Science > Physics > Shubert's Derivation of the Lorentz Transformations
| Topic: |
Science > Physics |
| User: |
"Shubee" |
| Date: |
18 Aug 2007 06:38:56 PM |
| Object: |
Shubert's Derivation of the Lorentz Transformations |
Understanding the Lorentz transformation through the most powerful and
insightful mathematical model possible requires being able to derive
the transformation from the weakest axioms available.
The usual approach to the Lorentz transformation is to presuppose no
absolute frame of reference. I believe that it's more instructive to
derive a more general theory.
The physicist J. H. Field also sees the importance of beginning with
less dogmatic assumptions. Consider his viewpoint:
"A much weaker statement of the Relativity Principle than Einstein's
first postulate is sufficient to derive the Lorentz Transformation."
"It was recognised at an early date by Ignatowsky and Frank and Rothe
that Einstein's second postulate was not necessary to derive the
Lorentz Transformation. The questions then arise: what are the weakest
postulates which are sufficient to derive it and what is their minimum
number?"
In fulfillment of Dr. Field's objective, I believe that I have
discovered the leanest derivation and the weakest postulate. It's easy
to see that all the empirical results from Einsteinian SR can be
obtained from a simple tautology. There are laws of physics that are
the same in all frames of reference and there may be laws of physics
that aren't.
I believe that Einstein was too religious. He assumed that all
physical laws are the same in all inertial frames of reference. My
weaker axiom set only requires that my definition of time be similarly
defined in all inertial frames of reference. Therefore, my simple
tautology is inviolable and consistent. There are laws of physics that
are Lorentz invariant and there may be laws of physics that aren't.
Ultimately, I believe that Einstein's theory is overly simplistic. My
generalization is obviously more complex. Every mathematician should
see the beauty of it. Einstein's theory could be false and my theory
could be true. But if my theory is false, then so is Einstein's
theory.
http://www.everythingimportant.org/relativity/special.pdf
Shubee
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 08:36:42 AM |
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Alfred Ziegler gives an overview of representative derivations of the
LT found in textbooks. See, The role of the two postulates of special
relativity (Dated: August 8, 2007).
http://www.arxiv.org/abs/0708.0988
VII. A SURVEY OF TEXTBOOKS
A. Linearity
The majority of textbooks sees no need to justify the linearity
of the transformation5,7,8,9,10,11,12,13,14,15. When given it
is derived either from homogeneity of time and space3,16,17,
18,19,20 or from Newton's first law which requires uniform and
thus straight motion to remain uniform and straight in another
system of reference4,21,22. Resnick3 is the most explicit text
in this respect. Actually a lot more "obvious" statements
regarding
the possible choices of coordinates have to be demonstrated which
among the texts listed here are only given in Rindler17.
Wow! According to this paragraph by Ziegler, it seems that the
majority of physicists that write for the purpose of explaining the
foundations of relativity are satisfied with voodoo physics. And when
physicists try to explain the foundations, they argue that linearity
follows from the homogeneity of time and space, which is totally
false, or they invoke a coordinate dependent version of Newton's first
law, which presupposes an unacknowledged law of physics. It is an
intrinsic law of nature that says that somehow the universe favors a
linear clock synchronization scheme or is it a law of the universe
that clock synchronization schemes are unconstrained and free be
arbitrary?
Shubee
http://www.everythingimportant.org/relativity/special.pdf
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 08:49:19 AM |
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Alfred Ziegler gives an overview of representative derivations of the
LT found in textbooks. See, The role of the two postulates of special
relativity (Dated: August 8, 2007). http://www.arxiv.org/abs/0708.0988
VII. A SURVEY OF TEXTBOOKS
A. Linearity
The majority of textbooks sees no need to justify the linearity
of the transformation5,7,8,9,10,11,12,13,14,15. When given it
is derived either from homogeneity of time and space3,16,17,
18,19,20 or from Newton's first law which requires uniform and
thus straight motion to remain uniform and straight in another
system of reference4,21,22. Resnick3 is the most explicit text
in this respect. Actually a lot more "obvious" statements
regarding the possible choices of coordinates have to be
demonstrated which among the texts listed here are only given
in Rindler17.
Wow! According to this paragraph by Ziegler, it seems that the
majority of physicists that write for the purpose of explaining the
foundations of relativity are satisfied with voodoo physics. And when
physicists try to explain the foundations, they argue that linearity
follows from the homogeneity of time and space, which is totally
false, or they invoke a coordinate dependent version of Newton's first
law, which presupposes an unacknowledged law of physics. Is it an
intrinsic law of nature that says that somehow the universe favors a
linear clock synchronization scheme or is it a law of the universe
that clock synchronization schemes are unconstrained and free to be
arbitrary?
