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Helmut Moritz said in e-mail: "... Quantum Theory was founded independently in two completely different mathematicalforms by two equally great physicists, Heisenberg and Schroedinger. The miracle was that both mathematical formulations were soon recognized to be equivalent: the underlying Hilbert space may be formulated in two equivalent ways: as a space of functions or a space if infinite-dimensional vectors. A periodic function is equivalent to the infinite vector formed by its Fourier coefficients.

If there had been a collaboration after World War II among




they might have formulated something similar to

the D4-D5-E6-E7-E8 VoDou Physics model:

The unified theory of


generalized Einstein's relativity by using 4-dimensional geometry with antisymmetric components and connections.


Feynman, in The Feynman Lectures on Gravitation (Addison-Wesley 1995), says (on pages 24, 30-32): "... what is the spin of the graviton?

If the spin were 1/2, or half integral, ... there could be no interference between the amplitudes of a single exchange, and no exchange ... a half unit of angular momentum cannot be emitted by an object that remains in the same internal state as it started in ... Thus the spin of the graviton must be integral ...

The rejection of spin-zero theories of gravitation is made on the basis of the gravitational behavior of binding energies. ... the interaction energy ... corresponding to the spin-0 field, would be proportional to sqrt( 1 - v^2/c^2 ). In other words, the spin-zero theory would predict that attraction between masses of hot gas would be smaller than for cool gas. ... the experimental evidence on gravity suggests that the force is greater if the gases are hotter ...

A spin-1 theory would be essentially the same as electrodynamics. ... one consequence of the spin 1 is that likes repel, and unlikes attract. This is ... a property of all odd-spin theories; ... even spins lead to attractive forces ...

the spin-2 theory leads to an interaction energy ...[between]... two masses of gas ... which has sqrt( 1 - v^2/c^2 ) in the denominator, in agreement with ... the experimental evidence on gravity ... that the force is greater if the gases are hotter ...

In the theories of scalar, vector,and tensor fields (another way of denoting spins 0, 1, and 2) the fields are described by scalar, vector, or tensor potential functions:

... assuming that the tensor is anti-symmetric ... would not lead to something resembling gravity, but rather something resembling electromagnetism; the six independent components of the antisymmetric tensor would appear as two space vectors. ...".

In Feynman's picture, if the fields are taken to be 4-dimensional covariant derivatives of the potentials, then there are 4 times as many fields as there are potentials, so that for spin 2 there are:

so that there are 16 general huv spin 2 potentials, 10 of which are symmetric and give Einstein gravity and 6 of which are antisymmetric and sort of like electromagnetism.

However, I think that Feynman's intuition is wrong about the 6 being like "two space vectors". From the point of view that 4x4 real antisymmetric matrices correspond to Spin(1,3) Lorentz transformations of 4-dim Minkowski space, which are the 6 bivectors of the 1 4 6 4 1 graded Clifford algebra structure

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

it appears that the 6 is like 3 bivector rotations of 1 3 3 1 which correspond to spatial rotations in 4-dim spacetime and to spatial angular momentum, plus 3 spatial vectors of 1 3 3 1 which correspond to boosts involving time in 4-dim spacetime and to spatial linear momentum, giving 4 corresponding to 3 space and 1 time (possibly special conformal) plus 2 others (possibly corresponding to dilatation and a phase).

A theory of Schroedinger is described in a 29 January 1947 AP newspaper article in part as follows:

"... DUBLIN, Eire, Jan 29 (AP) - ... Dr. Schroedinger said ... "We know that a rotating mass, such as the earth, has a magnetic field - now this theory should show why it has. ... The physicist said the Einstein theory was "purely geometrical" and that now electromagnetism must be included. ... Dr. Schroedinger said he had arrived at his theory by relying on a "general nonsymmetric affinity," whereas other scientists had failed to solve the problem because they used a "symmetrical affinity with only forty component parts instead of a general one with sixty-four [component parts]." ...".

The Feynman-Schroedinger picture and the Cl(8) Clifford algebra picture of the D4-D5-E6-E7-E8 VoDou physics model then correspond this way:
Feynman-Schroedinger Clifford Cl(8)   10 symmetric potentials 10 deSitter generators give the 4x10 = 40 components by MacDowell-Mansouri mechanism of Einstein's tensor gravity give Einstein gravity   6 antisymmetric potentials 4 special conformal GraviPhotons give Blackett-Sirag 1 scale dilatation graviphoton effect 1 U(1) propagator phase

Considering that Einstein also used the symmetric 40 of the 64 h_uv components to model gravity by Einstein's theory of relativity,

Did Einstein also use the antisymmetric 24 components to get the Blackett-Sirag graviphoton effect ?

