He starts by simplifying the problem to its bones. He assumes, quite properly, that the most distant items in universe together constitute the largest influence on the chosen subject. The increase of the number of contributing items with distance grows faster than the decrease of the influence of the separate items with that distance. Every variance in this background averages out. So this background acts as a uniform solid body. Sciama uses this in the computation of the Newton potential at the location of a chosen subject. The background uniformity is used in the form of a constant "charge" density. Charge is used here in the same sense as it is used in Noether’s theorems. In general, it is not electric charge.
The total potential at the location of the influenced subject is
Φ = −∫ρ/r dV = − ρ ∫ dV/r
Next Sciama gives the subject a uniform speed and interprets this as a current. Again he takes the volume integral over the whole universe. This time it delivers a vector potential.
If the subject moves relative to the universe with a uniform speed v, then a vector potential A is generated.
A = −∫ρv/(cr) dV
Both ρ and v are independent of r. Together with the constant c they can be taken out of the integral. Thus
A = Φv/c
Sciama does not say this in his article, but the two volume integrals are in fact the two components of a vector field that play their role in the Helmholtz decomposition theorem.
∇F(r) = 4πQ(r)
∇ × F(r) = 4πI(r)
F(r) = ∇^2 A(r) = F1(r) + F2(r)
F1(r) = ∇Φ(r)
F2(r) = −∇ × A(r)
∇ × F1(r) = 0
∇F2(r) = 0
Q(r) represents the local charge density ρ. I(r) stands for the current density represented by the product of v and ρ. I took this from an old math reference book ("Mathematical Handbook for Scientists and Engineers"; G.A. Korn and T.M. Korn; McGraw-Hill;1968; section 5.7-3. ) This puts Sciama’s approach in an interesting light. It is well known that Helmholtz decomposition divides the vector field in an irrotational vector point function and a solenoidal vector point function.
What makes the situation special is what we get when we reverse Sciama’s interpretation. The current in the field corresponds to the movement of physical items! Helmholtz decomposition only treats the stationary condition of the fields. It is related to the fact that the Fourier transform of a vector field can be split in a longitudinal and a transversal part. On its turn, this relates to the fact that the multidimensional Dirac function can be split in a longitudinal and a transversal part. As long as the whole situation stays stationary the two field components stay independent. However, as soon as a dynamic change happens, then the two fields get coupled. Maxwell’s equations show these facts.
Sciama uses one of Maxwell’s equations in order to show that the coupling of the fields goes together with an acceleration of the considered physical subject.
g = −∇Φ − (∂A/∂t)/c
Since the contribution of the first term on the right is practically negligible, the field g corresponds to a change in time of the speed v. Thus, it corresponds with an acceleration of the considered physical item.
This is his explanation of the origin of inertia. In the same article he relates this fact to the interaction between two physical items, which to my opinion describes gravity rather than inertia.
Helmholtz decomposition only treats the stationary situation. In a curved space the Helmholtz theorem must be replaced by the Hodge decomposition theorem. However, the main relations and reasoning stay the same.
So far the model applies to all fields for which at all locations a charge density and a current density can be defined. Thus, it may hold for different types of fields, such as the gravity field and the electromagnetic fields. Now we proceed by extending the ideas of Sciama.
There exits another set of laws that control the stationary relations between physical items. This is the set of axioms of traditional quantum logic. This logic has an atomic orthomodular lattice structure which is the same as the lattice structure of the closed subspaces of an infinite dimensional separable Hilbert space. It means that a proposition about a physical item can be represented by a closed Hilbert subspace. Traditional quantum logic states nothing about dynamics. It also states nothing about fields.
When a physical item moves, then the atomic propositions that describe its properties change. It means the enveloping proposition that says everything about that item that can be said, is redefined. If this is interpreted in the light of what is stated above, then it means that there must exist something in quantum logic that represents the fields that accelerated the physical item. With other words, the propositions in the lattice of quantum logic influence each other. This becomes apparent when the propositions get redefined. This influence depends on the distance between actor and subject.