Shubee
http://www.everythingimportant.org/relativity/special.pdf
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| User: "Eric Gisse" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 02:55:13 PM |
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On Sun, 19 Aug 2007 06:49:19 -0700, Shubee <e.Shubee@gmail.com> wrote:
[...]
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
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| User: "Bill Hobba" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 09:39:56 PM |
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"Eric Gisse" <jowr.pi.nospam@gmail-nospam.com> wrote in message
news:pv7hc3h9itgi0063lh806rk7uo7mb7a6s3@4ax.com...
On Sun, 19 Aug 2007 06:49:19 -0700, Shubee <e.Shubee@gmail.com> wrote:
[...]
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
To be specific, they are applied mathematicians where certain reasonable
mathematical properties are assumed by insight into what you are applying it
to. For example, when applying it infinite Markov chains, one usually
assumes the limit process in the chain states going to infinity can be
interchanged with the one raising it to a power. Interestingly a deeper
analysis using renewal theory shoes it is always justified. But that is
just in that case - often one makes such assumptions because the infinite
case is really an approximation to a finite one - infinite chains can not
actually physically occur. Same sort of considerations are made all the
time in physics.
Thanks
Bill
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| User: "Eric Gisse" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
20 Aug 2007 01:04:49 AM |
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On Aug 19, 6:39 pm, "Bill Hobba" <rubb...@junk.com> wrote:
"Eric Gisse" <jowr.pi.nos...@gmail-nospam.com> wrote in message
news:pv7hc3h9itgi0063lh806rk7uo7mb7a6s3@4ax.com...
On Sun, 19 Aug 2007 06:49:19 -0700, Shubee <e.Shu...@gmail.com> wrote:
[...]
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
To be specific, they are applied mathematicians where certain reasonable
mathematical properties are assumed by insight into what you are applying it
to. For example, when applying it infinite Markov chains, one usually
assumes the limit process in the chain states going to infinity can be
interchanged with the one raising it to a power. Interestingly a deeper
analysis using renewal theory shoes it is always justified. But that is
just in that case - often one makes such assumptions because the infinite
case is really an approximation to a finite one - infinite chains can not
actually physically occur. Same sort of considerations are made all the
time in physics.
I got a good example too.
Right now I'm playing around with Einstein-Cartan theory. I'm trying
to get the Schwarzschild solution that has spin. I'm toying with the
idea that the Kerr solution might be obtainable [not through
Schwarzschild - need an axially symmetric metric] as well. But
Schwarzschild first.
Without making some specific assumptions about the nature of the
solution, I wouldn't get very far.
If I didn't assume that spin is constant [w.r.t. the spherically
symmetric coordinate system...], it would be impossible to solve.
If I didn't assume reduction to Minkowski in the m,spin-->0 limit, I'd
be clueless.
If I didn't assume reduction to the Schwarzschild solution in the
spin-->0 limit, I would not have made the cute discovery that the g_tt
component splits as A(r)+B(t). Which was really ***** cool.
Ok, that's it because the third assumption *did* reduce the 5 PDE [4
unique] system to 3 [unique] PDEs of substantially lower difficulty.
But I'm still stuck. However, if I played physics like Eugene wants to
do, I could never do anything because I would be wasting my time going
for full generality regardless of physical utility.
Thanks
Bill
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 12:08:23 AM |
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Eric Gisse wrote:
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
One also needs isotropy of space. Then it is "good enough" for
mathematicians and physicists alike. And, of course, for SR one has both
(and for GR one in general has neither).
Tom Roberts
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| User: "Eric Gisse" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 12:15:56 AM |
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On Aug 20, 9:08 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Eric Gisse wrote:
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
One also needs isotropy of space. Then it is "good enough" for
mathematicians and physicists alike. And, of course, for SR one has both
(and for GR one in general has neither).
What's the operational difference between homogeneity and isotropy?
Tom Roberts
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 11:31:58 AM |
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Eric Gisse wrote:
On Aug 20, 9:08 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Eric Gisse wrote:
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
One also needs isotropy of space. Then it is "good enough" for
mathematicians and physicists alike. And, of course, for SR one has both
(and for GR one in general has neither).
What's the operational difference between homogeneity and isotropy?