I don't have a conclusive answer, but here are some indications that Einstein did NOT get the Blackett-Sirag graviphoton effect, but only attempted to get ordinary electromagnetism:

1 - upon hearing of Schroedinger's 1947 claim of having used the 24 anti-symmetric components to get the Blackett-Sirag effect, "... Einstein ... realised immediately that there was nothing of merit in Schrödinger's 'new theory' ... Einstein wrote immediately breaking off the correspondence [with Schroedinger] on unified field theory. ....", according to a Schroedinger web page.

2 - "... The nonsymmetric field extension of Einstein gravity has a long history. It was originally proposed by Einstein, as a unified field theory of gravity and electromagnetism ... A. Einstein, The Meaning of Relativity, fifth edition, Princeton University Press, 1956. A. Einstein and E. G. Straus, Ann. Math. 47, 731 (1946) ... But it was soon realized that the antisymmetric part g[uv] in the decomposition guv = g(uv) + g[uv] could not describe physically the electromagnetic field. ...", acccording to a paper by J. W. Moffat in which he (Moffat) describes his attempts at using the nonsymmetric field structure to describe, not electromagnetism, but a generalization of Einstein gravity denoted by NGT (Nonsymmetric Gravitational Theory). Moffat concludes that his NGT is unsatisfactory, and he abandons it for another (more complicated in my opinion) theory.

However, as far as I can see from looking at a few Moffat papers, Moffat never discusses (or even makes reference to) the 1947 theory of Schroedinger.

In Schroedinger's book Space-Time Structure (Cambridge 1950), Schroedinger says, of the tensor g_ik, that "... hoping that its skew part (1/2)(g_ik - g_ki) should have something to do with electromagnetism meets with a certain difficulty. ... The way out of this dilemma is ... due to Palatini ... Take the g_ik AND the ... Christoffel brackets ... GAMMA^i_kl as the independent functions, to be varied ...".

To me Schroedinger's approach looks a lot like Cartan-Einstein, which varies both metric and connection and has torsion. (It is interesting that Einstein had correspondence with both Cartan and Schroedinger, but apparently insofar as I understand it, neither set of correspondence produced any substantial shared understanding or progress.)

I think that Schroedinger failed to get ordinary electromagnetism, but may have gotten the Blackett-Sirag effect plus gravity.

If Schroedinger had been content to add in ordinary electromagentism by a U(1) gauge field, I think that he would have been more successful.

How could Schroedinger have gone beyond the 64-component theory to get SU(2) weak force and SU(3) color and U(1) electromagnetism ?

Schroedinger could have used the idea of


of a 256-component theory.


According to Uncertainty the life and science of Werner Heisenberg,
by David C. Cassidy (Freeman 1992) at pages 541-545:

"... By 1957, Heisenberg had modified his matter field to form an eight-component type of spinor field ... in February 1957 ... A mathematical ... battle ... broke out between Heisenberg and Pauli over ... the introduction of an indefinite space metric ... later [in 1957] Heisenberg ... stopped in Zurich for a ... visit with Pauli. Within a few weeks ... Heisenberg ... happened on a very simple field equation ...[with]... symmetry ... of ... both the relativistic Lorentz group and the isospin group. Pauli was elated ... The distribution ...[of a preprint]... was set for February 27, 1958. ... Three days before the preprint was to be distributed, Heisenberg announced the new formula .... at ... Gottingen ... One enthused press agent proclaimed, 'Professor Heisenberg and his assistant, W. Pauli, have discovered the basic equation of the cosmos!' ... Pauli had grown increasingly doubtful ... until he refused any further support of the theory ... later ... Heisenberg ... wrote 'He criticized many details of my analysis, some, I thought, quite unreasonably.' ... Pauli ... died suddenly of cancer at the age of 58. ... Heisenberg ... claimed that Platonism had dominated his thinking throughout his career. ... 'The particles of modern physics are representations of symmetry groups and to that extent they resemble the symmetrical bodies ofPlato's philosophy ...'. [ It would be natural for Platonic geometry to have been a dominant force in Heisenberg's way of thinking, because Heisenberg was a student of Sommerfeld, and Sommerfeld was a student of Felix Klein. ]... Heisenberg fell ill ... cancer of the kidneys and gall bladder ... died peacefully at home in Munich on Sunday, February 1, 1976. ...".

