## Thursday, October 14, 2010

### Riemann Hypothesis and Physical Connections

When I first read about connections as between the Riemann non-trivial zeros and certain physical energy states in quantum physics, I was not at all surprised.

For I had long believed that important physical applications of the Riemann zeros would necessarily exist.

Indeed I would go considerably further. Not alone do the zeros have implications for physics but equally they have a deep relevance for psychological understanding of various spiritual energy states. Furthermore in the holistic mathematical understanding of what is involved the two forms of understanding (physical and psychological) are fully complementary.

Though I have always recognised - in principle - the potential physical relevance of the zeros (and of course the associated Riemann Hypothesis) my main focus has been on the psychological relevance of the Riemann zeros.

And here there is an interesting twist! For in attempting to provide a coherent interpretation of what the Hypothesis actually entails (which then leads to a simple resolution of the Hypothesis), I have found the trivial zeros to be of greater significance.

Indeed my initial interest in the Riemann Hypothesis was sparked by the desire to provide a coherent explanation for the trivial zeros.

For example if the take the first of these for the zeta function (where s = -2) this would result in the series

1 + 4 + 9 + 16 +.....

Now clearly from a conventional linear quantitative perspective, this series diverges so that it has no finite sum.

However according to Riemann's Zeta Function, the sum of this infinite series = 0.

So it was through attempting to explain the nature of this unexpected value that I realised that it referred directly - not to a quantitative but rather - to a qualitative interpretation of number.

Basically in qualitative terms, 2-dimensional understanding refers to the complementarity of (real) polar opposites in experience such as internal and external. Though in linear (1-dimensional) terms these are clearly separated in experience enabling for example the unambiguous objective interpretation of mathematical calculations, in circular (2-dimensional) terms these are seen as complementary and interdependent (with ultimately no division possible between them).

Strictly 2-dimensional understanding (as positive) relates to the rational paradoxical interpretation of this circular type relationship.

However 2-dimensional understanding (as negative) relates to the purely intuitive appreciation of the same relationship (where secondary rational distinctions are negated).

Put another way when s = - 2, correct qualitative understanding relates to a pure contemplative i.e. spiritual energy state. (It would certainly have resonated with the Pythagoreans before the unfortunate subsequent split in mathematical understanding!) Though it is correctly represented as zero, it relates directly in this instance to a qualitative - rather than quantitative - meaning.

Therefore we give a coherent numerical explanation to the first of these trivial zeros through interpreting the relationship directly in qualitative terms. Equally all the other trivial zeros can be explained as "higher" intuitive contemplative states of understanding (that are nothing in phenomenal terms).

Riemann also provided a fascinating transformation formula enabling one to calculate from conventional values for s > 1, corresponding non-conventional values for s < 0. So the clear implication here is that the transformation formula - when correctly understood - provides a clear (indissoluble) relationship as between numerical results with a direct quantitative value (for s > 1) on the right hand side of his equation and corresponding results with a direct qualitative holistic value (for s < 0) on the left hand side.

We can also continue this procedure in the critical strip (0 < s < 1) where now values for the zeta series represent a hybrid mix of both quantitative and qualitative aspects. The exact matching of both left and right hand side values requires that s = 1/2 where - by definition - quantitative and qualitative aspects exactly match with each other.

And this in short is what the Riemann Hypothesis is all about i.e. the crucial condition enabling the consistency of both quantitative and qualitative aspects of mathematical interpretation.

Unfortunately in a Mathematics that only formally recognises the quantitative aspect of understanding the true nature of the Riemann Hypothesis will always prove elusive.

Incidentally attempts to "prove" the Riemann Hypothesis with respect to establishing an exact correspondence with certain physical energy states seems to me somewhat confused.

Though it would indeed be exciting and important to conclusively establish such a link this would only demonstrate an especially strong correlation as between corresponding mathematical and physical systems.
However perfect correlation - even if demonstrated to exist - does not constitute proof!