Heute zu Gast um 16:15 im Sitzungszimmer des Mathematischen Instituts für die #MathematischeGesellschaft #Göttingen: Ko Sanders von der @unihannover über
"Distributions of positive type and applications in quantum field theory"
#UniGöttingen #GöttingenCampus #Mathematics #QuantumFieldTheory #Colloquium
Today, I got to give one if my favorite lectures in quantum mechanics in my advanced undergraduate physics majors’ course.
The big concept is that the ultimate nature of reality is that particles are wave functions |Ψ> that exist outside of any coordinate system or physical space. The space that they occupy is instead a Hilbert Space. Their projection in what we percieve as our “space” is one representation of the fundamental nature of the particle — which is a field in a this abstract space. The properties of that field is what determines the properties and interactions of the particle.
This is where physics touches metaphysics — fun stuff!
In a magazine article [1] on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe.”
This article [2] provides a method for averaging an arbitrary collection of objects; however, the average can be any number in the extension of the range of these objects. (Note, an arbitrary collection of these objects is a function.)
Question: Suppose anything meaningful has applications in quantum field theory. Is there a way to meaningfully choose a unique, finite average of a function whose graph matches the description in Wood's quote?
For more info, see this post [3].
[2]: https://arxiv.org/pdf/2004.09103
[3]: https://math.stackexchange.com/q/5052005/125918
#PathIntegral #quantum #FeynmanPathIntegral #mean #average #expectedvalue #quantumfieldtheory
New theoretical #physics preprint https://arxiv.org/abs/2412.08617
We looked at the asymptotic growth rate of the beta function in #quantumFieldTheory , and the relative importance of subdivergence-free #Feynmangraph s. These graphs correspond to integrals, and the size of the graph is measured by its loop number, which also indicates how hard it is to solve the integral. State of the art computations in realistic theories are anywhere between 1 and 6 loops. The asymptotics of the perturbation series is known from instanton calculations. We now showed (in a model theory), that the leading asymptotics describes the true growth rate only for more than 25 loops, way beyond anything that can realistically be computed.
This is good news: It tells us that asymptotic instanton calculations provide non-trivial additional information that can not be trivially inferred from low-order perturbation theory.
In the plot, the red dots are numerical data points for the subdivergence-free graphs in phi^4 theory up to 18 loops, the green lines are the leading instanton asymptotics.
A longstanding conjecture in $ϕ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the exact asymptotics of Martin invariants. In 4D, each graph comes with a nontrivial Feynman integral, its period. We give bounds on the degree in $N$ for primitive and non-primitive graphs, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops. The qualitative behaviour is similar to the 0D series, with a small but systematic tendency for the 4D data to grow faster with $N$, indicating a correlation between periods and $O(N)$-symmetry factors. The zeros of the 4D primitive beta function approach their asymptotic locations quickly, but, like in 0D, the growth rate of the 4D primitive beta function does not match its asymptotics even at 17 loops. Our results improve on the knowledge of asymptotics in QFT by providing concrete analytic and numerical values, and putting individual observables into a broader context of $ϕ^4_4$ theory in 0D and 4D. We demonstrate that even if certain quantities are in agreement with the asymptotics already below 10 loops, this must not be mistaken as evidence that overall an asymptotic regime has been reached.
📣 Tiburtius Prize 2024 for Gustav Uhre Jakobsen 🏆
Recognition award for visiting postdoc at AEI Potsdam
Gustav Uhre Jakobsen, a postdoc at the Humboldt University of Berlin and in the Astrophysical and Cosmological Relativity Department at the @mpi_grav in the Potsdam Science Park, will be awarded a “Tirburtius Prize – Prize of the Berlin Universities” for his dissertation.
The reviewer praises not only the impressive wealth of topics in Jakobsen's doctoral thesis titled “Gravitational Scattering of Compact Bodies from Worldline Quantum Field Theory” and the quality of the research results, but also the impact it has had in the research community.
➡️ https://www.aei.mpg.de/1202051/tiburtius-prize-2024-for-gustav-uhre-jakobsen
#ResearchAward #Berlin #PhDThesis #PhDLife #QuantumFieldTheory #GeneralRelativity
Gustav Uhre Jakobsen will be awarded the Tiburtius Prize 2024 for his outstanding dissertation on gravitational scattering of compact bodies from worldline quantum field theory. Learn more about the award and his research at the Max Planck Institute for Gravitational Physics and the Humboldt University of Berlin.
🚨 New #preprint !
We study a stochastic PDE whose solutions want to be close to constant -1 or +1. But because it’s stochastic, the solutions occasionally jump between those two optima. How often, on average?
In technical terms, we study a certain nonlinear wave equation whose invariant measure is the \( \phi^4 \) #QuantumFieldTheory. The average transition time is called an Eyring–Kramers law, asymptotic in the low-temperature limit. It has already been derived for 2D stochastic heat equation and 1D wave; we extend it to 2D and 3D wave equations.
Joint work with my PhD advisor Nikolay Barashkov. #MathematicalPhysics #MathPhys
We study the expected transition frequency between the two metastable states of a stochastic wave equation with double-well potential. By transition state theory, the frequency factorizes into two components: one depends only on the invariant measure, given by the $ϕ^4_d$ quantum field theory, and the other takes the dynamics into account. We compute the first component with the variational approach to stochastic quantization when $d = 2, 3$. For the two-dimensional equation with random data but no stochastic forcing, we also compute the transmission coefficient.
MI Göttingen
Paul Balduf
Kev Abazajian
Bharath Krishnan
AEI has moved
Petri Laarne
Pustam | पुस्तम | পুস্তম🇳🇵
SubtleBlade ⚔️