A homogeneous manifold has a metric independent of position; in physics
one must also have all other tensors be independent of position. A
manifold that is isotropic at point p has a metric independent of
orientation around point p; ditto for other tensors in physics. Any
manifold that is isotropic at all points is homogeneous, but a
homogeneous manifold need not be isotropic.
The Euclidean n-space E^n is both homogeneous and isotropic everywhere.
The X-T plane of Minkowski spacetime (any one of them) is homogeneous
but is not isotropic at any point. One needs isotropy of space in the
quote above to prevent such an anisotropy in the X-Y plane (etc.).
Tom Roberts
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| User: "Eric Gisse" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 12:02:27 PM |
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On Aug 21, 8:31 am, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Eric Gisse wrote:
On Aug 20, 9:08 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Eric Gisse wrote:
Physicists are not mathematicians. That linearity follows from the
homogeneity of space and time is /good enough/.
One also needs isotropy of space. Then it is "good enough" for
mathematicians and physicists alike. And, of course, for SR one has both
(and for GR one in general has neither).
What's the operational difference between homogeneity and isotropy?
A homogeneous manifold has a metric independent of position; in physics
one must also have all other tensors be independent of position. A
manifold that is isotropic at point p has a metric independent of
orientation around point p; ditto for other tensors in physics. Any
manifold that is isotropic at all points is homogeneous, but a
homogeneous manifold need not be isotropic.
The Euclidean n-space E^n is both homogeneous and isotropic everywhere.
The X-T plane of Minkowski spacetime (any one of them) is homogeneous
but is not isotropic at any point. One needs isotropy of space in the
quote above to prevent such an anisotropy in the X-Y plane (etc.).
Tom Roberts
That certainly is a subtle point. Thanks!
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 08:41:57 AM |
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Alfred Ziegler gives an overview of representative derivations of the
LT found in textbooks. See, The role of the two postulates of special
relativity (Dated: August 8, 2007). http://www.arxiv.org/abs/0708.0988
VII. A SURVEY OF TEXTBOOKS
A. Linearity
The majority of textbooks sees no need to justify the linearity
of the transformation5,7,8,9,10,11,12,13,14,15. When given it
is derived either from homogeneity of time and space3,16,17,
18,19,20 or from Newton's first law which requires uniform and
thus straight motion to remain uniform and straight in another
system of reference4,21,22. Resnick3 is the most explicit text
in this respect. Actually a lot more "obvious" statements
regarding the possible choices of coordinates have to be
demonstrated which among the texts listed here are only given
in Rindler17.
Wow! According to this paragraph by Ziegler, it seems that the
majority of physicists that write for the purpose of explaining the
foundations of relativity are satisfied with voodoo physics. And when
physicists try to explain the foundations, they argue that linearity
follows from the homogeneity of time and space, which is totally
false, or they invoke a coordinate dependent version of Newton's first
law, which presupposes an unacknowledged law of physics. It is an
intrinsic law of nature that says that somehow the universe favors a
linear clock synchronization scheme or is it a law of the universe
that clock synchronization schemes are unconstrained and free be
arbitrary?
Shubee
http://www.everythingimportant.org/relativity/special.pdf
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| User: "Juan R." |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 09:13:47 AM |
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On Aug 19, 3:41 pm, Shubee <e.Shu...@gmail.com> wrote:
Alfred Ziegler gives an overview of representative derivations of the
LT found in textbooks. See, The role of the two postulates of special
relativity (Dated: August 8, 2007).http://www.arxiv.org/abs/0708.0988
VII. A SURVEY OF TEXTBOOKS
A. Linearity
The majority of textbooks sees no need to justify the linearity
of the transformation5,7,8,9,10,11,12,13,14,15. When given it
is derived either from homogeneity of time and space3,16,17,
18,19,20 or from Newton's first law which requires uniform and
thus straight motion to remain uniform and straight in another
system of reference4,21,22. Resnick3 is the most explicit text
in this respect. Actually a lot more "obvious" statements
regarding the possible choices of coordinates have to be
demonstrated which among the texts listed here are only given
in Rindler17.
Wow! According to this paragraph by Ziegler, it seems that the
majority of physicists that write for the purpose of explaining the
foundations of relativity are satisfied with voodoo physics. And when
physicists try to explain the foundations, they argue that linearity
follows from the homogeneity of time and space, which is totally
false, or they invoke a coordinate dependent version of Newton's first
law, which presupposes an unacknowledged law of physics. It is an
intrinsic law of nature that says that somehow the universe favors a
linear clock synchronization scheme or is it a law of the universe
that clock synchronization schemes are unconstrained and free be
arbitrary?
