@@ -74,7 +74,7 @@ hbar := 0.5
 e := 1.0
 csq := c * c
 mcOverHSq := (mass * mass * csq) / (hbar * hbar)
-heOver2mCSq := (hbar * e) / (2.0 * mass * csq)
+heOverMCSq := (hbar * e) / (mass * csq)
 
 var bphaseStr, massStr, hbarStr, msgStr string
 
@@ -89,15 +89,15 @@ chg := zeros(totalTime)
 
 func valUpdate() {
     mcOverHSq = (mass * mass * csq) / (hbar * hbar)
-    heOver2mCSq = (hbar * e) / (2.0 * mass * csq)
+    heOverMCSq = (hbar * e) / (mass * csq)
     bphaseStr = fmt.Sprintf("b phase: %4.0f", bphase)
     massStr = fmt.Sprintf("mass: %4.1f", mass)
     hbarStr = fmt.Sprintf("hbar: %4.1f", hbar)
     bi := ai * float64(math32.Cos(math32.DegToRad(float32(-bphase))))
-    bvi := math.Sqrt(mcOverHSq) * ai * float64(math32.Sin(math32.DegToRad(float32(-bphase))))
+    bvi := c * math.Sqrt(mcOverHSq) * ai * float64(math32.Sin(math32.DegToRad(float32(-bphase))))
     ##
     mf := array(mcOverHSq)
-    cf := array(heOver2mCSq)
+    cf := array(heOverMCSq)
     ap := array(ai)
     bp := array(bi)
     av := array(0.0)
@@ -111,10 +111,10 @@ func valUpdate() {
         db[t] = bv
         chg[t] = cf * (bp * av - ap * bv)
         
-        av -= mf * ap
+        av -= csq * mf * ap
         ap += av
         
-        bv -= mf * bp
+        bv -= csq * mf * bp
         bp += bv
         ##
     }
@@ -183,7 +183,7 @@ addSlider(&massStr, &mass, 0.1, 1.0)
 addSlider(&hbarStr, &hbar, 0.1, 1.0)
 ```
 
-[[#sim_cc]] demonstrates how this works, in terms of two simple [[harmonic oscillator]] variables _a_ and _b_, which are set to be a specific phase apart from each other (+90 degrees shifts _b_ to the _left_ (earlier) relative to _a_, while -90 shifts to the right, due to the trigonometric convention of 0 degrees being at 1,0 and proceeding counter-clockwise from there). Regardless of the phase relationship, the computed charge value remains constant across the cycles of oscillation. However, critically, the value of the charge is directly a function of this phase relationship, with a maximum of 0.5 when the _b_ value is +90 degrees in relation to the _a_ value, and a minimum of -0.5 for -90 degrees, and zero for 0 or 180 degrees. These relationships are fairly obvious once you appreciate the relationship between velocity and position for each of the variables (which are 90 degrees out of phase with each other, always), and how they enter into the charge equation.
+[[#sim_cc]] demonstrates how this works, in terms of two simple [[harmonic oscillator]] variables _a_ and _b_, which are set to be a specific phase apart from each other (+90 degrees shifts _b_ to the _left_ (earlier) relative to _a_, while -90 shifts to the right, due to the trigonometric convention of 0 degrees being at 1,0 and proceeding counter-clockwise from there). Regardless of the phase relationship, the computed charge value remains constant across the cycles of oscillation. However, critically, the value of the charge is directly a function of this phase relationship, with a maximum of 1 (in units of _e_, for a unit amplitude wave) when the _b_ value is +90 degrees in relation to the _a_ value, and a minimum of -1 for -90 degrees, and zero for 0 or 180 degrees. That maximum is not a coincidence: a wave at rest has $\rho = -e |\chi|^2$ exactly, which is what makes the charge of a lump of this stuff just its squared magnitude. These relationships are fairly obvious once you appreciate the relationship between velocity and position for each of the variables (which are 90 degrees out of phase with each other, always), and how they enter into the charge equation.
 
 Critically, complex numbers are _always_ 90 degrees out of phase with each other by the very nature of the complex plane. Thus, even though the separate real-valued wave functions are independently updated, it is critical that these two wave states are _initialized_ with the 90 degree phase relationship appropriate for complex numbers, which will then determine the sign of the charge value represented.
 
@@ -246,15 +246,15 @@ $$
 However, in Schrödinger's equation, external forces enter as a potential ($V$), in the first-order derivative  $\frac{\partial {}}{\partial t}$:
 
 $$
-i \hbar \frac{\partial {\chi}}{\partial t} = \frac{\hbar^2}{2m} \nabla^2 \chi + V \chi
+i \hbar \frac{\partial {\chi}}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \chi + V \chi
 $$
 
 This makes sense, because force is the derivative of a potential, so potential is a first-order factor, and force is a second-order factor.
