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Results tagged with mp.mathematical-physics
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user 20302
Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
11
votes
Laplacian on manifolds and random matrix theory
There are quite a few connections. I will mention a result of mine where the connection is explicit and essential. Fix the metric $g$. Set $m=\dim M$ and assume that ${\rm vol}_g(M)=1$.
Denote by $ …
- 34.7k
72
votes
Does Physics need non-analytic smooth functions?
Suppose that $P$ is a partial differential operator with constant coefficients. An old result of Petrowski (see Thm. 3.2 of Hörmander's 1955 Acta Math. paper On the theory of general partial differ …
- 34.7k
9
votes
Accepted
Uhlenbeck's theorem novelty
Denote by $A$ the connection and by $F_A$ its curvature. Then
$$dA=F_A-A\wedge A. $$
If $A$ is in Coulomb gauge we have an additional equation
$$d^*A=0. $$
The advantage is that the operator $d\op …
- 34.7k
4
votes
Gaussian measure on function spaces
You should have a look at the book by Gelfand and Vilenkin
Generalized functions. Vol. 4: Applications of harmonic analysis
where they describe how to construct Gaussian measures on (duals …
- 34.7k
3
votes
Differential calculus of functions of self-adjoint operators
Suppose that $A$ has discrete spectrum consisting of eigenvalues with finite multiplicities
$$ 0<\lambda_1 < \lambda_2<\cdots $$
with $\lambda_n\to\infty$ as $n\to \infty$. Denote by $(\psi_n)$ …
- 34.7k
7
votes
$A \wedge A \wedge A$ in Chern-Simons
For Lie algebras of matrices (which is what you really care about in Chern-Simons theory) think of $A$ as a form with matrix coefficients
$$ A=\sum_i A_i dx^i, $$
where $A_i$ are $r\times r$ matrice …
- 22.3k
1
vote
Can eta invariant be written in terms of topological data?
The eta invariant is typically defined for Dirac operators on odd-dimensional Riemann manifolds. They depend on the various geometric data needed to define them (metrics, connections etc) and for t …
- 34.7k
1
vote
Accepted
The space of holomorphic sections are finite dimensional?
$\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\Hom}{Hom}$ Suppose that $V$ is a finite dimensional real space equipped with an almost complex structure $J$. Let …
- 34.7k
8
votes
Validity of functional derivative using the Dirac delta function
In many variational problems one is given an action functional $f\mapsto S[f]$, described by an integral
$$ S[f]=\int_\Omega L\bigl(\;x,f(x),D f(x),\dotsc, D^k f(x)\;\bigr) dx $$
in which
$\O …
- 34.7k
3
votes
Fourier transforms of functions not in $L^2.$
To find the Fourier transform of this and many other functions I enthusiastically recommend volume 1 of the magnificent treatise Generalized Functions, by Gelfand and coauthors.
This monograph …
- 34.7k
2
votes
Hurwitz numbers and Frobenius manifolds
There are Frobenius algebras in the story. see
http://arxiv.org/pdf/1201.1273v1.pdf
- 34.7k
4
votes
References for classical Yang-Mills theory
Have you tried the book "the Geometry of physics" by Th. Frankel?
- 34.7k
3
votes
Where to start with research regarding maslov index/class
There are different incarnation of the Maslov index. The one that I prefer is the one proposed in Arnold's paper suggested by Igor Rivin. The paper by Cappell-Lee-Miller suggested by Greg Friedman …
- 34.7k