spin, what is it ?
Theresa Knott
I'm trying to visualise spin. What do you people see in you head when
you talk of spin of a fundamentsl particle ? I mean if a particle is
fundamental then even if it were not point like but had some size it
couln't have any structure, so it cant spin in any way that I can
visualise. Yet to say its just a quantum number seems to be a cop out
in my opinion because it does have the properties of angular momentum.
Theresa
Matt McIrvin
<the...@knott16.freeserve.co.uk> wrote:
What I see in my head is a circulating flow of energy and momentum density
around the borders of the particle's quantum-mechanical wave function.
Belinfante showed back in the 1930s that spin in the Dirac theory could be
interpreted this way, and H. C. Ohanian wrote a famous article about it in
the American Journal of Physics:
Hans C. Ohanian, "What is spin?" AJP 54 (6), 500-505 (1986)
I don't remember whether Ohanian mentions this in his article or not, but
you don't even need the full Dirac theory in order to see the spin emerging
this way. You can do everything nonrelativistically, using the Schrodinger
theory plus a little handwaving!
The trick is to write the del^2 operator as (del.sigma)^2.
(This is the same trick that Feynman once used to derive g=2 from the
Schrodinger theory by minimal coupling.) Apply the Noether procedure to
get the stress-energy tensor as the current associated with space-time
translations, and then symmetrize it. What you get has the Pauli matrices
in it.
In the momentum density there is a term that is the momentum density we
all know and love, and this weird *other* term that whirls in a loop
around every hump in the wave function. The angular momentum due to this
other term is hbar/2 and points in what we know as the spin direction.
(In much the same way, Feynman's minimal coupling gives a circulating
electric current.)
I even wrote up a paper about this. I was going to submit it to AJP,
but then I saw
Katsunori Mita, "Virtual probability current associated with the spin,"
AJP 68 (3), 259-264 (March 2000)
and I'm pretty sure that Mita beat me to it by about a year.
So I think I'm just going to write a more popularized version and make it
into a Web page someday.
--
Matt McIrvin http://world.std.com/~mmcirvin/
Mike York
Except for massless particles, which have no rest frame, spin is the
angular momentum in the rest frame.
For massless particles, spin is the maximum component of angular
momentum along the direction of motion.
The point is that angular momentum transforms as the observer moves
between frames. And is non-zero even in frames or directions where there
should be no "orbital" angular momentum.
So, to summarize, what I "see" is that spin is the residual angular
momentum in a frame of reference where the orbital angular momentum
vanishes.
Mike
Hans Aberg
Here is a mathematical, differential geometric, interpretation:
First visualize say a surface of some kind which is sufficiently smooth
that one can make analysis on it, say a sphere. Apart form being able to
draw curves on that sphere, at each point, one can also indicate at each
point the different directions. In fact, each sufficiently smooth curve
will have a direction at each point.
Then these vector directions will in fact be the differentials dx_i of the
surface, which are often though at as infinitesimal quantities present at
each point. And they serve as infinitesimal approximations of the analysis
at each point.
These differentials only serve as a first order linear approximation. If
one goes to higher order, one gets the differential tensor algebra at each
point. One can also see that the algebras at different points hang
together in a smooth way.
So, in fine, if we accept analysis at all, it is possible to
mathematically see that there are infinitesimal tensor algebra quantities
at each point. The picture is this:
| stalk = tensor algebra
----|----
/ p . / <-- surface.
---------
Now to spin: First, the manifold is not a surface, but the time-space with
the Lorentz metric. Then spin is simply the infinitesimal angular momentum
at each point. The particle spins, but infinitesimally, according to its
analysis, and not in the physical dimensions.
There are two ways to describe the spin: One can say that as it is
observed, it must exist infinitesimally, and hook it onto the tensor
algebra. This is the approach used in physics. One can then do this for
suitable Lie groups, and get a range of possible types of spin (which is
used in QED and quark theory). The thing that matters is then if this
approach agrees with physical experiments.
The other way would to somehow extract it directly from the tensor
algebra. This can be done using say Clifford algebras for the simple spin,
but I do not know any such approach for QED and quark theory.
Hans Aberg * Anti-spam: remove "remove." from email address.
* Email: Hans Aberg <remove...@member.ams.org>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>
Theresa Knott
On Tue, 20 Mar 2001 01:31:15 GMT, Mike York <mike...@home.com> wrote:
>So, to summarize, what I "see" is that spin is the residual angular
>momentum in a frame of reference where the orbital angular momentum
>vanishes.
Ok then what *is* angular momentum. [ I know the defn in terms of
commutation relations. I'm not looking for maths I'm looking for
imagery]
TIA
Theresa
>
>Mike
Theresa Knott
>What I see in my head is a circulating flow of energy and momentum density
>around the borders of the particle's quantum-mechanical wave function.
What do you mean by the borders of the wavefunction ?
>Belinfante showed back in the 1930s that spin in the Dirac theory could be
>interpreted this way, and H. C. Ohanian wrote a famous article about it in
>the American Journal of Physics:
>
>Hans C. Ohanian, "What is spin?" AJP 54 (6), 500-505 (1986)
Thanks I'll look it up
>
>I don't remember whether Ohanian mentions this in his article or not, but
>you don't even need the full Dirac theory in order to see the spin emerging
>this way. You can do everything nonrelativistically, using the Schrodinger
>theory plus a little handwaving!
>
>The trick is to write the del^2 operator as (del.sigma)^2.
>(This is the same trick that Feynman once used to derive g=2 from the
>Schrodinger theory by minimal coupling.) Apply the Noether procedure to
>get the stress-energy tensor as the current associated with space-time
>translations, and then symmetrize it. What you get has the Pauli matrices
>in it.
>
>In the momentum density there is a term that is the momentum density we
>all know and love, and this weird *other* term that whirls in a loop
>around every hump in the wave function.
V cool. I shall definitely have to look more into this thanks.
> The angular momentum due to this
>other term is hbar/2 and points in what we know as the spin direction.
>(In much the same way, Feynman's minimal coupling gives a circulating
>electric current.)
>
>I even wrote up a paper about this. I was going to submit it to AJP,
>but then I saw
>
>Katsunori Mita, "Virtual probability current associated with the spin,"
>AJP 68 (3), 259-264 (March 2000)
>
>and I'm pretty sure that Mita beat me to it by about a year.
>So I think I'm just going to write a more popularized version and make it
>into a Web page someday.
That would be nice. No time like the present eh ?
Thanks
Theresa
Cl.Massé
message news: i389btk9tbhl7tdec...@4ax.com...
> I'm trying to visualise spin. What do you people see in you head
> when you talk of spin of a fundamental particle ?
An arrow running along the border of a Mobius strip.
--
~~~~ %20cl...@free.fr%20 LPF
Liberty, Equality, Profitability.
[Moderator's note: we should probably not indulge in revealing our
private mental imagery unless we explain how it is helpful to
the task of understanding the concept of spin. - jb]
Matt McIrvin
<the...@knott16.freeserve.co.uk> wrote:
>On Mon, 19 Mar 2001 23:59:06 -0500, mmci...@world.std.com (Matt
>McIrvin) wrote:
[about spin]
>
>>What I see in my head is a circulating flow of energy and momentum density
>>around the borders of the particle's quantum-mechanical wave function.
>
>What do you mean by the borders of the wavefunction ?
The flow has the form of psi^dagger (sigma x grad) psi. So if,
for example, the electron is spin up, the circulating flow is strongest
where the wave function is most steeply sloped in directions perpendicular
to the z axis.
Gerard Westendorp
>
> In article <i389btk9tbhl7tdec...@4ax.com>, Theresa Knott
> <the...@knott16.freeserve.co.uk> wrote:
>
> >I'm trying to visualise spin.
Historically, the quantisation of angular momentum comes from
combining the classical concept of angular momentum with the notion
that particles are also waves, with the wavelength given by h/p
(p is linear momentum)
If a particle runs round in circles, its wave length must fit
an integral times in a circle, so
lambda = N 2pi r
L = pXr = h r/labda = N h/(2pi) = N h_bar
Unlike energy, which comes in discrete lumps *multiplied* by a
still variable frequency, angular momentum is truly quantised,
it really comes in lumps of h_bar. Planck's constant has the
dimension of angular momentum.
As you pointed out the article by Ohanian shows that "intrinsic"
angular momentum is really no different, it is implicit in
the wave function, so it is not really "intrinsic" in the sense
that it needs to be postulated independently.
Things start to get weird when you talk about fermions, which
can have half integer spin.
[..]
> The trick is to write the del^2 operator as (del.sigma)^2.
Hmm..
I wonder if an electron orbiting an atom actually has a half
integer angular momentum if you calculate it properly. Maybe
de Broglie's formula labda=h/p is a bit too simple for electrons
in a circular orbit.
Notably, the relation for angular momentum operator in the
z-direction (Lz) seems to be invalid:
Lz = h_bar d/dphi
Where phi is a spherical coordinate component.
It would be nice if we could derive a generalization of
this in terms of sigma matrices.
Gerard
Gerard Westendorp
>
> In article <i389btk9tbhl7tdec...@4ax.com>, Theresa Knott
> <the...@knott16.freeserve.co.uk> wrote:
>
> >I'm trying to visualise spin.
Historically, the quantisation of angular momentum comes from
combining the classical concept of angular momentum with the notion
that particles are also waves, with the wavelength given by h/p
(p is linear momentum)
If a particle runs round in circles, its wave length must fit
an integral times in a circle, so
lambda = N 2pi r
L = pXr = h r/labda = N h/(2pi) = N h_bar
Unlike energy, which comes in discrete lumps *multiplied* by a
still variable frequency, angular momentum is truly quantised,
it really comes in lumps of h_bar. Planck's constant has the
dimension of angular momentum.
As you pointed out the article by Ohanian shows that "intrinsic"
angular momentum is really no different, it is implicit in
the wave function, so it is not really "intrinsic" in the sense
that it needs to be postulated independently.
Things start to get weird when you talk about fermions, which
can have half integer spin.
[..]
> The trick is to write the del^2 operator as (del.sigma)^2.
Hmm..
Phil Gardner
>
> Theresa Knott <the...@knott16.freeserve.co.uk> wrote:
>
> >I'm trying to visualise spin. What do you people see in you head when
> >you talk of spin of a fundamentsl particle ? I mean if a particle is
> >fundamental then even if it were not point like but had some size it
> >couln't have any structure, so it cant spin in any way that I can
> >visualise. Yet to say its just a quantum number seems to be a cop out
> >in my opinion because it does have the properties of angular momentum.
>
> What I see in my head is a circulating flow of energy and momentum density
> around the borders of the particle's quantum-mechanical wave function.
What I see in my head is this. In the Schrodinger approximation the
wave function for an isolated, stationary, fundamental particle varies
as X = (1 + r/a) * exp(-r/a), to give del X = 0 at r = 0. At least in
the time-averaged sense the mass density of the particle varies as X^2.
Its spin momentum density varies as k x del(X^2), where the vector k
defines the spin axis. The spin velocity in the equatorial plane thus
varies just like the velocity in a low energy vortex in a macroscopic
fluid, with a smooth transition from a forced vortex at r = 0 to a free
vortex at large r.
Phil Gardner <pej...@oznetcom.com.au>
Gerard Westendorp
> I wonder if an electron orbiting an atom actually has a half
> integer angular momentum if you calculate it properly. Maybe
> de Broglie's formula labda=h/p is a bit too simple for electrons
> in a circular orbit.
I'm starting to figure out the answer.
Firstly, the "de Broglie" picture is actually wrong for the lowest
atomic orbital. With the de Broglie picture I mean than you start
with a classical an electron orbiting a nucleus, and then say the
electron is a wave of wavelength h/p.
The wave must fit an integer times around the atom, from which
you can derive that there is a discrete spectrum of energy states,
each of which has an angular momentum of an integer times h_bar.
It turns out however that the lowest orbital, according to the
Schrodinger equation, has the form
psi ~ exp(-r)
A *spherically symmetrical* wavefunction.
If you look at the momentum density of this orbital, you find
that the orbital angular momentum is zero. The electron is not
really "orbiting" then nucleus at all!
It is more like crashing into it (psi is infinite at r=0), but
due to the uncertainty in position it does not kill itself,
and continues to haunt the neighborhood of the nucleus.
According to Ohanian's article however, things should be
different for spin 1/2. Even the lowest orbital should have
a non-zero angular momentum.
Instead of the Schrodinger equation we have to look at the
Dirac equation. Unfortunately, there is not a nice closed
form solution to the Dirac equation in a 1/r coulomb field.
An approximation of a solution is:
( phi )
( sigma_i x_i/r phi / (E+m) ) exp(-r)
where phi is a Pauli 2-spinor.
Now this is indeed different. The solution is not spherical
symmetric, and it is easy to check that there are no solutions
with all 4 components spherically symmetric. So that makes sense,
tangential gradients mean tangential momentum, which means angular
momentum.
Another thing is that the momentum operator is NOT -i*hbar*d_x,
as in NRQM. We have to adapt the momentum operator anyway, because
we have 4 wave function components to operate on instead of one. But
there are other non-trivial changes. The momentum density (P) is
according to Ohanian:
P=(h_bar/(2i)) (psi d_x psi - (d_x psi) psi
+h_bar/4 curl(psi sigma psi )
If you work it all out, there seems little doubt you will find
the correct value of h_bar/2. The half integer value arises
as a consequence of the Dirac equation. If I try to visualize
this, I see 4 functions of ordinary space time, with momentum defined
in a way which is different from the de Broglie momentum picture.
I don't see any Mobius strips yet, but maybe you have to look at it
differently.
Gerard
Urs Schreiber
>
>Gerard Westendorp wrote:
>
>> I wonder if an electron orbiting an atom actually has a half
>> integer angular momentum if you calculate it properly. Maybe
>> de Broglie's formula labda=h/p is a bit too simple for electrons
>> in a circular orbit.
Is this about orbital angular momentum or about spin? An electron
alway has integral orbital angular momentum, which is correctly predicted
by de Broglie/Bohr. But Bohr, or Schroedinger for that matter, cannot say
anything about spin quantization, because this degree of freedom is not
even present in the model.
>It turns out however that the lowest orbital, according to the
>Schrodinger equation, has the form
>
> psi ~ exp(-r)
>
>A *spherically symmetrical* wavefunction.
