# 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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### Understanding the parabolic state of a quantum particle in the infinite square well

Consider a quantum particle in an infinite square well of width $2\pi$. In other words, the Hamiltonian is $H=p^2$, the position variable is restricted to $x\in[-\pi,\pi]$ and the wave function ...
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### Can a wave function discontinuous in the time variable be a solution of the Schrödinger equation?

It is well known that wave functions that are discontinuous in the space variable cannot be solutions of the Schrödinger equation, because the Schrödinger equation is a second-order differential ...
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### What is the cardinality of the set of all possible wave functions? [duplicate]

I was just thinking about set theory and somehow this question just comes up to my head. The cardinality of the set of all function from $\mathbb{R}^m$ to $\mathbb{R}^n$ is $2^{\mathfrak{c}}$. ...
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### Classification of Hamiltonians having the same set of solutions

Assume that we are given $2n$ the functions $q_k(t),p_k(t);k=1,2...,n$ all "sufficiently" smooth and invertible in a finite interval $t\in [0,t_{max}]$. My question is what class of ...
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### Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry?

From my notes I have that The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be realised if $A_\mu$ is an element of the Lie algebra. It can ...
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### Zeta Regularization

I am interested in applying the zeta regularization method to regularize $$\zeta_2(s=-1/2)=\sum_{l=1}^{\infty} (2l+1)(l(l+1))^{1/2}.$$ In section 4.5 of the book "Zeta Regularization Techniques ...
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### How are Schwinger functions defined as moments if they are actually operators?

Let $\mu$ be a measure on the space of tempered distributions. Assuming $\mu$ satisfies some other properties, then it is the measure of a quantum field in a Euclidean framework. The Schwinger ...
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### Quantum Mechanics book with good treatment on unbounded operators [duplicate]

I'm looking for a certain kind of approach to quantum mechanics. In most places I've looked, almost all of the exposition on how to employ construct observables in quantum mechanics is that of compact ...
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### Inverse problem for geodesic

If I know the expressions for geodesic distance between any points $x$ and $y$: $$L=L(x^\mu,y^\nu) \ .$$ How do I find the metric of the corresponding space?
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### Why can't there be an infinite number of simple objects in an anyon model?

It is a well-established fact that topological excitations (anyons) in 2D topologically-ordered systems are described by unitary modular tensor categories, see, e.g., Appendix E in Kitaev (2006). One ...
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### Equality of Hilbert subspaces

If $A,B\in \mathscr{L_H}$ in the lattice of subspaces of a Hilbert space $\mathscr{H}$, then is it always true that $$A\sqsubseteq B\ \&\ B\sqsubseteq A \implies A=B\ ~ ?$$ Or is there maybe an ...
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### Operator systems in functional analysis & quantum mechanics: intuition

I saw this concept of operator systems in here but I am not sure if I want to get deep into it before knowing roughly what it is used for in, say, quantum information or quantum mechanics. My very ...
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### Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?

I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
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### Angular momentum Lie algebra for infinite-dimensional Hilbert spaces

Let $V := \operatorname{span}{(J_1, J_2, J_3)}$ be a Lie algebra over the complex numbers such that $J_1$, $J_2$, and $J_3$ are essentially self-adjoint operators on some Hilbert space $\mathcal{H}$. ...
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### Electric field due to plane at constant potential and a cylinder with no flux on surface

There is a plate at a constant potential V and potential equal to zero far away. the problem is two-dimensional. For this case, the electric field lines will simply be straight lines. Now let there be ...
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### Rigorous definition of the average value $\langle a^{*}(f)a(g)\rangle$ on Fock spaces for arbitrary states [duplicate]

One of the axioms of quantum mechanics states that a quantum state $\rho$ is a positive (hence self-adjoint) trace class operator with trace one. Given an observable $A$, the expected value of $A$ in ...
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### Why does the finite trace of density matrix imply a discrete Schmidt decomposition? [closed]

In the paper defining an average Schmidt number for a particular entangled system, Law and Eberly say: Because density matrices always have finite trace, the Schmidt decomposition is always discrete, ...
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### What are the distinct mathematical formalisms of quantum mechanics?

Consider the physical theory called non-relativistic quantum mechanics. What are the distinct mathematical formalisms for this physical theory? That is, different mathematical frameworks for this ...
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### Non-orientability in electromagnetism

I'm currently studying E&M and I have a question related to the mathematical formalism of the theory. Electrodynamics depends heavily on the divergence and Stokes's theorem which in their ...
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### Concrete statement about QFT not being mathematically rigorous [duplicate]

It is often mentioned that QFT is ill-defined mathematically. I have seen this as stated that QFT can be defined on a lattice, but that it breaks down if the lattice spacing goes to zero. ...
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### Elasticity theory: homogeneous deformations of a perfect lattice

I want to understand how the macroscopic (linear) elasticity theory emerges from the microscopic properties of matter. My question is about the model of the "perfect lattice", which is used ...
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### Trouble understanding the experssion of gravity on a cube shaped earth

I'm a high school student working on a maths/geophysics project of which my goal is to try to mathmatically expressthe forces that apply on fluids, and then put them together to express geostrophic, ...
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### (Time-)Orientability in the Language of Fiber Bundles

I'm currently studying spin geometry through Hamilton's book Mathematical Gauge Theory. At a given point, Hamilton considers a pseudo-Riemannian manifold, which I'll take to be Lorentzian in $d=3+1$ ...
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### Azimuthal coordinate operator: Hermition or not? Self-adjoint or not?

I am told that the azimuthal coordinate operator $\hat{\phi}$ is not self-adjoint. I am told this by people who I am sure know much more about this stuff than I do. To my unsophisticated mind, "...