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From: Urs Schreiber (Urs.Schreiber@uni-essen.de)
Subject: Re: Physically understanding the Dirac equation and 4D.
Newsgroups: sci.physics.research
Date: 2002-03-28 18:06:52 PST

"Roman Arce" <shampoo@fibertel.com.ar> schrieb im Newsbeitrag
news:3ec0ba0c.0203261529.4352333e@posting.google.com...
> Does anybody have any idea of how to get a more physical understanding
> of Dirac's equation? I mean I can mess with the Dirac gamma matrices
> and do the math (specially with the help of the software Mathematica)
> but always loosing complete contact with the physical reality. Having
> four-vectors made of 4 components and each component being just a
> number it's physically understandable to me but when you need to
> replace each one of the 4 numbers with a 4x4 matrix it becomes
> something that can be played mathematically but it's ununderstandable
> physically (to me at least).
> Can the Dirac equation be understood physically, graphically, or I'm
> asking for something impossible like a human brain really watching 4
> dimensions graphically? With 4D for most things it's easy cos you can
> always think of only 1 space dimension and you're only left with 2D
> which is obviously easy to understand graphically and therefore
> physically but with the Dirac matrices you have spinors so you're
> stucked with 3D and with time it's 4D so any graphical thing is no
> longer an option, but could we still have a physical understanding
> some other way? I'm sure that if our brains could 'see' 4D all this
> wouldn't be a problem and the physical understanding would be easy (or
> at least possible).
> Any idea about how to 'see' the Dirac equation physically and any help
> about this subject is welcomed.

Dirac operators are quantized connections.

That's due to Quillen(see [2]).This simple statement contains in it the
key to all the intuition you may want about the meaning of Dirac
operators and Dirac equations in physics or elsewhere. I'll try
to give a first order approximation to an explanation of this
statement. But I will run out of time. To be continued...

First of all: I think your question is great. Almost all QM textbooks
give a treatment of the Dirac equation along the lines of Dirac's
original presentation. That was certainly a marvelous achievement,
then, but there has been a huge amount of progress - at least in the
mathematically oriented literature. Dirac operators are now recognized
as being intimately related to (semi-)Riemannian geometry and the
geometry of fiber bundles - that is: to the Geometry of Physics.

In fact, there is an approach due to Alain Connes [5] that goes the
other way round: Instead of starting with a geometry and deriving
a Dirac operator from there he starts with the algebra generated by
some function on a would-be manifold and an abstract operator
D. One can show that these ingredients alone define Riemannian
geometry as well as vast generalizations thereof. Connes identifies the
Dirac operator formally with the inverse of the infinitesimal length
element

D = 1/ds .

This may look a little hand-wavy, but can be given rigorous meaning
(see [5]). It shows what status the Dirac operator really has: It is
as fundamental as the notion of distance.This approach is immensely
powerful. (Very recently it has been generalized to semi-Riemannian
geometry: math-ph/0110001.)

But you write:
> I can mess with the Dirac gamma matrices [...]
> but always loosing complete contact with the physical reality .

That's a pity, because Dirac theory can get you much closer to
"physical reality." But it's probably not unusual, either, since the
underlying structure of Clifford algebra and Dirac theory is hardly
visisble in matrix representation. The key to a better understanding
is not to work (or at least not to think) in terms of explicit matrices
but in terms of "abstract" Clifford algebra. But don't worry, you'll
soon realize that that's far less "abstract" than you might think!
Quite the opposite: Doing away with the matrices reveals a
notational calculus well adapted to describe geometric and thus
physical notions.

At this point I have to mention the school of thought called
"Geometric Algebra". This originates with David Hestenes, who,
as a student, was troubled with similar questions as you raised
in your post. To his great delight (he describes that in some paper,
but I forget which one) he discovered how everything begins
to make sense when Clifford algebra is properly dealt with in
physics. He became very enthusiastic about this discovery and
started to rewrite huge wads of physics in a notation based on
Clifford algebra. (It might have helped if he had been a little less
enthusiastic.) There are lots of things to know and read about
Dirac theory, but you can give yourself a head start by first of all
looking at Hestenes' original book [6]. (A new version is in
preparation and online available [8]. Also see [7].) I'd say when
you feel familiar with his presentation, *then* look at the classic
texts on spinors and geometry [1]. With math textbooks it is
always helpful to know beforehand what they are *really* talking
about, because sometimes they are too shy to tell you frankly! :-)
I recommend to go to the Geometric Algebra/Geometric Calculus
homepage (I do not have the address at hand right now, search for
it via google) and have a look at the information to be found there.
The "geometric algebra"-school is often accused for overemphasizing
the importance of their work. That's because their practitioners are
so fond of the new insight they gain by rewriting existing results in
a neater way. But for someone troubled by questions like you asked
above this is a great source of information.

