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Hamilton's Quaternion
Classical Format
Hamilton's Quaternion Interpreted
Quantonics' Subjective Minus One

by Doug Renselle

First allow us to quote Dr. Paul Adrien Maurice Dirac himself:

"...We need a theory conforming to the principles of quantum mechanics, but it is quite a difficult
matter to get such a theory....One can get a much simpler theory if one goes over to the
corresponding classical mechanics, which is the form which quantum mechanics takes
when one makes Planck's constant tend to zero4."


"So I take the point of view that the Hamiltonian6 is really very important for quantum theory...In fact,
without using Hamiltonian methods one cannot solve some of the simplest problems in quantum theory,
for example the Balmer (, Johann Jakob) formula for hydrogen, which was the very beginning of quantum mechanics."

Both quotes are from Dirac's 1964 Lecture 1 at Yeshiva University, New York.
(Our italicization of Dirac's thelogos.)


You need to understand, using a quantum world view, basic issues of a classical world view of Hamilton's Quaternion.

Doug - 26Mar2014.

Here is a Hamiltonian quaternion in its classical hypercomplex1 form:

U =





I =





J =





K =





Here is a Hamiltonian quaternion with Quantonics substitutions
for quantum negational subjectivity:

First, realize that all operations in our script below are quantum. So our minus signs are quantum in our quantum~notation.
Compare negation and cancellation, classical vav quantum. Doug - 30Dec2012.

First we need quantum/Quantonic iq, -1q, 1q, 0q, and 2q analogues of classical i, -1, 1, 0, and 2.

If we assume that


iq -1q ei


where our uses of '-, e, , and i' are quantum subjective,
and if we assume that



-1q ei

where our power of quantum_two is a n¤nclassical quantum_square,
and our quantum '1' we represent as


1q =

Compare our usage
Classical Equals
Quantonic Equalings5



"No Bell Inequalities

"Bell Inequalities
expected as
quantum ubiquitous!"

"Suppes presented what he called
'the single most powerful argument for the use of
a non-classical logic in quantum mechanics.'"

"As Suppes explicitly declared,
a probability should be assigned to every element of
the [quantum]algebra of event[processe]s..."
p. 359, Max Jammer, The Philosophy of Quantum Mechanics.
(Our brackets and italicization of thelogos.)

Jammer rephrases from,
'The probabilistic argument for a non-classical logic of quantum mechanics,'
Philosophy of Science 33, 14-21 (1966), P. Suppes.

"Suppes' the opinion of Wolfgang Stegmüller is
'the only convincing argument' for quantum logic..." p. 360, Jammer. TPoQM.

We agree, but feel that Suppes did not go far enough.
Our approach is to assign potentia for unlimited probabilities and likelihoods
to each quanton's interrelationships. We do this using quantonics' fuzzons.
We offer a novel quantonics' fuzzons to fermion ontology for your consideration.

Added '...approach is to assign potentia...' phasement's link to
A Primer on Quantum~Cuneiform's
graphic entitled
'A Reservoir of Wave Functions.'

Doug - 23Mar2012.

and we represent quantum '0' as omnifference of two quantum '1s'


0q   -q

(also see our QELR of 'mihnus')

and we represent quantum '2' as quantum additi¤n of two quantum '1s'





Hamilton's hypercomplex quaternion in Quantonics notation

















Using our Quantonic approach2, keep on your quantum stages importance of quantum comtext.
Just above under quaternionKq both of those quantum 0qs are n¤t classically identical!
Assess Uq, Iq, and Jq similarly!
They have omniffering comtexts which we can/should show in some quantum realistic fashion.
Ditto our quantum iqs. For now, since we are breaking n¤vel ground, let's just assume Kij
subscripts as adequate quantum comtextual locales. It becomes rather complex at this juncture,
since quantum comtext is a n¤ntrivial topic for both discussion and innovation.
We need to include heterogeneous timings, momentumings, etc.
We see eidetically, heuristically how real3 quantum comtext issi pr¤cess.

Readers, amd students of Quantonics should also notice how
we can use our iq, -1q, 0q, 1q, and 2q to talk about amd
describe all areas of quantum science in which it is impossible to
describe nature's quantum subjectivity using classical mechanics.

As an example comsider quantum 0, quantum 1 and quantum 2 regarding quantum spin.
We achieve much more enlightening perspectives and memes of quantum nature when we view
odd (Fermi statistics) amd even (Bose statistics) multiples of fermionic ½ spin when
we view them as fermionic 1q/2q spin multiples. (Quantum divisi¤n. AKA quantum divide. J)

Another example is classical gravity as spin 2 vis-à-vis quantum gravity as spin 2q.
Our view is: antigravity is n¤nachievable using classical notions of spin 2.
Antigravity is (in our view) achievable using quantonic/quantum notions of spin 2q.


