A Unit-Derivation for the Vacuum Field
| Scientific Paper | |
|---|---|
| Title | A Unit-Derivation for the Vacuum Field |
| Read in full | Link to paper |
| Author(s) | Jeffrey N Cook |
| Keywords | Vacuum, Field |
| Published | 2012 |
| Journal | Proceedings of the NPA |
| Volume | 9 |
| No. of pages | 14 |
| Pages | 97-110 |
Read the full paper here
Abstract
I derive a Lagrangian for all fields of force known, as well as all that could possibly be discovered in the future, and show that the sum of the fields of force in space equals the vacuum field of force and that this field can be measured in dimensions of kilograms per second. Using Gauge Theory and the Euler-Lagrange Method, I show that that which interacts in the vacuum field of force is velocity itself, that it is an interaction of motion and the foundation (or result) of all other fields of force. The consequences of this paper should allow one to better study the known fields of force, and because of the results of these mathematics, a more accurate Gauge Theory for the nature of fields of force may be understood.
Overview
This is a dimensional-analysis paper. Cook asks what physical units the Vacuum field of force must carry if that field is understood, as it conventionally is, as the sum of the energies of every field in space — electric, magnetic, gravitational, gauge, fermionic, Higgs, and any field not yet discovered. He calls this the "Z Field", and is careful at the outset to distinguish it from the zero-point field: Zero Point Energy is the ground state of one particular quantum system, whereas his Z Energy is the total. His answer is that the Z Field's SI dimensions are kilograms per second, kg s−1, and that the quantity which interacts with it — its "Action", in the sense in which charge is what the electric field acts on — is velocity itself.
The departure from the standard account is less in the physics than in the method. Cook's complaint is that the four known forces are all measured in newtons, so comparing forces tells one nothing about how the underlying fields differ; only the fields have distinctive units. He therefore builds a sequence whose terms are field-unit combinations of the form M ta eb rc, indexed by a natural number w, sums it, and identifies the limit with the vacuum. Along the way he asserts — as a set of numbered conjectures rather than as derived results — that there are infinitely many unitary groups U(n), that each carries exactly three fields, and that all such fields are homotopic to one another and to Mass raised to the first power.
The argument
The four conjectures
The paper opens with definitions from gauge theory and then states four conjectures. First, an infinite number of unitary groups U(n) exists, extending past U(1), SU(2) and SU(3) to all integers n, including zero and negative n. Second, each group is "modular to three fields": U(1) describes exactly the pseudovector magnetic B field, the vector magnetic A field and the electric E field, and each higher group likewise carries three. Third, the number plane on which each field lives grows in complexity with n — U(1) isomorphic to the complex numbers, SU(2) to the unit quaternions, and so on. Fourth, the three field variables of each group correspond one-to-one to three unit exponents.
That last conjecture does the real work. Cook notes that the B field is in kg C−1 s−1, the A field in kg m C−1 s−1, and the E field in kg m C−1 s−2. Only distance, charge and time vary; mass appears in every one of them and always to the first power. From this he concludes that all fields must be "homotopic" to mass — deformable into it without tearing — and hence "contractible" to a single point, which is what licenses treating each field as a point in a larger space.
Two indexed families of fields
Cook then defines two sets. Set X collects the "Lie fields", elements of the form M t−a e−b rc, with the exponents constrained by a + b + c = w. Set Y collects the "Topological fields", of the form M ra eb tc, with a + b + c = −w + 1. The known fields fall out of X at the first three indices — B at w = 1, A at w = 2, E at w = 3 — while Y begins with mass itself (Y1, dimensions of kg) and then density (kg m−1) at Y2. Because X contains no smooth manifold structure past that point and Y converges almost immediately, the two families must be joined by a map, which Cook calls the net Θ.
Numerical values from the electron
To assign numbers rather than only dimensions, Cook takes three conserved electron parameters — rest mass, classical radius and charge — and manufactures a fourth, a time, by positing an arbitrary wave running from the electron's centre to its surface at v = −1 m s−1. The velocity is negative because the propagation is inward-referenced; its square is then v2 = 1 m2 s−2, positive, which he takes to put the Electron in resonance with every possible field at once. Since t = r / v with v = −1, the time value is numerically the radius.
The table of electron parameters he uses gives mass 9.10938×10−31 kg, charge −1.60217×10−19 C, and radius −2.18794×10−15 m. Running the sums produces X1 = −2.598×103 kg C−1 s−1, then X2 in kg m C−1 s−1, then 5.991×10−12 for the E-field entry, with the series thereafter running to 1022, 1041, 1059 and beyond. The Y series by contrast settles at 4.163×1016 by its second term.
The Z field theorem
The central claim is written as a comparison between Θ-mapped values of Y and X three indices back, involving an auxiliary function fx(w) = 5145×10w and a constant k = 6.271921. The "three arguments back" structure is deliberate: it encodes the claim that each unitary group's fields are built from the previous group's, and that the sums cancel every third step. Cook shows that the composition of the map vanishes whenever w mod 3 = 0, so the X contribution drops out and the vacuum reduces to the convergent Y sum.
