Quantum (Acausal) Determinism (1)

One of the less well-known areas of quantum mechanics research is in the area of quantum determinism, or more specifically quantum acausal determinism. This differs from classical determinism in a number of ways, but mainly by virtue of the fact we are looking at a multi-particle probabilistic ensemble which yields defined characteristics that are predictable over time.

Quantum acausal determinism differs from quantum acausal indeterminism in respect that the former is based on Louis deBroglie's concept of a physically real "pilot" wave which is associated with the wave function. Thus, a physically real wave solution satisfies Schrodinger's equation:

H_op(U) = E(U)

Here H_op is what we call a Hamiltonian or special mathematical operator (combining kinetic and potential energy functions) and U is the wave function. H_op operates on U (PSI) and yields the quantity shown on the right side which is the product of a constant (eigenvalue) by U. In one dimension (for simplicity) we have: H_op = [-i(h-bar) d/dx]^2/2m + kx^2/2. E will generally yield an energy eigenvalue. Energy eigenvalues for a particular configuration (infinite square well) are shown in the lower diagram (extreme right side). Note the far left side denotes specific wave functions, U1, U2, U3 etc. while the middle panel yields wave amplitudes squared or probability amplitudes.

Thus, the Schrodinger equation manifests as a deterministic descriptor in an x-y-z space (or x-y-z-t) if time-dependent. Meanwhile, the quantum indeterministic picture (after Max Born) views the wave as a mathematical statistical artifact only, not real. What the Schrodinger equation describes in that case is the evolution of the wave function in probability space.

In the deterministic quantum view we have a defined momentum:

p = (h/2 pi) [grad f]

where f is the phase of the wave function. The preceding suggests the associated particle (de Broglie hypothesized originally that every wave - 'matter wave' - defined by wavelength L = h/mv = h/p, had an associated particle) is "guided" by the background wave, hence the name "pilot wave". If it's guided, it must have a deterministic profile.

De Broglie's own description of the guiding process was articulate and concise: He proposed that inside the particle was a periodic process equivalent to a clock. In the rest frame this clock would exhibit frequency:

w(o) = m(o)c^2/ h-bar

where m(o) is the rest mass, and h-bar = h/2 pi, i.e. the Planck constant over two pi. By consideration of how the 'clock' would behave in other frames he derived what we now refer to as the Bohr-Somerfed relationship or:

INT p.dx = n(h-bar)

which is the condition for the clock to stay in phase with the pilot wave.

As history retells it, the fate of de Broglie's pilot wave hypothesis (and the quantum deterministic theory) was sealed at the Solvay Conference at which Wolfgang Pauli made strong objections on the basis that it didn't provide a consistent account in terms of a many-body system. Thus, the Copenhagen Interpretation (based on the wave function as purely statistical) won the day.

However, by 1952 David Bohm resurrected aspects of the deterministic theory which then were threaded into his own "Stochastic Intepretation of Quantum Mechanics" - and which is now often described as "Bohmian Quantum Mechanics". In 1952, Bohm published two papers on the topic. The first basically worked out the consequences for a one-body system using de Broglie's reall matter wave basis. The second paper extended the treatment ti the many body system which effectively neutralized many of Pauli's objections from Solvay.

The key insight, for practical purposes, is to look and see how Bohm dealt with a trajectory in a many-body system. He begins by writing the N body-wave function:

1) U(x1..........xn) = R(x1.......xn)exp(iS(x1.....xn)/h-bar

to define the momentum of the nth particle as:

2) p_n = grad_n S

where S is the 'action'

On substituting (2) into the many-bodied Schrodinger equation Bohm obtains the conservation equation in configuration space:

dP/dt + Sigma (grad _n) * (P grad_n S)/m = 0

where P = U*U is the probability density and the modified Hamilton -Jacobi equation is:

dS/dt + Sigma_n [grad_n S]^2/ 2m + V(x1...xn) _ Q(x1...xn) = 0


From the preceding,. Bohm deduced that each particle would be acted upon not only by a classical potential [1] but also an added quantum potential Q:

Q = - (h-bar)^2/2m {Sigma_n [grad_n ]^2 R/ R

In this view, novel features of QM are seen to arise from Q


Next: Making sense of the Einstein-Podolsky-Rosen thought experiment



[1] An example of the classical potential is that associated with the gravimetric change in 1-D space, given by: g = - dV/dx where V is the potential.