>If you look at the momentum density of this orbital, you find
>that the orbital angular momentum is zero. The electron is not
>really "orbiting" then nucleus at all!
>It is more like crashing into it (psi is infinite at r=0),
exp(-r) is equal to 1 at r = 0. In its full beauty, the ground state
is
<x |1,0,0> = (\pi a^3)^(-1/2) exp(-r/a)
(with a being the first Bohr radius), so at r=0 the probability
density is (\pi a^3)^(-1/2) . If one asks for the probability density
to find the electron at a given *radius*, one has to consider the
above density multiplied by r^2. Then the probability to find
the electron at r=0 even vanishes.
Urs Schreiber
Squark
<3ACE3777...@xs4all.nl>):
>It turns out however that the lowest orbital, according to the
>Schrodinger equation, has the form
>
> psi ~ exp(-r)
>
>A *spherically symmetrical* wavefunction.
>If you look at the momentum density of this orbital, you find
>that the orbital angular momentum is zero. The electron is not
>really "orbiting" then nucleus at all!
Right. This is why I dislike the kinder garden elliptic/circular orbit
explanation so much! :-)
>It is more like crashing into it (psi is infinite at r=0),
Not true. exp(-0) = 1 (not equal) infinity.
Best regards,
squark.
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Gerard Westendorp
> Is this about orbital angular momentum or about spin? An electron
> alway has integral orbital angular momentum, which is correctly predicted
> by de Broglie/Bohr. But Bohr, or Schroedinger for that matter, cannot say
> anything about spin quantization, because this degree of freedom is not
> even present in the model.
The whole point is that there is no difference between orbital and
spin. There are some previous threads on this that you probably
missed. A famous article by Ohanian, mentioned in this thread,
demonstrates that if you calculate the angular momentum of a
finite wave packet that satisfies the Dirac equation or the
Maxwell equation, you find the correct spin as field angular
momentum *without* postulating this separately.
So what I was wondering about was how this works out for
orbitals.
[...]
> exp(-r) is equal to 1 at r = 0.
[...]
> If one asks for the probability density
> to find the electron at a given *radius*, one has to consider the
> above density multiplied by r^2. Then the probability to find
> the electron at r=0 even vanishes.
Yes, I was wrong there...
Gerard
Urs Schreiber
>The whole point is that there is no difference between orbital and
>spin. There are some previous threads on this that you probably
>missed. A famous article by Ohanian, mentioned in this thread,
>demonstrates that if you calculate the angular momentum of a
>finite wave packet that satisfies the Dirac equation or the
>Maxwell equation, you find the correct spin as field angular
>momentum *without* postulating this separately.
I have read, and enjoyed, Ohanian's result that the
spin angular momentum manifests itself as a circulation
in the wave function in a way very analogous to the
situation in electromagnetism. But it should be noted
that one needs the Dirac equation to start with in order to
derive this fact, it does not follow from the Schroedinger
equation. From your previous posting I got the impression
that you claimed that spin is derivable from Schroedinger's
equation alone, which, I belive, is not true. Or am I missing
something?
The Dirac equation incorporates additional degrees of
freedom (the four spinor components) that are not present
in the Schroedinger equation. Only with these additional
degrees of freedom can one derive Ohanians result. Thus,
in a way, spin has still to be put in by hand. Of course,
Ohanian shows that what is put in as an abstract spinor
component can be seen to come out as a circulation in the
probability current of the Dirac particle.
BTW, the analogy between electromagnetic spin and
Dirac spin seems to be a special case of the general fact
that Maxwell's equations may be written in a Dirac-like
fashion and vice versa.
Urs Schreiber
Bennett Standeven
message news: i389btk9tbhl7tdec...@4ax.com...
> I'm trying to visualise spin. What do you people see in you head
> when you talk of spin of a fundamental particle ?
I imagine a set of points and circles around the axis of measurement,
representing the possible directions of the spin. For a spin 1/2 particle,
there are just two points, one at either "pole" of the sphere with radius
1/2.
For a spin 1 particle, there are two points at the "poles" the sphere of
radius 1, and a circle around the "equator". In the classical limit, spin
goes to infinity, and we get the entire sphere as the set of possible
directions.
Matt McIrvin
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>I have read, and enjoyed, Ohanian's result that the
>spin angular momentum manifests itself as a circulation
>in the wave function in a way very analogous to the
>situation in electromagnetism. But it should be noted
>that one needs the Dirac equation to start with in order to
>derive this fact, it does not follow from the Schroedinger
>equation. From your previous posting I got the impression
>that you claimed that spin is derivable from Schroedinger's
>equation alone, which, I belive, is not true. Or am I missing
>something?
Spin is not *derivable* from the Schroedinger equation, but it is
possible to see, using a form of it, that spin angular momentum arises
from a momentum density... with a little handwaving, but I'd argue
that it is really no less handwaving than happens in the relativistic
case.
It's possible to treat an electron with spin nonrelativistically by
using a two-component wave function with the Schrodinger equation (as
is often done in atomic physics, at least in undergraduate classes).
Feynman managed to derive the first-order term in the magnetic moment
of the electron (g=2) nonrelativistically by replacing the gradient^2
term in the Schrodinger equation with (gradient.sigma)^2 and using
minimal coupling (I read about this stunt somewhere in Sakurai's
_Advanced Quantum Mechanics_). If you work backwards and write down a
free Lagrangian that formally yields Feynman's version of the
Schrodinger equation, then find the Noetherian energy-momentum tensor
from that Lagrangian, then *symmetrize* it, you end up with the
nonrelativistic limit of Ohanian's current without having used the
Dirac equation.
The symmetrization is the one part that is sort of relativistic,
because to symmetrize between space and time components you need to
assume that the particle's energy has an mc^2 contribution. But this
is just the "zeroth-order" part of relativity rather than the
higher-order v^2/c^2, etc. contributions.
These formal choices are somewhat arbitrary and ultimately justified
only by such things as the experimental validation of the minimal
coupling result. But that's also true in the relativistic case;
after all, the free electron field obeys the Klein-Gordon equation
as well as the Dirac equation, but we use Dirac (with hindsight)
because it gives the right photon interaction under minimal coupling.
(This is where I'd say something deep about Foldy-Wouthuysen
transformations if I remembered how they worked.)
I really need to get around to writing my Web page about this.
Squark
>Imagine a set of points and circles around the axis of measurement,
>representing the possible directions of the spin. For a spin 1/2 particle,
>there are just two points, one at either "pole" of the sphere with radius
>1/2. For a spin 1 particle, there are two points at the "poles" the sphere
>of radius 1, and a circle around the "equator".
This picture is valid while you keep a warning in your head: it results
from an arbitrary choice of the axis along which you measure the
spin. Equally well, the two poles could be some two different antipodal
points etc. Moreover, there are specific transition matrices betweem
these different basises. Therefore, this picture is in a sense
incomplete, and may be misleading without the above warning. However, it
is still a useful one.
>In the classical limit, spin goes to infinity, and we get the entire
>sphere as the set of possible directions.
Yeah, but this doesn't explain the classical transition fully - how do
we get the separation between the points of the same "latitude"? To
imagine this transition, we need to build some sort of "coherent states"
for angular momentum... Has anyone done so? It sounds somewhat useful,
but I haven't seen any realization of the idea.
Best regards,
squark.
----------------------------------------------------------------------------
Write to me at:
[Note: the fourth letter of the English alphabet is used in the later
exclusively as anti-spam]
dSdqudarkd_...@excite.com
[Moderator's note: coherent states of angular momentum are discussed
in texts such as
J. Klauder, editor, Coherent States: Applications in Physics and
Mathematical Physics, World Scientific, Singapore, 1985.
and
A. Perelomov, Generalized Coherent States and Their Applications,
Springer, Berlin, 1986.
They are quite useful for understanding the relation between the
classical and quantum treatments of angular momentum. - jb]
Gerard Westendorp
[..]
> I have read, and enjoyed, Ohanian's result that the
> spin angular momentum manifests itself as a circulation
> in the wave function in a way very analogous to the
> situation in electromagnetism. But it should be noted
> that one needs the Dirac equation to start with in order to
> derive this fact, it does not follow from the Schroedinger
> equation.
I agree here. The Schrodinger equation describes a spin 0
particle. You can have angular momentum zero, as for example
in the lowest atomic orbital.
> From your previous posting I got the impression
> that you claimed that spin is derivable from Schroedinger's
> equation alone, which, I belive, is not true. Or am I missing
> something?
No, I was trying to see the impact of the replacing the
well known orbital solutions of the Schrodinger equation
with solutions to the Dirac equation, and look at how this
gives rise to an aditional net circulation of momentum density.
Gerard
Gerard Westendorp
> BTW, the analogy between electromagnetic spin and
> Dirac spin seems to be a special case of the general fact
> that Maxwell's equations may be written in a Dirac-like
> fashion and vice versa.
I have not seen this anywhere, but it turns out that the
Maxwell equations can be written as a special case of the
Dirac equation!
Suppose I rewrite the Maxwell equation by putting all
variables in a single vector. I'm also going to add
2 extra zero valued elements making a total of 8
elements.
Next, write out the Maxwell equations. The elements marked "a"
can be any value, since we are multiplying them by zero anyway.
( Ex ) ( 0 0 0 a 0 -d_z d_y a ) ( Ex )
( Ey ) ( 0 0 0 a d_z 0 -d_x a ) ( Ey )
( Ez ) ( 0 0 0 a -d_y d_x 0 a ) ( Ez )
d_t ( 0 ) = ( 0 0 0 a 0 0 0 a ) ( 0 )
( Bx ) ( 0 d_z -d_y a 0 0 0 a ) ( Bx )
( By ) (-d_z 0 d_x a 0 0 0 a ) ( By )
( Bz ) ( d_y -d_x 0 a 0 0 0 a ) ( Bz )
( 0 ) ( 0 0 0 a 0 0 0 a ) ( 0 )
Insert the relations:
d_t 0 = -div B
d_t 0 = div E
and put in convenient values for the "a" 's:
( Ex ) ( 0 0 0 0 0 -d_z d_y d_x) ( Ex )
( Ey ) ( 0 0 0 0 d_z 0 -d_x d_y) ( Ey )
( Ez ) ( 0 0 0 0 -d_y d_x 0 d_z) ( Ez )
d_t ( 0 ) = ( 0 0 0 0 -d_x -d_y -d_z 0 ) ( 0 )
( Bx ) ( 0 d_z -d_y -d_x 0 0 0 0 ) ( Bx )
( By ) (-d_z 0 d_x -d_y 0 0 0 0 ) ( By )
( Bz ) ( d_y -d_x 0 -d_z 0 0 0 0 ) ( Bz )
( 0 ) ( d_x d_y d_z 0 0 0 0 0 ) ( 0 )
We have now put the Maxwell equations in a somewhat unusual form,
one that will turn out to be a special case of Dirac.
************
Take the Dirac equation:
( p ) (-im 0 -d_z -d_x+id_y) ( p )
d_t ( q ) = ( 0 -im -d_x-id_y d_z ) ( q )
( u ) (-d_z -d_x+id_y im 0 ) ( u )
( v ) (-d_x-id_y d_z 0 im ) ( v )
Write out the complex numbers as 2-component real vectors:
( pr ) ( 0 m 0 0 -d_z 0 -d_x -d_y) ( pr )
( pi ) ( -m 0 0 0 0 -d_z d_y -d_x) ( pi )
( qr ) ( 0 0 0 m -d_x d_y dz 0) ( qr )
d_t ( qi ) = ( 0 0 -m 0 -d_y -d_x 0 d_z) ( qi )
( ur ) (-d_z 0 -d_x -d_y 0 -m 0 0 ) ( ur )
( ui ) ( 0 -d_z d_y -d_x m 0 0 0 ) ( ui )
( vr ) (-d_x d_y dz 0 0 0 0 -m ) ( vr )
( vi ) (-d_y -d_x 0 d_z 0 0 m 0 ) ( vi )
Now rearrange the elements by choosing a different basis:
( pi ) ( 0 -m 0 0 0 -d_z d_y d_x) ( pi )
( pr ) ( -m 0 0 0 d_z 0 -d_x d_y) ( pr )
(-qi ) ( 0 0 0 -m -d_y d_x 0 d_z) (-qi )
d_t (-qr ) = ( 0 0 m 0 -d_x -d_y -d_z 0 ) (-qr )
(-ur ) ( 0 d_z -d_y -d_x 0 m 0 0 ) (-ur )
( ui ) (-d_z 0 d_x -d_y -m 0 0 0 ) ( ui )
( vr ) ( d_y -d_x 0 -d_z 0 0 0 m ) ( vr )
(-vi ) ( d_x d_y d_z 0 0 0 -m 0 ) (-vi )
And, by taking the special case
m = 0 = qr = vi
We get the Maxwell equations!
Gerard
Hans Aberg
<wes...@xs4all.nl> wrote:
>> BTW, the analogy between electromagnetic spin and
>> Dirac spin seems to be a special case of the general fact
>> that Maxwell's equations may be written in a Dirac-like
>> fashion and vice versa.
>
>I have not seen this anywhere, but it turns out that the
>Maxwell equations can be written as a special case of the
>Dirac equation!
This is well known, simply use the Clifford algebra C(g) on the Lorentz metric:
If / is the Feynman slash, then the Dirac equation can be written
(1) i /d psi = (m + e /A) psi
whereas the Maxwell equations can be written
(2a) /d F = ~J/(epsilon_0 c)
(2b) /d A = F
where c is the speed of light in vacuum, epsilon_0 is the dielectric
constant in vacuum, and ~J is the metric dual of the source J.
Urs Schreiber
<remove.haberg-0...@du132-226.ppp.su-anst.tninet.se>):
>In article <3AF076ED...@xs4all.nl>, Gerard Westendorp
><wes...@xs4all.nl> wrote:
>>> BTW, the analogy between electromagnetic spin and
>>> Dirac spin seems to be a special case of the general fact
>>> that Maxwell's equations may be written in a Dirac-like
>>> fashion and vice versa.
>>I have not seen this anywhere, but it turns out that the
>>Maxwell equations can be written as a special case of the
>>Dirac equation!