The case w = 0 is treated separately and at length, because there the constraint a + b + c = −1 cannot be met by natural numbers. Cook reads this as a "signed zero" and glues x0 and y0 into a single wedge point — a quotient space — obtaining x0 − y0 = 0 and a null object Z that is simultaneously the initial and terminal element of the whole category of fields. Its dimensions are kg m0 C0 s−1, that is kg s−1. The Lagrangian follows in the classical form L = T − V with T the sum over X and V the sum over Y.
The Field-Action hypothesis
The second half generalises a pattern Cook reads off the known fields: force divided by field gives the quantity that field acts on. Newtons divided by the E-field units leave coulombs, i.e. charge; divided by the A-field units they leave C s−1, i.e. current; divided by the B-field units they leave C m s−1, the scalar magnetic potential. Applied to mass-as-a-field the same division leaves m s−2, acceleration — which prompts Cook's inversion of the usual relativistic reading: "if Relativity suggests Mass curves space and Time, affecting Acceleration, then this model would only suggest/allow the counter: Acceleration would curve space and Time, affecting Mass."
Applied to the Z field in kg s−1, the division leaves m s−1 — velocity. Cook then spends several pages establishing this by a bounding argument: he constructs an action function q(t) from the areas under the L-against-t graph (first values 1299.31, 3896.93, 5197.24, 5197.24, 1676995.98, 3348794.73 — note the repeat, the every-third-step cancellation showing up again), constructs a modular velocity function v(w) from a three-branch algorithm keyed to w mod 3, and argues that ln qZ(t) − v(w) = O(3). Since a bounded dimensionless difference cannot change dimensions, the action of the vacuum field is velocity. He notes explicitly that the velocity algorithm uses additive velocities rather than the Lorentz factor, to avoid complex values, and defends this on the ground that only dimensions are at stake.
Assessment
What is genuinely attractive here is the starting observation, which is correct and under-used: every force is measured in newtons, so force units carry no information about which interaction is at work, while field units are all distinct. Reading the interacting quantity off the ratio force / field is a clean piece of dimensional bookkeeping, and the three worked cases — charge for E, current for A, scalar magnetic potential for B — are right. The conclusion that the same ratio applied to a kg s−1 field yields a velocity is likewise simple arithmetic. Cook is also unusually honest about his own construction, repeatedly labelling steps "arbitrary, but directed" rather than passing them off as forced.
The difficulties are severe, and they are mostly difficulties of assertion rather than of arithmetic. The four conjectures carry the entire structure and none is argued for. That U(n) carries exactly three fields for every n is stated by inspection of U(1) and then extended; the claim that SU(2) and SU(3) each describe exactly three fields is simply false on the standard counting the paper itself cites — SU(2) has three generators but SU(3) has eight, and the paper acknowledges SU(3) involves "even more bosons" two sentences before asserting the three-field modularity. The homotopy language is likewise decorative: nothing in the paper constructs a continuous deformation between two field configurations, and "homotopic to Mass" is being used to mean no more than "mass appears to the first power in the unit string".
More fundamentally, the whole derivation operates on unit strings, not on fields. Adding X1 + X2 + X3 means adding quantities in kg C−1 s−1, kg m C−1 s−1 and kg m C−1 s−2, which are not commensurable; the sum has no dimensional meaning, and the appeal to the Buckingham π theorem does not rescue it, since that theorem constrains relations among dimensionless groups rather than licensing sums of dimensionally unlike terms. The auxiliary function fx(w) = 5145×10w and the constant k = 6.271921 are introduced without derivation or physical interpretation; they are fitted so that the tabulated πw values track the Xw values, which is why Table 4 shows agreement — the agreement is built in, not found. The velocity algorithm of equation (58), with its three hand-chosen branches and its magnitude factors of two and three described as "arbitrary but directed", has the same character.
There is also a numerical problem the paper does not flag. Table 1 lists the classical electron radius as 2.18794×10−15 m and calls these "the generally accepted SI Units values"; the accepted classical electron radius is 2.81794×10−15 m. The digits appear transposed. Because the manufactured time value is set equal to that radius and both then enter every term of both series, the transposition propagates into every number in Tables 3, 4 and 5. This does not touch the dimensional conclusion — kg s−1 follows from the exponents alone — but it does mean none of the tabulated magnitudes should be quoted.
Finally, the paper is careful never to claim a measurement. It offers no prediction that could be checked against, for instance, the Casimir Effect measurements of vacuum energy differences, or against the Cosmological Constant problem it cites in reference [22] — the 10120 discrepancy between the summed field zero-point energies and the observed cosmological energy density, which is precisely the arena a "sum of all fields" derivation would have to enter. Assigning the vacuum a unit of kg s−1 does not by itself say what magnitude it has, and Cook says as much: the calculation "only intends to solve for the units of the Z Field and not its true to nature numerical value if indeed it could have one". Judged as what it says it is — a units bookkeeping exercise offered as a road map for later experiment — it is internally consistent in its dimensional core and unsupported in almost everything built on top of that core. The closing appeal to "first hand, reproducible experiments" that "already backed up" the mathematics is not documented anywhere in the paper.