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Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

3 votes
Accepted

Why is the second order correction to energy zero for a fully degenerate eigensystem?

If $H$ is fully degenerate, all eigenvalues are identical, it means that $H$ is proportional to the unit matrix. The zeroth order eigenstates can be chosen as any orthonormal basis. Degenerate perturb …
Carlo Beenakker's user avatar
6 votes
Accepted

Are renormalizability and the criticality of a PDE synonymous?

The terms describe how the coupling terms of the theory change as one increases the energy. A theory is renormalizable = critical if the coupling terms remain unchanged, super-renormalizable = sub-cri …
Carlo Beenakker's user avatar
1 vote

Rigorous statistical mechanics: difficulty of realistic models

General remark on why one would study simple models: In the statistical mechanics of phase transitions one distinguishes relevant and irrelevant variables. A phase transition is associated with a dive …
Carlo Beenakker's user avatar
6 votes

Why is resonance such a widespread phenomenon?

A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\o …
Carlo Beenakker's user avatar
6 votes

Rigorous treatment of Ostrogradsky's instability theorem?

On the problem of stability for higher-order derivative Lagrangian systems in Letters in Mathematical Physics (1987) may have the desired level of rigor (see Theorem 1). The proof of the theorem is a …
Carlo Beenakker's user avatar
4 votes
Accepted

Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2)

You can find a fully worked-out derivation in these lecture notes. The formula you are looking for is equation (404), written in terms of the Wigner (small-)$d$ matrix. The relationship to the (large- …
LSpice's user avatar
  • 12.9k
6 votes

On the $\phi^4$-model on infinite lattice

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spaci …
Carlo Beenakker's user avatar
10 votes
Accepted

About Friedrichs historical contribution to QFT cited in Reed and Simon

Friedrichs' early contributions are discussed in On the Stone-von Neumann Uniqueness Theorem and Its Ramifications by S.J. Summers: In the early 1950's, K.O. Friedrichs undertook an influential atte …
Carlo Beenakker's user avatar
21 votes

Who says understanding physics helps mathematicians? (A reference request) [Take the word "w...

Michael Atiyah, On the Work of Edward Witten: In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to …
Carlo Beenakker's user avatar
1 vote
Accepted

Recursive relation to represent the last element of a matrix using determinant

Check out https://en.wikipedia.org/wiki/Minor_(linear_algebra) Cramer's rule says that $$R^{(N+1)}=\frac{1}{\det M^{(N+1)}}C^\top,$$ with $C$ the cofactor matrix of $M^{(N+1)}$. The $(N+1,N+1)$ elemen …
Carlo Beenakker's user avatar
4 votes
Accepted

Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT

You can just Fourier transform your signal $$\hat{x}(\omega)=A\int_{0}^\infty \sin(\omega_0 t)e^{-\alpha t}\,e^{i\omega t}\,dt=\frac{A\omega_0}{\alpha^2-2 i \alpha \omega+\omega_0^2-\omega^2}.$$ The p …
Carlo Beenakker's user avatar
2 votes

Algebra/Algebraic geometry in statistical mechanics

An algebraic approach is used in physics to develop a rigorous theory of systems with an infinite number of degrees of freedom, as they appear in quantum field theory and in the thermodynamic limit of …
Carlo Beenakker's user avatar
7 votes

Why computing $n$-point correlations?

Quite generally, three-point (and higher order) correlators are used to reveal the non-Gaussian (read: nonclassical) character of the fields, see for example Experimental characterization of a quantum …
Carlo Beenakker's user avatar
5 votes

Reference for rigorous interacting many-body quantum mechanics

A textbook that covers much ground in a mathematically rigorous way is Mathematical Methods of Many-Body Quantum Field Theory by Detlef Lehmann (2004). This book offers a comprehensive, mathematicall …
Carlo Beenakker's user avatar
6 votes

What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$

You need to solve the following equation for $z=e^{ix}$ $$\left(z^2+1\right) \left(z^{2 n}-1\right)-n \left(z^2-1\right) \left(z^{2 n}+1\right)=0.$$ Wolfram Alpha can do that for you. The answer is in …
Carlo Beenakker's user avatar

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