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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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2
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1answer
17 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 ...
1
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1answer
22 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 ...
5
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1answer
65 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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0answers
37 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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0answers
30 views

Is the incompressibility the only condition I could use for this problem? [closed]

The problem is In cylindrical coordinates, the velocity is $\mathbf{u}=u_{r} \hat{\mathbf{e}}_{r}+u_{\theta} \hat{\mathbf{e}}_{\theta}+u_{z} \hat{\mathbf{e}}_{z}$. Conside an incompressible flow ...
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1answer
58 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
52 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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33 views

Fractional differential equations and Physics [duplicate]

Are the "fractional differential equations" have any real significance in respect to physics? or are they just stilted math?
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0answers
13 views

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
96 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 ...
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0answers
22 views

Treatment of the trace as a Fredholm kernel, example with the interacting 1d bose gas

There is a confusion here that might be really obvious as each and every author just handwavingly glosses over it. I'm studying the Bethe Ansatz, and in the Algebraic Bethe Ansatz there appears to be ...
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0answers
33 views

Conceptional question about Quintic 3-fold and Calabi–Yau 3-fold

Has anyone an idea if I’m studying CY 3-fold in the context of string theory decomposition, for instance, from 10 dimensions to 4 dimensions supergravity , can I take the metric of the complex ...
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1answer
71 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
23 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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0answers
30 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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41 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
63 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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35 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
73 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
39 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.
2
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1answer
66 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 ...
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25 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
74 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
71 views

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
44 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 ...
3
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1answer
41 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 ...
-1
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1answer
130 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
79 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 ...
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0answers
22 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 ...
2
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1answer
67 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
68 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
63 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 ...
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0answers
70 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
604 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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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 ...
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0answers
73 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 ...
2
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0answers
45 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
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0answers
63 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 ...
1
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1answer
112 views

Orthochronous indefinite orthogonal group $O^+(m, n)$ forms 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
79 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?
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1answer
235 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 ...
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0answers
67 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 ...
2
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0answers
47 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
69 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
85 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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0answers
36 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 $\mathfrak g$ ...
3
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2answers
93 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
57 views

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

During field theory course the Källén-Lehmann propagator was defined as follows: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
1
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2answers
35 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 ...