Is this published in a serious journal?
It looks like a lot of technical jargon that doesn't make any sense. The quantization of the harmonic oscillator is important only for very small systems. The swing in the park is quantized, but the difference with a classical oscillator is negligible. Then financial market may be quantized, but the effect is ultra tiny. And to notice the quantum effects the system must be almost isolated, so there is no chance to see something like that in the financial market.
Perhaps there is a chance that the it's only a model that fit the data in spite there is absolutely no theoretical reason. It's possible, but it's much easier to overfit the model with the historic data. Do they have any prediction?
Relevant xkcd: https://xkcd.com/1240/
I don't think it's suggesting that quantum effects play a major role in the movement of the stock market, just that quantum models can be applied to finance with different parameters That said, I can't find a clear rationale for choosing this particular model.
I try to avoid reading this kind of papers, but I give it a chance. I only skimmed it, so I'm not sure my takeaway is totally accurate. If you can read it and make any corrections I'd be happy.
They model the financial market with 3 methods.
* geometric Brownian motion (GBM) that has 2 parameters
* Heston that has 1 parameter
* "Quantum" that in their case has 6 parameters
The quantum method is actually the decomposition of using Hermite polynomials multiplied by a Gaussian. This is the base that has the solutions of the quantum oscillator, but they pick the coefficients without any justification. They probably can use whatever smooth localize base of L2(R) they can find. I guess they can use some smooth localized wavelets and get almost the same result (and they can choose one with compact support).
It's possible to use these bases wavelets/HermiteXGaussian/whatever to decompose any signal, like sound or an image, but there is no relation with a background process. In particular that HermiteXGaussian is good doesn't prove that there is a quantum component of the market, only that orthonormal decomposition is a good mathematical tool.
The main differences between the models in their work is that the first one has 2 parameters, the second one has 1 parameter and the "quantum" model has 6 parameters. (Why 6? Why not 5 or 7?) It's easy to get a better fit using 6 parameters instead of 2 if your selection of the model is not horrible, and you can tweak the model a little, and you can pick the time frame, and you publish only the model that has the best fit. This is totally standard, nothing shady, but it may give the impression that the selection of the model has some meaning.