Shubeehttp://www.everythingimportant.org/relativity/special.pdf
You may find also interesting Eugene's discussion on section 10.3.1 on
http://www.arxiv.org/pdf/physics/0504062
I find beatiful the part
{BLOCKQUOTE
There are two common features of these derivations, which we nd
troublesome. First, they assume an abstract (i.e., independent on real
phys-
ical processes and interactions) nature of events occupying space-time
points
(t, x, y, z). Second, they postulate the isotropy and homogeneity of
space
around these points. It is true that these assumptions imply linear
univer-
sal character of Lorentz transformations, and, after some algebra,
speci c
expressions (J.2) - (J.5) follow. The main problem with these
approaches
is that in physics we should be interested in transformations of
observables
of real interacting particles, not of abstract space-time points. One
cannot
make an assumption that transformations of these observables are
completely
independent of what occurs in the space surrounding the particle and
what
are interactions of this particle with the rest of the observed
system. One
can reasonably assume that all directions in space are exactly
equivalent for
a single isolated particle [131], but this is not at all obvious when
the particle
participates in interactions.
Suppose that we have two interacting particles 1 and 2 at some
distance
from each other. Suppose that we want to derive boost transformations
for
observables of the particle 1. Clearly, for this particle di erent
directions in
space are not equivalent: For example, the direction pointing to the
particle
2 is di erent from other directions. So, the assumption of the spatial
isotropy
cannot be applied here.
}
Maxwell electrodynamics, QED, and GR are esentially one-body theories.
Those theories doe not modell the *full* two-body system. That is the
reason that textbooks you cite earlier introduce the one-body
Lagrangian and maybe some two-body approximated Lagrangians (e.g.
Darwin one) but never the *full* two-body system.
http://canonicalscience.blogspot.com/2007/08/relativistic-lagrangian-and-limitations.html
==========================================================
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 11:34:02 AM |
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On Aug 19, 7:13 am, "Juan R." <juanrgonzal...@canonicalscience.com>
wrote:
You may find also interesting Eugene's discussion on section 10.3.1
on http://www.arxiv.org/pdf/physics/0504062
Eugene Stefanovich wrote on p. 406 of that section:
"There is a significant number of publications which claim that
Lorentz transformation formulas (J.2) - (J.5) can be derived even
without using the Einstein's second postulate (see [127, 128, 129,
130, 131] and references cited there). There are two common features
of these derivations, which we find troublesome. First, they assume an
abstract (i.e., independent on real physical processes and
interactions) nature of events occupying space-time points (t, x, y,
z). Second, they postulate the isotropy and homogeneity of space
around these points."
Why should the isotropy and homogeneity of space be considered
troublesome? It's just a postulate. Can Eugene prove that this
postulate is inconsistent? Eugene might call it a naive postulate but
to the best of my understanding and belief, there is, at the moment,
no conclusive argument or experimental evidence that contradicts it.
Likewise, why should assuming the nature of space and time being
independent of real physical processes and interactions be troublesome
to anyone? Physicists are free to create any axiom set they wish. If
Eugene is smart enough to devise mathematical equations for a new
quantum theory where spacetime is changing while matter and energy are
undergoing physical processes and interactions, God bless him. But not
until Eugene gets there can his book be considered physics. Eugene
Stefanovich is merely in the pursuit of physics. The meaning of words
is important here. For my definition of axiomatic physics, see section
2 of http://www.everythingimportant.org/relativity/special.pdf
From what I know about real physics, the most troubling fact about
mainstream physics is that too many physicists are driven by hotly
pursuing grand theories while demonstrating a very poor understanding
of the simplest fundamentals.
Shubee
http://www.everythingimportant.org/relativity/special.pdf
{BLACKLIST
{ Bilge; Bill Hobba; Dono (once Karandash2); Eric Gisse; Tim
Shuba; }}
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
19 Aug 2007 11:31:50 AM |
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On Aug 19, 7:13 am, "Juan R." <juanrgonzal...@canonicalscience.com>
wrote:
You may find also interesting Eugene's discussion on section 10.3.1
on http://www.arxiv.org/pdf/physics/0504062
Eugene Stefanovich wrote on p. 406 of that section:
"There is a significant number of publications which claim that
Lorentz transformation formulas (J.2) - (J.5) can be derived even
without using the Einstein's second postulate (see [127, 128, 129,
130, 131] and references cited there). There are two common features
of these derivations, which we find troublesome. First, they assume an
abstract (i.e., independent on real physical processes and
interactions) nature of events occupying space-time points (t, x, y,
z). Second, they postulate the isotropy and homogeneity of space
around these points."
