Question about photon wavefunction

[Jay R. Yablon]

I have often seen an photon field quantum A^u written in terms of a
polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:

A^u = epsilon^u exp[ip^u x_u] (1)

1) Is p^u considered to be the momentum vector of that subject
photon, or an independent variable of integration unconnected to the
photon?

2) There is a discussion I had about the following with Igor a few
years ago, but I need to be refreshed: For a scalar (spinless) field
quantum psi, can one write that as:

psi = N exp[ip^u x_u] (2)

where N is a normalization factor and p^u is (presumably, as in
question 1) the momentum vector for the scalar field?

3) for a fermion phi, I have seen solutions to Dirac's equation
specified by:

phi = u(p^u) exp[ip^u x_u] (3)

where u(p) is a spinor function of momentum. Is this p^u similarly
the momentum of the fermion, or something else?

Thanks.

Jay.

[Igor]

On Jan 7, 4:29*pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> I have often seen an photon field quantum A^u written in terms of a
> polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>
> A^u = epsilon^u exp[ip^u x_u] * (1)
>
> 1) *Is p^u considered to be the momentum vector of that subject
> photon, or an independent variable of integration unconnected to the
> photon?

It's the former, since p = hbar k, where k is the wave number. This
can be easily derived from the Poynting vector for the propogating
wave integrated over all space.

> 2) *There is a discussion I had about the following with Igor a few
> years ago, but I need to be refreshed: *For a scalar (spinless) field
> quantum psi, can one write that as:
>
> psi = N exp[ip^u x_u] *(2)
>
> where N is a normalization factor and p^u is (presumably, as in
> question 1) the momentum vector for the scalar field?

Yes. How else would you want to interpret it?

> 3) for a fermion phi, I have seen solutions to Dirac's equation
> specified by:
>
> phi = u(p^u) exp[ip^u x_u] *(3)
>
> where u(p) is a spinor function of momentum. *Is this p^u similarly
> the momentum of the fermion, or something else?

Again, yes. The only real difference here is that the wave equation
and its solutions have internal degrees of freedom. But the de
Broglie relation still applies, relating particle momentum to the wave
number of the moving wave.

[Arnold Neumaier]

Jay R. Yablon schrieb:
> I have often seen an photon field quantum A^u written in terms of a
> polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>
> A^u = epsilon^u exp[ip^u x_u] (1)
>
> 1) Is p^u considered to be the momentum vector of that subject
> photon, or an independent variable of integration unconnected to the
> photon?

Yes. For further explanations, see Section S2f. What is a photon?
of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt


> 2) There is a discussion I had about the following with Igor a few
> years ago, but I need to be refreshed: For a scalar (spinless) field
> quantum psi, can one write that as:
>
> psi = N exp[ip^u x_u] (2)
>
> where N is a normalization factor and p^u is (presumably, as in
> question 1) the momentum vector for the scalar field?
>
> 3) for a fermion phi, I have seen solutions to Dirac's equation
> specified by:
>
> phi = u(p^u) exp[ip^u x_u] (3)
>
> where u(p) is a spinor function of momentum. Is this p^u similarly
> the momentum of the fermion, or something else?

Yes. But note that in all three cases, these only give a set of bais
vectors in a momentum eigenstates. The general state has unsharp
momentum, and is an arbitrary linear combinations of these, obtained
by a weighted sum over the degrees of freedom (i.e., integral over
momentum, sum over polarization/spin indices).


Arnold Neumaier

[Igor Khavkine]

On Jan 10, 5:58 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:
> Jay R. Yablon schrieb:
>
> > I have often seen an photon field quantum A^u written in terms of a
> > polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>
> > A^u = epsilon^u exp[ip^u x_u] (1)
>
> > 1) Is p^u considered to be the momentum vector of that subject
> > photon, or an independent variable of integration unconnected to the
> > photon?

[...]

> Yes. But note that in all three cases, these only give a set of bais
> vectors in a momentum eigenstates. The general state has unsharp
> momentum, and is an arbitrary linear combinations of these, obtained
> by a weighted sum over the degrees of freedom (i.e., integral over
> momentum, sum over polarization/spin indices).

A short follow up to Arnold's remark. In quantum mechanics,
observables are operators, including the components of the momentum.
For a wave function proportional to exp(ipx), the components of the
vector p are associated with the momentum only in as much as they are
the eigenvalues of the corresponding operator observables, with exp(ipx)
being the eigenvector.

Igor

[Jay R. Yablon]

"Igor" <thoovler@excite.com> wrote in message
news:f9c643ea-cdec-4a37-acd9-8f5d8d962825@k1g2000prb.googlegroups.com...
> On Jan 7, 4:29 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> I have often seen an photon field quantum A^u written in terms of a
>> polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>>
>> A^u = epsilon^u exp[ip^u x_u] (1)
>>
>> 1) Is p^u considered to be the momentum vector of that subject
>> photon, or an independent variable of integration unconnected to the
>> photon?
>
> It's the former, since p = hbar k, where k is the wave number. This
> can be easily derived from the Poynting vector for the propogating
> wave integrated over all space.

OK, Igor, that just as I thought and just as I had hoped.

So p^u is the momentum vector of the subject photon (or as you later
say, if "components of the vector p are associated with the momentum [of
the subject photon] only in as much as they are the eigenvalues of the
corresponding operator observables," and is not some disconnected
variable of integration.

So now, what about a Yang mills gauge field, call it

G^u = T^i G_i^u? (4)

Above, T_i are the group generators, and G_i^u contains an internal
symmetry index i=1,2,3..n^2-1 for SU(N). Let us write this with a
polarization vector and a plane wave as in (1):

G_i^u = epsilon_i^u exp[ip^u x_u] (5)

I assume I am correct in giving an internal symmetry index to
epsilon_i^u, since each gauge boson will have its own polarizationn
states?

Now, here is my question. Let's say to keep it easy, that we are using
SU(2). Then there are three G_i^u. Each one, it seems, should have an
associated momentum vector p^u (specifying momentum operator eigenvalues
of the operators therefor), that is, there should be three p^u, one for
each gauge boson. But p^u can only represent one gauge boson momentum,
not three. Put differently, there is nothing in (5) which tells us the
eigenvalues of the momentum operator p^u for G_1^u, versus G_2^u or
G_2^u.

So let me designate these three momenta a p_(i)^u, where for right now,
_(i) is just a label. p_(1)^u is to be associated with momentum
eigenvalues for G_1^u, and same for the other two p_(i)^u and G_i^u.

The question now becomes whether _(i) is just a label, or might be an
internal symmetry index subject to matematical treatment as such.

Now, I do not know if I can argue that this MUST be an internal symmetry
index rather than a mere label, but I do think is highly plausible if
not essential based on your answer above that as soon as we have more
than one gauge boson, we need to be able to talk in some fashion about
the momentum eigenvalues of each one.

If we consider the latter, i.e., that this is an internal symmetry
index, then

p_(i)^u --> p_i^u (6)

and one can then form hermitian SU(N) matrices:

p^u = T^i p_i^u (7)

because the p_i^u form a vector in the adjoint SU(N) representation.
Then, one might even take exp[ip^u x_u] in (5) to implicitly involve
(7).

Therefore, the differentiation operation:

d^u G_i^u = ip^u G_i^u (8)

with p^u as in (7), which will cause

(d^u d_u)^-1 --> -(p^u p_u)^-1 (9)

to show up on the right hand side of the path integral, in the
propagator.

Summing this all up, if p_u in (1) is indeed associated with the subject
photon, then how do we establish momentum vectors associated with the
gauge bosons in Yang-Mill gauge theory, unless we (at least) establish
labeled momentum vectors p_(i)^u, if not p_i^u with internal symmetry?
And, if we can in fact justify p_i^u with internal symmetry on order for
your answers above (and below) to be extended to Yang Mills theory and
give us an associated momentum for each gauge boson, then this has
ramifications all the way through how propagators are formed, since they
will have to involve matrix inverses akin to (9).

>
>> 2) There is a discussion I had about the following with Igor a few
>> years ago, but I need to be refreshed: For a scalar (spinless) field
>> quantum psi, can one write that as:
>>
>> psi = N exp[ip^u x_u] (2)
>>
>> where N is a normalization factor and p^u is (presumably, as in
>> question 1) the momentum vector for the scalar field?
>
> Yes. How else would you want to interpret it?

No other way. That is how I look at it too. But once you go to SU(N),
the same questions arise as above: you need more than one p^u to
associated with momentum eigenvalues with each scalar field. You may
break some symmetry and perhaps get rid of some of these in the end, but
before you break symmetry, this is what you would have to start out
with.

>
>> 3) for a fermion phi, I have seen solutions to Dirac's equation
>> specified by:
>>
>> phi = u(p^u) exp[ip^u x_u] (3)
>>
>> where u(p) is a spinor function of momentum. Is this p^u similarly
>> the momentum of the fermion, or something else?
>
> Again, yes. The only real difference here is that the wave equation
> and its solutions have internal degrees of freedom. But the de
> Broglie relation still applies, relating particle momentum to the wave
> number of the moving wave.

Your mentioning internal symmetry here is also consistent with the
above. If the phi form an SU(N) vector, then here too, it seems that we
need more than one momentum four-vector to associate with each of the
phi. For example, if phi is an "isospin up / isospin down" SU(2)
doublet, we need a momentum for the top member and a momentum for the
bottom member. This similarly, will migrate over to the fermion
propators.

Jay.

[Igor Khavkine]

On Jan 10, 5:58 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:
> Jay R. Yablon schrieb:
>
> > I have often seen an photon field quantum A^u written in terms of a
> > polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>
> > A^u = epsilon^u exp[ip^u x_u] (1)
>
> > 1) Is p^u considered to be the momentum vector of that subject
> > photon, or an independent variable of integration unconnected to the
> > photon?

[...]

> Yes. But note that in all three cases, these only give a set of bais
> vectors in a momentum eigenstates. The general state has unsharp
> momentum, and is an arbitrary linear combinations of these, obtained
> by a weighted sum over the degrees of freedom (i.e., integral over
> momentum, sum over polarization/spin indices).

A short follow up to Arnold's remark. In quantum mechanics,
observables are operators, including the components of the momentum.
For a wave function proportional to exp(ipx), the components of the
vector p are associated with the momentum only in as much as they are
the eigenvalues of the corresponding operator observables, with exp(ipx)
being the eigenvector.

