0answers
677 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
3
votes
0answers
784 views

What's the physical meaning of the integral of the squared temperature in a one-dimensional rod, used to prove uniqueness of soln. in heat eqn.?

Before getting to the question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation $...
3
votes
0answers
275 views

Geometric interpretation of Grassmann variable

Grassmann variables were introduced to make path-integral formalism easier to handle fermionic (anti-commutating) fields. Mathematically they represent the exterior algebra of forms (or exterior ...
3
votes
1answer
188 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 ...
3
votes
0answers
517 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 it....
3
votes
0answers
82 views

How obtain conserved quantities in integrable models in accordance with Liouville's theorem, via Sklyanin Poisson algebra?

In classical integrable models, in the discrete case we have the Sklyanin algebra, $$\lbrace T_{a}(u),T_{b}(v)\rbrace =[r_{ab}(u,v),T_{a}(u)T_{b}(v)].$$ How to prove that the conserved quantities are ...
3
votes
0answers
58 views

Scalar product of torsional forms - how are the standard identities modified?

It is known that for any smooth, orientable, compact manifold $X$ without boundary and $\alpha \in \Omega^{r}(X), \beta \in \Omega^{r-1}(X)$ it holds \begin{equation} (d\beta,\alpha)= (\beta, d^{\...
3
votes
1answer
96 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 ...
3
votes
0answers
146 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
3
votes
0answers
153 views

Holonomy twisting

There is Witten's topological twist of standard SUSY QFTs with enough SUSY into Witten-type TQFTs. What is a holonomy twist?
3
votes
0answers
161 views

Question about the HVZ theorem

In this paper1 the authors cite the HVZ theorem2 saying that it follows from the method used by M. Reed & B. Simon without modifications; I don't really understand this point. Is there anyone who ...
3
votes
0answers
196 views

Isospin and Hypercharge of the SU(2) bps monopole embedding

I am reading the paper Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups - Weinberg, Erick J . In appendix C of this paper the author states, that the solution ...
3
votes
0answers
109 views

Divergence calculation of a lie algebra valued quantity having spinor indices

I am reading this paper by E. Weinberg - Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups. I am having a problem with a calculation. I don't have much experience ...
3
votes
0answers
97 views

A doubt about fuchsian functions in physics?

I'm not sure if this is the right place (or math.stackexchange?) to ask the next What is the difference between fuchsian, theta-fuchsian, and kleinian functions? Please, suggest me an introductory ...
3
votes
0answers
502 views

How is the poincare conjecture(and perelman proof) helpful in studying the properties of the universe?

Can someone tell me how the poincare's famous conjecture or its proof by perelmen can be helpful in deciding some properties like the shape of the universe?
3
votes
0answers
312 views

What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?

What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & ...
3
votes
1answer
213 views

Simulation of oscillator with frequency dependent damping

What would be the equation for the frequency dependent damping of harmonic oscillator? Is there something like: $$ \ddot{x}+2\delta\dot{x}+\omega_0^2x = \frac{F}{m}f(t) $$ with frequency dependent ...
2
votes
0answers
42 views

Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
2
votes
0answers
20 views

Growth of apparant horizons and null convergence condition

An apparent horizon in general relativity is a surface where all null vectors are pointing "inwards", i.e. it is the location of a marginally outer trapped surface (for a review see here). It is well ...
2
votes
1answer
47 views

Partial trace over continuous degrees of freedom

Let us suppose we have some quantum system whose Hilbert space admits a bipartition $\mathscr{H}\simeq \mathscr{H}_A\otimes \mathscr{H}_B$. Let $|n\rangle_A$ be a basis of $\mathscr{H}_A$ and $|m\...
2
votes
0answers
43 views

Mathematically rigorours formulation of the Bogoliubov transform for bosons

Let $\mathfrak{H}$ denote the Hilbert space describing the single-particle states and $|k\rangle$ denote an orthonormal basis of $\mathfrak{H}$. Let $c_k$ denote the corresponding annihilation ...
2
votes
1answer
70 views

Relationship between boundary states and primary states of a Kazama-Suzuki model

In [1] and [2] the authors claim that the boundary states (not just the Ishibashi states) of a Kazama-Suzuki model are labelled in the same way as the primary states of the model, so that the boundary ...
2
votes
0answers
71 views

Is every operator a power series of creation and annihilation operators (in a rigorous mathematical sense)?

