Quantum Decision Theory Series: #5 Techniques from Quantum Field Theory

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By Sudip Patra

There are two stages of development in the basic quantum theory, one, the first quantization, which is dubbed as the ordinary quantum mechanics, and two, the second quantization, which is dubbed as the quantum field theory. The first wave of development happened in 1920s-30s, where great minds like Heisenberg, Bohr,  Schrödinger, Dirac, Born and others played critical roles. The main mathematical structures were provided by famous mathematicians like Neumann and Hilbert. Well the main take away from the first wave was that quantum mechanics is inherently a probabilistic theory based on a logic structure which is non-classical or non-Boolean, and certainly the wave-particle duality nature of all particles is the fundamental truth of the universe based on a deep uncertainty principle.

In this series we have discussed briefly earlier how basic mathematical structure of this first phase of quantum theory can be used and is used in decision making theory in general. However in the Physics world the story just began! The second phase of development was pioneered by next generation of great minds, like Dirac (still), Feynman, Gellman, Yukawa, Landau and others. This phase gave rise to incredibly accurate and beautiful theories like quantum electrodynamics (QED) which is the most accurate description of interaction between photons and electrons (Richard Feynman was the pioneer in this field, you can have a look at his popular books.).

However to come up with the new phase (called as second quantization since creation of particles are thought as to be excitation in fields), the math had to be worked out, and that is an understatement! The math was incredibly tough this time. Imagine performing infinite integrals when we struggle to solve simple integrals even in the ordinary calculus courses. However perhaps even the founders couldn’t have imagined that such strange mathematical tools could be used in social sciences too.

Let’s have a few examples. The first important tool in QFT is so called operators (mainly linear operators, please check any math book for definitions of  these terms), which when operates on a particular state of the system produces a change (there are so many examples: displacement operators, time reversal operators, parity operators etc) in the state. However most important are the so called creation and destruction operators.

Creation operators when operates on a ground state of a quantum system (by ground we mean the lowest energy state which can also be a state of vacuum with 0 particles, but please note according to quantum theory even a vacuum also contains fluctuations at the Plank scale due to Uncertainty principle) updates the system with adding a particle to the system, and destruction operators just do the opposite. Such operators are also called as the rising and lowering operators. These operators are absolutely critical for interaction or scattering theory of particles. Now back to finance.  Recently these operators with their mathematical commutation or anti commutation properties (called as their aljebras) have been used to explain price movements, and portfolio formulations in markets. Imagine a market at its ground state before trading starts, so when a trader buys or sells assets the price rise or falls which can be captured by operating creation (for price rise) or destruction (for price fall) operators on the market ground state.

You can then define other useful operators by products of these basic operators, for example if you have products of destruction followed by the creation operators (you can give a matrix form if you like) operating on the initial ground state you get back the nos of stocks traded in the market for that period, this is the so called number operator widely used in QFT. There can be plethora of other such descriptions, for example if you write in different orders a string of creation and destruction operators then you can get various interesting states from the initial ground state of the market. An entirely new picture of asset market dynamics will emerge.

Professor Sudip Patra is an Assistant Professor of Management Practice at O.P. Jindal Global University . His research interest encompasses dividend signaling theory under information asymmetry, game theory for applied corporate finance, econometric modeling, and allied areas.

Featured Image Source : Institute for Theoretical Physics

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