Igor

[Arnold Neumaier]

Jay R. Yablon schrieb:
> I have often seen an photon field quantum A^u written in terms of a
> polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>
> A^u = epsilon^u exp[ip^u x_u] (1)
>
> 1) Is p^u considered to be the momentum vector of that subject
> photon, or an independent variable of integration unconnected to the
> photon?

Yes. For further explanations, see Section S2f. What is a photon?
of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt


> 2) There is a discussion I had about the following with Igor a few
> years ago, but I need to be refreshed: For a scalar (spinless) field
> quantum psi, can one write that as:
>
> psi = N exp[ip^u x_u] (2)
>
> where N is a normalization factor and p^u is (presumably, as in
> question 1) the momentum vector for the scalar field?
>
> 3) for a fermion phi, I have seen solutions to Dirac's equation
> specified by:
>
> phi = u(p^u) exp[ip^u x_u] (3)
>
> where u(p) is a spinor function of momentum. Is this p^u similarly
> the momentum of the fermion, or something else?

Yes. But note that in all three cases, these only give a set of bais
vectors in a momentum eigenstates. The general state has unsharp
momentum, and is an arbitrary linear combinations of these, obtained
by a weighted sum over the degrees of freedom (i.e., integral over
momentum, sum over polarization/spin indices).


Arnold Neumaier

[p.kinsler@ic.ac.uk]

Igor Khavkine <igor.kh@gmail.com> wrote:
> > > I have often seen an photon field quantum A^u written in terms of a
> > > polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
> [...]
> A short follow up to Arnold's remark. In quantum mechanics,
> observables are operators, including the components of the momentum.
> For a wave function proportional to exp(ipx), the components of the
> vector p are associated with the momentum only in as much as they are
> the eigenvalues of the corresponding operator observables, with exp(ipx)
> being the eigenvector.

It is the mode function that has the "exp(ipx)" behaviour,
the wave function describes the state of the photons inside
that mode.
The wave function, therefore, does not extent over real space,
it instead extends over the excitation space of the mode.

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (PHOT) (ph) +44-20-759-47734 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2AZ, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/

[Hendrik van Hees]

p.kinsler@ic.ac.uk wrote:

> It is the mode function that has the "exp(ipx)" behaviour,
> the wave function describes the state of the photons inside
> that mode.
> The wave function, therefore, does not extent over real space,
> it instead extends over the excitation space of the mode.

The most safe way to define observables is by use of Noether's
theorem. The total momentum for free photons is easily written down:
It's simply given by the quantized Poynting vector:

\vec{P}=\int d^3 x :\vec{E}(t,x) \times \vec{B}(t,x):,

where \vec{E} and \vec{B} are the electric and magnetic field
operators and :...: denotes the normal ordering. With creation and
annihilation operators you get

\vec{P}=\sum_{\lambda=-1,1} \int d^3 \vec{p}/[(2pi)^3 2 |\vec{p}|]
\vec{p} a^{\dagger} (\vec{p},\lambda) a(\vec{p},\lambda),

where lambda denotes the helicity of the photon modes with definite
momentum.

--
Hendrik van Hees Institut für Theoretische Physik
Phone: +49 641 99-33342 Justus-Liebig-Universität Gießen
Fax: +49 641 99-33309 D-35392 Gießen
http://theory.gsi.de/~vanhees/faq/

[Igor Khavkine]

On Jan 22, 11:33 pm, p.kins...@ic.ac.uk wrote:
> Igor Khavkine <igor...@gmail.com> wrote:

> > [...] In quantum mechanics,
> > observables are operators, including the components of the momentum.
> > For a wave function proportional to exp(ipx), the components of the
> > vector p are associated with the momentum only in as much as they are
> > the eigenvalues of the corresponding operator observables, with exp(ipx)
> > being the eigenvector.
>
> It is the mode function that has the "exp(ipx)" behaviour,
> the wave function describes the state of the photons inside
> that mode.
> The wave function, therefore, does not extent over real space,
> it instead extends over the excitation space of the mode.

You're right, but so am I. There are two ways to represent a Fock
space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
each mode) and (2) as a sum of symmetrized (for bosons) tensor
products of the space of wave functions on the set of modes. I like to
call this equivalence the "fundamental theorem of second
quantization". Perhaps this terminology will catch on some day. :-)

You refer to method (1), while I referred to method (2).

Igor

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:d0c1dad1-cf9e-4f16-9cb6-907e051c4d11@w39g2000prb.googlegroups.com...
> On Jan 22, 11:33 pm, p.kins...@ic.ac.uk wrote:
>> Igor Khavkine <igor...@gmail.com> wrote:
>
>> > [...] In quantum mechanics,
>> > observables are operators, including the components of the
>> > momentum.
>> > For a wave function proportional to exp(ipx), the components of the
>> > vector p are associated with the momentum only in as much as they
>> > are
>> > the eigenvalues of the corresponding operator observables, with
>> > exp(ipx)
>> > being the eigenvector.
>>
>> It is the mode function that has the "exp(ipx)" behaviour,
>> the wave function describes the state of the photons inside
>> that mode.
>> The wave function, therefore, does not extent over real space,
>> it instead extends over the excitation space of the mode.
>
> You're right, but so am I. There are two ways to represent a Fock
> space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
> each mode) and (2) as a sum of symmetrized (for bosons) tensor
> products of the space of wave functions on the set of modes. I like to
> call this equivalence the "fundamental theorem of second
> quantization". Perhaps this terminology will catch on some day. :-)
>
> You refer to method (1), while I referred to method (2).
>
> Igor

But in either case, the momentum vector p is associated *with the
photon*, giving what Arnold calls "a set of bais vectors in a momentum
eigenstates" *of the photon*, and as you say, "p are associated with the
momentum only in as much as they are the eigenvalues of the
corresponding operator observables, with exp(ipx) being the eigenvector"
*of the photon.*

In all cases, p is associated *with the photon*, and is NOT some
free-standing variable of integration. Yes, or no?

To which my next question is, if the answer is yes, then what happens in
*non-Abelian Gauge theory*? If, for a set of gauge bosons, the gauge
field:

G^u = T^i G_i^u? (1)

where T_i are the group generators, and G_i^u contains an internal
symmetry index i=1,2,3..N^2-1 for SU(N), then it looks like:

G_i^u = epsilon_i^u exp[ip^u x_u] (5)

where epsilon_i^u for a given i represents the polarization vector for
G_i^u. But, with what physical entity is p^u associated? Above, it is
associated with the photon, but here, there are N^2-1 gauge bosons. How
do we associate a single p^u with N^2-1 of the G_i^u? Which G_i^u has
its state described by the mode function exp(ipx)? Don't we need to
have a corresponding set of p_i^u, one for each gauge boson, to describe
the harmonic oscillator modes / spaces of that particular gauge boson?

Jay.

[Igor Khavkine]

On Jan 24, 8:52 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

> But in either case, the momentum vector p is associated *with the
> photon*, giving what Arnold calls "a set of bais vectors in a momentum
> eigenstates" *of the photon*, and as you say, "p are associated with the
> momentum only in as much as they are the eigenvalues of the
> corresponding operator observables, with exp(ipx) being the eigenvector"
> *of the photon.*
>
> In all cases, p is associated *with the photon*, and is NOT some
> free-standing variable of integration. Yes, or no?

Unfortunately, this is *not* a yes or no question (a false dichotomy).
To be honest, it is not even clear what the two alternatives you
present mean mathematically or physically. Mathematically, it is not
clear what you mean by either "associated" or "free-standing
variable". And why would "of integration" be an important qualifier,
as opposed to say "of differentiation"? Physically, AFAIK, neither
being "associated" nor being a "free-standing variable" are measurable
properties of photons or momenta.

> To which my next question is, if the answer is yes, then what happens in
> *non-Abelian Gauge theory*? If, for a set of gauge bosons, the gauge
> field:
>
> G^u = T^i G_i^u? (1)
>
> where T_i are the group generators, and G_i^u contains an internal
> symmetry index i=1,2,3..N^2-1 for SU(N), then it looks like:
>
> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>
> where epsilon_i^u for a given i represents the polarization vector for
> G_i^u.

So far so good.

> But, with what physical entity is p^u associated? Above, it is
> associated with the photon, but here, there are N^2-1 gauge bosons.

Again, that's neither a mathematical nor physical question, hence not
answerable to your satisfaction.

> How
> do we associate a single p^u with N^2-1 of the G_i^u? Which G_i^u has
> its state described by the mode function exp(ipx)?

Same problem with this question as described above. However, if you do
manage to meaningfully define "associated", you'll find that the
answer to "how?" is exactly the same as with E&M. The number N^2-1
doesn't play a role when it comes to the mode functions exp(ipx). That
number could be 1 or it could be a million.

> Don't we need to
> have a corresponding set of p_i^u, one for each gauge boson, to describe
> the harmonic oscillator modes / spaces of that particular gauge boson?

No. All the information you need is already summarized in your
equation (5). You seem to have the impression that there's something
important that can't be accomplished with equation (5). But you've yet
to formulate this impression in a coherent physically or even
mathematically meaningful way. Maybe you can try to do that in your
next reply.

Igor

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:803f913e-8f5c-43e3-8b09-b9bfb69d14bf@t39g2000prh.googlegroups.com...
> On Jan 24, 8:52 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
.. . .
>> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>>
>> where epsilon_i^u for a given i represents the polarization vector
>> for
>> G_i^u.
>
> So far so good.
>
.. . .
>You seem to have the impression that there's something
> important that can't be accomplished with equation (5). But you've yet
> to formulate this impression in a coherent physically or even
> mathematically meaningful way. Maybe you can try to do that in your
> next reply.
>
> Igor
>
Very fair, let me give this a try.

Without going through all the details of how this occurs (which I can
fill in later in detail if need be), let's consider SU(2) as the
simplest example of Yang Mills. We spontaneously break symmetry in the
usual way bringing a gauge boson mass term m^2 into the action.
Additionally, the (d^u.d_u)G^v terms in the action turn
into -(p^u.p_u)G^v terms, directly from the p^u which is part of the
"mode function" discussion above in this thread. Following path
integration, all of this -- the p^u.p_u and the m^2 -- eventually ends
up in the propagator (modulo factors of -i):

D_uv = (g_uv - p_u p_v / m^2)/(p^u p_u - m^2) (1)

and all of the p^u in the above directly originate in the p^u which is
part of the mode function exp[ip^u.x_u]. Again, if need be, I can show
this whole development in detail, but I want to answer your question as
directly and broadly as possible without taking ten pages to do so.

