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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82 views

Which are the best mathematical methods books for which topic for a physics undergrad? [duplicate]

I am a physics undergraduate and I would be glad if you share your opinion about which books are best for which topics in mathematical methods, from very basic to advanced. (Like you some say Tom ...
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Initial value problem on $\mathcal{I}^-$ for Maxwell fields

In the paper "Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity" by Ashtekar and Streubel the authors state the following: Fix, as in § 2(a), a conformal completion $(...
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2answers
91 views

Stone’s Theorem and Time Ordered Exponentials

The time evolution operator of quantum mechanics seems (at least to me) to form a strongly continuous, one parameter group of unitary operators. Hence, by Stone’s theorem, we should have that $U(t) = \...
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Topological term in the Lieb-Liniger model: two different excitations map to the same state?

The many body Lieb-Liniger Hamiltonian is defined by ($c>0$ throughout ) \begin{equation} \hat H = -\sum_\ell \partial_{\ell}^2 + 2c\sum_{\ell}\sum_{m<\ell} \delta(x_{m}-x_\ell). \end{equation} ...
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1answer
423 views

Commutator of rotation matrices

How do you compute the commutator of rotation matrices in two different directions by different angles? Let $R_{x}(\alpha)$ be the rotation matrix about the $x$-axis and $R_{z}(\beta)$ be the ...
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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 ...
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257 views

Any real contribution of functional analysis to quantum theory as a branch of physics? [closed]

In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
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2answers
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Correspondence between integral transformations and differential operators in quantum mechanics

The translation operator in one dimension is defined as $$ \hat T \psi(x) = \psi(x-\alpha) . $$ This can be written as an integral transformation, $$ \begin{align*} \hat T\psi(x) = \langle x|\hat ...
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3answers
282 views

Wightman quantum field - Interpretation

I have a question regarding the interpretation of the Wightman quantum field in mathematical quantum field theory. A quantum field $\phi$ is a operator-valued distribution. This means that $\phi$ is ...
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2answers
132 views

How does one “invert” such infinite-dimensional sympletic form?

In the book "Lectures on the IR structure of gravity and gauge theories" by Strominger the author considers the sympletic form for free electrodynamics: $$\Omega_\Sigma[A;\delta_1 A,\delta_2A]=-\frac{...
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1answer
103 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\...
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59 views

The mathematical structure of $\widehat{su(2)}_k$

Some of my colleagues work on CFT's and quantum groups and I hear them talk a lot about $\widehat{su(2)}_k$ algebras. According to them (and the general physics literature) these are what ...
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1answer
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References of Deficiency indices theorem (von Neumann)

I am looking for proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula. I have already searched in papers and here but ...
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52 views

Recommended books for introduction to Quantum Mechanics for students who are mathematically aligned [duplicate]

I am a 4th-year undergraduate student and I have fully read R. Shankar's book on Quantum Mechanics and Griffiths book Quantum Mechanics. I have also done a bit of the Application of QM on ...
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92 views

Book recommendation on Quantum Mechanics which is a bit mathematically aligned and gives good introduction to Hilbert Space for beginners [duplicate]

I am a 4th-year undergraduate student and I have fully read R. Shankar's book on Quantum Mechanics and Griffiths book Quantum Mechanics. I have also done a bit of the Application of QM on ...
0
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1answer
36 views

Finding out the shape of the body from blackbody radiation spectrum

I have an idea similar to this, but I thought looking for an answer on this question might be a good start. Would it be possible to configure the geometry, i.e. shape of the body when we know the ...
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103 views

Regarding Rayleigh-Sommerfeld Diffraction Integral

While studying the Rayleigh-Sommerfeld diffraction formula I get the standard result for the following integration given at serial no. 11 under section 8.421 of "Table of Integrals" by Gradshteyn and ...
4
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1answer
111 views

Why do we need conformal compactification to define the global conformal group?

First I have the definition of a conformal map. Let $(M,g)$ and $(M',g')$ be two pseudo-Riemannian manifolds of same dimension. Let $U\subset M$ and $V\subset M'$, we say that a smooth map of maximal ...
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1answer
115 views

Why are Cauchy boundary conditions an over-specification of boundary conditions for solving Poisson’s equation?