Now that I have said all this in order to put your question and my
attempt at an answer in proper perspective, I should finally begin
with answering:

> Can the Dirac equation be understood physically, graphically,

Yes. But there may be several ways to look at it. One way is
this (from a recent thread called "Dirac for Dunces")

---- post from the thread "Dirac for Dunces":

"John Baez" <baez@galaxy.ucr.edu>, wrote in
news:a1b5fc$fcn$1@glue.ucr.edu...

[...]

> "But if he doesn't remember his tensors, how in the world will he
> ever understand spinors."

Apropos "understanding spinors":
Maybe it helps to chat about some fascinating and intuitively
accessible details (it sure helps me): Mathematically, there is a
close relation between spinors and reflections. The fascinating
implication for physics is that discrete checkerboard models,
where massive particles move around at the speed of light but
reverse direction (reflection!) every now and then, actually
describe *spinning* particles!

How does that come about?

The best way to talk about spinors and reflections is, once again,
Clifford algebra, so I'll stick to that. Adopting Danny Ross
Lunsford's ASCII notation for Clifford algebra I call

the i-th unit vector with square

yi yi = - 1 .

(the negative sign is conventional, see below)
The (Clifford) product of two such unit vectors

yi yj ,

called a bivector, is, as the Wizard has already pointed out,
the parallelogram spanned by these two vectors. Since
this is supposed to change sign when its orientation is reversed
we have

yi yj = - yj yi (if i is different from j).

The first fact to note here is that every vector, say y0, carries
with it the instruction to reflect any other vector at the
(hyper-)plane orthogonal to y0 by applying y0 from both
sides:

A simple example makes this clear: Consider the
2-dimensional euclidean plane with unit vectors y0 and y1.
An arbitrary vector v can be written as a linear combination
of these

v = v0 y0 + v1 y1 .

If one multiplies y1 on both sides of v one gets

y0 v y0 = v0 (y0 y0 y0) + v1 (y0 y1 y0) .

But by the two rules of Clifford multiplication above it
follows that

y0 y0 y0 = -y0

and

y0 y1 y0 = - y0 y0 y1 = y1 .

So that finally

y0 v y0 = -v0 y0 + v1 y1,

which is v reflected at the "hyperplane" orthogonal to
y0, i.e. reflected at the line spanned by y1.

Now for rotations: Every rotation can be constructed
from two successive reflections. In particular, reflecting
a vector in succession at two intersecting hyperplanes
is tantamount to rotating this vector by an angle that is
exactly *twice* the angle between the hyperplanes.
(I have emphasized "twice" here, because it is exactly
this innocent and easily visualized factor of 2 that can be
traced back to be at the root of the not quite so easily
visualizable sign-change behavior of spinors under rotation
by 2pi, see below.)

Another simple example will easily illustrate this: Consider
again the Euclidean plane with orthogonal unit vectors y0
and y1 and the arbitrary vector v = v0 y0 + v1 y1.
Now watch how v behaves when it is reflected first
at y1 and then at y0:

y1 (y0 v y0) y1
= y1 (y0 (v0 y0 + v1 y1) y0) y1
= y1 (-v0 y0 + v1 y1) y1
= - v0 y0 - v1 y1
= -v

Hence, by these two reflections v is rotated by pi around the
origin, which is exactly twice the angle of pi/2 between y0 and y1.

If one instead wants to rotate v by an arbitrary angle, then one
has to exponentiate the action of the discrete pi rotation above:

v' = v rotated by the angle phi
= exp(phi/2 y1 y0) v exp(phi/2 y0 y1),
= R v R~

where the factor 1/2 is inserted to compensate the above
mentioned factor 2 and the tilde indicates that y1 y0 has
to be "reversed" to give y0 y1.

