Introduction

            Preceding Units 1 thru 4 on mechanics and wave resonance are prerequisites to modern physics of spacetime relativity and quantum wave mechanics introduced here in Unit 8. A wave based geometric approach helps to understand special relativity (SR) and quantum mechanics (QM) while it shows that these two pillars of modern physics are actually belong to the same subject!

            SR and QM have been treated in separate texts as different and even inimical subjects. (SR is most often found in E&M texts.) While advanced quantum field theory treatments do integrate special relativity they do it in a mathematical way that lacks lucidity and physical intuition. The present development seeks to improve the situation by appealing to the detailed geometry of wave interference.

            Separated introductions to SR and QM lead to misconceptions for professional physicists as well as for their students. In spite of its simple algebra, SR is also regarded as mysterious. Student comments for SR and QM courses are typically, “Well, I didn’t understand it, but neither did the prof!” Comments on a QM and SR derivation by an editor of the Journal of Modern Optics in 2003 illustrate the problem:

“Even Schroedinger probably never claimed to have a derivation, and we certainly don't tell our students that we have one. (A) Hand-waving, inspired guess is more like it.”

 

A key problem has been a failure to clarify wave mechanics. Consider the editor’s next comment.

 

“It is quite arbitrary how one defines envelope and carrier parts of a wave. Usually this is done only when all frequencies are nearby and all k-vectors are nearby. Then something like the analytic signal formulation can be used to arrive at unique but still arbitrary definition.”

 

This statement exposes a pernicious blind spot in conventional wave analysis. Its resolution uses Unit 4 expo-cosine relations in (4.3.30) to clearly separate a wave envelope from its “carrier” phase.

     (4.3.30) _{repeat a}                      (4.3.30) _{repeat b}   

These are true regardless of how “nearby” are arguments a=(k_ax-ù_at) or b=(k_bx-ù_bt) or their constituent frequency-time ù_a,bt and wavevector-space k_a,bx terms. Identities (4.3.30) separate a wave’s modulus or group envelope embodied by the cosine or sine factor that defines the outside envelope or “skin” of a wave sketched in the figure below. The modulus is the factor that remains in the expression ø*ø for intensity while the phase part e^i(a+b)/2 of ø cancels the e^-i(a+b)/2 of ø* leaving real intensity |ø|² or MOD |ø|.

Envelope is thus distinguished from a wave’s argument or overall phase held by exponential factor e^i(a+b)/2.

The latter governs real (Re ø) and imaginary (Im ø) “carrier”parts that are the inside “guts” of the wave shown in the sketches below. (One may imagine a boa constrictor that has swallowed live prey.)

The speed of the  wave factor is called group velocity. This external “skin” of the wave is the only part visible to probability or intensity measurements of ø* ø. Meanwhile the speed of exponential phase factor inside the envelope is called mean phase velocity or just plain phase velocity. Internal phase “guts” may oscillate very rapidly and be difficult or impossible to measure directly.

 

Review and plan of attack: Relativity of pairs

            Our plan of attack in Unit 3 for relativity and quantum theory has similar philosophy to that of Unit 1 for classical Newtonian mechanics and Unit 4 for resonance. The idea is to develop the axioms, rules, or laws of physics using relativity of elementary pairs. It is a kind of Occam-razor philosophy.

            In Unit 1 we began with collisions between a pair of cars or a pair of bouncing super-balls and developed the rules of classical mechanics. In Unit 4 we used a pair of coupled pendulums to establish the rules of resonant energy transfer. Here in Unit 8 we use a pair of light waves to find the rules of relativity and quantum mechanics. Geometry is a key part of this analysis as before.

            At first, the approach seems almost childish in simplicity. Who hasn’t seen (or been) a child who puts two beetles together to see if they will fight (or whatever)? Relation between pairs is something that first comes to mind when we see new things. Unfortunately, human egocentricity steers us toward a more complicated pair, a single thing and you. That’s the adult analytic approach, isn’t it?

            Conventional “adult” mechanics books begin with axioms for a single mass or “particle” acted on by “outside forces” (presumably you) to move according to Newton laws 1 thru 3. Detailed treatment of all these laws at the start is usually not a favorite pastime for either instructor or student. The algebra or calculus is tedious and it is difficult at first to arrange neat compelling and demonstrative experiments.

            In contrast, Unit 1 starts off with just a single axiom (Newton law-1 of momentum conservation) for particle-pair mechanics and derives collision kinematics by simple geometry using velocity-velocity V₁-V₂ plots. This clearly exposes Galilean relativity symmetry and the logic of m₁V₁+m₂V₂ conservation. An almost child-like geometric simplicity of particle-pair relativity economizes the logic.

            Pair relativity easily finds results for neat first-day experiments by ignoring (until later) the “you” and your “outside force.” Energy conservation is then proved using V₁-V₂ geometry and time symmetry. An autonomous one-pair-at-a-time mechanics leads later to multi-mass force and potential relations that also have an Occam-cut-to-the-chase logic that may be derived using plane geometry.

 

            In Unit 4, resonance mechanics is based on autonomous pairs of coupled pendulums described by phasor-pair plots of x₁-V₁ versus x₂-V₂ or complex Ø₁ versus Ø₂ plots. Again, the key idea is pair-relative. Energy transfer rate is the product (|Ø₁||Ø₁|sin ñ) of phasor amplitudes and sine of relative phase angle ñ. Autonomous one-phasor-pair-at-a-time mechanics have a direct cut-to-the-chase geometric logic that then leads to multi-phasor wave mechanics, Fourier spectral analysis, and dispersion relations. Very important is a logic for complex pairs of numbers and for U(2) pairs of complex phasors in the study of resonance.

 

            In Unit 8, the protagonists are a pair of colliding (counter-propagating) laser continuous waves (CW). Here we contemplate what happens at the micron level in the collision region of two bluish green beams of 600 THz dye laser, and arrive at an extraordinary claim. A pair of CW light beams can show the rules of classical mechanics, quantum mechanics, and special relativity.

            What a claim! These green beams can expose the fundamental logic of three hard physics courses CM, QM, and SR all for a fraction of the price one of them. And, like the late night TV ad, that’s not all! For just a little more you can get EM (electro-magnetism) thrown into the bargain.

            How can it be possible to conjure rules about mechanics of mass particles from light waves? A 1939 theory by Dirac and a 1950 experiment by Anderson provide some advance motivation for such thought-experimentation. What Dirac theorized and Anderson demonstrated are results of colliding two high-energy light beams. Crossing two gamma-(ã)-ray beams can convert light to matter plus anti-matter in ã+ã electron-positron pair production reactions. This primal chemistry is denoted as follows.

                                                           

The idea that 0.51MeV ã-rays can produce positronium pairs is taken for granted by many high-energy physicists, but it should be a mind-boggling Genesis-moment for any thinking student of physics.

 

            Now green 600 THz lasers do not produce positronium matter. The ã-frequency mc²/h=0.1ExaHz is about a million times more than our lasers can go. (That’s just as well given current world politics!) But, any two lasers can produce something that obeys symmetry conditions for matter waves, and it is those symmetry “laws” and geometry that underlie all our mechanics whether classical, quantum, relativistic, or electromagnetic. CW laser beams may not be matter but they do expose some of its kinematics.

            Ideal CW-laser pairs help us derive fundamental relativistic or quantum concepts and formulas by ruler and compass in just a few steps. A number of concepts, quantities, and relations are exposed in Unit 3 by wave geometry. These include longitudinal Doppler shifts, Einstein-Lorentz-Minkowski frames, time dilation, length contraction, stellar aberration, transverse Doppler effects, mass-energy-momentum dispersion relations, Legendre-Lagrange-Hamilton-Poincare relativistic contact transformations, Compton recoil shifts, Compton scattering, polarization and spin transformation, acceleration by frequency chirps, Einstein wave-elevators, and quantum count-rate covariance.

            According to a historical footnote given to me by Dudley Herschbach, Einstein became fascinated with ruler and compass geometry when he was just five years old. We can only guess the age Euclid was when he first picked up a Babylonian compass. In any case, a ruler and compass is child’s play, first and foremost, and therein lies a certain pedagogical power.

            We cannot know if either of these gentlemen would welcome a geometric approach to relativity and quantum theory. I would like to think so, but it’s possible they might have taken a Bourbakian view and found all these pictures to be just so childish. If so then it’s their loss and our gain!

 

 

Chapter 1 Continuous Wave (CW) vs. Pulse Wave (PW) functions

            The standard units of time t and space x are seconds and meters. Pure waves are labeled by inverse units that count waves per-time or frequency í, which is per-second or Hertz (1Hz=1 s^-1) and waves per-meter that is called wavenumber ê  whose old units were Kaiser (1 K=1 cm^-1=100 m^-1). Inverting back gives the period ô=1/í  or time for one wave and wavelength ë=1/ê  or the space occupied by one wave.

Physicists like angular or radian quantities of radian-per-second or angular frequency ù=2ðí and radian-per-meter or wavevector k=2ðê in plane continuous wave (CW) functions ø.

.                               (1.1a)

Sine or cosine are circular functions of wave phase (kx-ù t) given in radians and defined here.

                                            (1.1b)                                                         (1.1c)

They relate time ô and space ë parameters to per-time ù or í and per-space k or ê wave parameters.

Phase velocity for 1-CW

            Spacetime plots of the real field for one CW laser light are shown in Fig. 1.1. The left-to-right moving wave in Fig. 1.1(a) has a positive wavevector k while k is negative for right-to-left moving wave in Fig. 1.1(b). Light and dark lines mark time paths of crests, zeros, and troughs of . A zero-phase line (where kx-ù t is zero) or crest line has slope c=V_phase.

                                                                                    (1.1d)

Each white line in Fig. 1.1 has a phase is an odd multiple (N=1,3,…) of π/2 and marks a ë/2-interval.

                                                           

            Slope or phase velocity V_phase of all lightwave phase line is a universal constant c=299,792,548m/s. (Note tribute to Ken Evenson’s c-measurement in Unit 4.) Velocity is a ratio of space to time (x/t) or a ratio of per-time to per-space (í/ê) or (ù /k), or a product of per-time and space (íë)=1/(ôê).

            The standard wave quantities of (1.1) are labeled for a long wavelength example (infrared light) in the lower part of Fig. 1.1. Note that the wave precedes the wave. A simple mnemonic is helpful, “Imagination precedes reality by one quarter.” and applies to combined waves, too.

Axioms for light: 2-CW vs. 2-PW

Beginning relativity courses paraphrase Einstein’s light speed axiom as in Fig. 1.2a, “Speed of a lightning flash is c according to passengers of any train,” or simply, “Pulse wave (PW) speed c is invariant.” For critically thinking students, that is a show-stopper. It boggles the mind that something of finite speed cannot ever be caught up to, indeed, cannot even begin to be caught.[i]

 

Fig. 1.1 Phasor and spacetime plots of moving CW laser waves. (a) Left-to-right. (b) Right-to-left.

            Occam’s razor can dissect the c-axiom into a less mind-boggling form. As Evenson viewed a frequency chain of multiple “colors” of continuous wave (CW) laser beams, he assumed that, “All colors have speed c.”  Had Einstein imagined trains viewing a 600THz (green) laser as in Fig. 1.2b, his c-axiom might be, “CW speed is c according to passengers of any train while frequency and wavelength vary by a Doppler effect that depends on velocity of the train,” or more simply, “All colors go c.”

A CW spectral component of a PW has a color variation with observer speed that a “white” PW does not. A colored wave (CW) will blue-shift if you approach its source or a red-shift if you run away from it. Doppler’s theory of acoustical wave frequency shift existed 200 years before radar, masers, and lasers showed the ultra-precise 1^st-order Doppler sensitivity of a coherent optical CW.

Also an optical Doppler shift depends on one relative velocity of source and observer while acoustical Doppler depends on three absolute (or three relative) velocities involving source, observer, and a “wind.”  This single-velocity simplicity of en vacuo optical Doppler shifts is crucial for relativity.

Consider a 600THz green wave from a 600THz source. One may ask, “Is it distinguishable from another 600THz green wave sent by a 599THz source approaching or a 601THz source departing at just the right speed? Or, could 600THz light, seen as we approach a fixed 599THz source, ever differ in speed from 600THz light seen as we depart a fixed 601THz source? How many kinds of 600THz light exist?”

Evenson’s axiom follows if one answers, “There is only one kind of each frequency (color) and only one speed independent of source or observer velocity.” An undesirable alternative is to have many different kinds of each color, corresponding to many ways to make each color by tuning source up (or down) while moving out (or in). (In fact, one color illuminating a gas, liquid, or solid may involve two or many varieties of mode dispersion with wave speeds ranging above or below c.) Evenson’s axiom demands that light in a vacuum be one speed for all frequency. In short, light is dispersion-free. 

If so, a PW must move rigidly at the speed c shared by its component CW colors. In this way one derives Einstein’s PW law as a theorem arising from Evenson’s CW axiom. Occam wins one! 

Astronomical view of CW axiom

It also relates to appearance of distant nebulae and the night sky. If any colors were even a fraction of a percent slower than other frequencies, they would show up thousands or millions of years later with less evolved images than neighboring colors. We might then enjoy a sky full of blurry colorful streaks but would lose the clarity of Hubble astronomical images of colliding galaxies billions of light years away.

Spectroscopic view of CW axiom

Astronomy is just one dependent of Evenson’s CW axiom. Spectroscopy is another. Laser atomic spectra are listed by frequency õ (s^-1) or period ô=1/õ (s) while early tables list atomic lines from gratings by wavenumber ê (m^-1) or wavelength ë=1/ê (m). The equivalence of time and space listings is a tacit

assumption in Evenson’s axiom. The axiom may be stated by the following summary of (1.1a-d).

                        c = õ ·ë  = ë/ô  = õ/ê   = 1/(ê ·ô) = c =299,792,548m/s                    (1.1)_summary

Fig. 1.2 Comparison of wave archetypes and axioms. (a) Pulse Wave (PW) peaks locate where a wave is. Their speed is c for all observers. (b) Continuous Wave (CW) zeros locate where it is not. Their speed is c for all colors (or observers.)

An atomic resonance is temporal and demands a precise frequency. Sub-nanometer atomic radii are thousands of times smaller than micron-sized wavelengths of optical transitions. Optical wavelength is not a key variable in atomic dipole approximations that ignore spatial dependence of light.

