No Split between Classical and Quantum Mechanics

The scales of the World without split.

À standard answer to a question about the difference between classical and quantum mechanics is that the former concerns macroscopic physics and the latter microscopic physics, and of course size makes a huge difference, right? But more precisely, what difference does size make? 

We know that an ant faces different conditions/forces than an elephant, but the same basic laws of physic apply. We know that the Moon follows the same all-size law of gravitation as ants and elephants. 

A standard physicist would now expand on the macro-micro split by informing us that macroscopic physics is deterministic while microscopic physics is non-deterministic or probabilistic, and that is huge difference: Atoms play dice while the macroscopic world is predictable or deterministic. 

You could react on this message with skepticism from a wealth of experience that a complex macroscopic world is unpredictable or non-deterministic, while an atom as an entity of utter simplicity must be expected to be predictable deterministic. So if determinism is to be what makes macro different from micro, it would be micro/simple which is deterministic and macro/complex which is non-deterministic, in direct opposition to standard physics. 

Let us now seek the reason why standard physics insists that microscopic physics is non-deterministic while macroscopic physics as meanvalues of microscopic physics is deterministic, even if that contradicts sound logic. We shall see that it all depends on the probabilistic interpretation of the wave function of quantum mechanics by Max Born, which has captured modern physics. Born proclaimed in 1926 that the square of the wave function would express the probability of the occurrence of the configuration acting as argument for the wave function as a function over configuration space, as the new buzz word which gave him the Nobel Prize in 1954 with a near 30 year delay. Let us now seek the origin of the concept of a wave function over a configuration space measuring configuration probabilities as the hall mark of quantum mechanics.  

The motion in 3d space over time $t$ of a classical mechanical system of $N$ particles is described by Newton's Laws of Motion or alternatively in terms of Lagrangian mechanics based on the Principle of Least Action for a Lagrangian

• $L(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$       (1)

where $x_i (t)$ is the position and $\dot x_i (t)\frac{dxi}{dt}$ the velocity of particle $i=1,...,N$ as function of time $t$ as a description of the evolution of the state of the system over time,  taking the form  of Newton/Lagrange's Laws of Motion. Various coordinates for 3d space may be used, such as Euclidean, spherical, cylindrical et cet.

It is here common to refer the set of all possible states $(x_1,x_2,...,x_N,\dot x_1,...\dot x_N)$ as configuration space, which thus has $6N$ dimensions. It is important to understand that all particles exist in the same 3d space and that the $x_i(t)$ with $i=1,2,,,N$ simply describe the locations of all the particles at some time $t$. The setting is a common 3d space for all particles and the $6N$ dimensional configuration space is a formality. In short (1) makes perfect sense. Out of all possible states in configuration space a specific trajectory $X(t)=(x_1(t),...,x_N(t)$ with coordinates $x_i(t)$ is determined by Lagrange's equations as a function of time $t$ only. 

We now turn to quantum mechanics with its wave function for $N$ electrons of the form

• $\Psi (x,t)=\Psi (x_1,x_2,...,x_N,t)$                            (2)

where now the $x=(x_1, ...,x_N)$ are coordinates for $S=R^3\times R^3\times ...\times R^3$ as $N$ different independent copies of 3d space $R^3$ with coordinate $x_i$. Superficially (2) resembles (1) and we may even speak of a $S$ as a configuration space for (2) with coordinates $x=(x_1,x_2,...,x_N)$.

The wave function $\Psi (x,t)$ satisfies Schrödinger's equation $i\dot\Psi =H\Psi$ with $H$ the Hamiltonian differential operator acting on the wave function $\Psi$ depending on $(x,t)$-coordinates with $x=(x_1,...,x_N)$ $3N$-dimensional. 

The hope of Schrödinger was that the generality of Lagrangian mechanics in terms of number of particles (and choice of coordinates), would allow a direct generalisation of his stunningly successful equation for the Hydrogen atom to atoms/molecules with many electrons. What could go wrong if (2) was similar to (1) both with formally a high-dimensional configuration space?

Let us compare (1) and (2) or more specifically $X(t)$ as a classical trajectory satisfying Lagrange's equations as a function depending on $t$, and $\Psi (x,t)$ satisfying Schrödinger's equations as a function of $(x,t)$. Ok, we see a clear difference in the coordinate dependence: $X(t)$ depends on a time coordinate, while $\Psi (x,t)$ depends on a time coordinate and a multidimensional spatial coordinate $x=(x1,...,xN)$. 

That is a monumental difference. Classical mechanics describes actual trajectories $X(t)$ in $R^3$ selected by Lagrange's equations from a pool of all possible trajectories, which is never covered.

On the other hand, the wave function $\Psi (x,t)$ with its multidimensional space variable $x$ freely sweeping configuration space $S$ independently of $t$, thus describes possiblities and not actuality and there are immensely many more possibilities than actualities. The wave function $\Psi (x,t)$ is thus a monster supposed to describe the probability of all possible configurations, which is such a huge undertaking that it swamps every conceivable effort by its complexity.

To reduce complexity the variation of $\Psi (x,t)$ is (ad hoc) reduced to consist of linear combinations of products of single electronic wave functions $\psi_i(x_i,t)$ each one depending on a single spatial coordinate $x_i$, typically in the form of so called Slater determinants, but even that is like seeking to emptying an ocean with a spoon as an uncomputable problem. 

On the other hand, in RealQM a different Schrödinger equations is formulated for a wave function of the form 

• $\Psi (y,t) = \sum_i\psi_i(y,t)$          

as a sum of one-electron wave functions $\psi_i(y,t)$ all depending on the same common physical 3d coordinate $y$, which is computable in the same sense as a classical deterministic continuum mechanics solid/fluid problem. 

In RealQM the physics on macro and microscope is the same, which is to expect since a split cannot be made.  

The split macro-micro of standard quantum mechanics is an artefact of the introduction of a configuration space of all possible configurations which is way too big to handle.

Another strange aspect of the standard Schrödinger equations are linear thus allowing superposition of states as something unexpected. Newton/Lagrange's equations in general are non-linear and superposition a rare effect of linearity only in very special cases, like a small amplitude vibrating string. The Schrödinger equation of RealQM is non-linear, just like the majority of similar equations in solid/fluid mechanics. 

In short, there are many serious reasons to consider the standard multi-dimensional linear Schrödinger equation as a too easily formed mathematical construct without physics. As compensation $\Psi$ is invited to belong to a mathematical Hilbert space $\mathbb{H}$ of maximal prestige and so the world can very neatly be described by a $\Psi$ in a $\mathbb{H}$, but without physics. 

One can argue that classical physics concerns real physics using concrete Calculus, while standard quantum physics rather is is a form of formal mathematics without real physics.