Mr. Yablon,
I get the impression from your lengthy Dec. 6 post that your
paper uses single-valued wave functions. If this is a wrong impression,
then please disregard the following, and state clearly that your wave functions
are multiple valued, or that they are single-valued in such-and-such
clearly specified context, and multiple-valued in some other clearly specified
contexts. It is impossible to discuss your paper when the meanings of
the symbols are not clearly specified.
The following will discuss the following points. The first
is the most important, and hasn't yet been raised explicitly.
(1) I think there may be a fundamental misunderstanding of what it means
to say that
`
[from your paper, attributed to Misner, Thorne, and Wheeler]
"...a spinor will reverse sign after any 2p = 360 degree rotation,
and will only regain its original sign after a
4p = 720 degree rotation."
The way you interpret this is not the way I interpret it.
(2) I think that your equation (4.8) definition of e_0 is circular,
and therefore not a valid definition. I do not know what your symbol
e_0 represents. A typo in my original analysis
prevented this from being as clear as it should have been.
BEGIN DISCUSSION OF POINT (1):
There are at least two conceivable meanings for
"a spinor will reverse sign after any 360 degree rotation:
(i) This could happen if the spinor wave function were multiple-valued.
But in my understanding of your paper, you don't assume
that it is.
(ii) This will require a bit of explanation, so please bear with me.
In describing a physical system, whether classically or quantum-
mechanically, one needs to first specify a coordinate system.
The description will depend on the coordinate system. Two descriptions
corresponding to two different coordinate systems will be related by
some transformation law.
Suppose we have a quantum mechanical system described in part by a wave
function \psi , and we want to ask what the wave function would look like
if we used a coordinate system rotated by a small angle, say 5 degrees,
about some axis relative to the original system.
If the wave function is a two-component spinor,
there are two possible answers. either U(5)\psi or U(365)\psi,
where U(angle) is a unitary operator (a 2x2 unitary matrix), and
U(365) = - U(5) .
It is a peculiarity of spinor representations that U(365) is not the
same as U(5). However, U(365)\phi does represent the same quantum state
as U(5)\phi. This is because when quantum states are represented by
wave functions, two (unnormalized) wave functions are considered the "same" if
each is a nonzero complex multiple of the other.
If we are considering a rotation through a very small angle,
it seems more natural to represent the wave function as seen in the
rotated coordinate system by U(5)\psi instead of U(365)\psi = - U(5)\psi,
because we would like for the wave function as seen from a slightly
rotated system to be close to the wave function as seen in the original
coordinate system.
Notice that WHAT IS BEING ROTATED IS THE COORDINATE SYSTEM.
The wave function never changes, only its mathematical representation
changes. An analogy is a classical particle not acted on by any forces.
We can find a coordinate system in which its momentum is constantly
zero or anything we please. The particle itself is always the same,
only its mathematical description changes as we change from one coordinate
system to another.
Now consider how our original wave function \psi will look in
a coordinate system rotated by 360 degrees from the original coordinate
system. Of course, it shouldn't change in any physically observable way,
because rotating a coordinate system by 360 degrees brings us back to
the original coordinate system. Remember, we are not performing some
kind of physical rotation; we are only changing coordinate systems.
However, if we conceive of performing the 360 degree coordinate
rotation as a succession of tiny rotations, say 72 rotations of five degrees
each, the resulting wave function will be described by
U(5)U(5) ... U(5) \psi = U(5)^72 \psi = U(360)\psi = - \psi .
The last equation U(360)\psi = -\psi is because of the two-valued nature
of spinor representations: U(angle + 360) = - U(angle)
for all angles including angle 0.
But -\psi represents the same quantum state as \psi because
in representating quantum states as wavefunctions, there is always
the ambiguity that for any nonzero complex constant *c*,
c \psi represents the same quantum state as \psi, even though considered
as functions, c \psi is different from \psi.
Physically, the above picture of rotating coordinate systems is
completely different from taking a wavefunction \psi = \psi(x) and
following its values as *x* traverses a small loop generated by a 360 degree rotation.
For single-valued \psi (which you seem to be assuming), the value \psi(x)
after traversing the loop will be the same as the value at the start of the
loop: the spinor function \psi DOES NOT CHANGE SIGN from the start to
the end of the rotation loop. If it's single-valued, it can't change sign!
I don't have Misner, Thorne, and Wheeler (MTW) at hand, but I
would bet that their interpretation of
"a spinor will reverse sign after any 360 degree rotation"
is (ii) rather than (i). I think this is the standard interpretation.
For example, the first paragraphs of the Wikipedia article on "Spinors"
unequivocally specifies this interpretation.
[Normally, I don't like to cite Wikipedia because some of its
articles are just plain wrong, but I don't have any better
reference at hand, and at least this reference shows that
my interpretation is a common one. If you rewrite
the Wikipedia article to change to interpretation (i), you
will probably generate howls of protest.]
I would bet that your paper was never read in any meaningful way
by Phys. Rev. D. I would also bet that if it ever *is* read carefully
by any knowledgeable referee for any mainstream journal,
he will cite the above apparent misunderstanding as reason to reject the paper.