However, optical grating diffraction demands precise spatial fit of micron-sized wavelength to micron grating slits. Optical frequency is not a key variable for time independent Bragg or Fraunhofer laws. Spatial geometry of a spectrometer grating, cavity, or lattice directly measures wavelength ë, and then frequency õ  is determined indirectly from ë by axiom (1.1). That is valid if the light speed c = õ ·ë is invariant throughout the spectrum (and throughout the universe.)

A spectroscopist expects an atomic laser cavity resonating at a certain atomic spectral line in one rest frame to do so in all rest frames. Each ë or õ  value is a proper quantity to be stamped on the device and officially tabulated for its atoms. Passersby may see output õ  Doppler red shifted to rõ  or blue shifted to bõ. Nevertheless, all can agree that the device and its atoms are actually lit up and working!

Moreover, Evenson’s CW axiom demands that õ and ë must Doppler shift inversely one to the other so that the product õ ·ë is always a constant c=299,792,458 m·s^-1. The same applies to ô and ê for which ê ·ô=1/c. Also, there is an inverse relation that exists between Doppler blue and red shifts seen before and after passing a source. This is our second CW axiom. It involves time reversal symmetry.

Time reversal axiom

Atoms behave like tiny radio transmitters, or just as well, like receivers. Unlike macroscopic radios, atoms are time-reversible in detail since they have no resistors or similarly irreversible parts. Suppose an atom A broadcasting frequency õ_A resonates an approaching atom B tuned to receive a blue shifted frequency õ_B = bõ_A. If time runs backwards all velocity values change sign. Atom B becomes a transmitter of its tuned frequency õ_B = bõ_A that is departing from atom A who is a receiver tuned to its frequency õ_A =  (1/b)õ_B. Atom A sees õ_A red-shifted from B’s frequency õ_B by an inverse factor r=1/b.

                                                             b=1/r                                                  (1.2)

Phase invariance axioms viewed in a classical way

Optical CW axioms may be based on deeper phase invariance principles. Elementary CW function Ø=A exp i(k·x-ù·t) or its real part Re Ø=A cos(k·x-ù·t) has a phase angle Ö=(k·x-ù·t) that is regarded as an invariant or proper quantity. Our rationale is that each space-time point of the wave has a phase clock or phasor (Re Ø, Im Ø) turning at angular frequency ù=2ð·õ. Each phasor reading Ö could be stamped or “officially” tabulated. All observers should agree on Ö even if Doppler shifts change frequency ù=2ð·õ and wavevector k = 2ðê to new values (ù′,k′) or if space x and time t also transform to x′,t′.

                                     k·x-ù ·t = Ö = k′·x′-ù′·t′                                                                                                                 (1.3)

(Lorentz-Einstein transformations for both space-time x,t to x′,t′ and inverse space-time (ù,k) to (ù′,k′) are derived in Ch. 2 using CW axioms (1.1) and (1.2) in a few algebraic or ruler-and-compass steps.)

Historically, invariance (1.3) relates to classical Legendre contact transforms of Lagrangian L to energy E or Hamiltonian H. Differential Ldt is Poincare’s action invariant dS or phase dÖ with an h factor.

              (1.4a)                                                 (1.4b)

Connecting (1.3) to (1.4b) requires quantum scaling relations p=hk of DeBroglie and E=hù of Planck. Ch. 3 shows how such relations arise from CW axioms (1.1-2). Exact relativistic quantum and classical mechanical relations are found in a few algebraic[ii] or ruler-and-compass steps. Elegant wave-geometric[iii] interpretations of momentum, mass, energy, and Poincare’s invariant are exposed in Ch. 4 and Ch. 5.[iv]

We surmised that Einstein might have liked geometric derivations since a compass first caught his theoretical attention at an age of five.[v] Perhaps, it might also appeal to Poincare who also discovered relativity around the time of Einstein’s 1905 annus mirabilis. Poincare phase invariance (1.3) underlies both CW lightspeed axiom (1.1) and time reversal axiom (1.2). Consider the Ö=0 point.

                                                k·x-ù ·t = 0                                                                         (1.5a)

Solving gives phase velocity x/t (meters-per-second) equal by (1.1) to õ/ê (per second)-per-(per meter).

                                               (1.5b)

Doppler shift leaves phase velocity invariant. Phase Ö=(k·x-ù·t) itself is invariant to time reversal ( and ) and that supports (1.2), the inverse-Doppler relation b=1/r.

            We find relativistic and quantum derivations based on classical mechanical laws to be clumsy at best and wrong-way-to at worst. Simple wave interference with axioms (1.1-2) can unite relativity and quantum theory. At the wave-phasor or “gauge” level, Nature may be seen as a big wave trick!

Comparing pulsed and continuous wave trains

            It is instructive to contrast two opposite wave archetypes, the Pulse Wave (PW) train sketched in Fig. 1.2a and the Continuous Wave (CW) train sketched in Fig. 1.2b. A CW is the more elementary theoretical entity, indeed the most elementary entity in classical optics since it has just one value of angular frequency ù=2ð·õ, one value of wavevector k = 2ðê, and one amplitude A.

                                                                                           (1.6)

The real part is the cosine waveshown in Fig. 1.2(b). Acronym CW fits cosine wave, as well. If frequency õ is in the visible 400-750THz range, then CW could also stand for colored wave.

In contrast, the PW is a less elementary wave function and contains N harmonic terms of CW functions where bandwidth N is as large as possible. Fig. 1.3 shows an example with N=12.

                     (1.7)

An infinite-N PW is a train of Dirac ä(x-a)-functions each separated by fundamental wavelength ë=2ð/k. The ä -spikes march in lockstep at light speed c=ù/k because of Evenson’s CW axiom (1.1).

                                                   

Delta functions have infinite frequency bandwidth and are thus impractical. Realistic PW trains apply cutoff or tapering amplitudes a_n to the harmonic so as to restrict frequency to a finite bandwidth Ä.

            (1.8)

One choice is the Gaussian taper that gives Gaussian PW functions .

            PW functions (1.8) involve an unlimited number of amplitude parameters a_n in addition to fundamental frequency ù, while a CW function has a single amplitude parameter A. Thus, theory based on CW properties is closer to an Occam ideal for axiomatic simplicity than one based on PW.

CW squares vs. PW diamonds in space-time plots

However, with regard to counter-propagating or colliding beams the PW appear in Fig. 1.4a to have simpler properties than CW in Fig. 1.4b. PW have a simple classical Boolean OFF (0) over most of space-time with an occasional ON (1) at a sharp pulse. On the other hand CW range gradually between +1 and –1 over most of space-time, but have sharp zeros (0) in between crest and trough. (A PW is designed to make precise peaks that show where it is. A CW naturally has precise zeros that show where it is not.)

Interference between two colliding CW makes a square (P, G)-zero-grid that is subtler and sharper than the left-right moving (L,R)-peak-diamond grid made by two colliding PW. One should understand how this wave interference works to make these two archetypical types of wave space-time geometry.

Interference of colliding PW in Fig. 1.4a or Fig. 1.5b is wysiwye (What you see is what you expect.). The pattern of interference for the sum of colliding CW in Fig. 1.4b and Fig. 1.5a is subtler. PW paths in space-time (x,ct) resemble baseline diamonds in Fig. 1.5b like paths in the American baseball sport. Meanwhile, CW zeros form Cartesian space-time squares in Fig. 1.5a with horizontal x-axial fixed time-lines (ct=…1,2…) and vertical temporal ct-axial lines of fixed location (x=…1,2…).

PW peak diamonds seem simple but hide intricate networks of zeros near each peak. CW squares make truly simple and precise lattices of standing wave zeros of given by (1.9), which is just a factored sum of two equal-but-opposite colliding CW. Note that the group envelope factoris zero on lines (kx/ð+1/2=…0,1,2…) parallel to the ct-axis. The phase factor has a zero real part on lines of simultaneous time (ct/ð+1/2=…0,1,2…) parallel to the x-axis. (At lattice corners, both factors are zero.)

                                (1.9)

CW wave-zeros vs. PW pulse paths

            Phase and group wave zeros of 2-CW interference define a space-time wave-zero (P,G)-coordinate grid for light waves in Fig. 1.5 and more general waves in Fig. 1.6. Vector P points along a phase zero path and vector G points along a group zero path. They complement PW pulse peak or peak-path (L,R)-grid based on vector L that points along a left moving peak path and a vector R that points along a right moving peak path. The half-sum-and-difference relation of (P,G) to (L,R) is as follows.

                                                            P =(L + R)                                                    (1.10a)

                                                            G =(L - R)                                                    (1.10b)

The peak-path vectors {L,R) are then given by sum-and-difference of wave zero vectors {P,G).

                                                            L = (P + G)                                                     (1.10c)

                                                            R = (P − G)                                                     (1.10d)

Sum-and-differences are due to phase sum-and-differences. (Recall discussion of (4.8.21) in Unit 4.)

 

Comparing wave-like vs particle-like behavior

Relations (1.10) highlight wave-particle duality. First, Newton saw light as particle-like. Then Young and Maxwell showed its wave-like nature. Finally, Planck, Einstein, and Compton found particle-like behavior of “photon” quanta. The label “photon” is reserved for quantum field eigenstates having decidedly more complicated behavior than is shown in semi-classical wave plots in Fig. 1.6 or colliding light waves in Fig. 1.5. Still the diamond left-and-right moving PW (L,R)-peak paths in Fig. 1.5b might be thought of as paths of fictitious particles or “photon bunches” that are well localized in space-time as they move at ±c in either direction. Each PW laser “spits” pulses (patooey! patooey!…) at 600Thz.

Optical pulse peaks do move like particles in between the points where “collisions” occur (with very complicated wave interference). After that the “particles” seem to pass through each other or recoil elastically. Newton wrote about optical interference behavior as crazy “light having fits.”

            Square 2-CW (P,G) zero-paths in Fig. 1.5a are due to counter-propagating 600Thz CW waves interfering wherever they exist in space-time. The wave between the zeros is delocalized in space-time compared to the PW peaks but the square white zero-lines are extremely sharp as are vectors L=( ck, ù) and R=(-ck, ù) that determine motion of left and right CW component laser beams while vectors P=(0,ù) and G=(ù,0) determine the real wave-zero lattice of their 2-CW interfering sum.

It is important to note that these vectors, appropriately scaled, describe both time-vs-space (x,t)-plots and Fourier inverse per-time-space or reciprocal space-time plots of frequency-vs-wavevector (ù,k). A general example of this is derived and shown in a following Fig. 1.6 where the two kinds of plots may be superimposed. We will see that a (ù,ck)(ck,ù) switch or else an (x,ct)(ct,x) switch to the Newtonian format is needed in order to make a CW lattice and reciprocal PW lattice coincide and that entails a (P,G)(P,G) switch. This is indicated in Fig. 1.5a to the right of the square space-time lattice.

PW (L,R)-peak paths are “particle-like” and stand out in space-time for N-CW wave trains. Then interference “fits” between pulses die off (to make Newton comfortable again.) But, CW (P,G)-zero paths, in contrast, are “wave-like” with very sharp lines in space-time for maximally interfering 2-CW beats.

           

Fig. 1.5 Space-time grids (a) 2-CW standing-wave-zero squares. (b) 2-PW diamond pulse peak paths.

 

           

 

Wave-zero (WZ) and pulse-peak (PP) space-time coordinate grids

The following Fig. 1.6 and Fig. 1.7 compare and superimpose time-vs-space (x,t)-plots of group and phase waves, on one hand, with their inverse per-time-space or reciprocal space-time plots of frequency-vs-wavevector (ù,k), on the other, and thereby mesh (x,t)-plots with their Fourier transforms.

            The plots apply to all waves and not just to light. The example in Fig. 1.6 begins by picking four random numbers, say, 1,2,4, and 4 to insert into frequency-wavevector K₂= (ù₂,k₂)=(1,2) of a mythical source-2 and frequency-wavevector K₄= (ù₄,k₄)=(4,4) of another mythical source-4. Velocity c₂=ù₂ /k₂ =1/2 of source-2 and c₄=ù₄ /k₄ =1 of source-4 are unequal. For light waves in Fig. 1.7, c₂ equals c₄ as required by an important axiom discussed in the following Chapter 2.

Let the continuous waves (CW) from the two sources interfere in a 2-CW sum.

                                                                                                    (1.11a)

To solve for zeros of this sum we first factor it into a phase-wave e^ip and a group-wave cos g factor.

                                               (1.11b))

Phase factor e^ip uses the half-sum (ù,k)-vector K_phase=(K₄+K₂)/2 in its argument. Group factor cos g has the half-difference (ù,k)-vector K_group=(K₄−K₂)/2 in its argument.

             (1.11c)                                   (1.11d)

The (ù,k)-vectors K_n define paths and coordinate lattices for pulse peaks and wave zeros in Fig. 1.6a.

Real zeros (ReØ=0) have velocity V_phase on K_phase paths. Group zeros (|Ø|=0) move at V_group on K_group.

      (1.12a)                (1.12b)

Zeros of phase factor real part lie on phase-zero paths where angle p is N(odd)·ð/2.

.

Zeros of group amp-factor lie on group-zero or nodal paths where angle g is N(odd)·ð/2.

.

Both factors are zero at wave zero (WZ) lattice points (x,t). This defines the lattice vectors in Fig. 1.6a.

                                                                                                         (1.13a)

Solving gives spacetime (x,t) zero-path lattice that are white lines in Fig. 1.6a. Each lattice intersection point is an odd-integercombination of wave-vectors and .

                                 (1.13b)

So space-time lattice points reuse the base lattice vectors K_group and K_phase of reciprocal per-space-time!Scaling factor 2D/ð =/ð converts (per-time, per-space) vectors K_group or K_phase into (space, time) vectors or . (Plot units are set so 2D/ð =1 or D=ð/2. This works only if D is non-zero.)

            Fig. 1.6b is a lattice of source vectors K₂ and K₄ (the difference and sum of K_group and K_phase).

                  (1.14a)                  (1.14b)

Source-2 has phase speed c₂ on K₂ paths of slope c₂. Source-4 has speed c₄ on K₄ paths of slope c₄.