>This is well known, simply use the Clifford algebra C(g) on the Lorentz metric:
>
>If / is the Feynman slash, then the Dirac equation can be written
> (1) i /d psi = (m + e /A) psi
>whereas the Maxwell equations can be written
> (2a) /d F = ~J/(epsilon_0 c)
> (2b) /d A = F
>where c is the speed of light in vacuum, epsilon_0 is the dielectric
>constant in vacuum, and ~J is the metric dual of the source J.
I had been thinking about this way to write down Maxwell:
with
| 0 |
\phi = | B_1 - iE_1 |
| B_2 - iE_2 |
| B_3 - iE_3 |
and j the usual 4-current Maxwell's equations are equivalent to
-i A^j d_j \phi = 4 pi/c j
with the following A^j:
1 0 0 0
A^0 = 0 1 0 0
0 0 1 0
0 0 0 1
0 -1 0 0
A^1 = -1 0 0 0
0 0 0 -i
0 0 i 0
0 0 -1 0
A^2 = 0 0 0 i
-1 0 0 0
0 -i 0 0
0 0 0 -1
A^3 = 0 0 -i 0
0 i 0 0
-1 0 0 0
[Equations aligned by moderator]
Hans Aberg
Schreiber<Urs.Sc...@uni-essen.de> wrote:
>Hans Aberg wrote :
>>If / is the Feynman slash, then the Dirac equation can be written
>> (1) i /d psi = (m + e /A) psi
>>whereas the Maxwell equations can be written
>> (2a) /d F = ~J/(epsilon_0 c)
>> (2b) /d A = F
>>where c is the speed of light in vacuum, epsilon_0 is the dielectric
>>constant in vacuum, and ~J is the metric dual of the source J.
I should have said that the Feynman slash is just the usual Clifford
multiplication. For the Dirac equation, I extracted it for the $\chi$-ral
representation in a spin manuscript I wrote; there is a link at my WWW
page. If somebody is interested in the Clifford algebra version of the
Maxwell equations, I can email a DVI file.
>I had been thinking about this way to write down Maxwell:
>
>with
> | 0 |
>\phi = | B_1 - iE_1 |
> | B_2 - iE_2 |
> | B_3 - iE_3 |
But what about coordinate transformations: The two-form formulation is
known to be coordinate independent on a Lorentz manifold. And when working
with QM, self adjoint operators correspond to observables, so you depart
from that picture.
Roland Franzius
>
> Urs Schreiber wrote:
>
> > BTW, the analogy between electromagnetic spin and
> > Dirac spin seems to be a special case of the general fact
> > that Maxwell's equations may be written in a Dirac-like
> > fashion and vice versa.
>
> Maxwell equations can be written as a special case of the
> Dirac equation!
Do it coordinate free. Look eg Maxwells own elegant quaternion
formulation at
http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=N095176&E=0
A Treatise .. Vol II Chapt. IX, sect 619-620
--
Roland Franzius
+++ exactly <<n>> lines of this message have value <<FALSE>> +++
[Moderator's note: Unnecessary quoted text trimmed. -MM]
Hans Aberg
<Roland....@uos.de> wrote:
>Do it coordinate free. Look eg Maxwells own elegant quaternion
>formulation at
>
>http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=N095176&E=0
And the quaternions are contained in the Clifford algebra, so one may well
end up with the latter anyhow.
Gerard Westendorp
Clifford algebra's to clarify the issue of the analogy between
the Maxwell and Dirac equations.
Since then, I have begun to like Clifford Algebra's. Let me
say something about them. I realize of course there are
experts on this subject at s.p.r., but for them it is
probably less fun to write about the first principles.
What is Clifford algebra, and what is so cool about it?
We already have lots of ways to multiply vectors: the
internal product, the external product, complex (2-vector)
multiplication, quaternion (4-vector) multiplication, the
Feynman slash product, etc. It seems we are hardly waiting
for yet another product. But the Clifford product actually
clarifies and classifies the other products, rather than
add extra complications. In fact, the basics of Clifford
algebra are very simple.
I will use a 2D example rather than write out the
nD case, which is easy to generalize from 2D.
Express a vector as a sum of real numbers a_i, and
elementary vectors e_i:
x = a_1 e_1 + a_2 e_2
y = b_1 e_1 + b_2 e_2
We can always write out the product z = x*y as:
z = a_1*b_1 e_11 + a_1*b_2 e_12
+ a_2*2_2 e_21 + a_2*b_2 e_22
where e_ij is shorthand for e_i * e_j.
Next, as with any product definition, we need a rule
to combine symbols like e_11 into others. Clifford (1876)
proposed that it would be nice if the square of a vector
produces its norm. This is true if and only if:
e_jj = 1 (or -1 for time-like indices)
e_jk = -e_kj (j ^= k)
That's all there is to Clifford algebra's!
The rest is just exploring the consequences. The product
is, with the new rules:
z = (a_1*b_1 + a_2*b_2) + (a_1*b_2 - a_1*b_2)e12
This result looked a bit disappointing to me at first. A
product of 2 vectors is not a vector, but the sum of a scalar
and a new quantity e12, called a bi-vector. The introduction
of these new "dimensions" seems to make life more complicated.
In higher dimensions, things get worse. A Clifford algebra based
n dimensional vectors, called Cl(n), has dimension 2^n, for example,
Cl(4) has the basis:
1
e_1 e_2 e_3 e_4
e_12 e_13 e_14 e_23 e_24 e_34
e_234 e_134 e_124 e_123
e_1234
Note that combinations with repeated indices, like e_223, can be
written eliminated using our rules. So for any dimension, there
is a finite basis for the Clifford algebra. Note also the symmetry
in the above table. It gives rise to the "Hodge dual" construction,
mentioned in the thread on "Electromagnetic energy".
Later I realized that the "extra dimensions", or multivectors as
they are called, actually have a distinct meaning. Because in our
familiar 3D world, a bi-vector can be uniquely associated with
a vector (its Hodgwe dual), we sometimes overlook the fact that
they are really different things than vectors. This can be visualized
in the discrete world of circuit diagrams.
I am thinking specifically about grid-like circuits, as the ones
on my site (link at the end). If you take a piece of squares-paper,
how many nodes, edges and squares does it have, if there are m
nodes on it? The answer, ignoring the complications at the edge of
the paper, is:
nodes: m
edges: 2 m
squares: m
compare this with the basis for Cl(2)
1
e_1 e_2
e_12
See the pattern emerging?
The scalars of Clifford correspond to nodes on the grid.
The edges, of which there are n in an nD grid, correspond
to the vectors e_i.
The bivectors e_ij, one for each combination ij, correspond
to the squares, or loops, in the diagram.
This correspondence continues, as trivectors e_ijk correspond
to 3D boxes, and so on. There is a one to one correspondence
between the basis of the Clifford algebra and topological
features in the circuit.
As I describe on my site, you can assign energy to each of
these topological features, by inserting a capacitor or
an inductance. For vector, this is done in the normal way:
the edge contains a circuit element.
For scalars, you put a circuit element from the node to
ground potential. For bi-vectors, you can couple a transformer
yoke to the loop (A magnetic field element). I don't know any
electric element that couples in the right way to a volume. But we
could just formally define one, and invent a suitable symbol for it.
Thus, we can describe the dynamics of a grid-like circuit in terms
of fields that are naturally described by Clifford algebra's!
The dynamic equations will look particularly compact and
elegant when witten using the Clifford product.
I have a circuit equivalent for the Maxwell equations, but not
for the Dirac equation yet. With these new insights, I think
I should have one soon.
Gerard
Some stuff on circuit diagrams:
http://www.xs4all.nl/~westy31/Electric.html#Maxwell
Hans Aberg
The Clifford algebra is so cool, because it can be defined simply in terms
of the Lorentz vector space, which is the data available in QM and GR.
>We already have lots of ways to multiply vectors: the
>internal product, the external product, complex (2-vector)
>multiplication, quaternion (4-vector) multiplication, the
>Feynman slash product, etc. It seems we are hardly waiting
>for yet another product.
The Clifford algebra can be realized as the exterior algebra with
multiplication the sum of the exterior and the (metric dual) interior
multiplication.
> But the Clifford product actually
>clarifies and classifies the other products, rather than
>add extra complications.
Right, that makes it even cooler. :-)
Mark William Hopkins
>I have not seen this anywhere, but it turns out that the
>Maxwell equations can be written as a special case of the
>Dirac equation!
In article <remove.haberg-0...@du132-226.ppp.su-anst.tninet.se>
(Hans Aberg) writes:
>This is well known, simply use the Clifford algebra C(g) on the Lorentz
>metric [...]
The correspondence between the two is purely formal and is a consequence of
the fact that both are instances of the general spin N/2 wave equation
(which can be represented in unified form using the algebra of Kemmer matrices,
the equations 2a and 2b of Hans Aberg are the components of the spin-1
10x10 Kemmer matrix rep of the spin-1 wave equation).
But they have completely different transformation properties and it's
the transformation properties where all the Physics resides, not the
correspondence.
So nothing Physically relevant comes of the correspondence.
Hans Aberg
William Hopkins) wrote:
>>I have not seen this anywhere, but it turns out that the
>>Maxwell equations can be written as a special case of the
>>Dirac equation!
>>This is well known, simply use the Clifford algebra C(g) on the Lorentz
>>metric [...]
>
>The correspondence between the two is purely formal and is a consequence of
>the fact that both are instances of the general spin N/2 wave equation
>(which can be represented in unified form using the algebra of Kemmer matrices,
>the equations 2a and 2b of Hans Aberg are the components of the spin-1
>10x10 Kemmer matrix rep of the spin-1 wave equation).
>
>But they have completely different transformation properties and it's
>the transformation properties where all the Physics resides, not the
>correspondence.
If it is the covariance principle that is applied to the Dirac equation,
then that is a somewhat fishy principle:
The Dirac equation appears to only describe a particle-antiparticle in
some kind of coupling, producing from the physical point of view some kind
of strange phenomenons in the solutions.
In addition, when applied to higher degree spin not all spin combinations
can appear if the covariance principle is imposed (I recall it must be of
equal spin and anti-spin).
(The covariance transformations of the Dirac equation are the same as the
Lorentz coordinate transformations except for an inverse put in on the
transformation matrix. This is thus, from the mathematical point of view,
a form of invariance when the Lorentz group is acting on the Dirac
equation.)
There one has abandoned those simple field equations in favour of QFT.
If one should move forward on those simple field equations to get some
kind of relativistic Schrodinger equation (many particle theory), it
appears that one has to find some reinterpretation of the covariance
principle.
>So nothing Physically relevant comes of the correspondence.
So one should not be too sure of that.
John Baez
Hans Aberg <remove...@matematik.su.se> wrote:
>The Dirac equation appears to only describe a particle-antiparticle in
>some kind of coupling, producing from the physical point of view some kind
>of strange phenomenons in the solutions.
What does this mean? To me, the first-quantized Dirac equation
describes a free spin-1/2 particle/antiparticle. I don't know
what this stuff about "some kind of coupling" or "strange phenomena"
refers to. There's nothing very strange about it.
Or are you talking about what happens when you introduce a
potential...?
Ralph E. Frost
John Baez <ba...@galaxy.ucr.edu> wrote in message
news:9e3ll7$pkn$1...@news.state.mn.us...
Just curious, but HOW does one introduce a potential?
That is, does that involve arranging groups of particles and antiparticles
into various kinds of structures, or must one stir in other components?
Gerard Westendorp
[..Maxwell/Dirac analogy..]
> The correspondence between the two is purely formal and is a consequence of
> the fact that both are instances of the general spin N/2 wave equation
> (which can be represented in unified form using the algebra of Kemmer
> matrices,
> the equations 2a and 2b of Hans Aberg are the components of the spin-1
> 10x10 Kemmer matrix rep of the spin-1 wave equation).
>
> But they have completely different transformation properties and it's
> the transformation properties where all the Physics resides, not the
> correspondence.
>
> So nothing Physically relevant comes of the correspondence.
I don't agree here. Transformations are not physical processes,
but changes in the conventions of the observer. The real physics
is in the dynamic equations, so if these are the same, that is
something at least as relevant as a transformation property.
For example, a linear polarized em. plane wave in the z-direction:
(Using the notation I introduced in my first mail, with the extra
dummy components)
( Ex )
( 0 )
( 0 )
( 0 ) *cos(kz+wt)
( 0 )
( By )
( 0 )
( 0 )
Has the massless Dirac spinor equivalent:
( iEx )
( 0 ) *cos(kz+wt)
( iBy )
( 0 )
This represents a superposition of a fermion and an anti
fermion plane wave, both with spin in the z-direction. Similarly,
any em. plane wave has a massless Dirac spinor equivalent. We can
build any wave packet outof these plane waves, and the Dirac and Maxwell
wave packets will behave identically. But I am not sure about angular
momentum and energy.
Gerard
btw, I'd be interested in learning a bit more on Kemmer matrices.
[Moderator's note: in this context "transformations" include
everything in the Poincare group: in particular, time translations,
which is exactly what the dynamical equations describe. So
"dynamics" is all about how a system appears different to a
time-translated observer. This blurs the distinction Westendorp
is trying to make above, though it may not invalidate his point. - jb]
Gerard Westendorp
the Maxwell equations can be written as a special case of the Dirac
I should add that the analogy between the Maxwell equation
and the Dirac equation is more than just "looks like". A Maxwell
vacuum solution *is* also a solution to the Dirac equation.
For example, a linear polarized plane wave in the z-direction:
( Ex )
( 0 )
( 0 )
( 0 ) *cos(kz+wt)
( 0 )
( By )
( 0 )
( 0 )
With the Dirac spinor equivalent:
( iEx )
( 0 ) *cos(kz+wt)
( iBy )
( 0 )
This represents a superposition of a fermion and an anti
fermion plane wave, both with spin in the z-direction. I am still
confused about the role of spin in this story.
Gerard
Gerard Westendorp
> In article <3AF076ED...@xs4all.nl>, Gerard Westendorp
> <wes...@xs4all.nl> wrote:
[..]
> >I have not seen this anywhere, but it turns out that the
> >Maxwell equations can be written as a special case of the
> >Dirac equation!
>
> This is well known, simply use the Clifford algebra C(g) on the
> Lorentz metric:
I am not used to working with Clifford algebra's, but I
have a copy of the book "Clifford Algebra's and Spinors" by Lounesto,
so I don't mind pulling them into the discussion, especially if
it helps to visualize the Dirac equations.
btw, with C(g) you mean Cl_3,1 ?