Let $\mathscr{H}$ be a Hilbert space denoting the single-particle states and $c_k^*,c_k$ denote creation and annihilation operators of orthonormal basis $\phi_k\in \mathscr{H}$. Let $\mathscr{F}$ ...
2
votes
0answers
49 views

Physical interpretation of biharmonic operator

In the book Mathematics of Classical and Quantum Physics, the authors give an (enlightening) interpretation of the Laplace Operator $\nabla^{2}$ of a field $f(\mathbf{x})$, $\nabla^{2}f(\mathbf{x})$ ...
2
votes
0answers
64 views

Is there a commonly accepted definition of a quantum phase definition for a finite lattice/set of particles?

As noted by Sachev, and in a previous question, https://www.physicsoverflow.org/41602/, there cannot be quantum phase transitions for finite systems (with bounded local Hilbert space dimension). The ...
2
votes
0answers
81 views

Proper path integral of a field theory

I have been trying to find out the sweet middle ground of describing path integration of field theories, in between the physicist way and the mathematician way, but it seems hard to find something ...
2
votes
0answers
48 views

Rigorous derivation of the ground state projector using euclidean time evolution

Usually one argues that the euclidean path integral is able to recover the ground state of a system along the following lines: Take the time evolution operator $U(t,t_0)=e^{-iH(t-t_0)}$. Transform to ...
2
votes
0answers
50 views

Exotic perturbative anomaly captured only by higher-loop Feynman graphs, but not by any 1-loop Feynman graph?

My question: Are there any perturbative anomaly captured by higher-loop but not by captured at the 1-loop Feynman graph (say, not enough)? We are familiar with the text book example of a ...
2
votes
0answers
53 views

Can an arbitrary spin state be written uniquely in a Dicke state basis?

Consider a system of e.g. $N=3$ spin-1/2 particles. The state of the system $\vert\psi\rangle$ lives in a Hilbert space of dimension $2^N=8$. Now, consider the collective spin operator $$\mathbf{J} = ...
2
votes
0answers
37 views

List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
2
votes
0answers
63 views

Positive frequency definition in general spacetime for general fields

In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles. In Minkowski spacetime we have a ...
2
votes
0answers
101 views

Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...
2
votes
1answer
103 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
2
votes
0answers
55 views

Pictures of Different Coordinate Systems in General Relativity

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
2
votes
0answers
68 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
2
votes
0answers
101 views

Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
2
votes
0answers
108 views

In what sense are solutions to the Dirac equation and solutions to the Laplace equation equivalent in string theory?

I have come across statements like elementary particles on a Calabi-Yau correspond to harmonic forms (or to cohomology classes, which is equivalent for a compact Kähler manifold, since every ...
2
votes
0answers
101 views

Is there an intrinsic physical meaning for characteristic curves of a PDE?

For partial differential equations (such as those that govern many physical phenomena), there exist characteristic curves, along which the equations can be reduced to total derivatives and solved. The ...
2
votes
0answers
68 views

How to make sense of this uncountable tensor product construction?

The reference for this discussion is this paper. It is a paper related to the Unruh effect and QFT. The problem is the following: as is well-known, when we quantize a KG field, we get a Fock space ...
2
votes
0answers
52 views

Phase transition between two CFTs

If we start with a CFT (say CFT$_1$) and deform it by some relevant operator, in the IR we can get another CFT (say CFT$_2$). This is flowing from one CFT to another. I was wondering whether there ...
2
votes
1answer
81 views

How to carry out this kind of analysis in Relativity?

In the paper arxiv.org/abs/0801.0926 the authors propose the following simple system of particles in Newtonian mechanics. Four particles of same mass placed as follows The $a(t)$ and $\theta(t)$ can ...
2
votes
0answers
328 views

Invariance of a vector under parallel transport along an infinitesimal orthogonal loop

Suppose I have a curved space-time with metric $g_{\mu \nu}$ and with the connection coefficients $\Gamma^\alpha_{\mu \nu}$. Given some vector $v$, the expression for parallel transport of this vector ...
2
votes
0answers
113 views

Deriving the longitudinal sound wave from the transverse string vibration

I am trying to derive the longitudinal sound wave (i.e. either the pressure $\psi_P(x,t)$ or the displacement $\psi(x,t)$, where both $\psi$s are in the direction of $x$) produced by the vibrations of ...
2
votes
0answers
66 views

Spinors, punctured plane and principle frame bundle

I am reading Applied Conformal Field Theory by Ginsparg. On page 72, while describing different boundary conditions on fermion he states the following. We shall choose to consider periodic $(P)$ ...
2
votes
1answer
137 views

Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?

In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
2
votes
0answers
172 views

From Number Theory to Physics.

I have asked a question here: I want to see an example which is related to (integral) quadratic forms or theta series. @Kiryl Pesotski answered me in some comments as following: For ...