Thursday, July 29, 2010

Lorentz Force in Binary Mechanics

Updated: July 30, 2011
At relatively low bit densities, the Lorentz force is consistent with binary mechanics[1] (BM), with which this note assumes familiarity.

BM predicts that experimental data for particle events approaching the level of fineness of single BM bits will tend to show anomalies when evaluated with conventional quantum mechanics (QM), which assumes the components of the electromagnetic four-potential (Φ,A) can be defined at arbitrary spatial points in continuous space-time. On the other hand, the BM model quantizes both space and time and assigns each component to slightly different spatial locations (Figs. 1A-1C, 2B-2D).

[Note: Fig. and Table references refer to those in the original paper[1].]

In BM, primary forces have been defined as transitions in bit state which alter unconditional bit motion which each act independently of the others. One might say that the set of BM forces are orthogonal, which simplifies understanding physical processes. The four fundamental bit operations -- unconditional, scalar, vector and strong, are independent insofar as each is thought to be implemented in a separate quantized time tick in a specific order completing 4-tick bit operations cycles.

The conventional electromagnetic force combines two independent BM forces, designated as scalar and vector primary forces (Table 4), similar to familiar expressions such as the Lorentz force, typically written as

F = q(Φ + v X A) (1)

where the force F is the product of charge q and the sum of scalar potential Φ and the cross-product of the velocity and vector potential A components. F may be evaluated in each of three spatial dimensions in frame, Si, i = 1, 2, 3, chosen with position in S expressed by integer coordinates, Xi, as in a cubic lattice (Fig 1C). Spot unit bits (Fig 1A), lite L and mite M, are oriented parallel to each axis of Si at each coordinate location. At each Xi, three spot units form a spot (Fig 1B).

Fig. 1

Fig. 1C


For one dimension, eq. 1 becomes

F = q(Φ + A') (2)

omitting subscript i = 1, 2, 3 for F, Φ and A', where A' is the vector result of cross product v X A. Since the scalar and vector bit transitions (forces) act independently in BM as in quantum mechanics, the Lorentz force F may be linked to BM for each independent component:

F = qΦ + qA' (3)

Since a mite M is a binary bit (M = 0,1), the mass in the volume of its position in the spot unit is zero if there is no mite M, or a positive scale factor if a mite is present.

Substituting the product of mass m and acceleration a for F in eq. 3,

ma = qΦ + qA' (4)

Scalar Force. If a mite is present in the spot unit (M = 1), let mass m = 1 and q = -1, 1 (eq. 5 in [1] which describes a sign function of parity in S). If Φ = 1 then acceleration a = q = -1, 1. On the other hand, if M = 0 then q and m = 0, regardless of the value (0,1) of Φ. Finally, if both q and Φ are non-zero, then F = a = -1, 1.

For acceleration a, recalling that both space and time are quantized, we try description of the spot unit as two cubes with side length d, a BM primary constant, one each for the M and L bits, sharing a common side to form a 2d x d x d volume.

Recalling that the primary constant t is the time interval for one tick of a BM system, we have

v = d/t = 1 (5)

where velocity v is the ratio of two primary constants.

In any bit motion in one tick, a mite or lite moves distance d in time t. That is, all M or L bit motion occurs at maximum velocity v.
With the cubic lattice architecture, it is presently thought that v is greater than the measured speed of light c, since for distances of large integer n multiples of d, a bit must typically "travel" a greater distance than the origin point to destination point, much like a person might move in New York City, traveling streets arranged at right angles and up and down elevators.

This consideration would seem to predict that for the shortest distance nd where integer n = 1, bit velocity v = 1, and as n increases, apparent bit velocity, as measured on a point-to-point line, would decrease to light speed c.
Results as shown in Fig. 2B where the scalar potential Φ accelerates mites of like sign may be interpreted as reducing density of mites of like sign.

Vector Force. One may apply the A' component for dimension i, instead of the Φ component, using the exact same logic concerning initial conditions at t=0 for Lorentz F and acceleration a results.

Discussion

Given unconditional bit motion is the default time development of BM states in the absence of any of the primary forces, it might seem awkward to insert BM quantities into the Lorentz force which assumes continuous space-time and in particular, that the electromagnetic four-potential is presumed to be definable at arbitrary spatial points.

Further, conventional particles such as electron and proton (listed in Table 3 in [1]) are thought to consist of more than one bit. For example, the electron and positron consist of at least 2 bits in a spot which has a maximum capacity of 6 bits. Also, conventional use of the Lorentz force expression applies to particles not individual mite bits as presented above.

Nonetheless, unpublished simulation data suggests that mite acceleration associated with the scalar potential desribed above favors displacement of like-signed bits away from each other, so the parallel presented above at the mite, not particle, level may be relevant after all.

However, caution might be observed in over-interpreting the treatment above of the Lorentz force in BM. There are parallels, which may be somewhat superficial. For example, in BM, Φ and A' each have three independent components, so one might better speak of an electromagnetic six-potential. For consistency, Φ was subscripted for the applicable spatial dimensions i = 1,2,3 in eq. 2 above.

The situation gets worse or better depending on viewpoint when one considers that the fields of bits which "cause" the potentials are located at nominal 90 degree angles to F direction, for both A' and Φ. For vector A' direction, this angle agrees with conventional physical interpretation of the Lorentz force F direction.

These "field" bits are a concurrent mite MJ for Φ (eq. 9 in [1]) and the "oncoming lite" L for A' (eq. 13 in [1]).

The Lorentz force is discussed further in "Quantized electromagnetism" [2], which through the miracle of the internet was published after the present paper -- a citation from the future which would not be possible in print media.

References
[1] Keene, J. J. "Binary mechanics" J. Bin. Mech. July, 2010.
[2] Keene, J. J. "Quantized electromagnetism" J. Bin. Mech. May, 2011.
© 2010 James J Keene