Dirac spinors: dumb question?

#1 February 2nd 04 posted to sci.physics

(Gregory L. Hansen) wrote in message ...
In the usual textbook representations, the Dirac spinor has the particle
spin and magnitude in the top two components and antiparticle spin and
magnitude in the bottom two components. But I've never really understood
(nor needed to understand in class) what that means. A stationary
particle is all upper. Boost it and you put some magnitude in the lower
components. Does that mean a likelihood of detecting an antiparticle?
Suppose we shot electrons into a magnetic field that's as weak as we like
but will sufficiently separate charge, do we expect most of them to turn
left but some to turn right? I'm sure that can't be quite right.

COMMENT:

I'm a out of my league on the math, but can give you some language and
handwaving. You shouldn't be wary of the lower spinor components
showing up when you give some KE to an electron (or drop it into some
E or B potential, also). That's just the Dirac equation's "way" of
accounting for the change in the "relativistic mass" of the electron.
Or the increase in total energy of the thing, as we say today. The
extra KE has no charge (obviously), so what kind of "stuff" is it made
of, particle-wise? According to the Dirac theory the extra stuff over
the rest mass (rest energy) of the electron is composed of half
virtual electron and half virtual positron. These travel along with
the original electron, and appear and disappear in a ghostly was as
you view the electron from frames other than the rest frame. So if you
boost an electron to total energy 2mc^2 it's composed of 1.5 electrons
and 0.5 positron. But the positron component is virtual since it's off
mass shell and hasn't got enough energy to be real.

So long as you haven't boosted the electron to a total energy higher
then 3mc^2, however, the virtual positron traveling along with it
never has enough energy to be real, so there's no danger of detecting
it directly as a free particle. It does, as a virtual field, has some
mild effects on the interaction of the whole ensemble on stationary
potentials, however. Over energies of 3mc^2 or kinetic energies of
2mc^2, of course, you can have pair production of the virtual
components if you can figure out a way to offload the momentum through
another interaction.

If you think the Dirac solutions for freely propagating particles are
strange, consider what happens when you drop an electron into a
positive potential well, as with a nucleus, and the well is really
deep. Say a 1s orbital for a nucleus of a couple of hundred Z (if
there was such a thing). At classical Z=137, the negative potential
energy of binding is equal to mc^2, so the total energy of the
electron is zero! It doesn't exist! Make the positive binding
potential even stronger and now the energy of the electron starts to
go negative, which means that you start to get more positronic
components than not. At some point, you have enough energy that this
positron will have enough energy to be real and pop out of the
potential, leaving only the electron's negative charge behind like the
smile of the Cheshire cat, to neutralize some of the positive nuclear
charge. This is a lot like polarization of a black hole's horizon by
the strong G field there (except this is a strong electric field). You
get pair production with one component of the pair being eaten and the
other popping free. So that counts as detection. In the black hole
when that happens, the mass of the hole decreases. For the Dirac case,
the charge of the potential decreases. So you can't have + charge
greater than Z= 180 or so in the Dirac theory, since vacuum breaks
down, and positrons are emitted until you get back down to critical
field (of course the same is true for - fields, where electrons are
emitted). The difference between Z=137 and 180 is due to corrections
resulting from the 2 center Dirac equatiion, and the one you usually
see, which is for only one particle (the electron).

SBH

**Gregory L. Hansen** · #2 February 2nd 04 posted to sci.physics

In article ,
Steve Harris wrote:
(Gregory L. Hansen) wrote in message
...
In the usual textbook representations, the Dirac spinor has the particle
spin and magnitude in the top two components and antiparticle spin and
magnitude in the bottom two components. But I've never really understood
(nor needed to understand in class) what that means. A stationary
particle is all upper. Boost it and you put some magnitude in the lower
components. Does that mean a likelihood of detecting an antiparticle?
Suppose we shot electrons into a magnetic field that's as weak as we like
but will sufficiently separate charge, do we expect most of them to turn
left but some to turn right? I'm sure that can't be quite right.

COMMENT:

I'm a out of my league on the math, but can give you some language and
handwaving. You shouldn't be wary of the lower spinor components
showing up when you give some KE to an electron (or drop it into some
E or B potential, also). That's just the Dirac equation's "way" of
accounting for the change in the "relativistic mass" of the electron.
Or the increase in total energy of the thing, as we say today. The
extra KE has no charge (obviously), so what kind of "stuff" is it made
of, particle-wise? According to the Dirac theory the extra stuff over
the rest mass (rest energy) of the electron is composed of half
virtual electron and half virtual positron. These travel along with
the original electron, and appear and disappear in a ghostly was as
you view the electron from frames other than the rest frame. So if you
boost an electron to total energy 2mc^2 it's composed of 1.5 electrons
and 0.5 positron. But the positron component is virtual since it's off
mass shell and hasn't got enough energy to be real.

Trippy. I just got done watching Indiana Jones, with the angels floating
around at the end, so I sort of imagined positrons picking people up and
making their heads melt.

