Dirac's paper on QM of Electron

[Stephen Montgomery-Smith]

I'm trying to read Dirac's paper "The Quantum Theory of the Electron."
I am really struggling with the first displayed equation on 611. Is
there a sign error? Should it read something like:

F = -(W/c - eA_0/c)^2 + (p + eA/c)^2 + m^2c^2?

I googled the Klein-Gordon equation, which Dirac is motivating in this
section, and it does suggest the same sign error occurs in equation (1).

I don't think this error significantly effects the main results. But I
am curious to know if I am misunderstanding something.

Here is the paper:
http://rspa.royalsocietypublishing.org/content/117/778/610.full.pdf

[Hendrik van Hees]

On 05/12/2013 02:05 AM, Stephen Montgomery-Smith wrote:

> I'm trying to read Dirac's paper "The Quantum Theory of the Electron."
> I am really struggling with the first displayed equation on 611. Is
> there a sign error?

> [...]

I can't help with your specific question (sorry), because I've never
studied his theory of the electron. But I HAVE spent a LOT of time with
both his "QM Principles" book and his GR book. And for those books,
I've NEVER seen ANY other author who was so careful about the accuracy
of both the logic and the printing ... he set the standard ... he never
said one word too many, or one word too few. But it's not impossible
that a mistake slipped by him, or perhaps there was an error between his
proofing and the final printing.

--
Mike Fontenot

[Mike Fontenot]

On 05/12/2013 02:05 AM, Stephen Montgomery-Smith wrote:

> I'm trying to read Dirac's paper "The Quantum Theory of the Electron."
> I am really struggling with the first displayed equation on 611. Is
> there a sign error?
> [...]

[Moderator's note: This posting has been sent before falsely under my
name due to a mistake in my moderation process. Apologies to Mike for
that.
HvH.]

[Hendrik van Hees]

Well, as far as I can see in Eq. (1) a minus sign is missing at the
first term. In Eq. (3) it's correct, but later he switches to the ict
metric convention, which was quite common at this time. Perhaps in the
first paragraph there got something mixed up with the i's. That's indeed
very strange, because as Mike wrote, Dirac's paper are the most
carefully written ones ever, I'd say .


--
Hendrik van Hees
Frankfurt Institute of Advanced Studies
D-60438 Frankfurt am Main
http://fias.uni-frankfurt.de/~hees/

[jcordes]

[[Mod. note -- Please limit your text to fit within 80 columns,
preferably around 70, so that readers don't have to scroll horizontally
to read each line. I have manually reformatted this article. -- jt]]

Stephen:

Apologies to all if this is a repeat post. I already wrote a message
but some sort of glitch in posting seems to have occurred (this is
at least the 3rd time I've tried to get the post out).

Surprisingly (since I agree with Mike's sentiments in his response)
it appears you are right about the sign error in the 1928 Dirac
paper. I believe the external minus sign in front of the first term
in parentheses on the RHS, which you show, is correct. However, I
don't know why you have introduced an *internal* minus sign in that
term. Dirac's electron has charge -e, I think, and I believe the
internal signs in his original equation are correct. Caution: I
am very rusty at this, having been away from it for a good many
years!

John Cordes

[Stephen Montgomery-Smith]

John Cordes just emailed me the reference
http://www.philoscience.unibe.ch/documents/TexteFS10/Hanson1961dpI.pdf - on
page 205, in the footnote, it does indicate a sign error. My sign change
inside the parentheses is wrong. I had taken e to be a negative number,
but then to be consistent I should have made a similar sign change inside
the second parentheses.

So it should be:

F = -(W/c + eA_0/c)^2 + (p + eA/c)^2 + m^2c^2

[Roland Franzius]

The introduction states a "wrong" classical "Hamiltonian" and a wrong
wave equation. The paper itself, starting with equ (3) is of course
completely consistent and correct.

Maybe the young P.A.M. Dirac was confused about different concepts of
relativistic mechanics, like so many people in his times (amd later so on).

Dirac had been working in general relativity and just started his
lifelong research on the completely new concepts of quantum mechanics
from a purely mathematical point of view.

Confer the citing of current operator moments as constituents of
electric currents in radiation processes as the only link to reality in
that times.

I assume, that, in the introduction, he used the classical quadratic
Hamiltonian and the quadratic Lagrangian density for unclear reasons.

The Lagranian density is a Lorentz covariant scalar density

dL = 1/2 (W^2 - m^2 - p^2 ) d^4x

from wich one gets for a wave pair (psi*, psi) by mumbling some words
about independent variation of psi and psi* (or Re psi and Im psi)

W^2 -> (i d_t psi)^* (i d_t psi)

The momentum as the cofactor of the velocity in L is

P = dL/d(i psi/dt)* =i d psi/dt

and fraom this we get the Legendre transform

H = P v - L = 1/2 (W^2 + m^2 + p^2 )

But then the equations of motion are of course

V = dH/p = W

and

d_t P = -dH/dpsi

In this way the metric -sign in the wave equation here is entering via
the symplectic structure aka canonical equations for variables (psi at
each point) and momenta (d_t psi at each point).

