Draft paper deriving a non-empty, stationary, axisymmetric solution solution of Einstein's Equations, based on the Lorentz Force Law

Jay R. Yablon

Dear friends at SPR,

I have just posted a DRAFT paper to:

http://home.nycap.rr.com/jry/Papers/...ce%20Paper.pdf

This paper is titled: "Is the Lorentz Force Law Based Upon a Relation
Between rho=mu+p Perfect Fluids and alpha=1 Kerrighan-Type Electromagnetic
Energy Tensors?"

I would appreciate your review and comment on this draft before I consider
next steps.

The abstract is as follows:

It is demonstrated how the Lorentz force law is a direct consequence of
relating a perfect fluid tensor T^uv_Euler for which the rest mass
density rho is related to the energy density me and pressure p according
to rho=mu+p, with an electromagnetic energy tensor T^uv with certain
uniqueness conditions established by Kerrighan in the early-1980s, and
by in turn relating both of these tensors with the Einstein tensor R^uv
- ½ g^uvR. We then use these relationships -- which are effectively the
first integral of the Lorentz force law -- to first establish the metric
tensor g_uv using the known general solution for a non-empty stationary
axisymmetric perfect fluid, and then, to specify the electromagnetic
fields underlying the structure of this perfect fluid for which the
equation of motion is the Lorentz force law. The key advance, is
showing that a solution does exist to the Einstein equations which is
fully compatible with, and indeed is based upon, the Lorentz force law.

I do want to emphasize that this is a work in progress. But, it is now
developed far enough that a posting seeking input is warranted at this time.

Very truly yours,

Jay R. Yablon
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com

Igor Khavkine

On 2006-04-05, Jay R. Yablon <jyablon@nycap.rr.com> wrote:

> I have just posted a DRAFT paper to:
>
> http://home.nycap.rr.com/jry/Papers/...ce%20Paper.pdf
>
> This paper is titled: "Is the Lorentz Force Law Based Upon a Relation
> Between rho=mu+p Perfect Fluids and alpha=1 Kerrighan-Type Electromagnetic
> Energy Tensors?"
>
> I would appreciate your review and comment on this draft before I consider
> next steps.

Dear Jay, your paper draft is 30 pages long. Which is too long for me,
and I'm sure many other s.p.r patrons, to read in detail. If you really
want feed back, you may be interested in the following exercise in
efficient communication.

Can you summarize the technical (mathematical) content of your work in
one paragraph? In advance, I can't necessarily say that this will be
possible. But you won't know unless you try. It is also a good test of
whether your terminology is standard enough to be recognizable by people
familiar with the mathematical background.

Perhaps in another paragraph, can you summarize the physical
interpretation of your technical results? Usually, this involves
relating mathematical objects to things we can measure in the lab. At
this stage, you should also consider consistency with known physics and
possible measurable predictions.

Igor

tessel@um.bot

On Wed, 5 Apr 2006, Jay R. Yablon wrote:

> I would appreciate your review and comment on this draft before I
> consider next steps.

Easy to say... unfortunately you do not seem to be a good listener.

Critiquing your paper is a job for referees, but to judge from the
abstract you have not yet studied basic gtr (despite your references to
MTW, which is one of the best).

I think you should study a good textbook before rushing off to write your
own papers, but somehow I doubt you will listen to this advice, since you
seem to have established yourself as an irrepressible engine of junk
physics, to what purpose I don't know.

> The key advance, is showing that a solution does exist to the Einstein
> equations which is fully compatible with, and indeed is based upon, the
> Lorentz force law.

This makes a very bad impression, because if you had studied a textbook
you would know that any exact solution to the EFE which involves an EM
field is automatically "fully compatible with the Lorentz force law".

Your abstract is in fact incomprehensible. For all I could tell, you
might be

1. trying to find some kind of exact solution in gtr,

2. trying to show that the Lorentz force law is a consequence of something
"more fundamental",

3. trying to show that perfect fluids are really electromagnetic fields or
vice versa (really bad idea, and you ought to know why),

or none of the above. But I don't care whether any of my guesses are
close to what you may have had in mind. I mention them only to point out
that the fact that these guesses are so wildly divergent should itself be
a compelling indication that your abstract must be completely inadequate.
Alas, in my experience, a paper with a miserably garbled abstract rarely
becomes a model of clarity in the body of the text.

> I do want to emphasize that this is a work in progress. But, it is now
> developed far enough that a posting seeking input is warranted at this
> time.

Not at all, in fact you've impressed me with your poor judgement.

"T. Essel" (too disappointed to continue with this)

spardr@netscape.net

Igor Khavkine has offered some sage advice. I would add the following.