> If / is the Feynman slash, then the Dirac equation can be written
> (1) i /d psi = (m + e /A) psi
> whereas the Maxwell equations can be written
> (2a) /d F = ~J/(epsilon_0 c)
> (2b) /d A = F
> where c is the speed of light in vacuum, epsilon_0 is the dielectric
> constant in vacuum, and ~J is the metric dual of the source J.
I don't understand these equations. For example, in (2b), a 4-component
real valued vector seems to be turned into a 16 component real valued
tensor, by the operation of /d. How does that work? I can imagine that
this makes sense using some Clifford product, but I would need more
details here.
Gerard
Charles Francis
<whop...@alpha2.csd.uwm.edu> writes
>In article <3AF076ED...@xs4all.nl>, Gerard Westendorp
><wes...@xs4all.nl> wrote:
>>I have not seen this anywhere, but it turns out that the
>>Maxwell equations can be written as a special case of the
>>Dirac equation!
>So nothing Physically relevant comes of the correspondence.
I think a great deal comes of it. First that we formulate fundamental
physical theory by assuming the interacting Dirac equations, and
deriving Maxwell's equations for the expectations of operators, and we
can do this without canonical quantisation, and second that the value of
charge used in the interacting Dirac equation is the same as that used
in Maxwell's equations - i.e. all renormalisation of charge cancels in
this classical limit, which is important to define the counter terms in
Feynman rules.
Consider the general solution of
d.f(x) = 0 [1]
i.e.
f_a(x) = integral_dp F(p,r) w_a(p,r) e^-ip.x [2]
where
2.1) p^2 =0, and
2.2) w are orthonormal vectors such that
w(p,0) = (1,0,0,0) and for r=1,2,3 w(p,r) = (0,w(p,r))
where w(p,1) and w(p,2) are transverse, and w(p,3) is
proportional to p (3-vector p)
2.3) F satisfies F(p,0) = F(p,3)
Then from [2]
Box(f) = Integral_dp F(p,r) w_a(p,r) p^2 e^-ip.x
= 0 [3]
by 2.1.
That seems to be all right for wave functions, but it is expectations
which appear in classical law, so what we actually want is Box(<A>), so
we want to reinterpret [1] and [2] as operator equations. But
Box(<A>) = i d_a { [H, d_a A] + <d_a A> }
= i [H, Box(A)] + i [H, d_a A] + <Box(A)>
= i [H, d_a A] [4]
by [3]. Now because of the equal time commutator, [A(x),A(y)] = 0, [4]
reduces at once to
Box(<A>) = i [H, d_0 A] [4]
So for an interaction density H = e A j we obtain the form of Maxwell's
equations under the constraint [1] (Lorentz gauge), i.e.
Box(<A>) = - e <j>
- --
Charles Francis
Hans Aberg
<wes...@xs4all.nl> wrote:
>... Transformations are not physical processes,
>but changes in the conventions of the observer. The real physics
>is in the dynamic equations, so if these are the same, that is
>something at least as relevant as a transformation property.
This is also my interpretation of the situation:
The covariance principle is way to introduce a change-of-observer
principle. It is incompatible with the way GR (general relativity)
describe change of observers.
Further, I have found that mathematicians may get into the trap to confuse
the covariance principle with the change-of-coordinate principle used when
introducing a manifold and data on its analysis (like group bundles and
the like).
Using the covariance principle on some kind of manifold or group bundle
imposes severe restrictions on the manifold, and one could not really
expect to do any kind of GR that way. (This is the empiricism I arrived at
when looking into the matter several years ago, but I could not provide
you with any details at this point in time.)
Gerard Westendorp
"Ralph E. Frost" wrote:
> Just curious, but HOW does one introduce a potential?
>
> That is, does that involve arranging groups of particles and antiparticles
> into various kinds of structures, or must one stir in other components?
There is a cool way to introduce a potential into the Dirac equation,
called the gauge principle. You just substitute:
d_mu -> d_mu + i e A_mu
e is the charge, A_mu is the potential 4 vector.
To get quantum electrodynamics, you also have to put a source
term in the Maxwell equations, (that govern dynamics of the potentials
A_mu), and then second quantize the whole thing. But the non-quantized
theory of a Dirac wave interacting with electromagnetic waves, as
described by combining the Dirac and Maxwell equations is also quite
interesting.
Gerard
[Moderator's note: Frost may have desired a less technical answer,
such as: the Dirac equation describes electrons and positrons,
and we "introduce a potential" when we put these electrons and
positrons in an electromagnetic field, e.g. by shining a light
on them. - jb]
Hans Aberg
Baez) wrote:
>>The Dirac equation appears to only describe a particle-antiparticle in
>>some kind of coupling, producing from the physical point of view some kind
>>of strange phenomenons in the solutions.
>
>What does this mean? To me, the first-quantized Dirac equation
>describes a free spin-1/2 particle/antiparticle. I don't know
>what this stuff about "some kind of coupling" or "strange phenomena"
>refers to. There's nothing very strange about it.
>
>Or are you talking about what happens when you introduce a
>potential...?
It relates to the solutions of the Dirac equation described in the book by
Itzykson & Zuber, "Quantum Field Theory":
One gets (or so I recall) some solutions that looks like that of a coupled
particle-antiparticle, called "Bremsstrahlung" or something. (I have only
a vague memory of this.)
Anyway, both the spin and anti-spin are always present in the Dirac
equation, and it does not seem possible to get a field equation for just
one of them. (It is certainly not possible if you impose the covariance
principle.)
Gerard Westendorp
>
> Hans Aberg wrote:
[..]
>
> > If / is the Feynman slash, then the Dirac equation can be written
> > (1) i /d psi = (m + e /A) psi
> > whereas the Maxwell equations can be written
> > (2a) /d F = ~J/(epsilon_0 c)
> > (2b) /d A = F
> > where c is the speed of light in vacuum, epsilon_0 is the dielectric
> > constant in vacuum, and ~J is the metric dual of the source J.
>
> I don't understand these equations. For example, in (2b), a 4-component
> real valued vector seems to be turned into a 16 component real valued
> tensor, by the operation of /d. How does that work? I can imagine that
> this makes sense using some Clifford product, but I would need more
> details here.
The quoted message took some time to get through, so in the meanwhile
I can partly answer this question myself. Hopefully this will help to
clear things up.
Using as a basis:
1
e_x e_y e_z e_t
e_xy e_xz e_xt e_yz e_yt e_zt
e_xyz e_xzt e_xyt e_xyz
e_xyzt
with rules:
e_t^2 = -1
e_x^2 = e_y^2 = e_z^2 = 1
e_i e_j = - e_j e_i
We can write out the 4-vector A and the differential operator d as
A = A_t e_t + A_x e_x + A_y e_y + A_z e_z
d = d_t d_t + d_x e_x + d_y e_y + d_z e_z
The product d*A leads to a scalar and a 6 component bivector.
The scalar is related to the Lorentz condition, the bivector
is the antisymmetric em 4-tensor F.
The Maxwell equations then turn out quite nicely as d*F=J. This
equation contains all 4 Maxwell equations when you write out
the Clifford stuff.
So that is nice.
But how about the Dirac equation? I want to write it out
in real numbers, because I want a one-to one correspondence
in real dynamic variables. The problem is that when you write the
differential operator the same way as with Maxwell, and try
psi = psi_t e_t + psi_x e_x + psi_y e_y + psi_z e_z
ie. a 4 component complex valued vector.
The product d*psi = 0 gives a scalar and a 6 component
bivector, so that does not fit in with the Dirac equation.
One thing that would seem to work, is build a 16 component psi
multivector, with a component in each element of the Clifford
algebra. If you then write
i d*psi = m psi
or in real numbers:
e_xyzt d*psi = m psi
Then you should indeed have a representation of the Dirac equation,
as the gamma matrices (gamma_x, gamma_y, gamma__z, gamma__t) obey
the same algebra as the elements (e_x, ey e_z, e_t) of the
Clifford algebra.
Unfortunately, this equation has 16 real dynamic components, instead
of the 8 (=4 complex) in the usual representation of the Dirac
equation. I suspect we can rewrite things to cure this, but I have
not yet figured it out.
Gerard
Hans Aberg
<wes...@xs4all.nl> wrote:
>> This is well known, simply use the Clifford algebra C(g) on the
>> Lorentz metric:
>
>I am not used to working with Clifford algebra's, but I
>have a copy of the book "Clifford Algebra's and Spinors" by Lounesto,
>so I don't mind pulling them into the discussion, especially if
>it helps to visualize the Dirac equations.
>
>btw, with C(g) you mean Cl_3,1 ?
I don't know what notation they use in that book. :-)
But my guess is what they write as Cl_3,1 _or_ Cl_1,3 are what I write as
C(g) (depending on whether g is +--- or -+++ in some order and conventions
for defining the Clifford algebra).
Note that there is a problem here the real algebras Cl_3,1 and Cl_1,3 are
not isomorphic; their complexifications might be isomorphic. This problem
has been discussed here before: Perhaps the electromagnetic potential is
fishy quantity, and the real operators are all in the even part of C(g)
which does not depend on the sign of g. Or some other solution to the
problem.
>> If / is the Feynman slash, then the Dirac equation can be written
>> (1) i /d psi = (m + e /A) psi
>> whereas the Maxwell equations can be written
>> (2a) /d F = ~J/(epsilon_0 c)
>> (2b) /d A = F
>> where c is the speed of light in vacuum, epsilon_0 is the dielectric
>> constant in vacuum, and ~J is the metric dual of the source J.
>
>I don't understand these equations. For example, in (2b), a 4-component
>real valued vector seems to be turned into a 16 component real valued
>tensor, by the operation of /d. How does that work? I can imagine that
>this makes sense using some Clifford product, but I would need more
>details here.
One way to realize the Clifford C(g) algebra is by taking the algebra of
differential Omega, with multiplication the sum of the exterior and
interior multiplications. The Feynman slash just becomes this
multiplication. The ordinary differential d has exterior multiplication as
symbols (raising degrees by one on Omega), and its dual "divergence"
$\partial$ has the interior multiplication as symbol (lowering degrees by
one).
So (2) can be written $(d + \partial)A = F$, and as F is a two form, it
follows that dA = F and that \partial A = 0, that is the divergence is
zero. Etc.
I can send you a DVI with the rest of the details.
Gerard Westendorp
> such as: the Dirac equation describes electrons and positrons,
> and we "introduce a potential" when we put these electrons and
> positrons in an electromagnetic field, e.g. by shining a light
> on them. - jb]
If you haven't already done so, you might want to read
Feynman's book "QED". This explains it all in a pretty nice
way. But Feynman does not use the Maxwell and Dirac
equations, he uses the picture of particles that take
multiple paths, and carry mysterious clocks with them
that keeps track of the phase of the particle's path.
Gerard
Ralph E. Frost
> such as: the Dirac equation describes electrons and positrons,
> and we "introduce a potential" when we put these electrons and
> positrons in an electromagnetic field, e.g. by shining a light
> on them. - jb]
I appreciate both clarifications and can understand jb's less abstract and
thus (for me) more comprendable expression better that the other
approximation. I thought the beginning point, though, was beginning just
with particles and antiparticles (or as stated, electrons and positrons)
what happens when those found themselves in a potential. So I was
thinking that the gradient or potential would also have to be related back
to SOME clump of particles and antiparticles. Thus, that led me to wonder
specifically HOW one arranges and stacks up particles and antiparticles so
as to fabricate the potential.
That is, it looks to me like nobody has any means of acquiring an
electromagnetic field except by arranging particles and antiparticles -- so
I am wondering if this is "just" a structures problem. If is it, hen it
sseems like someone ought to be able to state which angles and distances are
impotant, etc.
Is that what "guage" refers to?
Thanks for any help you can provide.
[Moderator's note: Quoted text trimmed...
As to Ralph's question, a single charged particle is enough to
produce a Coulomb electric potential, with a corresponding effect
on electrons and positrons; the theory of the hydrogen atom deals
with this. More complex arrangements can be produced; see the
chapters on electrostatics in any electromagnetism text for
examples. This is with the usual potential conventions of
electrostatics; "gauge" here refers to our freedom to change those
conventions without affecting electromagnetic fields. -MM]
Gordon D. Pusch
> [...] One thing that would seem to work, is build a 16 component
> psi multivector, with a component in each element of the Clifford
> algebra. If you then write
>
> i d*psi = m psi
>
> or in real numbers:
>
> e_xyzt d*psi = m psi
>
> Then you should indeed have a representation of the Dirac equation,
> as the gamma matrices (gamma_x, gamma_y, gamma__z, gamma__t) obey
> the same algebra as the elements (e_x, ey e_z, e_t) of the
> Clifford algebra.
I suspect this equation is equivalent to what David Hestenes called the
``Dirac-Gursey Equation'' in his book ``Spacetime Algebra.'' In terms of
two four-component spinors \xi and \chi, it has the form:
\dslash \xi = m \gamma_5 \chi,
\dslash \chi = -m \gamma_5 \xi.
where \dslash = \gamma^\mu \partial_\mu. It describes a doublet of spin-1/2
particles with an isospin-like SU(2) symmetry and a discrete symmetry
equivalent to ``G-parity.'' (``Isospin'' transformations are the result
of a Clifford rotor acting on \psi from the right instead of the left.)
> Unfortunately, this equation has 16 real dynamic components, instead
> of the 8 (=4 complex) in the usual representation of the Dirac
> equation. I suspect we can rewrite things to cure this, but I have
> not yet figured it out.
Alternatively, one could look on this doubling of the number of degrees
of freedom as saying something ``deep:'' It is most natural within the
Clifford-algebraic treatment of spinors for spin-1/2 particles to occur
in doublets having an isospin-like SU(2) symmetry. The question then
becomes how to identify these doublets with the doublets in the
Standard Model; Hestenes appears to have made a beginning of this
in Chap. 14 of ``SpaceTime Calculus,'' a draft of which is available
at <http://modelingnts.la.asu.edu/pdf/SpaceTimeCalc.pdf>.
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Mark William Hopkins
>What does this mean? To me, the first-quantized Dirac equation
>describes a free spin-1/2 particle/antiparticle. I don't know
>what this stuff about "some kind of coupling" or "strange phenomena"
>refers to.
It probably refers to this.
