The following are Heisenberg's goals for a unified field theory of elementary particles:
[Werner Heisenberg, Rev. Mod. Phys., v. 29, n. 3, p. 270]
1. The field operators necessary for formulating the equations shall not refer to any specified particle like proton, meson, etc.; they shall simply refer to matter in general.
2. The particles (elementary or compound) should be derived as eigensolutions of the field equations.
3. The fundamental field equations must be nonlinear in order to represent interaction. The masses of the particles should be a consequence of this interaction. Therefore the concept of a "bare particle" has no meaning.
4. Selection rules for creation and decay of particles should follow from symmetry properties of the fundamental equations. Therefore the empirical selection rules should provide the most detailed information on the structure of the equations.
5. Besides the selection rules and the invariance properties, the only other guiding principle available seems to be the simplicity of the equations.
First of all, I will say something about #1, above. Heisenberg states explicitly in other places that what he intends is that the unified field should create and annihilate not elementary particles but matter in general. But, what does that mean? One thing it could mean is that the continuity equation in Bohmian mechanics is violated. Since the Hamilton-Jacobi equation is especially applicable to bulk matter, that would seem to be the most appropriate method of implementing the material aspect of Heisenberg's first goal.
To get a sense of the second goal, it is useful to consider the following early work, before Heisenberg even believed that what later became his "world formula" could be quantized. The next year, in 1954, he published a paper on its quantization, and of course in 1958 he believed that he had a workable theory. But this early work reflects the search for an Ansatz, and is in that sense of great value.
[Werner Heisenberg, Physica 19, 1953, pp. 906-7] ...if we have so many different particles and if we have to introduce a wave-function for every kind of particle, then the theory becomes so complicated that we can never hope to get a consistent scheme. Now I think it is not at all as difficult as that and I will now write down a formula which will certainly not give the correct theory of the elementary particles but which is only meant as a kind of foundation for optimism in this respect:
gamma_mu p-d Psi/p-d x_mu = A gamma_lambda Psi(Psi^+ gamma_lambda Psi)
Here [in equation (3)] the constant A has the dimension of the square of a length. Let us assume (what is rather doubtful) that this equation can be quantized and gives convergent results. If that would be the case, the results of this equation would qualitatively (not quantitatively) agree with all what we know about the elementary particles. Say we have a state called vacuum, the relativistically invariant state of lowest energy; let us further assume that we have one discrete level. This discrete level would mean an elementary particle of a given mass. So we can write down the following equation
PSI_K,0 = O_K,0 PSI_0
The operator O_K,0 makes from the Hilbert-vector of the vacuum state PSI_0 a one-particle state with energy K and momentum 0. From this one-particle state we can by Lorentz-transformation form another state where the same particle is now moving with a certain velocity. We can superpose such Hilbert vectors and get a new Hilbert-vector which represents that particle in a certain region of space, a kind of wave-packet in the region R. Then we can write
PSI_R_1 = O_R_1 PSI_0
where PSI_R_1 corresponds to a particle in the region R_1. Of course we can do that for any region in space. Finally we can have two particles of the same kind, one in region R_1 and the other in region R_2 with the Hilbert-vector
PSI_R_1R_2 = O_R_1.O_R_2.PSI_0
If R_1 and R_2 are sufficiently distant from each other the operators O_R_1 and O_R_2 commute; therefore there is no difficulty in writing down the state-vector and we see at once that we have not only a solution corresponding to one particle but to infinitely many of these particles. Furthermore we see that equation (3) will probably have solutions with different masses, because we will probably have different discrete levels. The equation will lead to Fermi particles and Bose particles because whenever the operator O_K,0 is an odd function of Psi it will give a Fermi particle, when it is an even function it will give a Bose particle. So it appears that such a simple equation may lead to quite a number of different particles and obviously the ratio of the masses of these particles will be determined just as well as the different stationary states of the H-atom are determined in ordinary quantum mechanics. Also the quantity e^2/hc would be determined by such a theory for the equation would describe the interactions between the different particles in the same way as the Schrodinger equation of a molecule determines the interactions between different atoms in the molecule. It is also important to note that Psi(x) itself, the wave function, is definitely not the operator O_K,0 but that this operator O_K,0 is a complicated function of Psi...
