Questions tagged [quantum-mechanics]
For questions about mathematical problems arising from quantum mechanics, a branch of physics describing the behaviour of nature at very small scales, at the level of atoms and subatomic particles.
389
questions
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28
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Moment generating function for product states
In the sequel $B=M_\ell(\mathbb{C})$.
For $M\in\mathbb{N}$ fixed and $N\geq M$ I consider the symmetrizer $\pi_{M,N}(x_M)\in B^{\otimes N}$, which is the symmetrized tensor product of $a_1$,...,$a_M$ ...
4
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0
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121
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Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?
Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group
of automorphisms of $\...
1
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0
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134
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Recommendation to understand mean field theorem
I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
2
votes
1
answer
163
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Poisson quantization vs quantization in atomic physics
Is it possible to interpret quantization in atomic physics ( e.g. the quantization condition in hydrogen atom stated as exponential decay of wave functions at infinity and analogously for n-electron ...
-2
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1
answer
122
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Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]
Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
1
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0
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81
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Definition of second quantization
The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
4
votes
4
answers
403
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Why computing $n$-point correlations?
I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE.
In axiomatic QFT,...
4
votes
2
answers
191
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Reference for rigorous interacting many-body quantum mechanics
Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics:
Second ...
1
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0
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143
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Reconstructing the manifold from space of functions in quantum mechanics
Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$.
But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the ...
6
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0
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90
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Conditions for completely positive maps to act homomorphically across multiple subalgebras
For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
1
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1
answer
105
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Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime
This is a bit of a qualitative question, but I have great difficulty finding a reference that clarifies the point I have been confused about. So, I guess I need to ask here..
Let us restrict atttetion ...
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1
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147
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+100
Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
0
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0
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114
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Generalized operator norm triangle inequality
Let $O_1, \cdots, O_n$ be Hermitian operators and $c_1, \cdots, c_n$ be complex numbers. If $\| \cdot \|$ denotes the operator norm, does the following inequality hold?
$$\| \sum_{i=1}^N c_i O_i \| \...
12
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0
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Does this matrix norm inequality have interesting application in other areas of mathematics?
In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices:
Theorem 3. Let $A=[a_{ij}]$ be a real symmetric ...
1
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0
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77
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Definition of this formula for the $2p$ functions
I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$...
0
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1
answer
115
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From superoperator of unitary to the unitary itself
I have a question about super operators. Let's say I have the super operator of some unitary matrix $u$ called $SU$ where $SU = u^\ast\otimes u$ (here $u^\ast$ is the complex conjugate of $u$). If I ...
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0
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110
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Sudden drop in complexity class due to the more general correlations
Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
5
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0
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243
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Lie algebras, root systems and qubits
This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...
1
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0
answers
112
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What is $H^*(\mathbb{CP}^{2^N-1}/\Sigma_n;\mathbb{Z})$ when $N=\binom{n}{2}$?
$H^*((S^3)^N/\Sigma_n;\mathbb{Q})$ is computed here.
It makes a little more sense to compute $H^*((S^2)^N/\Sigma_n;\mathbb{Q})$ given that global phase is irrelevant. The proof is exactly the same.
...
4
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1
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544
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What is a particle in the context of QFT with interactions?
I'm a bit of a novice, so bear with me.
My understanding of the story is as follows.
From Lagrangians to Irreducible Representations
The story of the types of possible particles begins with the ...
1
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0
answers
226
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Riemann hypothesis and zero of Mellin transform of eigenfunction of Schrödinger equation
The paper [1] shows that the eigenfunction of the Schrödinger equation
$$
\left(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2}\right)f_n=\frac{2n+1}{2\pi}f_n
$$
satisfies the same functional equation as the ...
5
votes
1
answer
240
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von Neumann algebra of canonical commutation relations
In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $...
0
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0
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60
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How to derive the formulas of the spin-weighted spheroidal eigenvalues (2.16a)-(2.16g) in arXiv:gr-qc/0511111?
I am reading the article "Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics
in four and higher dimensions", which is on https://arxiv.org/abs/gr-qc/0511111.
I want to ...