 
 Our KG (Klein-Gordon) charge wave equation is a second-order equation, expressed in terms of $\frac{\partial^2 {}}{\partial t^2}$, and therefore we need to include external driving forces, not potentials. However, for various reasons, it is necessary to derive such an equation starting from the potential. To do this, we can re-derive a second-order wave equation by replacing the first-order derivative with the following first-order _covariant_ derivative operator $D$, that subtracts the external driving potential:
 
 $$
-D_\mu \def \partial_\mu - i \frac{e}{c} A^\mu
+D_\mu \def \partial_\mu - i \frac{e}{\hbar c} A_\mu
 $$
 
 This derivative can be derived from the principle of _local gauge invariance_, in [[gauge theory]], as a way to compensate for the introduction of a local phase factor, which is parameterized by the $A^\mu$ potential field. In other words, local gauge invariance means that the EM field potential might be different at every different point in space, and gauge theory shows you how to have the EM field and charge field interact in a way that makes this situation mutually compatible, so that charge and energy end up being conserved over time.
@@ -371,26 +371,11 @@ $$
 
 At this point, we have reached an important milestone --- if you take the equations just presented above, this describes a particle as a distributed wave of charge that gets pushed around by the electromagnetic field potentials ($A_0$ and $\vec{A}$). Furthermore, this wave of charge produces electromagnetic fields, in terms of charge and current densities $\rho$ and $\vec{J}$. Thus, we finally have a complete system of equations that can potentially simulate charged particles whizzing around and interacting with each other. In other words, we finally have the potential to make direct contact with observable physics! Indeed, you can explore the behavior of this system in the model, by using the Complex Coupled KG wave equations setting.
 
-## Numerical Issues with Coupling: Symmetry Breaking
+## Self-field interactions
 
-When you actually simulate these equations on the computer, something very interesting (and initially distressing) happens --- they blow up! As you run the equations over time in the presence of a fixed electromagnetic field, the total charge value, far from being a constant, increases steadily, and eventually you end up with numbers approaching infinity. This is not because the math is wrong (after very thorough checking!), but because of the coupling between the two elements of the complex variable that occurs in the update equation. Specifically, the update of $\phi_a$ depends on $\dot \phi_b$ and $\vec{\nabla} \phi_b$, and vice-versa. This interdependency creates numerical instabilities when we simply substitute in the discrete computed values at each time step. In particular, because the change in $\phi_a$ depends on the change in $\phi_b$, this cycle of dependency can get out of whack.
+When you actually simulate these equations on the computer, the self-coupling interactions between these two fields has a tendency to result in positive feedback loops that quickly go to infinity. There is an important numerical integration fix, known as the _Boris push_, which deals with the fact that the electromagnetic potentials drive a rotation through the $\phi_a$ and $\phi_b$ variables, which is evident in the fact that they subtract from $\phi_b$ but add to $\phi_a$ --- these opposite signs are the signature of a rotation (and are caused by the presence of the imaginary $i$ numbers in the equations). The Boris push implements this rotation explicitly, instead of using discrete integration steps, and helps manage the numerical issues ([[@Boris70]]; [[@QinZhangXiaoEtAl13]]). It is also important to ensure that the numerical values do not exceed critical thresholds, which are within the constraints of the known physical parameters. 
 
-Intuitively, the electromagnetic potentials drive a rotation through the $\phi_a$ and $\phi_b$ variables, which is evident in the fact that they subtract from $\phi_b$ but add to $\phi_a$ --- these opposite signs are the signature of a rotation (and incidentally are caused by the presence of the imaginary $i$ numbers in the equations). To the extent that the potentials are pushing the $\phi_a$ variable up, there should be an equal and opposite pushing of the $\phi_b$ variable down, causing the rotation. However, if the $\phi_b$ variable only has the "old" data from the previous time step about how much $\phi_a$ got pushed up, then it doesn't compensate correctly in how much it gets pushed down. Thus, you end up with a "leak" in the system, where instead of rotating nicely in place, the system starts to fly out of control, spinning wider and wider circles each time.
-
-The solution to this problem is to _break the symmetry_ between $\phi_a$ and $\phi_b$ in these update equations. Instead of updating each of them at the same time, based on the prior values of the other, we choose one variable ($\phi_a$, arbitrarily) and update its values first. Then, when we compute $\phi_b$, we use the _current value_ of $\dot \phi_a$ in the update equation for $\phi_b$. This prevents the rotation between these variables from getting out of whack, and restores numerical stability to the system.
