Einstein’s equation for a free particle in special relativity E = m'c² is 5-dimensional². Displays in 3-dimensions are easy enough in Cartesian space; and the mathematical tesseract is sometimes used [1] to simulate 4-dimensional structures. The last is generally unenlightening because unintuitive, and the tesseract is also short of one variable. The only easy solution is found by combining physical constraints within Cartesian geometry. This solution is here developed to display not only the conjugate variables of space-time, x,y,z,t, p_x, p_x, p_x and E and ω, (as in the free particle wave function below), but also m_o, phase and group velocities v_p, v_g, plus other variables that describe the dynamics of internal motion [2]-[5].

We begin by expanding the two scalers for rest mass m_o and energy E into two concentric, symmetric, 3-dimensional spheres, before cutting them through to a single plane that contains the two concentric radii in common with the 3-dimensional momentum p in the free particle (Figure 1). A legend for the symbols used is given in Appendix1.

The two scalers are thus represented by concentric circles whose surfaces contact the vector p, at both ends. The radii at the points of contact form a Pythagorean triangle with right angle at the point of contact between tangential p and m_oc. It is convenient to choose the orientation p = p_x as indicated in the figure. The triangle is a 2-dimensional cut from the spheres into the triangular plane bounded by the lines p_x −E/c− m_oc.The construction is cylindrically symmetric as indicated, and the rest mass and energy can be treated as vectors m_oc and E within the planar physical constraint.

Figure 1. The momentum vector p is shown displaced from its Cartesian coordinate axes (right). The vector is tangential to a sphere S m o that represents the scalar mass m_oc, which touches the vector at its base. At its point of contact, p is perpendicular to the radius m_oc.Similarly, a concentric spherical surface S_E represents the scalar energy E/c. This contacts the other end of p. Physics constrains the three sides of the right-angled triangle to co-planarity.

Figure 2. Band structures are constructed by physical constraints: firstly as p → 0, a tangential and horizontal p_y band at the intersection of a vertical m_o and nearly parallel E; secondly asp,E ≫ m_o, almost vertical tangential bands; and thirdly intermediate solutions derived as in Figure 3.

Figure 3. Construction of bands at various tangents on m_oc that plot the constrained intersections of p with E/c as the latter changes with increasing concentric radius. Notice that the 3-D construction (including spheres expanded from the m_o scaler and various E scalers) is reduced (by physical constraint due to the relativity equation) to a plane dictated by the vector p. This is the planar section displayed by broad lines in the figure.

Energy bands occur on the locus of intersections of p with E, consistent with varying tangents at m_oc (Figure 2, Figure 3). The bands are cylindrically symmetric, with particle bands mirrored by anti-particle bands where m_oc, E/c and p are all negative in dispersion dynamics [4]. The particle to anti-particle band gap is, obviously, 2m_oc. These figures describe, simply and intuitively, the display of 5-D structures onto 2-D displays. Next, we extend their application by simplifying to variables in dispersion dynamics with natural units.

To define the point of our discussion, it is necessary to outline the perspective of dispersion dynamics. This transforms the variables we have been using to their equivalent wave properties in order to associate them with their origins: in emission, in their resonant interactions, in their measurement, and in their quantum values. The free-particle wave-function in Figure 4 serves not only the classical optics of Huygens, Fresnel and Fraunhoffer; and furthermore the modern, quantized waves of Schrödinger and de Broglie; but the wave also explains Heisenberg’s uncertainty with advanced precision.

Figure 4. The electromagnetic quantum due to dipole emission is derived from Maxwell’s equations. The electric vector is shown in the vertical plane with the magnetic vector horizontal out of phase π/2. The phase velocity of a free photon is equal to its group velocity.

The wave group is the product of the normal distribution with Euler’s wave:

where X = i ( ω ¯ t − k ¯ x ) when ω ¯ coincides with peak ωat x = 0;

while k ¯ coincides with peak k at t = 0.

Then referring to Figure 4, it is obvious that when x = 0:

i.e. the group width at height φ_max/e. The transform of a Gaussian is Gaussian. The Fourier transform of Equation (1) is written:

This clear, wave result is 16x greater than Heisenberg’s ‘limits’, whatever they may be.

In optics, the wave is real for both the electric field and magnetic field, while their phase relationship is expressed by making the magnetic field imaginary in Euler’s wave formulation. The wave also applies to electron optics when allowance is made for mass, charge and internal motion that replace polarization in massless light optics. The finite mass of the electron results in complicated internal motion [2]. By differentiating Einstein’s formula in special relativity, the phase velocity in a massive free particle is shown and measured to be faster than the speed of light within the wave group. In free space typically, v_g < c and v_p > c (Figure 5) [4]: This is demonstrated in 6 quick steps:

Figure 5. Functions of relativistic free particles. The velocities in massless light is represented by the dashed diagonal through the origin; in free electrons the internal phase velocity is faster than light and v_p∙v_g = c².

1) In wave mechanics, phase velocity is the product of wavelength and frequency:

2) The phase velocity is proportional to λ. Proceeding from Newton to relativity to quanta as below:

3) Differentiate relativity: ω 2 = k 2 + m o 2 in natural units to find v_g∙v_p = 1:

where v_g is the beat velocity in the normal distribution, Equation (3).

4) Semi-Classically, p → 0:

5) Substituting the variable E in relativity for the classical constantE_o:

6) The group velocity v_g is proportional to λ⁻¹ cf. Equation (7)!

The five variables in special relativity, one vector and two scalers, are recognized as co-planar in a Pythagorean triangle. This fact transforms the scalers to vector look-alikes in many cases that are represented in 2-dimensional drawings. These are easily recognized by perspectives on Cartesian axes. The variables are conjugated, as in wave equations and in quantum mechanics: energy with time, and momentum with space, velocity and mass. Further co-planarities are discovered in time-velocity-space properties, and in dispersion dynamics with internal motion. At least 26 dimensions are translated to 2-dimensional drawings by these procedures.

Generally, multi-dimensional structures in Hilbert space can often be represented in 2-dimensional space when they are subject to physical constraints. The structures and dynamics are readily understood by a common sense of Cartesian space with time as a static variable. The drawings provide not only proper understandings of temporal change and energy constraints, but also representations of the dynamics of internal motion and of the de Broglie hypothesis in modern physics [3]: Planck’s energy depends on phase velocity; de Broglie’s momentum on group velocity. The number of possible applications, for 2-dimensional reductions of other physical constraints onto multi-variate dimensions, is unlimited. The mathematics of dimensionality is more informative when combined with physical constraints.

¹Typically 10 keV - 2 MeV.

²Rest mass m_o is constant in Special Relativity but not necessarily in General Relativity.m_o is normal to the momentum p in the Pythagorean triangle. In General Relativity, space is locally Euclidean as it is generally in Special Relativity.

³As in chamber photographs of particle-antiparticle creation.