...Equation (3) may be too simple, and it is quite doubtful whether local theories of this type can be quantised. I only wanted to point out that so far physicists have looked for too complicated schemes. We have been accustomed to introduce one wave-function for the proton, one for the neutron, one for the pi-meson, one for the electron, etc., and so we got hopelessly complicated schemes and have made things much more difficult than they are. I think there is a good chance that a wave equation of a very simple kind may be so rich as comprise quite a number or perhaps all of the elementary particles. One should try not to extend the present foundations of physics and to study very complicated schemes unless one is absolutely forced to do so...
What Heisenberg is doing here is giving the momentum representation for a particle picture that will be equivalent to Bohm's position representation. PSI_0 is the zero-point energy as a state, PSI_K,0 is a discrete energy state, PSI_K,L would be an energy-momentum eigenstate. When these are superposed, we get PSI_R_1 in a certain region of time and space. For our Vedic friends, the Brahmic mantra for this is: OM KRIYA BABAJI NAMA RUPA KALA DESA AUM. This is spiritually the mantra for the ascension in quantum matter, even as its abbreviated version, OM KRIYA BABAJI NAMA AUM, is the mantra for the resurrection from the tomb of classical matter.
The essential point is that the quantum state is projected ontologically before the etiological particle state, mutually configured by observer and observed, is superimposed on the ontological reality. Even then, the particle state refers to a generic Schrodinger/Born or Bohm type particle - it does not have the detailed Platonic form, which is added last, in this conception, as a result of the oroboric connection in the etiological system of worlds. In the Vedic terminology, the ADI|ANU-PA-DA-KA particle emerges from the astral into the physical and then is "dressed" in the "skins" of Atma-Buddhi-Manas-Kama-Prana, but the detailed form, as expressed by the quantum numbers including mass, is only realized in Linga-Sarira and Sthula-Sarira. It is well known, however, that Linga signifies the spiritual totality in the highest world of Platonic forms, bordering on formlessness. This is the oroboric connection from the lowest world to the highest in the etiological system of worlds.
The coherent representation of the unified field, it will turn out, is the right description of the embodied Observer-Observed construct, O, in von Neumann's all-quantum composite system. The Bohm concept of Classical Universe Particle in relation to its Quantum Ground will prove to be the correct description of the Measuring Apparatus, M, while the Whiteheadian development of the Subject-Superject will prove to be the correct description of the System, S. This latter conception of the quantum System, S, forms a supersymmetry with the momentum representation described above by Heisenberg and, by virtue of its intimate connection to the Bohm Ansatz for M, supersedes the position representation as the post-quantum wave-particle synthesis. It will turn out that the Classical Universe Particle, along with the Bohm Point, has the additional significance of being a particle representation for the unified field operator.
Heisenberg's second goal, that the particles should be eigensolutions of the field equations, is interpreted by Heisenberg himself in one place to mean that there will be effective field operators in addition to the unified field operators. One may say in perhaps a more profound way that the Gestalt field operator supersedes the quantum field operator and gives the final form to the quantum states via the patterns of geometric algebra. Another way of fulfilling the second goal is to make the radical supposition that the unified field operators are observables and the particle states indicated by the Bohm Point or the Classical Universe Particle are its eigenvectors. In even ordinary quantum field theory, the coherent states are eigenvectors of the destruction operators, and we are using the same concept here to treat the embodied Observer-Observed construct, O, in terms of the coherent states of the unified field. My further proposal, however, is that the particle state representing the classical-like Bohm particle will be an eigenvector of the full unified field operator. It is also true that the Schrodinger-Born particle, which is intimately connected in various ways to the Bohm particle, is an eigenvalue of the primal particle state, ANU. And since the Bohm particle, being a solution of the Hamilton-Jacobi equation, lends itself to being generalized to bulk matter, we have then a further explanation of the "creation and annihilation of matter in general", which is essential to Heisenberg's first goal.