         (1.15a)                                              (1.15b)           

One may view the K₂ and K₄ paths from a classical or semi-classical viewpoint if pulse waves (PW) were wave packets (WP) that mimic particles. Newton took a hard-line view of nature and ascribed reality to “corpuscles” but viewed waves as illusory. He misunderstood light if it exhibited interference phenomena and complained that its particles or “corpuscles” were having “fits.”  Newtonian corpuscular views are parodied here by imagining that frequency  (or) is the rate at which source-2 (or 4) emits “corpuscles” of velocity c₂ (or c₄). Then the wavelengths (or) are just inter-particle spacing of K₂ (or K₄) lines in Fig. 1.6a. Since wavelength ë₂ (ë₄ ) separates K₂ (K₄) lattice lines in Fig. 1.6b, one can imagine them as “corpuscle paths.” The paths are diagonals of the K_group(K_phase) wave-zero lattice in time vs space (x,t) of Fig. 1.6a.

 

Fig. 1.6 “Mythical” sources and their wave coordinate lattices in (a) Spacetime and (b) Per-spacetime.

CW lattices of phase-zero and group-node paths intermesh with PW lattices of pulse, packet, or “particle” paths.

 

This development shows wave-particle, wave-pulse, and CW-PW duality in the cells of each CW-PW wave lattice. Each (K₂ ,K₄)-cell of a PW lattice has a CW vector 2P or 2G on each diagonal, and each (P,G)-cell of the CW lattice has a PW vector K₂ or K₄ on each diagonal. This is due to sum and difference relations (1.11d) or (1.14b) between (P,G)=(K_phase, K_group) and (K₂, K₄).

In order that space-time (x,t)-plots can be superimposed on frequency-wavevector (ù,k)-plots or (õ,ê)-plots, it is necessary to switch axes for one of them. The space-time t(x)-plots in Fig. 1.6a follow the convention adopted by most relativity literature for a vertical time ordinate (t-axis) and horizontal space abscissa (x-axis) that is quite the opposite of Newtonian calculus texts that plot x(t) horizontally. However, the frequency-wavevector k(ù)-plots in Fig. 1.6b switch axes from the usual ù(k) convention so that t(x) slope due to space-time velocity x/t or Äx/Ät (meter/second) in Fig. 1.6a matches that of equal per-time-per-space wave velocity ù/k or Äù/Äk (per-second/per-meter) in Fig. 1.6b.

Superimposing t(x)-plots onto k(ù)-plots also requires that the latter be rescaled by the scale factor ð /2D derived in (1.13b), but rescaling fails if cell-area determinant factor D is zero.

                                                    (1.16)

Co-propagating light beams K₂= (ù₂,k₂)=(2c,2) and K₄= (ù₄,k₄)=(4c,4) in Fig. 1.7b have D=0 since all K-vectors including K_phase=(ù_p,k_p)=(3c,3) and K_group= (ù_g,k_g)=(c,1) lie on one c-baseline of speed c that has unit slope (ù/ck=1) if we rescale (ù,k)-plots to (ù,ck) and (x,t)-plots to (x,ct).

            In summary, co-propagating light waves absolutely fail to make coordinate grids! However, counter-propagating (right-left) light waves are another “matter” altogether.

Fig. 1.7 Co-propagating laser beams produce a collapsed wave lattice since all parts have same speed c.

 

In Ch. 2 counter-propagating (right-left) light wave vectors (R,L)= (K₂,-K₄) are used to make CW bases (P= K_phase, G= K_group) with a non-zero value for area D =|GxP|. Opposing PW base vectors are sum and difference (R,L)=(P+G,P-G) of CW bases so a PW cell area |RxL| is twice that of CW cell |GxP|.

                                    |RxL|= |(P+G) x (P-G)|=2|GxP|                                              (6.22)

Wave cell areas due to colliding CW are key geometric invariants for relativity and quantum mechanics as will be shown. As waves pass us bye and bye, then will the spray go high and fly.

 

 

 

Chapter 2 When Light Waves Collide: Relativity of waves in spacetime

            Let us represent counter-propagating frequency-ù laser beams by a baseball diamond in Fig. 2.1a spanned by CW vectors for waves moving left-to-right (R on 1^st base) and right-to-left (L on 3^rd base).

                        R=K₁=(ck₁,ù₁)= ù(1,1)    (2.1a)             L=K₃=(ck₃,ù₃)= ù(-1,1)              (2.1b)

Fig. 2.1 uses conventional (ck,ù)-plots for per-space-time and (x,ct)-plots for space-time. Both beams have frequency õ=ù/2ð=600THz(green), the unit scale for ù and ck axes. For the L-beam, ck equals -ù.

Phase vector P=K_phase and group vector G=K_group are also plotted in (ù,ck)-space in Fig. 2.1b.

             (2.2a)                        (2.2b)

Phase and group velocities of counter-propagating light waves may vary from c. These surely do!

        (2.3a)                                  (2.3b) 

The extreme speeds account for the square (Cartesian) wave-zero (WZ) coordinates plotted in Fig. 2.1c. As noted for Fig. 1.5, the group zeros or wave nodes are stationary and parallel to the time ct-axes, while the real-zeros of the phase wave are parallel to the space x-axes. The latter instantly appear and disappear periodically with infinite speed (2.3a) while standing wave nodes have zero speed (2.3b).

            Fig. 2.1d shows 2-way pulse wave (2-PW) trains for comparison with the 2-CW WZ grid in Fig. 2.1c. As noted for Fig. 1.3, a PW function is an N-CW combination that suppresses its amplitude through destructive interference between pulse peaks that owe their enhancement to constructive interference.

Colliding PW’s show no mutual interference in destroyed regions. Generally one PW is alone on its diamond path going +c parallel to 1^st baseline R=K₁ or going –c parallel to 3^rd baseline L=K₃.

          (2.4a)                                                       (2.4b) 

But wherever two PW peaks collide, each of the CW pairs will be seen trying to form a square coordinate grid that 2-CW zeros would make by themselves. This begins to explain the tiny square “bases” seen at the corners of the space-time “baseball diamonds” in Fig. 2.1d simulation.

CW-Doppler derivation of relativity

Evenson’s CW razor-cut of Einstein’s PW axiom improves relativity development. However, quantifying Einstein’s popular (and still common) derivation is difficult as is a step-by-step count for the CW derivation that follows. Let us just say that several steps are reduced to fewer and clearer steps. Most important is the wave-natural insight that is gained and the wave mechanics that follows.

Fig. 2.1. Laser lab view of 600Thz CW and PW light waves in per-space-time (a-b) and space-time (c-d).

 

In fact, we could claim that a CW derivation takes zero steps. It is already done by a 2-CW wave pattern in Fig. 2.2c that automatically produces an Einstein-Lorentz-Minkowski[vi] grid of space-time coordinates. Still we need logical steps drawn in Fig. 2.2a-b that redo the Cartesian grid in Fig. 2.1 just by Doppler shifting each baseline one octave according to c-axiom (1.1) (“Stay on baselines!”) and t-reversal axiom (1.2) (“If 1^st base increases by one octave, 3^rd base decreases by the same.”)

So Fig. 2.2 is just Fig. 2.1 seen by atoms going right-to-left fast enough to double both frequency õ=ù/2ð and wavevector ck of the vector R on 1^st base (while halving vector L on 3^rd base to obey (1.2).)

                        R=K₁=( ck′₁,ù′₁)= ù(2,2)    (2.5a)                      L=K₃=( ck′₃,ù′₃ )= ù(-1/2, 1/2)  (2.5b)

The atom sees head-on R-beam blue-shift to frequency õ₁′=2õ=ù₁′/2ð=1200THz(UV) by doubling green õ₁=ù/2ð=õ₃=600THz. It also sees the tail-on L-beam red-shift by half to õ₃′=õ/2=ù₃′/2ð=300THz(IR).

The phase vector K_phase and group vector K_group are plotted in (ck′,ù′)-space in Fig. 2.2b.

             (2.6a)          (2.6b)

Phase velocity is the inverse of group velocity in units of c, and V′_group is minus the atoms’ velocity!

           (2.7a)              (2.7b) 

Velocity u=V′_group =3c/5 is the atoms’ view for a lab speed of zero had by laser standing nodes. It is the speed of the lasers’ group nodes (and its supporting lab bench!) relative to the atoms. Phase velocity V′_phase =5c/3 is the atoms’ view for a lab speed of infinity had by lasers’ real wave zeros. The x-zero lines are simultaneous in the laser lab but not so in the atom-frame. x-lines tip toward ct-lines in Fig. 2.2c.

            Eqs. (2.5-7) use a Doppler blue-shift factor b=2. If each “2” is replaced by “b” then Eq. (2.7b) yields a relation for the laser velocity u=V′_group relative to atoms in terms of their blue-shift b.

                                                                                                   (2.8a)

Inverting this gives the standard relativistic Doppler b vs. u/c relations.                            (2.8b)

First things first

It is remarkable that most treatments of relativity first derive second order effects, time dilation and length contraction. Doppler and asimultaneity shifts are first order in u but treated second. Setting 2=b in (2.6) using (2.8) gives vectors and  with dilation factor  and asimultaneity factor a=u·d/c. (So a and d may be derived first here, too, but in a wavelike way.)

(2.9a)       (2.9b)

K-vector components d and a (in ù units) are Lorentz-Einstein (LE) matrix coefficients relating atom- values (ck′,ù′) or x′,t′ to lab-values (ck,ù) or x,t based on lab unit vectors = and = in (2.2).

            The new K-vectors define the new coordinate grid of white-line wave-zero paths in space-time of Fig. 2.2c and, perhaps more importantly, the new (ck′,ù′) coordinates in per-space time of Fig. 2.2b.

Fig. 2.2 Atom view of 600Thz CW and PW light waves in per-spacetime (a-b) and space-time (c-d) boosted to u=3c/5.

            Einstein’s PW axiom “PW speed c is invariant,” might give the impression that pulses themselves are invariant, but finite-Ä pulses in Fig. 2.2d clearly deform. Pulse speed is invariant but each CW square in Fig. 2.3a deforms into a Minkowski-like rhombus in Fig. 2.3b simply due to Doppler detuning beats.

Lorentz-Einstein transformations

The Lorentz[vii]-Einstein[viii] per-spacetime and spacetime transformations follow from K-vectors (2.9).  

 (2.10a)         (2.10b)

Wave K-vectors are bases for space-time and per-space-time. One symmetric LE matrix, invariant to axis-switch (ù,ck)(ck,ù), applies to both. Conventional ù−ordinate vs. ck-abscissa per-space-time and ct-ordinate vs. x-abscissa space-time plots are used in Fig. 2.2 where =P=K_phase and =G=K_group vectors serve as x-space and ct-time bases, respectively, and then also serve as ù−and-ck-bases.

            The left and right pulse wave (PW) vectors L and R in per-space-time Fig. 2.2a also define left and right PW paths in space-time Fig. 2.2d. This holds in either convention because L and R lie on 45° reflection planes that are eigenvectors of an axis-switch (ù,ck)(ck,ù) with eigenvalues +1 and –1 while half-sum-and-difference vectors and simply switch (PG).

Geometry of Lorentz-Einstein contraction-dilation

            Fig. 2.3 compares wave path space-time coordinate lines for the laser lab in top figure (a) and for the atom going right-to-left at speed u=3c/5 in bottom figure (b).

The fast wave-phase zeros define the space-x axis and gridlines in either view where they go at a speed of 5c/3 in the atom view and at infinite speed ∞ in the lab view.

The slow wave-group zeros define the time-ct axis and gridlines in either view where they go at a speed of 3c/5 in the atom view and at zero speed 0 in the lab view.

The spatial separation of the slow wave-group zeros in Fig. 2.2c is 4/5 of the original 1/4ìm shown separating the stationary wave zeros in Fig. 2.1c or Fig. 1.5a. That is the Lorentz contraction factor

.

            The inverse time dilation factor d=5/4 is the vertical height of the new “pitcher’s mound” P in Fig. 2.2a that was originally of unit height in Fig. 2.1a. In space-time diamond of Fig. 1.5b the pitcher’s mound is 5/6 fs from origin or “home plate” and that dilates by factor d=5/4 to 25/24 fs in Fig. 2.2c.

            Detailed geometry of relativistic quantities is given in later figures. (Fig. 5.1, 5.4, and 5.5.)

 

Fig. 2.3 Lasers make Cartesian (x,ct)-wave frame for themselves and Minkowski (x′,ct′)-frame for atom.

 

            Should relativity continue to be taught by imagining monstrous frames, mirrors, and smoke to trace bouncing “photon bunches” using clanking clocks carefully synchronized by Swiss gnomes?

      Perhaps, that works as a humorous historical aside but current GPS systems and ultra high precision pioneered by Evenson and coworkers begs our attention and critical thought. Now as his students are achieving better than 17-figure time and frequency measurements, it is time for theoretical pedagogy to sharpen Occam’s razor accordingly. And, if there is history to review, it is first of Galileo and Euclid.

 

 

Chapter 3. Invariance and Relative Phase: Galileo’s revenge

            Einstein relativity shows Galilean relativity, based on simple velocity sums and differences, to be a 400 year-old approximation that fails utterly at high speeds. Einstein also dethrones infinite velocity that is the one invariant velocity shared by Galilean observers regardless of their (finite) velocity. In its place reigns a finite velocity limit c=299,792,458ms^-1 that is now the Einstein-Maxwell-Evenson invariant speed.

            So it is remarkable that frequency sums and differences (1.10) simplify relativity by using Galilean-like rules for angular velocities of light phases . Frequency sums or differences  from interference terms like between wave pairs and  are relative frequencies (beat notes, overtones, etc.) subject only to simple addition and subtraction rules that are like Galileo’s rules for linear velocity. Simple angular phase principles deeply underlie modern physics, and so far there appears to be no c-like speed limit for an angular velocity ù.

            Phase principles have electromagnetic origins. Writing oscillatory wave functions using real and imaginary parts is used to study AC phenomena or harmonic oscillators in Unit 4. Real part q of oscillator amplitude q+ip= is its position q=Acosùt. Imaginary part p=Asinùt is oscillator velocity v =-Aù sinùt in units of angular frequency ù. Positive ù gives a clockwise rotation like that of classical phase space or analog clocks, so a minus sign in a conventionalphasor serves to remind us that wave frequency ù defines our clocks and wavevector k=ù/c defines our meter sticks. (Recall Fig. 1.10.5 and Fig. 4.2.1.)