As I have mentioned in another recent post, the object
R, sometimes called a "rotor", which is "one
half" (the "left half" in this case) of a rotation operator, is
essentially a spinor. Why? Because under rotation it behaves
"just like" a vector, except that it picks up a sign when
rotated by 2pi. To see this, rotate the v' once more, this
time by an angle psi:

v'' = v' rotated by an angle psi
= exp(psi/2 y1 y0) v' exp(psi/2 y0 y1)
= exp(phi/2 y1 y0) R v R~ exp(psi/2 y0 y1)
= (exp(phi/2 y1 y0) R) v (exp(phi/2 y1 y0) R)~

Hence, while the *vector* v is rotated around by applying
rotation operators on *both* sides, the *spinor* R
is transformed under rotation by applying
exp(phi/2 y1 y0) only from the left. But because of the
factor 1/2 in the exponent of the latter (which was
introduced in order to cancel the factor of 2 that arose
because rotations were constructed from reflections)
the operator exp(phi/2 y1 y0) is equal to -1 for
a full rotation of phi = 2pi, not equal to 1. This does not
affect the transformation of the vector v, since it
is multiplied under 2pi rotations by -1 from both sides,
giving a total of 1. But it means that under a 2pi rotation
the object R transforms as

R -> -R .

Hence R is a spinor, qed. ... Well, ok, this is not really
precise and thus not really correct. I'll make it a little
more precise and thus a little more correct, but only so much
that I can discuss the physics that I actually set out to
discuss:

The somewhat more precise version, and I'll make a long story
short, is that which involves "minimal left ideals": The simple
point is that one wants the "algebraic spinor" R (as it is called) to
only notice what is going on on its left side. Hence, loosely
speaking, one projects its right side onto some subspace and
forgets about it.

I'll explain that again in terms of an example: To get closer to
the physics let the Euclidean plane from the previous examples
now be equipped with the lorentzian metric (1,-1), i.e.

y0 y0 = 1, y1 y1 = -1 .

In order to project something we need a projector constructed
from y0 and y1 and I'll let that projector be

P = (1 + y0)/2

which implies

y0 P = P
P P = P ,

so that P is recognized as the projector onto the +1 eigenspace
of y0.

The point is that it is quite correct to call the rotator R from
above a spinor when it is applied on the projector P:

RP is a spinor.

But remember, the point is merely that RP is transformed
by acting on it from the left, the P is essentially only there
to make precise the idea that we do not care about what
happens on the right hand side.

But since now R is applied to a projector, several different
Rs give the same result RP. A simple inspection shows that the
space of all possible RPs is spanned by only two elements,
which can for example be chosen as follows:

r = (1 + y0 y1)/2 P
l = (1- y0 y1)/2 P

The reason why I call these elements "r" and "l" will arise
shortly.

Observe that from general considerations we know that
spinors in 2n dimensions have 2^n components. Incidentally,
in 1+1 dimensions considered here, 2^1 = 2 are two
components of a spinor, which complies with the two
components r and l found above. Hence, a spinor valued
wave function psi in 1+1 dimensions can be written as

psi = rho r + lambda l

where rho and lambda are complex coefficients that
give a weight (or rather "amplitude") to the components
r and l.

This finally puts us in position to do some interesting
physics: The Dirac equation in 1+1 Minkowski space reads

( y0 d0 + y1 d1 ) psi = i m psi .

(Where d0 and d1 are partial derivatives in the y0 and y1
direction (i.e. time and space direction, respectively)
acting on the components lambda and rho.)

As is often done, I multiply this equation by y0 on
both sides for convenience

( d0 + y0 y1 d1 )psi = i m y0 psi

and reorder a little to make the time evolution transparent:

d0 psi = - y0 y1 d1 psi + im y0 psi .

This is nice, because with the above discussion of reflections
and rotations we can now do what is most fun in physics: We
can point at each term in an equation and give it a fancy
physical interpretation. First observe that

y0 y1 r = r
y0 y1 l = -l
y0 r = l
y0 l = r .

y0 y1 is, by the above discussion, the generator of rotations
in the space-time plane, that is: of Lorenz boosts in the
y1 direction. y0 y1 r = r says that r is invariant under
such boosts. But this must mean that r is a lightlike object
pointing in the y1 direction. Similarly, l is a lightlike
object pointing in the opposite direction. This is consistent
with y0 r/l = l/r, which says, that under time reflection
r and l interchange roles.