1 - Hamilton's "hypercomplexity" illustrates quantum reality's own heterogeneity. It shows a second set of complex variables. Both are what we call "conjugates." It does n¤t show Quantonics' quantum comjugation, which requires, as we show above, introducing real quantum subjectivity. Of course, introducing quantum subjectivity breaks all classical rules which depend upon classical immutability/stability/stoppability (except for unitemporal objective motion), and classical Aristotelian/EEMD/Sheffer-stroke-binary-alternative-denial/lisr classically-negatable, contradictable objective independence.

2 - What we have done here is apply Quantonic n¤mbærs to Hamilton's quaternions in their representation (each of U, I, J, K are two-space planes and hyperplanes; their combination is three-space hyper volumes). Our approach is applicable in too, where matrices become 4 by 4. Dirac used to develop his relativistic version of QED. Indeed, that was Dirac's own epiphany: to use to solve QED relativity problems which flummoxed Pauli. See Tomonaga's Lecture 3 in his The Story of Spin, 1997 (translation), UChicP. Please realize that all of Pauli's and Dirac's accomplishments were purely classical. Perhaps what is most interesting to us is a general classical bias that physicists and mathematical physicists then felt it necessary to relativize quantum theory, but not to quantize relativity theory... Is that bias because relativity came first? We think so. Beware unitemporal classical cohesion (essentially classical cause-effect induction). Einstein's celebrity pulled lots of wool and paradigmatically disciplined lots of blinders. Beware celebrity! Too, beware Einsteinian classical relativity!

3 - As we have shown elsewhere in Quantonics, when we say "real" that statement issi n¤t an analogue of saying "reality." Simply, real issi n¤t reality. Why? We can use Quantonic script to show you: realityquantons(reals'_complements,reals) which is very much like saying realityquantons(n¤nactuality,actuality), latter which most students of Quantonics are exceptionally familiar by now. See our real table entry in our review of EPR's paper EPR (large page).

4 - As we have impressed on students of Quantonics elsewhere and frequently, classicists tend to use classical concepts to make quantum reality go away, and make quantum reality classically 'real.' Taking to zero is taking quantum reality to classical reality. See our relevant July, 2001 Flash commentary on classical 'gauge invariance.'

Least energy action e issi 2qqq. Said Hamiltonian quaternion is an action representation. Zeroing zeroes and disables quantum action. It removes quantum flux from all classical notation and thing-king. It, simply, is n¤n-quantum-real. Notice how our better (yet still wanting, always wanting) Quantonics approach anihmatæs our representation of quantum action in its Hamiltonian quaternion emerqancy. Doug - 28Jun2003.

5 - Students of Quantonics may find it worthwhile to experience an epiphany that Quantonics' equalings are all "Bell inequalities." Indeed, all quantum n¤mbærs, all quantons, according to Quantonics' beliefs, are essentially quantum c¤mplementary "Bell inequalities." Bell's inequalities, as he propounded them, are classically presumed inanimate, excluded-middle, and state-ic. Naïve AKA "local" realists (classicists) insist upon a classical notional ideal 'equality' akin Aristotle's first syllogism, A=A. So finding inequalities, to a classicist, is a BIG surprise! Of course our equalings are anihmatæ, EIMA quantons. What makes "Bell's inequalities" inequalities? Quantum reality's intrinsic anihmatæ EIMA. Why is this quantum notion ironic and paradoxical to classicists? They view reality as inanimate, immutable, state-ic, excluded-middle, and everywhere-independent-dissociative!

We created another n¤vel graphic which students may use to visualize any quanton's EIMA:

That sphere issi a sphere of p¤tentially (an) ¤mnivalent (ensehmble of) quantum included-middle c¤mma-n¤-space c¤pula. It shows that when we depict quanton(c¤mplementqi,c¤mplementqj), its quantum EIMA c¤mma-n¤-space issi massively heter¤gene¤us. Imagine our quanton's c¤mma-n¤-space c¤pulum as only an iceberg tip of that c¤mma-n¤-space sphere. Then imagine all that as abs¤lutely anihmatæ quantum flux.