Of course, the many ways of choosing variational principles, derivation
of covariant equations of motions, signs of t, omega, x and p as well as
the analytic extensions in the momentum representations to complex
values of k in the wave equations, all that emerged as being centrally
for the development of symmetry concepts of QED and was completely
unforseen in 1927.

Just read the paper as the transition from the sloppy handwaving
arguments presented in the introduction in the way of Schr�dinger and
Heisenberg to one of the fine first papers in mathematical physics.

Would be interesting to know what Dirac thought about that sigen, when
Atiyah presented the euclidean Dirac operator on manifolds

D = (d_i) D_i

with a euclidean Clifford base (d_i) and a covariant derivative D_i on
spin manifolds.

--

Roland Franzius

[Vladimir Kalitvianski]

Dirac says in his "Hopes and fears" (Eureka, 1969, N32, pages 2-4) that
there was no error, but an implicit use of the imaginary unit "i" in
time variable.

[Alfred Einstead]

On May 12, 3:05 am, Stephen Montgomery-Smith <step...@missouri.edu>
wrote:

There's no sign error. It's specifically intended to go that way.

Using normal MKS units, the 1-body equations for electromagnetism may
be written:
d/dt (E + e phi) = d_t U
d/dt (p + e A) = -del U
with a velocity-dependent potential U(r, t, v) given by
U = phi - v.A.

The signs are determined by the respective 1-forms:
canonical 1-form: theta = E dt - p.dr
potential 1-form: A.dr - phi dt.
Under operator correspondence, theta <-> i h-bar d (where d is the
total differential). So,
E = i h-bar d_t, p = -i h-bar del.

The relation to Dirac's notation is my (A, phi, E) = Dirac's (A/c,
A_0, W).

Here's an interesting aside to put this in perspective. Dirac
mentioned non-relativistic theory briefly at the head of the article.
You can, in fact, pose the same starting conditions that lead
ultimately to the Dirac equation even in non-relativistic theory
(contradicting one of his main points, in fact, that his formalism is
somehow specific to relativity).

Since the kinetic energy (which I'll call H, instead of E), the
momentum p = (p_x, p_y, p_z) and mass (m) transform under a boost (v)
in, say the x-direction, as:
E -> E - v p_x + v^2/2 m
p_x -> p_x - v m
p_y -> p_y
p_z -> p_z
m -> m
then, they comprise a *5* vector. The invariants of the 5-vector are:
mu = m
rho = p_x^2 + p_y^2 + p_z^2 - 2mH.

The relativistic form of this is found by separating mc^2 from E, and
writing H = E - mc^2. Then you have the relativistic versions of the
above:
mu = M = E/c^2 = m
rho = p_x^2 + p_y^2 + p_z^2 - 2mH - H^2/c^2.

For ordinary non-interacting bodies, mu = m and rho = 0.

The Dirac equation comes as a result of converting rho to a
differential equation, while the conversion of mu to a differential
equation yields a linear eigenvalue equation which allows you to
eliminate the coordinate that goes with the 5th component of the 5
vector.

The general quadratic invariant
rho = p_x^2 + p_y^2 + p_z^2 - 2MH + alpha H^2
(replacing m by M = m + H/c^2) and generalizing (1/c^2) to alpha), has
a ++++- signature, for all alpha ... including alpha = 0 (non-
relativistic theory).

This leads to *5* gammas: gamma_1, gamma_2, gamma_3, gamma_4, gamma_5
and an algebra composed from them by taking all 32 of their products
(including the unit 1) with *real* coefficients.

The "Dirac algebra" you're used to seeing is the one with 4 gammas
(gamma_0, gamma_1, gamma_2, gamma_3). However, the *real* algebra
formed from 4 gammas, is *not* the one actually used in the Dirac
equation. Rather, it is the complex algebra that is used. This may be
derived from the real algebra by any of the following ways:
(a) allow complex coefficients with the 16 products formed from
(gamma_0, gamma_1, gamma_2, gamma_3), or
(b) add in "i" as an extra basis element that commutes with the 4
gammas, or
(c) add in some imaginary multiple of the gammas, such as gamma_5.
The result is (surprise) equivalent to the algebra obtained from the
*five* dimensional geometry.

5-D geometry ... gamma_5 ... coincidence, you think? Or possibly the
remnant of a strand of thought Dirac had, exactly along the lines
described above.