One can now read off the little story that the above equation is
telling:

In each time step (d0 psi) three different things happen:
The r component of psi is translated at the speed of light
a little in the -y1 direction (-d1 rho) while the l component is
translated at the speed of light a little in the +y1 direction
(d1 lambda). Furthermore, part of the r component is
switched to the l component (i m y0 r = i m l) and
vice versa (im y0 l = i m r).

This "story of the zig-zagging electron" has first been discovered
by Feynman for 1+1 dimensions, who called it the "chessboard
model", and has been generalized to 1+3 dimensions by
Ord, a few years ago (but in ugly matrix notation not in nice
Clifford algebra). Remarkably, this story contains no mention of
spin, only of reflections, and yet it describes "spinning"
electrons.
I hope the above derivation makes this at first puzzling connection
a little transparent.

Although I have not done it yet, by means of the Clifford
algebra approach above it should be easy to generalize
the checkerboard model to higher dimensions and
arbitrary complex interaction terms. There are two important
points here:

The y0 yi which multiply the partial derivatives of the
free Dirac operator have eigenvalues +/- 1. Hence
the free Dirac operator explicitly says: Translate all +1
eigenstates in one direction and all -1 eigenstates in the
other (at the speed of light!), and then swap some
eigenstates (in higher dimensions).

The interaction terms, like i m y0 psi, in general
swap the "directional" eigenstates. By the above discussion
one can easily see how each term of a vector potential

A = Ai yi

affects the checkerboard model (i.e. what exactly is does
to a zig-zagging electron at each discrete instance of time.)

This is fun if interaction is gravity: Writing the Dirac equation
for curved space-time

(yi di + 1/4 yi omega_ijk yj yk )psi = i m psi

(where omega is the spin connection), one sees that the
connection influences the weights of the left and right
going components of the electron path. For example, the
simple case of 1+1 dimensions has the general affine
connection

omega
= yi omega_ijk yj yk
= omega_001 y1 + omega_101 y0

(because of antisymmetry in jk).

Multiplying by y0 as above gives the interaction

omega_001 y0 y1 + omega_101 .

which increases the left going amplitude by

omega_001 - omega_101

and the right going amplitude by

-omega_001 - omega_101 .

More information on the checkerboard model can be
found at

http://www.cbloom.com/physics/2d_dirac.html#new

(which apparently originates from old postings here on
s.p.r) and references given there.

You may have noticed that the interactions I discussed in my above
discussion, the vector potential and gravity, did non introduce kinks into the
electron path, but merely influenced the amplitudes of the left and right
moving components, thereby luring the electron in one direction or
another. But wouldn't one expect that a potential in the checkerboard model
instead makes itself felt by directly influencing the probability for
reflections of the electron's velocity? Playing ping-pong with the particle?
I would, and here is how:

Instead of a vector-potential, I now introduce a super-potential (as
in the Witten model of supersymmetric quantum mechanics). I can
provide the details if desired, but the result is that the 1+1 dimensional
massless Dirac operator then looks like

D = y0 d0 + y1 d1 + Y0 W,0 + Y1 W,1

where

y0 and y1 are the generators of Cl(1,1) as before
d0 and d1 are the partial derivatives
Y0 and Y1 are generators of *another* copy of Cl(1,1), anticommuting
with y0 and y1
W,0 and W,1 are the d0 and d1 derivatives of a scalar function W.

Because of the new generators, Y0 and Y1, this Dirac operator
does not act on a single spin bundle S any more, but on the product
of two spin bundles SxS*. This can be represented by introducing
a "vacuum" element |0> and the rule

y0 |0> = Y0 |0>
y1 |0> = Y1 |0> .

When the spinor states of my previous post

r/l = (1 +/- y0 y1) (1 + y0)/4

are applied to this vacuum state,

r/l -> (1 +/- y0 y1) (1 + y0) /4 |0>

they also span a representation space for the action of Y0 and Y1.
One finds:

Y0 (1 +/- y0 y1) (1 + y0) /4 |0>
= (1 +/- y0 y1) (1 - y0) /4 Y0 |0>
= (1 +/- y0 y1) (1 - y0) /4 y0 |0>
= - (1 +/- y0 y1) (1 - y0) /4 |0>

and

Y1 (1 +/- y0 y1) (1 + y0) /4 |0>
= (1 +/- y0 y1) (1 - y0) /4 Y1 |0>
= (1 +/- y0 y1) (1 - y0) /4 y1 |0>
= (y1 -/+ y0) (1 + y0) /4 |0>

Note that Y0 switches the projector (1+y0)/2 to its complement, hence
it annihilates the (1+y0)/2 particle that has been considered so far and
instead creates a new kind of particle living in the (1-y0)/2 representation!