So, there must exist a notion of distance in quantum logic. This can be similar to the notion of distance that exists in Hilbert space. When the redefinition occurs such that it conforms to a uniform speed (inside a geodesic in a curved space) then the combined influences do not change the enveloping proposition. However when an acceleration occurs, then this goes together with a change of the enveloping proposition. This corresponds with a move in Hilbert space of the representation of the considered item with respect to the representations of the atomic propositions. These insights can guide the way to transform the current stationary traditional quantum logic into a more dynamic version of logic.
The movement treated here, is a movement of subspaces in Hilbert space. It is not yet an observed movement. Our usual notion of time, the coordinate time does not yet play a role there. The spaces that we talk about do not yet have a Minkowski signature.
The distribution of items in the neighborhood of the considered subject is not uniform. As a consequence the fields in this environment are curved. We have to find a thing that couples the stationary fields and the moving Hilbert subspaces. If you investigate it deeply then you will conclude that unitary transforms are not sophisticated enough to perform that job.
Let us introduce a new concept and call this thing a redefiner. At each step it redefines the closed subspace that represents a physical item and the propositions about this item. The proposition is redefined in terms of atomic propositions. The subspace is redefined in terms of the eigenvectors of observables. The redefiner is not a unitary transform, but like a unitary transform has eigenvalues and eigenvectors, the redefiner has unit sized values that belong to characteristic vectors. The characteristic vector specifies the precise location of the redefined subspace. The redefiner does not touch the eigenvectors of observables. The redefiner is a stepper. Its steps define a progression parameter. It is still wrong to interpret that parameter as "time". It is possible to let the redefiner act universe wide. That is over the whole Hilbert space. It steps from one stationary condition to the next.
What we described here is a manipulators world. Let us now consider what is being manipulated. First try to find a means to define a useful topology in the Hilbert space such that we can use the distances between representations of propositions to account for the dependence of the influences/fields on that parameter. It means that a characteristic vector inside each representing subspace must be selected in order to get a sufficiently precise notion of distance. Next we can use a normal operator with an orthonormal set of eigenvectors that span the Hilbert space and that has an evenly distributed set of eigenvalues, such that we can use it as a kind of Hilbert GPS device. We will not stick to real or complex numbers for the eigenvalues of operators. We may even tolerate the 2^n-on hyper complex numbers of Warren Smith for this purpose. They have the nice property that in their lower 2^m dimensions they act as 2^m-ons. Thus in tiny local situations they act as complex numbers. With these tools we can compute what happens during the steps.
Somewhere the transfer to an observed space with a Minkowski signature must be made. To my opinion the vehicle that does this is the number waltz (c=ab/a). It is only noticeable in Hilbert spaces where the number field is non-commutative. The stepper action transforms the local GPS position such that it is transformed by an infinitesimal number waltz action q_new = uq/u, where u is an infinitesimal unitary transformer u ≈ 1 + Δs. Δs represents an infinitesimal action. The transformation does not affect the real part of q. If the quaternions are taken as the number field then the imaginary part of q gets a partial precession. Δq = q_new - q is perpendicular to Δs. When we close the rectangular triangle and call Δt = Δq + Δs the (coordinate) time step, then the resulting space has a Minkowski signature.
This procedure introduces special relativity and a maximum speed of information transfer. (Here c = 1). It means that the number waltz splits the manipulating part from the manipulated part and at the same time the selected procedure creates an observable space with a Minkowski signature and a Lorentzian metric. The coordinate time is not the same as the progression parameter. The position is transformed. Its original real part does not play a role in the dynamical model. It is replaced by the coordinate time.
The action step Δs represents the combined influence of all fields.
After the step again a stationary status quo between fields and Hilbert subspaces is reestablished. However, the conditions have changed with respect to the situation before the step. The subspace has been moved and due to that fact the fields have been reconfigured. This holds for all representations of items in the Hilbert space.
When I take the number waltz and a corresponding time step selection as responsible for the Minkowski signature of observable spacetime, then I also say that the shift to another expected value of the position operator by the move of the representation of the considered item causes a far less significant effect. Thus the current eigenvalue of the redefiner plays a more prominent role than its shifting properties. These effects will never be uncovered in a complex Hilbert space.
This is a bare bone vision on the origin of physical dynamics.
Reference: "On the origin of inertia", by Denis Sciama (http://www.adsabs.harvard.edu/abs/1953MNRAS.113...34S)