Physics Asked by mrmemo on March 13, 2021
Disclaimer: I am an amateur, and an experimentalist and don’t understand the equations involved.
Intro: Fermions and bosons are (we’re almost definitely sure this time) fundamental particles. Fermions have mass and make everything up so I don’t trust them. Bosons are field particles like the photon. Quantum Field Theory describes their interactions. Both fermions and bosons travel through spacetime. Spacetime gets warped thanks to General Relativity, so the path that those particles take changes based on the shortest actual distance through warped spacetime. This appears like acceleration; the object has a new velocity, so an energy change has occurred.
You see this acceleration on fermions every time you drop your phone. It’s a little more exotic to see the effect on bosons, but gravitational lensing warps light around massive objects. You can see a star behind another star!
Trickier stuff: The equations describing Quantum Field Theory can’t be solved in warped spacetime, which happens due to General Relativity. Dang. BUT, I’m curious if we couldn’t just measure the impact of warping spacetime on the quantum energy of bosons. Wouldn’t that basically be a brute-force transfer function between spacetime and quantum chromodynamics?
So let’s say you’ve got two planck-length-second volumes of spacetime, $V_1$ and $V_2$, and a 500nm photon is traveling from one to the other. $V_1$ is flat spacetime, but $V_2$ is warped. The path of the photon changes based on the layline of the new volume, which appears to us (outside the new volume) as acceleration.
But what happens to the photon while it’s inside that volume? Wouldn’t it gain or lose energy based on the compression along its wave-axis? The reference frame for the original wavelength changed, so the 500nm photon has a different wavelength to an observer inside the volume. Has spacetime not only accelerated the boson, but also given it some quantum energy as well?
Overall question: If the "theory of everything" is blocked by our inability to solve closed-form equations, could we try a brute force approach? Could an option for this look like: bend spacetime some known amount, fire a known wavelength boson into that volume, measure the observed wavelength differential, now you know how much energy spacetime lends bosons due to curvature. Do this enough, and wouldn’t you have a transfer function between QFT and GR?
$newcommand{D}{mathrm{D}}$ Quantum field theory in curved spacetime can very well be studied. The fact that we don't know many fundamental things is not due to lack of understanding, but due to the fact that real-life models, such as the Standard Model, are way too complicated to perform calculations. If we restrict to toy models, e.g. lower or higher dimensions, conformal symmetry, supersymmetry etc. we can "solve"$^{(*)}$ QFT, either perturbatively, or numerically, or even analytically. For the rest, the distinction between fermions and bosons will not play a role, since the question is on a very abstract level, so I will refer to both as "fields" or "particles".
The keyword in your question is the word "known". Whenever you know how the spacetime is bent, you can write down the metric of your universe: $g$. Then you can in principle go out and calculate things at the presence of this $g$. For example you can calculate the partition function, written schematically as $$ Z(g,text{sources}) := int D[text{fields}] exp!Big(iS[g,text{fields}]+text{fields}cdot_gtext{sources}Big). tag{1}$$ From here you can ask everything that you could in flat spacetime QFT and the partition function will answer it for you. In the above formula, the fields are quantum (which is indicated by the $D[text{fields}]$ and the rest of the data are classical.
Quentum gravity is a whole different story. This is when the metric is also quantum. In this case, most of the times you can't naively write down the partition function as $$Z(text{sources}) = int D[text{fields}],D g exp!Big(iS[g,text{fields}]+text{fields}cdot_gtext{sources}Big),$$ except for some very special and interesting cases.
In your question you are concerned about QFT in curved spacetime, not quantum gravity. The setup you are proposing, translated to manageable form is that you are giving me an empty finite volume of space. Then you bend it by a known amount, i.e. you now gave me a metric $g$ of that space. Magically a particle is now in your hands. The particle has known properties (you mentioned wavelength, but let's be general). Let's call the particle with these properties $phi$. You fire it in the curved space, and it comes having different properties; it's a $phi'$ now. What happens in such a scenario in QFT is encoded in the S-matrix, which can be inferred from the various correlation functions, which can, in turn, be derived from the partition function. Depending on the details of your question, the hands-on calculation of this can range from trivial to immensely complicated.
Regarding the last part of your question, it would not give you a "transfer function" between QFT and GR. It gives you just a GR-modified version of QFT. To have a "transfer function" you would have to calculate how the mere existence of the particle inside the curved space affects the curvature, and update the curvature before sending in another guy. This is also a well-known framework, known as semiclassical gravity.
$^{(*)}$ Solve is not the best word to use here. A better phrase would be "calculate the partition function at the presence of sources"
Answered by ɪdɪət strəʊlə on March 13, 2021
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