The above is taken to be the propagator *for the gauge boson* and the
p^2=p^u.p_u is taken to be the square four momentum *for the gauge
boson*. That is propagators, insofar as I understand, are "associated
with" individual field quanta. Or, at least, people often talk about
the "photon propagator" or the "vector boson propagator" or the "fermion
propagator" and unless that is some sloppy shorthand from something
else, the propagator, including the p_u which appears in the propagator
and which originates in the mode function exp(ipx), is "associated with"
individual particles / fields quanta. If the p_u is associated with a
field quantum in the propagator, and it originates in the mode function,
then I do not see how one can draw a sharp line between the p_u as it
appears in the mode function and the p_u as it ends up in the propagator
as a direct outgrowth of taking the path integral. The p^u still, it
seems to me, is associated with a field quantum and is the "four
momentum" of that field quantum, recognizing as you say that that
exp(ipx) is to be taken as an eigenvector.

Now, let me come to your statement that "You seem to have the impression
that there's something important that can't be accomplished with
equation (5)." Yes, I do. Let me lay it out the best I can, looking at
the "+i epsilon prescription" and "off-shell" particles, both of which I
have a quarrel with for reasons to be discussed, and both of which can
be avoided if the p^u in equation (5) is taken to be the Hermitian
matrix:

p^u = T^i p_i^u (2)

In particular, (1) above then must become the slightly modified:

D_uv = (g_uv - p_u p_v / m^2)x(p^u p_u - m^2)^-1 , (3)

and the matrix inverse (p^u.p_u-m^2)^-1 in (3) sidesteps +i epsilon and
off-shell particles. Particularly, in (2), if a particle is on shell
(as observable particles are thought to be),

p^u p_u = m^2 , (4)

then (1) hits a pole unless one engages in the *ad hoc expedient* of
adding a +i epsilon term to the denominator in (1) and thus introducing
an imaginary dimension (associated with particle half life) to sidestep
the pole in a complex plane rather than being forced to remain on a real
line.

Now, with all due respect to Weinberg's discussion of polology Section
10.2 that you referred me to several weeks ago, you may believe that
there is nothing wrong with doing this. However, I find it very
difficult to swallow that nature introduces +i epsilon in such an ad hoc
manner to sidestep a pole, *or that nature has poles which need to be
sidestepped in the first place*. More to the point: when p^u in the
mode function is taken "as is," rather than being defined as in (2), one
hits a pole for on-shell particles unless one introduces +i epsilon.
But when p^u is defined as in (2), which then yields an NxN "unitary"
mode function for SU(N):

exp(ip^u x_u) = exp(i T_i p_i^u x_u) (6)

*the poles never arise in the first place.* I can show the details
separately, but it is never necessary to introduce +i epsilon, because
on-shell particles never run into poles in the first instance. That is,
the need to avert poles never arises at all, because particles can stay
on mass shell without +i epsilon.

Now, physics is at bottom an effort to represent how nature behaves, and
derive results from those representations which accord precisely with
observed phenomenology. While I am admittedly about to make an appeal
to aesthetics, I find it hard to believe that nature herself really
encounters poles for on-shell particles, and then "adds" +i epsilon to
her propagator denominators in order to avoid the poles. All of this is
human calculation only -- which to my mind reflects an imperfection in
the way we understand nature at the present day, and not a reflection of
nature herself. With all things observational being equal (a
fundamental caveat), if one has a choice between two approaches -- one
where we hit a pole on shell and have to introduce + i epsilon as a
"patch" to fix this problem and render our theory calculable, and the
other where we never encounter on-shell poles in the first place -- it
seems to me that the approach that never hits the pole in the first
instance is to be preferred, and is far more likely by appeal to
physical aesthetics to be in accord with how nature herself would choose
to operate than an approach where one hits a pole and is required to
introduce an ad hoc term to avert the pole.

Now, for anyone who has never looked at the +i epsilon prescription and
said to themselves "that sure seems like an ad hoc solution," my
argument is likely to fall flat. But for anyone who has ever learned
about the +i epsilon and made a mental note that this seems like a
provisional patch that needs to be better understood and one day left
behind in the dustbin of scientific history, the prospect that a unitary
non-Abelian mode factor (6) can avoid +i epsilon and simultaneously
leave particles on shell, would seem to have a certain attractiveness
worthy of further exploration.

Of course, in the end -- back to our caveat -- the aesthetics must defer
to what is observed. So, if (6) leads to contradiction with
observation, than +i epsilon would have to prevail. So the entire
question becomes whether results flowing from (6) can be sustained
against measured experimental observation. I can show (and have shown
in previous posts) that for SU(2), an approach based on (6) is
absolutely consistent with observation of the W+/- vector bosons. It
has occurred to me in the past month that I should do a detailed
examination of SU(2)xU(1) electroweak theory, particularly to see what
happens to the photon propagator, because the SU(2)xU(1) symmetry is
broken to yield a *massless* photon (which is my next project).
Obviously, if I cannot reconcile this approach to a massless photon,
then that would be cause for "hanging it up." And for SU(3), there is
indeed "something important that can't be accomplished with equation
(5)":

If uses (6) for the mode function in SU(3), and places all gauge bosons
on shell after using Higgs-Kibble to spontaneously break symmetry, it
turns out the that theory becomes numerically *predictive*
of dozens of meson masses in the propagator denominators which ensure
after path integration, and that these masses do not need to be
introduced "by hand" or from observational data. Further, while I am
still developing this, the mass spectrum turns out in many way to
resemble that which is actually observed. And, if one takes SU(3) to
be, not color, but the old flavor model based on the states u,d,s, the
predicted ratio of the mass of the u-bar s (K) mesons to the mass of the
u-bar d (pi) mesons matches experimental observation to just over one
part in 1000 (0.1%). And, imaginary numbers enter the masses
automatically, which may provide a handle not only on particle masses,
but also on particle widths.

Does this means that I have "proved" that (6) ought to be employed as a
unitary mode function for non-Abelian gauge theory? No. But, I believe
there are enough positive aspects to this approach that is it worthy of
being explored and developed, to see if this in fact the route that
nature herself chooses.

Finally, I will point out that the approach I am advocating for is a
very minimalist, conservative approach. All the formalisms of
Yang-Mills theory remain unchanged, insofar as every appearance of a
four-momentum p^u remains unchanged. The only difference, is that in
Yang Mills, p^u is interpreted to be the Hermitian matrix T^i.p_i^u as
in (2), and so the mode functions exp(ip^u.x_u) are interpreted to be
unitary matrices exp(i.T_i.p_i^u.x_u) as in (6). With this simple,
minimalist change, particles can be kept on shell without giving rise to
poles or the need for +i epsilon, and larger groups such as SU(3),
following Higgs-Kibble symmetry breaking, become predictive in terms of
the meson masses which appear in the propagator denominators to which
they give rise following path integration.

Thanks for your comments, Igor, which are always helpful, challenging,
and thought provoking.

Jay.

[Igor Khavkine]

On Jan 27, 1:58*pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:803f913e-8f5c-43e3-8b09-b9bfb69d14bf@t39g2000prh.googlegroups.com..
> > On Jan 24, 8:52 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> . . .
> >> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>
> >> where epsilon_i^u for a given i represents the polarization vector
> >> for G_i^u.
>
> > So far so good.
> . . .
> > You seem to have the impression that there's something
> > important that can't be accomplished with equation (5). But you've yet
> > to formulate this impression in a coherent physically or even
> > mathematically meaningful way. Maybe you can try to do that in your
> > next reply.

[...]
> Now, let me come to your statement that "You seem to have the impression
> that there's something important that can't be accomplished with
> equation (5)." *Yes, I do. *Let me lay it out the best I can, looking at
> the "+i epsilon prescription" and "off-shell" particles, both of which I
> have a quarrel with for reasons to be discussed, and both of which can
> be avoided if the p^u in equation (5) is taken to be the Hermitian
> matrix:
>
> p^u = T^i p_i^u *(2)

OK, IIRC, summarized above is your view of what is wrong with equation
(5) and why you think equation (2) should hold instead. The rest of
your post was a bit long winded, so I haven't read through it
carefully, although I may at a later point.

Paraphrasing a bit, in my understanding, you are making two
statements: (a) that equation (5) can't be right since it leads to the
introduction of the "+i epsilon prescription" and (b) that equation
(2) must be right because it leads to what you consider desirable
physical consequences.

Unfortunately, you are wrong in both statements. Your first statement
is wrong because, as I've demonstrated in other posts, it is a simple
consequence of the standard calculations in field theory. Your second
statement is wrong because equation (2), even by your own account, is
incompatible with equation (5). This also implies that the
consequences of (2) that you find physically desirable are actually
not.

Equally unfortunate is the fact that you seem to have convinced
yourself of the verity of statements (a) and (b) only with the help of
two common logical fallacies: argument from ignorance (again, that's a
technical term) and argument from final consequence.

You appear to like the idea of avoiding using the "+i epsilon
prescription" and because equation (5) leads to this prescription, you
reject it. However, wishing something to be true (or false) does not
make it so. This is the argument from final consequence fallacy. For
whatever reason, you have cooked up equation (2) as a substitute to
equation (5) and argue it to be true because you don't know of a
better way to avoid using or to explain the "+i epsilon prescription".
But, just because *you* don't have a good explanation for the "+i
epsilon prescription" does not mean that one doesn't exist. That's the
argument from ignorance.

Now, I think, the core of the problem here is that you are not
comfortable with using the "+i epsilon prescription" for the momentum
space expression of the propagator. That's perfectly understandable,
as this is a subtle mathematical point that is often glossed over QFT
texts and lectures. However, if you are interested, I can give you
some elementary exercises for the simple harmonic oscillator that will
make you more comfortable with the "+i epsilon prescription", explain
why it's used, why it's +epsilon and not -epsilon, and any other
questions you might have. I believe that would be a much more
productive use of your time than the alternatives you've been
considering.