We'll also need the actions of y0 Y0 and y0 Y1. These are now readily seen
to be

y0 Y0 (1 +/- y0 y1) (1 + y0) /4 |0>
= - y0 (1 +/- y0 y1) (1 - y0) /4 |0>
= (1 -/+ y0 y1) (1- y0)/4 |0>

and

y0 Y1 (1 +/- y0 y1) (1 + y0) /4 |0>
= y0 (y1 -/+ y0) (1 + y0) /4 |0>
= -(1 -/+ y0 y1) (1 + y0) /4 |0> .

Hence both, y0 Y0 and y1 Y1, reverse the direction of the particle. The
former also switches the representation while the latter introduces a sign.

Ok, so this is what the terms in the above Dirac operator do. As before,
I now write out the Hamiltonian, i.e. the generator of time translations:

D psi = 0
<=> (y0 d0 + y1 d1 + Y0 W,0 + Y1 W,1) psi = 0
<=> d0 psi = ( -y0 y1 d1 - y0 Y0 W,0 - y0 Y1 W,1) psi

The operators on the right each do something to the "electron" in
a small unit of time, namely:

y0 y1 d1 translates the r component in the -y1 direction and the l component
in the +y1 direction a little bit, at the speed of light (as before)

y0 Y0 W,0 annihilates the particle and creates a corresponding particle
going in the opposite direction with amplitude W,0

y0 Y1 reverses the direction of the particle with amplitude W,1

Physical intuition is rather delighted:

There is a particle moving around at the speed of light, stochastically
reversing direction as on Feynman's checkerboard, but with a probability
that varies with the strength of the spatial potential W,1: The higher the
potential, the more the particle's path wiggles and the slower the particle
propagates (in the mean). Also, because the probability to return when heading
in a direction of lower potential decreases, the particle will (in the mean)
travel
in the direction opposite to the potential's gradient. (There is a subtlety
here,
because the proper potential is the square of W,1, but the qualitative idea is
correct.) If, in addition, the potential depends on time, i.e. when W,0 =/= 0 ,
so that the energy content of the system is not constant, the particle may
"decay" into its partner representation and back again.

This is, however, not the story of the electron, anymore.

--------------

> or I'm
> asking for something impossible like a human brain really watching 4
> dimensions graphically?

No. Of course, you can have Dirac equations in all dimensions and
on all kinds of rather exotic geometries, and these may be difficult
to handle by themselves, but for ordinary setups there is an ordinary
visualization of the Dirac equation. It goes like this (see also [3]):

Let's first concentrate on the homogeneous case, e.g. when a
massless particle (like a neutrino) is described. The the Dirac
equation reads

D psi = 0

where

D = ym nabla_m

is the Dirac operator.

Here ym are the generators of the Clifford
algebra

{ym,yn} = -2 gmn

and gmn are the components of the metric on your manifold. In flat
Minkowski space this usually looks like

{ym,yn} = 2 eta_mn

where

eta = diag(+1,-1,-1,-1) .

All Dirac operators always look this way (in local coordinates).
nabla_m is a covariant derivative

nabla_m = @m + omega_m

where @m is the partial derivative by x^m and omega_m are the
components of a connection on some fiber bundle. Really, nabla_m
are components of the connection 1-form

nabla = dx^m (x) nabla_m

that, when applied to a tangent vector v = v^n @_n, returns the
operator v^n nabla_n. This, in turn, is a differential operator that,
when applied to any field, say S, returns the first order
approximation of the change of S when parallely transported along
v.

This is the reason why Quillen says that Dirac operators are
"quantized connections". All Dirac operators arise by replacing
in the connection 1-form

dx^m (x) nabla_m

the differential form dx^m by the Clifford generator ym. This
replacement is sometimes called "quantization", because

{dx^m, dx^n} = 0

and

{ym, yn} = 2 eta^mn

differ like commutators of classical phase space functions and
quantum operators. But "quantized connections" have nothing to
do with quantization in the sense of quantum physics, it's just a
formal analogy. Dirac operators have their place in classical physics,
too.