            A plane wave of wavevector k in Fig. 3.1 is drawn as a phasor array, one  for each location x. A plane wave advances in time according to  at phase velocity V=ù/k. Similar convention and notation are used for light waves and for quantum matter waves, but only light waves have physical units, vector potential A and electric E-field, defining their real and imaginary parts. While classical laser wave phase is observable, only relative phase of a quantum wave ø appears to be so.

            The concept of relative phase (and frequency) arises in classical or quantum interference where a sum of two waves and may be represented at each position x by a vector sum of a phasor-A with a phasor-B as in Fig. 3.1a. (Fig. 3.1 has a sum of 12 phasors, one for each each x-point.)  The result is a clockwise race around a track between the faster one, say A-phase  of angular speed -ù_A, and the slower B-phase  of angular speed -ù_B as sketched in Fig. 3.1b.

            Galilean relativity of phase angular velocity holds if the phase wave is governed by linear equations of motion such as Maxwell’s equations. Very precise measurements of en vacuo light have verified this so far and Einstein relativity is a consequence. You might say this is Galieo’s revenge!

 

Fig. 3.1 Wave phasor addition. (a) Each phasor in a wave array is a sum (b) of two component phasors.

(c) In phasor-relative views either A or else B is fixed. An evolving sum-and-difference rectangle is inscribed in the (dashed) circle of the phasor moving relative to the fixed one.

           

Geometry of relative phase

When A passes B the sum is a maximum or beat that then subsides to a minimum or node when A is on the opposite side of the track from B. If amplitude magnitudes |A| and |B| are equal as they are in Fig. 3.1, then the wave node is a wave zero that defines one of the group G-lines in WZ coordinates of Fig. 1.4 through Fig. 2.2. The relative angular velocity (beat angular frequency) is the angular rate at which A passes B. A-B passings occur ä times (per sec.) where ä is Ä divided by track length 2ð.

                                                (3.1)

            If one could ride in an angular Galilean frame of phasor-B, then A would be seen passing at angular speed Ä with frequency ä. Suppose instead, one could ride at their average angular speed .

                                                                                                                       (3.2)

Then Galilean arithmetic (which lasers given no reason to doubt in these matters) implies that phasor A or B would each appear with a relative speed of plus-or-minus half their relative velocity.

                                                                                                                 (3.3)

A point of view relative to phasor B is shown by the first of Fig. 3.1c. A dashed circle represents moving phasor A with on one diagonal of an inscribed rectangle whose sides are the resultant sum  and difference . The other diagonal appears fixed. A companion figure has  appear fixed instead. Resultants in either figure begin and end on a dashed circle traced by the phasor that is moving relative to the other. A rectangle-in-circle is a key Euclidian element of wave physics and is a key feature of a later figure (Fig. 3.3) that shows the essence of wave interference geometry.

The half-sum and half-difference angles in Fig. 3.1b and frequencies (3.2) and (3.3) appear in the interference formulas (1.10) that lead to relativistic Lorentz-Einstein coordinate relations (2.10) and their WZ grid plots of Minkowski coordinates in Fig. 2.2c. One key is the arithimetic mean  of phases that gives the geometric mean of wave phasor amplitudes.  The other key is the difference mean  and that is the phase angle of a cross mean.

Euclidian means and rectangle-in-circle constructions underlie relativistic wave geometry as is shown below.  This geometry also leads to the geometry of contact transformations in classical mechanics that exposes relations between classical and quantum mechanics in Ch. 5. 

Geometry of Doppler factors

Any number N of transmitter-receivers (“observers” or “atoms” previously introduced) may each be assigned a positive number b₁₁, b₂₁, b₃₁, …that is its Doppler shift of a standard frequency ù₁ broadcast by atom-1 and then received as frequency ù_m1= b_m1 ù₁ by an atom-m. By definition a transmitter’s own shift is unity. (1= b₁₁) Also, coefficient b_m1 is independent of frequency since such geometric relations work as well on 1THz or 1Hz waves as both waves march in lockstep to the receiver by Evenson’s CW axiom (1.1). The production times of a single wavelength of the 1Hz-wave and 10¹² wavelengths of the 1THz wave must be the same (1sec.), and so must be reception time for the two waves since they arrive in lock step, even if ô is shortened geometrically by 1/ b_m1. Doppler is a geometric and multiplicative effect.

Fig. 3.2 Doppler shift b-matrix for a linear array of variously moving receiver-sources.

 

If atoms travel at constant speeds on a straight superhighway, then b_m1 in (2.8a) tells what is the relative velocity u_m1 of the m^th atomic receiver relative to the number-1 transmitter.

                                                                                                    (3.4)

The velocity u_m1 is positive if the m^th atom goes toward transmitter-1 and sees a blue (b_m1>1) shift, but if it moves away u_m1 is negative so it sees a red (b_m1<1) shift. Transmitter-1 has no velocity relative to itself. (u₁₁=0) Infinite blue (or red) shift b_m1=∞ (or b_m1=0) gives u_m1=c (or u_m1=-c) and this defines the range of parameters. The b_m1 are constant until atom-m passes atom-1 so relative velocity flips sign (). Doppler shift then inverts () as is consistent with axiom (1.2).

            Suppose now b₁₂, b₂₂, b₃₂, …are Doppler shifts of frequency ù₂ transmitted by the second atom and received by the m^th atom as frequency ù_m2= b_m2 ù₂. (Any atom (say the n^th) may transmit, too.)

                                                   ù_mn= b_mn ù_n                                                    (3.5a)

Recipients don’t notice if atom-n just passes on whatever frequency ù_nm came from atom-m. If frequency ù_n in (3.5a) is ù_n1= b_n1 ù₁ that atom-n got from atom-1 then atom-m will not distinguish a direct ù_m1 from a perfect copy b_mn b_n1 ù₁ made by atom-n from atom-1 and then passed on to atom-m.

                                                ù_m1 = b_m1 ù₁= b_mn b_n1 ù₁                         (3.5b)

A multiplication rule results for Doppler factors and applies to light from atom-1 or any atom-p.

                                                ù_mp/ù_p= b_mp = b_mn b_np                                        (3.5c)

An inverse relation results from atom-p comparing its own light to that copied by atom-n.

                                                1= b_pp = b_pn b_np or:  b_pn =1/b_np                          (3.5d)

            Notice that copying or passing light means just that and does not include reflection or changing +k to –k or any other direction. This presents a problem for a receiver not in its transmitter’s (+k)-beam and certainly for atom-p receiving its own beam. The relations (3.5) depend only on relative velocities and not positions (apart from the problem that a receiver might be on the wrong side of a transmitter).

An obvious solution is to let the receiver overtake its transmitter or failing that delegate a slave transmitter or receiver on its right side. Fig. 3.2 shows N=5 receivers of a ù₃=600THz source whose various speeds produce a matrix of N(N-1)=20 Doppler shifted frequencies ù_mn and factors b_mn.

Doppler rapidity and Euclid means business

            Composition rules (3.5c) suggest defining Doppler factors b=e^ñ in terms of rapidity ñ=ln b.

                                    b_mp = b_mn b_np     implies: ñ_mp = ñ_mn +ñ_np  where:          (3.6)

Rapidity parameters ñ_mn mimic Galilean addition rules as do phase angles ö of wavefunctionss e^{i ö}. Both ñ and ö are the parameters that underlie relativity and quantum theory. In fact, by (3.4) rapidity ñ_mn approaches the relative velocity parameter u_mn /c between atom-m and atom-n for speeds much less than c. Rapidity is also convenient for astronomically large Doppler ratios b_ab since then the numerical value of ñ_ab =ln b_ab is much less than b_ab while u_mn /c approaches 1 in a way that is numerically inconvenient.

            At intermediate relativistic speeds the geometric aspects of Doppler factors provide a simple and revealing picture of the nature of wave-based mechanics. Pairs of counter moving continuous waves (CW) have mean values between a K-vector R=K₁=(ck₁,ù₁) going left-to-right and an L=K₃=(ck₃,ù₃) going right-to-left. A key quantity is the geometric mean ϖ of left and right frequencies.

                                                                                                                              (3.7)

In Fig. 3.2a frequency ù₁=1 or ù₃=4 is a blue (b=e^+ñ=2) or red (r=e^−ñ=1/2) shift of mean .

                                                            (3.8a)                                      (3.8b)

In units of 2ð ·300THz, frequency values ù₃=1 and ù₁=4 were used in Fig. 2.2. Their half-sum 5/2 is their arithmetic mean. That is the radius of the circle in Fig. 3.2b located a half-difference (3/2) from origin.

  (3.9a)                          (3.9b)

By (2.8) the difference-to-sum ratio is the group or mean frame velocity-to-c ratio u/c=3/5 for b=2.

              (3.9c)                                                                   (3.9d)

Fig. 3.3a Euclidian mean geometry for counter-moving waves of frequency 1 and 4. (300THz units).

Fig. 3.3b Geometry for the CW wave coordinate axes in Fig. 2.2.

            The geometric mean () in units of 2ð ·300THz is the initial 600THz green laser lab frequency used in Fig. 2.1. Diamond grid sections from Fig. 2.2b are redrawn in Fig. 3.3b to connect with the geometry of the Euclidian rectangle-in-circle elements of interfering-phasor addition in Fig. 3.1c.

Various observers see the single continuous wave frequencies ù₁ or ù₃ shifted to ù′₁=e^+ñù₁ and ù′₃=e^−ñù₃, that is, to values between zero and infinity. But, because factor e^−ñ cancels e^+ñ, all will agree on the 2-CW mean value ϖ =[ù₁ù₃]^1/2=[ù′₁ù′₃]^1/2. A 2-CW function has an invariant ϖ of its rest frame (Recall Fig. 2.2c) seen at velocity u=c(ù₁-ù₃)/( ù₁+ù₃). A single CW has no rest frame or frequency since all observers see it going c as in Fig. 1.1. To make a home frame, a single CW must marry another one!

Invariance of proper time (age) and frequency (rate of aging)   

            Space, time, and frequency may seem to have an out-of-control fluidity in a wavy world of relativism, so it is all the more important to focus on relativistic invariants. Such quantities make ethereal light billions of times more precise than any rusty old meter bar or clanking cuckoo clock.

            It is because of the time-reversal (1.2) and Evenson axiom (1.1) that product ù₁ù₃=ϖ² is invariant to inverse blue-and-red Doppler shifts b=e^+ñ and r=e^−ñ. It means the blue-red shifted diamond in Fig. 3.3b or Fig. 2.2 has the same area R′xL′ as the original green “home field” baseball diamond area RxL drawn below it and in Fig. 2.1. Constant products ù₁ù₃=const. give families of hyperbolas.

                                    |RxL|=2|GxP|=2|K_groupxK_phase|=2|ϖ²cosh²ñ  -ϖ²sinh²ñ|=2ϖ²

One hyperbola in Fig. 3.4a intersects bottom point B=ϖ (“pitchers’mound”). The other hits 2B (2^nd base). Each horizontal P -hyperbola is defined by the phase vector P=K_phase or some multiple of P.

                       (3.10a)

Each vertical G -hyperbola is defined by the wave group vector G=K_group or some multiple of G.

              (3.10b)

The G-vectors serve as tangents to P-hyperbolas and vice-versa. The tangent slope to any ù(k) curve is a well known definition of group velocity. Fig. 3.4b shows how of a P-hyperbola is equal to secant slope in Fig. 3.4a as defined in the u=V_group equation (2.7b) based on CW axioms.

Phase velocity =V_phase and its P-vector is an axis-switch (ù,ck)(ck,ù) of and its G-vector. In conventional c-units V_group/c<1 and 1<V_phase/c are inverses according to (2.7). (V_phase·V_group=c²)

 

    

Fig. 3.4 (a) Horizontal G-hyperbolas for proper frequency B=ϖ and 2B and vertical P-hyperbolas for proper wavevector k. (b) Tangents for G-curves are loci for P-curves, and vice-versa. Note: secant Äù/Äk and tangent dù/dk are always exactly equal.

 

Features on per-space-time (ck,ù) plots of Fig. 3.3-Fig. 3.4 reappear on space-time (x,ct) plots as noted in Fig. 2.1 and Fig. 2.2. A space-time invariant analogous to (3.10) is called proper-time ô.

                                                                                            (3.11)

It conventional to locate oneself at (0,ct) or presume one’s origin x=0 is located on oneself. Then (3.11) reduces to time axis ct=cô. A colloquial definition of proper time is age, a digital readout of one’s computer clock that all observers may note. By analogy, ϖ is proper-frequency, a rate of aging or a digital readout on each of the spectrometers in Fig. 3.2. Each reading is available to all observers.

                                                                                          (3.12)

The same hyperbolas (3.12) mark tics on the laser lab (ù,ck), the atom frame (ù′,ck′), or any other frame.

            The proper frequency of a wave is that frequency observed after one Doppler shifts the wave’s kinks away, that is, the special frequency ϖ seen in the frame in which its wavevector is zero (ck=0) in (3.12). Hence a single CW has a proper frequency that is identically zero (ϖ =0) by Evenson’s axiom (ù=ck), so single CW light cannot age. If we could go c to catch up to light’s home frame then its phasor clocks would appear to stop. Someone moving along a line of phasor clocks in Fig. 1.1c would always see the same reading, but that would be an infinite Doppler shift that one can only approach.

To produce a nonzero proper frequency ϖ ≠0 requires interference of at least two CW entities moving in different directions and this produces a standing wave frame like Fig. 2.1c moving at a speed less than c as shown in Fig. 2.2c. Matched CW-pairs of L and R baselines frame a “baseball diamond” for which the phase wavevector k_p in (2.2a) is zero. Then frame velocity u=V_group in (2.3b) is zero, too.

Fig. 3.5 shows the plots of per-spacetime “baseball diamond” coordinates for comparison of lab and atom frame views. While Fig. 3.5a is a “blimp’s-eye view” of the lab-frame diamond in Fig. 2.1, the atom frame view in Fig. 3.5b looks like the baseball field seen by a spectator sitting in the stands above the dugout. Nevertheless, identical hyperbolas are used to mark grids in either view.

