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.

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8 views

Inertial Forces while analysing forces on a Piston in a Slider - Crank Mechanism?

I get that we are analysing the Piston in a Non - Inertial Frame of Reference, but the point of my question is that according to D'Alemberts Principle, whenever an Inertial Force comes in the line of ...
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46 views

Maurer-Cartan form in Physics

I am just reading about the Maurer-Cartan form in the context of Lie Groups, although the mathematical definition: $$\Theta(g)({\bf v}) = (L_{g^{-1}})_{*g}({\bf v})$$ for $g\in G$, $G$ a Lie group, ${\...
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What would I need to pickup to understand the theory of topological insulators? [on hold]

I am a mathematician who does mostly analysis, so excuse me for my ignorance. I was wondering where I could start learning about topological insulators. I have finished reading up on the operator ...
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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 $\mathscr{F}$ ...
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What am I missing here? (How do we know the universe has a cause?) [closed]

I apologise if this has been asked before or is otherwise an ill-formed question. Consider the following predicates: $B(x)$: "$x$ began to exist". $C(x)$: "$x$ has a cause". Let $U$ be the ...
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How can we deduce that a hydrogen atom is stable in relativistic QED?

Consider relativistic quantum electrodynamics (QED) with three quantum fields: the electromagnetic field $A_\mu$, one fermion field $\psi$ for electrons/positrons, and one fermion field $\psi'$ for ...
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588 views

Is the evolution operator well-defined mathematically?

We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is ...
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Why is the Bogoliubov transform unitary of $H\oplus H$

In Bach, V., Lieb, E.H. & Solovej, J.P. J Stat Phys (1994) 76: 3. https://doi-org.stanford.idm.oclc.org/10.1007/BF02188656, page 10, the Bogoliubov transform on the Fock space $\mathscr{F}$ is a ...
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36 views

System in Lieb and Yngvason's paper [closed]

I'm reading The Physics and Mathematics of the Second Law of Thermodynamics and have a question. In A. Basic concepts 1. Systems and their state spaces, the term system is formally introduced and one ...
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What parts of “Geometry, Topology and Physics” by Mikio Nakahara is typically studied in a 1 semester course in graduate school? [closed]

I have some months in my hand before i head to graduate school. I would like to learn and strengthen my grasp on mathematical physics. I would like to do high energy physics (not necessarily just ...
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38 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 $f(\mathbf{x})$, $\nabla^{2}f(\mathbf{x})$ ...
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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 ...
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1answer
77 views

Orthochronous indefinite orthogonal group $O^+(m, n)$ form a group

My question is based on Qmechanic's answer here which proves that $O^+(m, 1)$ forms a group -- that if two Lorentz transformations have positive time-time co-ordinate, so does their product. The key ...
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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 ...
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45 views

Why are only functions discussed in physics and not relations? [closed]

Why are only functions discussed in physics and not relations?
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The use of Helmholtz decomposition

Examining the article on Wikipedia Helmholtz decomposition, compatible with the explanations of the book Introduction to Electrodynamics $4^{\mathrm{th}}$ edition David J. Griffiths §1.6 the theory of ...
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Physicist path integral and cylinder set measures

Path integral via discretization So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with ...
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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 ...
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Relation between functional measures and states in AQFT

Let $(M,g)$ be a globally hyperbolic spacetime and $\phi$ a KG field. In AQFT we consider the algebra of observables $\mathfrak{A}$ generated by $\phi(f)$ where $f\in C^\infty_0(M)$ is a test function....
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69 views

Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
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30 views

Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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2answers
85 views

Velocity-Dependent Potential and Helmholtz Identities

I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
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1answer
45 views

How to get the imaginary part from the Källén-Lehmann propagator

During field theory course the Källén-Lehmann propagator was defined as follows: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
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Operators that act on the edge of a quantum spin chain with periodic boundaries

Consider a quantum spin chain of length $N$. Each site/spin has the local Hilbert space $\mathbb{C}^d$ and so for the whole chain the Hilbert space is $(\mathbb{C}^d)^{\otimes N}$. Now for periodic ...
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55 views

Is it there any theory or model in theoretical physics that is akin to Tegmark's Mathematical Universe Hypothesis?

Physicist Max Tegmark proposed a hypothesis that asserts that all mathematical structures do exist as universes. (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) But this hypothesis ...
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1answer
56 views

Is this in fact an equation for the normal vector of the null surface?

I'm reading the paper "Area, Entanglement Entropy and Supertranslations at Null Infinity" and there is a point that I have a doubt on what the authors are actually doing. First, let me summarize the ...
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2answers
80 views

Completeness condition involving continuum states

Consider a potential $V(x)$ in 1d. Suppose that $V(|x| > a )= 0$ for some positive $a$. We then know that the hamiltonian $H = - \frac{\partial^2}{\partial x^2 } + V(x)$ has non-normalizable or ...
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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 ...
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Proof that the boundary of the causal past of a Cauchy surface is the Cauchy surface

Let $(M,g)$ be a globally hyperbolic spacetime. Let $\Sigma$ be a Cauchy surface in $(M,g)$. In this paper, page 9, Lemma A.1, the author says that if we take $D = J^{-}(\Sigma)\setminus \Sigma$ then $...
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1answer
45 views

Compactness of product

Let $Q$ be a position operator and $P$ its conjugate momentum. We would like to show that $f(P)g(Q)$ is compact if $f$ and $g$ are sufficiently smooth operator-valued functions and have compact ...
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1answer
36 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
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1answer
82 views

Normal ordering by contour integral in CFT

In chapter 6 of Di Francesco, they introduce the normal ordering $$ (AB)(w) = \oint_w \frac{ dz }{ 2\pi i (z-w) }A(z) B(w)\ .\tag{6.130}$$ So far so good. But then starting eq (6.139) $$ \oint_w \...
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Why shouldn't I choose my boundary limits corresponding to the direction I'm integrating?

I have a question regarding the choice of boundary limits when it comes to vector integrals. Why shouldn't I always choose the boundary limits corresponding to the direction I'm integrating. I.e why ...
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When do the solutions of combinatorial Dyson-Schwinger equations generate a Hopf subalgebra?

Say I have a set of combinatorial Dyson-Schwinger equations of the form $$\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha ...
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1answer
70 views

Condition for finitely many bound states in one dimension

This came up in the context of the inverse scattering transform for the KdV equation. My primary reference, a set of lecture notes on integrable systems by Maciej Dunajski, makes the claim that the ...
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Optimising equation with 3 or more changable variables

In order not to bother you with technical details I've laid my therms in plain math. The sets are experimentally obtained mechanical characteristics, and unknowns are some empirical parameters from ...
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56 views

Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
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1answer
78 views

If a team of Engineers Made a Laser From Scratch When Would They Need to Use the Schrodinger Equation? [closed]

I know the Schrodinger Equation is a key part of quantum mechanics. I am trying to understand it’s applications. Let’s say a team of engineers wants to build a laser from scratch, assuming they have ...
2
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1answer
89 views

How can tempered distributions be paths?

I'm reading the Appendix A of Glimm and Jaffe book "Quantum Physics: a functional integral point of view", and there is something that I'm missing In section A.4 the authors talk in a very general ...
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2answers
26 views

Angular velocity is $\dot{g}$ carried to the identity element of the group

I was reading the example below from Arnolds book I can't really understand why the angular velocity is $\dot{g}$ carried to the identity element of the group. I would appreciate if someone who ...
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2answers
149 views

Mathematical rigorous definition for an electrical dipole

I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
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1answer
384 views

Srednicki Eqs. (6.22) and (9.6). How to get rid of $i\epsilon$ in the interaction term?

I'm studying qft from Srednicki's book. If one writes down the full $i\epsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields $$\left<0|...
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1answer
85 views

Lefschetz and Witten indices$.$

I couldn't help but notice a formal similarity between the Lefschetz index $$ \mathrm{ind}(f)=\sum_k (-1)^k\operatorname{tr}(f_*|H_k) $$ and the Witten index $$ Z=\operatorname{tr}((-1)^Fe^{-\beta H}) ...
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58 views

Why do we choose the operator or supremum norm while proving unboundedness of the momentum operator?

In most sources, I've noticed that while proving the unboundedness of the momentum operator $\left(-i\hbar \frac{\partial}{\partial x}\right)$ the operator norm (or supremum norm) $\lVert\ .\rVert_\...
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42 views

How shall we show the surface integral approaches a limit (or does not blow up) at a field point near $S'$

Consider the electric field due to volume charge distribution in volume $V'$: $\mathbf{E}=\displaystyle \int_{V'} \rho' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$ The integrand ...
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1answer
45 views

Discretization of path integral and linear interpolations

Consider the evaluation by discretization of the path integral $$\int e^{iS[x(t)]}\mathfrak{D}x(t),\quad S[x(t)]=\int_{t}^{t'}\left[\frac{m}{2}\dot{x}(\tau)^2-V(x(\tau))\right]d\tau.$$ One ...
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How are local observables encoded in this formulation of quantum field theory as a functor?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]....
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157 views

Legal values of quantum field can take? $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

Main issue: What are the legal and possible values of the quantum field can take? Clarify by examples: (1) For example, for the spin-0 Klein Gordon field $\phi$, we may choose it to be: real $\...
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39 views

What, if any, is the relation between the Lie derivative and the Legendre transform?

I don't remember where I read it, or even if my memory serves me correctly, but I think that I read somewhere that the Lie derivative amounts to a Legendre transform. Is this true and, if it is, what ...
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21 views

Arranging objects by mass [closed]

I have n objects with different masses . Say mass of n1=m1, n2=m2 .... so on. I want to use concept of centrality in physics to arrange these objects in a field (computer simulation ) . Where object ...