Theoretical Physics Benchmark (TPBench) -- a Dataset and Study of AI Reasoning Capabilities in Theoretical Physics
Paper • 2502.15815 • Published
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A 3-State QM Problem | QM | 2 | The Hamiltonian of a three-level system is given as $H = \begin{pmatrix}
E_a & 0 & A \\
0 & E_b & 0 \\
A & 0 & E_a \\
\end{pmatrix}$ where $A$ is real. The state of the system at time $t=0$ is (in this basis) $\psi(t=0) = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \\
1\\
0\end{pmatrix}$ What is the expectation valu... | The eigenstates are easily found to be $\frac{1}{\sqrt{2}}(1,0,\pm 1)^T$ and $(0,1,0)^T$ with corresponding energies $E_a\pm A$, $E_b$. Let us denote them as $|1\rangle$, $|2\rangle$ and $|3\rangle$. Given state $\psi$ is decomposed as $\frac{1}{2}(|1\rangle +|2\rangle) + \frac{1}{\sqrt{2}}|3\rangle$, the expectation o... | \begin{equation*}
\boxed{\langle E\rangle = \frac{1}{2}(E_a+E_b)}
\end{equation*} | Provide the answer in the form of \texttt{python} code. Implement the following function
\begin{python}
def expectation_value(A: float, E_a:float, E_b:float, t:float) -> float:
pass
\end{python} | \begin{python}
def expectation_value(A: float, E_a:float, E_b:float, t:float) -> float:
return 0.5*(E_a+E_b)
\end{python} |
Bias of a Sampled Halo Field | Cosmology | 5 | In cosmology, large-scale cosmological dark-matter halo fields are biased tracers of the underlying Gaussian matter density $\delta_m$. Assume we have a sample $\delta_m$. We simulate a halo number density field by taking $n(\mathbf{x}) = \bar{n}\max(0,1+b\delta_m(\mathbf{x}))$, where bare number density $\bar{n}$ and ... | \textbf{Detailed Steps:}
The solution to this question involves some domain knowledge, parts of which were given in the problem's statement, some approximations sourced by the domain knowledge, and some mathematical calculations. The domain knowledge is very basic and should be known to anyone in the field. Approximati... | The bias of the sampled halo field is given by:
\begin{equation}
\boxed{
b^{'} = \frac{b \Phi_1\left(\frac{1}{|b|\sigma}\right)}{\Phi_1\left(\frac{1}{|b|\sigma}\right)+|b|\sigma\phi_1\left(\frac{1}{|b|\sigma}\right)}}
\end{equation}
where $\Phi_1$ is the normal cumulative distribution function, $\phi_1$ is the standard... | Provide the answer in the form of the \texttt{python} code. Implement the following function.
\begin{python}
#let b_in stand for bare bias
def b_eff(sigma: float, b_in:float) -> float:
pass
\end{python} | \begin{python}
from scipy.stats import norm
#let b_in stand for bare bias
def b_eff(sigma: float, b_in:float) -> float:
alpha = sigma*abs(b_in)
return b_in*norm.cdf(1/alpha)/(norm.cdf(1/alpha)+alpha*norm.pdf(1/alpha))
\end{python}
\newpage
\newpage |
Blackbody in d Dimensions | Stat Mech | 1 | Assume we live in a 4+1 dimensional spacetime. How does the total energy density of a black body scale with temperature T. Find the exponent $n$ in the expression $u \propto T^{n}$. | The density of states scales as $k^{D-1}dk$ in D spatial dimensions giving
$T^{D+1}$ scaling for the total energy density. Hence, $\boxed{n=5}.$ | $\boxed{n=5}.$ | Provide the answer in a form of \texttt{python} code. Implement the following function
\begin{python}
def answer() -> float:
pass
\end{python} | \begin{python}
def answer() -> float:
return 5
\end{python} |
Boosted Parabolic Trajectory | Classical Mechanics | 1 | Consider a situation where a space-probe very briefly fires its rockets while passing a planet of mass \(M\) at periapsis, its nearest point to the planet. Suppose that the probe is on a parabolic trajectory and at periapsis, when travelling at velocity $v_e$, it results in a boost of $\delta v$. What will be its speed... | Conservation of energy gives $\frac{1}{2}m(v_e+\delta v)^2-\frac{mMG}{r_p} = \frac{1}{2}mv^2_\infty$. We also know that $\frac{1}{2}m(v_e)^2-\frac{mMG}{r_p} = E = 0$ for the parabolic trajectory. We can solve for $v_e$: $v_e = \sqrt{\frac{2MG}{r_p}}$. Then we can substitute it in the first equation and get:
\begin{equa... | \begin{equation*}
\boxed{v_\infty = \delta v\sqrt{1+\frac{2v_e}{\delta v}}}
\end{equation*} | Provide the answer in the form of \texttt{python} code. Implement the following function
\begin{python}
def speed(v_e: float, delta_v:float) -> float:
pass
\end{python} | \begin{python}
from math import sqrt
def speed(v_e: float, delta_v:float) -> float:
return delta_v*sqrt(1+2*v_e/delta_v)
\end{python} |
Dark Matter Capture as a Function of Time | Cosmology | 2 | Suppose $C$ is the capture rate of dark matter in an astrophysical
body. Let $C_{A}$ be the dark matter annihilation rate per effective
volume. Then an approximate Boltzmann equation governing the number
$N$ of dark matter particles in the astrophysical body is
\[
\frac{d N}{dt}=C-C_{A}N^{2}.