4
votes
0
answers
95
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Frobenius norm bounds on exponentials of anti-Hermitian matrices
Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
1
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0
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149
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A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
0
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0
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99
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A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
10
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2
answers
445
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Does approximate equality of quantum states imply operator inequality in a large subspace?
Let the trace norm of $X$ be
$$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$
and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
4
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1
answer
1k
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What is Quantum Geometry supposed to be about?
There are meaningful questions we can ask about Euclidean geometry which could not have been posed in the time of Riemann or even of Hilbert, and which would have made no sense at all to Euclid. For ...
7
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2
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515
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Weak convergence related to Hermite polynomial?
I am reading Griffiths's quantum mechanics book; in the section about harmonic oscillators, he wrote out the amplitude of wave function, and compared with the classical harmonic oscillators. He ...
2
votes
1
answer
156
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Manifold of entangled states
Given a specific density matrix $\rho$ that corresponds to an entangled quantum state, I would like to find a class of operators (that might be $\rho$ dependent) that tranform (with high probability) $...
2
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0
answers
46
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Coefficient growth upper bound of a recurrence relation
Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
...
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0
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118
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Probabilistic interpretation of von Neumann's approach to quantum mechanics
One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
0
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0
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149
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About the proof of Lebesgue decomposition theorem for Hilbert spaces
Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
1
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0
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180
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Characterization of the Hamiltonian's spectrum in quantum mechanics
This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some ...
3
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1
answer
110
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Time-dependent quantum dynamics as dynamical system and ergodic therorems
In full glory, a dynamical system is defined as a tuple $(T,M,\Phi)$ where $T$ is a monoid, written additively, $M$ is a set, and $\Phi$ is a function.
$$ \Phi : U \subset T \times M \to M$$
with
$ I(...
13
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4
answers
2k
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Meaning of a quantum field given by an operator-valued distribution
I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me.
Let $\mathcal{H}$ be a Hilbert space in which ...
2
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0
answers
45
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Moduli spaces of 'generalized mutually unbiased bases'
Mutually unbiased bases in $\mathbb{C}^n$ with a chosen inner product are collections of orthonormal bases such that for each pair of orthonormal bases $e_i,f_i$, $i=1,\ldots,n$ we have $|\langle e_i, ...
8
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2
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518
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Interpretation of spectral measures in quantum mechanics
Let us define a pure vector state of a quantum system as a vector $\psi$ in a Hilbert space $\mathscr{H}$ with norm $\|\psi\| = 1$. Let $\mathscr{B}(\mathscr{H})$ be the Banach space of bounded linear ...
2
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0
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80
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Evolution equation in renormalization group for infinitely-many variables
Let $\varepsilon > 0$, $L \gg 1$ and define the torus $\mathbb{T} = \varepsilon \mathbb{Z}^{d}/L\mathbb{Z}^{d}$. Let $K$ be a smooth, strictly decreasing function. To make things easier, consider ...
6
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2
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418
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Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
1
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0
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63
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Moyal products of exponentials
I have trouble verifying a claim made by Sharan in his paper: https://doi.org/10.1103/PhysRevD.20.414.
What he essentially claims is that the sequence of Moyal $*$-products
$$\underbrace{\exp(-itH/n)*\...
6
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0
answers
115
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Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
4
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0
answers
102
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Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators
I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
0
votes
1
answer
267
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Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy?
von-Neumann entropy
I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum_i p_i\ln p_i$ in classical mechanics. And I want to get Bose-...
3
votes
1
answer
491
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References for applications of Young diagrams/tableaux to Quantum Mechanics
I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "...
4
votes
1
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348
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Motivation for Heisenberg's modeling of observables
What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
0
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0
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219
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Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)
I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows:
$$\hat{X}^{r}=\hat{x}-i(r-...
0
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0
answers
52
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Approximating e-bits by CHSH black boxes
In the CHSH game two parties Alice and Bob independently get a bit $x$ and $y$ which is $0$ or $1$ with probability $\frac{1}{2}$.
Without communicating each of them has to send a referee a bit $a$ ...
2
votes
1
answer
310
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On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
3
votes
1
answer
259
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Generators of polynomial invariant ring of compact Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$I'm a PhD student in physics working in the broad area of photonic quantum computing. My current project looks at the ...