Suppose I posted a 30-page paper, obscurely written and packed
with complicated, unmotivated calculations. Suppose the paper ended
with a statement to the effect that I hadn't bothered to think carefully
about its physical implications, but I felt that there might be some.
("The physical viability of these results will need to be considered
over time.") Would Mr. Yablon invest the months which would be required
in working through it? If not, is it realistic to expect anyone to do
so?

Nevertheless, I did take a look at it. I think it is full of
questionable statements, including errors which are both elementary and
serious.

The questionable points are too numerous to catalog here, so I'll
just mention one. Anyone thinking of investing the time to read it
carefully should check this one out first because it is a definite error
at the heart of his argument and easy to check. It occurs on p. 8, just
before equation (5.1).

To explain this error, I quote from another post of his:

> But again, the approach I am taking -- and again, it may take some
> getting used to and I certainly want to make sure it is at most
> unusual but not "illegal" -- is not to add tensor properties, but to
> set one tensor equal to another which thereby constrains each one.
> In fact, I am setting three tensors all to one another: 1) The
> Maxwell tensor with an extra "variable Lambda" term based on
> Kerrighan, 2) the perfect fluid tensor with rho = mu + p (i.e.,
> specific enthalpy = 1), and 3) G^uv. The first derivative of this
> equality is the Lorentz force law; the solution to Einstein's
> equations for this equality yields the spacetime geometry of "a
> perfect fluid which moves according to the Lorentz force law."
> Perhaps when fully developed, we will find that such "a perfect
> fluid which moves according to the Lorentz force law" actually has
> the properties of some real physical entities already known in
> nature.

What he calls the "Maxwell tensor" is the usual energy-momentum
tensor for an electromagnetic field plus a multiple of the identity.

[It probably won't be obvious to many readers that the tensor of
equation (5.1) really is of this form, but if you work it out, you
will see that it is. I happen to be particularly familiar with such
tensors, so I was able to spot this easily. That is the reason for
this post---to save others time by sharing my limited expertise.]

This tensor is a linear transformation which has two distinct
eigenvalues, each with multiplicity 2 (except in the trivial case in
which the tensor vanishes). His "perfect fluid tensor" is diag{mu, p,
p, p}, and so has an eigenvalue p with multiplicity either 3 or 4. Thus
the "Maxwell tensor" cannot equal the "perfect fluid tensor" except in
the trivial case in which both vanish.

Note also that a "perfect fluid tensor" singles out a distinguished
timelike direction, namely the direction of the eigenvector for
eigenvalue mu (except in the degenerate case in which mu = p). So his
analysis contains the hidden assumption that at each point of spacetime,
there is a distinguished timelike direction at any point. Such a strong
assumption, which is contrary to the spirit of general relativity,
should be mentioned in the paper. Because the exposition is so obscure,
the reader has to dig these things out for himself.

This is the kind of paper which gives conscientious referees
nightmares. If one is lucky, one might spot an obvious error like the
above without investing too much time, but a subtle error buried in 30
pages of dense tensor calculations might take months to find.


Stephen Parrott

Jay R. Yablon

I have had a chance to carefully consider the posts by Dr. Khavkine,
Dr. Parrott, and T. Essel. I have concluded that Dr. Parrott's
objection is persuasive. I have also concluded that I have a serious
communication problem, which I will work hard to address in the future
as well as here. Here, I address both Dr. Parrott's and T. Essel's
replies. I will provide a reply to Dr. Khavkine separately, under his
original reply.

Referring to my equation (5.1), not *all* of the components vanish in
the relationship between the electromagnetic (with extra term) and the
perfect fluid tensors. But, the Poynting vector components *do*
vanish, and that carries the day. Because the energy tensor I use in
(5.1) must apply everywhere in spacetime if it is of be of general use
in defining the metric tensor throughout spacetime, my equation (5.1)
effectively asserts that there is no free propagation of
electromagnetic energy as described by the Poynting vector, anywhere in
spacetime. Such an assertion, Houston, clearly is a problem.

This answers clearly for me, the discussion I was having with T. Essel
in the thread "Question about Algebraic classification and
Compatibility of Energy Momentum Tensors." He and I were talking
about what happens when we set two energy tensors with different
Algebraic classification to one another. He was talking about adding
(superposing) two or more energy tensors, and I was bent on setting
them to one another as I did in (5.l). I now know that setting them to
one another will not work, because it zeros out the free propagation of
EM field energy, not to mention as does Dr. Parrott, it establishes a
preferred timelike direction. My apology to T. Essel: sometimes I can
be thick. I do try to listen, but sometimes I do not understand what I
am hearing until I manage to discover it for myself. I do very much
appreciate the earlier discussions we had and hope we can resume some
good discussions in the future.