\]
If initially, $N(0)=0$,... | We can integrate by quadrature.
\begin{equation}
\int\frac{dN}{C-C_{A}N^{2}}=t.
\end{equation}
We can express the integrand as a sum of two fractions:
\begin{align*}
\frac{1}{C-C_{A}N^{2}} & = \frac{1}{\sqrt{C}-\sqrt{C_{A}}N}\frac{1}{\sqrt{C}+\sqrt{C_{A}}N}\\
& = \frac{1}{2\sqrt{C}}\left[\frac{1}{\sqrt{C}-\sqrt{C_{A}}... | \begin{equation}
\boxed{N=\frac{\sqrt{C}}{\sqrt{C_{A}}}\frac{\left(e^{2\sqrt{C C_A}t}-1\right)}{\left(e^{2\sqrt{C C_A}t}+1\right)}}.
\end{equation} | Provide the answer in the form of the \texttt{python} code. Implement the following function.
\begin{python}
def answer(C: float, C_A: float, t: float) -> float:
pass
\end{python} | \begin{python}
from math import sqrt, exp
def answer(C: float, C_A: float, t: float) -> float:
return sqrt(C/C_A) * (
(exp(2*sqrt(C*C_A)*t) - 1)
/ (exp(2*sqrt(C*C_A)*t) + 1)
)
\end{python}
\newpage
\newpage |
One-Pole Problem | Cosmology | 5 | Consider the conformally coupled scalar field $\phi$
\begin{equation}
\mathcal{L}=\frac{1}{2}\left[g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\left(m^{2}-\frac{1}{6}R\right)\phi^{2}\right]
\end{equation}
in curved spacetime
\[
ds^{2}=a^{2}(\eta)\left(d\eta^{2}-|d\vec{x}|^{2}\right)
\]
where the Ricci scalar is
\... | \begin{figure}
\begin{centering}
\begin{tikzpicture}
\draw[->] (-5,0) -- (5,0);
\draw[blue,ultra thick, ->] (-4,-0.1) -- (4,-0.1);
\draw plot[only marks, mark=x, mark size=4pt](0,-1);
\draw[ultra thick, orange, ->] (-4,-.1) arc (180:225:4);
\draw[ultra thick, orange, ->] (-2.82843, -2.92843) -- (-.2,-1);
\draw[ultra... | \begin{equation}
\boxed{|\beta|\approx\frac{\pi}{3}\exp\left(-\frac{4}{3}\sqrt{2\pi}\frac{\Gamma(5/4)}{\Gamma(3/4)}\frac{(k/a_{e})^{3/2}}{H_{I}\sqrt{m}}\right)}.\label{eq:exampleresult}
\end{equation} | Provide the answer in the form of the \texttt{python} code. Implement the following function.