The semiclassical electron can be consistently modelled as a kind of
self-interacting bump of some sort in the electromagnetic field, which
lies on a helical lightlike worldline. This is Hestenes' Zitterbewegung
Interpretation.
The essential core of this interpretation lies in decomposing the
Dirac spinor into
psi = (rho exp(i gamma5 beta))^{1/2} R (1 0 0 0)^T
where the 6-component bivector R (the "Lorentz rotor") yields an invariant
representation a comoving Lorentz frame, e^i = R gamma^i. The Dirac equation
induces a set of equations for rho, beta and the e^i which yields the desired
interpretation and contains the conservation equation as a special case. The
current vector
J = (-e) gamma^m (psi-bar gamma_m psi)
is just (-e) e^0. The worldlines generated by the e^0 vector field yield the
axes of the helical motion and e^2 points along the radius. The angular
motion along the helix yields the correct angular momentum and gyromagnetic
ratio corresponding to the electron "spin" and everything else works
out correctly, and there is even an interpretation and representation of the
Pauli exclusion principle without the need for 2nd quantization; and
supposedly also there is a deterministic interpretation of the Uncertainty
Principle as relates to the particle's worldlines.
He's gone way off with this stuff and even has a web site now.
modelingnts.la.asu.edu.
The underlying programme (aside from the desire to assimilate all of
physics within his mathematical framework) was, like that of the rest of
the Foundations of Physics crowd, to dispense with the need for
2nd quantization by providing a purely particle-theoretic interpretation
of all the key phenomena (e.g. particle identity, pair processes,
radiative corrections, etc.) and resolving the (still open) problem
of arriving at a mathematically consistent particle-based relativistic
quantum mechanics. He, Bohm, De Broglie and Barut were the chief
progenitors of this family of approaches.
Ironically, there is a recent book that came out which provides a
treatment of an entirely non-perturbative quantum field theory, in which
bosons are fermion composites, by 2nd quantizing de Broglie's 'fusion' theory.
John Baez
Hans Aberg <remove...@matematik.su.se> wrote:
>In article <9e3ll7$pkn$1...@news.state.mn.us>, ba...@galaxy.ucr.edu (John
>Baez) wrote:
>>>The Dirac equation appears to only describe a particle-antiparticle in
>>>some kind of coupling, producing from the physical point of view some kind
>>of strange phenomenons in the solutions.
>>What does this mean?
>It relates to the solutions of the Dirac equation described in the book by
>Itzykson & Zuber, "Quantum Field Theory":
>
>One gets (or so I recall) some solutions that looks like that of a coupled
>particle-antiparticle, called "Bremsstrahlung" or something. (I have only
>a vague memory of this.)
Hmm. Bremsstrahlung, Zitterbewegung, it's all so confusing....
>Anyway, both the spin and anti-spin are always present in the Dirac
>equation, and it does not seem possible to get a field equation for just
>one of them. (It is certainly not possible if you impose the covariance
>principle.)
You're right, this is true for particles with mass.
For massless spin-1/2 particles we can have Weyl spinors, which
come in only one handedness. We can also have Dirac spinors,
which come in both handedness. Both these sorts of particles
are not their own antiparticle. Besides these two, we can
also have Majorana spinors, which come in both handedness
but are their own antiparticle!
Massive spin-1/2 particles can be either Dirac spinors
(i.e., not their own antiparticle) or Majorana spinors
(i.e., their own antiparticle).
For example, in the old version of the Standard Model -
back in the good old days when we thought neutrinos
had no mass, and there were no right-handed neutrinos -
the neutrinos were treated as Weyl spinors.
Now that we think the neutrinos have mass, they could
be either a Dirac or Majorana spinors. People like to
argue about which, and do experiments to settle this
question:
The neutrino oscillation industry,
http://www.hep.anl.gov/NDK/hypertext/nu_industry.html
Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/~www/neutrino/jhu/jhu.html
Mathematical note:
All these facts I just stated about the classification of spinors
are special to 4d spacetime. In other dimensions it follows this
pattern:
n Dirac Majorana Weyl Majorana-Weyl
1 C R
2 C^2 R^2 C R
3 C^2 R^2
4 C^4 R^4 C^2
5 C^4
6 C^8 C^4
7 C^8
8 C^16 R^16 C^8
When there are blanks here, the relevant sort of spinor doesn't
exist. The pattern repeats after n = 8 in a period-8 way. Note
that Majorana-Weyl spinors don't exist in dimension 4. These are
spin-1/2 particles that have only one handedness and are their own
antiparticle! Majorana-Weyl spinors are only a mathematical
possibility in dimension 2, dimension 10, dimension 18, dimension
26, dimension 34, dimension 42, and so on. These include some of
dimensions string theorists used to like most: 2, 10, and 26.
Here is some more information about this business...
Also available at http://math.ucr.edu/home/baez/week130.html
February 27, 1999
This Week's Finds in Mathematical Physics (Week 130)
John Baez
All sorts of cool stuff is happening in physics - and I don't mean
mathematical physics, I mean real live experimental physics! I feel
slightly guilty for not mentioning it on This Week's Finds. Let me
atone.
Here's the big news in a nutshell: we may have been wrong about four
fundamental constants of nature. We thought they were zero, but maybe
they're not! I'm talking about the masses of the neutrinos and the
cosmological constant.
Let's start with neutrinos.
There are three kinds of neutrinos: electron, muon, and tau neutrinos.
They are closely akin to the charged particles whose names they borrow -
the electron, muon and tau - but unlike those particles they are
electrically neutral and very light. They are rather elusive, since
they interact only via the weak force and gravity. I'm sure you've all
heard how a neutrino can easily make it through hundreds of light years
of lead without being absorbed.
But despite their ghostly nature, neutrinos play a very real role in
physics, since radioactive decay often involves a neutron turning into a
proton while releasing an electron and an electron antineutrino. (In
fact, Pauli proposed the existence of neutrinos in 1930 to account for a
little energy that went missing in this process. They were only
directly observed in 1956.) Similarly, in nuclear fusion, a proton may
become a neutron while releasing a positron and an electron neutrino.
For example, when a type II supernova goes off, it emits so many
neutrinos that if you're anywhere nearby, they'll kill you before
anything else gets to you! Indeed, in 1987 a supernova in the Large
Magellanic Cloud, about 100,000 light years away, was detected by four
separate neutrino detectors.
I said neutrinos were "very light", but just how light? So far most
work has only given upper bounds. In the 1980s, the Russian ITEP group
claimed to have found a nonzero mass for the electron neutrino, but this
was subsequently blamed on problems with their apparatus. As of now,
laboratory experiments give upper bounds of 4.4 eV for the electron
neutrino mass, .17 MeV for the muon neutrino, and 18 MeV for the tau
neutrino. By contrast, the electron's mass is .511 MeV, the muon's is
106 MeV, and the tau's is a whopping 1771 MeV.
For this reason, the conventional wisdom used to be that neutrinos were
massless. After all, the electron neutrino is definitely far lighter
than any known particle except the photon - which is massless. The
larger upper bounds on the other neutrino's masses are mainly due to
the greater difficulty in doing the experiments.
Having neutrinos be massless would also nicely explain their most
stunning characteristic, namely that they're only found in a left-handed
form. What I mean by this is that they spin counterclockwise when
viewed head-on as they come towards you. It turns out that this
violation of left-right symmetry comes fairly easily to massless
particles, but only with more difficulty to massive ones. The reason is
simple: massless particles move at the speed of light, so you can't
outrun them. Thus everyone, regardless of their velocity, agrees on
what it means for such a particle to be spinning one way or another as
it comes towards them. This is not the case for a massive particle!
There was, however, a fly in the ointment. Since the sun is powered by
fusion, it should emit lots of neutrinos. In fact, the standard solar
model predicts that here on earth we are bombarded by 60 billion solar
neutrinos per square centimeter per second! So in the late 1960s, a
team led by Ray Davis set out to detect these neutrinos by putting a
tank of 100,000 gallons of perchloroethylene down into a gold mine in
Homestake, South Dakota. Lots of different nuclear reactions are going
on in the sun, producing neutrinos of different energies. The Homestake
experiment can only detect the most energetic ones - those produced when
boron-8 decays into beryllium-8. These neutrinos have enough energy to
turn chlorine-37 in the tank into argon-37. Being a noble gas, the
argon can be separated out and measured. This is not easy - one only
expects about 4 atoms of argon a day! So the experiment required
extreme care and went on for decades.
They only saw about a quarter as many neutrinos as expected.
Of course, with an experiment as delicate as this, there are always many
possibilities for error, including errors in the standard solar model.
So a Japanese group decided to use a tank of 2,000 tons of water in a
mine in Kamioka to look for solar neutrinos. This "Kamiokande"
experiment used photomultiplier tubes to detect the Cherenkov radiation
formed by electrons that happen to be hit by neutrinos. Again it was
sensitive only to high-energy neutrinos.
After 5 years, they started seeing signs of a correlation between
sunspot activity and their neutrino count. Interesting. But more
interesting still, they didn't see as many neutrinos as expected.
Only about half as many, in fact.
Starting in the 1990s, various people began to build detectors that
could detect lower-energy neutrinos - including those produced in the
dominant fusion reactions powering the sun. For this it's good to use
gallium-71, which turns to germanium-71 when bombarded by neutrinos.
The GALLEX detector in Italy uses 30 tons of gallium in the form of
gallium chloride dissolved in water. The SAGE detector, located in a
tunnel in the Caucasus mountains, uses 60 tons of molten metallic
gallium. This isn't quite as scary as it sounds, because gallium has a
very low melting point - it melts in your hand! But still, of course,
these experiments are very difficult.
Again, these experiments didn't see as many neutrinos as expected.
By this point, the theorists had worked themselves into a full head of
steam trying to account for the missing neutrinos. Currently the most
popular theory is that some of the electron neutrinos have turned into
muon and tau neutrinos by the time they reach earth. These other
neutrinos would be not be registered by our detectors.
Folks call this hypothetical process "neutrino oscillation". For it
to happen, the neutrinos need to have a nonzero mass. After all,
a massless particle moves at the speed of light, so it doesn't experience
any passage of time - thanks to relativistic time dilation. Only particles
with mass can become something else while they are whizzing along minding
their own business.
If in fact you posit a small mass for the neutrinos, oscillations happen
automatically as long as the "mass eigenstates" are different from the
"flavor eigenstates". By "flavor" we mean whether the neutrino is an
electron, muon or tau neutrino. For simplicity, imagine that the state
of a neutrino at rest is given by a vector whose 3 components are the
amplitudes for it to be these three different flavors. If all but one
of these components are zero we have a neutrino with a definite
flavor - a "flavor eigenstate". On the other hand, the energy of a
particle at rest is basically just its mass. Thus in the present
context the energy of the neutrino is described by a 3 x 3 self-adjoint
matrix H, the "Hamiltonian", whose eigenvectors are called "mass
eigenstates". These may or may not be the same as the flavor
eigenstates! Schroedinger's equation says that any state psi of the
neutrino evolves as follows:
d psi/dt = -iH psi
Thus if psi starts out being a mass eigenstate it stays a mass eigenstate.
But if it starts out being a flavor eigenstate, it won't stay a flavor
eigenstate - unless the mass and flavor eigenstates coincide! Instead, it
will oscillate.
I bet you were wondering when the math would start. Don't worry, there
won't be much this time.
Anyway, for other particles, like quarks, it's well-known that the mass
and flavor eigenstates *don't* coincide. So we shouldn't be surprised
at neutrino oscillations, at least if neutrinos actually have nonzero
mass.
Actually things are more complicated than I'm letting on. In addition
to oscillating in empty space, it's possible that neutrinos oscillate
*more* as they are passing through the sun itself, thanks to something
called the MSW effect - named after Mikheyev, Smirnov and Wolfenstein.
And there are two different ways for neutrinos to have mass, depending
on whether they are Dirac spinors or Majorana spinors (see "week93").
But I don't want to get caught up in theoretical nuances here! I want
to talk about experiments, and I haven't even gotten to the new stuff
yet - the stuff that's getting everybody *really* confused!
First of all, there's now some laboratory evidence for neutrino
oscillations coming from the Liquid Scintillator Neutrino Detector at
Los Alamos. What these folks do is let positively charged pions decay
into antimuons and muon neutrinos. Then they check to see if any muon
neutrinos become electron neutrinos. They claim that they do! They
also claim to see evidence of muon antineutrinos becoming electron
antineutrinos.
Secondly, and more intriguing still, there are a bunch of experiments
involving atmospheric neutrinos: Super-Kamiokande, Soudan 2, IMB, and
MACRO. You see, when cosmic rays smack into the upper atmosphere, they
produce all sorts of particles, including electron and muon neutrinos
and their corresponding antineutrinos. Cosmic ray experts think they
know how many of each sort of neutrino should be produced. But the
experimenters down on the ground are seeing different numbers!
Again, this could be due to neutrino oscillations. But what's REALLY
cool is that the numbers seem to depend on where the neutrinos are
coming from: from the sky right above the detector, from right below the
detector - in which case they must have come all the way through the
earth - or whatever. Neutrinos coming from different directions take
different amounts of time to get from the upper atmosphere to the
detector. Thus an obvious explanation for the experimental results is
that we're actually seeing the oscillation process AS IT TAKES PLACE.
If this is true, we can try to get detailed information about the
neutrino mass matrix from the numbers these experiments are measuring!
And this is exactly what people have been doing. But they're finding
something very strange. If all the experiments are right, and nobody is
making any mistakes, it seems that NO choice of neutrino mass matrix
really fits all the data! To fit all the data, folks need to do
something drastic - like posit a 4th kind of neutrino!
Now, it's no light matter to posit another neutrino. The known
neutrinos couple to the weak force in almost identical ways. This
allows one to create equal amounts of neutrino-antineutrino pairs of
all 3 flavors by letting Z bosons decay - the Z being the neutral
carrier of the weak force. When a Z boson seemingly decays into
"nothing", we can safely bet that it has decayed into a neutrino-
antineutrino pair. In 1989, an elegant and famous experiment at CERN
showed that Z bosons decay into "nothing" at exactly the rate one would
expect if there were 3 flavors of neutrino. Thus there can only be
extra flavors of neutrino if they are very massive, if they couple very
differently to the weak force, or if some other funny business is going
on.