In terms of my research on this topic, it is clear to me now that I
must consider a *superposition* of the perfect fluid and the EM field
tensor, plus the extra E dot B term I have been using, and *cannot*
merely set them to one another. That is what I was trying to figure
out in the earlier thread. I can use a superposition to obtain the
Lorentz force law the same as I do from (5.1), with some minor
differences in the mathematical development. But, solving the EFE in
this circumstance appears, at first view, practically speaking, to be
very difficult. Stephani et al. are clear that they do not in general
do not consider solutions for superposed energy tensors. T. Essel
provides the best information I have thus far about how to obtain
superposed solutions when he says: "possible contributions arising
from the matter in a perfect fluid and from the field energy-momentum
of an EM field will have different algebraic properties. Such
contributions can be added to yield legal stress-energy tensors in
appropriate exact solutions of the EFE (e.g. for charged dusts),
although it may be difficult to find examples of such models which
'solve the EFE' simply by guessing. Fortunately, systematic and
elementary approaches are available, if sometimes delicate or
challenging to realize."

Can someone please point me to some good references which show how to
approach EFE solutions in the case where one in fact *superposes* known
energy tensors. Is there something buried somewhere in Stephani where
they discuss how to do this? In MTW? I could not find it in either.
Obviously, I'd prefer to avoid having to "brute force" a
solution, and so would be very interested in these "systematic and
elementary approaches." I am especially looking for solutions where
the usual EM field tensor is superposed with a perfect fluid, e.g.,
so-called charged dust (but a solution generalized to pressure is even
better). Then, in hte specific context of what I am considering, the
question would become how that solution changes when one further
introduces the identity times a constant multiple of E dot B.

Short of obtaining a specific exact solution for two superposed energy
tensors, is there any sort of theorem which says that if an exact
solution exists for energy tensor A^uv, and an exact solution exists
for tensor B^uv, that an exact solution *does also* exist for tensor
A^uv plus B^uv? To lay out the problem clearly: suppose we have a
tensor A^uv for which we can obtain solutions when we set A^uv_;u=0 and
tensor B^uv which yields solutions for B^uv_;u=0. Does this mean that
there will exist solutions when we set (A^uv+B^uv)_;u =0? Are there
ways to take the separate A^uv_;u=0 and B^uv_;u=0 and migrate over to
the (A^uv+B^uv)_;u =0 solution in a systematic way? The difficulty, of
course, is that the equations we are talking about are G^uv = A^uv and
then G^uv = B^uv, and finally G^uv = A^uv + B^uv, where G^uv, the
Einstein tensor, is a second order differential equation for the metric
tesnor. In more general mathematical terms, the question is this: if I
obtain a solution to "second order non-linear differential
equation" = A, and then a solution to "same differential
equation" = B, what is the solution to "same differential
equation" = A+B? I see it is not an easy problem; I am asking how
well this has been explored and where I may find the best explorations
to date of this problem.

Jay R. Yablon

Below is a two paragraph summary as suggested, which accounts for Dr.
Parrott's "persuasive objection" which I discuss in reply to his
post. I have taken the liberty of separating the question of "things
we can measure in the lab" into a third, separate paragraph from
physical interpretation:

Technical (mathematical) content: When one sets the perfect fluid
tensor with rho = mu + p to the usual electromagnetic energy tensor
plus a supplemental E dot B term, and sets each to the Einstein tensor,
as shown in (5.1), the perfect fluid is found to have an equation of
motion which is equivalent to the Lorentz force plus a term
proportional to the gradient of the rest mass density which can
alternatively be formulated in terms of temperature, entropy and
pressure. Above, rho is rest mass density, mu is energy density and p
is pressure. When the metric tensor is calculated for the example of a
rigidly-rotating axially symmetric perfect fluid, and simultaneous
equations subsisting between the two former tensors including this
metric tensor are decoupled, we obtain a non-trivial, non-vanishing
relationship between the rest mass density rho and the field
combination E dot B. If and only if E dot B = 0, do we find that these
two tensors vanish, and at the same time, that the rest mass density
rho becomes zero. Otherwise, they do *not* vanish, rho is not zero,
and the solution is not trivial. Further, it is found that the current
density coupled to the electromagnetic field strength tensor, is
directly related to the *gradient* of E dot B. E dot B is thus
integrally and inextricably linked to *sources*, both gravitational and
electrodynamic. However, because the perfect fluid tensor with
diag{mu, p, p, p} has an eigenvalue p with multiplicity either 3 or 4
and the electromagnetic tensor has two distinct eigenvalues, each with
multiplicity 2, and although some of the equations (5.1) do not vanish,
the Poynting vector *does* vanish everywhere. Additionally, because a
"perfect fluid tensor" singles out a distinguished timelike direction,
namely the direction of the eigenvector for eigenvalue mu except in the
degenerate stiff matter case in which mu = p, the analysis here does
appear to be yield a distinguished timelike direction at any event in
spacetime. As such, setting (constraining) the two energy tensors to
one another as in (5.1) is not viable. To continue this approach, one
would need to *superpose* the two energy tensors, rather than constrain
them to one another. Lorentz motion can still be obtained in a similar
way; the practical solution to the Einstein equations is made more
challenging because of the energy tensor superposition.