Each point on the lower hyperbola is a bottom point ù′=B=2 (600THZ) for the frame whose relative velocity u′ makes it a ù′-axis (k′=0)-point, and every (k′=0)-point on the upper hyperbola is its bottom point ù′=2B=4 (1200THZ), and so on for hyperbolas of any given proper frequency value ϖ.  The same applies to space-time plots for which time ct′ takes the place of per-time ù′ and space x′ takes the place of per-space ck′. Then bottom points are called proper time or ô-values from (3.11).

For single CW light the proper time must be constant since a single CW cannot age. It is a convention to make the baselines or light cone intersect at the origin in both time and space. This sets the baseline proper time constant ô to zero. Then invariants (3.11) reduce to baseline equations x=±ct or x′=±ct′ for all frames. The space-time light cone relations are in direct correspondence with the per-space-time light cone relations ù=±ck or ù′=±ck′ for zero proper frequency in all frames and are concise restatements of the Evenson CW axiom (1.1).

Fig. 3.5 Dispersion hyperbolas for 2-CW interference (a) Laser lab view. (b)Atom frame view.

 

 

Chapter 4. Mechanics based on CW axioms

            Each of the 2-CW structures or properties discussed so far are due to relative interference effects between pairs of 1-CW entities that, by themselves, lack key 2-CW properties such as a proper invariant frequency ϖ, a rest frame, or any speed below the mortally unattainable velocity of c. To acquire “mortal” properties requires an interference encounter or pairing of 1-CW with another.

            Now we see how 2-CW interference endows other “mortal” properties such as classical mass and relativistic mechanics of energy-momentum that characterize a quantum matter wave. Such endowment lies in P-hyperbola phase relations (3.10a) that in turn are due to CW axioms (1.1) and (1.2).

            (4.1a)   (4.1b)         (4.1c)

Hyperbola in Fig. 3.4 has bottom B=ϖ and P-vector components (ù_p,ck_p) with tangent slope u/c at P. At low group velocity (u=c) the rapidity ñ approaches u/c. Then ù_p and k_p are simple functions of u.

                          (4.2a)                                                               (4.2b)

The ù_p and k_p fit Newtonian-energy E and Galilean-momentum p. Is that a coincidence? Perhaps, not!

                                 (4.3a)                                                                          (4.3b)

Wave ù and k results (4.2), scaled by a single factor s=Mc²/B, match classical E and p definitions (4.3).

               (4.4a)                                                       (4.4b)

In Newton’s mechanics, only energy difference ÄE counts, so he might ignore the term E=const. (4.3a). But, in (4.4a) that const.=sB is the proper phase carrier-frequency value B=ϖ at hyperbola bottom B in Fig. 3.4b. That is scaled by s=Mc²/B to sB=sϖ in Fig. 4.1. It is Einstein rest energy and not ignorable!

const.= sB = Mc² = sϖ                                                                         (4.4c)

            ϖ-mass-energy equivalence is a huge idea due to Einstein (1905) and Planck (1900). k-vector-momentum equivalence by DeBroglie came later (1920). CW results (4.1) give both directly and exactly.

     (4.5a)                                    (4.5b)

            Scale factor s in Planck[ix] E=sù or DeBroglie[x] p=sk laws is found experimentally. The lowest observed s-value is Planck angular constant h=1.05·10^-34J·s. That is Planck’s axiom E=hù_=õ for N=1. Integer  is Planck’s optical quantum number later called photon-number. At first, Planck regretted his 1900 axiom E=õ. It seems inconsistent with  ù ²-dependence of classical oscillator energy E=A²ù². In 1905, Einstein resolved this. A key idea is quantized amplitude A_N=√(hN/õ). (Even amplitude is wavy!)

Fig. 4.1 Energy vs. momentum dispersion functions including mass M, photon, and tachyon.

(a)  Relativistic (Einstein-Planck-deBroglie) case: (^)²=E²-(cp)² = 1 or  ì²=ù²-(ck)² = 1/h².

(b)  Non-relativistic (Bohr-Schrodinger-deBroglie) case: Å =-(1/2M)p²  or ù =hk²/2M

 

Quantized cavity modes and “fuzzy” hyperbolas

Cavity boundary conditions “1^st-quantize” classical wave mode variables (ù_n,k_n) so as to have discrete numbers n=1,2,3,… of half-wave anti-nodes that fit in a cavity of length-l as shown at the top of Fig. 4.2.

                        k_n=π/ë_n= n·π/l              (4.6a)                           ù_n=c k_n = c n·π/l           (4.6b)

Planck’s axiom “2^nd-quantizes” each fundamental mode frequencyù_n to have discrete quantum numbers N_n=0,1,2,3,… of photons. Each level E_N(n)=hN_nù_n labels a hyperbola in Fig. 4.2 whose number n of anti-nodes and N of photons is invariant. This lends object-permanence to cavity “light particles” or photons.

            As discussed in Ch. 6, laser waves are coherent state combinations of N-photon states that have semi-classical properties that include well-defined wave phase. One “fuzzy” hyperbola of uncertain N and mass-energy replaces the ladders in Fig. 4.2. This is a kind of 2^nd Occam-razor cut after the 1^st cut of PW into CW. As discussed in Ch. 6, it resolves CW into coherent combinations of “2^nd-quantized” photons.

Fig. 4.2 Optical cavity energy hyperbolas for mode number n=1-3 and photon number N=0, 1, 2,....

 

Alternative definitions of wave mass

            If mass or rest energy is due to proper phase frequency ϖ, then a quantum matter wave has mass without invoking hidden Newtonian “stuff.” With Occam logical economy, 2-CW light led to exact mass-energy-momentum (ù,k) relations (4.5) and not just low-speed classical ones (4.3). Now we see how 2-CW results expose some salient definitions of mass or matter that a classical theory might overlook.

            First, the Einstein-Planck wave frequency-energy-mass equivalence relation (4.4c) ascribes rest mass M_rest to a scaled proper carrier frequency sϖ /c². The scale factor s is Planck’s s=h for N quanta.

                                                                                                          (4.7)

For rest electron mass m_e =9.1·10^-31kg or M_p =1.67·10^-27kg of a proton, the proper frequency times N=2 is called zwitterbevegun (“trembling motion”) and is as mysterious as it is huge. (Electron rest frequency ϖ_e = m_e c²/h =7.76·10⁺²⁰(rad)s^-1 is the Dirac (e⁺e⁻)-pair production[xi] threshold as discussed in Ch. 8.)

            Second, we define momentum-mass M_mom by ratio p/u of momentum (4.5b) to velocity u. (Galileo’s p=M_momu) Now M_mom varies as  at high rapidity ñ but approaches invariant M_rest as.

                                                      (4.8)

Frame velocity u is wave group velocity and the Euclid mean construction of Fig. 3.3a shows u is the slope of the tangent to dispersion function ù(k). A derivative of energy (4.5a) verifies this once again.

                                                                                                     (4.9)

            Third, we define effective-mass M_eff as ratio=F/a=dp/du of momentum-change to acceleration. (Newton’s F=M_effa) M_eff varies as at high rapidity ñ but also approaches M_rest as.

                                                                       (4.10)

            Effective mass is h divided by the curvature of dispersion function ù(k), a general quantum wave mechanical result. Geometry of a dispersion hyperbola ù=Bcoshñ is such that its bottom (u=0) radius of curvature (RoC) is the rest frequency B=M_restc²/h, and this grows exponentially toward ∞ as velocity u approaches c. The 1-CW dispersion (ù=±ck) is flat so its RoC is infinite everywhere and so is photon effective mass M_eff(ã)=∞. This is consistent with the (All colors go c)-axiom (1.1). The other extreme is photon rest mass M_rest(ã)=0. Between these extremes, photon momentum-mass depends on CW color ù.

            M_rest(ã)=0      (4.11a)             M_mom(ã)=p/c=hk/c=hù/c²   (4.11b)     M_eff(ã)=∞       (4.11c)

For Newton this would confirm light’s “fits” to be crazy to the point of unbounded schizophrenia.            A 2-CW 600THz cavity has zero total momentum p, but each photon adds a tiny mass M_ã to it.

                        M_ã=hù/c²=ù  (1.2·10^-51)kg·s=  4.5·10^-36kg                        (for: ù = 2ð·600THz )

 In contrast, a 1-CW state has no rest mass, but 1-photon momentum (4.5b) is a non-zero value p_ã=M_ã c.

                        p_ã=hk=hù/c=ù  (4.5·10^-43)kg·m=1.7·10^-27kg·m·s^-1                     (for: ù = 2ð·600THz )

This p=Mc resembles Galilean relation p=Mu in (4.3b) and is perhaps another case of Galileo’s revenge!

Absolute vs. relative phases: Method in madness

            Probably Newton would find a CW theory to be quite mad. Claiming that heavy hard matter owes its properties to rapid hidden “carrier” phase oscillations would not elicit a Newtonian invitation to the Royal Society but rather to a lunatic asylum. Even though CW results (4.2) give Newtonian axioms (4.3) at low speeds, the result would seem to fail at high speeds where exact results (4.5) sag below Newton’s. Also, having an enormous constant Mc² be part of energy would, in 1670, seem insanely meaningless.

            But, in 1905[xii] Einstein relations appear with both Mc² and energy sag. Now Einstein’s classical training left him leery of hidden quantum wave phases with dicey interpretations of intensity Ø^∗Ø as probability. Also, he may have asked why observable results depend on a square Ø^∗Ø=|Ø|^ that kills that overall phase frequency, seemingly losing the one quantity that represents (or is) the total mass-energy.

            Square |Ø|^ of a 2-CW Ø=e^ia+e^ib loses phase factorleaving group functions of differences or of 1^st or 3^rd base frequencies or k-vectors. Group beat frequency is zero in the rest frame of Fig. 2.1c where it is a stationary wave. In Fig. 2.2c or any other frame, |Ø|^ is not stationary but is observed to have velocity V_group≠0. Fourier sums of m=3 or more terms may have multiple beats in as in Fig. 2.2d.

                                                                        (4.12)

            With observable difference or beat notes, P cannot rest in any frame. Differences or derivatives are observable while absolute Ø-frequency stays hidden until two quantum objects interfere. Then new beats arise from differences between the two absolute frequencies and others. A new absolute phase (not in |Ø|^) is the sum of all. But, we can only observe beats of relative frequency! That may be a quantum version of Einstein’s popularized saw, “It’s all relative.” Phase velocity escapes with its Galilean arithmetic intact in Fig. 3.1, but here it finally surrenders its absolutes to relativism.

            Total phase gives total energy E or momentum p, but differentials are what one feels through work ÄE or impulse Äp. Invariant quantities like ϖ and M_rest depend on total phase but intensity (4.12) has only differentials Äk_ij or relative beats Äù_ij. Among frame-dependent relative quantities are group velocity u (4.9), M_mom (4.8), and M_eff (4.10), but rest mass M_rest (4.4c) is a frame-invariant absolute quantity. Also note that M_mom and M_eff approach M_rest at zero velocity. Now |Ø|^ may register an ϖ beat with a DC (static ù₀=0) wave, but lack of resonance confines (ù₀=0)-carrier waves to beat only locally.

            Phase frequency ù_p in a quantum wave is fast and silent like a carrier frequency of radio wave. Group frequency ù_g is like the audible signal, much slower and heard in resonant beats ù_a −ù_b involving carrier and receiver. Atomic “carrier” frequencies ù_p=M_c²/h due to rest mass are enormous as are those of atomic measuring devices that play the role of “receivers” in quantum experiments. Measurement involves resonant contact of an atom and devices that horse-trade beats at truly huge frequencies.

One way to avoid huge Mc²/h-related phase frequencies is to ignore them and approximate the relativistic equation E=Mc²coshñ of (4.5a) by the Newtonian approximation (4.4a) that deletes the big rest-energy constant sB=Mc². The exact energy (4.5a) that obeys CW axioms (1.1) is rewritten in terms of momentum (4.5b) below to give a Bohr-Schrodinger (BS) approximation (4.14) with Mc² deleted. 

                              (4.13)

                                       (4.14)

If only frequency difference affect observation based on |Ø|^ (4.12), the BS claim is that energy origin may be shifted from (E=Mc², cp=0) to (E=0, cp=0). (Frequency is relative!) Hyperbola (4.13) in Fig. 4.1a, for u way less then c, approaches the BS parabola (4.14) in Fig. 4.1b. That is the only E(p) Newton knew.

Group velocity u=V_group of (4.9) is a relative or differential quantity so origin shifting does not affect it. However, phase velocity=V_phase is greatly reduced by deleting Mc² from E=hù. It slows from V_phase=c²/u that is always faster than light to a sedate sub-luminal speed of V_group/2. Having V_phase go slower than V_group is an unusual situation but one that has achieved tacit approval for BS matter waves.[xiii] The example used in Fig. 1.6 of Ch. 1 is a 2-CW BS matter wave exhibiting this low V_phase.

Standard Schrodinger quantum mechanics, so named in spite of Schrodinger‘s protests[xiv], uses Newtonian kinetic energy (4.14) or (4.3) with potential ϕ (as the const.-term) to give a BS Hamiltonian.

H=p²/2M + ϕ    or:        hù= h²k²/2M + 〈ϕ〉_                           (4.15)

The CW approach to relativity and quantum exposes some problems with such approximations.

First, a non-constant potential ϕ must have a vector potential A so that (ϕ,cA) transform like (ù,ck) in (2.10a) or (ct,x) in (2.10b) or as (E,cp) with scaling laws p=hk and E=hù. Transformation demands equal powers for frequency (energy) and wavevector (momentum) such as the following.

(E- ϕ)²=(p-cA)²/2M+Mc²         or:        (hù-〈ϕ〉_)²= (hk-cA)²/2M+Mc²                     (4.16)

Also, varying potentials perturb the vacuum so single-CW’s may no longer obey axioms (1.1-2).

            Diracs’s elegant solution obtains ±pairs of hyperbolas (4.13) or (4.16) from avoided-crossing eigenvalues of 4x4 Hamiltonian matrix equations with negative frequency hyperbolas. The negative-ù hyperbolas in Fig. 4.1 are (conveniently) hidden by the BS approximate dispersion parabola.