I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics: This rather surprising result [the fact that the potential within a charge-free volume is ...
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1answer
33 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 ...
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1answer
239 views

Feynman diagrams as topology

When we talk about Feynman diagrams we know they are tools to make calculations easier and more intuitive. Moreover, it's said that they are "topological" representations of the interactions. But, ...
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56 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 ...
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1answer
71 views

Magnetization of the Ising model for an asymptotic vanishing magnetic field

I am considering the following ferromagnetic Hamiltonian for the 2-d Ising Model, say with periodic boundary condition in the torus $\Lambda_n=\mathbb{T}^2_n := (\mathbb{Z}/ \mathbb{Z}_n)^2$: $$ H_n(\...
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1answer
66 views

Asympototic analysis for the following series sum

I am wondering is there one way to extract the asymptotic behavior of $x$ in the following expression near $x=0$? $$\sum_{n=1}^{\infty} n\log(1-\exp(-n x))$$ where $x $ is real.
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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(\...
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1answer
112 views

Can there be an **essential topic** in physics which cannot be archimedean? [closed]

In physics it seems everything is explained with $\mathbb R$ or $\mathbb C$ typed entitites. Is there anything in or that would be in future in physics that would need the utility of $p$-adics in an ...
2
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1answer
76 views

Symmetry v.s. isometry of Minkowski and AdS or dS spacetime

We know some nice spacetime have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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1answer
92 views

How Quintic 3-fold is a Calabi–Yau manifold and has non-vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.wikipedia.org/wiki/Quintic_threefold Now the main ...
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1answer
43 views

Choice of conjugate momentum for Ostrogradsky instability

I was reading this post and I don't understand why chosing: $Q_1=q\ $ and $\ Q_2=\dot{q}$ implies that $$P_1=\dfrac{\partial L}{\partial \dot{q}}-\dfrac{\mathrm{d}}{\mathrm{d}t}\dfrac{\partial L}{\...
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43 views

Conformal weight of a coset model, and a specific case

Given a coset model $(G\times SO(2d))/H$, what is the expression for its conformal weight (in terms of its central charge or, alternatively, in terms of the highest weights of irreducible ...
3
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54 views

Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
2
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1answer
74 views

Lax Pairs In Integrability

I am working through Dr. Beiserts notes (https://people.phys.ethz.ch/~nbeisert/lectures/IntHS16-Notes.pdf) and have difficulty obtaining the second step in (2.9): $$\{{\rm tr}L^{k},{\rm tr}L^{\ell}\} ...
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82 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. $...
3
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1answer
134 views

Eigenspaces of the hydrogen atom as representations of $SO(3)$

When we computing the discrete spectrum of the hamiltonian of the hydrogen atom $$H=\Big(-\frac{\hbar^2}{2m} \Delta - \frac{e^2}{r} \large),$$ by some explicit computation we get that eigenspace $...
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0answers
40 views

Given a positive element, $a$, of a $C$*-algebra, why does there exists a pure state, $p$, on $A$ such that $p(a)=||a||$? [duplicate]

I'm reading secondary literature where they make this claim, however, I cannot see why it holds true. This is a reformulation from a previous question that I didn't specify good enough.
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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 ...
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31 views

Surface integral of vector field over cone, vertex not at origin

I have a vector field (originally given in Cartesian form). I need to find its integral over a cone with equation something like:$$1-z=\sqrt{x^2+y^2}, z>0$$ How do I proceed? It is not possible in ...
2
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1answer
100 views

Is there an integral form of the equations of QM or GR?

Maxwells equations and also the equations of fluid dynamics can be formulated as integral equations. These equations allow so called weak (non-differentiable) solutions, e.g. shock waves in fluid ...
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1answer
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Different levels of physical model solvability and why reality doesn't care [closed]

In studying physics, one may get the impression that there exists some underlying or even physical difference between models, which solution - motion of a body, wave function - can be found explicitly,...
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1answer
150 views

Which matrices represent unitary projective representations of ${\rm SO(3)}$?

I was reading this post which triggered the following question. The group ${\rm SO(3)}$ is real orthogonal. However, it is possible to consider representations of ${\rm SO(3)}$ on a complex vector ...
2
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1answer
70 views

Are special conformal transformations continuous?

My understanding of special conformal transformations (SCTs) is fairly limited, but I believe that they are composed of an inversion, a translation and another inversion. Since inversions are discrete ...
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1answer
142 views

Statistical physics is unable to prove that $TdS=d\overline{E}$

I will pose $k_B=1$. Suppose a system of statistical physics with the constraints: $$ \begin{align} 1&=\sum_{q\in\mathbb{Q}}\rho(q)\\ \overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...
3
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1answer
93 views

Degenerate ground state of Hamiltonian from analytical perspective

Suppose I have a Hamiltonian that depends on the continuous vector parameter $\boldsymbol{\theta}$, and the ground state corresponds to line/plane or some other $1$ to $p-1$ dimensional subspace of ...
2
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1answer
145 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
104 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
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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 ...
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213 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 ...
8
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1answer
634 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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48 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 ...
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504 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 ...

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