Physical interpretation: Fundamentally, I am attempting to understand
how the General Theory of Relativity may be applied to understand
"the structure of the elementary particles of matter," that is, to
apply general relativity inside the atom and inside the nucleus. The
simplest starting point for such an endeavor, as it was in 1919, is to
try to understand spacetime inside the electron, and so this is where
we begin. The electron, and presumably worldlines within the electron,
move according to the Lorentz force law, or perhaps a variant thereof,
i.e., the Lorentz force law with additional terms. Thus, the first
goal of this research is to understand what we characterize as
"regions of spacetime where worldlines exhibit Lorentz motion in the
presence of an electromagnetic field." This means 1) finding a
vanishing-divergence energy tensor for which the equation of motion is
naturally the Lorentz force law or a suitable variant thereof, and 2)
solving the Einstein equations for such an energy tensor to obtain the
metric tensor for this region of spacetime. Such a metric tensor would
therefore describe the spacetime geometry inside a body which exhibits
Lorentz motion, such as the electron. By setting the energy tensors
together as in (5.1), we are indeed able to derive Lorentz force motion
as a consequence, and also to derive the g_uv which obtain in regions
where that Lorentz motion also obtains. But, by setting these energy
tensors to one another, the Poynting vector vanishes, which is a
non-starter. Instead, we must from here explore *superpositions* of
the tensors in (5.1). These also can be used to obtain Lorentz motion,
but obtaining solutions of the Einstein equations for such superposed
tensors is, practically, much more difficult than for tensors equated
as in (5.1). *If* such solutions exist, then the g_uv of these
solutions would describe the structure of spacetime inside the
electron. The g_uv so-obtained, because they would describe a region
of spacetime where Lorentz force motion obtains, might then be used to
*define* the Dirac matrices (1/2)(gamma^u gamma^v + gamma^v gamma^u =
g^uv) and perform index contractions in the Dirac (relativistic
Schroedinger) equation for the electron (e.g., in the term, id^u g_uv
gamma^v, and for an interacting electron, J^u g_uv A^v where A^v is the
four-vector potential for the photon). This would be instead of using
the Minkowski tensor of flat spacetime which is typically employed.
Perhaps the deepest question, is whether the Einstein equation, so
employed, can in any way end up constraining the possible
configurations for the charge density distribution of the electron to a
very restricted set of discrete configurations, such as, for example,
those shown at http://www.orbitals.com/orb/orbtable.htm, and
characterized by the quantum numbers n, l, m. This is a question, not
an answer.

Experimental confirmation: The most direct way to confirm a results
based on this approach would appear at this juncture to be based on any
extra terms which appear in the Lorentz force law. When viewed
thermodynamically, such terms describe a quantitative relationship
which exists whereby the motion of a charge in an electromagnetic field
is altered as the ambient temperature and / or pressure are made
different. By injecting electrons into an electromagnetic field and
recording their movement under tightly controlled circumstances where
the only difference from one run to the next entails a controlled
difference in temperature or pressure, slight variations in the path of
movement may be observed.

Igor Khavkine

Jay R. Yablon wrote:
> Below is a two paragraph summary as suggested, which accounts for Dr.
> Parrott's "persuasive objection" which I discuss in reply to his
> post. I have taken the liberty of separating the question of "things
> we can measure in the lab" into a third, separate paragraph from
> physical interpretation:

These paragraphs are still too long. However, one need not read past
the first few lines to discover problems. Some of these comments have
already been made.

> Technical (mathematical) content: When one sets the perfect fluid
> tensor with rho = mu + p

If rho is the time-time component of the stress-energy tensor, then
this is a very unusual equation of state. For a non-relativistic ideal
gas, it is rho ~ mu + 3p/2, while for an ultrarelativistic ideal gas,
it is rho ~ 3p. The 3 in either case is important because it tells us
about the dimensionality of space.

> to the usual electromagnetic energy tensor

That's a big No. The perfect fluid tensor is in general not traceless,
while the electromagnetic stress-energy tensor is. Besides, if you do
this, you are saying that your perfect fluid *is* incoherent radiation.
Such an equality will also have many other undesirable consequences.

