## Quantum Mechanics and Market Observables

Quantum mechanics has quite powerful notion of observables. They are just properties of system state that can be determined by measurement process. Let’s recall main principles of quantum mechanics and how they could be applied to markets.

### Quantization principle

Observables in quantum mechanics are represented by operators on a Hilbert space. However, not all mathematical operators are suitable for representing observables. Since the values of observables have to represent real quantities, like coordinate and momentum in physics, price and volume in markets, the operator associated to a quantum observable must be a Hermitian operator. An operator said to be Hermitian when

Hermitian operator could be represented by it’s spectrum or eigenvalues and eigenvectors:

where is eigenvalue and is eigenvector.

Price representation is subject of different regulations. Notably, price decimalization is one of things had influenced markets last time. But price itself remains discrete but theoretically infinite. Therefore, quantum operator of fair price must be defined on infinite-dimensional Hilbert space and must have discrete spectrum.

Non-injective operators have only discrete spectrum. Operator is said to be injective if and only if for all and from operator domain, if then This property should be used to construct fair price operator.

The following observables are present at market:

### Statistical algorithm

Given that a quantum system is completely defined by a vector (Eq. (2.15)), the probability of having a determinate measurement result – an eigenvalue of the measured observable – is given by

What it gives us is a method for establishing the link between order book properties and observable. Thus order book provides buy/sell distribution by price .

### Simultaneous measurability

The necessary and sufficient condition for two observables and to be simultaneously measurable with arbitrary precision is that they commute:

As order book is visible for any market participant and fair price and implied volatility are not measurable, then their respective observables must not commute:

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