Igor

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:7f2dd2a9-15af-49fd-94f0-f2b0f65e08db@v5g2000prm.googlegroups.com...
> On Jan 27, 1:58 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> "Igor Khavkine" <igor...@gmail.com> wrote in message
>>
>> news:803f913e-8f5c-43e3-8b09-b9bfb69d14bf@t39g2000prh.googlegroups.com..
>> > On Jan 24, 8:52 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> . . .
>> >> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>>
>> >> where epsilon_i^u for a given i represents the polarization vector
>> >> for G_i^u.
>>
>> > So far so good.
>> . . .
>> > You seem to have the impression that there's something
>> > important that can't be accomplished with equation (5). But you've
>> > yet
>> > to formulate this impression in a coherent physically or even
>> > mathematically meaningful way. Maybe you can try to do that in your
>> > next reply.
>
> [...]
>> Now, let me come to your statement that "You seem to have the
>> impression
>> that there's something important that can't be accomplished with
>> equation (5)." Yes, I do. Let me lay it out the best I can, looking
>> at
>> the "+i epsilon prescription" and "off-shell" particles, both of
>> which I
>> have a quarrel with for reasons to be discussed, and both of which
>> can
>> be avoided if the p^u in equation (5) is taken to be the Hermitian
>> matrix:
>>
>> p^u = T^i p_i^u (2)
>
> OK, IIRC, summarized above is your view of what is wrong with equation
> (5) and why you think equation (2) should hold instead. The rest of
> your post was a bit long winded, so I haven't read through it
> carefully, although I may at a later point.
.. . .

I just submitted a new post in reply, so I do not know the order in
which they will come through. But to supplement that first reply, let
me take a bit different approach.

Rather than continue into what is now our almost two month discussion of
(2) versus (5) above, let me take this from a different angle.

Suppose, just for the sake of argument, that one was to utilize a non
Abelian mode function:

exp[i p^u x_u] (1)

where:

p^u = T^i p_i^u (2)

and the T^i are non-Abelian group generators. Forgetting "argument from
final consequence," or whether we think is is or is not a good idea,
let's just ask the question, what would that mean, mathematically and
physically, on its very own terms? Can this be understood in a context
that makes both mathematical and physical sense, given our present
pedagogical approaches to the standard techniques we use to do
calculation in field theory?

More to the point, what does it even mean to have a non-Abelian mode
function, on its own terms, as you say elsewhere, "in a coherent
physically or even mathematically meaningful way"?

I have tried to address these questions in the brief set of calculations
linked below:

http://jayryablon.files.wordpress.com/2008/12/non-abelian-mode-functions.pdf

Thanks,

Jay.

[p.kinsler@ic.ac.uk]

Igor Khavkine <igor.kh@gmail.com> wrote:
> You're right, but so am I. There are two ways to represent a Fock
> space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
> each mode) and (2) as a sum of symmetrized (for bosons) tensor
> products of the space of wave functions on the set of modes. I like to
> call this equivalence the "fundamental theorem of second
> quantization". Perhaps this terminology will catch on some day. :-)

> You refer to method (1), while I referred to method (2).

OK. Is there any work you can cite that uses (2) to do
some sort of generic quantum optical calculation -- e.g.
excitation of a two-level atom in a cavity mode? I'd like
to see how it works in practice.

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (PHOT) (ph) +44-20-759-47734 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2AZ, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:7f2dd2a9-15af-49fd-94f0-f2b0f65e08db@v5g2000prm.googlegroups.com...
> On Jan 27, 1:58 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> "Igor Khavkine" <igor...@gmail.com> wrote in message
>>
>> news:803f913e-8f5c-43e3-8b09-b9bfb69d14bf@t39g2000prh.googlegroups.com..
>> > On Jan 24, 8:52 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> . . .
>> >> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>>
>> >> where epsilon_i^u for a given i represents the polarization vector
>> >> for G_i^u.
>>
>> > So far so good.
>> . . .
>> > You seem to have the impression that there's something
>> > important that can't be accomplished with equation (5). But you've
>> > yet
>> > to formulate this impression in a coherent physically or even
>> > mathematically meaningful way. Maybe you can try to do that in your
>> > next reply.
>
> [...]
>> Now, let me come to your statement that "You seem to have the
>> impression
>> that there's something important that can't be accomplished with
>> equation (5)." Yes, I do. Let me lay it out the best I can, looking
>> at
>> the "+i epsilon prescription" and "off-shell" particles, both of
>> which I
>> have a quarrel with for reasons to be discussed, and both of which
>> can
>> be avoided if the p^u in equation (5) is taken to be the Hermitian
>> matrix:
>>
>> p^u = T^i p_i^u (2)
>
> OK, IIRC, summarized above is your view of what is wrong with equation
> (5) and why you think equation (2) should hold instead. The rest of
> your post was a bit long winded, so I haven't read through it
> carefully, although I may at a later point.

I will certainly appreciate that if you should do so.

> Paraphrasing a bit, in my understanding, you are making two
> statements: (a) that equation (5) can't be right since it leads to the
> introduction of the "+i epsilon prescription" and (b) that equation
> (2) must be right because it leads to what you consider desirable
> physical consequences.

Let me clarify a bit more. The "+i epsilon prescription" is the symptom
not the problem. The problem is that (5), when a particle is on shell,
leads to singular behavior which them motivates +i epsilon. With
equation (2), the singular behavior never results in the first place. .

[p.kinsler@ic.ac.uk]

Jay R. Yablon <jyablon@nycap.rr.com> wrote:
> > You're right, but so am I. There are two ways to represent a Fock
> > space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
> > each mode) and (2) as a sum of symmetrized (for bosons) tensor
> > products of the space of wave functions on the set of modes. I like to
> > call this equivalence the "fundamental theorem of second
> > quantization". Perhaps this terminology will catch on some day. :-)
> >
> > You refer to method (1), while I referred to method (2).

> But in either case, the momentum vector p is associated *with the
> photon*, [...]

I emphatically disagree. In case (1), the the photon is associated
with[1] the mode; the possible momenta are parameters of the mode
functions.

The momentum is not an intrinsic property of this or that photon,
it's an intrinsic property of the mode which photon(s) may happen
to inhabit.

Statements such as "p is associated with the photon" yours omits
the fundamental role of the mode in the description, and therefore
highly misleading.

[1] i.e. is an excitation of

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (PHOT) (ph) +44-20-759-47734 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2AZ, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/

[Igor Khavkine]

Jay R. Yablon previously wrote:

> >>>> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
> >>>> where epsilon_i^u for a given i represents the polarization vector
> >>>> for G_i^u.
[...]
> >> p^u = T^i p_i^u (2)

On Jan 29, 10:59 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:7f2dd2a9-15af-49fd-94f0-f2b0f65e08db@v5g2000prm.googlegroups.com...

> > Paraphrasing a bit, in my understanding, you are making two
> > statements: (a) that equation (5) can't be right since it leads to the
> > introduction of the "+i epsilon prescription" and (b) that equation
> > (2) must be right because it leads to what you consider desirable
> > physical consequences.
>
> Let me clarify a bit more. The "+i epsilon prescription" is the symptom
> not the problem. The problem is that (5), when a particle is on shell,
> leads to singular behavior which them motivates +i epsilon. With
> equation (2), the singular behavior never results in the first place. .

I'm sorry to say, but this kind of statement only confirms that you
don't understand the "+i epsilon prescription". It is neither a
symptom nor a problem in the sense that you mean. My offer to help you
get comfortable with it still stands.

As to what (2) means mathematically, I have no idea. But I can tell
you that unless whatever way you introduce (2) is equivalent to
standard calculations (singularities in the propagator and all), it's
not real physics and has no physical meaning whatsoever.

Igor

[Igor Khavkine]

On Jan 29, 10:50 pm, p.kins...@ic.ac.uk wrote:
> Igor Khavkine <igor...@gmail.com> wrote:
> > You're right, but so am I. There are two ways to represent a Fock
> > space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
> > each mode) and (2) as a sum of symmetrized (for bosons) tensor
> > products of the space of wave functions on the set of modes. I like to
> > call this equivalence the "fundamental theorem of second
> > quantization". Perhaps this terminology will catch on some day. :-)
> > You refer to method (1), while I referred to method (2).
>
> OK. Is there any work you can cite that uses (2) to do
> some sort of generic quantum optical calculation -- e.g.
> excitation of a two-level atom in a cavity mode? I'd like
> to see how it works in practice.

The reference which in my opinion demonstrates the equivalence between
pictures (1) and (2) in a clear and direct way is Dirac's _The
Principles of Quantum Mechanics_. Specifically sections 59-65 treat
the two different ways of representing Fock space for both bosons and
fermions, with photons as a specific example.

Unfortunately, my knowledge of the quantum optics literature is sorely
lacking. So I don't know of a specific reference that treats a
standard problem in quantum optics, as you suggest, using the
symmetrized wave function representation as opposed to the oscillator
occupation number representation. However, it's straightforward to
change from one representation to the other, as described for instance
in Dirac's book. Then any standard calculation is convertible from
picture (1) to (2). In the process, the notation may become more
cumbersome than usual, which might explain the prevalence of picture
(1) in such calculations.

BTW, if you are willing to post a concise version of a standard
quantum optical calculation, done in the way you are familiar with,
then I can try to convert it to a different picture.

Mahan's book _Many Particle Physics_ also has a very clear and concise
explanation of second quantization in the context of condensed matter
theory. He doesn't treat photons in the manner of (2), but he does
that for phonons and electrons.

Also, the last chapter of Weinberg's _Quantum Theory of Fields v.I_
explicitly shows how to recover wave functions from field theory.
Though he's talking about electron wave functions, rather than photon
ones, his discussion may be generalized.

Hope this helps.