\begin{python}
def abs_beta(k:float, a_e:float, m:float, H_I:float) -> float:
pass
\end{python} | \begin{python}
from numpy import sqrt, exp, pi
from scipy.special import gamma
def abs_beta(k:float, a_e:float, m:float, H_I:float) -> float:
return pi/3*exp(-4/3*sqrt(2*pi)*gamma(5/4)*(k/a_e)**(3/2)/gamma(3/4)/H_I/sqrt(m))
\end{python}
\newpage
\newpage |
Scalar Particle Scattering | HET | 3 | Consider
\begin{equation}
\mathcal{L} = \left\{ \sum_{i=1}^2 \left[ \frac{1}{2} (\partial_\mu \phi_i)(\partial^\mu \phi_i) - \frac{m_i^2}{2} \phi_i \phi_i \right] - \frac{\lambda}{4} \phi_1^2 \phi_2^2 \right\}
\end{equation}
What is the differential cross section \( \frac{d\sigma}{d\Omega} \) for \( \phi_1 (\vec{k}_1) ... | \textbf{Detailed Steps:}
The amplitude for this process is
\begin{equation}
i \mathcal{M} = -4 i \frac{\lambda}{4} = -i \lambda
\end{equation}
In the CM frame, energy conservation gives
\begin{equation}
2 \sqrt{|\vec{k}_1|^2 + m_1^2} = 2 \sqrt{|\vec{k}_1'|^2 + m_2^2}
\end{equation}
A standard formula for differential c... | \[
\left( \frac{d\sigma}{d\Omega} \right)_{\text{CM}} = \frac{\lambda^2}{64 \pi^2 s} \frac{\sqrt{s - 4 m_2^2}}{\sqrt{s - 4 m_1^2}}
\] | Provide the answer in the form of the \texttt{python} code. Implement the following function.
\begin{python}
def dsigma_domega(lam: float, s_m: float, p_m: float, u_m: float,
m1: float, m2: float) -> float:
pass
\end{python} | \begin{python}
from math import sqrt, pi
def dsigma_domega(lam: float, s_m: float, p_m: float, u_m: float, m1: float, m2: float) -> float:
return lam**2/(64*pi**2*s_m)*sqrt(s_m-4 * m2**2)/sqrt(s_m-4*m1**2)
\end{python}
\newpage |
SHO Vacuum Entanglement | QM | 4 | Consider a coupled simple harmonic oscillator governed by the Hamiltonian
\begin{align}
H & =\sum_{i=1}^{2}\frac{1}{2}\left(\frac{p_{i}^{2}}{m}+kx_{i}^{2}\right)+g\frac{(x_{1}-x_{2})^{2}}{2}.
\end{align}
If the ground state is $|\Omega\rangle$ and the operator $\hat{\rho}$
is the vacuum density matrix partially traced ... | Diagonalize the original Hamiltonian
\begin{align}
H & =(x_{1}\quad x_{2}\quad p_{1}\quad p_{2})\left(\begin{array}{cccc}
\frac{k+g}{2} & -\frac{g}{2}\\
-\frac{g}{2} & \frac{k+g}{2}\\
& & \frac{1}{2m}\\
& & & \frac{1}{2m}
\end{array}\right)\left(\begin{array}{c}
x_{1}\\
x_{2}\\
p_{1}\\
p_{2}
\end{array}\right).
\... | \begin{equation}
S = \boxed{-\ln\left(\frac{4\sqrt{\omega_{1}\omega_{2}}}{\left(\sqrt{\omega_{1}}+\sqrt{\omega_{2}}\right)^{2}}\right)-\left(\frac{(\omega_{2}-\omega_{1})^{2}}{4\sqrt{\omega_{1}\omega_{2}}\left(\sqrt{\omega_{1}}+\sqrt{\omega_{2}}\right)^{2}}\right)\ln\left(\frac{(\omega_{2}-\omega_{1})^{2}}{\left(\sqrt{... | Provide the answer in the form of the \texttt{python} code. Implement the following function
\begin{python}
def entropy(k:float,g:float,m:float)->float:
pass
\end{python} | \begin{python}
from math import sqrt, log
def entropy(k:float,g:float,m:float)->float:
w_1 = sqrt(k/m)
w_2 = sqrt((k+2*g)/m)
expr_1 = 4*sqrt(w_1*w_2)
expr_2 = (sqrt(w_1)+sqrt(w_2))**2
expr_3 = (w_2-w_1)**2
return - log(expr_1/expr_2)-(expr_3/expr_1/expr_2)*log(expr_3/expr_2**2)
\end{python} |
Slow-Roll Inflation | Cosmology | 3 | For the action
\begin{equation}
S = \int dt a^3(t) \left\{ \frac{1}{2} \dot{\phi}^2 - V_0 \exp \left[ - \sqrt{\frac{2}{q}} \left( \frac{\phi}{M_P} \right) \right] \right\}
\end{equation}
where \( q \) and \(V_0\) are constants, derive and solve (integrate) the equation of motion for the field $\phi$ assuming slow-roll ... | The equation of motion is
\begin{equation}
\ddot{\phi} + 3 H \dot{\phi} - \sqrt{\frac{2}{q}} \left( \frac{1}{M_P} \right) V_0 \exp \left[ - \sqrt{\frac{2}{q}} \left( \frac{\phi}{M_P} \right) \right] = 0.