In 1959, the Heisenberg paper from which Pauli withdrew was extended and published by Durr, Heisenberg, Mitter, Schlieder, and Yamazaki (Z. Naturforschg. 14a (1959) 441).

Durr and some others have continued to work on it since then, for example Durr and Saller (Phys. Rev. D22 (1980) 1176) and some other publications by Durr, Saller, et. al.

They begin with U(2) = U(1)xSU(2).

Then they construct a fundamental field (urfield) X
whose 4 components are the elements of U(2) = U(1)xSU(2).
For example (my examples, not theirs),
you might think of them as the identity plus 3 Pauli matrices,
or as 4 complex Dirac gammas. Denote the 4 things by:

X1   X2   X3   X4

Then take the duals of those 4 things:

                    X1*  X2*  X3*  X4*

These 8 things form their 8-element fundamental urfield

X1   X2   X3   X4   X1*  X2*  X3*  X4*

Those 8 things form the following 256 antisymmetric
(spinor-like, fermion-like) combinations:

0-grade with 1 element: 1 (unity, zero) 1-grade with 8 elements: Xi i arbitrary from 1 to 4 Xi* i arbitrary from 1 to 4 2-grade with 6+16+6 = 28 elements: Xi Xj i =/= j Xi X*j i,j arbitrary from 1 to 4 X*i X*j i =/= j 3-grade with 4+24+24+4 = 56 elements: Xi Xj Xk i =/= j =/= k Xi Xj X*k i =/= j, k arbitrary Xi X*j X*k i arbitrary, j =/= k X*i X*j X*k i =/= j =/= k 4-grade with 1+16+36+16+1 = 70 elements: Xi Xj Xk Xm i =/= j =/= k =/= m Xi Xj Xk X*m i =/= j =/= k, m arbitrary Xi Xj X*k X*m i =/= j, k =/= m Xi X*j X*k X*m i arbitrary, j =/= k =/= m X*i X*j X*k X*m i =/= j =/= k =/= m 5-grade with 56 elements: ...(dual to 3-grade)... 6-grade with 28 elements: ...(dual to 2-grade)... 7-grade with 8 elements: ...(dual to 1-grade)... 8-grade with 1 element: X1 X2 X3 X4 X*1 X*2 X*3 X*4
The structure has 1+8+28+56+70+56+28+8+1 = 256 complex dimensions, They call it the Proliferated Urfield, and it has the graded structure of the complexified Clifford algebra Cl(8;C). (In this and in what follows, I will sometimes ignore signature complications, such as whether I should use the Clifford algebras Cl(0,8) or Cl(4,4) etc.)

The D4-D5-E6-E7-E8 VoDou Physics model is based on the 256-real-dimensional Clifford algebra Cl(8), using 4 real Dirac gammas,

Radical Unification uses 4 complex Dirac gammas.

I think that Heisenberg and Durr use complex Dirac gammas
because complex Dirac gammas were generalizations of
complex Pauli matrices, and were found to be useful in
simpler, less complete, physics models such as QED.

The grade-2 part of real Cl(8) is
the 28-real-dimensional D4 Lie algebra Spin(8),
and (prior to dimensional reduction of physical spacetime
from 8-dim to 4-dim) it is the gauge symmetry group
in the Lagrangian of my physics model.

If Radical Unification were only a complexification
of my physics model, it would have a Lagrangian
with 28-complex-dimensional gauge symmetry group Spin(8,C).

As Durr says in his paper Radical Unification
(which paper is a continuation of the paper by Durr, Heisenberg,
Mitter, Schlieder, and Yamazaki in Z. Naturforschg. 14a (1959) 441):

"... Symmetry of the urfield Lagrangian
The Lagrangian ... has an extremely high symmetry.
It is not only invariant under Poincare transformations and dilatations,
... but, in fact, under the full 15-parameter conformal group. ...
... In addition ... it appears to be invariant under the huge
gauge-type group U(1) x SL(4,C) ...".

SL(4.C) has 15 complex dimensions,
and there is a local Lie algebra isomorphism SL(4,C) = Spin(6,C).