D = ym nabla_m

is a *very* general construction. Physics enters by specifying a
fiber bundle and a specific connection on that bundle so that
nabla gets a definite meaning.

Unfortunately, I am running somewhat out of steam and also out of
time. I'll leave it at that for the moment, which is, admittedly, a very
incomplete state of things. But I'd be glad to discuss things further.

> Screw operators, screw differential equation, hail the path integral!

A Japanese proverb says (or so I have been told):

When you do not understand, mountains are mountains.
When you begin to understand, mountains are no longer mountains.
When you have understood, mountains will again be mountains.

----- references:

[1]

@book{
BerlineGetzlerVergne:1992,
author = {N. Berline and E. Getzler and M. Vergne},
title = {Heat Kernels and {D}irac Operators},
publisher = {Springer},
year = {1992}
}

@book{
LawsonMichelsohn:1989,
author = {H. Lawson and M. Michelsohn},
title = {Spin Geometry},
publisher = {{P}rinceton {U}niversity Press},
year = {1989}
}

@book{
BennTucker:1987,
author = {I. Benn and R. Tucker},
title = {An introduction to Spinors and Geometry},
publisher = {{A}dam {H}ilger},
year = {1987}
}

[1b]
@book{
Frankel:1997,
author = {T. Frankel},
title = {The Geometry of Physics},
publisher = {Cambridge University Press},
year = {1997}
}

[2]
@article{
RoepsdorffVehns:1999,
author = {G. Roepsdorff and Ch. Vehns},
title = {Generalized {D}irac Operators and Superconnections},
journal = {math-ph/9911006},
year = {1999}
}

[3]
@article{
RodriguesVazPavsic:1996,
author = {W. Rodrigues and J. Vaz and M. Pavsic},
title = {The {C}lifford bundle and the dynamics of the superparticle},
journal = {{B}anach Center Publications},
year = {1991},
volume = {37},
pages = {295-315}
}

[4]
@article{
RodriguesDeSouzaVazLounesto:1996,
author = {{W. Rodrigues, Jr.} and {Q. De Souza} and {J. Vaz, Jr.} and P.
Lounesto},
title = {{D}irac-{H}estenes spinor fields in {R}iemann-{C}artan spacetime},
journal = {\tt hep-th/9607073},
year = {1996}
}

[5]

@article{
Varilly:1997,
author = {J. Varilly},
title = {An introduction to Noncommutative Geometry},
journal = {\tt physics/9709045},
year = {1997}
}

@book{
Connes:1994,
author = {A: Connes},
title = {Noncommutative Geometry},
publisher = {Academic Press},
year = {1994}
}

[6]
@book{
Hestenes:1966,
author = {D. Hestenes},
title = {Space-time algebra},
publisher = {Gordon and Breach science publishers},
year = {1966}
}

[7]
@book{
HestenesSobczyk:1984,
author = {D. Hestenes and G. Sobczyk},
title = {{C}lifford Algebra to {G}eometric {C}alculus},
publisher = {D. Reidel Publishing Company},
year = {1984}
}

[8]
@book{
Hestenes:2002,
author = {D. Hestenes},
title = {Spacetime Calculus},
publisher = {(in preparation, draft available at {\tt
modelingts.la.asu.edu/HTML/STC.html})},
year = {2002}
}

--
Urs.Schreiber@uni-essen.de

From: Urs Schreiber (Urs.Schreiber@uni-essen.de)
Subject: Re: Dirac for Dunces
Newsgroups: sci.physics.research
Date: 2002-01-12 12:34:01 PST

"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:a1euhh$js0$1@rs04.hrz.uni-essen.de...

[...]

> Mathematically, there is a
> close relation between spinors and reflections. The fascinating
> implication for physics is that discrete checkerboard models,
> where massive particles move around at the speed of light but
> reverse direction (reflection!) every now and then, actually
> describe *spinning* particles!

[...]

> Actually, I am also too tired to continue writing this post,
> so I'll stop here.

Now that I'm fresh again, I can't refrain from adding one more interesting
detail. You may have noticed that the interactions I discussed in my above
post, the vector potential and gravity, did non introduce kinks into the
electron path, but merely influenced the amplitudes of the left and right
moving components, thereby luring the electron in one direction or
another. But wouldn't one expect that a potential in the checkerboard model
instead makes itself felt by directly influencing the probability for
reflections of the electron's velocity? Playing ping-pong with the particle?
I would, and here is how: