05
Jun

(For a fuller account of the subjects discussed in this post see **Ch. 4** and **Sec. 12.4 **of *The Principle of True Representation*.)

**I.** Although originating in metaphysics, the Law of Laws (LL), once geometrized, predicts a variety of physical effects attributable to the presence of extra dimensions *x *^{5 }and *x *^{6}. The fundamental fermions are a good place to start, because in extra dimensions we have a good chance of resolving the twin problems of fermion flavor and family replication. Consider a self-imaging fermion field *ψ* propagating in direction* x *^{5}, driven by *pseudoenergy p *^{6}, so-called because although acting as energy, *p *^{6} is metrically space-like. Because it is self-imaging, *ψ* is expandable in eigenstates of the fifth momentum operator *i *(*h */ 2 π ) ∂ / ∂ *x *^{5 }. We identify the expansion coefficients *ψ _{n }*(

*p *^{6 }= ( *p *_{5 n }^{2 }+* m _{n }*

Here *p *_{5 n }denotes (because it is metrically time-like) the *pseudomomentum* of the *n *th flavor particle of family *f* and *m _{n}*

Now pseudomomentum* p *_{5 }, like ordinary 3-momentum** p **, is conserved in any process; and since dimension *x *^{5 }is orthogonal to Minkowski *M *^{4 }, it is conserved separately from **p**. *Only those processes can occur that conserve the overall* *p *_{5 }. Let us apply this conservation law to the lepton decay reaction *ℓ *^{ − }→ *ν _{ℓ }+ ν _{e}^{ }**

*p *_{5 ℓ n} = *L _{ℓ }p *

where *L _{ℓ}* denotes lepton number.

Eliminating *p *_{5 n }between (1) and (2) and solving for we get for the *p *^{6} value of family* ℓ*

*p *^{6 }* _{ℓ}* = (

where *m _{ℓ}* is the mass of the charged member of lepton family

- Each lepton family
*ℓ*is distinguished in momentum space by the invariant pseudoenergy*p*^{6 }_{ℓ }^{ }of its members. Determining the number of families*ℓ*via the mass spectrum of the charged members is a separate issue, a long-standing problem known as the*family problem*. - The charged leptons mu and tau are sometimes characterized as simply heavier versions of the electron. From (2) we see that that characterization is wrong. They are distinguished from the electron, not only by mass, but by their respective values of pseudomomentum
*p*_{5 }in momentum space. - Flavor neutrinos
*ν*, though massless (see below) and electrically neutral, are distinguished from each other in momentum space by their respective values of_{ℓ }*p*_{5 }(*ν*) ._{ℓ }

For the quarks, the formula for pseudomomentum comparable to (2) is

*p *_{5 q} = *B R* + ( B + Q * _{q }*)

where *B* = 1 / 3 , *Q _{u }*

Δ *L _{ℓ}* = 0 , Δ

*In all processes, lepton family numbers L _{ℓ} and baryon number B are separately conserved*. Thus, according to the Law of Laws,

- Proton decay, as in for example
*p*→*e*^{+ }+ π^{0}or*ν*+ π_{e}^{ }*^{+}( Δ*L*= Δ_{e}*B*= − 1), is forbidden. - Neutrinoless double beta decay (A, Z) → (A, Z + 2) +
*e*^{− }+*e*^{+}(Δ*L*= 2, Δ_{e}*B*= 0) is likewise forbidden.

Should either of these reactions be observed, the PTR and its Law of Laws would be falsified.

**II.** Let us now turn to the question of neutrino mass. The discovery that neutrinos oscillate between flavors has been interpreted by the particle physics community to mean that neutrinos must have mass. But the theory used to arrive at this conclusion depends *not* on the mass of the neutrino, but on *differences* between the squared masses of the interfering mass eigenstates. The notion that flavor neutrinos have mass is thus a mere inference, dependent on the theory employed.

But the standard theory is surely wrong, as it takes no account of the electric charge neutrality of the neutrino. To see the significance of this, we note that for any given lepton, the standard account based on the 4-D Dirac equation assumes charge current density *j ** ^{μ}* and probability current density

*j ** ^{μ}* =

where *q* is the total charge of the particle. Now the spatial integrals over the temporal components *j *^{0} and *s *^{0 }yield, respectively, *q c* and *c*, giving from (6) the useless tautology *q* = *q *. Equation (6) is thus purely ad hoc. Clearly the connection between probability and charge requires for its expression something beyond the standard 4-D description of leptons. This extra something is supplied by the Law of Laws in the form of *two* Dirac equations for each lepton: in six dimensions, the probability equation transforms into a separate one for electric charge. For a lepton of mass *m* belonging to family* ℓ*, the connection between the two corresponding current densities, *j ** ^{μ}* and

*j ** ^{μ}* = −

where again *m _{ℓ}* is the mass of the charged member of family

*m *_{ν }^{2 }= 0 . (8)

That is, on the present theory deriving from the PTR, and contrary to the prevailing view, *all flavor neutrinos are massless*.

Note that our zero-mass prediction refers specifically to *inertial* mass, for which the only true test is the direct, model-independent measurement based on the kinematics of beta-decay. Indirect estimations of neutrino mass from cosmological observables indeed suggest a non-zero mass total for the three neutrino flavors. But the mass determined in this way is *gravitational*, not inertial. This apparent violation of the equivalence principle traces to the circumstance that, to achieve zero inertial mass, the mass eigenstates must provide in equal measure both positive and negative squared-masses, a requirement made possible by the fact that, in 6-space, one can have *p *^{6} < *p *_{5} [see Eq. (1)]. Assuming that our zero-mass prediction is confirmed, what the two mass determinations (kinematic and cosmological) show is that the gravitational field produced by a tachyonic mass eigenstate is negligible in comparison to that of its bradyonic counterpart. It is not a little surprising that strategies to measure neutrino mass should yield information on the gravitational properties of the tachyon, a particle apparently existing nowhere but in confinement in the wave function of the oscillating flavor neutrino.