Now, electron or muon neutrinos are unlikely to oscillate into a very
*massive* sort of neutrino - basically because of energy conservation.
So if we want an extra neutrino to explain the experimental results
we find ourselves stuck with, it'll have to be one that couples to the
weak force very differently from the ones we know. A simple, but
drastic, possibility is that it not interact via the weak force at all!
Folks call this a "sterile" neutrino.
Now, sterile neutrinos would blow a big hole in the Standard Model,
much more so than plain old *massive* neutrinos. So things are getting
very interesting.
Wilczek recently wrote a nice easy-to-read paper describing arguments
that *massive* neutrinos fit in quite nicely with the possibility that
the Standard Model is just part of a bigger, better theory - a "Grand
Unified Theory". I sketched the basic ideas of the SU(5) and SO(10)
grand unified theories in "week119". Recall that in the SU(5) theory,
the left-handed parts of all fermions of a given generation fit into two
irreducible representations of SU(5) - a 5-dimensional rep and a
10-dimensional rep. For example, for the first generation, the
5-dimensional rep consists of the left-handed down antiquark (which
comes in 3 colors), the left-handed electron, and the left-handed electron
neutrino. The 10-dimensional rep consists of the left-handed up quark,
down quark, and up antiquark (which come in 3 colors each), together
with the left-handed positron.
In the SO(10) theory, all these particles AND ONE MORE fit into a single
16-dimensional irreducible representation of SO(10). What could this
extra particle be?
Well, since this extra particle transforms trivially under SU(5), it
must not feel the electromagnetic, weak or strong force! Thus it's
tempting to take this missing particle to be the left-handed electron
antineutrino. Of course, we don't see such a particle - we only see
antineutrinos that spin clockwise. But if neutrinos are massive Dirac
spinors there must be such a particle, and having it not feel the
electromagnetic, weak or strong force would nicely explain *why* we
don't see it.
Grotz and Klapdor consider this possibility in their book on the weak
interaction (see below), but unfortunately, it seems this theory would
make the electron neutrino have a mass of about 5 MeV - much too big!
Sigh. So Wilczek, following the conventional wisdom, assumes the missing
particle is very massive - he calls it the "N". And he summarizes some
arguments that this massive particle could help give the neutrinos very
small masses, via something called the "seesaw mechanism". Unfortunately
I don't have the energy to describe this now, so for more you should look
at his paper (referred to below).
To wrap up, let me just say one final thing about the cosmic significance
of the neutrino. Massive neutrinos could account for some of the "missing
mass" that cosmologists are worrying about. So there's an indirect
connection between the neutrino mass and the cosmological constant!
The cosmological constant is essentially the energy density of the vacuum.
It was long assumed to be zero, but now there are some glimmerings of
evidence that it's not. In fact, some people are quite convinced that
it's not. The fate of the universe hangs in the balance....
Unfortunately I am too tired now to say much more about this.
So let me just give you a nice easy starting-point:
1) Special Report: Revolution in Cosmology, Scientific American,
January 1999. Includes the articles "Surveying space-time with
supernovae" by Craig J. Horgan, Robert P. Kirschner and Nicholoas B.
Suntzeff, "Cosmological antigravity" by Lawrence M. Krauss, and
"Inflation in a low-density universe" by Martin A. Bucher and David N.
Spergel.
How can you learn more about neutrinos? It can't hurt to start here:
2) Nikolas Solomey, The Elusive Neutrino, Scientific American Library,
1997.
If you want to dig in deeper, you need to learn about the weak force,
since we've only seen neutrinos via their weak interaction with other
particles. The following book is a great place to start:
3) K. Grotz and H. V. Klapdor, The Weak Interaction in Nuclear, Particle
and Astrophysics, Adam Hilger, Bristol, 1990.
Then you'll be ready for this book, which examines every aspect of
neutrinos in detail - complete with copies of historical papers:
4) Klaus Winter, ed., Neutrino Physics, Cambridge U. Press, Cambridge,
1991.
And then, if you want to study the possibility of *massive* neutrinos,
you should try this:
5) Felix Boehm and Petr Vogel, Physics of Massive Neutrinos, Cambridge
U. Press, Cambridge, 1987.
But neutrino physics is moving fast, and lots of the new stuff hasn't
made its way into books yet, so you should also look at other stuff.
For links to lots of great neutrino websites, including websites for
most of the experiments I mentioned, try:
6) The neutrino oscillation industry,
http://www.hep.anl.gov/NDK/hypertext/nu_industry.html
For some recent general overviews, try these:
7) Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/~www/neutrino/jhu/jhu.html
8) Boris Kayser, Neutrino mass: where do we stand, and where are we
going?, preprint available as hep-ph/9810513
For information on various experiments, try these:
9) GALLEX collaboration, GALLEX solar neutrino observations: complete
results for GALLEX II, Phys. Lett. B357 (1995), 237-247.
Final results of the CR-51 neutrino source experiments in GALLEX,
Phys. Lett. B420 (1998), 114-126.
GALLEX solar neutrino observations: results for GALLEX IV, Phys.
Lett. B447 (1999), 127-133.
11) SAGE collaboration, Results from SAGE, Phys. Lett B328 (1994),
234-248.
The Russian-American gallium experiment (SAGE) CR neutrino source
measurement, Phys. Rev. Lett. 77 (1996), 4708-4711.
12) LSND collaboration, Evidence for neutrino oscillations from
muon decay at rest, Phys. Rev. C54 (1996) 2685-2708, preprint available
as nucl-ex/9605001.
Evidence for anti-muon-neutrino -> anti-electron-neutrino oscillations
from the LSND experiment at LAMPF, Phys. Rev. Lett 77 (1996), 3082-3085,
preprint available as nucl-ex/9605003.
Evidence for nu_mu -> nu_e oscillations from LSND, Phys. Rev. Lett. 81
(1998), 1774-1777, preprint available as nucl-ex/9709006.
Results on nu_mu -> nu_e oscillations from pion decay in flight,
Phys. Rev. C58 (1998), 2489-2511.
13) Super-Kamiokande collaboration, Evidence for oscillation of
atmospheric neutrinos, Phys. Rev. Lett 81 (1998), 1562-1567,
preprint available as hep-ex/9807003.
14) MACRO collaboration, Measurement of the atmospheric neutrino
induced upgoing muon flux, Phys. Lett. B434 (1998), 451-457,
preprint available as hep-ex/9807005.
15) IMB collaboration, A search for muon-neutrino oscillations with
the IMB detector, Phys. Rev. Lett 69 (1992), 1010-1013.
For a fairly model-independent attempt to figure out something about
neutrino masses from the latest crop of experiments, see:
16) V. Barger, T. J. Weiler, and K. Whisnant, Inferred 4.4 eV upper
limits on the muon- and tau-neutrino masses, preprint available as
hep-ph/9808367.
For a nice summary of the data, and an argument that it's evidence
for the existence of a sterile neutrino, see:
17) David O. Caldwell, The status of neutrino mass, preprint available
as hep-ph/9804367.
For a very readable argument that massive neutrinos are evidence for a
supersymmetric SO(10) grand unified theory, see
18) Frank Wilczek, Beyond the Standard Model: this time for real,
preprint available as hep-ph/9809509.
Finally, with all this talk of cracks in the Standard Model, it's nice
to think again about the rise of the Standard Model. The following book
is packed with the reminiscences of many theorists and experimentalists
involved in developing this wonderful theory of particles and forces,
including Bjorken, 't Hooft, Veltman, Susskind, Polyakov, Richter,
Iliopoulos, Gell-Mann, Weinberg, Lederman, Goldhaber, Cronin, and
Kobayashi:
19) Lilian Hoddeson, Laurie Brown, Michael Riordan and Max Dresden,
eds., The Rise of the Standard Model: Particle Physics in the 1960s and
1970s.
It's a must for anyone with an interest in the history of physics!
-----------------------------------------------------------------------
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For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
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If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
Did you read down this far? Get a free prize by sending email
to g.w....@whitehouse.us-government.com
Gerard Westendorp
>
> Gerard Westendorp <wes...@xs4all.nl> writes:
>
> > [...] One thing that would seem to work, is build a 16 component
> > psi multivector, with a component in each element of the Clifford
> > algebra. If you then write
> >
> > i d*psi = m psi
> >
> > or in real numbers:
> >
> > e_xyzt d*psi = m psi
> >
> > as the gamma matrices (gamma_x, gamma_y, gamma__z, gamma__t) obey
> > the same algebra as the elements (e_x, ey e_z, e_t) of the
> > Clifford algebra.
>
> I suspect this equation is equivalent to what David Hestenes called the
> ``Dirac-Gursey Equation'' in his book ``Spacetime Algebra.''
I've downloaded the book by David Hestenes. It quite interesting and
very readable, although I have seen on the web that it also a bit
controversial. One thing that was very interesting is that there is
indeed a representation of the Dirac equation in terms of real-valued
multivectors, that has 8 instead of 16 components. Hestenes uses the
even multivector. This is the multivector with only even grades, ie.
( scalar, e_xy, e_xz, e_yz, e_xt, e_yt, e_zt, e_xyzt )
The Dirac equation then becomes the "Dirac Hestenes" equation:
d psi e_yx = m psi e_t
This all seems very nice, a theory based on geometric objects
rather than the abstract complex valued bi-spinors.
However, there is something that worries me. The 8 component
multivector should also transform like a multivector. But that
means that for example one of the components is a scalar. So
if you rotate the spinor, one of the components will remain
constant. That is something that a spinor in conventional
Dirac theory can never do. But there must be a 1-to-1
relation between conventional spinors and multivector spinors.
This relation cannot be a linear map. I suspect that the 8
components in the multivector are bilinear covariants of the
conventional spinor. But that is something Hestenes does not
explain very well, or perhaps even forgets about. I'm trying
to figure this out, but any comments or help would be
appreciated.
Gerard
Gerard Westendorp
[..]
> I appreciate both clarifications and can understand jb's less abstract and
> thus (for me) more comprendable expression better that the other
> approximation. I thought the beginning point, though, was beginning just
> with particles and antiparticles (or as stated, electrons and positrons)
> what happens when those found themselves in a potential. So I was
> thinking that the gradient or potential would also have to be related back
> to SOME clump of particles and antiparticles. Thus, that led me to wonder
> specifically HOW one arranges and stacks up particles and antiparticles so
> as to fabricate the potential.
One thing to remember is that there are various interpretations,
or pictures, to talk about electrodynamics. They are of course
all ultimately related. Quantum electrodynamics (qed) describes
the behavior of charged particles extremely accurately. But
qed isn't usually a very convenient theory. At the moment,
it can only be solved "perturbatively", meaning that you
use successive approximations, which hopefully converge to the
quantity you want to calculate. In various circumstances, there
are various good simpler approximations. The problem is that these
approximations seem to give rise to different pictures. This
can lead to some confusion.
In all pictures, one can distinguish between the "matter"
component, which carries charge, and the "force" component.
Matter can be pictured as:
1. No matter present
2. particles (e.g. electrons, positrons)
3. Bodies with charge distribution
4. Waves (Dirac equation, Schrodinger equation)
5. Collection of oscillators (quantum field)
em. force fields can be pictured as:
1. No force fields present
2. particles (photons)
3. Static fields (time independent solutions to Maxwell)
4. waves (Maxwell equations)
5. Collection of oscillators (quantum field)
Combining these pictures, there are at least 5*5 = 25
different ways to look at electrodynamics. (1&1 is
of course a bit trivial)
The most complete is 5&5, quantum electrodynamics.
In the thread "Maxwell and Dirac" we compared 4&1 with
1&4, and contemplated on going from 4&1 to 4&3.
(Putting a potential in the Dirac equation)
Your question seems to relate to how to go from 1&3 to
2&3. In other words, how do charged particles give rise to
fields. Basically, this is just the coulomb law. You add
the coulomb contributions from each particle. Magnetic
fields are a bit more complicated, but more or less
the same story.
Gerard
Gerard Westendorp
[..]
> All these facts I just stated about the classification of spinors
> are special to 4d spacetime. In other dimensions it follows this
> pattern:
>
> n Dirac Majorana Weyl Majorana-Weyl
>
> 1 C R
> 2 C^2 R^2 C R
> 3 C^2 R^2
> 4 C^4 R^4 C^2
> 5 C^4
> 6 C^8 C^4
> 7 C^8
> 8 C^16 R^16 C^8
>
Just wondering, what do spinors in for example 1 dimension
mean? Is there a 1 dimensional Dirac equation?
Gerard
Jonathan Scott
> However, there is something that worries me. The 8 component
> multivector should also transform like a multivector. But that
> means that for example one of the components is a scalar. So
> if you rotate the spinor, one of the components will remain
> constant. That is something that a spinor in conventional
> Dirac theory can never do. But there must be a 1-to-1
> relation between conventional spinors and multivector spinors.
> This relation cannot be a linear map. I suspect that the 8
> components in the multivector are bilinear covariants of the
> conventional spinor. But that is something Hestenes does not
> explain very well, or perhaps even forgets about. I'm trying
> to figure this out, but any comments or help would be
> appreciated.
I've not followed up Hestenes' recent work on this, but I have
my own document on this subject at
http://pws.prserv.net/jonathan_scott/physics/diraceqn.pdf
which shows how to describe the Dirac equation in what I
called the complex four-vector algebra, which is
equivalent to the Pauli algebra, and in that formulation
the relationship between the bispinor notation, the two
by two Pauli algebra notation and the complex four-vector
equivalent is explicitly discussed in section 7, "Comparison
with bispinors", and section 8, "The probability current".
(I need to update that document to correct a typo or two
and to acknowledge that William Baylis has published a
paper containing the same result, but I never seem to find
the time now I have two kids to keep my occupied when I'm
not working on my day job).
Jonathan Scott
Cl.Massé
3B10C02C...@xs4all.nl...
> One thing that was very interesting is that there is
> indeed a representation of the Dirac equation in terms of real-valued
> multivectors, that has 8 instead of 16 components. Hestenes uses the
> even multivector. This is the multivector with only even grades, ie.
>
> ( scalar, e_xy, e_xz, e_yz, e_xt, e_yt, e_zt, e_xyzt )
>
> The Dirac equation then becomes the "Dirac Hestenes" equation:
>
> d psi e_yx = m psi e_t
>
> This all seems very nice, a theory based on geometric objects
> rather than the abstract complex valued bi-spinors.
