On explanation of negative energy states and electron pair creation

[zak]

Dirac's theory of the electron predicted that there were identical
particles of equal mass but of negative energy.

He appealed to the Pauli exclusion principle and proposed that there
was a negative energy 'sea' of electrons that was full up to -2mc^2 in
order to answer critics that positive energy electrons described by the
Dirac electron theory would simply decay down to -infinity.
With this description pair creation is described by absortion of a
photon (where the energy of the photon E_p > 2mc^2) by a negative
energy electron that scatters up to a poitive energy state leaving
behind a hole.

This hole is the negative energy 'sea' has equal but opposite charge to
the electron and is commonly known as a positron.

I believe that this description is somewhat old hat and not used anymore
in modern QFT circles. Can anybody give me a not too technical
explanation to why the 'old hat' qualitative explanation is
unsatisfactory ?

Zak

 

[Hendrik van Hees]

zak wrote:

> I believe that this description is somewhat old hat and not used
> anymore in modern QFT circles. Can anybody give me a not too technical
> explanation to why the 'old hat' qualitative explanation is
> unsatisfactory ?

I have had always difficulties in understanding this old concept of
Dirac's sea. First of all, if there is a sea of infinitely many
electrons in the universe, why isn't there a tremendous negative
background charge with some observable effects?

Further, there are not only fermions but also bosons, and then the very
concept of the Dirac sea does not work at all.

IMHO, modern quantum field theory is much more comprehensive, although
not completely formulated in a strict mathematical sense: There one
assumes that the Hamiltonian is bounded from below, i.e., there exists
a stable ground state (which may be degenerated). The apparent problem
of states with negative energy from naive (canonical) "first
quantization" in old-fashioned relativistic quantum mechanics simply
does not exist, but one formulates it right from the beginning as a
many-body (i.e., quantum field) theory ("second quantization") which is
adequate in the relativistic context because there particles can be
created and destroyed.

If one assumes in addition that the field theory is local, i.e., the
Hamiltonian (Lagrangian) is a polynomial of the field operators and
their derivatives and micro-causality, i.e., local observables (like
energy-momentum, angular momentum and various charge densities) commute
for space-like separated arguments, one can derive phenomenologically
successful properties from the fact that the theory must admit a ray
representation of the Poincare group to be consistent with the
space-time structure of special relativity, that there is the usual
connection between spin and statistics, i.e.,

particles with half-integer spin are fermions,
particles with integer spin are bosons,

and that for each particle there must exist a corresponding antiparticle
(which can be identical with the particles like, e.g., for photons),
and that the model is automatically also CPT invariant, i.e., for any
process in nature there exists also the process, where all particles
are substituted by their corresponding antiparticles (C=charge
conjugation), the whole thing is looked at in a mirror (P=parity) and
all momenta and spins are inverted (T=time reversal).

In the mathematical formalism of the theory the trick is that the field
operators are superpositions of annihilation and creation operators
(and not simply annihilation operators as in the non-relativistic
quantum theory in "second quantization"). This is known as the
Feynman-Stückelberg trick, but can also be derived from the above
mentioned assumptions and representation theory of the Poincare group.

For details, you may look at my quantum-field theory script on my
homepage

http://cyclotron.tamu.edu/hees/publ/lect.pdf --
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:hees@comp.tamu.edu

 

[Arnold Neumaier]

zak schrieb:
> Dirac's theory of the electron predicted that there were identical
> particles of equal mass but of negative energy.
>
> He appealed to the Pauli exclusion principle and proposed that there
> was a negative energy 'sea' of electrons that was full up to -2mc^2 in
> order to answer critics that positive energy electrons described by the
> Dirac electron theory would simply decay down to -infinity.
> With this description pair creation is described by absortion of a
> photon (where the energy of the photon E_p > 2mc^2) by a negative
> energy electron that scatters up to a poitive energy state leaving
> behind a hole.
>
> This hole is the negative energy 'sea' has equal but opposite charge to
> the electron and is commonly known as a positron.
>
> I believe that this description is somewhat old hat and not used anymore
> in modern QFT circles. Can anybody give me a not too technical
> explanation to why the 'old hat' qualitative explanation is
> unsatisfactory ?

