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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References about Quantum Theory (for mathematicians) [duplicate]
I’m interested in Quantum Theory. Can anyone recommend a good book (for mathematicians) that tackles varios topics (not necessarily formally).
I have in mind books that have the same idea of books ...
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Question on ordered exponential explanation in Wikipedia
Let $\gamma:[0,1] \to \mathbb R^2$ be a path describing a rectangle with vertices $x$, $x+u$, $x+u+v$, $x+v$, where $x, u, v \in \mathbb R^2$ ($u, v$ linearly independent).
Let $J:\mathbb R^2 \to \...
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Sensor Array Position Calibration in Anisotropic Media
Problem.
I have a sensor array consisting of $n \gg 4$ receivers at unknown locations $\langle x_n, y_n, z_n\rangle$ embedded in an anisotropic medium whose index of refraction varies as a known ...
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Construction of the Klein-Gordon field theory - what is missing?
Many references I know on QFT start the discussion of the Klein-Gordon field theory with some discussion about harmonic oscillators. One such reference is Folland's Quantum Field Theory book. The idea ...
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An elementary random walk model to incorporate non-Gaussianity
I am preparing a talk for young students to introduce heterogenous dynamics in complex fluids and give them a flavour of non-Gaussianity in displacements which are defined by,
$$
\alpha (t) = \frac{\...
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How do I self-study physics at the undergrad level? [closed]
I'm a new physics undergrad worried that I won't be able to learn everything I want at the university I'm going to.
Basically the Institute I'm going to is applied sciences focused, and all electives ...
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Non-distributivity of quantum logic according to C. Piron
I'm trying to understand this highlighted sentence in Piron's "Foundations of Quantum Physics" on p. 21:
I know that distributivity of a lattice means $a\land (b\lor c)=(a\land b)\lor(a\...
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Square root of number operator for quantum harmonic oscillator
Let $a$, $a^{\dagger}$ denote the standard annihilation and creation operators for the quantum harmonic oscillator, with $[a, a^{\dagger}] = \mathbb{I}$. The number operator is then defined as $a^{\...
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On the Bogoliubov-de Gennes (BdG) equation
I'm a graduate student majoring in mathematics, in particular nonlinear PDEs.
So I know very little about physics, including quantum mechanics.
I'm interested in the Bogoliubov-de Gennes (BdG) ...
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Compactification in String Theory and Compactification in Topology are they the same thing?
In topology, there is a concept of compactification which is defined as follows.
A space $Z$ is a compactification of $X$ if $Z$ is compact Hausdorff and there exists an
embedding $j:X \rightarrow Z $ ...
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Doubt on the continuity factor of Dyson mega-spheres
I) Dyson Mega-Spheres
In a nice and cool recent paper, $[1]$, the authors constructed another interresting solution of general relativity; they constructed a thin-shell around a star: a dyson sphere ...
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In what sense is string theory not expected to be a QFT?
This question came to mind while reading about Weinberg's folk theorem that any quantum theory that is Poincare covariant and satisfies cluster decomposition will look like a quantum field theory at ...
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Intuition for the differences between two notions of quantum ergodicity: One given by weak-* convergence and one by pseudodifferential operators
Consider the two notions of quantum ergodicity of the Laplacian operator $\Delta$.
(Phase space): $\Delta$ is said to be quantum ergodic (in the phase space) in a compact Riemannian manifold if there ...
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Mathematical equivalent of Fundamental nature of charge [closed]
How to mathematically represent the fact that electric charge is a fundamental quantity? i.e. that it cannot be explained in terms of other things, for example, the normal force can be explained as ...
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Whats is a large fields problem in RG?
I was advised on MO to link this question and reproduce it here, so here it goes.
I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify.
...
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I'm getting imaginary eigenvalues of $X^2-P^2$
This is hermitian but I'm getting imaginary eigenvalues. Start with the commutator:
$$[X^2-P^2,X+P]$$
$$=[X^2,P]-[P^2,X]$$
$$=X[X,P]+[X,P]X-(P[P,X]+[P,X]P)$$
$$=2i(X+P)$$
Now consider the ket $|E\...
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Quasifree states vs Gaussian states
Does anyone know the origin of the term "quasifree states" in algebraic formulation of QFT? As far as I am aware of, the definition is effectively equivalent to Gaussian states centred at ...
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Do continuous wavefunction form a Hilbert space?
In quantum mechanics we are told that
the wavefunctions live in Hilbert space.
the wavefunctions are continuous.
It recently came to my notice that in Mathematics, there is a theorem which says ...
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Are mathematical physics and theoretical physics the same thing at highest levels? [duplicate]
To my understanding Mathematical physics is about how one could find a rigorous basis to understand physics/ study the mathematics used in physics. However, high level theoretical modern physics like ...
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Motivation behind defining new measures?
Suppose we are given a measure $d\mu$. I have seen in some physics textbooks that we often define a new measure in terms of this measure, let's say
$$d\rho = f\,d\mu,$$
where $f$ is some integrable ...
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Two questions regarding Spivak's Configuration Space
The following is from the fifth Chapter Rigid Bodies of Spivak's Physics for Mathematicians. The post consists of a statement Spivak makes -with no proof- that I do not understand. For clarity, I've ...
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Dynamical system model representation for the geometric motion of soap bubbles
I have been trying to describe the dynamical system model for soap bubbles proliferating, coming into existence and moving and pushing around other bubbles (i.e. water gushing into a sink, full of ...
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Follow-up to the question of why the momentum operator is Hermitian on $\mathbb{R}$
I have a follow-up question to the following post: Imaginary Eigenvalue Of A Hermitian Operator.
In the post, the question reads,
The eigenfunctions of a Hermitian operator are real. But consider a ...