END DISCUSSION OF POINT (1)
BEGIN DISCUSSION OF POINT (2):
This will be essentially the same as the original discussion of
your equations (4.7) and (4.8), with a silly typo corrected.
The paper's equation (4.7) states the "Dirac Quantization Condition":
(4.7) 2eg = n .
Here *e* is the electronic charge, *g* the monopole's "magnetic charge",
and *n* an arbitrary nonzero integer. The paper goes on:
"From (4.7), defining the n=1 charge units as e_0 = 1/2g
and g_0 = 1/2e, we see that the respective electric
and magnetic charge strengths are reciprocally quantized by
(4.8) e = n/2g = n e_0
g = n/2e = n g_0 . "
[In the original, g = n/2e was replaced by g = n/23.]
This seems to be circular logic: e is defined in terms of g while
g is defined in terms of e. Morever, I think that the preceding sentence
should read:
e_0 = 1/2g_0 and g_0 = 1/2 e_0 .
The effect is that neither e_0 nor g_0 are actually defined,
making it impossible to follow later discussion which references them.
END DISCUSSION OF POINT (2)
Some of the posts in this group have been viewed thousands of times,
which suggests that there could be hundreds of members. I have to wonder
if there is *no one* in the group sufficiently knowlegeable about spinors
to comment on whether interpretation (i) or (ii) above is the standard
interpretation. So far, no one other than I has volunteered any
substantive comment on your paper.
You want to avoid rejection for cause (e.g., incorrectness
or unclarity) because that will impact your credibility with the journal
for further submissions. Your future papers could be editorialy rejected
without even being refereed. Indeed, I suspect that this may have been
the case with Phys. Rev. D.
In my opinion, the paper is probably incorrect and surely unclear.
But I am only one fallible individual.
You can submit with more confidence if you can find even one qualified
person who will publicly assert that the paper is clearly written and
conceivably correct. If you can't find such a person in
sci.physics.founations, you might try sci.physics.research, which is
a moderated discussion group accessible through Google groups.
At one time, that group hosted many high-level discussions
read by many physicists and mathematicians. It used to be a reliable
source of inspiration for me. Of late, it has fallen on hard times,
with few posts, and fewer knowledgeable ones. But it may still be read
occasionally by competent people. The moderators are competent and
might well give reliable opinions on issues as straightforward as
the meaning of spinor sign reversals.
Mr. Yablon,
I get the impression from your lengthy Dec. 6 post that your
paper uses single-valued wave functions. If this is a wrong impression,
then please disregard the following, and state clearly that your wave functions
are multiple valued, or that they are single-valued in such-and-such
clearly specified context, and multiple-valued in some other clearly specified
contexts. It is impossible to discuss your paper when the meanings of
the symbols are not clearly specified.
The following will discuss the following points. The first
is the most important, and hasn't yet been raised explicitly.
(1) I think there may be a fundamental misunderstanding of what it means
to say that
`
[quote][from your paper, attributed to Misner, Thorne, and Wheeler]
"...a spinor will reverse sign after any 2p = 360 degree rotation,
and will only regain its original sign after a
4p = 720 degree rotation."
[/quote]
The way you interpret this is not the way I interpret it.
(2) I think that your equation (4.8) definition of e_0 is circular,
and therefore not a valid definition. I do not know what your symbol
e_0 represents. A typo in my original analysis
prevented this from being as clear as it should have been.
BEGIN DISCUSSION OF POINT (1):
There are at least two conceivable meanings for
"a spinor will reverse sign after any 360 degree rotation:
(i) This could happen if the spinor wave function were multiple-valued.
But in my understanding of your paper, you don't assume
that it is.
(ii) This will require a bit of explanation, so please bear with me.
In describing a physical system, whether classically or quantum-
mechanically, one needs to first specify a coordinate system.
The description will depend on the coordinate system. Two descriptions
corresponding to two different coordinate systems will be related by
some transformation law.
Suppose we have a quantum mechanical system described in part by a wave
function \psi , and we want to ask what the wave function would look like
if we used a coordinate system rotated by a small angle, say 5 degrees,
about some axis relative to the original system.
If the wave function is a two-component spinor,
there are two possible answers. either U(5)\psi or U(365)\psi,
where U(angle) is a unitary operator (a 2x2 unitary matrix), and
U(365) = - U(5) .
It is a peculiarity of spinor representations that U(365) is not the
same as U(5). However, U(365)\phi does represent the same quantum state
as U(5)\phi. This is because when quantum states are represented by
wave functions, two (unnormalized) wave functions are considered the "same" if
each is a nonzero complex multiple of the other.
If we are considering a rotation through a very small angle,
it seems more natural to represent the wave function as seen in the
rotated coordinate system by U(5)\psi instead of U(365)\psi = - U(5)\psi,
because we would like for the wave function as seen from a slightly
rotated system to be close to the wave function as seen in the original
coordinate system.