The eigen-value of the unified field operator will be the body-world schema of the observer as an organism. The meaning of the eigenvalue equation is expressed in the ontological system of worlds, since it has to do with the fundamental projection of classical phenomenal reality onto quantum noumenal reality, but the meaning of the unified field itself can best be understood in the field theoretic system of worlds. There, the observer as an organism is in the astral world and it stands apart as an individuating entity from its environment in the physical world. The field is the relationship between organism and environment. The field is introjected as an informing field by the organism, so that it can function in its individuated separation from environment. The field informs the internal construction of reality known as the Wave Function in the inner mental body of the astral organism. The Bohm Point is a projection onto physical reality of the Unified Field Object in the inner mental body of the organism.
So, from all of this it becomes clear that the unified field is not primarily a field of matter and elementary particles, as Heisenberg believed, at least as of 1966. It is rather a unified field of consciousness, as Maharishi Mahesh Yogi and John Hagelin have been preaching for some time. Unfortunately the Maharishi and Hagelin falsely employ the additive unified field theory rather than the radical unified field theory of Heisenberg and Duerr. The former is by no means deep enough metaphysically to describe consciousness. It is peculiar, however, that Heisenberg seemed to abandon his deeper concerns, as expressed in "Physics and Philosophy", for instance, when it came to the unified field theory.
I have shown how even ordinary quantum dynamics can be interpreted as either a unitary development in matter or in consciousness. I will present the argument below and then show how it ties into this discussion:
Psi(x,t) = <x|[1+(iHt/hbarN)]^N|PSI(0)>, as N -> infinity;
where S(0) = |PSI(0)><PSI(0)|
[PS(t)P + (1-P)S(t)(1-P)]
where there is a measurement, P, at t and then a unitary evolution to T. We can of course put indices on P and T and add more terms vertically to get a whole string of reductions and unitary evolutions.
S(t) = e^-iHt/hbar S(0) e^iHt/hbar =
[-iHt/hbar, S(0)] +
1/2! [-iHt/hbar, [-iHt/hbar, S(0)]] +
1/3! [-iHt/hbar, [-iHt/hbar, [-iHt/hbar, S(0)]]] + ... =
S(0) + t S'(0) + 1/2! t^2 S''(0) + 1/3! t^3 S'''(0) + ...
With regard to the meanings of the first two forms, in the realistic sense of Bohm's Wave Function, one can take only the lowest power of H, but one must then divide the time interval into an infinite number of parts and multiply by an infinite number of such infinitesimal operations, while in the algorithmic or phenomenal sense of von Neumann, one may retain a finite time interval, but one must then use all powers of H. This implies that the Quantum continuum, Star Trek's Q-continuum, is on the realistic side, while the spacetime continuum, as a more or less classical entity, is on the phenomenal side of metaphysical reality.
The basic Ansatz, which explains the meaning of the realistic and phenomenal realms, is that the quantum theory supersedes the classical theory *as a description of the physical world*, but the classical theory does not disappear - far from it, it is still required as a basic metaphysical reference that gives meaning to the quantum description of the physical world (see Bohm in "Quantum Theory", 1951). So, I put this classically described world, which I call the meta-physical world, seven worlds removed from the physical world, in order to make room for intermediate worlds, like the emotional world, the mental world, the etheric world, the phenomenal world, and the causal world. There is then an indirect connection between the classical world and the quantum world, passing through the intermediate worlds, and also an oroboric connection, directly from the physical world to the meta-physical world, across what Bohr called the abyss. This oroboric connection is also related to Chew's idea of the bootstrap.
The meaning of the third form can be readily understood from the following quote from David Bohm, at http://www.DouglasHospital.qc.ca/fdg/kjf/34-TABOH.htm, 5.4:
"Thus, our thoughts may contain a whole range of information content of different kinds. This may in turn be surveyed by a higher level of mental activity, as if it were a material object that one were 'looking at'. Out of this may emerge a more subtle level of information, whose meaning is an activity, virtual or actual, that is able to organize the original items of information into a single greater whole. But even more subtle information of this kind can in turn be surveyed by a yet more subtle level of mental activity. And at least in principle, this can evidently go on indefinitely."