Igor

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:6b1daf0f-4136-489d-ba67-4380218162a0@a39g2000prl.googlegroups.com...
> Jay R. Yablon previously wrote:
>
>> >>>> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
>> >>>> where epsilon_i^u for a given i represents the polarization
>> >>>> vector
>> >>>> for G_i^u.
> [...]
>> >> p^u = T^i p_i^u (2)
>
> On Jan 29, 10:59 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> "Igor Khavkine" <igor...@gmail.com> wrote in message
>> news:7f2dd2a9-15af-49fd-94f0-f2b0f65e08db@v5g2000prm.googlegroups.com...
>
>> > Paraphrasing a bit, in my understanding, you are making two
>> > statements: (a) that equation (5) can't be right since it leads to
>> > the
>> > introduction of the "+i epsilon prescription" and (b) that equation
>> > (2) must be right because it leads to what you consider desirable
>> > physical consequences.
>>
>> Let me clarify a bit more. The "+i epsilon prescription" is the
>> symptom
>> not the problem. The problem is that (5), when a particle is on
>> shell,
>> leads to singular behavior which them motivates +i epsilon. With
>> equation (2), the singular behavior never results in the first place.
>> .
>
> I'm sorry to say, but this kind of statement only confirms that you
> don't understand the "+i epsilon prescription". It is neither a
> symptom nor a problem in the sense that you mean. My offer to help you
> get comfortable with it still stands.
>
> As to what (2) means mathematically, I have no idea. But I can tell
> you that unless whatever way you introduce (2) is equivalent to
> standard calculations (singularities in the propagator and all), it's
> not real physics and has no physical meaning whatsoever.
>
> Igor
>
Thanks Igor,

Two things, with more to follow as time permits.

1) I would be pleased to take you up on your offer to develop a comfort
with +i epsilon, and thank you for the offer. I will need to return to
that once I finish working through the present discussion.

2) As a result of our discussions together with Paul Kinsler's latest
post, I have undergone a change of view. I now agree that it does not
make sense to try to associate p_i^u with individual gauge bosons any
more than to associate p^u with an individual photon. I have dropped
the notion of such an association entirely. But, I am still keeping
p_i^u and p^u=T^i.p_i^u, and I'll now explain how.

You say "As to what (2) means mathematically, I have no idea." I think
I now do have an idea. This all has to do with the delta function. The
file which I separately posted at
http://jayryablon.files.wordpress.com/2008/12/non-abelian-mode-functions.pdf
has the basic idea, though equation (4) is wrong because that only
applies to commuting exponents. This equation can be corrected (and the
calculation fixed as I have now done but need to write up) and it does
lead to a delta function which, in the case of SU(2) isospin -- which
has 3 generators and an associated O(3) geometric space of internal
symmetry -- is found to be:

I_(2x2)delta^(4x3)(x_u-y_u)=$d^(4x3)exp[i 4pi T^i p_i^u(x_u-y_u)] (1)

where I_(2x2) is a 2x2 identity matrix, delta^(4x3)(x_u-y_u) is a delta
function defined over four spacetime dimensions and three internal
symmetry dimensions, $ is an integral from -oo to +oo, d^(4x3) is the
differential element taken over over four spacetime dimensions and three
internal symmetry dimensions, and exp[i.4pi.T^i.p_i^u(x_u-y_u)] is the
factor we have been talking about for the past two months which includes
p_i^u and p^u=T^i.p_i^u.

Delta functions are used to define densities in spaces.
Delta^4(x_u-y_u) helps define densities in 4-dimensional spacetime. The
internal symmetry space of SU(2), which is the simplest example we can
use, since it has an associated O(3) space, can also be thought to have
density functions. More to the point: if the O(3) space of our everyday
experience can have densities, than the O(3) "isospace" of SU(2) isospin
can also have densities. If we consider the left side of equation (1)
to be something that helps specify densities both in spacetime and in
internal symmetry space (physical quantity per unit of spacetime per
unit of internal symmetry space), then it can be shown that the right
hand side is mathematically equivalent to the left hand side as I will
shortly do.

So, we can use p_i^u in a way that makes mathematical sense to define
densities in the internal symmetry space, but the p_i^u have nothing to
do with the momentum of the gauge bosons. Rather, they are used as
variables of integration on the right side of (1), to specify density
functions in internal symmetry space. Of course, you may ask why one
needs to take densities in internal symmetry space, but I don't think
you can say that such an idea is any longer mathematically undefined.

I also wonder if the very fact that (1) does contain singular density
functions (infinitely high and thin with an area of 1), might satisfy at
least in part your statement that "whatever way you introduce (2) [must
be] equivalent to standard calculations (singularities in the propagator
and all)?

Thanks,

Jay.

[Jay R. Yablon]

"Jay R. Yablon" <jyablon@nycap.rr.com> wrote in message
news:6uibj1Ffhsp6U1@mid.individual.net...
.. . .
> You say "As to what (2) means mathematically, I have no idea." I
> think
> I now do have an idea. This all has to do with the delta function.
> The
> file which I separately posted at
> http://jayryablon.files.wordpress.com/2008/12/non-abelian-mode-functions.pdf
> has the basic idea, though equation (4) is wrong because that only
> applies to commuting exponents. This equation can be corrected (and
> the
> calculation fixed as I have now done but need to write up) and it does
> lead to a delta function which, in the case of SU(2) isospin -- which
> has 3 generators and an associated O(3) geometric space of internal
> symmetry -- is found to be:
>
> I_(2x2)delta^(4x3)(x_u-y_u)=$d^(4x3)exp[i 4pi T^i p_i^u(x_u-y_u)] (1)
>
> where I_(2x2) is a 2x2 identity matrix, delta^(4x3)(x_u-y_u) is a
> delta
> function defined over four spacetime dimensions and three internal
> symmetry dimensions, $ is an integral from -oo to +oo, d^(4x3) is the
> differential element taken over over four spacetime dimensions and
> three
> internal symmetry dimensions, and exp[i.4pi.T^i.p_i^u(x_u-y_u)] is the
> factor we have been talking about for the past two months which
> includes
> p_i^u and p^u=T^i.p_i^u.
.. . .
I have now completed and posted write-up for the corrected derivation of
the above, and posted it at:

http://jayryablon.files.wordpress.com/2009/01/non-abelian-mode-functions-2.pdf

Jay.

[Igor Khavkine]

Jay R. Yablon previously wrote:

> >>>> G_i^u = epsilon_i^u exp[ip^u x_u] (5)
> >>>> where epsilon_i^u for a given i represents the polarization vector
> >>>> for G_i^u.
[...]
> >> p^u = T^i p_i^u (2)

On Jan 29, 10:59 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:7f2dd2a9-15af-49fd-94f0-f2b0f65e08db@v5g2000prm.googlegroups.com...

> > Paraphrasing a bit, in my understanding, you are making two
> > statements: (a) that equation (5) can't be right since it leads to the
> > introduction of the "+i epsilon prescription" and (b) that equation
> > (2) must be right because it leads to what you consider desirable
> > physical consequences.
>
> Let me clarify a bit more. The "+i epsilon prescription" is the symptom
> not the problem. The problem is that (5), when a particle is on shell,
> leads to singular behavior which them motivates +i epsilon. With
> equation (2), the singular behavior never results in the first place. .

I'm sorry to say, but this kind of statement only confirms that you
don't understand the "+i epsilon prescription". It is neither a
symptom nor a problem in the sense that you mean. My offer to help you
get comfortable with it still stands.

As to what (2) means mathematically, I have no idea. But I can tell
you that unless whatever way you introduce (2) is equivalent to
standard calculations (singularities in the propagator and all), it's
not real physics and has no physical meaning whatsoever.

Igor

[Igor Khavkine]

On Jan 29, 10:50 pm, p.kins...@ic.ac.uk wrote:
> Igor Khavkine <igor...@gmail.com> wrote:
> > You're right, but so am I. There are two ways to represent a Fock
> > space: (1) as a sum of harmonic oscillator Hilbert spaces (one for
> > each mode) and (2) as a sum of symmetrized (for bosons) tensor
> > products of the space of wave functions on the set of modes. I like to
> > call this equivalence the "fundamental theorem of second
> > quantization". Perhaps this terminology will catch on some day. :-)
> > You refer to method (1), while I referred to method (2).
>
> OK. Is there any work you can cite that uses (2) to do
> some sort of generic quantum optical calculation -- e.g.
> excitation of a two-level atom in a cavity mode? I'd like
> to see how it works in practice.

The reference which in my opinion demonstrates the equivalence between
pictures (1) and (2) in a clear and direct way is Dirac's _The
Principles of Quantum Mechanics_. Specifically sections 59-65 treat
the two different ways of representing Fock space for both bosons and
fermions, with photons as a specific example.

Unfortunately, my knowledge of the quantum optics literature is sorely
lacking. So I don't know of a specific reference that treats a
standard problem in quantum optics, as you suggest, using the
symmetrized wave function representation as opposed to the oscillator
occupation number representation. However, it's straightforward to
change from one representation to the other, as described for instance
in Dirac's book. Then any standard calculation is convertible from
picture (1) to (2). In the process, the notation may become more
cumbersome than usual, which might explain the prevalence of picture
(1) in such calculations.

BTW, if you are willing to post a concise version of a standard
quantum optical calculation, done in the way you are familiar with,
then I can try to convert it to a different picture.

Mahan's book _Many Particle Physics_ also has a very clear and concise
explanation of second quantization in the context of condensed matter
theory. He doesn't treat photons in the manner of (2), but he does
that for phonons and electrons.

Also, the last chapter of Weinberg's _Quantum Theory of Fields v.I_
explicitly shows how to recover wave functions from field theory.
Though he's talking about electron wave functions, rather than photon
ones, his discussion may be generalized.

Hope this helps.

Igor

[Jay R. Yablon]

What I had posted to 2/1/2009 at 6:46 PM had a minor typographical error
flagged by kp in spf, where the matrices were transposed in (3)-(6) and
(9). This is corrected in:

http://jayryablon.files.wordpress.com/2009/02/non-abelian-mode-functions-3.pdf

I also added an new paragraph at the end further tying the non-Abelian
four-momenta to internal symmetry space densities. (Per my earlier
post, Igor and Paul both finally convinced me that my original effort to
associate these with gauge bosons momenta was not correct and I have
abandoned that association, nor is this association necessary to justify
continuing to use these non-Abelian momenta.)

Thanks,

Jay.

[Igor Khavkine]

On Feb 1, 5:05 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

> 1) I would be pleased to take you up on your offer to develop a comfort
> with +i epsilon, and thank you for the offer. I will need to return to
> that once I finish working through the present discussion.

My suggestion would be to give these exercises a higher priority than
other topics covered in this thread.

Here's the equation for a simple harmonic oscillator with frequency w
and a force term:

(eq.1) x''(t) + w^2 x(t) = f(t) .