>
> However, there is something that worries me. The 8 component
> multivector should also transform like a multivector. But that
> means that for example one of the components is a scalar. So
> if you rotate the spinor, one of the components will remain
> constant. That is something that a spinor in conventional
> Dirac theory can never do.
Isn't there also 8 real components in the Dirac spinor, i.e. 4 complex?
It seems to me that the Dirac Hestenes equation rather describes the
electromagnetic field strength plus two additional components, since
this latter obeys a Dirac-like equation. The linear space spanned by
the e_ab is a vector representation space of the Lorenz group, namely
[1, 0] + [0, 1], while the representation for the Dirac spinors is
[1/2, 0] + [0, 1/2].
--
~~~~ %20cl...@free.fr%20 LPF
Liberty, Equality, Profitability.
John Baez
Gerard Westendorp <wes...@xs4all.nl> wrote:
>John Baez wrote:
>> All these facts I just stated about the classification of spinors
>> are special to 4d spacetime. In other dimensions it follows this
>> pattern:
>>
>> n Dirac Majorana Weyl Majorana-Weyl
>>
>> 1 C R
>> 2 C^2 R^2 C R
>> 3 C^2 R^2
>> 4 C^4 R^4 C^2
>> 5 C^4
>> 6 C^8 C^4
>> 7 C^8
>> 8 C^16 R^16 C^8
>Just wondering, what do spinors in for example 1 dimension
>mean?
See below for the whole story.
>Is there a 1 dimensional Dirac equation?
Sure - there's a Dirac equation for any dimension and any
signature of the metric: it looks like
gamma^i d_i psi = 0
where psi is a spinor field, d_i is the partial derivative
in the ith direction, and gamma^i are the "gamma matrices", or
more elegantly, generators of our Clifford algebra... below
I call them e_i, just to confuse the unwary.
However: don't expect the 1-dimensional one to be very exciting!
Unless I'm getting senile, in 1-dimensional spacetime the
Dirac equation just says our spinor field is constant as a
function of time. It gets more interesting in higher dimensions.
For example, in 2 dimensions it's a close relative of the
Cauchy-Riemann equations.
..................................................................
October 27, 1996
This Week's Finds in Mathematical Physics - Week 93
John Baez
[stuff deleted]
To understand fermions in different dimensions we need to understand
Clifford algebras. As far as I know, when Clifford originally invented
these algebras in the late 1800s, he was trying to generalize Hamilton's
quaternion algebra by considering algebras that had lots of different
anticommuting square roots of -1. In other words, he considered
an associative algebra generated by a bunch of guys e_1,...,e_n,
satisfying
e_i^2 = -1
for all i, and
e_i e_j = - e_j e_i
whenever i is not equal to j. I discussed these algebras in "week82"
and I said what they all were --- they all have nice descriptions in terms
of the reals, the complexes, and the quaternions.
These original Clifford algebras are great for studying rotations in
n-dimensional Euclidean space --- please take my word for this for now.
However, here we want to study rotations and Lorentz transformations
in n-dimensional Minkowski spacetime, so we need to work with a slightly
Different kind of Clifford algebra, which was probably invented by Dirac.
In n-dimensional Euclidean space the metric (used for measuring distances)
is
dx_1^2 + dx_2^2 + ... + dx_n^2
while in n-dimensional Minkowski spacetime it is
dx_1^2 + dx_2^2 + ... - dx_n^2
or if you prefer (it's just a matter of convention), you can
take it to be
- dx_1^2 - dx_2^2 - ... + dx_n^2
So it turns out that we need to switch some signs in the definition
of the Clifford algebra when working in Minkowski spacetime.
In general, we can define the Clifford algebra C_{p,q} to be the algebra
generated by a bunch of elements e_i, with p of them being square roots
of -1 and q of them being square roots of 1. As before, we require that
they anticommute:
e_i e_j = - e_j e_i
when i and j are different. Physicists usually call these guys "gamma
matrices". For n-dimensional Minkowski space we can work either
with C_{n-1,1} or C_{1,n-1}, depending on our preference. As Cecile
DeWitt has pointed out, it *does* make a difference which one we use.
With some work, one can check that these algebras go like this:
C_{0,1} R + R C_{1,0} C
C_{1,1} R(2) C_{1,1} R(2)
C_{2,1} C(2) C_{1,2} R(2) + R(2)
C_{3,1} H(2) C_{1,3} R(4)
C_{4,1} H(2) + H(2) C_{1,4} C(4)
C_{5,1} H(4) C_{1,5} H(4)
C_{6,1} C(8) C_{1,6} H(4) + H(4)
C_{7,1} R(16) C_{1,7} H(8)
I've only listed these up to 8-dimensional Minkowski spacetime, and
the cool thing is that after that they sort of repeat --- more precisely,
C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
and C_{1,n+8} is just 16 x 16 matrices with entries in C_{1,n}!
This "period-8" phenomenon, sometimes called Bott periodicity, has
implications for all sorts of branches of math and physics. This is
why fermions in 2 dimensions are a bit like fermions in 10 dimensions
and 18 dimensions and 26 dimensions....
In physics, we describe fermions using "spinors", but there are
different kinds of spinors: Dirac spinors, Weyl spinors, Majorana
spinors, and even Majorana-Weyl spinors. This is a bit technical but
I want to dig into it here, since it explains what's special about
8k + 2 dimensions and especially 10 dimensions.
Before I get technical, though, let me just summarize the point for
those of you who don't want all the gory details. "Dirac spinors"
are what you use to describe spin-1/2 particles that come in both
left-handed and right-handed forms and aren't their own antiparticle
--- like the electron. Weyl spinors have half as many components,
and describe spin-1/2 particles with an intrinsic handedness that
aren't their own antiparticle --- like the neutrino. "Weyl spinors"
are only possible in even dimensions!
Both these sorts of spinors are "complex" --- they have complex-valued
components. But there are also real spinors. These are used for describing
particles that are their own antiparticle, because the operation of
turning a particle into an antiparticle is described mathematically
by complex conjugation. "Majorana spinors" describe spin-1/2 particles
that come in both left-handed and right-handed forms and are their
own antiparticle. Finally, "Majorana-Weyl spinors" are used to describe
spin-1/2 particles with an intrinsic handedness that are their own
antiparticle.
As far as we can tell, none of the particles we've seen are Majorana
or Majorana-Weyl spinors, although if the neutrino has a mass it
might be a Majorana spinor. Majorana and Majorana-Weyl spinors
only exist in certain dimensions. In particular, Majorana-Weyl spinors
are very finicky: they only work in dimensions of the form 8k + 2.
This is part of what makes supersymmetric string theory work in 10
dimensions!
Now let me describe the technical details. I'm doing this mainly
for my own benefit; if I write this up, I'll be able to refer to
it whenever I forget it.
So: part of the point of these Clifford algebras is that they give
representations of the double cover of the Lorentz group in different
dimensions. In "week61" I explained this double cover business,
and how the group SO(n) of rotations of n-dimensional Euclidean space
has a double cover called Spin(n). Similarly, the Lorentz group
of n-dimensional Minkowski space, written SO(n-1,1), has a double cover
we could call Spin(n-1,1). The spinors we'll discuss are all
representations of this group.
The way Clifford algebras help is that there is a nice way to
embed Spin(n-1,1) in either C_{n-1,1} or C_{1,n-1}, so any
representation of these Clifford algebras gives a representation
of Spin(n-1,1). We have a choice of dealing with real representations or
complex representations. Any complex representation of one of
these Clifford algebras is also a representation of the *complexified*
Clifford algebra. What I mean is this: above I implicitly wanted
C_{p,q} to consist of all *real* linear combinations of products of
the e_i, but we could have worked with *complex* linear combinations
instead. Then we would have "complexified" C_{p,q}. Since the
complex numbers include a square root of minus 1, the complexification
of C_{p,q} only depends on the dimension p + q, not on how many minus
signs we have.
Now, it is easy and fun and important to check that if you complexify R
you get C, and if you complexify C you get C + C, and if you complexify
H you get C(2). Thus from the above table we get this table:
dimension n complexified Clifford algebra
1 C + C
2 C(2)
3 C(2) + C(2)
4 C(4)
5 C(4) + C(4)
6 C(8)
7 C(8) + C(8)
8 C(16)
Notice this table is a lot simpler --- complex Clifford algebras
are "period-2" instead of period-8.
Now the smallest complex representation of the complexified Clifford
algebra in dimension n is what we call a "Dirac spinor". We can figure
out what this is using the above table, since the smallest complex
representation of C(n) or C(n) + C(n) is on the n-dimensional complex
vector space C^n, given by matrix multiplication. Of course, for
C(n) + C(n) there are *two* representations depending on which copy
of C(n) we use, but these give equivalent representations of Spin(n-1,1),
which is what we're really interested in, so we still speak of "the"
Dirac spinors.
So we get:
dimension n Dirac spinors
1 C
2 C^2
3 C^2
4 C^4
5 C^4
6 C^8
7 C^8
8 C^16
The dimension of the Dirac spinors doubles as we go to each new
even dimension.
We can also look for the smallest real representation of C_{n-1,1}
or C_{1,n-1}. This is easy to work out from our tables using
the fact that the algebra R has its smallest real representation
on R, while for C it's on R^2 and for H it's on R^4.
Sometimes this smallest real representation is secretly just the
Dirac spinors *viewed as a real representation* --- we can view C^n
as the real vector space R^{2n}. But sometimes the Dirac spinors
are the *complexification* of the smallest real representation ---
for example, C^{2n} is the complexification of R^n. In this
case folks call the smallest real representation "Majorana spinors".
When we are looking for the smallest real representations, we get
different answers for C_{n-1,1} and C_{1,n-1}. Here is what we get:
n C_{n-1,1} smallest C_{1,n-1} smallest
real rep real rep
1 R + R R Majorana C R^2
2 R(2) R^2 Majorana R(2) R^2 Majorana
3 C(2) R^4 R(2) + R(2) R^2 Majorana
4 H(2) R^8 R(4) R^4 Majorana
5 H(2) + H(2) R^8 C(4) R^8
6 H(4) R^16 H(4) R^16
7 C(8) R^16 H(4) + H(4) R^16
8 R(16) R^16 Majorana H(8) R^32
I've noted when the representations are Majorana spinors. Everything
repeats with period 8 after this, in an obvious way.
Finally, sometimes there are "Weyl spinors" or "Majorana-Weyl"
spinors. The point is that sometimes the Dirac spinors, or
Majorana spinors, are a *reducible* representation of Spin(1,n-1).
For Dirac spinors this happens in every even dimension, because the
Clifford algebra element Gamma = e_1 ... e_n commutes with everything
in Spin(1,n-1) and Gamma^2 is 1 or -1, so we can break the space of
Dirac spinors into the two eigenspaces of Gamma, which will be smaller
reps of Spin(1,n-1) --- the "Weyl spinors". Physicists usually call this
Gamma thing "gamma_5", and it's an operator that represents parity
transformations. We get "Majorana-Weyl" spinors only when we have
Majorana spinors, n is even, and Gamma^2 = 1, since we are then working
with real numbers and -1 doesn't have a square root. You can work out
Gamma^2 for either and C_{n-1,1} or C_{1,n-1}, and see that we'll
only get Majorana-Weyl spinors when n = 8k + 2.
Whew! Let me summarize some of our results:
n Dirac Majorana Weyl Majorana-Weyl
1 C R
2 C^2 R^2 C R
3 C^2 R^2
4 C^4 R^4 C^2
5 C^4
6 C^8 C^4
7 C^8
8 C^16 R^16 C^8
When there are blanks here, the relevant sort of spinor doesn't
exist. Here I'm not distinguishing Majorana spinors that come from
C_{n-1,1} and those that come from C_{1,n-1}; you can do that with
the previous table. Again, things continue for larger n in an obvious
way.
Gordon D. Pusch
Hestenes claims that a Dirac-Hestenes spinor (i.e., an even multivector)
transforms as \psi' = L\psi, where L is a ``rotor'' (i.e., a multivector
such that LL~ == 1, where L~ denotes the ``reverse'' of L obtained
by reversing the order of each factor in its multivector expansion).
By contrast, an ``ordinary'' multivector transforms as A' = LAL~,
so it is apparently a different type of geometric object.
How (or even whether!) this is consistent with the ``usual''
Lorentz-group spin decomposition is a question I can't answer;
however, IIRC Hestenes and one of his students have reformulated
the ``standard'' Lorentz-group theory in terms of multivectors,
so presumably if there were an inconsistency between them,
they would have noticed...
Gerard Westendorp
[..]
> Sure - there's a Dirac equation for any dimension and any
> signature of the metric: it looks like
>
> gamma^i d_i psi = 0
In most textbooks, the Dirac equation is:
i gamma^j d_j psi = m psi
For the massless case, you can get rid of the complex factor i,
but it seems that because of the mass term in combination with
i, you force the psi to be complex.
[..]
> However: don't expect the 1-dimensional one to be very exciting!
> Unless I'm getting senile, in 1-dimensional spacetime the
> Dirac equation just says our spinor field is constant as a
> function of time. It gets more interesting in higher dimensions.
> For example, in 2 dimensions it's a close relative of the
> Cauchy-Riemann equations.
The (0+1)D case would be:
d_t psi = -i m psi
A harmonic oscillator.
The (1+1)D case is:
d_t psi_1 = -d_x psi_2 - im psi_1
d_t psi_2 = -d_x psi_1 - im psi_2
This is a 1D Klein Gordon equation, written as a coupled set
of 1rst order equations. The massless case reduces the
number of real-valued parameters by getting rid of i. Then the
equation looks like Cauchy-Riemann, but with a minus sign changed.
This minus sign is the difference between a (1+1) D wave equation
and a (2+0)D potential equation.
A representation of the (2+1)D case is:
d_t psi_1 = -d_x psi_2 - id_y psi_2 - im psi_1
d_t psi_2 = -d_x psi_1 + id_y psi_1 - im psi_2
I don't have an interpretation for this. There must be
Majoranas and Weyls lurking in it somewhere, because they
are supposed to exist in 2D.
[..]
> In physics, we describe fermions using "spinors", but there are
> different kinds of spinors: Dirac spinors, Weyl spinors, Majorana
> spinors, and even Majorana-Weyl spinors. This is a bit technical but
> I want to dig into it here, since it explains what's special about
> 8k + 2 dimensions and especially 10 dimensions.
