history update, schrodinger name fix, etc

Author: rcoreilly

citedrefs.json

@@ -4,13 +4,13 @@ bibfile = "mechphys.json"
 
 <!--- TODO: intro, also [[@MallickChandrashekar16]] -->
 
-We have mentioned that a major problem with the scalar KG equation developed so far is that it doesn't represent any kind of conserved value: you cannot compute some constant, unchanging value from the $\varphi$ state variables under this equation. Why is this a problem? If you want to develop a probabilistic interpretation of the wave, as in Schrodinger's equation, then you need this. But the conserved value that emerges naturally from the KG equation comes with two different signs, positive and negative, whereas Schrodinger's equation always produces a positive value. This is one of the major reasons why the KG equation is not widely discussed in quantum physics: it doesn't quite fit with the standard probabilistic framework.
+We have mentioned that a major problem with the scalar KG equation developed so far is that it doesn't represent any kind of conserved value: you cannot compute some constant, unchanging value from the $\varphi$ state variables under this equation. Why is this a problem? If you want to develop a probabilistic interpretation of the wave, as in Schrödinger's equation, then you need this. But the conserved value that emerges naturally from the KG equation comes with two different signs, positive and negative, whereas Schrödinger's equation always produces a positive value. This is one of the major reasons why the KG equation is not widely discussed in quantum physics: it doesn't quite fit with the standard probabilistic framework.
 
-Instead, it seems to make much more sense to interpret the KG waves as **waves of charge**, because charge is also strictly conserved, and it comes in both positive and negative varieties. Indeed, the authors that do write extensively about the KG equation adopt this interpretation (Greiner, 2000; Gingrich, 2004; Mandl and Shaw, 1984). Furthermore, we will see that this charge interpretation fits naturally with the coupling of this KG equation to the electromagnetic field, where the conserved charge value acts just like the electric charge in driving the field. Interestingly, this idea was pursued initially by Schrodinger in 1926, and has been pursued more recently in neoclassical self-coupled field theory (Jaynes & Cummings, 1963; Crisp & Jaynes, 1969; Barut & Van Huele, 1985; Barut & Dowling, 1990; Crisp, 1996; Finster, Smoller & Yau, 1999c; Radford, 2003; Masiello, Deumens & Ohrn, 2005). We discuss this approach, which is essentially an analytic version of our computational model, in more detail later.
+Instead, it seems to make much more sense to interpret the KG waves as **waves of charge**, because charge is also strictly conserved, and it comes in both positive and negative varieties. Indeed, the authors that do write extensively about the KG equation adopt this interpretation (Greiner, 2000; Gingrich, 2004; Mandl and Shaw, 1984). Furthermore, we will see that this charge interpretation fits naturally with the coupling of this KG equation to the electromagnetic field, where the conserved charge value acts just like the electric charge in driving the field. Interestingly, this idea was pursued initially by Schrödinger in 1926, and has been pursued more recently in neoclassical self-coupled field theory (Jaynes & Cummings, 1963; Crisp & Jaynes, 1969; Barut & Van Huele, 1985; Barut & Dowling, 1990; Crisp, 1996; Finster, Smoller & Yau, 1999c; Radford, 2003; Masiello, Deumens & Ohrn, 2005). We discuss this approach, which is essentially an analytic version of our computational model, in more detail later.
 
 Our emerging picture of a particle is therefore that it is a distributed wave-packet of charge that flows through space according to some variant of the KG equation. This conflicts with the standard view that charge is contained within single discrete point particles, that somehow float around in clouds of moving probability waves. Indeed, even such a quasi-mechanistic view is too strong for the classical Copenhagen interpretation of quantum physics, which says that the particles simply do not have any definite existence until they are measured by some kind of measuring device. We return to such issues later.
 
-Now, let's see how we can get this conserved charge value out of a KG equation. Fortunately it is quite simple, and somewhat magical. By just computing the same basic KG equation on a complex state variable ($\phi$) instead of a scalar state variable ($\varphi$), a conserved quantity emerges. The magic of this result is magnified by the fact that, as a second order wave equation without an $i$ term, the KG equation on a complex variable is identical to just computing two separate KG equations on each of the two scalar values represented by the complex state variable (i.e., $\varphi_a$ and $\varphi_b$). Note that this was not true of Schrodinger's wave equation, which is first order and has an $i$ term that causes the $a$ and $b$ terms to intermix as the wave unfolds.
+Now, let's see how we can get this conserved charge value out of a KG equation. Fortunately it is quite simple, and somewhat magical. By just computing the same basic KG equation on a complex state variable ($\phi$) instead of a scalar state variable ($\varphi$), a conserved quantity emerges. The magic of this result is magnified by the fact that, as a second order wave equation without an $i$ term, the KG equation on a complex variable is identical to just computing two separate KG equations on each of the two scalar values represented by the complex state variable (i.e., $\varphi_a$ and $\varphi_b$). Note that this was not true of Schrödinger's wave equation, which is first order and has an $i$ term that causes the $a$ and $b$ terms to intermix as the wave unfolds.
 
