Often, the study of the natural world reveals unexpected connections between seemingly disparate topics, and researchers can find themselves working in an area very far removed from their usual work. This is especially prevalent in theoretical physics, where ingenious applications of mathematical models can open up hidden connections and deliver surprising new insights.
Such was the case for Dorje Brody, professor of mathematics at Brunel University London, who has just been appointed to the Editorial Board of JPhysA. A specialist in quantum theory, statistical mechanics and mathematical finance, Brody has made some interesting discoveries at the intersection of these fields.
I spoke to Brody to find out more about his work on a description of quantum mechanics that is stochastic. Stochastic systems are random, and are modelled using probabilities.
‘Quantum mechanics is intrinsically stochastic, but many physicists have a psychological barrier to accepting a stochastic evolution equation that models fundamental laws of physics,’ he says. ‘I certainly was such a person.’
The Schrödinger equation, which models the continuous evolution of the quantum state, is deterministic: it gives an explicit and unique picture of the wave function with time. To model quantum mechanics stochastically, you relax the requirement for the system to be deterministic. This gives a variant of the Schrödinger equation called the energy-based stochastic Schrödinger equation, originally introduced by Nicolas Gisin, where random movements like Brownian motion can affect the evolution of the state.
According to this equation, a given initial state will spontaneously evolve into one of the possible stable energy states, subject to certain probabilities, on time scales that are proportionally smaller to how uncertain the intial energy measurement is. So while the regular Schrödinger equation makes no dynamical prediction when the system is coupled to a measurement apparatus, according to its stochastic counterpart the state will almost instantaneously ‘collapse’ into one of the possible states, since the energy uncertainty of the system is significantly amplified by the coupling.
In other words, Brody says, ‘if one accepts the stochastic equation as representing the fundamental evolutionary equation, then there is no need to make a distinction between the evolution part and the reduction part of quantum mechanics: the two aspects can be unified in a single equation.’
The problem of finding a solution to this equation was where a background in mathematical finance proved useful. When Brody was a graduate student, many of his collaborators were working for investment banks, and he became aware of some of the problems they were facing. ‘Since I am generally interested in understanding various phenomena that we observe, both natural and social, I started to think about some isolated problems in finance,’ he says. This led to him teaching a financial mathematics course to physics students, as well as a few papers published in addition to his usual work. Apart from free trips to places like New York, Tokyo, and Paris, ‘perhaps the most significant benefit was that it forced me to learn the basics of modern probability theory and stochastics—one area of mathematics often left out in physics education.’
Brody and his collaborator Lane Hughston found the solution to the energy-based stochastic Schrödinger equation by borrowing the mathematical techniques of nonlinear filtering theory. ‘The basic idea is that there is a signal one wishes to detect, but the observation is obscured by background noise. Thus, there are two unknown quantities, signal and noise, but only one known, observation; what filtering theory offers is the best estimate of the unknown signal, given the observation.
‘We imagine that the ‘signal’ in question is not necessarily ‘transmitted’ as such, but rather constitutes the ‘true’ stable configuration in which the system should settle. The particle is ‘observing’ (by abusing the terminology) this under Brownian noise, and continuously re-adjusts its own wave function in line with the best estimate under the noisy observation.’
The differential equation satisfied by this estimate turned out to be the same as the stochastic equation that they were trying to solve.
But this didn’t just give insight into quantum mechanics. Human behaviour can also be modelled by the same filtering techniques. ‘We were able to develop a completely new framework for asset pricing theory in finance, based on the mathematical formalism we used to solve the energy-based stochastic Schrödinger equation,’ said Brody.
Now, in physics he’s working on certain aspects of controlled quantum dynamics, and also on equilibrium properties of quantum systems. In finance his focus is on problems he thinks are important for society.
‘Little is presently understood about how to model and control the risks [of long-term financial contracts],’ he said. These contracts, such as pension plans or mortgages, are vital to the stability of society. ‘Our aim is to address these issues directly with the development of new mathematical models in finance suitable for the valuation, simulation, and risk management of long-term contracts.’
I’m looking forward to working with Brody on JPhysA, as he will bring a unique perspective to our Editorial Board.
This work is licensed under a Creative Commons Attribution 3.0 Unported License
Image copyright Dorje Brody; used with permission.