Spin(6,C) is a subgroup of the Spin(8,C) symmetry group of the complexified version of the D4-D5-E6-E7-E8 VoDou Physics model, so the Lagrangian of a complexified version of the D4-D5-E6-E7-E8 VoDou Physics model has a larger Spin(8,C) symmetry than the symmetry of the Lagrangian of Radical Unification.

If you were to look at only the real part of
the complex Radical Unification structure, and its Lagrangian,
you would see that the symmetry group would be SL(4,R)
which has a local Lie algebra isomorphism SL(4,R) = Spin(3,3).
Since 15-real-dimensional Spin(3,3) = SU(2,2) is
the conformal group of non-linear conformal transformations
of 4-real-dimensional Minkowski space,
it seems to me that Durr's remark
"... The Lagrangian ... is ... invariant under ...
... the full 15-parameter conformal group ...".
could be interpreted as referring to the symmetry of
the real part of Radical Unification and its Lagrangian.

Since the compact version of the conformal group, Spin(6) = SU(4), is a 15-real-dimensional subgroup of the 28-real-dimensional Spin(8) gauge symmetry group of the D4-D5-E6-E7-E8 VoDou Physics model, and since I use the non-compact conformal group Spin(2,4) = SU(2,2) as a component of the U(2,2) = U(1) x SU(2,2) subgroup of Spin(4,4) to (again ignoring some signature matters) describe gravity by the MacDowell-Mansouri mechanism, leaving 28 - 16 - 12 real dimensions to form the standard model SU(3)xSU(2)xU(1).

The Heisenberg-Durr Lagrangian is described by Durr in Radical Unification,
in terms of the 8 complex basis elements
              X1  X2  X3  X4  X*1 X*2 X*3 X*4
"... an expression which has essentially the structure of
the 4x4 determinant constructed from X and X* ...
... involving also derivatives of the fields
(... up to the third derivative) ...".

As Durr says,
their full Lagrangian appears on its face to have symmetry U(1) x SL(4,C),
which is locally isomorphic to U(1) x Spin(6,C),
Durr says that anticommutator structures involving both X and X*
cause their Lagrangian symmetry to be reduced to U(1) x SU(2).

U(1)xSU(2) is fine for the electroweak force,
but is not big enough to include the SU(3) color force. 

As to the color force problem, Durr says:
"... whether soliton-type solutions are possible or not it is,
of course, by no means obvious that they will offer a chance
for a dynamical interpretation of the colour property ...".

In the D4-D5-E6-E7-E8 VoDou Physics model, you have a natural SU(3) color force, but the unconfined particles that we see (protons and pions) are soliton structures made up of confined quarks and gluons, which is a similar approach to that advocated by Durr.

As to gravity, Durr says:
"... There is an extension of the gauge invariance group ...U(1)xSU(2)...
which includes the local Lorentz group
if one explicitly introduces the vierbein as independent field.
Here then the ... additional "gauge fields" ... connected with
the SL(4,C) invariance of the naive Lagrangian ... are
really "connections" and relate to torsion. ...".

Durr's description is very similar to the MacDowell-Mansouri mechanism that I use in the D4-D5-E6-E7-E8 VoDou Physics model, so we both get gravity with torsion etc. by effectively gauging "connection" "gauge fields".

3 main points of comparison between Heisenberg-Durr Radical Unification and the D4-D5-E6-E7-E8 VoDou Physics model are:

1 -

2 -
3 -
Both approaches get gravity plus torsion, etc., by something like a MacDowell-Mansouri mechanism gauging of "connection" "gauge fields".

If Schroedinger and Heisenberg had gotten together and used the structure of a real 256-dimensional Cl(8) Clifford algebra, with a real 8-dimensional vector space, then the picture described by Feynman might have been:

The 8 vector components would have been 4 dimensions of physical spacetime plus 4 dimensions of internal symmetry space

The 28 bivector components would correspond to the 28 dimensions of Spin(8).

 Also, you might visualize the 16, not as antisymemtric bivectors, but as general 4x4 Feynman-Schroedinger potentials, with 10 asymmetric potentials leading to the 40 components of Einstein gravity and 6 antisymmetric potentials including conformal graviphotons and producing the Blackett-Sirag effect.


How could Schroedinger and Heisenberg have quantized such a theory ?


Schroedinger and Heisenberg could have used the idea of


of a Quantum Potential.