> plus a supplemental E dot B term,

NEVER EVER, use thee-dimensional language when speaking in a
four-dimensional context (fine print: unless you've given everyone
enough information to pick a special reference frames, which is
certainly not the case here). You must also watch out for gauge
invariance, which is again not obvious as soon as you write that term.

> and sets each to the Einstein tensor,
> as shown in (5.1), the perfect fluid is found to have an equation of
> motion which is equivalent to the Lorentz force

It is completely mysterious what you mean by this. The Lorentz force is
an extra term in the hydrodynamic equations for matter, a term
depending on an external EM field. However, from the above, your matter
density and not independent from the EM field. So, on its own, the
above statement is meaningless.

It is hardly woth going through the rest given the number of problems
already found.
> [...]

These problems are elementary yet important. One lesson here is to not
write 30 page papers before being able to summarize your approach in a
short paragraph.

> Physical interpretation: Fundamentally, I am attempting to understand
> how the General Theory of Relativity may be applied to understand
> "the structure of the elementary particles of matter," that is, to
> apply general relativity inside the atom and inside the nucleus.

This is possibly a doomed affair. The best currently known theory of
elementary particles is the quantum field theory of the Standard Model.
Which, in the very least, means that any original description of
elementary particles must describe quantum effects. If you don't know
how hard that is (which is quite possible from your previous posts),
then you need to study up on your quantum mechanics and QFT. Another
lesson here is to not seek a solution until you understand the problem.

Igor

strangerep@yahoo.com.au

Jay Yablon wrote:

>> [...] Fundamentally, I am attempting to
>> understand how the General Theory of
>> Relativity may be applied to understand "the
>> structure of the elementary particles of
>> matter," that is, to apply general relativity
>> inside the atom and inside the nucleus.

Igor Khavkine replied:

> This is possibly a doomed affair. The best
> currently known theory of elementary
> particles is the quantum field theory of the
> Standard Model. Which, in the very least,
> means that any original description of
> elementary particles must describe quantum
> effects. If you don't know how hard that is
> (which is quite possible from your previous
> posts), then you need to study up on your
> quantum mechanics and QFT. Another lesson
> here is to not seek a solution until you
> understand the problem.

Jay, I just want to reinforce what Igor is
telling you...

I get the impression from all your writings that
your knowledge of QFT is inadequate for the
sort of research areas that you're attempting.

You need to put aside your current "research"
efforts for several months and concentrate heavily
on mastering at least the basics of Quantum
Electrodynamics. By "basic", I mean reaching the
point where you can perform loop integrals in
Feynman diagrams for yourself and compute
cross-sections for various physical processes.

One way to achieve this is to self-study chapters
1-7 from Peskin & Schroeder, verifying all
calculational steps in the text, and completing
the exercises in full. (Many solutions can be
obtained off the web if you get stuck, and there's
plenty of people here on spr who would help you.)

[Peskin & Schroeder lists for USD 77.00 new on
Amazon, and around USD 57.00 used. It is a good
investment, provided one also downloads the errata
list from Prof Peskin's website:
http://www.slac.stanford.edu/~mpeskin]

Jay R. Yablon

>
> It is completely mysterious what you mean by this. The Lorentz force is
> an extra term in the hydrodynamic equations for matter, a term
> depending on an external EM field. However, from the above, your matter
> density and not independent from the EM field. So, on its own, the
> above statement is meaningless.
>

Please take a read through Dr. Einstein's 1916 GR paper, section 19, Dover
pp 152-153, less for what is says about the Euler tensor specifically, and
more for what it says about equations of motion and the metric tensor g_uv.
Then, I have the following questions which I hope can clarify what we are
discussing. I ask these questions in terms of CLASSICAL field theory:

1) Start out not knowing the metric tensor g_uv. Take an energy tensor
T^uv (the Euler tensor or any other energy tensor for which there exist
solutions to the AE field equations), and posit that this energy tensor
describes the matter in a region r of spacetime under study. (Maybe the
energy tensor is different somewhere else in spacetime, we are just looking
at a specified region of spacetime for this discussion.) As soon as we set
this energy tensor T^uv to the Einstein tensor G^uv = R^uv - (1/2) g^uv R,
have we not at the same time implicitly established the four equations of
motion k_0 T^uv_;u = (R^uv - (1/2) g^uv R)_;u = 0 for that same region r?

2) Again, assuming solutions can be found to the AE field equations for
T^uv, have we not at the same time we set T^uv to the Einstein tensor, also
established the metric tensor g_uv which applies in that region r?