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What makes coming up with a mathematically solid, non-shaky relativistic quantum field theory (RQFT) so hard?
This is something I know of but I'm not quite sure I understand the details. Particularly, when it comes to interacting RQFTs, such as even QED, where some posts here have pointed out that it cannot ...
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As written, how is the spectral theorem useful?
I am self-teaching myself QM using Brian Hall's Quantum Theory for Mathematicians. He devotes a good chunk of the book to motivating, explaining, and proving the spectral theorem in multiple ways. The ...
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$q$-dilogartihm function power series
I was reading the $q$-dilogarithm function (Faddev-Kashaev)
https://arxiv.org/abs/hep-th/9310070
How can I derive $$\frac{1}{\Psi(x)}=\sum_{n=0}^{\infty} \theta^{n} x^{n} /(\theta)_{n}$$
by using the ...
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How fast can the spectral gap of a translation-invariant Hamiltonian close?
Consider an arbitrary local Hamiltonian defined on some lattice of size L where the local Hilbert space dimension associated with each site on the lattice is finite. If there is no constraint on the ...
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Some questions related to the energy of a viscoelastic bar
Let's consider a problem of free longitudinal vibrations of an one dimensional infinitely long elastic relaxing bar (Maxwell material) with constant cross section. The bar's displacement $u$ and ...
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Gauge Theory determined by Gauge Group and Representation: What about specifying the bundle?
I have the following question. In physics, when one talks about (Yang-Mills) gauge theories, one often states that it is enough to specify the following data:
The gauge group $G$, which is usually a ...
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Doubt on the geometry of "quantum phase space"
In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is ...
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The abstract state of a particle
I recently started learning about quantum physics. In the book, Quantum physics by H.C. Verma, the author explains that there are many ways to represent the state of a particle. The wave function $\...
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Quantum Particle in a Fractal Box
I was thinking about particle in a 2D box the other day, and I realize that it shapes actually affect its energy and wavefunction. Therefore I thought to myself, what if a particle is inside a fractal ...
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Resources on Post-Einsteinian Results in GR
What are some good books, lecture notes, articles, etc. that can be used as introduction to the landscape of major results in general relativity since Einstein? In terms of the timeline, I'm thinking ...
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How was the minimal model with a boundary related to the D brane?
Quote my advisor:
The D brane was the boundary of the CFT
However, in the development of the rational CFT, such as the minimal model, the D brane was not realized. Thus, when the boundary CFT was ...
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Green's function existence vs explicit description
Are there examples in physics where the mere existence of a Green's function on some domain (for some PDE) has useful applications? Or is it true that in literally all applications of Green's ...
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Contractivity of trace distance in infinite dimensions
The trace distance is heavily used in physical applications and is defined being half of the trace norm $T(\rho, \sigma) := \frac{1}{2} Tr\left[\sqrt{(\rho-\sigma)^{\dagger}(\rho-\sigma)}\right]$.
It ...
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Interacting QFT construction on curved spacetime
As far as I can tell, most of the concrete models considered in (rigorous) QFT on curved spacetime are either free or perturbative. In fact the only construction of an interacting QFT on curved ...
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Euler equation of motion for fluids
I was seeing some proofs of Euler equation of motion for fluids online and most of the videos drew this figure in which they consider infinitesimal cylindrical element.
My question:-
Now it mentions ...
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It is possible to have a drag force which is non-Lipschitz?
When working with the Drag Force is typical used on classical mechanics systems the following:
For high speeds it is used the Drag equation which says the drag is proportional to the squared of the ...
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Are there physics models that accurately handle the assumption of having solutions that achieve finite ending times?
Are there physics models that accurately handle the assumption of having solutions that achieve finite ending times?
Intro
Recently I learned on the answers and comments of this QUESTION that the ...
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Coefficient of effective chiral Lagrangian of $\pi\pi$ scattering
I have been suffering from the coefficient in the expansion of chiral lagrangian.
Consider $$L=\frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U),$$
where $$U=\exp(i\frac{\phi}{F}).$$ ...
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On the properties of a projection map from points in the bulk to qubits on the boundary
Note: I have completely re-edited my post, so as to make it much clearer. I apologize for the previous version of this post, which I had written rather fast.
Let $H^3$ denote hyperbolic $3$-space and ...
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Local algebra of AQFT vs Bisognano Wichmann Theorem
Maybe I am misunderstanding something really stupid, but I am finding it hard to think of local algebras in terms of wedge algebras. One of the claims (see, e.g., Section 3 and 4 of this paper) is ...
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Random matrix theory and the singularities of the Weingarten function
In the random matrix theory literature, one often encounters identities associated with averages over ensembles of random unitaries. For a simple example let's say we're interested exclusively in $2\...
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Path integral on many-body quantum mechanics
Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
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Question about the kinetic energy operator
The Kinetic Energy Operator is essentially self-adjoint. Under what circumstances does it have a unique extension?
user335943
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Resolution of the identity of operator with mixed spectrum
In most quantum mechanics text books, the resolution of the identity or completeness relation is stated in the following (or similar) form
$$ \mathbb I_\mathcal H = \sum\limits_n |\lambda_n\rangle \...
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Representations of minimal model primary fields in the Coulomb-gas Formalism
This question is a cross-post from MO (link).
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? As far as I can ...
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Is higher or lower specific heat better for cooling an object via contact?
Suppose you want to cool an object by putting it into contact with another object, much colder object, and the transferal of joules to an intermediary equilibrium temperature is instantaneous.
If this ...
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Uncertainty of waves
Here in the pictures I have written about some question I have been thinking about a long time, what do you think?
Link to the chapter I am talking about:
http://www.its.caltech.edu/~matilde/...
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