Dirac’s ideas require three-dimensional wavevectors and momenta. But first, fundamental Lagrangian-Hamiltonian geometric relations of quantum phase and frequency relate relativistic classical and quantum mechanics in the following Ch. 5. These relations expose more of the logic of phase-based Evenson axiom (1.1), Doppler T-symmetry axiom (1.2), and Euclid frequency means in Fig. 3.3.

 

 

Chapter 5. Classical vs. quantum mechanics

            The CW-spectral view of relativity and quantum theory demonstrates that wave phase and in particular, optical phase, is an essential part of quantum theory. If so, classical derivation of quantum mechanics might seem about as viable as Aristotelian derivation of Newtonian mechanics.

            However, the 19^th century mechanics of Hamilton, Jacobi, and Poincare developed the concept of action S defined variously by area in phase-space or a Lagrangian time integral . The latter action definition begins with the Legendre transformation of Lagrangian L and Hamiltonian H functions.

                                                                                                                                      (5.1a)

L is an explicit function of x and velocity  while the H is explicit only in x and momentum p.

               (5.1b)                           (5.1c)                           (5.1d)                      (5.1e)

Multiplying by dt gives the differential Poincare invariant dS and its action integral .

                                (5.2a)                                (5.2b)

Planck-DeBroglie scaling laws p=hk and E=hù (4.5) identify action S as scaled quantum phase hÖ.

                        (5.3a)                                           (5.3b)

            If action dS or phase dÖ is integrable, then Hamilton-Jacobi equations or (k,ù) equivalents hold.

                  (5.4a)            (5.4b)                        (5.4c)                     (5.4d)

Phase-based relations (5.4c-d) define angular frequency ù and wave number k. The definition (3.8) of wave group velocity is a wave version of Hamilton’s velocity equation (5.1e).

                                                               equivalent to:

The coordinate Hamilton derivative equation relates to wave diffraction by dispersion anisotropy.                                                                       equivalent to:    

Classical HJ-action theory was intended to analyze families of trajectories (PW or particle paths), but Dirac and Feynman showed its relevance to matter-wave mechanics (CW phase paths) by proposing an approximate semi-clasical wavefunction based on the Lagrangian action as phase.

                                                                                                                     (5.5)

The approximation symbol () indicates that only phase but not amplitude is assumed to vary here. An x-derivative (5.4a) of semi-classical wave (5.5) has the p-operator form in standard BS quantum theory.

               (5.6a)                                                                            (5.6b)

The time derivative is similarly related to the Hamiltonian operator. The HJ-equation (5.4b) makes this appear to be a BS Hamiltonian time equation.

                      (5.7a)                                                       (5.7b)

However, these approximations like the BS approximations of (4.14) ignore relativity and lack economy of logic shed by light waves. The Poincare phase invariant of a matter-wave needs re-examination.

Contact transformation geometry of a relativistic Lagrangian

            A matter-wave has a rest frame where x′=0=k′ and its phase Ö = kx-ù t reduces to −ìô, a product of its proper frequency ì =(or Mc²/h) with proper time t ′=ô. Invariant differential dÖ is reduced, as well, using the Einstein-Planck rest-mass energy-frequency equivalence relation (4.4c) to rewrite it.

                                                 dÖ = kdx−ù dt=−ì  dô = -(Mc²/h) dô.                                                (5.8)

ô-Invariance (2.21) or time dilation in (2.10b) gives proper dô in terms of velocity and lab dt.

                                                             dô = dt √(1-u²/c²) )=dt sech ñ                                                (5.9)

Combining definitions for action dS=Ldt (5.2) and phase dS = hdÖ  (5.3) gives the Lagrangian L.

                                                              L =−hìô  = -Mc²√(1-u²/c²)= -Mc²sech ñ                              (5.10)

Fig. 5.1 plots this free-matter Lagranian L next to its Hamiltonian H using units for which c=1=M.

 

Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian

 

            Relativistic matter Lagrangian (5.10) is a circle (Fig. 5.1b). L-values and in Fig. 5.1 are contact Legendre transforms of H-values andof Hamiltonian hyperbola in Fig. 5.1a. Abscissa p and ordinate H of a point P in plot (a) gives negative intercept -H and slope p of tangent HQ contacting the transform point Q in plot (b) and vice-versa. (Contact geometry is really wave-action-energy mechanics.)

            If , Lagrange kinetic energy is Hamilton . Then circle L and hyperbola H both approximate a Newtonian parabola at low speed u<<c. But, asthe L-circle rises above the parabola and the H-hyperbola sags below it and instead approaches contacting c-asymptote in Fig. 5.1.

            Action integral S=∫Ldt is to be minimized. Feynman’s interpretation of S minimization is depicted in Fig. 5.2. A mass flies so that its “clock” ô is maximized. (Proper frequency is constant for fixed rest mass, and so minimizing −ìô means maximizing +ô.) An interference of Huygen wavelets favors stationary and extreme phase. This favors the fastest possible clock as is sketched in Fig. 5.3.

            Feynman described families of classical paths or rays fanning out from each space-time point on a wavefront of constant phase Ö or action S. Then, according to an application of Huygen's principle to matter wave, new wavefronts are continuously built in Fig. 5.3 through interference from “the best” of all the wavelets emanating from a multitude of source points on each preceding wavefront. Thus classical momentum p=∇S by (5.4a) for the “best” ray ends up normal to each wavefront.

            The “best” are so-called stationary-phase rays that are extremes in phase and thereby satisfy Hamilton's Least-Action Principle requiring that ∫Ldt is minimum for “true” classical trajectories. This in turn enforces Poincare' invariance by eliminating, by de-phasing, any “false” or non-classical paths because they do not have an invariant (and thereby stationary) phase. “Bad rays” cancel each other in a cacophonous mish-mash of mismatched phases. Each Huygen wavelet is tangent to the next wavefront being produced. That contact point is precisely on the ray or true classical trajectory path of minimum action and on the resulting “best” wavefront. Time evolution from any wavefront to another is thus a contact transformation between the two wavefronts described by the geometry of Huygens Principle.

            Thus a Newtonian clockwork-world appears to be the perennial cosmic gambling-house winner in a kind of wave dynamical lottery on an underlying wave fabric. Einstein’s God may not play dice, but some persistently wavelike entities seem to be gaming at enormous Mc²/h-rates down in the cellar! 

            It is ironic that Evenson and other metrologists have made the greatest advances of precision in human history, not with metal bars or ironclad classical mechanics, but by using the most ethereal and dicey stuff in the universe, light waves. This motivates a view of classical matter or particle mechanics that is more simply and elegantly done by its relation to light and its built-in relativity, resonance, and quantization that occurs when waves are subject to boundary conditions or otherwise confined. While Newton was complaining about “fits” of light, that crazy stuff was just trying to tell him something!

            Derivation of quantum phenomena using a classical particle paradigm seems silly now. If particles are made by waves, optical or otherwise, rather than vice versa as Newton believed, the case is closed. Also, CW trumps PW as CW symmetry axioms (1.1-2) derive classical results (4.4) while giving exact relations (4.5) for relativity and quantum theory tossed into the bargain. Such Occam economy is found lacking on a PW path from Newton to Einstein and Planck.

            Thus basic CW sum-and-difference phase relations seem to underlie the physics of Poincare contact geometry. This in turn is based on circular and hyperbolic geometry described next.

 

Fig. 5.2 “True” paths carry extreme phase and fastest clocks. Light-cone has only stopped clocks.

 

 

Fig. 5.3 Quantum waves interfere constructively on “True” path but mostly cancel elsewhere.

Geometry of circular and hyperbolic functions

            Geometry of half-sum and half-difference phase P=(R+L)/2 and group G=(R-L)/2 vectors is based on trigonometric exponential identities that are crown jewels of 18^th century mathematics and have Euclidian geometric origins shown in Fig. 5.4. Phase angle-ö identities apply to Fig. 5.4a.

                   (5.11a)                         (5.11b)

            Circular function tanö is named for a tangent to a unit circle shown in Fig. 5.4(a). Its incline (sine) elevation is sinö. The complimentary tangent or cotangent cotö completes the tangent distance between axes where ö is circle arc-length-ö or subtended area-ö. Hyperbolic functions use area ñ for “angle.”

                (5.12c)                         (5.12d)

Fig. 5.4b shows how hyperbolic functions relate to circular ones in Fig. 5.4a. The circular sine equals the hyperbolic tangent (sinö =tanhñ) and vice versa (tanö =sinhñ). Each circular function has a segment that matches one for a hyperbolic function, for example (cosö =sechñ) matches (secö =coshñ). These relations recap the CW view of the Legendre contact transformation in Fig. 5.1 that underlies classical and quantum theory that is in the algebra and geometry for every bit of light-and-matter in and around us! 

            In Fig. 5.4, circular area ö and hyperbolic area ñ have been chosen so that tanö =1.15=sinhñ and sinö =0.75=tanhñ, that is for u=3c/4. The tangent to the circle in Fig. 5.4a-b is like the one that contacts the Lagrangian circle in Fig. 5.1b to contact-transform it to the Hamiltonian hyperbola in Fig. 5.1a, and vice versa the hyperbolic tangent in Fig. 5.4b is like the one that transforms the Hamiltonian hyperbola in Fig. 5.1a to the Lagrangian circle in Fig. 5.1b.

            The hyperbolic tangent u/c=tanhñ of (2.19) corresponds to frame rapidity ñ and group velocity u=V_group in (2.8), (4.9) and in Fig. 3.3a-b. The circular tangent angle ö or inclination sinö belongs to Lagrangian velocity function (5.10) in Fig. 5.1b. (The horizontal axis of the latter in the vertical axis of Fig.11. This geometry is symmetric to axis-switching.) As u and ñ approach c and ∞, respectively, the circular angle ö approaches ð/2.

            This angle ö is the stellar velocity aberration angle, that is, the polar angle that vertical starlight is seen by a horizontally moving astronomer to tip into her direction of motion. Aberration angle ö, like rapidity ñ, is 1^st-order in velocity u and both ñ and ö equal u/c at low speeds. (See the discussion of Fig. 5.6 near the end of this chapter. This deepens the development to include 4-vector space-time.)

            Many of the twelve circular-hyperbolic trigonometric ratios in Fig. 5.1 belong to one or more physical or geometric effects shown in prior diagrams beginning with Euclid’s rectangle-in-circle mean construction of Fig. 3.3. Prior ratio constructions are overlapped in the form of Fig. 5.1 and results in Fig. 5.5a that might be described as a global ratio riot. This riot is simplified and labeled in zoom-in views of Fig. 5.5b-d and they are the basis of the following discussion of the role of tangent-contact geometry in CW analysis of Poincare contact transformation and relativistic quantum waves.

Fig. 5.4 Trigonometric geometry (a) Unit circular area ö=0.86 and (b) Unit hyperbolic area ñ=0.99.

Hyper-circular contacts

            Beginning with the Euclidian mean diagram of Fig. 3.3, three mean frequencies arise from an interfering pair of left-moving “red” and right-moving “blue” beams of frequency ù_L and ù_R. First is a half-sum phase frequency ù_p=(ù_R+ù_L)/2 (arithmetic mean) that defines the circle radius in Fig. 3.3. Second is a half-difference group beat frequency ù_g=(ù_R-ù_L)/2 (difference mean) that is radial distance of circle center to origin. Third is a root-product proper frequency ϖ =( ù_R·ù_L)^1/2 (geometric mean) that is the base radius or bottom of a ù(k) hyperbola of rest energy B=h ϖ =Mc² above origin in Fig. 3.3.

            Phase and group frequencies are defined as ratios or shifts of the geometric mean frequency ϖ, and this begins with the Doppler shift definition of the red ù_L=e^−ñϖ and blue ù_R=e^+ñϖ CW components. Ratio values ù_p=ϖcoshñ and ù_g=ϖsinhñ define each point on a ϖ-hyperbola dispersion curve in Fig. 5.5.

Fig. 5.5 is based on circles with three different radii, one for each mean frequency. The base circle-b drawn centered at origin has radius B=h ϖ =Mc² of the Lagrangian circle in Fig. 5.1b. A smaller circle-g has group radius hù_g=Bsinhñ. A larger circle-p has phase radius hù_p=Bcoshñ of the Euclidean circle in Fig. 3.3 and is drawn with dashed lines in Fig. 5.5. (Base value B is scaled for energy here.)

Circle-p of larger radius hù_p=Bcoshñ is centered at cp=hù_g=Bsinhñ, a horizontal distance equal to the radius of the smaller circle-g, while the latter is centered at E=hù_p=Bcoshñ, a vertical distance equal to the radius of the larger circle-p. Tangents that contact circles or hyperbolas define many of the physical quantities labeled in the zoom-in view of Fig. 5.5b. Intersections and chords shared by two of the circles also provide the key quantities as seen in Fig. 5.5a.

            So far the CW development has emphasized the Doppler ratio as a starting point beginning with Fig. 2.2 and culminating with the Euclidean means of Fig. 3.3. However, most developments of relativity start with velocity u, and that geometric approach is excerpted in a simplified construction of Fig. 5.5c where u/c=45/53 and Fig. 5.5d where u/c=3/5. (Fig. 5.5a-b and most other figures use u/c=3/5.)  Once the velocity u/c line intersects the basic b-circle and its horizontal tangent of unit-energy (B=1=Mc²), it only takes three more lines to derive Lagrangian -L=Bsechñ, then momentum cp=Bsinhñ, and finally the Hamiltonian H=Bcoshñ. Then a compass is used to check accuracy with the phase p-circle by making sure it goes from (cp,H) to the (0,B)-point on top of the b-circle. The p-circle goes on to intersect the negative cp-axis at the Doppler red shift rB=Be^+ñ. Finally, the group g-circle in Fig. 5.5a-b has a chord intersection with the p-circle that is the hyperbolic contact tangent, and it grazes the ö-angle normal to the Lagrangian circle tangent in Fig. 5.5b. This helps to clarify geometry of H-L contact transformations of Fig. 5.1 for reciprocal space-time (ù,ck) and (Ö,u/c). The constructions also apply to space-time.

