When we talk about markets we often use terms like equilibrium or even *market force*. We choose this terminology for a reason. The analogy to the well established theories of mechanics and quantum mechanics is intended and the pictures we have in mind are a pendulum or even a simple spring. Their restoring forces seem to model the *market forces* and therefore we frequently observe argumentations very similar to:

if prices increase, then demand decreases and vice versa finally, because of some process still to be described, the market settles down in an equilibrium (called Walrasian price equilibrium).

As a start, that sounds convincing. There just remains one big question. Is that a good picture? Or, even more to the point:

Are there any justifications for the existence of *market forces*?

Rather than answering this question (regular readers know my standpoint anyway) I would like to justify why this question is actually reasonable and should be asked and answered. In physics this question is answered to the positive, in economics the situation is a little blurry to say the least. I continue by comparing mechanics with economics in catchwords. Thereby pointing out similarities, but also discrepancies and, in a way, recalling ‘the story so far’.

**Basic notions**

Let me start with two of the fundamental notions in mechanics, namely *position* and *momentum*. In earlier posts we have identified their counterparts in economics as *price* and *demand*.

**Symmetries**

In mechanics the intuition is that *momentum* is invariant under translation of *position*. In economics we need *demand* invariance under *price*-scaling.

**Commutation relations**

These symmetries lead to commutation relations of the form in quantum mechanics and in economics (cf. here). This difference is essential and has a huge impact, albeit not immediately.

**Bounded representations**

Both commutation relations imply that the symmetry groups do not have representations on a finite-dimensional vector space (cf. here).

**Unbounded representations**

While there are no bounded representations, we get unbounded representations on the Hilbert space of square integrable functions. *Momentum* and *demand* operators are differential operators, whereas *position* and *price* are (different) multiplication operators (cf. here).

**Uncertainty principle**

The uncertainty principle of quantum mechanics is well-known. So far I didn’t write about that here in the blog, but in economics the commutation relations imply inequalities which can also be interpreted as some sort of uncertainty principle. I shall come back to this later.

**Time evolution**

As described in scientific laws to get the time evolution in quantum mechanics one chooses an *action*, one uses Legendre transform to obtain the *energy*, one derives the canonical equations and essentially plugs in the above representation to obtain SchrĂ¶dingers equation governing the time evolution of a quantum system. That surely sounds more complicated than it actually is.

Why can’t we just do that for markets and obtain market equations governing their time evolution? Now, there are a couple of technical difficulties. The most prominent probably is that the Legendre transform of a market action is **not** invariant under time translation. Hence, in markets there is **no** conservation of energy. This fact alone makes the usage of a term like *market force* a little obscure. What is meant by *force* if there is no *energy* or at least no *energy conservation*?

That essentially is the programme for the rest of the year. I shall spell out the maths behind the uncertainty principle for markets and then delve into the technical details of obtaining a time evolution for markets.

Stay tuned …