-
-<!--- todo: run KG invr5 case, plot figure! -->
-
-`\begin{figure}`
-` \centering\includegraphics[height=2in]{fig.dirac_invr5_50.eps}   \caption{\small Total charge for updating of the charge wave     equation over 100,000 time steps in the presence of a fixed $1/r$ potential with magnitude .5, in a universe of $50^3$ cubes.  The`  
-`   instantaneous total charge varies considerably over time, but the average     across time is constant, demonstrating that total charge is conserved on     average, but not at each moment.  If no potential is present, then charge     is identically conserved from one time step to the next.}`  
-` \label{fig.kg_invr5_50} \end{figure}`
-
-However, if you run this equation in a static electromagnetic field, it is clear that the total amount of charge in the model at any one time changes over time (Figure~\ref{fig.kg_invr5_50}). This flies in the face of the fancy math that says that these equations conserve charge! Somewhat amazingly, however, if you run the system long enough, it becomes clear that the _average_ amount of charge never changes.
-
-Overall, we have now happened upon a very interesting situation. The breaking of symmetry between the two variables in the complex wave state, forced upon us by implementational considerations, actually fits at least qualitatively with a known and otherwise very puzzling property of physics. The weak force also breaks symmetry in a very similar way: there is a preferred direction of rotation in the weak force. Although it is not yet clear (to me at least) that this preferred rotational direction in the weak force maps identically onto this preferred rotation direction, it is nevertheless a tantalizing possibility. The weak force has been integrated with the electromagnetic force, as the _electroweak_ force --- it is possible that the effects described by the electroweak force correspond in some way to the oscillations in charge value that are observed in our model. We will return to these issues later.
-
-Meanwhile, we nee need to introduce just a bit more complexity into our KG wave equation before we have a fully satisfactory model of a fundamental particle of nature: the electron (and its antiparticle, the positron). This extra bit of complexity extends the phenomenon of rotation that we've just been discussing, to account for the strange quantum mechanical property of **spin**. The resulting equation goes by the name of the second-order Dirac equation. Once we have that, we will have a complete system that, if all the math is correct, should make direct and numerically accurate contact with observable phenomena!
+Meanwhile, we nee need to introduce just a bit more complexity into our KG wave equation before we have a fully satisfactory model of a fundamental particle of nature: the electron (and its antiparticle, the positron). This extra bit of complexity extends the phenomenon of rotation that we've just been discussing, to account for the strange quantum mechanical property of [[spin]]. The resulting equation goes by the name of the second-order Dirac equation. Once we have that, we will have a complete system that, if all the math is correct, should make direct and numerically accurate contact with observable phenomena!
 
 ## Summary
 
 
@@ -0,0 +1,60 @@
++++
+Categories = ["Standard Model"]
+bibfile = "mechphys.json"
++++
+
+<!--- TODO: WaveC, Schrodinger, WaveCDir, Weyl logic. -->
+<!--- claude todo: try the idea of 2nd order complex wave?? -->
+
+The _second-order_ [[wave]] equation is the prototypical version of a wave, which corresponds to macroscopic water waves and other such familiar phenomena. The core dynamic in such waves is the continuous bidirectional conversion of energy between wrinkles in _space_ (potential energy) and movement through _time_ (kinetic energy). However, standard quantum mechanics is based on [[Schrodinger]] waves, which obey a _first-order_ equation that uses the rotational properties of [[complex number]]s instead of the second-order integration of kinetic-energy to accomplish the core oscillatory property of waves.
+
+The ability to capture oscillations through either a second-order kinetic system, or a first-order complex-number system, plays out in various ways throughout the larger space of quantum wave equations. Here, we develop a systematic understanding of this space, and the relative tradeoffs between these different wave functions.
+
+Of primary interest is the extent to which the different wave equations tend to spread out over time (known as **dispersion**, which is also related to **diffraction**), because that relates to the [[epistemic]] vs _ontological_ aspect of what the wave is describing: as a wave propagates, the uncertainty in where it is going should generally increase, especially to the extent it exhibits [[stochastic motion]]. This is purely epistemic: our knowledge of it becomes less precise, but, under the [[pilot wave]] framework, a discrete particle should be localized precisely at all times, and its surrounding [[quantum wave]] should presumably _not_ spread out.