3) The g_uv and the T^uv_;u=0 are three differential orders apart, of
course, but they do both derive from and are tied together by the same T^uv.
Can't we therefore say that the metric tensor g_uv so-obtained describes the
geometry of the region r of spacetime for which the energy tensor is T^uv
and for which the equation of motion is T^uv_;u=0?

4) Therefore, suppose we can find some T^uv for which the four equations
T^uv_;u=0 are equivalent to the Lorentz force law, and for which it is
possible to obtain a solution to the AE field equation and thus obtain the
g_uv. (Yes, I know, I am assuming that a solution exists -- so I am asking
this hypothetically for the moment). (Here, we have in mind the Lorentz
force law written in terms of the current density four vector J^u and the
density of rest mass, rho, as opposed to written in terms of the unit
quantum of charge e and the rest mass m of the entire electron.) Can't we
say that the g_uv so-obtained describe the spacetime geometry in this region
r of spacetime where the Lorentz force law is the equation of motion?

5) If we can obtain the g_uv which describe the geometry of spacetime where
the equation of motion is the Lorentz force law as set forth above, have we
not, classically speaking, effectively specified the spacetime geometry
inside, for example, an electron?

Again, I realize I have not yet said anything about quantum field theory.
Let's walk before we run. Let's line up the classical concepts in a
sensible way and then work on the QFT requirements. (PS, thank you Mike /
strangerep for the QFT reference in the other post, which I plan to obtain
and study.)

Please let me know, CLASSICALLY speaking, whether the above is a good line
of reasoning, or if not, specifically where the pitfalls may be. Depending
on those answers, perhaps we can then ask a few more questions about how to
next dip our toes into QFT.

Thanks.

Jay.

Jay R. Yablon

> One way to achieve this is to self-study chapters
> 1-7 from Peskin & Schroeder, verifying all
> calculational steps in the text, and completing
> the exercises in full. (Many solutions can be
> obtained off the web if you get stuck, and there's
> plenty of people here on spr who would help you.)
>
> [Peskin & Schroeder lists for USD 77.00 new on
> Amazon, and around USD 57.00 used. It is a good
> investment, provided one also downloads the errata
> list from Prof Peskin's website:
> http://www.slac.stanford.edu/~mpeskin]
>

One question about this book. My personal "bible" for particle physics has
long been Halzen and Martin's Quarks and Leptons, and I know most of the
material in that book quite thoroughly.

For those familiar with Halzen and Martin as well as Peskin and Schroeder,
how do they compare? What will Schroeder have which Halzen and Martin does
not, or, how does Peskin & Schroeder perhaps amplify or delve more deeply
into the subject matter covered by Halzen and Martin?

Thanks again, Mike / SRep.

Jay.

Igor Khavkine

Jay R. Yablon wrote:

> One question about this book. My personal "bible" for particle physics has
> long been Halzen and Martin's Quarks and Leptons, and I know most of the
> material in that book quite thoroughly.
>
> For those familiar with Halzen and Martin as well as Peskin and Schroeder,
> how do they compare? What will Schroeder have which Halzen and Martin does
> not, or, how does Peskin & Schroeder perhaps amplify or delve more deeply
> into the subject matter covered by Halzen and Martin?

Halzen & Martin is a cookbook. Its goal is to teach you particle
physics, the kind that is done is high energy accelecrators and
colliders. QED and the Standard Model happen to describe the outcomes
of such experiments quite well, that's why they are included in this
book.

A book on Quantum Field Theory proper, such as Pesking & Schroeder,
aims to teach you how to apply quantum mechanics to fields (mechanical
systems with infinitely many degrees of freedom). This includes
directing your attention to many subtleties that may not be apparent if
you are only interested in one specific application. In principle, it
should also teach you how to construct a QFT of your own. Of course,
P&S do aim their contents toward application to particle physics.
However, there are many QFT books that aim instead toward condensed
matter or nuclear physics. QFT is a very general subject, and it's good
to be aware of its breadth.

Igor

Igor Khavkine

Jay R. Yablon wrote:
> >
> > It is completely mysterious what you mean by this. The Lorentz force is
> > an extra term in the hydrodynamic equations for matter, a term
> > depending on an external EM field. However, from the above, your matter
> > density and not independent from the EM field. So, on its own, the
> > above statement is meaningless.
>
> Please take a read through Dr. Einstein's 1916 GR paper, section 19, Dover
> pp 152-153, less for what is says about the Euler tensor specifically, and
> more for what it says about equations of motion and the metric tensor g_uv.

Whatever Einstein says in his 1916 paper can be found in more modern
references. These, in addition, provide of benefit correcting mistakes
that may not have initially been noticed, use clearer and more modern
notation, and provide references to work that has been done in any
particular area since 1916. Learn to take advantage of these benefits.