            If Fig. 5.5 is in space-time, the segment -L=Bsechñ is Lorentz contraction . The H=Bcoshñ and cp=Bsinhñ segments are, respectively Einstien time dilation and asimultaneity a=ud/c coefficients. Node-to-node or peak-to-peak gaps contract by l= in Fig. 2.2d-e. As speed reduces in Fig. 5.5c-d from u/c=45/53 to u/c=3/5 or to lower values, the Lagrangian velocity angle ö and Hamiltonian rapidity ñ approach the velocity ratio u/c. Galilean velocity addition rules resume. In the opposite ultra-relativistic regime, ö approaches ð/2, ñ approaches ∞, and u/c nears unit slope or 45° in Fig. 5.5c. But, Galilean-like rules (3.6) apply to rapidity ñ at all speeds (so far).

 

Fig. 5.5 Relativistic wave mechanics geometry. (a) Overview. (b-d) Details of contacting tangents.

 

Transverse vs. longitudinal Doppler: Stellar aberration

            A novel description of relativity by L. C. Epstein[xv] in Relativity Visualized introduces a "cosmic speedometer" consisting of a telescope tube tipped to catch falling light pulses from a distant overhead star. A stationary telescope points straight up the x-axis at the apparent position S of the star. (Fig. 5.6a) But, with velocity u=u_ze_z across to the star beam x-axis, the telescope has to tip to catch the starlight, so the apparent position S' tips toward u. (Fig. 5.6b).

The telescope tips by a stellar aberration angle ó(ö in (5.11a) or Fig. 5.4a.). The sine of angle ó is velocity ratio â= uz /c which is the hyper-tangent of relativistic rapidity õz (ñ in (5.12a) or Fig. 5.4b.)

                                                  â= uz /c =sin ó = tanh õz                                                    (5.13)

            Proper time ô and frequency invariance (3.10) forces 4-vector components normal to velocity u of a boost to be unchanged. That is, a boost along z of (ct,z) to (ct',z') (or (ù,ckz) to (ù',ckz') ) must preserve both (x,y)=(x',y') and (ckx,cky)=(ckx',cky') just as a rotation in the xy-plane of (x,y) to (x',y') leaves unaffected the components (ct,z)=(ct',z') and (ù,ckz)=(ù',ckz') transverse to the rotation.

 

Fig. 5.6 Epstein’s cosmic speedometer  with aberration angle ó and transverse Doppler shift coshõ.

 

Fig. 5.7 CW version of cosmic speedometer showing transverse and longitudinal k-vectors.

 

            Invariant (3.10) demands light-speed conservation as sketched in Fig. 5.6b. Starlight speed down the ó-tipped telescope is c, so the x-component of starlight velocity reduces from c to

                                                 cx'=c cos ó=c√(1- uz2/c2) = c/cosh õz .                                (5.14)

Transformation (5.17a) below assures that x-or-y-components of k↓ are unchanged by uz-boost.

                                                  (ckx,cky)=(ckx',cky')                                                               (5.15)

So the length of k↓ increases by a factor cosh õ as shown in Fig. 5.7 as does the frequency ù'↓.

                                                 c|k′↓| = c|k↓| coshõz = ù0 cosh õz =ù0/√(1-u2/c2)                  (5.16)

            If the observer crosses a star ray at very large velocity, that is, lets uz approach c, then the star angle ó approaches 90° and the frequency increases until the observer sees an X-ray or ã-ray star coming almost head on! The coshõz factor is a transverse Doppler shift. For large õz, it approaches e^õz, which is the ordinary longitudinal Doppler shift upon which the CW relativity derivations of Ch. 2 are based. Relations (5.13-16) are summarized in a 4-vector transformation: ù0 has a transverse Doppler shift to ù0coshõz, so ckz=0 becomes ckz' = -ù0 sinhõz , but the x-component is unchanged: ckx' = ù0 = ckx.

  

                                                                                                                                                (5.17a)

If starlight had been k← or k→ waves going along u and z-axis, the usual longitudinal Doppler blue shifts e^+õz or red shifts e^−õz would appear on both the k-vector and the frequency, as stated by the following.

                        (5.17b)

            The Epstein speedometer tracks light pulses and particles in space and time. Instead of space-x and time-ct coordinates of a Minkowski graph, he plots space coordinate-x against proper time-cô. This view has all things, light ã and particle P included, moving at the speed of light as shown in Fig. 5.8. Light never ages, so its “speedometer” is tipped to the maximum along x-axis.

Fig. 5.8 Space-proper-time plot makes all objects move at speed c along their cosmic speedometer.

 

            One cute feature of the Epstein space-proper-time view is its take of the Lorentz-Fitzgerald contraction of a proper length L to L′=L√1-u²/c². (Recall discussion around (2.11).) As shown in Fig. 5.9 below, L′ is simply the projection onto the x-axis of a length L tipped by ó.

Fig. 5.9 Space-proper-time plot of Lorentz contraction as geometric projection of rotated line L.

 

            The problem with the (x,cô) view is that a space-time event is not plotted as a single point for all observers. Since the time parameter ô is an invariant, the (x,cô) graph it is not a metric space.

 Graphical wave 4-vector transformation

            Geometric constructions combining Fig. 5.6 and Fig. 5.7 help to quantitatively visualize 4-wavevector transformations. One is shown in Fig. 5.10. The c-dial of the “speedometer” is first set to the desired u-speed which determines angle ó. The top of the c-dial (which may also represent a transverse ck↑-vector in units of Lab frequency ù0) is projected parallel to the velocity axis until it intersects the c-dial vertical axis. A transformed ck'↑-vector of length ù'↑=ù0 cosh õ results, similar to ck'↓ in (5.17a). Both ck'↑ and ck'↓ have a projection on the velocity axis of ù0sinh õ while maintaining their transverse components ù0 and -ù0 , respectively, in order to stay on the light cone.

            A dashed circle of radius cosh õ is drawn concentric to the c-dial and determines the longitudinal vectors ck'→ and ck'← of Doppler shifted length and frequency ù0e-õ and ù0eõ, respectively, as required by transformation (5.17b). This construction is part of Fig. 5.4 and Fig. 5.5.

 

Fig. 5.10 CW cosmic speedometer. Geometry of Lorentz boost of counter-propagating waves.

Symmetry and conservation principles

            In Newtonian theory the first law or axiom is momentum conservation. Physical axioms, by definition, have only experimental proof. Logical proof is impossible unless a theory like Newton’s becomes sub-summed by a more general theory with finer axioms. Proof of an axiom then undermines it so it becomes a theorem or result of more basic axioms. (Or else an axiom might be disproved or reduced to an approximate result subject to certain conditions.)

            The logic of axioms yielding results or theorems in mathematical science probably goes back two thousand years to the time of Euclid’s Elements. Also, axiomatic approaches to philosophy and natural science show up in writings as early as that of Occam or even Aristotle, but it is not until the European Renaissance that experiments began to be precise enough to support mathematical models. By the European Enlightenment period, mathematical logic of physical science had become more effective and productive than any preceding philosophy due in no small part to increasingly precise evidence.

            As stated by introduction, current time and frequency measurements have achieved almost unimaginable precision. In celebration of this, two continuous wave (CW) axioms (1.1-2) have been used to undermine Newtonian axioms for mass, energy, and momentum. They then became approximate results (4.4) and give rise to exact equivalents of Newtonian concepts in Einstein and Planck relativity and quantum theory in (4.5). It is a non-trivial example of undermining axioms by Occam razor-cutting.

            The undermining of Newton’s first axiom (momentum conservation) by the shaved CW axioms is a good example to expose the logic involved. CW logic leads to the DeBroglie scaling law (4.5b) that equates momentum p to wavevector k scaled in h units. A rough statement of how CW axioms undermine or “prove” p-conservation axioms is that k-conservation is required by wave coherence and so p=hk must be conserved, as well. However, that oversimplifies a deeper nature of what is really symmetry logic.

            A strength (and also, weakness) of CW axioms (1.1-2) is that they are symmetry principles due to the Lorentz-Poincare isotropy of space-time that invokes invariance to translation in the vacuum. Operator has plane wave eigenfunctions with roots-of-unity eigenvalues .

                 (5.18a)                           (5.18b)

This also applies to 2-part or “2-particle” states where exponents add (k,ù)-values of each constituent to K=k₁+k₂ and Ù=ù₁+ù₂, and -eigenvalues also have the form of (5.1). Matrix of T-symmetric evolution U is zero unless  and .

                              (5.19)

T-symmetry requires total energy  and total momentum be conserved for archetypical CW states, but laboratory CW have momentum uncertainty Äk=1/Äx due to finite beam sizeand energy uncertainty due to time limits. So, Newton’s 1^st law or axiom is verified but only as an ideal limit.

            Symmetry is to physics what religion is to politics. Both are deep and grand in principle but roundly flaunted in practice. Both gain power quickly by overlooking details. In Ch.4 relativistic and quantum kinetic properties of a massive “thing” arise from those of an optical 2-CW function in one space dimension. This means that mass shares symmetry with 2-CW light, not that mass is 2-CW light. Massive “things” do not vanish if a laser turns off, but our tiny optical mass hNù/c² is quickly gone!

            Puzzling questions remain. Why do simple wave optics lead directly to general properties (4.5) of relativity and quantum mechanics of a massive particle? How does a cavity of counter-propagating green light waves act like it holds particles of mass M=hù/c²?

            A short answer to one question is that particles are waves, too, and so forced by Lorentz symmetry to use available hyperbolic invariants  for dispersion. To answer the second question entails further loss of classical innocence. In Ch. 6 Occam’s razor is again applied to cut semi-classical CW laser fields down to single field quanta or photons. So the second short answer is that waves are particles, too, even for optical dispersion .

            By many accounts, quantum theory begins with Planck axiom E=hNù. This is distinguished from the scaling law E=sù derived in (4.5a) since its scale factor s=hN is not an obvious consequence of CW phase axioms (1.1-2) that lead to (4.5). CW logic involves additional axioms for Maxwell electromagnetic energy E and field amplitude quantization to render Planck’s axiom. This is discussed shortly.

1^st and 2^nd Quantization: phase vs. amplitude

            Waves resonate at discrete wave numbers  in a ring or cavity of length L. Then relations (4.5b) between k and momentum p force p-quantization  so momentum quantum numbers[xvi] m=0, ±1, ±2,… count waves on ring L as in Bohr electron orbitals or for cavity modes in (4.6a).

Then Planck dispersion (4.5a) gives electron energy levels for the BS approximation  or for cavity fundamental frequency levels (4.6b). Wave-fitting in x-space is called 1^st quantization. Related fitting in wave amplitude space is called 2^nd quantization.

            Heisenberg[xvii] showed quanta or arise from eigenvalues (literally “own-values”) of matrix operators or  whose eigenvectors (“own-vectors”) or  may be superimposed.

                                                …                                 (5.20)

(Dirac’s bra-ket[xviii] notation came later.) Allowing things to be at (or in) m places (or states) allows mean values  to range continuously from lowest quantum levels  to the highest .

                                    …                              (5.21)

For classicists, the notion that each multiple-personality-k has a probabilityseems, if not crazy, then at least dicey in the sense of Einstein’s skeptical quote, “God does not play dice…” [xix]

            But, superposition is an idea borrowed from classical waves. Resulting interference makes them ultra-sensitive to relative position and velocity, a first order sensitivity that leads elegantly to relativity transformation (2.10) and kinematic relations (4.5) by geometry of optical phase kx-ùt of ø=Ae^i(kx-ùt).

Amplitude “A” of wave (1.6) or (1.9) is set arbitrarily since only real wave zeros were needed. It is ignored in (5.5). Without Maxwell and Planck rules, CW amplitude or wave quantity is undefined and un-quantized while wave quality (frequency and phase) may be well defined and quantized. Amplitudes need a similar treatment that is begun in Ch. 6.

 

Chapter 6. Variation and quantization of optical amplitudes

What is deduced from wave phase alone? Wave amplitude has so far been skirted for Occam economy: “Pluralitas non est ponenda sine neccesitate” (Assume no plurality without necessity.) CW phase axioms (1.1-2) give Lorentz-Doppler and Planck-DeBroglie symmetry relations yet 2-CW amplitudes (1.10) are not defined beyond assuming their 1-CW amplitudes match. Standing wave grid reference frames in Fig. 2.1 and Fig. 2.2 are just points where amplitude is zero, that is, loci of real wave function roots.

Discussion of non-zero amplitude variation begins with counter-propagating 2-CW dynamics involving two 1-CW amplitudes  and  that we now allow to be unmatched.  

                             (6.1a)

Half-sum mean phase rates and half-difference means appear here as in (1.10).

                             (6.1b)                                           (6.1c)  

Also important is amplitude mean  and half-difference. Wave motion depends on standing-wave-ratio SWR or the inverse standing-wave-quotient SWQ.

                              (6.2a)                                             (6.2a)  

Recall mean frequency ratios for group velocity (2.3b) or its inverse that is phase velocity (2.3a).

                (6.3a)                               (6.3b) 

A 2-state amplitude continuum is bounded by a pure right-moving 1-CW  of SWR=1 and a pure left-moving 1-CW of SWR=-1. A 2-CW standing-wave  has SWR=0.

            Wave paths for various SWR values are drawn in Fig. 6.1 for 600THz 2-CW pairs and in Fig. 6.2 for Doppler shifted 300THz and 1200THz 2-CW pairs at the same SWR values. The SWQ is the ratio of the envelope peak (interference maximum) to the envelope valley (interference minimum), and vice versa for SWR=1/SWQ. Single frequency 2-CW paths of nonzero-SWR in Fig. 6.1 do a galloping motion. Each wave speeds up to peak speed c/SWR=c·SWQ as it first shrinks to squeeze through its envelope minima and then slows to resting speed c·SWR as it expands to its maximum amplitude. Only at zero-SWR do 2-CW zero-paths appear to travel at a constant group speed (6.3a) and phase speed (6.3b) as in Fig. 6.1c or 6.2c. (For 1-CW paths or unit SWR=±1 there is just one speed ±c by axiom (1.1).)

            The real and imaginary parts take turns. One gallops while the other rests and vice versa and this occurs twice each optical period. If frequency ratio (6.3) and amplitude ratio (6.2) have opposite signs as in Fig. 6.1c (±0 or ±∞) and in Fig. 6.2e (±3/5 or ±5/3), wave zero paths will follow a right angle staircase. 1-frequency staircase (V_group=0=SWR) in Fig. 6.1c is a Cartesian grid like Fig. 2.1c. 2-frequency waves (V_group≠0) have Minkowski grids like Fig. 2.2c for SWR=0 or quasi-Cartesian stair steps like Fig. 6.2e for V_group=-cSWR. To broadcast Cartesian grids to a u-frame one tunes both V_group and cSWR to u.