* It is superseded by quantum field theory, which has a much wider
domain of applicability. For example, a hole description applies only
to Fermions and not to Bosons (since these have no exclusion principle,
the sea would not be stable).

* In quantum field theory, positrons and electrons appear on a symmetric
footing, which reflects a symmetry (CPT) in the laws of Nature.
In the old picture, positrons are completely different objects than
electrons.

* The hole theory does not work well for positrons and electrons in
strong electromagnetic fields.Arnold Neumaier

 

[Oh No]

Thus spake zak <b.zarychta@googlemail.com>
>Dirac's theory of the electron predicted that there were identical
>particles of equal mass but of negative energy.
>
>He appealed to the Pauli exclusion principle and proposed that there
>was a negative energy 'sea' of electrons that was full up to -2mc^2 in
>order to answer critics that positive energy electrons described by the
>Dirac electron theory would simply decay down to -infinity.
>With this description pair creation is described by absortion of a
>photon (where the energy of the photon E_p > 2mc^2) by a negative
>energy electron that scatters up to a poitive energy state leaving
>behind a hole.
>
>This hole is the negative energy 'sea' has equal but opposite charge to
>the electron and is commonly known as a positron.
>
>I believe that this description is somewhat old hat and not used anymore
>in modern QFT circles. Can anybody give me a not too technical
>explanation to why the 'old hat' qualitative explanation is
>unsatisfactory ?
>

It is conceptually unsatisfying, invoking an infinite sea before we can
do physics, but more importantly perhaps, it is not a concept which
works at all well when one introduces interactions between particles.

I find the Stuckelburg-Feynman interpretation better, that negative
energy particles are particles for which proper time is reversed. They
are literally particles going backwards in time. Then the creation of a
negative energy particle appears from the point of view of macroscopic
time as the annihilation of a positive energy one.

The more modern approach is to think that particles are not fundamental
at all, that they are knots in a quantum field. This generally goes hand
in hand with a denial of interpretational issues altogether, which I
find equally unsatisfying.Regards

--
Charles Francis
substitute charles for NotI to email

 

[Hendrik van Hees]

Oh No wrote:


> I find the Stuckelburg-Feynman interpretation better, that negative
> energy particles are particles for which proper time is reversed. They
> are literally particles going backwards in time. Then the creation of
> a negative energy particle appears from the point of view of
> macroscopic time as the annihilation of a positive energy one.

They go forward in time with reversed momentum. By definition, there is
nothing going backward in time in physics!--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:hees@comp.tamu.edu

 

[John (Liberty) Bell]

Oh No wrote:

> Thus spake zak <b.zarychta@googlemail.com>
> >Dirac's theory of the electron predicted that there were identical
> >particles of equal mass but of negative energy.

What I think you are actually talking about here is Dirac's
relativistic wave equation for the electron, which is sort of like
Schrodinger's wave equation, except fully relativistic. It is thus not
really Dirac's 'theory of the electron', but Dirac's application of
Einstein's theory of relativity to the pre-relativistic wave mechanics
of early 20th century quantum theory.

Dirac was astute enough as a physicist, and competent enough as a
mathematician to recognise that this relativistic wave equation
predicted negative as well as positive energy states.

> >He appealed to the Pauli exclusion principle and proposed that there
> >was a negative energy 'sea' of electrons that was full up to -2mc^2 in
> >order to answer critics that positive energy electrons described by the
> >Dirac electron theory would simply decay down to -infinity.
> >With this description pair creation is described by absortion of a
> >photon (where the energy of the photon E_p > 2mc^2) by a negative
> >energy electron that scatters up to a poitive energy state leaving
> >behind a hole.
> >
> >This hole is the negative energy 'sea' has equal but opposite charge to
> >the electron and is commonly known as a positron.
> >
> >I believe that this description is somewhat old hat and not used anymore
> >in modern QFT circles. Can anybody give me a not too technical
> >explanation to why the 'old hat' qualitative explanation is
> >unsatisfactory ?

