The Principle of True Representation (PTR) is no respecter of supposedly settled scientific doctrine. In **Post 5** of this series I showed that, in a world of 4 + 2 spacetime dimensions—*our* world, according to the PTR—the graviton does not exist, implying not only that string theory is wrong, but that quantum mechanics and gravitation likely cannot be unified. Then in **Post 6** I showed that, in that same ( 4 + 2 ) – dimensional world, flavor neutrinos are necessarily massless, contradicting the orthodox claim that oscillating neutrinos must have (inertial) mass. In the present post I outline yet another heterodox implication of the PTR, namely, that the masses of the charged leptons are attributable, *not* to interaction with the vacuum Higgs field as suggested by the Standard Model (SM) of particle physics, but to excitations of an electron trapped in a 1 – D well in an inner Minkowskian spacetime,* N *^{2}. The quark masses in turn are determined, in part, by coupling to the masses of the charged leptons. For complete details, see *The Principle of True Representation *(*PTR*), **Chapters 13 **and **14**.

As argued in* PTR*, the Higgs field enables the presence in the SM Lagrangian of particles of finite mass but does not determine the values of those masses, much as the stem of a plant enables the display of a flower at its top, but does not determine the bloom’s form or color. The charged leptons and quarks, in other words, come to the Lagrangian already endowed with mass. But what is the source of those masses and what determines their values? Unsurprisingly, as lepton mass is flavor-dependent, and as flavor is defined in terms of pseudoenergy *p* ^{6} (see **Post 6**), the determination of lepton mass *m* begins in 4 + 2 dimensions. Our objective there is to express *m* in terms of the particle’s electromagnetic self-energy, for in extra dimensions the self-energy will involve flavor-dependent terms not present in ordinary 4 – D spacetime, and these can account for mass over and above that of the electron. To carry this out we first write down the *total* self-energies (mechanical plus electromagnetic) of the same generic lepton observed at rest in two different reference frames, *K* and *K* ′, separated by a finite (circular) rotation angle *ϑ* in the *x* ^{0} – *x* ^{5 }plane. Angle *ϑ *is chosen to provide vanishing pseudomomentum *p *_{5 } in *K* ′. Equating the two expressions for total self-energy we obtain for mass *m*:

*m* = *m* * _{e}* + (

where *U* and *U* ′ denote the *electromagnetic* self-energies of the particle evaluated in frames *K* and *K* ′, respectively. Calculation of these self-energies is accomplished by a straightforward extension of electromagnetic potential theory to 4 + 2 dimensions. We obtain

*U* = 5 *α m c *^{2 }/ 3 + *p *_{5 ∙ }*p *_{5 }^{e m} / 2 *m* (2)

*U* ′ = 5 *α p *^{6 }*c */ 3 (3)

where *α *is the Sommerfeld fine-structure constant, *p *_{5 }^{e m} is the fifth component of the electromagnetic momentum, and *p *_{5 }and *p *^{6 } are the invariant fifth and sixth momenta already defined in **Post 6**: *p *_{5} = *p*^{6} – *m _{e }c *and

However, before solving it we need to put the unknown *p *_{5 }^{e m} in terms of observable quantities. More pointedly, it is clear that, if (1) is to yield a mass spectrum, *p *_{5 }^{e m} must be quantized. But quantization implies confinement, impossible in 4 + 2 spacetime, where our generic lepton runs free. It is at this point that we look to the inner manifold,* N *^{2 }.

The decisive indicator of its existence is to be found in the form of the internal structure function of the electron, *D *( **x **) . This function, already described in **Post 4**, represents the position probability density of a massless point *P *circulating in Zitterbewegung at light speed *c* about the electron’s center of mass *C*, generating among other things, the spin and magnetic moment of the electron. Significantly, distribution *D* exhibits both positive and negative densities, a binary feature uninterpretable in either ordinary 4 – D spacetime ( *M *^{4 }) or the (4+2) – D one implied by the PTR ( *M *^{4 + 2 }) . No less strange, owing to the fact that the electron is a spin – ½ particle, point *P* must orbit about *C* through 4 π radians to return to its original position, ensuring reëntrance of its associated de Broglie wave. It would be natural to associate one of the two successive orbits of 2 π radians with a positive value of *D*, and the other with a negative value. However, such a picture is obviously unrealizable in either *M *^{4 }or* M *^{4 + 2}. Enter* N *^{2}. This is a 2-D internal spacetime consisting of one infinite temporal dimension *u *^{0} and one *bounded* spatial dimension *u *of width *W* equal to one-half the de Broglie wavelength of the electron. Total spacetime now becomes the 8 – D product space *M *^{4 + 2} x *N *^{2} , in which all dimensions are mutually orthogonal. Now imagine a 7-D subspace consisting of a slab of thickness *W* created by the cross product of hyperplane *M *^{4 + 2 }and dimension *u*. Then, on the assumption that point *P* can enter the slab, to explain the occurrence of negative probability density one has only to assume coordination between two successive circular orbits of *P* in *M *^{4 + 2} and linear motion back and forth in the slab. Inward motion against positive *u* codes as negative density, and outward motion back to its starting point as positive density. As the interior of the slab is inaccessible to observers in *M *^{4 + 2}, and as the Zitterbewegung is itself unobservable, the presence in distribution *D* of negative density poses no formal or practical difficulty.

It is easy to imagine the electron, as it is already in the slab, becoming trapped there in a 1 – D well of width *W *and finite depth *V *_{0 }. The internal momentum operator corresponding to the positive mass constant *m _{e} c* must be quadratic in the differential operator

*E* = *n* ^{4} *m _{e} c *

Identifying *E* with *p *_{5 }^{e m} *c*, we can now solve cubic equation (1) for *m*. One root is *m* = *m _{e }*; electron mass is determined

Let us now turn to the quarks. Their masses *m _{q}* are found from Eq. (1) of

*m _{q }c *= (

where

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

Here *B* = 1/3 is the baryon number, *Q _{q}* = electric charge, constant

*p *^{6 }_{t} ≡ p ^{6 }* _{σ}* = (

Putting the above calculated value of *m _{σ }*into this quadratic form, and using it in (5), we obtain