> Then, I have the following questions which I hope can clarify what we are
> discussing. I ask these questions in terms of CLASSICAL field theory:
>
> 1) Start out not knowing the metric tensor g_uv. Take an energy tensor
> T^uv (the Euler tensor or any other energy tensor for which there exist
> solutions to the AE field equations), and posit that this energy tensor
> describes the matter in a region r of spacetime under study. (Maybe the
> energy tensor is different somewhere else in spacetime, we are just looking
> at a specified region of spacetime for this discussion.) As soon as we set
> this energy tensor T^uv to the Einstein tensor G^uv = R^uv - (1/2) g^uv R,
> have we not at the same time implicitly established the four equations of
> motion k_0 T^uv_;u = (R^uv - (1/2) g^uv R)_;u = 0 for that same region r?

Let [T] be the set of all symmetric rank (2,0) tensors T^uv
for which there exists a metric g_uv satisfying Einstein's
equations G^uv = k_0 T^uv. Then, using this particular metric, the
Bianchi identity T^uv_;u is automatically satisfied. Hence, provided
you select T^uv from [T], "imposing" the equation of motion T^uv_;u
gives you absolutely no new information.

> 4) Therefore, suppose we can find some T^uv for which the four equations
> T^uv_;u=0 are equivalent to the Lorentz force law, and for which it is
> possible to obtain a solution to the AE field equation and thus obtain the
> g_uv.

Again, you've yet to say with certainty what you mean by the "Lorentz
force law". When I hear "Lorentz force", I immediately think of an
electromagnetic field interacting with matter (say a charged fluid).
In that case, the total stress-energy tensor is made up of two parts,
one for the EM field and one for the charged fluid. However, it is only
the total stress tensor that is divergence free, neither of the
individual ones is.

The vanishing of the total stress tensor's divergence does give
equations of motion for the transport of the joint energy and momentum
densities of the EM field and the charged fluid. However, it tells you
nothing about the manner in which the fluid and the EM field exchange
energy and momentum between themselves (which is the content of the
Lorentz force law).

If you forsake the presence of an EM field, and suppose the vanishing
divergence of the fluid's stress tensor in some way "automatically"
provides the same equations of motion as if it were interacting both
with an EM field and gravity, then it can be quickly checked that this
situation is impossible. Simply consider the equation of motion for a
single fluid element. When under the influence of pure gravity, its
equation of motion is the geodesic one with an extra pressure-gradient
term:

(*) du/dtau = grad' p,

with u being the fluid elements 4-velocity, tau its world line's proper
time parametrization, and p the pressure, is the gradient transverse to
u. On the other hand, when interacting with an EM field, there is yet
an extra term corresponding to the Lorentz force:

(**) du/dtau = grad' p + F.u,

where now F.u is the contraction of the Faraday field tensor F with the
4-velocity u. It is straightforward to check that there is no way that
equation (*) could reproduce equation (**) just by itself. This is
owing to the different kinds of dependence on u of the grad' p and F.u
terms.

> Can't we
> say that the g_uv so-obtained describe the spacetime geometry in this region
> r of spacetime where the Lorentz force law is the equation of motion?

Now, I still do not know whether what you are trying to do corresonds
to either situation that I've just described (in deed you haven't even
clearly said what you are trying to do yet). However, any
interpretation that I can come up with for your claim or ambition
makes them impossible.

> 5) If we can obtain the g_uv which describe the geometry of spacetime where
> the equation of motion is the Lorentz force law as set forth above, have we
> not, classically speaking, effectively specified the spacetime geometry
> inside, for example, an electron?

No. Ignoring the ambiguities in your descriptions, the best you can
hope for is a solution for the equations of motion of some kind of
charged fluid (macroscopic), together with the EM field, together with
the metric. There is no way to directly identify this scenario with the
description of a single electron (especially given that hydrodynamic
equations are not expected to hold exactly at the microscopic level).
On the other hand, there are models of a classical electron as an
extended object. They usually aim at removing some of the pathologies
associated with the assumptions of its point particle nature. The
literature on this subject is fairly large and goes back all the way to
Abraham and Lorentz. AFAIK, none of these models have a claim to being
"fundamental". To the best of our knowledge, this title goes to the QED
treatment of the electron.

Igor

Hendrik van Hees

Jay R. Yablon wrote:


> For those familiar with Halzen and Martin as well as Peskin and
> Schroeder,
> how do they compare? What will Schroeder have which Halzen and Martin
> does not, or, how does Peskin & Schroeder perhaps amplify or delve
> more deeply into the subject matter covered by Halzen and Martin?