Relativistic mass is an interesting point of view. I'll have to think on
that one.

So long as you haven't boosted the electron to a total energy higher
then 3mc^2, however, the virtual positron traveling along with it
never has enough energy to be real, so there's no danger of detecting
it directly as a free particle. It does, as a virtual field, has some
mild effects on the interaction of the whole ensemble on stationary
potentials, however. Over energies of 3mc^2 or kinetic energies of
2mc^2, of course, you can have pair production of the virtual
components if you can figure out a way to offload the momentum through
another interaction.

If you think the Dirac solutions for freely propagating particles are
strange, consider what happens when you drop an electron into a
positive potential well, as with a nucleus, and the well is really
deep. Say a 1s orbital for a nucleus of a couple of hundred Z (if
there was such a thing). At classical Z=137, the negative potential
energy of binding is equal to mc^2, so the total energy of the
electron is zero! It doesn't exist! Make the positive binding
potential even stronger and now the energy of the electron starts to
go negative, which means that you start to get more positronic
components than not. At some point, you have enough energy that this
positron will have enough energy to be real and pop out of the
potential, leaving only the electron's negative charge behind like the
smile of the Cheshire cat, to neutralize some of the positive nuclear
charge. This is a lot like polarization of a black hole's horizon by
the strong G field there (except this is a strong electric field). You
get pair production with one component of the pair being eaten and the
other popping free. So that counts as detection. In the black hole
when that happens, the mass of the hole decreases. For the Dirac case,
the charge of the potential decreases. So you can't have + charge
greater than Z= 180 or so in the Dirac theory, since vacuum breaks
down, and positrons are emitted until you get back down to critical
field (of course the same is true for - fields, where electrons are
emitted). The difference between Z=137 and 180 is due to corrections
resulting from the 2 center Dirac equatiion, and the one you usually
see, which is for only one particle (the electron).

SBH

I had thought about trying to calculate the vacuum current that would be
associated with a given electrical field. But then I got distracted by
shiny objects and never got around to it.

--
"There's nary an animal alive that can outrun a greased Scottsman!" --
Groundskeeper Willy

**FrediFizzx** · #3 February 2nd 04 posted to sci.physics

"Steve Harris " wrote in
message m...
| (Gregory L. Hansen) wrote in message
...
| In the usual textbook representations, the Dirac spinor has the particle
| spin and magnitude in the top two components and antiparticle spin and
| magnitude in the bottom two components. But I've never really
understood
| (nor needed to understand in class) what that means. A stationary
| particle is all upper. Boost it and you put some magnitude in the lower
| components. Does that mean a likelihood of detecting an antiparticle?
| Suppose we shot electrons into a magnetic field that's as weak as we
like
| but will sufficiently separate charge, do we expect most of them to turn
| left but some to turn right? I'm sure that can't be quite right.
|
| COMMENT:
|
| I'm a out of my league on the math, but can give you some language and
| handwaving. You shouldn't be wary of the lower spinor components
| showing up when you give some KE to an electron (or drop it into some
| E or B potential, also). That's just the Dirac equation's "way" of
| accounting for the change in the "relativistic mass" of the electron.
| Or the increase in total energy of the thing, as we say today. The
| extra KE has no charge (obviously), so what kind of "stuff" is it made
| of, particle-wise? According to the Dirac theory the extra stuff over
| the rest mass (rest energy) of the electron is composed of half
| virtual electron and half virtual positron. These travel along with
| the original electron, and appear and disappear in a ghostly was as
| you view the electron from frames other than the rest frame. So if you
| boost an electron to total energy 2mc^2 it's composed of 1.5 electrons
| and 0.5 positron. But the positron component is virtual since it's off
| mass shell and hasn't got enough energy to be real.

I think it is more than 1.5 electrons and .5 positrons. It is a mix of all
the charged fermion pairs. But the mix is for sure dominated by virtual
electrons and positrons at lower energies.

| So long as you haven't boosted the electron to a total energy higher
| then 3mc^2, however, the virtual positron traveling along with it
| never has enough energy to be real, so there's no danger of detecting
| it directly as a free particle. It does, as a virtual field, has some
| mild effects on the interaction of the whole ensemble on stationary
| potentials, however. Over energies of 3mc^2 or kinetic energies of
| 2mc^2, of course, you can have pair production of the virtual
| components if you can figure out a way to offload the momentum through
| another interaction.