 To be explicit, the KG wave equation for a complex state variable $\phi$ is:
 
@@ -30,7 +30,7 @@ $$
 
 where again the $\varphi_a$ indicates a scalar state variable representing the real $a$ component of $\phi$, and $\varphi_b$ represents the imaginary $b$ value.
 
-Ok, so how do you get a charge out of that, so to speak? As with Schrodinger's equation, the procedure involves multiplying by the complex conjugate ($\phi^* = \varphi_a - i \varphi_b$), which generally produces the overall magnitude or length of the vector represented by the two components of the complex number: $\varphi_a^2 + \varphi_b^2$. As the derivation presented in detail in Appendix B shows, if you compute the sum of this value across all of space (actually an integral, using continuous equations), and set it equal to zero (so that it never changes), you end up with an expression for the density and motion (current) of a quantity that is conserved (i.e., the charge).
+Ok, so how do you get a charge out of that, so to speak? As with Schrödinger's equation, the procedure involves multiplying by the complex conjugate ($\phi^* = \varphi_a - i \varphi_b$), which generally produces the overall magnitude or length of the vector represented by the two components of the complex number: $\varphi_a^2 + \varphi_b^2$. As the derivation presented in detail in Appendix B shows, if you compute the sum of this value across all of space (actually an integral, using continuous equations), and set it equal to zero (so that it never changes), you end up with an expression for the density and motion (current) of a quantity that is conserved (i.e., the charge).
 
 The resulting expression for computing the density of charge (typically written as $\rho$, which is the Greek letter "rho"), which is to say, the amount of charge per cubic state unit, is:
 
@@ -50,7 +50,7 @@ This is directly computable for each cubic cell $i$ in the system.
 
 It becomes very clear when explicitly written out in this manner that charge represents a coupling of the two otherwise independent variables in the complex number, and this suggests why a single scalar number cannot represent a conserved charge value. The fact that these variables are coupled here, but not in the actual wave equations that drive their updating, seems magical.
 
-Perhaps the most important feature of this equation is that it can be either positive or negative. For example, if ${\varphi_a}_i \dot {\varphi_b}_i$ happens to be larger than ${\varphi_b}_i \dot {\varphi_a}_i$ (and there is nothing preventing this from being the case), then it will be negative. This is not true of the corresponding expression for Schrodinger's equation, which is "definitely positive", or, in mathematical terminology, "positive definite". As noted earlier, this is one of the major reasons why standard quantum physics has strongly embraced Schrodinger's equation, and not KG: KG does not fit with the standard probabilistic framework, where the wave describes a probability, and a probability is always positive.
+Perhaps the most important feature of this equation is that it can be either positive or negative. For example, if ${\varphi_a}_i \dot {\varphi_b}_i$ happens to be larger than ${\varphi_b}_i \dot {\varphi_a}_i$ (and there is nothing preventing this from being the case), then it will be negative. This is not true of the corresponding expression for Schrödinger's equation, which is "definitely positive", or, in mathematical terminology, "positive definite". As noted earlier, this is one of the major reasons why standard quantum physics has strongly embraced Schrödinger's equation, and not KG: KG does not fit with the standard probabilistic framework, where the wave describes a probability, and a probability is always positive.
 
 Interestingly, the Dirac equation, which standard physics has adopted as a model of the electron (and we'll cover later), also produces negative "probabilities", but these have been (correctly, in our framework) reinterpreted as representing antiparticles (i.e., particles with an opposite charge). The antiparticle of the electron is the **positron**, and it is just like an electron, except it has the opposite charge. Historically, this antiparticle nature of the Dirac equation was regarded as a major problem, until positrons were subsequently discovered, and then Dirac looked like a genius for having made such a bold prediction. Nevertheless, there seems to be some residual discomfort in all this, and many treatments of quantum electrodynamics marginalize the Dirac equation in favor of a largely particle-based treatment. We return to these issues later.
 