But anyway, when you expand the canonical form to include all 5
components of momentum/mass/energy, it takes the form in non-
relativistic theory:
theta = p_x dx + p_y dy + p_z dz - H dt + mu du
with the extra coordinate u.

In relartivity, it takes either of the two equivalent forms:
theta = as above, or
theta = p_x dx + p_y dy + p_z dz - H ds + M du
where ds = dt + du/c^2 plays the role of an "absolute time".

When using the coordinate representation in the linear form (mu) and
quadratic form (rho), you have the operator correspondence
p = -i h-bar del, H = (d_t)_u, mu = (d_u)_t
or
p = -i h-bar del, H = (d_s)_u, M = (d_u)_s
depending on what sets of variables you use for coordinates #4 and #5.

And you get the Dirac equation (using the (t,u) coordinates):
(alpha.(-i h-bar del) - delta (i h-bar d_t) + epsilon (-i h-bar
d_u)) psi = 0,
(-i h-bar d_u) psi = m psi,
where alpha = (alpha_x, alpha_y, alpha_z), delta and epsilon are
suitable combinations of (gamma_1, gamma_2, gamma_3, gamma_4 and
gamma_5) from the algebra with the ++++- signature.

When alpha = (1/c^2), this yields the equivalent of the Dirac
equation. When alpha = 0 it yields the Schroedinger equation.

For electromagnetism, the potential form becomes a 5-form:
A = A.dr - phi dt + B du
The scalar B component is one and the same as the "B" field used in
the B field formalism in QED. When this field is inserted into the
coupled Dirac equation, with B = 0, the result is the Pauli-
Schroedinger equation for non-relativistic theory when alpha = 0, and
the Dirac-Maxwell equation for relativity.

Finally, since the Dirac equation -- which was motivated by the de
Broglie correspondence -- has a non-relativistic form, then so does
the de Broglie correspondence! So, this is not specific to relativity
either, but applies across the board to both relativity and non-
relativistic theory. I'll leave that as an exercise for you to work
out.

[Alfred Einstead]

On May 12, 3:05 am, Stephen Montgomery-Smith <step...@missouri.edu>
wrote:

> I'm trying to read Dirac's paper "The Quantum Theory of the Electron."
> I am really struggling with the first displayed equation on 611. Is
> there a sign error?

Oops. Sorry. I took another look at the paper. You're referring to the
signs *outside* the parentheses. Yes, equation (1) is an error. Dirac
(indirectly) said just as much by (wrongly) stating that (3) is a
consequence of (1). But (3) has the other sign.

He would have noticed it, had he considered the non-relativistic limit
as a consistency check.

> F = -(W/c - eA_0/c)^2 + (p + eA/c)^2 + m^2c^2?

I was focusing on the wrong thing, the different signs used *inside*
the parentheses in:
(+i h d/d(ct) + e/c A_0)^2
versus the signs used in
|-i h d/dx + e/c A|^2.

To expand on my previous discussion, you need only pass over to the
non-relativistic form to see the error. First replace E by H = E -
mc^2 (where I'm using H to denote the kinetic energy), replace (1/c)^2
by the parameter alpha and write:
0 = -p_0^2 + p^2 + m^2 c^2
= -(E/c)^2 + p^2 + m^2 c^2
= -(H/c + mc)^2 + p^2 + m^2 c^2
= p^2 - 2mH - alpha H^2.
The non-relativistic form, with alpha = 0, has p^2 = 2mH. If, in
contrast, you use a + sign with p_0 you get a mess:
(H/c + mc)^2 + p^2 + m^2 c^2
= p^2 + 2mH + alpha H^2 + 2m^2/alpha
which diverges in the limit alpha -> 0.

As I mentioned, Dirac made a grave error (here and elsewhere in many
other respects) in asserting at the outset that his developments were
something that were "specific and unique to relativity" (to
paraphrase). As a result of that error, he disrupted the natural
linkage between the two forms; and, thus, the natural sanity and error
check that comes from having a non-relativistic form to check
everything against.

The conceptions he raised at the outset are (yet another) case of bad
information that eventually get handed down in the literature as
community folklore and finds its way into textbooks. In this case, it
has taken a long time since to excise it and has never been fully
excised. We now know today that:

(1) Spin has nothing per se to do with quantum theory. It is firmly
rooted in *classical* theory and is only present in quantum theory by
way of inheritance from classical theory, not something that emerges
as a consequence of quantum theory. It arises classically as a direct
result of the symplectic classification of the Poisson manifold
associated with the Lie algebras that define the respective kinematic
symmetry groups (be it Poincare' for relativity or Bargmann for non-
relativistic theory).

(2) Spin has nothing per se to do with relativity, either. The
symplectic classes for the Bargmann group also include representations
corresponding to elementary systems of non-zero mass and spin. This is
the non-relativistic version of spin.

