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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6
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1answer
153 views

What classifies gaugings?

Gauging a global symmetry $G$ introduces several free parameters to the theory. For example, In $d=4$, gauging a simple and simply-connected Lie group introduces a coupling constant and a theta term, ...
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14answers
9k views

Why can't $ i\hbar\frac{\partial}{\partial t}$ be considered the Hamiltonian operator?

In the time-dependent Schrodinger equation, $ H\Psi = i\hbar\frac{\partial}{\partial t}\Psi,$ the Hamiltonian operator is given by $$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V.$$ Why can't we ...
5
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2answers
391 views

Combining rotations for orbital and rotational motion of a planet

If I have a planet orbiting the Sun (assuming circular orbit) at angular velocity $\Omega$ and rotating about its axis at $\omega$. I also have a normal to the surface of the planet $\vec{n}_{\rm surf}...
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0answers
18 views

Current-carrying wire in a magnetic field. Cross product, vectors and scalars

We have a wire with cross-sectional area $A$, length $L$ and current $I$. If the wire is in a magnetic field $\vec B$, the magnetic force on each charge is $\vec F =q\vec v_d \times \vec B$. $\vec ...
0
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0answers
45 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 ...
-2
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1answer
218 views

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 ...
0
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0answers
8 views

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

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 ...
2
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1answer
55 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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0answers
34 views

What would I need to pickup to understand the theory of topological insulators? [closed]

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 ...
15
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3answers
741 views

Is a system Liouville integrable if and only if its Hamilton-Jacobi equation is separable?

I am asked to show that, a system is completely integrable Liouville if and only if its Hamilton-Jacobi equation is completely separable. I get the idea and understand that is very related to the ...
2
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0answers
63 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 $\mathscr{F}$ ...
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1answer
60 views

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 ...
23
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2answers
5k views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
7
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0answers
60 views

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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0answers
39 views

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 ...
8
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1answer
591 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 ...
2
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1answer
111 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 ...
16
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1answer
525 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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0answers
62 views

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 ...
8
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1answer
462 views

Is $SU(2)\times U(1) = U(2)$?

In the many textbook of the Standard Model, I encounter the relation \begin{align} SU(2)_L \times U(1)_L = U(2)_L. \end{align} Here the subscript $L$ means the left-handness (i.e., the chirality of ...
1
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0answers
87 views

Are there some websites for self-learning of advanced mathematics? [duplicate]

Are there some websites for self-learining of advanced mathematics? For example there is perimeter scholars for self study of theoretical physics, but I haven't found some good websites providing ...
2
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0answers
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 $f(\mathbf{x})$, $\nabla^{2}f(\mathbf{x})$ ...
14
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6answers
2k views

In coordinate-free relativity, how do we define a vector?

Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change ...
3
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1answer
155 views

Do translation formulae for generalised solid spherical harmonics exist?

I'm aware of the solid spherical harmonics functions, which are basically the surface spherical harmonics $Y^m_{\ell}(\theta,\varphi)$ with an additional monomial term along the radial direction: $R^...
12
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4answers
8k views

How to prove that orthochronous Lorentz transformations $O^+(1,3)$ form a group?

Orthochronous Lorentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$) $$ \Lambda^0{}_0 \geq +1.$$ How to prove they form a subgroup of ...
1
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1answer
78 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 ...
2
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0answers
54 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
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0answers
73 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 ...
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1answer
46 views

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

Why are only functions discussed in physics and not relations?
4
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2answers
278 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
3
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1answer
178 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 ...
0
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1answer
117 views

Can one test an octonionic interpretation for a quantum-information conjecture, apparently valid in the real, complex and quaternionic settings?

For the values $\alpha = \frac{1}{2},1, 2$, corresponding to real, complex and quaternionic scenarios, the formulas (https://arxiv.org/abs/1301.6617, eqs. (1)-(3)) \begin{equation} \label{Hou1} P_1(\...
2
votes
1answer
389 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|...
5
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0answers
65 views

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 ...
1
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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 ...
3
votes
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 ...
2
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0answers
44 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 ...
4
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0answers
67 views

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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0answers
71 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 $\...
6
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2answers
153 views

Physical significance of no self-adjoint momentum operator on half line?

I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
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0answers
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 $\mathfrak g$ ...
0
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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: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
0
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1answer
199 views

Finding the maximum value of electric field

Suppose you have a surface of finite area with a fixed surface charge distribution. Does a maximum electric field magnitude $|\vec{E}|_{max}$ exist for each and every possible surface area? If yes, ...
0
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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 ...
13
votes
1answer
489 views

Electric potential of a spheroidal gaussian

I'm looking for results that compute the electrostatic potential due to a spheroidal gaussian distribution. Specifically, I'm looking for solutions of equations of the form $$ \nabla^2\Phi=N\exp\left({...
1
vote
2answers
34 views

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 ...
0
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0answers
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 ...
2
votes
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 ...
2
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0answers
45 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 ...
0
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0answers
44 views

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 $...