So, the Hamiltonian, as the ascending element, represents mental activity, and the Density Operator represents the physical-astral object of that mental activity. The Hamiltonian is like the Homunculus, and, if we regard the Homunculus on the same level as its object, we must consider that there is an infinite regress of Homunculus perceivers. However, in quantum mechanics, we can synthesize the infinite expansion into a single form on the higher mental level: e^-iHt or e^iHt. This signifies that the progression of thought is in a sense a unified transcendental process and not just an infinite regress.
However, Bohm's explanation of this third form may be a bit of a blind. His explanation occurs not in the ontological system of worlds, with the first two forms, but in the field theoretic system of worlds, dedicated to Heisenberg's world formula for the unified field. There is alot of Bohm stuff going on there too, including this explanation from Bohm and Sarfatti's mental Wave Function with physical Bohm Point.
But logic dictates that this third form should have to do with the third major quantum ontology, namely the Everett interpretation. The way to see that is to transport Bohm's notions about mental and physical from the field theoretic system of worlds, which happens to be the Order of Melchizedek, to the phenomenological system of worlds. The ascending Hamiltonian refers to cogito as thought in the mental sense of Bohm, but cogito in the phenomenological sense refers to the absolute consciousness of the transcendental ego. The Density Operator goes from representing external physical-astral reality to representing the internal causal matrix (go Keanu!) in the causal world of the phenomenological system of worlds.
In the ontological system of worlds, it is the particle that is mutually configured by Observer and Observed in the oroboric abyss between them. The oroboric connection in the phenomenological system of worlds, however, is occupied not by an individual particle, but by an entire quantum-classical world. It is for this reason that Heisenberg's "world formula", describing matter in general, is especially applicable to the phenomenological perspective, and it is also for this reason that DeWitt's characterization of Everett's "relative state" approach as the "many-worlds" ontology is *very* correct, but if and only if one uses not the Schrodinger Equation, as did Everett, to implement it, but the Heisenberg world formula. The degeneracy of the vacuum, which represents in the Heisenberg approach the classical world, is totally indicative of the many-worlds character of the unified field theory. At the level of algorithm, however, there can still be definite results, as even Everett averred.
Now we have a continuity of mathematical forms, descending from the third form and the second form in the phenomenological system of worlds to the second form and the first form in the ontological system of worlds. In the phenomenological system of worlds, the algorithm of the second form operates in the sixth or causal world, while in the ontological system of worlds it spans the three upper worlds, which constitute von Neumann's III, centered upon the causal world.
Now, the point is that the field theoretic system of worlds is to the Bohm particle system of worlds as the phenomenological system of worlds is to the ontological system of worlds. This means that the unified field definitely corresponds to the unitary development in consciousness, while the Bohm particle treatment definitely corresponds to the unitary development in matter. The Bohm particle should be understood as the material basis for the unified field of Heisenberg and Duerr. The unified field of the observer becomes involved in its material basis and this leads to a complete reconception of the physical dynamic known as physics. We return to an ontological type of physics in which the objective observers and their material bases investigate the subjective primordial quantum systems known as actual entities. Thus revealing to consciousness who it is through investigation of nature, the detailed form of the material/spiritual identity of consciousness is then seen to be a product of the mind in the causal world of the phenomenological system of worlds.
Bohm's 1952 theory was the correct foundation for Heisenberg's unified field theory, and Heisenberg's unified field theory, developed after 1953, was the correct description of the implicate order of Bohm. If there was really an unfortunate historical contingency in physics, it was that these two theories, which fit like hand in glove, passed like ships in the night. But now, in the hour that the ship comes in, they must come together again. Heisenberg's plea for simplicity, as quoted above, gives a more than heuristic significance to Bohm's discovery, which even Bohm did not appreciate due to his failure to link up with the other fundamental theory developing synchronistically with his own.