Quote: “Someday we’ll understand the whole thing as one single marvelous vision that will seem so overwhelmingly simple and beautiful that we may say to each other; ‘Oh, how could we have been so stupid for so long? How could it have been otherwise!’ (J. A. Wheeler)”

I will expand my thoughts on the issue of wavefunction symmetry based on various papers from David Bohm, Louis deBroglie, Dürr, Holland, Bell, Goldstein, Tumulka, Rovelli, and Struyve.

The wavefunction is used in quantum mechanics to describe physical systems. It is a function of space that maps the possible states of the system into probability amplitudes — elements in a complex Hilbert space — as the squares of the absolute values that give the probability distribution that the system will be in any of the possible states; either a complex vector with finite or infinite number of components or a complex function of one or more real variables. For systems with multiple particles, the underlying space represents the possible configurations of all the particles and the wave function describes the probabilities of those configurations.

A paradox appears when one now tries to measure position and momentum and the probability is turned into a real numbers. The decisions where to measure violates the temporal causality of the particle being measured. Also the particle or wave duality is based on the same issue. To avoid causality paradoxes, the wave function collapse is required in the Copenhagen interpretation. Louis de Broglie offered in 1927 at the Solvay Congress in Brussels a solution which he called pilot-wave theory but abandoned it after it was criticized. In the pilot-wave theory the particle is guided by a field that is represented by the wave-function. David Bohm’s ontological interpretation is similar to de Broglie’s pilot-wave and based on hidden variables or – as Bell called them – beables of defined particle position and momentum.

Quantum theory explains the behavior by means of ‘microscopic’ systems represented by a wavefunction ψ, which is defined by the microscopic initial wavefunction and its eventual collapse caused by a ‘macroscopic observer’ (human or not). Definitions of ‘microscopic’ and ‘macroscopic’ systems, or ‘observer’ and ‘system’ are unfortunately ambiguous and present a serious ﬂaw. If the macroscopic system is regarded as a collection of microscopic systems, then the wavefunction of the total system should also evolve according to Schrödinger equations without collapse. The problem has finally been shown by Bell, that any realistic theory which leads to the same statistical predictions as standard quantum theory must be non-local. Non-locality refers to the effectiveness of fields in quantum theory. Just because a field is non-local and is effective anywhere at the same time, that does not imply that signals or energy can be transferred at higher than the speed of light. The non-locality manifests itself by the fact that the position of one-particle beable may depend on the positions of other particle beables. This dependence is instantaneous no matter how far apart the other particle beables may be located. In this way the problem that the ‘observing macroscopic’ system is magically defining the causal reality of the ‘observable microscopic’ is avoided and it becomes clear that both entities are wavefunctions.

Problems arise for a transcript of the quantum mechanical interpretation of non-relativistic quantum theory to relativistic wave equations: a) identifying a future-causal current which can be interpreted as a particle probability current; b) deﬁning a positive deﬁnite inner product; and c) particles and anti-particles must be freely created and annihilated without the need of considering negative energy states. The above led to quantum ﬁeld theory, where ﬁeld operators take over the role of the particle operators in non-relativistic quantum theory. Therefore, ﬁelds rather than particles seem to be the most natural beables in the pilot-wave approach.

I propose that the pilot-wave field must be represented in space topology (space-time geometry) in the same way as spin-foam models are used for Rovelli’s non-perturbative quantum gravity. Imagine it as a space topology that exhibits non-local density changes that because of general relativity (GR) are not relative to a background time. Rovelli also defined the concept of a partial observable as any quantity that can be measured even without prediction. What GR can predict is the correlations between partial observables, which are manifest as ‘propagators’ between possible and actual Hilbert spaces. I see the pilot-wavefunction as describing propagator ‘density grooves’ in space topology that the energy between observer and measurer can travel in. Feynman therefore described particle paths as an integral over all possible ‘grooves’.

If the complete microscopic system is not only defined by the wavefunction, but also by some extra variables that have an objective existence, they should also determine the outcome of experiments. The Bohm interpretation is causal but non-local and non-relativistic: Every particle travels in a definite path with definite position and velocity. We do not know what that path is but can find out what path (density groove) a particle traveled while uncertainty of position and momentum remains.

I propose that the use of topology to describe the particle causality phenomenon is appealing because it is simple. It allows to disregard all sorts of other information that confuses the issue. Connectivity and tensors are all that remains. Like others I propose that demanding classical causality at all times, is most likely reducing the possible solutions to a problem substantially. Einstein has offered us many great advances but that does not mean that all his propositions are valid conclusions, such as that energy must at all times be a cause of gravity. Einstein failed to distinguish between the active gravitational mass of matter, the contribution to the density of matter, and the inert mass of matter. A particle can clearly exhibit an inert momentum and not have mass or cause gravity. That gravity changes the space-time tension and makes photons follow the tensors sounds dramatically like the ‘density grooves’ of the wavefunction. And what about the phase transitions exhibited by for example the Bose-Einstein condensate? Are those not plausibly connected to a change of topological degrees of freedom? Meaning that the number of possible patterns that the space topology can take is reduced as the amount of potential energy in space-time tensors is reduced.

To me that sounds as if a timeless, non-local topology of space is the only thing that is common when we talk about any kind of field or particle!

Filed under: Cosmology, Quantum Field, Space Topology