This equation can be specialized for the free oscillation
(homogeneous)
case

(eq.2) x''(t) + w^2 x(t) = 0 ,

and for the impulsive force case

(eq.3) x''(t) + w^2 x(t) = delta(t) .

Here are some calculations to get started.

(ex.1) The solutions of the differential equation (eq.2) form a vector
space. What is its dimension? (Hint: it's 2. Why?) Find a basis
for the solution space.

(ex.2) What is the dimension of the space of solutions of the now
inhomogeneous (eq.3)? (Hint: adding any homogeneous solution
to
an inhomogeneous solution, is still a solution.)

(ex.3) Characterize the solutions of (eq.3) near t=0. (Hint: assume
x(t) is continuous at t=0; what can you say about the
continuity
of x'(t)?)

(ex.4) Find the unique solution of (eq.3) with the boundary conditions
(R) x(t) = 0 for t <= 0, (A) x(t) = 0 for t >= 0,
(F) x(t) = exp(iwt)/(2w) for t >= 0, (AF) x(t) = exp(-iwt)/(2w)
for t >= 0. Call these solutions G_R(t), G_A(t), G_F(t), and
G_AF(t), respectively.

These are fairly elementary exercises. But it's good to get the
fundamentals out of the way before moving on to something more
complex.

The next batch of exercises will involve taking Fourier transforms of
(eq.1) and of the various solutions specified above.

> 2) As a result of our discussions together with Paul Kinsler's latest
> post, I have undergone a change of view. I now agree that it does not
> make sense to try to associate p_i^u with individual gauge bosons any
> more than to associate p^u with an individual photon. I have dropped
> the notion of such an association entirely. But, I am still keeping
> p_i^u and p^u=T^i.p_i^u, and I'll now explain how. [...]

That's too bad; the very last part, that is.

> I also wonder if the very fact that (1) does contain singular density
> functions (infinitely high and thin with an area of 1), might satisfy at
> least in part your statement that "whatever way you introduce (2) [must
> be] equivalent to standard calculations (singularities in the propagator
> and all)"?

Not even close. :-(

Igor

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:db12cbd6-54f5-42dd-92a4-8777f13ede99@t26g2000prh.googlegroups.com...
> On Feb 1, 5:05 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>
>> 1) I would be pleased to take you up on your offer to develop a
>> comfort
>> with +i epsilon, and thank you for the offer. I will need to return
>> to
>> that once I finish working through the present discussion.
>
> My suggestion would be to give these exercises a higher priority than
> other topics covered in this thread.
>
> Here's the equation for a simple harmonic oscillator with frequency w
> and a force term:
>
> (eq.1) x''(t) + w^2 x(t) = f(t) .
>
> This equation can be specialized for the free oscillation
> (homogeneous)
> case
>
> (eq.2) x''(t) + w^2 x(t) = 0 ,
>
> and for the impulsive force case
>
> (eq.3) x''(t) + w^2 x(t) = delta(t) .
>
> Here are some calculations to get started.
>
> (ex.1) The solutions of the differential equation (eq.2) form a vector
> space. What is its dimension? (Hint: it's 2. Why?) Find a basis
> for the solution space.

OK, Igor, let's give it a try. This is my first physics "exam" since
the 1970s, so I hope this old geezer is not too rusty ;-):

The solution to (2) has the general form

x(t) = A cos wt + B sin wt (2a)

Because A and B may be independently chosen, this is a two dimensional
vector space. Because A and B can each be complex numbers consistent
with (1), this may actually be a two-dimensional complex space with four
degrees of freedom total, two per dimension. (Which I think the math
gurus would say has bears a relationship to (maybe is?) a Hilbert
space?) We may wish to represent this as:

(A Cos wt , B sin wt )

We can then use the basis vector:

u_1 = (A,0); u_2 = (0,B) (2b)

and then write:

(A Cos wt , B sin wt ) = u_1 cos wt + u_2 sin wt (2c)

> (ex.2) What is the dimension of the space of solutions of the now
> inhomogeneous (eq.3)? (Hint: adding any homogeneous solution
> to
> an inhomogeneous solution, is still a solution.)

I take it that delta(t) is an "ideal impulse." This means that prior to
t=0 the oscillator will be oscillating along at some A cos wt + B sin
wt. Then, the impluse occurs which causes an instantanous "damping" or
"enhancement" (opposite of damping whatever word that is) to the motion.
After the impulse, A-->A' and / or B-->B', so there is a new
2-dimensional vector space:

u'_1 = (A',0); u'_2 = (0,B') (3a)

I would guess that if one puts t<0 together with t>0, it may be that
this is a 2+2=4 dimensional vector space. But I would tend to think
that it remains a two dimensional space, with a different basis brought
about by the force f(t)=delta(t), which more generally, is a source
term. I'd say, the force / impulse / source changes the basis of the
2-D solution space. (To be dimensionally pedantic in (1), your f(t)
should really be an acceleration a(t), or, (1) should have a mass m and
be mx''(t) + mw^2 x(t) = f(t). The former is in the nature of
gravitational acceleration, the latter in the nature of a acceleration
under force.)

>
> (ex.3) Characterize the solutions of (eq.3) near t=0. (Hint: assume
> x(t) is continuous at t=0; what can you say about the
> continuity
> of x'(t)?)

In (3), delta(t) is in units of acceleration = d/t^2. x'(t) at t=0 is
discontinuous, because the velocity instantaneously changes down or up
by 1 unit of velocity:

$delta(t)dt=1 (3b)

based on the impulse (and whether it is against (damping) or with
(enhancing) the oscillation at the time of impulse).

>
> (ex.4) Find the unique solution of (eq.3) with the boundary conditions
> (R) x(t) = 0 for t <= 0, (A) x(t) = 0 for t >= 0,
> (F) x(t) = exp(iwt)/(2w) for t >= 0, (AF) x(t) = exp(-iwt)/(2w)
> for t >= 0. Call these solutions G_R(t), G_A(t), G_F(t), and
> G_AF(t), respectively.

G_R(t):

x'(t) jumps from 0 to 1 at t=0, and just after t=0, (3) reverts to (2).
So the solution is:

x'(t) = 1 cos (wt)

(1 from the impulse integral) thus

G_R(t) = sin (wt) + 0 (0 from the the boundary conditions) for t>0


G_A(t):

Just flip G_R(t) about the t=0 axis:

G_A(t) = -sin (-wt) for t<0


G_AF(t): (I''ll do this first)

The implulse adds +cos(wt) to x'(t) at t=0. So the solution becomes:

G_AF(t) = x(t) = exp(-iwt)/(2w) + sin(wt)/w


G_F(t):

The implulse adds +cos(wt) to x'(t) at t=0 so for t<0 we subtract
cos(wt). Thus, refferring to G_AF(t):

G_F(t) = exp(-iwt)/(2w) - sin(wt)/w

In all of these cases, the impulse only affects one of the dimensions in
the vector space, and for the affected dimension, only the real degree
of freedom. I'll need to think about the more general case, but don't
want to hold up this reply while I do so.

Jay.

>
> These are fairly elementary exercises. But it's good to get the
> fundamentals out of the way before moving on to something more
> complex.
>
> The next batch of exercises will involve taking Fourier transforms of
> (eq.1) and of the various solutions specified above.
>

[Igor Khavkine]

On Feb 4, 12:37 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:db12cbd6-54f5-42dd-92a4-8777f13ede99@t26g2000prh.googlegroups.com...

> > Here's the equation for a simple harmonic oscillator with frequency w
> > and a force term:
>
> > (eq.1) x''(t) + w^2 x(t) = f(t) .
>
> > This equation can be specialized for the free oscillation
> > (homogeneous) case
>
> > (eq.2) x''(t) + w^2 x(t) = 0 ,
>
> > and for the impulsive force case
>
> > (eq.3) x''(t) + w^2 x(t) = delta(t) .
>
> > Here are some calculations to get started.
>
> > (ex.1) The solutions of the differential equation (eq.2) form a vector
> > space. What is its dimension? (Hint: it's 2. Why?) Find a basis
> > for the solution space.
>
> OK, Igor, let's give it a try. This is my first physics "exam" since
> the 1970s, so I hope this old geezer is not too rusty ;-):

Think of it more as learning by doing. :-)

> The solution to (2) has the general form
>
> x(t) = A cos wt + B sin wt (2a)
>
> Because A and B may be independently chosen, this is a two dimensional
> vector space. Because A and B can each be complex numbers consistent
> with (1), this may actually be a two-dimensional complex space with four
> degrees of freedom total, two per dimension. (Which I think the math
> gurus would say has bears a relationship to (maybe is?) a Hilbert
> space?)

Lets allow the coefficients A and B to be complex. So, for
definiteness, the solution space is a 2-dimensional complex vector
space.

> We may wish to represent this as:
>
> (A Cos wt , B sin wt )
>
> We can then use the basis vector:
>
> u_1 = (A,0); u_2 = (0,B) (2b)
>
> and then write:
>
> (A Cos wt , B sin wt ) = u_1 cos wt + u_2 sin wt (2c)

To be slightly pedantic, it is cos(wt) and sin(wt) that you want to
call your basis vectors. Say, u_1 = cos(wt) and u_2 = sin(wt). Then
an arbitrary solution is

A cos wt + B cos wt = A u_1 + B u_2

is a linear combination of the two basis vectors u_1 and u_2, with
possibly complex coefficients A and B.

> > (ex.2) What is the dimension of the space of solutions of the now
> > inhomogeneous (eq.3)? (Hint: adding any homogeneous solution
> > to an inhomogeneous solution, is still a solution.)
>
> I take it that delta(t) is an "ideal impulse." This means that prior to
> t=0 the oscillator will be oscillating along at some A cos wt + B sin
> wt. Then, the impluse occurs which causes an instantanous "damping" or
> "enhancement" (opposite of damping whatever word that is) to the motion.
> After the impulse, A-->A' and / or B-->B', so there is a new
> 2-dimensional vector space:
>
> u'_1 = (A',0); u'_2 = (0,B') (3a)
>
> I would guess that if one puts t<0 together with t>0, it may be that
> this is a 2+2=4 dimensional vector space. But I would tend to think
> that it remains a two dimensional space, with a different basis brought
> about by the force f(t)=delta(t), which more generally, is a source
> term. I'd say, the force / impulse / source changes the basis of the
> 2-D solution space.

