Skip to main content# The equation x'=x^3 has a non-deterministic solution

# Abstract

**General picture**

# Example

# Conjecture

We show that the classical dynamical system x'=x^3 has a non-deterministic time-evolution and we argue that for most real-world cases the so-called general solution with a singularity is not valid.

Published onOct 05, 2022

The equation x'=x^3 has a non-deterministic solution

We show that the classical dynamic system x’=x^3 has a non-deterministic time-evolution and we argue that for most real-world cases the so-called general solution with a singularity is not valid. We also show that time-independent quantum Hamiltonians are expressive enough, even in the presence of singularities and time-dependent differential equations at the cost of choosing a larger sample space.

Consider a sample space defined by the real line. Since we can always define a wave-function by taking the square-root of the probability density function, the Koopman-von Neumann version of classical statistical mechanics[1][2][3][4] defines classical statistical mechanics as a particular case of quantum mechanics where the algebra of observable operators is necessarily commutative (because the time-evolution is deterministic).

However, it is well known that for two unbounded self-adjoint operators, the fact that they commute does not imply that their corresponding spectral measures commute[5], then we would say that they *strongly* commute.

Thus in principle, it is possible to have a classical dynamical system defined by an *ordinary differential equation* where the unitary time-evolution is non-deterministic, that is, the algebra of observable operators is non-commutative.

Since the real line is a simply connected group (for addition), then by Bargmann’s theorem any strongly continuous projective representation of the real line is induced by some strongly continuous representation of the real line on the corresponding Hilbert space[6][7]. Then, Stone’s theorem implies that such representation on the Hilbert space is the exponential of a time-independent Hamiltonian[6][7]. Thus, if there is a solution in a standard probability space of a differential equation without singularities for all times (including negative times), then there is a quantum Hamiltonian corresponding to such solution which is time-independent. Moreover, if such solution is deterministic then the fact that there are more Hamiltonians that generate the same differential equation does not change the unique solution, up to initial conditions: suppose that

On the other hand, if there is no deterministic solution without singularities for all times (including negative times), then a solution without singularities must be defined by a self-adjoint Hamiltonian (by Bargmann’s and Stone’s theorems) and it must be non-deterministic. Moreover, since any solution in a standard probability space is given by a time-dependent self-adjoint Hamiltonian in a larger probability space, then the singularities in a standard probability space are consequence of a time-dependent Hamiltonian which is non-integrable. But since any time-dependent Hamiltonian can be converted into a time-independent Hamiltonian in a even larger sample space, then we can always resolve the singularities in a larger sample space which better defines the dynamical system. Thus, solutions without singularities for all times (including negative times) defined by time-independent Hamiltonians are expressive enough, even to deal with singularities at the cost of choosing a larger sample space.

Consider the differential equation[8]:

$x(t)=\pm \frac{1}{\sqrt{\frac{1}{(x(0))^2}-2t}} \tag{1}$

With the plus or minus sign chosen such that the solution is continuous at

On the other hand, the corresponding quantum Hamiltonian operator *strongly *commute for all times. If they would *strongly *commute for all times, then they could be simultaneously diagonalized, thus there would be a deterministic solution without singularities. This would be in contradiction with the fact that there are no deterministic solutions without singularities[8].

If we now assume that the deterministic solution Equation 1 exists for some

We could think that a unitary time-evolution is somehow too demanding, that only with restrictions to the allowed initial probability measures we can find all solutions to the differential equation defined above. But as a matter of principle that should not be correct since we always work with such restrictions under the assumption that we will somehow be able to recover a corresponding result in the space of probability measures, otherwise such result cannot be tested experimentally or even in most numerical simulations (due to approximation uncertainties). Indeed, if we consider the space

Now we can compare both unitary solutions in different sample spaces and choose the one we need.

For most real-world problems, we need

We can easily see that for instance, all time-independent polynomial quantum Hamiltonians such that the domain is the dense set of smooth functions with compact support, can be proved to be essentially self-adjoint[9][10], using