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Results tagged with mp.mathematical-physics
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user 11260
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 …
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 …
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 …
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 $\o …
6
votes
Rigorous treatment of Ostrogradsky's instability theorem?
On the problem of stability for higher-order derivative Lagrangian systems in Letters in Mathematical Physics (1987) may have the desired level of rigor (see Theorem 1).
The proof of the theorem is a …
4
votes
Accepted
Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2)
You can find a fully worked-out derivation in these lecture notes. The formula you are looking for is equation (404), written in terms of the Wigner (small-)$d$ matrix. The relationship to the (large- …
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 …