@@ -107,7 +107,7 @@ $$
 \frac{\partial^2 {A_0}}{\partial t^2} = \nabla^2 A_0 + \frac{1}{\epsilon_0} \rho
 $$
 
-However, in Schrodinger's equation, external forces enter as a potential ($V$), in the first-order derivative  $\frac{\partial {}}{\partial t}$:
+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 {\phi}}{\partial t} = \frac{\hbar^2}{2m} \nabla^2 \phi + V \phi
@@ -304,7 +304,7 @@ The final step in our long journey is to come to terms with the full glory of th
 
 The third basic property of the electron is known as its spin. It is known as a spin $\frac{1}{2}$ particle, along with all of the other fundamental particles know (e.g., quarks). Unfortunately, our wave equations so far do not support this spin property, and so we'll need to do a little bit more work. However, once we're done, we'll find that our equations capture all of the fundamental properties of the electron: we should have a 100% complete description of it! Actually there is one last thing, which is that the electron is a member of the _lepton_ family, whereas the other fundamental particles are quarks, and they have other fundamental properties in addition to those carried by leptons. But, this is presumably because quarks live in some other set of state variables separate from the lepton state variables we're simulating here in our model. So, with that assumption, we might have captured everything about the electron.
 
-So what exactly does it mean for an electron to have spin $\frac{1}{2}$? The quantum mechanical concept of spin is perhaps one of the most difficult to grasp. Sometimes people try to think of a little point particle spinning about like a top on its axis, but this doesn't actually fit the facts very well. In the end, the best strategy may be to just see what the equations we derive next actually do, and call that spin. Indeed, in the computer simulations, one can clearly see a spinning motion. This spin is very much like the first-order Schrodinger equation dynamics, where the two different elements of the complex variable rotate into each other. We also saw this kind of rotation earlier in the complex coupled KG equation, where the electromagnetic potential introduced a rotation among the complex variables. In the present case, we're going to have four different variables, and they will all rotate amongst themselves to produce this mysterious spin.
+So what exactly does it mean for an electron to have spin $\frac{1}{2}$? The quantum mechanical concept of spin is perhaps one of the most difficult to grasp. Sometimes people try to think of a little point particle spinning about like a top on its axis, but this doesn't actually fit the facts very well. In the end, the best strategy may be to just see what the equations we derive next actually do, and call that spin. Indeed, in the computer simulations, one can clearly see a spinning motion. This spin is very much like the first-order Schrödinger equation dynamics, where the two different elements of the complex variable rotate into each other. We also saw this kind of rotation earlier in the complex coupled KG equation, where the electromagnetic potential introduced a rotation among the complex variables. In the present case, we're going to have four different variables, and they will all rotate amongst themselves to produce this mysterious spin.
 
 Incidentally, quantum physics holds that photons (which we think of as wave packets of the electromagnetic field that we've already characterized above) have a spin of 1. Furthermore, the charged complex KG wave equation is described as having a spin of 0. This latter case makes sense to me, in that the two components of the complex number do not rotate into each other, and thus they do not spin at all. However, the electromagnetic field case is a bit more confusing, because as we saw, the four components of this field do not interact with each other in the basic wave equations either! Therefore, it would seem that it should have a spin of 0 as well. Countering this are two considerations. First, the observable variables of the electric and magnetic fields $\vec{E}$ and $\vec{B}$, which are derived from these non-interacting electrical potentials, do rotate around each other as the wave propagates. Second, when these potentials interact with our charge wave, the do so in a way that ends up coupling (and rotating) the two independent scalar values in the complex number, and thus they impart some spin on our otherwise spinless particle.
 

content/dirac.md

@@ -15,7 +15,7 @@ Surprisingly, one still observes the interference effect in this case ([[#figure
 The [[pilot-wave]] framework of de Broglie and Bohm provides the most natural, intuitive explanation of these effects: the wave goes through both slits, and the particle goes through one, but it is influenced by the wave.
 
 {id="figure_double-slit-deb" style="height:20em"}
-![Trajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot-wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrodinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don't happen to observe them.](media/fig_double_slit_debroglie_bohm.png)
+![Trajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot-wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrödinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don't happen to observe them.](media/fig_double_slit_debroglie_bohm.png)
 
 [[#figure_double-slit-deb]] shows what the underlying trajectories of particles under the pilot-wave framework look like in a double-slit experiment, and [[#figure_double-slit-kocsis]] shows some recent data from an experiment where _weak measurements_ that minimally disturb the system allow one to infer particle trajectories, which look remarkably similar to those predicted by the pilot-wave model ([[@KocsisBravermanRavetsEtAl11]]).