Hilbert Spaces: Cheating at Reality (1219 words)

Disclaimer

These are my opinions.

This is not a beginner friendly piece.

If you are interested but have no physics background, please first check out:

https://churchofentropy.wordpress.com/2015/11/25/quantum-chemistry-teaser/

and

https://churchofentropy.wordpress.com/2015/12/23/why-general-relativity-and-quantum-mechanics-are-incompatible/

Commentary on Modern Physics

According to Wikipedia:

“A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.”

https://en.wikipedia.org/wiki/Hilbert_space

What are the techniques of calculus? Calculus is all about measurement: slopes, areas, densities… Calculus is the most accurate way to make these measurements. Calculus says that in the infinitely small spatial limit, the area bounded by a smooth curve is equivalent to the sum of infinitely thin inscribed rectangles.

Is this approach reasonable? It certainly feels true and is perfectly reasonable to use under circumstances where space can be modelled as continuous. That is, it works until we reach the limits set forth by the Uncertainty Principle, at which point spacetime becomes decidedly discrete.

The Rules of Engagement

From: www.web.mit.edu/edbert/GR/gr1.pdf

“[…] the fundamental object in quantum mechanics is the state vector, represented by a ket |ψi in a linear vector space (Hilbert space).”

and

“It is possible to formulate the mathematics of general relativity entirely using the abstract formalism of vectors, forms and tensors. […] To simplify calculations it is helpful to introduce a set of linearly independent basis vector and one-form fields spanning our vector and dual vector spaces. In the same way, practical calculations in quantum mechanics often start by expanding the ket vector in a set of basis kets, e.g., energy eigenstates.

Why is it so important to have a Hilbert Space versus an ordinary Euclidean Space in QM? We can be pretty sure it has to do with measurement: the HS imposes the condition of measurability on all of its lengths and angles (Wikipedia). Measurements must be well-defined for things like the Centre of Mass coordinate system transformation to follow suit.

The only difference between Euclidean and Hilbert spaces is the number of dimensions: the former being a special case of the (potentially) infinite dimensional latter. The Hilbert Space is a way to generalize all that can be known in advance of measurement for a QM system without being limited to 3 spatial coordinates.

Next I will show that although there exist 4 spacetime dimensions on the fine scale, they cannot be basis vectors in a geometric sense because they are maximally dependent.

The Fundamental Spacetime Event

In relativistic mechanics, points are referred to as events, possessing 3 spatial and one temporal coordinate.

The MIT paper says:

  1. “We then need the proper four-volume element, i.e., the physical volume element that is invariant (a scalar) under coordinate transformations.” (for a volume integration).
  2. “By definition, the dimensionality of spacetime (four) equals the number of linearly independent basis vectors and one-forms.”

The existence of such a volume element is central to GR by (1). We also have the condition that the dimensionality of spacetime is equal to the number of l.i. basis vectors by (2). We will next show that the atomic volume cannot possibly satisfy (1), because it contradicts (2). A Hilbert space is not an accurate model of reality on the quantum scale under the coordinate transformations of General Relativity.

The Smear-iodic Table

The Periodic Table is the most extreme demonstration that the 3 physical dimensions are not independent of acts of observation on the fine scale. Below represents the periodic table (two elements per enclosure) as it truly occurs. It starts at the bottom with Hydrogen and Helium. Mass then swirls up through the micro-physical (entangled) dimensions before “running out of room” once it saturates R3 (there are no more elements after the 7p orbital).

postulates of quantum chemistry 3jp

Stylized R’s represent the physical dimensions on the atomic scale

All seven rows of the periodic table are represented here: there is one in R0 and 2 per physical dimension. We can interpret this drawing as mass/energy entering space in an initiating instance (R0). This mass then moves through the 3 dimensions of space, which exist atop this initial instance.

This formulation explains why we observe radioactivity in higher-numbered elements. The Uncertainty Principle indicates that the position and momentum cannot be known to arbitrary precision. As mass saturates the physical dimensions, more becomes known about the atomic system (in this case it is mass-energy that is knowing/observing space), until the limit of what is knowable is breached (by Uncertainty).

The fact that the atomic system is bounded in space and that Uncertainty precludes arbitrary measurement precision forces nuclear decay in higher numbered elements. It is not a coincidence that all elements in the seventh row of the periodic table are radioactive. Stability for these species would imply arbitrary precision to their position and momentum values of their constituent waveforms: precluded by Uncertainty!

th chain
The Thorium Chain (P vs. N)

It is interesting to note that since the atom is a QM system (in potential), that the UP is only violated in a probabilistic sense (decay events aren’t happening all the time).

It is clear that on this fine scale, the physical dimensions are not independent of the mass that is experiencing (observing) them. In fact, on this scale, we can see that the spacetime dimensions are maximally dependent. Every atom vibrates within the same configuration space and so it stands to reason that subsequent dimensions would not be independent of previous ones! R2 is larger than R1 because it is on top of R1. Thus, we can be certain that the proper “four-volume” element cannot be any smaller than the atomic volume, since the latter’s basis vectors are entangled: the opposite of linearly independent, a must-have for a basis vector!

Conclusion

Although the 4 dimensions of spacetime may appear to be connected only by the “event” phenomena of special relativity, they are interconnected on a much deeper level in the atom. Thus the correct four-volume element (necessitated by GR to integrate a scalar field over a 4-dimensional spacetime volume) must be larger than the atomic volume (arguably it must be large enough that the effects of the entaglement of space and time in the atom are negligible).

The GR model is thus incompatible with the QM model.

We model space as continuous when it isn’t, then wonder why we cannot find convergence between quantum mechanics and general relativity. I wish I could say I am surprised, but it’s par for the course in the “land of illusion” 😉 .

Thank you.

Epilogue – Singularities Irritate Me

According to Wikipedia:

“In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.”

https://en.wikipedia.org/wiki/Singularity_(mathematics)

I always found these to be the biggest cop-out of physics. My book will present a model with no singularities. It is nearing completion and I expect to publish before 2017.

In the meantime, I encourage everyone to consider the possibility that mass is everywhere interconnected and that gravitons do not exist. This is permissible if we accept that all mass acts instantaneously at a distance on all other mass and all fields.

This is not a popular viewpoint in physics, but I figured any discipline putting out comics this desperate:

fixion

(http://m.xkcd.com/1621/)

Could probably use a pick-me-up.

Thank you.

 

 

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