I think this is what happens in accelerated e+e- scattering. When the
energies are high enough, the heavier charged fermion pairs in the mix can
become more dominate.

| If you think the Dirac solutions for freely propagating particles are
| strange, consider what happens when you drop an electron into a
| positive potential well, as with a nucleus, and the well is really
| deep. Say a 1s orbital for a nucleus of a couple of hundred Z (if
| there was such a thing). At classical Z=137, the negative potential
| energy of binding is equal to mc^2, so the total energy of the
| electron is zero! It doesn't exist! Make the positive binding
| potential even stronger and now the energy of the electron starts to
| go negative, which means that you start to get more positronic
| components than not. At some point, you have enough energy that this
| positron will have enough energy to be real and pop out of the
| potential, leaving only the electron's negative charge behind like the
| smile of the Cheshire cat, to neutralize some of the positive nuclear
| charge. This is a lot like polarization of a black hole's horizon by
| the strong G field there (except this is a strong electric field). You
| get pair production with one component of the pair being eaten and the
| other popping free. So that counts as detection. In the black hole
| when that happens, the mass of the hole decreases. For the Dirac case,
| the charge of the potential decreases. So you can't have + charge
| greater than Z= 180 or so in the Dirac theory, since vacuum breaks
| down, and positrons are emitted until you get back down to critical
| field (of course the same is true for - fields, where electrons are
| emitted). The difference between Z=137 and 180 is due to corrections
| resulting from the 2 center Dirac equatiion, and the one you usually
| see, which is for only one particle (the electron).

This looks like Gribov's vacuum scenario.

FrediFizzx

#4 February 3rd 04 posted to sci.physics

"FrediFizzx" wrote in message ...
"Steve Harris " wrote in
message m...
| (Gregory L. Hansen) wrote in message
...
| In the usual textbook representations, the Dirac spinor has the particle
| spin and magnitude in the top two components and antiparticle spin and
| magnitude in the bottom two components. But I've never really
understood
| (nor needed to understand in class) what that means. A stationary
| particle is all upper. Boost it and you put some magnitude in the lower
| components. Does that mean a likelihood of detecting an antiparticle?
| Suppose we shot electrons into a magnetic field that's as weak as we
like
| but will sufficiently separate charge, do we expect most of them to turn
| left but some to turn right? I'm sure that can't be quite right.
|
| COMMENT:
|
| I'm a out of my league on the math, but can give you some language and
| handwaving. You shouldn't be wary of the lower spinor components
| showing up when you give some KE to an electron (or drop it into some
| E or B potential, also). That's just the Dirac equation's "way" of
| accounting for the change in the "relativistic mass" of the electron.
| Or the increase in total energy of the thing, as we say today. The
| extra KE has no charge (obviously), so what kind of "stuff" is it made
| of, particle-wise? According to the Dirac theory the extra stuff over
| the rest mass (rest energy) of the electron is composed of half
| virtual electron and half virtual positron. These travel along with
| the original electron, and appear and disappear in a ghostly was as
| you view the electron from frames other than the rest frame. So if you
| boost an electron to total energy 2mc^2 it's composed of 1.5 electrons
| and 0.5 positron. But the positron component is virtual since it's off
| mass shell and hasn't got enough energy to be real.

I think it is more than 1.5 electrons and .5 positrons. It is a mix of all
the charged fermion pairs. But the mix is for sure dominated by virtual
electrons and positrons at lower energies.

COMMENT:

Well, yes, in our mature theories. But we're talking about the 1928
Dirac equation. It specifies the electron, because it has only enough
variable room to describe the quantum states of a relativistic
spin-1/2 particle in external field, and no more. And the rest mass is
added by hand. So nothing but electrons and positrons come out of it,
and what it specifies is what's above.

Now, Dirac wasn't expecting positrons to come out of it, for sure.
Positrons weren't known in 1929 when Dirac was sweating over the
meaning of those other two spinor components. Only two components are
necessary for the quantum state of a spin 1/2 particle in a field, one
for spin up and the other for spin down. Ala Pauli. But to make
Pauli's equation covarient so that it describes relativistic mass
increase, Dirac found he had to add, at minimum, two more
wavefunctions for mathematical reasons. And these functions had
significant values whenever the electron had a lot of energy. So what
did these represent? When Dirac looked at them, they seemed to have
negative energy.

Now we know they represented wavefunctions for a particle of opposite
charge, which acts to interfere with the electron wavefunction, so
that its mass/energy can increase relativistically without its CHARGE
increasing. So a doubling of possible wavefunctions can not only be
expected for the electron, but for every other charged particle which
is moving relativistically (and for non-charged particles, too, if
they spin, since you have the same problem with angular momentum that
you do for charge). And the particle represented can actually come
into reality if there is enough kinetic energy to give it one electron
rest mass.

Dirac at first thought that the negative energy solutions might
represent holes in the fabric of space. But since they were positively
charged (sort of like holes in a semiconductor), Dirac thought that
maybe normal vacuum was composed of such holes filed up with
electrons. A free hole would then "look" like a positive electron,
since it would take the same push to move a hole as it would to move
an electron (since you'd have to move the hole BY moving an electron
into it).

But then Dirac wimped out in 1930 and finally suggested that the holes
might be more massive than electrons, and might in fact be protons.
Later when the positron was found in 1932, Dirac commented that his
equation was smarter than he was. Einstein said much the same about
his general relativity theory, when the universe was found to be
expanding G.

SBH

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