I don't know Halzen and Martin very well, but I think it is more a
particle physics text book, while P&S is a (somewhat overrated)
textbook on quantum field theory. One should always read it very
critically. The erratum page is good, but corrects only for typos not
for conceptually wrong statements (as one can find in the section on
the renormalization of the linear sigma model and in the chapter on the
renormalization group). If one likes to learn QFT, I'd recommend
Weinberg's Quantum Theory of Fields which is much better in the
conceptual development, although it's missing a little bit the more
technical details of doing calculations.

--
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/ mailto:hees@comp.tamu.edu

Jay R. Yablon

"Igor Khavkine" <igor.kh@gmail.com> wrote:

> Halzen & Martin is a cookbook. Its goal is to teach you particle
> physics, the kind that is done is high energy accelecrators and
> colliders. QED and the Standard Model happen to describe the outcomes
> of such experiments quite well, that's why they are included in this
> book.
>
> A book on Quantum Field Theory proper, such as Pesking & Schroeder,
> aims to teach you how to apply quantum mechanics to fields (mechanical
> systems with infinitely many degrees of freedom). This includes
> directing your attention to many subtleties that may not be apparent if
> you are only interested in one specific application. In principle, it
> should also teach you how to construct a QFT of your own. Of course,
> P&S do aim their contents toward application to particle physics.
> However, there are many QFT books that aim instead toward condensed
> matter or nuclear physics. QFT is a very general subject, and it's good
> to be aware of its breadth.

"Hendrik van Hees" hees@comp.tamu.edu> wrote:

> I don't know Halzen and Martin very well, but I think it is more a
> particle physics text book, while P&S is a (somewhat overrated)
> textbook on quantum field theory. One should always read it very
> critically. The erratum page is good, but corrects only for typos not
> for conceptually wrong statements (as one can find in the section on
> the renormalization of the linear sigma model and in the chapter on the
> renormalization group). If one likes to learn QFT, I'd recommend
> Weinberg's Quantum Theory of Fields which is much better in the
> conceptual development, although it's missing a little bit the more
> technical details of doing calculations.
>

I have been looking at the reviews on Google and am thinking of starting out
studying Zee's "Quantum Field Theory in a Nutshell." How does that compare
to P&S?

Jay
____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com

sr

Jay Yablon asked:

> I have been looking at the reviews on Google
> and am thinking of starting out studying
> Zee's "Quantum Field Theory in a Nutshell."
> How does that compare to P&S?

Opinions differ, but here's my $0.02 worth...

Zee emphasizes concepts, and the functional
integral method. But you will not learn how to do
detailed calculations of cross-sections for
various processes from Zee. P&S is better for
that, but one must remember it is only an
introductory book. Weinberg is advanced and can be
quite frustrating for those who don't already
have a reasonable understanding of basic QFT.

I acquired all 3, (and several others besides).
I don't regret the investment in any of them. I
studied P&S first, then Zee. That was ok
because I'd already learned a lot of the gory
calculation skills from P&S so that when Zee
skipped over them I didn't mind too much. Zee boils
a lot of tricky ideas down to a simpler essence.
I got the hang of the functional integral method
much more easily from Zee than P&S.

I think the serious student should study both P&S and
Zee. Then confront the weighty tomes of Weinberg.

Whichever book you start with, you'll find some
good things, and some things to complain about.
The important thing is to get stuck into one
of them, and eventually study the others too,
for a more well-rounded education.

Igor Khavkine

Jay R. Yablon wrote:

> I have been looking at the reviews on Google and am thinking of starting out
> studying Zee's "Quantum Field Theory in a Nutshell." How does that compare
> to P&S?

Any introductory book is good to start with. However, once you try to
understand any particular topic, you'll find that every book has its
advantages and deficits. One book that I know devotes a lot of effort
to guiding the reader through the elementary steps is Quantum Field
Theory by Ryder. Other books, like Weinberg's or de Witt's, are much
more high brow about the elementary steps and devote more time to
generalities and fine points. P&S falls somewhere in between. It covers
many topics and tries to get the reader to do calculations as early as
possible. I would imagine that Zee's book, like many other QFT books,
is somewhere on par with P&S but with greater emphasis on path integral
techniques. If you have access to a university library, you'll be able
to find all these books there. So don't spend too much time fretting
about which book to start with. Start somewhere, then look up
alternative references when you get stuck.

Ideally, all these books should be teaching you the same thing:

* Canonical field quantization (Fock space)
* Path integral quantization
* Relation between the Heisenberg and Interaction pictures
* Time evolution or the S-matrix through the Dyson formula
* Perturbative expansion, propagators, and Feynman diagrams
* Renormalization

It usually takes graduate students at least one semester to go through
these basic topics and an additioinal semester to get through a few
other topics including applications. There is a long road ahead.

Igor