In Lounesto's book, he describes a 4th type called "flag-dipole"
spinors. I haven't figured out yet in which dimensions they exist.
Gerard
Gerard Westendorp
[..]
>
> Hestenes claims that a Dirac-Hestenes spinor (i.e., an even multivector)
> transforms as \psi' = L\psi, where L is a ``rotor'' (i.e., a multivector
> such that LL~ == 1, where L~ denotes the ``reverse'' of L obtained
> by reversing the order of each factor in its multivector expansion).
> By contrast, an ``ordinary'' multivector transforms as A' = LAL~,
> so it is apparently a different type of geometric object.
Yes, by now I have caught up with this. So the "scalar" component
in the multivector is not really a scalar in the complete sense
of the word. ie it can be changed by a rotation.
Gerard
Gerard Westendorp
[..]
> I've not followed up Hestenes' recent work on this, but I have
> my own document on this subject at
> http://pws.prserv.net/jonathan_scott/physics/diraceqn.pdf
> which shows how to describe the Dirac equation in what I
> called the complex four-vector algebra, which is
> equivalent to the Pauli algebra, and in that formulation
> the relationship between the bispinor notation, the two
> by two Pauli algebra notation and the complex four-vector
> equivalent is explicitly discussed in section 7, "Comparison
> with bispinors", and section 8, "The probability current".
It seems there is a relationship between the multivector
representation and the complex 4vectors that you use in
your article.
In C_3, there are 8 geometrical objects, the right number
for the Dirac equation. To make a relationship with the
even multivector in C_1,3, that also has 8 components,
you can just add a "time" component to make odd multi's
even:
e00 -> e00 (I write scalar as e00)
e1 -> e01
e2 -> e02
e3 -> e03
e123 -> e0123
e12 -> e12
e13 -> e13
e23 -> e23
The first 4 components can be regarded as a 4vector, and
the components 5to8 as a pseudo 4vector. It seems OK to
put these 2 together as a single complex 4 vector. The
pseudoscalar e0123 is often called i, and one has
e0123( e00, e01, e02, e03) = ( e0123, e23, -e13, e12)
The relation you give between complex 4vectors and bi-spinors
is very similar to the stuff Hestenes does. But I think you
explain it better, especially the point that the complex
4vector does not always transform as a conventional 4vector,
and that there is a hidden choice of axis in the representation.
The use of multivectors suggests that spinors are "ordinary"
geometrical objects, but this is somewhat misleading, because
the multivector
> but I never seem to find
> the time now I have two kids to keep my occupied when I'm
> not working on my day job).
same here.
Gerard
Jim Jastrzebski
> Now, electron or muon neutrinos are unlikely to oscillate into a very
> *massive* sort of neutrino - basically because of energy conservation.
Some time ago, when I asked how to show that energy is conserved
when something drops and gains kinetic energy in the process, you
said that energy is not conserved adding that the best way to
understand nature is not to ask questions. So I absorbed the news
without questioning it.
Now you are saying that "energy conservation" might be an obstacle
for neutrino to oscillate into a very massive one. If energy is really
not conserved as you said before how it can be an obstacle for
creation of a very *massive* sort of neutrino?
Sorry that I'm asking a question but I see a contradiction here which
can't be solved without asking.
John Baez
Gerard Westendorp <wes...@xs4all.nl> wrote:
>John Baez wrote:
>> Sure - there's a Dirac equation for any dimension and any
>> signature of the metric: it looks like
>>
>> gamma^i d_i psi = 0
>In most textbooks, the Dirac equation is:
>
> i gamma^j d_j psi = m psi
For some reason I was considering the massless case - don't ask
me why!
>For the massless case, you can get rid of the complex factor i,
>but it seems that because of the mass term in combination with
>i, you force the psi to be complex.
You are right that to include a mass term we need to use the
correct sort of spinors. There are a couple of ways to do
it:
1) The most common way is to work with Dirac spinors, which are
complex (not their own antiparticles). This lets you put the i
in the equation you wrote down.
2) A subtler way is to use Majorana spinors, which are real
(their own antiparticles). Right now I'm confused about how
you put in the mass term in this case. This option is one of
many that people have considered when pondering the confusing
data concerning massive neutrinos. For more try this:
Theory of Neutrino Mass
Paul Langacker
http://dept.physics.upenn.edu/neutrino/jhu/node2.html
It's worth adding the following note:
3) In the Standard Model, spin-1/2 particles get their mass not
from a mass term like you wrote, but from interaction with the Higgs.
However, the way this works is rather similar. Roughly speaking,
the Higgs field times a coupling constant takes over the role of
"m" in the above equation.
(It's not really that simple, because the Higgs is not a scalar
under weak isospin, and the Higgs coupling to spin-1/2 particles
is really described not by a number but a matrix. But it's still
just a fancier version of the idea I described.)
>> Unless I'm getting senile, in 1-dimensional spacetime the
>> Dirac equation just says our spinor field is constant as a
>> function of time. It gets more interesting in higher dimensions.
>> For example, in 2 dimensions it's a close relative of the
>> Cauchy-Riemann equations.
Again here I was talking about the massless case for some reason.
John Baez
Jim Jastrzebski <Jim...@aol.com> wrote:
>John Baez wrote:
>> Now, electron or muon neutrinos are unlikely to oscillate into a very
>> *massive* sort of neutrino - basically because of energy conservation.
>Some time ago, when I asked how to show that energy is conserved
>when something drops and gains kinetic energy in the process, you
>said that energy is not conserved, adding that the best way to
>understand nature is not to ask questions.
I never said "the best way to understand nature is not to ask
questions"! I presume you're referring to my remarks in
http://www.lns.cornell.edu/spr/2001-04/msg0032116.html
Here's what I actually said:
"It is always surprising when it happens, but sometimes to learn
more about the world we must stop asking certain questions...
... namely, those based on false assumptions."
>Now you are saying that "energy conservation" might be an obstacle
>for neutrino to oscillate into a very massive one. If energy is really
>not conserved as you said before how it can be an obstacle for
>creation of a very *massive* sort of neutrino?
In our previous discussion, you seemed to assume that energy conservation
holds in general relativity. It does not - at least, not in the sense you
expected. I was telling you to drop this assumption. I was telling you
that questions based on this false assumption would lead you astray. I
was NOT telling you to throw energy conservation out the window in all
contexts! In fact, I said quite explicitly that energy is *approximately*
conserved to an extremely high degree of accuracy in situations where the
gravitational field is weak - the Newtonian limit.
In this neutrino business, the gravitational fields are EXTREMELY weak.
Therefore, we can use energy conservation - it's approximate, but it's
an EXTREMELY good approximation. That's good enough for the argument to
go through. Remember, the business of physics consists of using the
relevant approximations to deal with the situation at hand. When you're
operating a forklift, you don't worry about the quantum uncertainty in
the position of the pile of dirt you're moving.
Bagnoud Maxime
> 2) A subtler way is to use Majorana spinors, which are real
> (their own antiparticles). Right now I'm confused about how
> you put in the mass term in this case.
You have to be in a Majorana representation of the Gamma matrices,
i.e. purely imaginary (or real, depending on the choice of signature),
to see a real equation with real spinors solution. In other
representations, less convenient for the Majorana case, the Majorana
spinos aren't strictly speaking real, but rather satisfy the slightly
obscure Majorana condition.
I'm not sure if that was your confusion, but this might enlighten the
question in any case.
Maxime.
Gerard Westendorp
>
[..]
> Gerard Westendorp <wes...@xs4all.nl> wrote:
[..]
>
> >In most textbooks, the Dirac equation is:
> >
> > i gamma^j d_j psi = m psi
>
> For some reason I was considering the massless case - don't ask
> me why!
>
> >For the massless case, you can get rid of the complex factor i,
> >but it seems that because of the mass term in combination with
> >i, you force the psi to be complex.
>
> You are right that to include a mass term we need to use the
> correct sort of spinors. There are a couple of ways to do
> it:
>
> 1) The most common way is to work with Dirac spinors, which are
> complex (not their own antiparticles). This lets you put the i
> in the equation you wrote down.
>
> 2) A subtler way is to use Majorana spinors, which are real
> (their own antiparticles). Right now I'm confused about how
> you put in the mass term in this case.
Actually, I think I' ve figured it out now. I wrote that
i gamma^j d_j psi = m psi
implies that psi must be complex, but I overlooked that
all the gamma^j matrices could be imaginary. In that case,
psi can be real.
It is probably neater to absorb the i into the gamma matrices,
and then redefine the algebra accordingly. You then get that
all gamma's anticommute, and the squares of the space gamma's
are +1, and [gamma_0]^2 = -1. The Dirac equation then looks like
gamma_j d_j psi = m psi
The Dirac equation for Majorana spinors looks the same as an
ordinary Dirac equation, but the gamma matrices are different,
eg.
(3+1)D:
gamma_t gamma_x
( 0 0 0 1 ) (-1 0 0 0 )
( 0 0 -1 0 ) ( 0 1 0 0 )
( 0 1 0 0 ) ( 0 0 -1 0 )
(-1 0 0 0 ) ( 0 0 0 1 )
gamma_y gamma_z
( 0 0 0 -1 ) ( 0 1 0 0 )
( 0 0 1 0 ) ( 1 0 0 0 )
( 0 1 0 0 ) ( 0 0 0 1 )
(-1 0 0 0 ) ( 0 0 1 0 )
(2+1)D:
gamma_t gamma_x gamma_y
( 0 -1 ) ( 1 0 ) ( 0 1 )
( 1 0 ) ( 0 -1 ) ( 1 0 )
Gerard
Gerard Westendorp
Gerard Westendorp wrote:
[..]
> The Dirac equation for Majorana spinors looks the same as an
> ordinary Dirac equation, but the gamma matrices are different,
> eg.
>
> (3+1)D:
> gamma_t gamma_x
> ( 0 0 0 1 ) (-1 0 0 0 )
> ( 0 0 -1 0 ) ( 0 1 0 0 )
> ( 0 1 0 0 ) ( 0 0 -1 0 )
> (-1 0 0 0 ) ( 0 0 0 1 )
>
> gamma_y gamma_z
> ( 0 0 0 -1 ) ( 0 1 0 0 )
> ( 0 0 1 0 ) ( 1 0 0 0 )
> ( 0 1 0 0 ) ( 0 0 0 1 )
> (-1 0 0 0 ) ( 0 0 1 0 )
>
> (2+1)D:
> gamma_t gamma_x gamma_y
> ( 0 -1 ) ( 1 0 ) ( 0 1 )
> ( 1 0 ) ( 0 -1 ) ( 1 0 )
>
I should add that these equations have weird
solutions. If you substitute plane waves, you find that
the mass of the particles is propagation direction dependent.
It would be really weird if nature allows Majorana spinors.
Gerard
[Moderator's note: the usual version of the Dirac equation
with massive Majorana spinors is rotation-covariant, so the
mass of the particles is *not* direction-dependent. I think
you've made a mistake somewhere. - jb]
Gerard Westendorp
[..]
> > The Dirac equation for Majorana spinors looks the same as an
> > ordinary Dirac equation, but the gamma matrices are different,
> > eg.
> >
> > (3+1)D:
> > gamma_t gamma_x
> > ( 0 0 0 1 ) (-1 0 0 0 )
> > ( 0 0 -1 0 ) ( 0 1 0 0 )
> > ( 0 1 0 0 ) ( 0 0 -1 0 )
> > (-1 0 0 0 ) ( 0 0 0 1 )
> >
> > gamma_y gamma_z
> > ( 0 0 0 -1 ) ( 0 1 0 0 )
> > ( 0 0 1 0 ) ( 1 0 0 0 )
> > ( 0 1 0 0 ) ( 0 0 0 1 )
> > (-1 0 0 0 ) ( 0 0 1 0 )
> >
> > (2+1)D:
> > gamma_t gamma_x gamma_y
> > ( 0 -1 ) ( 1 0 ) ( 0 1 )
> > ( 1 0 ) ( 0 -1 ) ( 1 0 )
> >
>
> I should add that these equations have weird
> solutions. If you substitute plane waves, you find that
> the mass of the particles is propagation direction dependent.
[..]
> [Moderator's note: the usual version of the Dirac equation
> with massive Majorana spinors is rotation-covariant, so the
> mass of the particles is *not* direction-dependent. I think
> you've made a mistake somewhere. - jb]
>
Yes, I did make a mistake.
I substituted a real spinor and then looked at
different plane wave solutions with wave vector (w,kx,ky,kz).
But not all combinations (a,b,c,d) / (w,kx,ky,kz) are allowed
by the Dirac/Majorana equation.
This is true already for ordinary Dirac bi-spinors. Usually
we choose the first spinor component of the bi-spinor,
then a wave vector. The other second component of the bi-spinor
is then no longer free to choose.
But with Majorana spinors, there are less degrees of freedom
to start with, so the restrictions the Dirac equation poses
are more severe.
You cannot for example use (1,0,0,0) and say it is stationary.
An general solution for a stationary particle is:
( a cos(wt+alfa) )
( b cos(wt+beta) )
( -b sin(wt+beta) )
( a sin(wt+alfa) )
So some of the informaiton is stored in the fases of the
time evolution. This related to of the quantity
( a exp(i alfa) )
( b exp(i beta) )
which is a Pauli Spinor.
By applying a Lorentz transform, we can construct a solution
from this for a wave propagating in the x-direction:
( a cos(wt+kx+alfa+phi) )
( b cos(wt+kx+beta-phi) )
( -b sin(wt+kx+beta+phi) )
( a sin(wt+kx+alfa-phi) )
phi depends on the velocity of the Lorentz transform.
Now, the relation between the Majorana spinor and a
Pauli spinor is obsured by phase shifts (phi).
Somehow, Majorana spinors must be important, because
of their economical use of variables. On the other hand,
it is hard to follow what they mean in geometrical terms.
There is also not much literature on them, at least not the
sort of literature that is helpful in understanding the
meaning of the equations.
Interesting is the fact that the 4 4X4 real matrices
generate a 16 element group of matrices, that is a
basis for all 4x4 matrices. And they form a representation
of the Clifford algebra Cl(3,1). I get the feeling that
there must be some enlightening way to talk about this in
terms of Clifford algebra's.
Gerard