## Scientific Laws

September 2, 2009

As I have told you earlier, my guest is very sceptical about our scientific achievements. What follows are the notes I took, when he gave me a short summary of what he considers ‘our strategy’.

In modern understanding of science, the fundamental laws seem to be consequences of various symmetries of quantities like time, space or similar objects. To make this idea more precise scientists often use mathematical arguments, thereby choosing some set ${X}$ as state space encoding all necessary information on the considered system. The system then is thought to evolve in time on a differentiable ${n}$-dimensional path ${x_i(t)\in X}$ for all ${t\in\mathbb{R}}$ and ${1\leq i \leq n\in\mathbb{N}}$. Quite frequently there is a so-called Lagrange function ${L}$ on the domain ${ X^n \times X^n \times \mathbb{R} }$ and a constraint function ${W}$ on the same domain. The path ${x(\cdot)}$ is required to minimizes or maximizes the integral

$\displaystyle \int_0^T L\left(x(s),\dot{x}(s),s\right)ds$

under the constraint

$\displaystyle W\left(x(s),\dot{x}(s),s\right)=0.$

(Under some technical assumptions) a path does exactly that, if it satisfies the Euler-Lagrange equations

$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}-\frac{\partial L}{\partial x_i}=\lambda \frac{\partial W}{\partial \dot{x}_i}$

for some function ${\lambda}$ depending on ${X^n \times X^n \times \mathbb{R}}$.

Define ${y_i:=\frac{\partial L}{\partial \dot{x_i}}}$ and observe that (under suitable assumptions) this transformation is invertible, i.e. the ${\dot{x}_i}$ can be expressed as functions of ${x_i, y_i}$ and ${t}$. Next, define the Hamilton operator

$\displaystyle H(x,y,t) = \sum_{i=1}^n \dot{x}_i(x,y,t) y_i - L(x,\dot{x}(x,y,t),t)$

as the Legendre transform of ${L}$. The Legendre transformation is (under some mild technical assumptions) invertible.

Now, (under less mild assumptions, namely holonomic constraints) two things happen. The canonical equations

$\displaystyle \frac{d x_i}{d t} = - \frac{\partial H}{\partial y_i} \left(=[x_i, H]\right), \frac{d y_i}{d t} = \frac{\partial H}{\partial x_i}\left(=[y_i, H]\right),\frac{d H}{dt} = -\frac{\partial L}{\partial t}$

are equivalent to the Euler Lagrange equations. Here ${[\cdot,\cdot]}$ denotes the commutator bracket ${[a,b]:= ab-ba}$. Furthermore, if ${L}$ does not explicitly depend on time, then ${H}$ is a constant. That is the aforementioned symmetry. ${H}$, the energy, is invariant under time translations.

Given all that, the solution of the minimisation or maximisation problem can then be given (either in the Heisenberg picture) as

$\displaystyle x(t) = e^{t H} x(0) e^{-t H}, y(t) = e^{t H} y(0) e^{-t H}$

or (in the in this case equivalent Schrödinger picture,) as an equation on the state space

$\displaystyle u(t)= e^{t H}u(0).$

This description is equivalent (under mild technical assumptions) to the following initial value problem:

$\displaystyle \dot{u}(t)=H u(t), u(0) = u_0\in X.$

where the operator ${H}$ is the ‘law’. More technically, the law is the generator of a strongly continuous (semi-)group of (in this case linear and unitary) operators acting on (the Hilbert space) ${X}$. As an example of this process he mentioned the Schrödinger equation governing quantum mechanical processes.

His conclusion was that the frequently appearing ‘technical assumptions’ in the above derivation make it highly unlikely for laws to exist even for systems with, what he calls, no emergent properties. ‘If that was true’, I thought ‘then … bye bye theory of everything!’ He explained further, that under no reasonable circumstances it is possible to extrapolate these laws to the emergent situation. I am not sure, whether I understand completely what he means by that, but his summary on how we find scientific laws is in my opinion way too simple. It can’t be true and I told him.

With just a couple of ink strokes he derived the commutation relations for exchange markets from microeconomic theory. That left me speechless, since I always thought, that there cannot be ‘market laws’. Markets are on principle unpredictable! They are, or?