Good question. It certainly seems to be true that Dirac has fallen out
of favour in the mainstream physics community, of late. When I was
still an undergrad, I asked my personal tutor if we could do Dirac
then, and he responded that Dirac was "too difficult". However, I
personally found Dirac beautifully simple conceptually, at the same
time as being advanced for its era (which I think was the 1920's). I
thus found Dirac's negative energy solution of his own relativistic
wave equation to be a good conceptual "stepping stone".

> I find the Stuckelburg-Feynman interpretation better, that negative
> energy particles are particles for which proper time is reversed.

I am inclined to agree.

> They are literally particles going backwards in time.

Fine. This ties in with the Feynman diagram.

> Then the creation of a
> negative energy particle appears from the point of view of macroscopic
> time as the annihilation of a positive energy one.

I am not sure what you mean by this.

Consider a point in spacetime that is identified in arbitrary Gaussian
coordinates as the point of interaction between a photon, a positron,
and an electron.

We have 2 distinct situations to consider here.

A) Creation of a particle/antiparticle pair from the destruction of a
photon.
B) Creation of a photon from the interaction between (hence anihilation
of) a particle/antiparticle pair

Consider (A) first. Within the context of the Stuckelburg-Feynman
interpretation, the simlest way to look at this is probably as a
collision between a positron and a photon, where the photon's energy
and momentum are 'just right' to reflect that positron in the dimension
of time, thus turning that positron into an electron, with the
resultant annihilation of the photon.
I see no difficulties with that iviewpoint.

Now, the really neat thing about the Feynman diagram is you can just
rotate your coordinate system by 180 degrees, to obtain situation (B).

John (Liberty) Bell
http://global.accelerators.co.uk
(Change John to Liberty to respond by email)

 

[Oh No]

Thus spake Hendrik van Hees <hees@comp.tamu.edu>
>Oh No wrote:
>
>
>> I find the Stuckelburg-Feynman interpretation better, that negative
>> energy particles are particles for which proper time is reversed. They
>> are literally particles going backwards in time. Then the creation of
>> a negative energy particle appears from the point of view of
>> macroscopic time as the annihilation of a positive energy one.
>
>They go forward in time with reversed momentum. By definition, there is
>nothing going backward in time in physics!
>
>
One has to distinguish between time as we define it, and time as it is
defined by nature. I distinguish here between macroscopic time, which we
measure and which goes forward by our empirical definition, and proper
time, which applies to any matter, but which we cannot measure directly
- e.g. we cannot measure proper time for a distant galaxy, although we
can very reasonably define cosmic time, meaning proper time from the big
bang.

It is reasonable to think that a concept of proper time applies to
particles in the quantum domain. Clearly this is not the same as
macroscopic time. E.g. it is measureably not the same for particles at
high momenta in accelerators. Nor could it be in a relativistic theory.

If you define proper time of an elementary particle to be always
forwards you are making a non-empirical assumption about the behaviour
of matter. That is not science, it is metaphysics and it is speculative.
Feynman and Stuckelburg have pointed out that if you make no such non-
empirical assumption then you get a very simple and easy to understand
explanation for antimatter, which makes complete sense of the appearance
of such states in the Dirac equation. In the absence of any such
explanation based on the assumption that proper time is always forwards,
there is a strong reason to believe that your definition is inconsistent
with nature.

Regards

--
Charles Francis
substitute charles for NotI to email

 

[Arnold Neumaier]

Hendrik van Hees schrieb:

> Further, there are not only fermions but also bosons, and then the very
> concept of the Dirac sea does not work at all.
>
> IMHO, modern quantum field theory is much more comprehensive, although
> not completely formulated in a strict mathematical sense:

The Dirac also has no strict mathematical sense, so one does not need to
excuse QFT for that...