Bohm's Quantum Potential Quantum Theory is equivalent to Many-Worlds Quantum Theory. To make a concrete model of Bohm's Quantum Potential, consider:

The space of Possibilities would have the following physical degrees of freedom:

Particle-antiparticle pairs of those fermions would have represented the 8/\8 = 28 gauge bosons.

The mathematical interpretation of those degrees of freedom would have been similar to that of the D4-D5-E6-E7-E8 VoDou Physics model:

Due to Triality, +/- pairs of those half-spinors could represent the vector /\ vector = 8 /\ 8 = 28 bivectors.

Therefore, the Quantum Potential should be described by the physics of World Lines moving in 8+8+8 = 24-dim space.

If the World Lines are regarded as (closed unoriented bosonic) strings, and if 1+1 dimensions are added to the 24-dim space to give it Minkowski-like structure, then

the Quantum Theory would hve been the String Theory of closed unoriented bosonic strings in 1+25 = 26-dim space.


As noted by Rey, and by Horowitz and Susskind, that theory may have a 27-dimensional M-theory, related to the 27-dimensional Jordan algebra J3(O) and its 26-dimensional traceless subalgebra J3(O)o.


If Schroedinger, Heisenberg, and Bohm had collaborated actively immediately after World War II, they might have produced a truly unified theory of everything.

Instead, they remained in their separate worlds:


Unified Theories and Hilbert, Kaluza, et al

According to Thall's History of Quantum Mechanics: "... David Hilbert ... professor of mathematics at the University of Gottingen ... suggested to Heisenberg that he find the differential equation that would correspond to his matrix equations. Had he taken Hilbert's advice, Heisenberg may have discovered the Schrodinger equation before Schrodinger. When mathematicians proved Heisenberg's matrix mechanics and Schrodinger's wave mechanics equivalent, Hilbert exclaimed, "Physics is obviously far too difficult to be left to the physicists ..." ...".

According to Peter Woit: "... By 1925, Hilbert was getting old (63). ...".

According to Hilbert's Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity, by Jü\urgen Renn and John Stachel, Max Planck Institute for the History of Science, preprint 118 (1999), ISSN 0948-9444: "... In 1924 Hilbert published another revised version ... Hilbert again claims ... that there is a necessary connection between the theories of Mie ... electromagnetism ... and Einstein ... gravitation ... In spite of the reassertion of ... Hilbert['s] ... programmatic goal of providing foundations for all of physics, this theory now was, in effect, transformed into a variation on the themes of general relativity. ... The nature of this source term can be expressed on the level of the Lagrangian ... but this relation is in no way peculiar Mie's theory. ... Mie's original theory is in fact not gauge invariant ... the field equations can only hold ... if ... the theory is gauge invariant, i.e. the potentials themselves do not enter the field equations ... Hilbert ... did not derive the identity for gauge-invariant electromagnetic Lagrangians ...".

According to Modern Kaluza-Klein Theories, by Applequist, Chodos, and Freund (Addison-Wesley 1987) and papers reprinted therein: "... In 1914 ... Nordstrom ...[ based on ]... theories developed by ... Mie and ... Nordstrom ... proceeded to unify ... gravitation with Maxwell's theory ...[ by assuming ]... scalar gravity in our four-dimensional world to be a remnant of an abelian gauge theory in a five-dimensional ... space-time. ... Kaluza in 1919 ... proposed that one pass to an Einstein-type theory of gravity in five dimensions, form which ordinary four-dimensional Einstein gravity and Maxwell electromagnetism are to be obtained upon imposing a cylindrical constraint. ... [In] 1938 ... Einstein and ... Bergmann ...[said]... two fairly simple and natural attempts to connect gravitation and electricity by a unitary field theory have been made, one by Weyl [ gauge theory ], the other by Kaluza [ higher dimensions ]....".

In his paper Extra gauge field structure uncovered in the Kaluza-Klein framework, Class. Quantum Grav. 3 (1986) L99-L105, N. A. Batakis says: "... In a standard Kaluza-Klein framework, M4 x CP2 allows the classical unified description of an SU(3) gauge field with gravity. However, the possibility of an additional SU(2) x U(l) gauge field structure is uncovered. ... As a result, M4 x CP2 could conceivably accommodate the classical limit of a fully unified theory for the fundamental interactions and matter fields. ...".


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