Galloping is a fundamental interference property that may be clarified by analogy with elliptic orbits of isotropic 2D-harmonic oscillators and in particular with elliptic polarization of optical wave amplitudes. Fig. 6.3 relates polarization states and wave states of Fig. 6.1 beginning with left (right)-circular polarization that is analogous to a left (right)-moving wave in Fig. 6.3g (Fig. 6.3a). As sketched in Fig. 6.3(b-e), galloping waves are general cases analogous to general states of elliptic polarization or general 2DHO orbits obeying a Keplerian geometry shown in Fig. 6.3h. Standing waves correspond to plane-polarization. Polarization in the x-plane of Fig. 6.3d corresponds to a standing cosine wave and y-plane polarization (not shown) would correspond to a standing sine wave.

            Isotropic oscillator orbits obey Kepler’s law of constant orbital momentum. Orbit angular velocity slows down by a factor b/a at major axes or aphelions ±a and then speeds up by a factor a/b at minor axes or perihelions ±b just as a galloping wave, twice in each period, slows down to SWR·c and speeds up to SWQ·c. The galloping or eccentric motion of the eccentric anomaly angle ö(t) in Fig. 6.3h is a projection of a uniformly rotating mean anomaly (phase angle ù·t) of the isotropic oscillator, and this gives a Keplarian relation of the two angles seen in the figure.

                                                                                                                               (6.4a)

The eccentric anomaly time derivative of ö (angular velocity) gallops between ù ·b/a and ù ·a/b.

       (6.4b)

The product of angular moment r² and is orbital momentum, a constant proportional to ellipse area.

                                   

Consider galloping wave zeros of a monochromatic wave (6.1a) having SWQ (6.2b).

                       

Space k₀x varies with time ù₀t in the same way that eccentric anomaly varies in (6.4a).

                                                         (6.5a)

Speed of galloping wave zeros is the time derivative of root location x in units of light velocity c.

                               (6.5b)

Single frequency 2-CW paths in Fig. 6.1 have a constant product of instantaneous wave velocity and wave amplitude analogous to the constant product of orbital velocity and radius. So vacuum optical amplitude and phase motion obey a funny version of Kepler and Galileo’s rules. The extent to which 14^th century geometric relations underlie basic wave physics has not been fully appreciated.

 

Fig. 6.3 (a-g) Elliptic polarization ellipses relate to galloping waves in Fig. 6.1. (h-i) Kepler anomalies.

Maxwell amplitudes and energy

Classical Maxwell field amplitudes and are derivatives of vector potential A. Maxwell energy U per volume V or total energy U·V is a sum of amplitude squares E•E and c²B•B.

                                                              (6.7)

Fourier analysis of A into amplitudes and leads to a harmonic oscillator sum over each plane CW mode frequency, k_m-vector allowed by a large-cavity, and polarization á=x,y normal to k_m.

                                                                                               (6.8)

Harmonic oscillator frequency is independent of amplitude. This is consistent with CW phase axiom (1.1) and dispersion relations (3.5) derived from 2-CW superposition, but such a simple axiom seems unable to derive the Maxwell vector amplitude structure of 2-dimensional polarization normal to k_m of each wave mode or even to establish that its wave variables A, B, E, or k_m are, in fact, 3D vectors.

            The CW axiom (1.1) gives what is effectively a 2-dimensional harmonic oscillator (2DHO) with two complex amplitudes (a_L, a_R) for the two longitudinal propagation directions, but each comes with two transverse polarization amplitudes (a_x, a_y) that describe the second 2DHO in Maxwell light, namely polarization ellipsometry used in Fig. 6.3 as an analogy for propagation left-and-right along z.

Quantized optical fields

Mode amplitudeor  in classical electromagnetic energyare replaced by oscillator operators or for a field Hamiltonian with explicit linear frequency dependence of Planck.

                                                 (6.9)

The H-eigenstates for exactly quantized photon numbers  fix a definite energy value for each mode-k_m but has quite uncertain field phase. Average energy of one mode is

                                                                                        (6.10a)

where a 1-CW-1-photon E-field and vector potential A-amplitude is as follows.

                          (6.10b)                                             (6.10c)

            Field quantization is called 2^nd-quantization to distinguish 1^st-quantization k_m mode numbers m, used for classical light, from “purely quantum” photon numbers for wave amplitude. This may be a prejudice that waves (particles) are usual (unusual) for light but unusual (usual) for matter.          Amplitudes involve relations (6.7) to (6.10) that are more complex than axioms (1.1-2) for wave phase. While Maxwell-Planck relations lack the simplicity of the latter, they do derive the linear dispersion (1.1) by Fourier transform of the Maxwell wave equations, and they show optical wave amplitude has an internal symmetry analogous to that of wave frequency. The following discussion of this analogy involves a Doppler shift of wave amplitude with invariance or covariance of photon number N_k and standing wave ratio (SWR) (6.5). Also, one begins to see how Born quantum probability formulas  arise and are consistent with Dirac amplitude covariance.

Relativistic 1-CW covariance of Poynting flux

Maxwell-Planck energy density U(Joule/m³) in (6.10a) leads to a related Poynting flux S[Joule/(m²·s)].

               where:       (6.11)

Flux S contains two frequency factors, the fundamental laser frequency ù_k and the photon count rate n_k per[ (m²·s)]. Frequency ù_k is quantum quality of a laser beam and rate n_k is its quantum quantity. The product hù_k·n_k is Poynting flux. Rate n_k and frequency ù_k both Doppler shift by an exponential e^±ñ of rapidity ñ in (2.16). So do 1-CW fields E_±k as may be shown by Lorentz transforming them directly.

                        (6.12a)                                                              (6.12b)           

            Thus both electric field polarization E-amplitudes E_x an E_y of a 1-CW field undergo the same e^±ñ Doppler shift that the frequency ù_k or wavevector k experience. If E in (6.11) scaled by 1-photon factor (6.10) a probability wave ø follows whose square ø∗ø is a volume photon count N/(m³).

            ø_k= ø_k* ø_k                                               (6.13a)

Or, a flux probability wave ø is defined so its square ø∗ø  is an expected flux photon count n/(m²·s).

                                                                     (6.13b)

Due to the scaling of (6.13) the Doppler factor of drops an e^±ñ/2 factor from E_k in (6.12).

                                                      (6.14)

This is a starting point for the spinor form of Lorentz transformation for Dirac amplitudes.

 Relativistic 2-CW invariance of cavity quanta

Mean photon numberof a 2-CW cavity mode, unlike a 1-CW flux quantum n_k, is invariant to cavity speed. By analogy, 2-CW modes have variant group-phase velocity (V_group, V_phase), energy-momentum (hck,hù), but invariant mean velocity and frequency.

                                       (6.15a)                            (6.15b)

Linear dispersion ù_±k=±ck and (1.11) or (2.7) are used. Note the analogy to SWR relations (6.2).

                                            (6.15c)                                 (6.15d)

Each ratio (6.15) is a wave velocity that Doppler-transforms like relativistic (non-Galilean) velocity.

                                       (6.16a)                         (6.16b)

Velocity u_AB/c=tanhñ_AB is a hyperbolic sum since rapidity is a simple sum ñ_AB= ñ_A+ ñ_B by (3.6).

                                     (6.17)

The energy and momentum flux values are found for counter-k 2-CW beam functions .

                                                                                      

Lab 1-CW flux number expectation values  give 2-CW flux expectations in lab.          

                                   

The relation (6.13b) of quantum field  and classical Maxwell -field expectation is used.

                                                         (6.18a)

                                                        (6.18b)

Values and  lie on an invariant hyperbola of constant geometric means or ².

                       

                              (6.19)

                                                (6.20a)

The geometric mean frequency , mean quantum number , and mean field  are defined.

                           (6.20b)           (6.20c)                           (6.20d)

            Doppler relations imply Lorentz invariance for the mean number  and for the mean frequency  as well as their geometric mean that is 2cå₀ times the mean fieldand applies to a general 2-CW beam function Ø. A factor 2 on  or  in (6.20a) is consistent with 1-photon 2-CW states having equal average number and total1-photon Planck energy expectation E=hù.

            Ideal cavities balance field , frequency , and number. But, a general beam with , , and  has a center-of-momentum CoM-frame of zero flux where  by (6.18b), an isochromatic IsoC-frame with , and an IsoN-frame with balanced photon count. Frame speeds u^á may be distinct as sketched in Fig. 6.4. 

            (6.21a)         (6.21b)      (6.21c)

Flux invariant  is maximized by balanced amplitude but is zero if  or  is zero.

Thus optical rest mass (6.20a) decreases continuously as a 2-CW beam is unbalanced toward 1-CW.

 

Fig. 6.4. Cavity 2-CW modes. (a) Invariant “mass” hyperbolas. (b) COM frame. (c) ISOC frame.

 

It is argued in Ch. 4 that mass is a coherent 2-CW interference effect that is not possible for a 1-CW beam. If we replace Planck energy relation  by a Maslov form it has a tiny zero-point energy minimum. Does a tiny mass exist for 1-CW and even 0-CW beams in all frames in spite of the incoherence of such zero-point fluctuations? Such a presence in (6.20) may be ruled out if the speed-of-light axiom (1.1) is exact. There may still be much to learn about zero-point effects in QED and cosmology but this seems to indicate that their direct effects are effectively non-existent.

N-Photon vs Coherent-á-states

Optical fields A or E have quantum expectation values of field operators based on mode amplitudes or  in classical energy. Eachor  is replaced by oscillator boson operator or in a quantum field Hamiltonian whose eigenstates have exact quantized photon numbers for each mode-k_m.

            Each mode phase quanta m and amplitude quanta N_m are invariant constants that define another hyperbola with Einstein-Planck proper frequency as sketched in Fig. 6.4a and Fig. 4.2. The problem is that absolute certainty of photon number N_m implies totally uncertain field phase just as absolutely certain k_m of 1-CW symmetry implies totally uncertain position in space and time.

            Space-time position coordinates were defined by taking 1-CW combinations to make 2-CW coordinates of Fig. 2.1c or Fig. 2.2c. Ultimately an n-CW pulse-wave (PW) of Fig. 2.1d or Fig. 2.2d was localized with as low a space-time uncertainty Äô as desired but it acquires per-space uncertainty or bandwidth Äõ according to Fourier-Heisenberg relation Äõ · Äô >1.

            So also must photon-number states be combined if amplitude and phase uncertainty are to be reduced to the point where wave space-time coordinates can emerge. Such combinations are known as coherent states or á-states of harmonic oscillation. Sharper wave zeros require fuzzier hyperbolas.

Fuzzy hyperbolas vs. fuzzy coordinates

            Model micro-laser states are coherent statesmade of single-mode eigenstates with amplitudes . Variable  is average mode phase, and , rescaled by a quantum field factor f, are field averages .

                                                                 (6.22)

Amplitude factor f makes Planck’s  equal Maxwell field energy .

                                                                               (6.23)

            A fundamental laser mode in a 0.25ìm cubic cavity (See E-wave sketched in a strip of Fig. 2.2c.) has green light with  or 2.5eV per photon. The average photon number  models a laser with mean energy  in a volume . Photon number uncertaintyvaries inversely to phase uncertainty.

                                          (6.24a)                                 (6.24b)           

            Amplitude expectation value  is zero for states due to incoherence of phase, but number value  is exact as is proper frequency  due to the phase factor of .

            A volume V with -photons has energy or mass-equivalent  on a hyperbola quanta above the N=1 hyperbola. A coherent-state has a mass  with uncertainty so its phase uncertainty  is low enough to make an (x,ct)- grid (Fig. 6.6a) but a low-á state (Fig. 6.6c) has too few photon counts-per-grid to plot sharply. Photon-number eigenstate  in Fig. 6.6d is a total wash even for high-N since implies maximal phase uncertainty (ÄÖ=∞>>2ð).

 

Fig. 6.6 Simulated spacetime photon counts for coherent (a-c) and photon-number states (d).

 

Deeper symmetry aspects of pair creation

            Discussion of relativity and quantum theory of wave amplitude requires further details. This includes Dirac’s extraordinary theory that 2-CW light of certain frequencies in a vacuum may create “real” matter that does not vanish when the light is turned off. For example, we know that two 0.51MeV ã-ray photons of frequency ù_e=m_ec²/h may create an electron and positron “hole” that form positronium  pairs. Also, 0.94GeV ã-rays with ù_p=m_pc²/h may create proton-anti-proton pairs, and so on.

            Dirac creation processes raise questions, “What “cavity” traps 0.51MeV ã-pairs into stable  pairs?” The discussion so far has only begun to define 2-CW symmetry properties by phase rates in per-spacetime (K,Ù)-quantum variables. Conservation (5.2) of these kinetic (K,Ù)-values implies that  orpairs have the same (K,Ù)-values as the 2-CW light that “creates” them.

            However, space-time symmetry arguments by themselves seem unable to derive internal lepton or baryon structure that might show how light becomes “trapped.” That question still lies beyond the scope of this discussion, and indeed, still largely beyond what is presently known. In fact, the current standard Weinberg-Salam model of high energy electroweak and strong quantum-chromo-dynamics (QCD) has abandoned the Dirac picture almost entirely. Pauli’s apparent dislike for Dirac may have had an effect.

            In its place have there has arisen a large and controversial area known as super-symmetric-string-theory or “superstrings” that has generated over 10,000 publications in about 40 years and promised a “theory of everything” that would include quantum gravity. However, this flurry of mathematical activity has not yet yielded new experimental or physical insight nor has it provided a better way to study or teach existing areas of classical mechanics, relativity or quantum theory.

            Two books give well written history of super-strings and related philosophy. One is by Lee Smolin and the other by Peter Woik. They show the presence at the highest academic levels of a rather pernicious group-think or make-believe that seems to have long given up the logical ideals of William of Occam.

 

L. Smolin, Trouble in Physics: The rise of string theory and the fall of a science, Houghton-Miflin (New York 2006)

P. Woik, Not even GNORW: The Failure of String Theory…, Persius Basic Books (New York 2006).