OK, here's where you are starting to get into trouble. The dimension
remains 2, as the parts of the solution for t<0 and t>0 are not
completely independent. Let me first get this subtle point out of the
way before moving on to the next exercise.

The space of solutions to (eq.1) when f(t) is not 0 is no longer a
vector space. If you take two solutions x_1(t) and x_2(t) of (eq.1),
then their sum will not be a solution of the same equation; it will
satisfy equation, but with f(t) replaced by 2f(t). However, adding a
solution x_0(t) of the homogeneous (eq.2) to either x_1(t) or x_2(t)
will yield, a solution of the same (eq.1). Technically speaking, the
solutions of (eq.1) form an affine space (it doesn't contain 0),
rather than a vector space. Any solution of (eq.1) can be represented
by picking some particular solution x_p(t) of (eq.1) and adding to it
some homogeneous solution of (eq.2). Thus, the space of solutions is
still of dimension 2 and can be parametrized as

(eq.1b) x(t) = x_p(t) + A cos wt + B sin wt .

> > (ex.3) Characterize the solutions of (eq.3) near t=0. (Hint: assume
> > x(t) is continuous at t=0; what can you say about the
> > continuity of x'(t)?)
>
> In (3), delta(t) is in units of acceleration = d/t^2. x'(t) at t=0 is
> discontinuous, because the velocity instantaneously changes down or up
> by 1 unit of velocity:
>
> $delta(t)dt=1 (3b)
>
> based on the impulse (and whether it is against (damping) or with
> (enhancing) the oscillation at the time of impulse).

Your answer is essentially correct. However x'(t) doesn't change up or
down. As (eq.3) is written it always changes "up". The impulse added
is
in the positive direction.

Here's my "official" solution. It's not significantly different,
except
in the amount of detail, there for whoever is interested.

Integrate both sides of (eq.3) from t=-eps to t=+eps,
for some small eps > 0:

(eq.3b) int_(-eps)^(+eps) [x''(t) + w^2 x(t)] dt = 1 .

The 1 on the RHS comes from integrating over delta(t). Assuming that
x(t) is continuous at t=0, the second term on the LHS will be of order
2*eps*w^2*x(0), which -> 0 as eps -> 0. The first term on the LHS can
be integrated explicitly and yields in the limit eps -> 0

(eq.3c) x'(+0) - x'(-0) = 1 .

This means that the velocity, x'(t), will be discontinuous at t=0 and
the jump in the value of the velocity from t<0 (that's x'(-0)) to t>0
(that's x'(+0)) will be 1. Visually, you will see this as a kink in
the
graph of the solution x(t) at t=0. The difference of the slopes on
either side of the kink will be 1.

To reason about this situation physically, imagine a mass on a spring
that's suddenly hit by baseball bat at t=0, which imparts one unit of
velocity to the mass, in the positive direction. While the position of
the mass does not change abruptly (it does not instantaneously jump
from one place to another), its velocity does. This is equivalent to
the continuity of x(t) and the jump discontinuity of x'(t) at t=0.

> > (ex.4) Find the unique solution of (eq.3) with the boundary conditions
> > (R) x(t) = 0 for t <= 0, (A) x(t) = 0 for t >= 0,
> > (F) x(t) = exp(iwt)/(2w) for t >= 0, (AF) x(t) = exp(-iwt)/(2w)
> > for t >= 0. Call these solutions G_R(t), G_A(t), G_F(t), and
> > G_AF(t), respectively.
>
> G_R(t):
>
> x'(t) jumps from 0 to 1 at t=0, and just after t=0, (3) reverts to (2).
> So the solution is:
>
> x'(t) = 1 cos (wt)
>
> (1 from the impulse integral) thus
>
> G_R(t) = sin (wt) + 0 (0 from the the boundary conditions) for t>0

Let me state that more explicitly and make a correction:

G_R(t) = { 0 for t<=0
{ sin(wt)/w for t>0 .

Note the factor of 1/w. It ensures that the jump in the derivative is
of exactly 1 unit.

> G_A(t):
>
> Just flip G_R(t) about the t=0 axis:
>
> G_A(t) = -sin (-wt) for t<0

Again, more precisely:

G_A(t) = { -sin(wt)/w for t<0
{ 0 for t>=0 .

Again, the jump in the derivative must be exactly 1 as t=0 is crossed
in the positive direction.

> G_AF(t): (I''ll do this first)
>
> The implulse adds +cos(wt) to x'(t) at t=0. So the solution becomes:
>
> G_AF(t) = x(t) = exp(-iwt)/(2w) + sin(wt)/w
>
> G_F(t):
>
> The implulse adds +cos(wt) to x'(t) at t=0 so for t<0 we subtract
> cos(wt). Thus, refferring to G_AF(t):
>
> G_F(t) = exp(-iwt)/(2w) - sin(wt)/w
>
> In all of these cases, the impulse only affects one of the dimensions in
> the vector space, and for the affected dimension, only the real degree
> of freedom. I'll need to think about the more general case, but don't
> want to hold up this reply while I do so.

Sorry, Jay. You are still being a bit sloppy. I again refer you to my
discussion of the dimension of the solution space above. The main
reason I brought it up is that knowing the dimension of the solution
space is equivalent to knowing how many initial/boundary conditions
you
need to get a unique solution. As the dimension is 2, you need two
conditions, e.g. x(0) and x'(0). To fully specify a solution to
(eq.3), you need x(+0), x(-0), x'(+0) and x'(-0). But they are not all
independent, as x(+0) = x(-0) and x'(+0) = x'(-0) + 1.

OK, for (F), we have G_F(+0) = 1/2w and G_F'(+0) = -i/2. Therefore,
G_F(-0) = 1/2w still, while G_F'(-0) = ... Hmm, I've just realized
that
I've been a bit sloppy as well. So, with apologies, let me change the
(F) and (AF) conditions to

(F) G_F(t) = i exp(-iwt)/2w for t>=0 , and
(AF) G_AF(t) = -i exp(iwt)/2w for t>=0 .

Then G_F(+0) = G_F(-0)= i/2w, while G_F'(+0) = 1/2 and
G_F'(-0) = -1/2. Similarly, G_AF(+0) = G_AF(-0) = -i/2w, while
G_AF'(+0) = 1/2 and G_AF'(-0) = -1/2. Solving the homogeneous (eq.2)
with the given boundary conditions gives

G_F(t) = i exp(-iw|t|)/2w , and
G_AF(t) = -i exp(iw|t|)/2w .

I hope that clears things up and that you can check through the
calculations without problems.

Now, I'll tip my cards a bit and reveal that the labels R, A, F, and
AF
stand respectively for Retarded, Advanced, Feynman and anti-Feynman
(although I may have mixed up G_F and G_AF). And the G functions we
calculated are the corresponding Green functions for a simple harmonic
oscillator. And finally, the next set of exercises:

(ex.5) Take the Fourier transform of both sides of (eq.1). Solve for
the Fourier transform of x(t)?

(ex.6) Find the Fourier transform of each of the Green functions of
(ex.4): G_R(t), G_A(t), G_F(t), G_AF(t).

For definiteness, use [int dt exp(iwt)] as the forward Fourier
transform and [int dw/2pi exp(-iwt)] as the inverse transform.

Igor

[Igor Khavkine]

On Feb 1, 5:05 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:

> 1) I would be pleased to take you up on your offer to develop a comfort
> with +i epsilon, and thank you for the offer. I will need to return to
> that once I finish working through the present discussion.

My suggestion would be to give these exercises a higher priority than
other topics covered in this thread.

Here's the equation for a simple harmonic oscillator with frequency w
and a force term:

(eq.1) x''(t) + w^2 x(t) = f(t) .

This equation can be specialized for the free oscillation
(homogeneous)
case

(eq.2) x''(t) + w^2 x(t) = 0 ,

and for the impulsive force case

(eq.3) x''(t) + w^2 x(t) = delta(t) .

Here are some calculations to get started.

(ex.1) The solutions of the differential equation (eq.2) form a vector
space. What is its dimension? (Hint: it's 2. Why?) Find a basis
for the solution space.

(ex.2) What is the dimension of the space of solutions of the now
inhomogeneous (eq.3)? (Hint: adding any homogeneous solution
to
an inhomogeneous solution, is still a solution.)

(ex.3) Characterize the solutions of (eq.3) near t=0. (Hint: assume
x(t) is continuous at t=0; what can you say about the
continuity
of x'(t)?)

(ex.4) Find the unique solution of (eq.3) with the boundary conditions
(R) x(t) = 0 for t <= 0, (A) x(t) = 0 for t >= 0,
(F) x(t) = exp(iwt)/(2w) for t >= 0, (AF) x(t) = exp(-iwt)/(2w)
for t >= 0. Call these solutions G_R(t), G_A(t), G_F(t), and
G_AF(t), respectively.

These are fairly elementary exercises. But it's good to get the
fundamentals out of the way before moving on to something more
complex.

The next batch of exercises will involve taking Fourier transforms of
(eq.1) and of the various solutions specified above.

> 2) As a result of our discussions together with Paul Kinsler's latest
> post, I have undergone a change of view. I now agree that it does not
> make sense to try to associate p_i^u with individual gauge bosons any
> more than to associate p^u with an individual photon. I have dropped
> the notion of such an association entirely. But, I am still keeping
> p_i^u and p^u=T^i.p_i^u, and I'll now explain how. [...]

That's too bad; the very last part, that is.

> I also wonder if the very fact that (1) does contain singular density
> functions (infinitely high and thin with an area of 1), might satisfy at
> least in part your statement that "whatever way you introduce (2) [must
> be] equivalent to standard calculations (singularities in the propagator
> and all)"?

Not even close. :-(

Igor

[Jay R. Yablon]

Dear Igor,

As regards the below, thank you!

In the next few days I will go back through and transcribe these
exercises and their answers into a document with "real" non-ascii
equations just to create an easier-to-track document going forward, and
will embark on the additional exercises.