Arnold Neumaier

 

[Oh No]

Thus spake "John (Liberty) Bell" <john.bell@accelerators.co.uk>
>> Then the creation of a
>> negative energy particle appears from the point of view of macroscopic
>> time as the annihilation of a positive energy one.
>
>I am not sure what you mean by this.
>
>Consider a point in spacetime that is identified in arbitrary Gaussian
>coordinates as the point of interaction between a photon, a positron,
>and an electron.
>
>We have 2 distinct situations to consider here.
>
>A) Creation of a particle/antiparticle pair from the destruction of a
>photon.
>B) Creation of a photon from the interaction between (hence anihilation
>of) a particle/antiparticle pair

>
>Consider (A) first. Within the context of the Stuckelburg-Feynman
>interpretation, the simlest way to look at this is probably as a
>collision between a positron and a photon, where the photon's energy
>and momentum are 'just right' to reflect that positron in the dimension
>of time, thus turning that positron into an electron, with the
>resultant annihilation of the photon.
>I see no difficulties with that iviewpoint.
>
>Now, the really neat thing about the Feynman diagram is you can just
>rotate your coordinate system by 180 degrees, to obtain situation (B).
>
>

It's also really neat that you can do a little more with it than that
even. You can simply rotate your positron line and it becomes an
electron line, or your electron line and it becomes a positron line. As
you rotate the one line individually, creation turns to annihilation or
annihilation to creation for that particle, and at the same time matter
becomes antimatter or antimatter becomes matter.

Regards

--
Charles Francis
substitute charles for NotI to email

 

[basically yes]

"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> a écrit dans le message de
news: 45A610CE.60105@univie.ac.at

> * In quantum field theory, positrons and electrons appear on a symmetric
> footing, which reflects a symmetry (CPT) in the laws of Nature.
> In the old picture, positrons are completely different objects than
> electrons.

But Dirac's description is anyway symmetric. It may as well mean that
electrons are holes in the sea of negative energy positrons.

 

[basically yes]

"Hendrik van Hees" <hees@comp.tamu.edu> a écrit dans le message de news:
yckph.26683$Dy2.829@newsfe20.lga

> I have had always difficulties in understanding this old concept of
> Dirac's sea. First of all, if there is a sea of infinitely many
> electrons in the universe, why isn't there a tremendous negative
> background charge with some observable effects?

Because that charge is uniform, and from the laws of electromagnetism
produces no observable effect. There is one exception: if there is a hole
in that sea. Proper calculation then shows that the electric filed is
identical to the one produced by a positively charged particle. The same is
true for the magnetic field, since all the momentum states are present.

 

[Arnold Neumaier]

basically yes schrieb:
> "Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> a écrit dans le message de
> news: 45A610CE.60105@univie.ac.at
>
>> * In quantum field theory, positrons and electrons appear on a symmetric
>> footing, which reflects a symmetry (CPT) in the laws of Nature.
>> In the old picture, positrons are completely different objects than
>> electrons.
>
> But Dirac's description is anyway symmetric. It may as well mean that
> electrons are holes in the sea of negative energy positrons.
>
No. This would require that Nature fills the states of large energy first.

Arnold Neumaier

 

[basically yes]

J'ai écrit :

>> But Dirac's description is anyway symmetric. It may as well mean that
>> electrons are holes in the sea of negative energy positrons.

"Arnold Neumaier" <Arnold.Neumaier@univie.ac.at> a écrit dans le message de
news: 45D2E262.4040203@univie.ac.at

> No. This would require that Nature fills the states of large energy first.

That doesn't answer what I wrote. And I don't know this 'Nature' who makes
some housekeeping. The world is as it is, and it's all that we humans can
say about it. As the two different possibilities can't be told
experimentally, the symmetry is perfect from a logical point of view.
Trying and going beyond that get out of the frame of that NG.

 

[wofsy]

I just read that the electron in the ionized hydrogen molecule, two protons one electron, has negative energy. If the protons are far apart, then the potential is small over most of the space between the protons so the electron approximates a free particle. Its kinetic energy is therefore negative and its momentum is imaginary.

I could use an explanation of why the energy is negative in this case and what it means to have imaginary momentum.