The 2-dimensional Feynman checkerboard49,50 uses a lattice whose nearest neighbors to the origin are light-cone vertices (±1 ±i)/Ã2 to describe a fermion propagating forward in time, therefore moving from each vertex in its path either by (+1+i)/Ã2 or by (+1-i)/Ã2.

The 2-dimensional Feynman checkerboard amplitude for the fermion to propagate along a path is weighted by a multiplicative factor of ime for each turn in the path, where m is the mass of the fermion and e is the length of a path segment. In the 2-dimensional Feynman checkerboard, there is only one type of massive fermion (the electron particle and positron antiparticle), only one type of imaginary i, and only one lattice. In the 4-dimensional F4 model there are 7 massive Dirac fermions in three generations. Let m denote the fermion mass, of whatever type and generation since they all use the same D4 lattice, and e the length of a path segment.

In the 4-dimensional F4 model there are 3 types of imaginaries

i, j, and k. A light-cone path segment is of the type (+1±i±j±k)/2. A turn in the path is a change in sign of one or more of the imaginaries from the preceding light-cone path segment. The multiplicative weighting factor for the fermion propagator amplitude is therefore of the form (ime)ni (jme)nj (kme)nk where ni, nj , and nk are the number of turns involving sign changes of the imaginary i, j, and k . By quaternion multiplication, the end result is a multiplicative factor ±i(me)N, where i is one of the imaginary quaternions and N = ni+nj+nk is the total number of turns in the path. For each calculation, i is taken to correspond to the complex imaginary i, so that the fermion propagator has effectively a complex phase.

A full quantum field theory of the 4-dimensional Feynman checkerboard comes from sum over histories of amplitudes of three types of processes:

The amplitude for fermion propagation from place to place is given by a complex number whose length is the standard probablity amplitude of moving one lattice step (Planck length) in any spacetime direction and whose phase is the standard phase shift due to moving one lattice step.

The amplitude is a complex number, corresponding to a local U(1) symmetry in the complex bounded domain D8+ of which Q8+ = S7 x RP1 is the Silov boundary.

That local U(1) symmetry is also found in the complex bounded domains corresponding to the spinor part of each of the four forces after dimensional reduction:

D5+ , of which Q5+ = S4 x RP1 is the Silov boundary, for gravitation;

D1,3+ , of which Q1,3+ = S5 is the Silov boundary, for the color force;

D3+ , of which Q3+ = S2 x RP1 is the Silov boundary, for the weak force;

and S1 for electromagnetism.

The amplitude for gauge boson propagation from place to place is given by a complex number whose length is the standard probablity amplitude of moving one lattice step (Planck length) in any spacetime direction and whose phase is the standard phase shift due to moving one lattice step.

Gauge boson amplitudes are complex numbers, because the Lie algebra elements of complex semisimple Lie algebras (corresponding to compact semisimple Lie algebras and compact Lie groups) are complex.

The amplitude for emission (or absorption) of gauge bosons by fermions or gauge bosons is given by the product of the square root of the force strength constant of the force carried by the emitted gauge boson times the charge of emitting fermion or gauge boson with respect to that force.

Fermion history sums are carried out with Grassmann variables, so that the Pauli exclusion principle is obeyed. Paths in a history sum of a fermion in a given state that intersect, cancel out. Therefore, the sum over histories for a fermion is carried out over non-intersecting paths. The probability of an event whose amplitude y to propagate from a past point to a future point (retarded propagator) is a real number y(r2,t2)*y(r2,t2), where *y is the complex conjugate of y.

As *y naturally represents the amplitude to propagate from the future point to the past point (advanced propagator), the probability y(r2,t2)*y(r2,t2) can be interpreted as a transaction51 between the past and future consummated in the present. Time asymmetry of the quantum theory is present because both y and *y are evaluated at the future point (r2,t2).

The transaction theory explains why the probability is the square of the norm of the amplitude rather than the norm of the amplitude, the real part of the amplitude, or some other real quantity.

In the transaction theory, the future can affect the past by offering possibilities for transactions, but no direct advanced-wave signalling is possible because the transaction erases all advanced effects.

The many-worlds interpretation of quantum theory52,53 can be used with the transaction picture, leading to a world of parallel universes branching both forward and backward in time. Our present is a transaction between our past branch, one of many past branches, and one future branch. A nearby and highly probable future branch might influence our present by being the dominant offeror of advanced waves.

Branching in both past and future directions could obviate causality objections to altering history by time travel to a past.