This exercise is especially important to me, now that I have completed
the calculation tying the non-Abelian four-momenta to internal symmetry
space densities, posted at:

http://jayryablon.files.wordpress.com/2009/02/non-abelian-mode-functions-4.pdf

At SPF, kp, who is a tough but fair critic like you, said of the draft
above:

"OK, now that this trivial calculation seems complete, the hard part is
related this to any physics. The reason why plane-waves appear so often
and thus Fourier transforms is the fact that they are eigenstates of the
momentum operator with eigenvalue p, i.e. measurable. In your
substitution this is no longer the case. So you have to relate what you
have done to a real measurable quantity, in which case it would agree
with +i\epsilon calculation or you have to reformulate/extended quantum
mechanics. good luck! kp"

It is now very important for me to understand how the "+i epsilon"
prescription arises so that I can a) become comfortable with it origins
and b) if I feel thereafter a critique of and alteration to this
prescription is warranted, I can articulate that critique better with a
full understanding of what underlies this prescription in the first
place.

Perhaps the biggest problem I have at the outset, is the way in which
this is widely presented as a "prescription." One goes to most texts,
sees propagator poles on shell at p^2=m^2, and is told "here, just
thrown in this extra term, and that will fix everything." Every part of
my scientific being reacts adversely to that. If a term is in an
equation, it should emerge naturally into the equation, and not just be
a term that one puts in by hand. But I must entertain and understand
the rationale behind this, beyond just the fact that it fixes the pole.
I can take just about any result I don't like, and add some term that
makes things better, but that is just not natural to me.

I have also done a bit of poking around and see that your three
equations below are just about how to use Green's functions with
specified boundary conditions to solve inhomogeneous differential
equations. That gives me a good context, so as I do the exercises, I
will also try to "frame" them as best as I can. And, of course, I am
keenly interested in Fourier analysis and the Fourier kernel as it is at
the center of the research I am presently attempting.

But this really is all converging around +i epsilon, and this is what I
need to understand as deeply as I can. In addition to those exercises,
is there a good online reference you can point me to that explains this
prescription, and that goes beyond the usual very inadequate
presentations which simply say "just throw in this term and all will be
well?"

I notice that the delta function, which is central to the exercise upon
which you have been so kind as to guide me, and is central to the
calculation in the link I posted above, can be expressed in a number of
different ways, including various "limits" as (perhaps a different,
perhaps the same) epsilon approaches zero. Am I correct in suspecting
that the "limiting" definition of one of these delta definitions is what
may give rise to the +i epsilon prescription, from a more "natural"
standpoint?

I note in this regard that the Dirac delta(x-y) "function," while it
arises perfectly naturally as an integral over the Fourier kernel
exp[ipx], it itself a singular, discontinuous pulse which cannot in and
of itself have any physical reality since no matter how strong and short
a physical impulse, one can never concoct a real, physical impulse which
is infinitely sharp and infinitely narrow with an area integral of 1. A
physically-possible pulse must have some smoothness to it, i.e., it must
be not the delta function, but at best, the delta "function" in one of
its epsilon-definition incarnations, with epsilon small but finite.

Thanks,

Jay

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:ea465ce6-2519-489d-a6fe-796ff10ddd44@p23g2000prp.googlegroups.com...
> On Feb 4, 12:37 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> "Igor Khavkine" <igor...@gmail.com> wrote in message
>> news:db12cbd6-54f5-42dd-92a4-8777f13ede99@t26g2000prh.googlegroups.com...
>
>> > Here's the equation for a simple harmonic oscillator with frequency
>> > w
>> > and a force term:
>>
>> > (eq.1) x''(t) + w^2 x(t) = f(t) .
>>
>> > This equation can be specialized for the free oscillation
>> > (homogeneous) case
>>
>> > (eq.2) x''(t) + w^2 x(t) = 0 ,
>>
>> > and for the impulsive force case
>>
>> > (eq.3) x''(t) + w^2 x(t) = delta(t) .
>>
>> > Here are some calculations to get started.
>>
>> > (ex.1) The solutions of the differential equation (eq.2) form a
>> > vector
>> > space. What is its dimension? (Hint: it's 2. Why?) Find a
>> > basis
>> > for the solution space.
>>
>> OK, Igor, let's give it a try. This is my first physics "exam" since
>> the 1970s, so I hope this old geezer is not too rusty ;-):
>
> Think of it more as learning by doing. :-)
>
.. . .

[Jay R. Yablon]

Dear Igor,

As regards the below, thank you!

In the next few days I will go back through and transcribe these
exercises and their answers into a document with "real" non-ascii
equations just to create an easier-to-track document going forward, and
will embark on the additional exercises.

This exercise is especially important to me, now that I have completed
the calculation tying the non-Abelian four-momenta to internal symmetry
space densities, posted at:

http://jayryablon.files.wordpress.com/2009/02/non-abelian-mode-functions-4.pdf

At SPF, kp, who is a tough but fair critic like you, said of the draft
above:

"OK, now that this trivial calculation seems complete, the hard part is
related this to any physics. The reason why plane-waves appear so often
and thus Fourier transforms is the fact that they are eigenstates of the
momentum operator with eigenvalue p, i.e. measurable. In your
substitution this is no longer the case. So you have to relate what you
have done to a real measurable quantity, in which case it would agree
with +i\epsilon calculation or you have to reformulate/extended quantum
mechanics. good luck! kp"

It is now very important for me to understand how the "+i epsilon"
prescription arises so that I can a) become comfortable with it origins
and b) if I feel thereafter a critique of and alteration to this
prescription is warranted, I can articulate that critique better with a
full understanding of what underlies this prescription in the first
place.

Perhaps the biggest problem I have at the outset, is the way in which
this is widely presented as a "prescription." One goes to most texts,
sees propagator poles on shell at p^2=m^2, and is told "here, just
thrown in this extra term, and that will fix everything." Every part of
my scientific being reacts adversely to that. If a term is in an
equation, it should emerge naturally into the equation, and not just be
a term that one puts in by hand. But I must entertain and understand
the rationale behind this, beyond just the fact that it fixes the pole.
I can take just about any result I don't like, and add some term that
makes things better, but that is just not natural to me.

I have also done a bit of poking around and see that your three
equations below are just about how to use Green's functions with
specified boundary conditions to solve inhomogeneous differential
equations. That gives me a good context, so as I do the exercises, I
will also try to "frame" them as best as I can. And, of course, I am
keenly interested in Fourier analysis and the Fourier kernel as it is at
the center of the research I am presently attempting.

But this really is all converging around +i epsilon, and this is what I
need to understand as deeply as I can. In addition to those exercises,
is there a good online reference you can point me to that explains this
prescription, and that goes beyond the usual very inadequate
presentations which simply say "just throw in this term and all will be
well?"

I notice that the delta function, which is central to the exercise upon
which you have been so kind as to guide me, and is central to the
calculation in the link I posted above, can be expressed in a number of
different ways, including various "limits" as (perhaps a different,
perhaps the same) epsilon approaches zero. Am I correct in suspecting
that the "limiting" definition of one of these delta definitions is what
may give rise to the +i epsilon prescription, from a more "natural"
standpoint?

I note in this regard that the Dirac delta(x-y) "function," while it
arises perfectly naturally as an integral over the Fourier kernel
exp[ipx], it itself a singular, discontinuous pulse which cannot in and
of itself have any physical reality since no matter how strong and short
a physical impulse, one can never concoct a real, physical impulse which
is infinitely sharp and infinitely narrow with an area integral of 1. A
physically-possible pulse must have some smoothness to it, i.e., it must
be not the delta function, but at best, the delta "function" in one of
its epsilon-definition incarnations, with epsilon small but finite.

Thanks,

Jay

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:ea465ce6-2519-489d-a6fe-796ff10ddd44@p23g2000prp.googlegroups.com...
> On Feb 4, 12:37 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> "Igor Khavkine" <igor...@gmail.com> wrote in message
>> news:db12cbd6-54f5-42dd-92a4-8777f13ede99@t26g2000prh.googlegroups.com...
>
>> > Here's the equation for a simple harmonic oscillator with frequency
>> > w
>> > and a force term:
>>
>> > (eq.1) x''(t) + w^2 x(t) = f(t) .
>>
>> > This equation can be specialized for the free oscillation
>> > (homogeneous) case
>>
>> > (eq.2) x''(t) + w^2 x(t) = 0 ,
>>
>> > and for the impulsive force case
>>
>> > (eq.3) x''(t) + w^2 x(t) = delta(t) .
>>
>> > Here are some calculations to get started.
>>
>> > (ex.1) The solutions of the differential equation (eq.2) form a
>> > vector
>> > space. What is its dimension? (Hint: it's 2. Why?) Find a
>> > basis
>> > for the solution space.
>>
>> OK, Igor, let's give it a try. This is my first physics "exam" since
>> the 1970s, so I hope this old geezer is not too rusty ;-):
>
> Think of it more as learning by doing. :-)
>
.. . .

[Jay R. Yablon]

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:dcdcef52-7bbd-4a84-bb57-864b73e71c24@w24g2000prd.googlegroups.com...
> On Jan 10, 5:58 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
> wrote:
>> Jay R. Yablon schrieb:
>>
>> > I have often seen an photon field quantum A^u written in terms of a
>> > polarization vector epsilon^u and plane wave exp[ip^.ux_u] as:
>>
>> > A^u = epsilon^u exp[ip^u x_u] (1)
>>
>> > 1) Is p^u considered to be the momentum vector of that subject
>> > photon, or an independent variable of integration unconnected to
>> > the
>> > photon?
>
> [...]
>
>> Yes. But note that in all three cases, these only give a set of bais
>> vectors in a momentum eigenstates. The general state has unsharp
>> momentum, and is an arbitrary linear combinations of these, obtained
>> by a weighted sum over the degrees of freedom (i.e., integral over
>> momentum, sum over polarization/spin indices).
>
> A short follow up to Arnold's remark. In quantum mechanics,
> observables are operators, including the components of the momentum.
> For a wave function proportional to exp(ipx), the components of the
> vector p are associated with the momentum only in as much as they are
> the eigenvalues of the corresponding operator observables, with
> exp(ipx)
> being the eigenvector.
>
> Igor
>
For a matrix A, the eigenvalues L and eigenvectors u are given by:

(A-LI)u=0

where I is a unit matrix.

In that context, can you please show a bit more explicitly the manner in
which the p^u components are Eigenvalues L, the manner in which
exp(ip^ux_u) is an eigenvector u, as well as show what is the subject
matrix A.

Thanks,

Jay.