2

I have heared that there is connection between number theory and physics.
I specially ask that:

  • Can anyone give a concrete example which shows this connection?
    I want to see an example which is related to (integral) quadratic forms or theta series.

Let $f$ to be a quadratic form; i.e a homogeneous polynomial of degree two (in any number of variables), for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.
Also let's assume that all of the co-efficeints of $f$ are integral.

Now let's fix a positive definite (integral) quadratic form $f$ in $m$ variables, and let $n$ varies in $\mathbb{N}$; let's define:

$$ N(n) := N(f,n) = N\big(f(x_1, ..., x_m),n\big) = \# \{(x_1, ..., x_m) \in \mathbb{Z}^m : f(x_1, ..., x_m)=n \} ; $$
note that $N(n)$ is the number of integral solutions to the equation $f=n$; now by a theta series associated to the quadratic form $f$ we mean the following:

$$ \Theta(q) := \Theta_f(q) = \sum_{n\in \mathbb{N}_0}N(n)q^n =\sum_{(x_1, ..., x_m) \in \mathbb{Z}^m}q^{f(x_1, ..., x_m)} ; $$

also if we assume that $q=e^{2\pi i z}$; then there is theorem which states that $\Theta_f(q)$ is a modular form in $z$.

I have asked my question here again.

Davood
  • 177
  • 2
    For example, you may want to compute the partition function for the pure point spectrum of the Hamiltonian where the eigenenergies are quadratic in the quantum numbers. This is e.g. particle in a box or the leading order correction due to the $x^{3}$ and $x^{4}$ powers to the spectrum of the harmonic oscillator. These series can be written in terms of the Jacobi theta functions. – Kiryl Pesotski Oct 03 '17 at 20:46
  • 2
    This post (v4) seems like a list question. For starters, there is modular invariance in string theory. – Qmechanic Oct 03 '17 at 20:52
  • 1
    I mean you have the energies being something like $E_{n}=\alpha+\beta{n}+\gamma{n^{2}}$, where $n\in\mathbb{Z}$, or $\mathbb{N}$. The partition function is given by the seres $Z=\sum_{\forall{n}}e^{-\beta{E_{n}}}$, such sreis are used to represent Jacobi theta functions. – Kiryl Pesotski Oct 03 '17 at 21:01
  • 1
    The other example is the heat equation. The Jacobi theta function solves it. E.g. $\partial_{t}u=\frac{1}{4\pi}\partial^{2}_{x}u$ is solved by $u(x, t)=\theta(x, it)$, where $\theta(x, \tau)$ is the jacobi theta function. – Kiryl Pesotski Oct 03 '17 at 21:03
  • @Kiryl Pesotski ; I have asked my question here again: https://math.stackexchange.com/q/2457644/ I have ask for more explanation. – Davood Oct 04 '17 at 17:11
  • @Kiryl Pesotski ; That question migrated to physics-stack-exchange again! https://physics.stackexchange.com/q/361650/ . – Davood Oct 08 '17 at 09:17
  • More here https://physics.stackexchange.com/q/805997/226902 – Quillo Mar 12 '24 at 08:16

0 Answers0