+
+Thus, if we can find wave equations that differ in their tendency to spread out, this may have implications for the kinds of waves we might want to be using for a pilot-wave model. One interesting conclusion from this analysis is that the [[neutrino]] could be seen as a kind of necessary byproduct of a type of first-order electron waves that don't spread out as much as they do in the second-order wave function (described by the [[Weyl]] equation, which is what is necessary for the [[weak#electroweak]] framework). Furthermore, the Schrödinger wave function lies at the other extreme end of the spectrum in terms of its tendency to spread out, which would seem to have major implications for all of the abstract quantum math based on it.
+
+Before proceeding, it is particularly useful to first read the [[harmonic oscillator]] page, which provides a simple and direct comparison between a second-order kinetic version of an oscillator, and a first-order complex-number version of the same oscillator dynamics. This harmonic oscillator has a single spatial value (i.e., within a single isolated [[cellular automaton]] cell), so its spatial dimensionality is much simpler than the wave, especially when it spreads over 3D space. That makes it easier to see the fundamental difference in the way the time dimension works, which is the key difference between the first and second-order wave equations.
+
+The natural starting point with respect to dispersion is with a second-order wave equation configured instead to implement the **diffusion equation**. This is a very simple change in the equation, which also reveals the essential role of the second-order acceleration in driving the oscillatory behavior of the wave equation. You simply directly couple the spatial curvature (potential energy) directly to the _velocity_ (first order), without going through the acceleration (second order).
+
+{id="eq_wave" title="wave equation"}
+$$
+\frac{\partial^2 \phi}{\partial t^2} = c^2 \frac{\partial^2 \phi}{\partial x^2}
+$$
+
+{id="eq_diffuse" title="diffusion equation"}
+$$
+\frac{\partial \phi}{\partial t} = c^2 \frac{\partial^2 \phi}{\partial x^2}
+$$
+
+When you run this equation (you can do it in the [[waves simulation]]) you see any disturbance melt away over time -- any "concentration" of wave "stuff" simply diffuses down into a uniform splat.
+
+This is an interesting starting point, because the first-order Schrödinger equation is effectively the same as the diffusion equation, except it operates on complex state variables, and it sticks an $i$ in there to drive the rotation.
+
+{id="eq_schrod" title="Schrödinger essentially"}
+$$
+\frac{\partial \chi}{\partial t} = -i \frac{\partial^2 \chi}{\partial x^2}
+$$
+
+where $\chi = \phi_a + i \phi_b$ is a complex-valued wave state with the two underlying values, which are just ordinary numbers. Although everyone somehow gets hung up on the $\phi_b$ number being "imaginary", it is just a regular number that happens to be located on an orthogonal plane to the "real" number, so that these two numbers can rotate around each other _in qualitatively the same way that velocity and the state position value do_ in the second-order version of the wave equation. 
+
+Is the velocity in a second-order equation "imaginary"? No. It is just another degree of freedom -- a different value that you can use to make things oscillate. The connection between rotation and oscillation is basic trigonometry: the sine and cosine are waves that arise in any orthogonal basis coordinates as you rotate around a circle, and the complex plane just provides a 2D space for this rotation to occur.
+
+To understand exactly what is happening in [[#eq_schrod], we can write it in terms of the two ordinary numbers. The key algebraic step is that you only keep the factors _without_ an $i$ for the $\phi_a$ factor, and those _with_ an $i$ for the $\phi_b$ factor. Furthermore, the convenient fact that $i^2 = -1$ makes the signs work out correctly for the rotation. The net result is that the $-i$ factor makes $\phi_a$ depend on $\phi_b$ and vice-versa -- it causes the two values to rotate into each other:
+
+$$
+\frac{\partial \phi_a}{\partial t} = - \frac{\partial^2 \phi_b}{\partial x^2}
+$$
+
+$$
+\frac{\partial \phi_b}{\partial t} = \frac{\partial^2 \phi_a}{\partial x^2}
+$$
+
+From the perspective of the second-order wave equation ([[#eq_wave]]), the Schrödinger equation is a bit strange, because the spatial factor on the right-hand side is a _second-order_ spatial derivative (also written as $\nabla^2 \phi$ where $\nabla^2$ is the Laplacian).
+
+The critical advantage of this second-order spatial derivative is that _it doesn't care about direction:_ a spatial disturbance in any direction surrounding a given point in the wave state will give rise to a change in that wave state, proportional to the magnitude of the disturbance. This means that this equation supports waves traveling in any direction, just like the standard second-order wave equation, because that directional property depends entirely on the nature of the spatial derivative factor, which is shared with the second-order wave equation.
+
+However, the coupling of this second-order spatial factor directly to the first-order temporal derivative in the Schrödinger equation results in an extremely high level of dispersion
+
+<!--- todo: directional wave, then helical. -->