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I have asked a question here:

  • I want to see an example which is related to (integral) quadratic forms or theta series.

@Kiryl Pesotski answered me in some comments as following:

  • 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.

  • 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.

  • 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.

I have not any physical knowledge;

Can any one explain his answer for me in more details?

Also I have list other related questions from the highest vote to the lowest:
[None of them answers my question; except the $6^{\text{th}}$ question; which I feel a connection.]

  1. Number theory in Physics

  2. Examples of number theory showing up in physics

  3. Why is there a deep mysterious relation between string theory and number theory, elliptic curves, $E_8$ and the Monster group

  4. $p$-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

  5. Where do theta functions and canonical Green functions appear in physics

  6. Number of dimensions in string theory and possible link with number theory

  7. Are there any applications of elementary number theory to science?

  8. Algebraic number theory and physics

Davood
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  • I think this question should be posted in the physics-site. – Peter Oct 04 '17 at 17:25
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    @Peter ; My dear Peter; as I have been mentioned, I have asked this question in physics-stack-exchange.
    I have asked this here because there are tags like "physics" and "heat-equation" in math-stack-exchange.     – Davood Oct 04 '17 at 17:31
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    You can create toy "physics problems" where modular functions are part of the solution. Modular functions (and modular forms ?) should have a good physics meaning in an hyperbolic universe ? And in general if $f \in \mathbb{C}(X)$ is in the function field of some compact Riemann surface $X$ then (since $\mathbb{C}(X)$ is finitely generated as a field) there is a non-linear differential equation between the powers of $f^{(k)}$. –  Oct 04 '17 at 17:36
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    I'd say in maths we create problems, but in physics the problems are already there and it makes little sense to create new abstract ones (ie. without physical realization, $p$-adic string theory is a good example) –  Oct 04 '17 at 17:40
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    By the way I made some progress on your problem on binary quadratic forms, send to me if you have some references (did you read Cox ?) –  Oct 04 '17 at 18:48
  • @reuns ; I have read the first chapter of cox and a about half of the 3-rd chapter. I have been so excited. What do you mean by the following: ** I made some progress on your problem on binary quadratic forms**? What reference do you request? – Davood Oct 04 '17 at 19:10
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    I meant I have little to achieve to construct a quadratic number field and a sum of Hecke L-functions (multiplicative coefs) whose coefficients are of $\Theta_Q$ –  Oct 04 '17 at 19:14
  • @reuns ;It makes me so excited; I will waiting for your answers. – Davood Oct 04 '17 at 19:19
  • Talking about the second point: the problem is that the partition function is often given by a sum over non negative numbers, so unless you have a model with $\beta = 0$, you can't introduce Jacobi theta functions. – Matteo Oct 08 '17 at 10:25

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