Notice that WHAT IS BEING ROTATED IS THE COORDINATE SYSTEM.
The wave function never changes, only its mathematical representation
changes. An analogy is a classical particle not acted on by any forces.
We can find a coordinate system in which its momentum is constantly
zero or anything we please. The particle itself is always the same,
only its mathematical description changes as we change from one coordinate
system to another.
Now consider how our original wave function \psi will look in
a coordinate system rotated by 360 degrees from the original coordinate
system. Of course, it shouldn't change in any physically observable way,
because rotating a coordinate system by 360 degrees brings us back to
the original coordinate system. Remember, we are not performing some
kind of physical rotation; we are only changing coordinate systems.
However, if we conceive of performing the 360 degree coordinate
rotation as a succession of tiny rotations, say 72 rotations of five degrees
each, the resulting wave function will be described by
[quote]U(5)U(5) ... U(5) \psi = U(5)^72 \psi = U(360)\psi = - \psi .[/quote]
The last equation U(360)\psi = -\psi is because of the two-valued nature
of spinor representations: U(angle + 360) = - U(angle)
for all angles including angle 0.
But -\psi represents the same quantum state as \psi because
in representating quantum states as wavefunctions, there is always
the ambiguity that for any nonzero complex constant *c*,
c \psi represents the same quantum state as \psi, even though considered
as functions, c \psi is different from \psi.
Physically, the above picture of rotating coordinate systems is
completely different from taking a wavefunction \psi = \psi(x) and
following its values as *x* traverses a small loop generated by a 360 degree rotation.
For single-valued \psi (which you seem to be assuming), the value \psi(x)
after traversing the loop will be the same as the value at the start of the
loop: the spinor function \psi DOES NOT CHANGE SIGN from the start to
the end of the rotation loop. If it's single-valued, it can't change sign!
I don't have Misner, Thorne, and Wheeler (MTW) at hand, but I
would bet that their interpretation of
[quote]"a spinor will reverse sign after any 360 degree rotation"[/quote]
is (ii) rather than (i). I think this is the standard interpretation.
For example, the first paragraphs of the Wikipedia article on "Spinors"
unequivocally specifies this interpretation.
[Normally, I don't like to cite Wikipedia because some of its
articles are just plain wrong, but I don't have any better
reference at hand, and at least this reference shows that
my interpretation is a common one. If you rewrite
the Wikipedia article to change to interpretation (i), you
will probably generate howls of protest.]
I would bet that your paper was never read in any meaningful way
by Phys. Rev. D. I would also bet that if it ever *is* read carefully
by any knowledgeable referee for any mainstream journal,
he will cite the above apparent misunderstanding as reason to reject the paper.
END DISCUSSION OF POINT (1)
BEGIN DISCUSSION OF POINT (2):
This will be essentially the same as the original discussion of
your equations (4.7) and (4.8), with a silly typo corrected.
The paper's equation (4.7) states the "Dirac Quantization Condition":
[quote](4.7) 2eg = n .[/quote]
Here *e* is the electronic charge, *g* the monopole's "magnetic charge",
and *n* an arbitrary nonzero integer. The paper goes on:
[quote]"From (4.7), defining the n=1 charge units as e_0 = 1/2g
and g_0 = 1/2e, we see that the respective electric
and magnetic charge strengths are reciprocally quantized by
(4.8) e = n/2g = n e_0
g = n/2e = n g_0 . "[/quote]
[In the original, g = n/2e was replaced by g = n/23.]
This seems to be circular logic: e is defined in terms of g while
g is defined in terms of e. Morever, I think that the preceding sentence
should read:
[quote]e_0 = 1/2g_0 and g_0 = 1/2 e_0 . [/quote]
The effect is that neither e_0 nor g_0 are actually defined,
making it impossible to follow later discussion which references them.
END DISCUSSION OF POINT (2)
Some of the posts in this group have been viewed thousands of times,
which suggests that there could be hundreds of members. I have to wonder
if there is *no one* in the group sufficiently knowlegeable about spinors
to comment on whether interpretation (i) or (ii) above is the standard
interpretation. So far, no one other than I has volunteered any
substantive comment on your paper.
You want to avoid rejection for cause (e.g., incorrectness
or unclarity) because that will impact your credibility with the journal
for further submissions. Your future papers could be editorialy rejected
without even being refereed. Indeed, I suspect that this may have been
the case with Phys. Rev. D.
In my opinion, the paper is probably incorrect and surely unclear.
But I am only one fallible individual.
You can submit with more confidence if you can find even one qualified
person who will publicly assert that the paper is clearly written and
conceivably correct. If you can't find such a person in
sci.physics.founations, you might try sci.physics.research, which is
a moderated discussion group accessible through Google groups.
At one time, that group hosted many high-level discussions
read by many physicists and mathematicians. It used to be a reliable
source of inspiration for me. Of late, it has fallen on hard times,
with few posts, and fewer knowledgeable ones. But it may still be read
occasionally by competent people. The moderators are competent and
might well give reliable opinions on issues as straightforward as
the meaning of spinor sign reversals.