\end{equation}
For the slow-roll inflation, the following must hold:
\begin{equation}
\ddot{\phi}\ll 3H\dot{\ph... | \[
\phi = \sqrt{2q} M_P \ln \left\{ \exp \left[ \sqrt{\frac{1}{2q}} \left( \frac{\phi_0}{M_P} \right) \right] + \frac{1}{M_P q} \sqrt{\frac{V_0}{3}} t \right\}.
\] | Provide the answer in the form of the \texttt{python} code. Implement the following function
\begin{python}
import numpy as np
def phi(q: float, M_p: float, phi_0: float, V_0: float, t: np.ndarray)->np.ndarray:
pass
\end{python} | \begin{python}
import numpy as np
from numpy import sqrt, log, exp
def phi(q: float, M_p: float, phi_0: float, V_0: float, t: np.ndarray):
answer = sqrt(2*q)*M_p*log(exp(sqrt(1/(2*q))*(phi_0/M_p))+1/(M_p*q)*sqrt(V_0/3)*t)
return answer
\end{python}
\newpage
\newpage |
SUSY-Symmetry | HET | 4 | Consider the theory
\begin{equation}
\mathcal{L}=i\bar{\xi}\bar{\sigma}^{\mu}\partial_{\mu}\xi+|\partial\phi|^{2}-|F|^{2}
\end{equation}
where $\xi$ is a 2-component Weyl spinor while $\phi$ and $F$ are
complex scalar fields. Suppose you want to make the following infinitesimal
transformation a symmetry of this theory:... | Denoting the variation $\left(\delta_{\eta}\phi\right)^{\dagger}$
as $\delta_{\eta}\bar{\phi}$, we write
\begin{align*}
\delta_{\eta}\mathcal{L} & = i\delta_{\eta}\bar{\xi}\bar{\sigma}^{\mu}\partial_{\mu}\xi+i\bar{\xi}\bar{\sigma}^{\mu}\partial_{\mu}\delta_{\eta}\xi+\partial_{\mu}\delta_{\eta}\bar{\phi}\partial^{\mu}\... | \begin{equation}
\boxed{\delta_{\eta}\phi=-\sqrt{2}\eta\xi,\quad\left(\delta_{\eta}\phi\right)^{\dagger}=-\sqrt{2}\bar{\xi}\bar{\eta}}.\label{eq:L4-susy}
\end{equation} | Provide the answer in the form of the \texttt{python} code. Implement the following function
\begin{python}
from math import sqrt
def find_delta_phi(eta:float, xi:float, bar_eta:float, bar_xi:float) -> Tuple[float, float]:
"""
Returns the SUSY transformation rules for phi and its Hermitian conjugate:
a tu... | \begin{python}
from math import sqrt
def find_delta_phi(eta:float, xi:float, bar_eta:float, bar_xi:float):
"""
Returns the SUSY transformation rules for phi and its Hermitian conjugate
Returns
-------
A tuple (delta_phi, delta_phi_dagger)
"""
delta_phi = -sqrt(2)*eta*xi
delta_phi_d... |
TP Bench – Theoretical Physics Benchmark for AI
TPBench is a curated dataset and evaluation suite designed to measure the reasoning capabilities of AI models in theoretical physics. Our test problems span multiple difficulty levels—from undergraduate to frontier research—and cover topics such as cosmology, high-energy theory, general relativity, and more. By providing a unified framework for problem-solving and auto-verifiable answers, TPBench aims to drive progress in AI-based research assistance for theoretical physics.
Dataset Sources
- Paper: https://arxiv.org/abs/2502.15815
- Website: https://tpbench.org/
Uses
Citation
If you do find our dataset helpful, please cite our paper.
BibTeX:
@misc{chung2025theoreticalphysicsbenchmarktpbench, title={Theoretical Physics Benchmark (TPBench) -- a Dataset and Study of AI Reasoning Capabilities in Theoretical Physics}, author={Daniel J. H. Chung and Zhiqi Gao and Yurii Kvasiuk and Tianyi Li and Moritz Münchmeyer and Maja Rudolph and Frederic Sala and Sai Chaitanya Tadepalli}, year={2025}, eprint={2502.15815}, archivePrefix={arXiv}, primaryClass={cs.LG}, url={https://arxiv.org/abs/2502.15815}, }Dataset Card Contact
Please email us at research@tpbench.org if you have questions or concerns.
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