(3) The Dirac equation (and, as I pointed out) even the de Broglie
correspondence itself are not specific to relativity, either. Both
have non-relativistic forms. Therefore, neither can be used as a way
of qualitatively distinguishing relativity from non-relativistic
theory.

(4) The linearity of d/dt in the Dirac equation is a red herring. You
can write the Klein-Gordon equation for *arbitrary* spin in this form
as the Dirac-Kemmer equations.

(5) The "asymmetry" of the roles played by t and (x, y, z) in the
Schroedinger equation is also a red herring, because there is no
asymmetry, when the equations are written the right way (as I pointed
out). They BOTH enter into the equation within a *bilinear quadratic
form*, when rendered in the following form:
(p^2 - 2 mu H - alpha H^2) psi = 0
mu psi = m psi
or re-written in form as the following differential equations:
-h-bar^2 (del^2 + 2 d_t d_u - alpha d_t^2) psi = 0
-i h-bar d_u psi = m psi
These equations yield results that are equivalent to the Klein-Gordon
equation when alpha = (1/c)^2, and equivalent to the Schroedinger
equation when alpha = 0.

(6) At the same time, the absence of any coordinate entering the
equations lineraly is not specific to non-relativistic theory! Both
the relativistic and non-relativistic forms have the linear equation.

So, whatever problem that Dirac thought the Schroedinger equation had
that he thought he had to remedy, its apparent linearity in t was
obviously not it, and was never anything more than a red herring. The
asymmetric appearance of t versus (x, y, z) was obviously not it
either and was nothing more than a red herring. The emergence of spin
was clearly not it either, since it was already there in both
classical and non-relativistic theory in the first place (and also
emerges in the non-relativistic form of the Dirac equation). So, that
too was a red herring.

And the errors in all cases come from not asking the right question at
the outset: what is the Correspondence Limit of all this?

The first axiom of Relativity, before even posing the other "two"
axioms is that all that *was* valid and established under the old
paradigm (even including those things that we only came to be aware of
*after* 1905 as having been part of the old paradigm all along, such
as the fact that the Bargmann group is the symmetry group of non-
relativistic theory and not the Galilei group) should be grandfathered
into the new paradigm, apart from corrections that, as functions of
alpha, go to 0 as alpha -> 0; and vice versa. In other words: the
Correspondence Limit has to be checked, and checked in both
directions.

Because he failed to check the Correspondence Limit of equation (1)
(even going as far as to assert that his entire line of development
had no correspondence limit at all, by virtue of it being
"specifically and uniquely relativistic" to paraphrase), he wrote down
equation (1) that made absolutely no sense that could have easily been
checked, verified and corrected by seeing what their non-relativistic
limits were.

I just did that, for equation (3). I'll leave it as an exercise for
(the correction of) equation (1), along with the exercise of deriving
the Pauli-Schroedinger equation from as the non-relativistic form of
(1), after the correction is made.

This is why you should be keeping the c's in equations,
notwithstanding what the current conventions are; and keeping the c's
in the *right* places. There is no c in A/c, as Dirac wrote it. A is
the potential that goes with momentum and is what Maxwell called the
"electromagnetic momentum", so it's p + eA, not p + eA/c. So this way,
you don't end up losing contact with the real world by descending too
far into the Ivory Tower of "everything is c = 1, h-bar = 1 and G = 1
or 8 pi."

When equation (4) is written in a form within a universal framework
that encompasses both the relativistic and non-relativistic forms, it
would take the form:
(a^x p_x + a^y p_y + a^z p_z - d H + e M) psi = 0
(M - alpha H) psi = m psi
To recover the quadratic form
rho := p^2 - 2MH + alpha H^2 = 0
from the first of these requires:
a^i a^j + a^j a^i = delta^{ij}, for i, j = x, y, z
a^i d + d a^i = 0, a^i e + e a^i = 0, for i = x, y, z
d^2 = alpha
de + ed = 2
e^2 = 0
This generates a Clifford algebra with *real* coefficients that is the
same -- for *all* alpha -- as either:
(a) the Clifford algebra with complex coefficients generated from
{ gamma_0, gamma_1, gamma_2, gamma_3 }, or
(b) the Clifford algebra with real coefficients generated from
{ gamma_0, gamma_1, gamma_2, gamma_3, gamma_5 }.

As mentioned before, for alpha = 0, this is the Schroedinger equation.
For alpha > 0, it is the Dirac equation with light speed set to c =
alpha^{-1/2}. Finally, what was not mentioned is that for alpha < 0,
it is the *Euclidean* form of the Dirac equation.

An interesting exercise, as alluded to before, is to write (a^x, a^y,
a^z, d, e) in terms of the gamma's in (b).