Here is an analogous ~circular GIF showing what we intend, in 2D, less abstractly (coarse 30° increments for readability; imagine that disk of c¤mplements being rotated ecliptically too, to make a quasi-sphere, and then anihmatæd):

Students should assume c¤mplementqi is varying due absolute quantum flux.
Further assume c¤mplementsqj are each unique and varying even
though we do not show their quantum-unique comtexts.
Click graphic's link to see a comtextual interpretation.

Here you can see heter¤gene¤us, ensehmble quantum c¤mplements without having to infer their presence only by a spherical 'hive' of c¤mma-n¤-space anihmatæ EIMAs. This is a sort of sledge-hammer way of saying, "There are n¤ classical dyads in quantum reality, and there are n¤ Aristotelian syllogisms in quantum reality. Too, if you fathom what we intend, Bell's inequalities are hologral and thus perpetually~ubiquitous in quantum reality. There are n¤ 'equalities' in quantum reality!" See identity. See equals.

6 - At beginning of section 79 of Dirac's The Principles of Quantum Mechanics he says, "The complete Hamiltonian for electrons and positrons interacting with the electromagnetic field is

H = HF + HP + HQ;"

where F is electromagnetic field 'alone,' P is positrons and electrons 'alone,' and Q is the interaction energy 'alone;'
Dirac's use of 'alone' here exhibits his fundamental, dialectical, analytic, Aristotelian lisr habits.

T¤ sh¤w this ihn Quantonics scrihpt wæ d¤ this


where ¤ur c¤mma~nospacæ ræpræsænts Quantonics' is¤flux REIMAR ihnterrelati¤nshipings twixt quantum~n¤nahctualihty amd ~ahctualihty

It's just that simple!

Doug - 10Jul2003-17Feb2005

To contact Quantonics write to or call:

Doug Renselle
Quantonics, Inc.
Suite 18 #368 1950 East Greyhound Pass
Carmel, INdiana 46033-7730

©Quantonics, Inc., 2003-2026 Rev. 26Mar2014  PDR Created 9Feb2003  PDR
(11Feb2003 rev - Finish bottom of page comments and heuristics.)
(12Feb2003 rev - Add some apropos links.)
(13Feb2003 rev - Minor text clarifications.)
(15Feb2003 rev - Add note 2.)
(16Feb2003 rev - Clarify 2D hyperplanes and 3D hypervolumes.)
(26Feb2003 rev - Change 'EIMA' to 'EEMD' in Note 1. Change classical 'equals' to Quantonic "equalings." Repair typos.)
(12Mar2003 rev - Widen some tables so our script aligns well.)
(24Mar2003 rev - Repair off page links.)
(18Jun2003 rev - Minor table format changes.)
(19Jun2003 rev - Add Quaternion Iq link to our recent 'Hamiltonian Quantum Interrelationships.')
(28Jun2003 rev - Add page top 1964 Dirac-Yeshiva comments plus Note 4. Add some missing 'q' subscripts.)
(10Jul2003 rev - Add Note 5 on "Bell Inequalities.")
(18Aug2003 rev - Add "Bell Inequalities" remarks to our quantum descriptions of quantum 1.)
(24Sep2003 rev - Repair EPR Table link.)
(14Oct2003 rev - Typo.)
(14Feb2004 rev - Reset some legacy red text.)
(19Mar2004 rev - Add anchor to our quote of Dirac on zeroing h-bar.)
(28May2004 rev - Add Jammer~Suppes comment~quote under our Quantonics' '1' quanton.)
(2Sep2004 rev - Reset legacy red text box. Add extension red text box.)
(29Oct2004 rev - Reset red text.)
(7Jan2005 rev - Show Balmer's full name.)
(17Feb2005 rev - Add Dirac sec. 79 TPrioQM quote.)
(8Apr2006 rev - Reset legacy red text. Minor typos and page reformat.)
(21Aug2006 rev - Minor reformating. Massive respell.)
(21Jan2007 rev - Add quantum 'one' and 'zero' anchors.)
(30Jul2008 rev - Reformat.)
(12Jan1010 rev - Make page current.)
(3May2010 rev - Adjust colors.)
(2Jun2010 rev - Repair 'included~middle' typo.)
(23Mar2012 rev - Added '...approach is to assign potentia...' phasement's link to A Primer on Quantum~Cuneiform's graphic entitled 'A Reservoir of Wave Functions.')
(20Oct2012 rev - Add "...hologral and thus perpetually~..." text and a hologral link near end of page.)
(30Dec2012 rev - Add 'Quantum Script Caveat' anchor and text near page top.)
(26Mar2014 rev - Add page top link to CE of Quantum Systems which exegetizes issues of any classical Hamiltonian.)