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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Index notation to matrix notation in supersymmetry
Let us consider the barred sigma matrices defined as $$(\overline{\sigma}^\mu)^{\dot{\alpha}\alpha}:=\epsilon^{\alpha \beta} \epsilon^{\dot{\alpha} \dot{\beta}} (\sigma^\mu)_{\beta \dot{\beta}} \tag{1}...
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Non-trivial "energy coverings" of reciprocal space
My question is about sheets of the Fermi surface and their mathematical properties.
As far as I understand, in the one-electron approximation with a weak periodic potential (Bloch approximation) you ...
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Hellmann-Feynman theorem and the derivation of the Lippmann-Schwinger equation
When deriving the Lippmann-Schwinger equation, one denotes
$$H_\text{free}|\phi\rangle = E|\phi\rangle \tag{1}$$
with $H_\text{free}$ as the free Hamiltonian and
$$H|\psi\rangle = E|\psi\rangle \tag{2}...
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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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Can the Froissart bound be improved?
The Froissart bound is a fundamental result in (axiomatic) quantum field theory, an introduction to which from Froissart himself can be found here.
Given a reaction of scalar particles $1 + 2 \...
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What's the relevance of geometric rigidity/flexibility to physics?
I'm currently working on a mathematics research problem in differential geometry that deals with the rigidity of closed manifolds described by non-trivial induced metrics. I'm curious what the ...
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What is the qualitative difference between the (generalized) Israel theorem and the no-hair theorem?
I know that the Israel theorem accounts for only non-rotating, non-electrically charged black holes, but as I understand the theorem was then generalized for rotating and charged black holes. And, as ...
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Why can non-differentiable solutions to the Schrödinger equation be ignored?
To clarify the question, let's consider the particle in a box (infinite potential $V$ outside [0,1], potential 0 inside [0,1]). (But the problems illustrated here also apply to particles in a non-...
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Global Hyperbolicity of Spacetimes implying Connectedness
I am currently working on a problem and right now I want to show that the global hyperbolicity of a spacetime M implies, that M is connected. Therefore, I assumed the following:
We could write $M$ as ...
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References discussing renormalizability via Sobolev norms
Hawking & Hertog's paper Living With Ghosts has a nice introduction in which the authors discuss the issue of renormalizability of a field theory in terms of Sobolev norms. More specifically, they ...
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Index theorem of Callias operator in physics
In the article "On the index type of Callias-type operator" (https://doi.org/10.1007/BF01896237) Anghel study the index of a Callias type operator over an odd dimensional complete ...
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Do optimal Lieb-Thirring constants have physical meaning?
In their proof of stability of matter Lieb and Thirring used a particular set of inequalities. Namely, if $H=-\Delta+V(x)$ is a Schrödinger operator, then the sum of (powers of the absolute value of) ...
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The meaning of a representation in one-dimensional quantum mechanics
In many places, one reads about chosing a representation for studying a particular one-dimensional quantum system. Usual representations are the position representation, the momentum representation or ...
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On which bundle do QFT fields live?
In QFT, there is a vector field of electromagnetism, usually notated by $A$, which transforms as a 1-form under coordinate changes. Since quantum fields are operator-valued, I thought it is a section ...
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Effects of Localized Medium Changes on Field Propagation
I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
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Why we use trace-class operators and bounded operators in quantum mechanics?
The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
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Jensen's inequality on (super)operator exponential
Let us define the expectation value $\langle A\rangle_{\rho}$ of a superoperator $A$ over a density matrix $\rho$ as $(\rho, A(\rho))$, where the scalar product between operators reads $(O_1,O_2):= Tr[...
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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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Question about the identity operator and the bosonic ladder operators
Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
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Determining Bound States from Møller Operator
Hello I came across an interesting property of the Møller operator, which I summarize below:
The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
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Equivalent definitions of Wick ordering
Let $\phi$ denote a field consisting of creation and annihilation operators. In physics, the Wick ordering of $\phi$, denoted $:\phi:$, is defined so that all creation are to the left of all ...
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Scattering matrix vs. unitary transformations
In quantum optics, the input/output bosonic modes at a beam splitter transform according to the scattering matrix
$$
\begin{pmatrix}
a_1 \\
a_2 \\
\end{pmatrix} = \dfrac{1}{\sqrt{...
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Multimode squeezed operator
Given CCR (bosonic) algebra, with creation / annihilation operators $a_{i}^{\dagger}, a_i$ acting on a single particle Hilbert space $\mathbb{h}$, let's introduce the multimode squeezed operator for $...
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Why does a singularity imply the need for a distribution?
I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says:
...for $d = 1$, the Green's function $G(x)$ is continuous at $...
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When do two state functions represent the same quantum state?
According to the standard quantum mechanics, quantum states are one-dimensional subspaces of a separable Hilbert space. In practice, this Hilbert space is $L^2(M)$ where $M$ is the classical ...
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Correlation length in a 3d Ising slab with one dimension much smaller than the other two
Suppose I have a 3d Ising model on a cubic lattice, but one of its dimensions is much smaller than the other two. That is, I have an $L$ by $L$ by $L'$ slab with $L' << L$; in particular, $L'$ ...
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Why is $H = J \sum_i (S^x_i S^x_{i+1} + S^y_iS^y_{i+1})$ always gapless for any spin $S$?
In the following I have in mind antiferromagnetic spin chains in periodic boundary conditions on a chain of even length $L$.
Consider the spin-$S$ spin chain
$$H = J \sum_{i=1}^L (S^x_i S^x_{i+1} + S^...
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
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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, "...
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