Why should the isotropy and homogeneity of space be considered
troublesome? It's just a postulate. Can Eugene prove that this
postulate is inconsistent? Eugene might call it a naive postulate but
to the best of my understanding and belief, there is, at the moment,
no conclusive argument or experimental evidence that contradicts it.
Likewise, why should assuming the nature of space and time being
independent of real physical processes and interactions be troublesome
to anyone? Physicists are free to create any axiom set they wish. If
Eugene is smart enough to devise mathematical equations for a new
quantum theory where spacetime is changing while matter and energy are
undergoing physical processes and interactions, God bless him. But not
until Eugene gets there can his book be considered physics. Eugene
Stefanovich is merely in the pursuit of physics. The meaning of words
is important here. For my definition of axiomatic physics, see section
2 of http://www.everythingimportant.org/relativity/special.pdf.
From what I know about real physics, the most troubling fact about
mainstream physics is that too many physicists are driven by hotly
pursuing grand theories while demonstrating a very poor understanding
of the simplest fundamentals.
Shubee
http://www.everythingimportant.org/relativity/special.pdf
{BLACKLIST
{ Bilge; Bill Hobba; Dono (once Karandash2); Eric Gisse; Tim
Shuba; }}
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 01:47:23 AM |
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On Aug 19, 9:31 am, Shubee <e.Shu...@gmail.com> wrote:
Why should the isotropy and homogeneity of space be considered
troublesome? It's just a postulate. Can Eugene prove that this
postulate is inconsistent? Eugene might call it a naive postulate but
to the best of my understanding and belief, there is, at the moment,
no conclusive argument or experimental evidence that contradicts it.
I think it is important to distinguish two very different meanings of
the expression "isotropy and homogeneity of space".
The first meaning is just the basic principle of relativity: Identical
experiments performed in different regions of space (here in the Solar
system or in the Andromeda galaxy) or at different times (today or
tomorrow) yield identical results. I fully agree with this postulate.
Although there are theories which challenge these assumptions (e.g.,
claiming that physical constants change with time), I don't think
these theories have any credible support.
It is important to realize that "derivations" of Lorentz
transformations (quoted as references [127-130] in my book) do not
use the above definition of the "isotropy and inhomogenity of space".
They use the same words, but apply them in a completely different
situation. To show the absurdity of these "derivations", let us take a
system of two classical particles A and B that interact with each
other. (For example, these could be two point charges) Suppose that
observer at rest O measures particle A position as r_A. Suppose that
we want to find what is the position of A from the point of view of
the moving oberver O' (r_A'). In the above references it is (wrongly)
assumed that when calculating r_A' from r_A we can think that all
directions from the point A are physically equivalent. This would be
correct if particles A and B didn't interact with each other. However,
they do interact, and this interaction certainly makes the direction
from A to B somewhat different from other directions.
To make it even more clear, let us apply the same idea of "isotropy"
to time translations (dynamics) rather than to boosts. Suppose that we
want to find a relationship between the position of A in the reference
frame O and the position of A in the reference frame O'' displaced in
time with respect to O. If we apply the same logic about "isotropic
space" we would reach an absurd conclusion that for the particle A
there is no preferred direction of movement. This is wrong, because
there is a vector of force acting on A from the direction of B. So,
for calculations of trajectories of interacting particles we cannot
use the "isotropy" argument. If this argument doesn't work for time
translations, why should it work for boosts?
Eugene.
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 02:09:52 PM |
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wrote:
I think it is important to distinguish two very different meanings of
the expression "isotropy and homogeneity of space".
I think first it is important to resolve your wildly unconventional
usage of he word "space".
The first meaning is just the basic principle of relativity: Identical
experiments performed in different regions of space (here in the Solar
system or in the Andromeda galaxy) or at different times (today or
tomorrow) yield identical results. I fully agree with this postulate.
Although there are theories which challenge these assumptions (e.g.,
claiming that physical constants change with time), I don't think
these theories have any credible support.
This has NOTHING WHATSOEVER to do with isotropy or homogeneity of space.
This has to do with the invariance of physical laws under spatial and
temporal translations.
For instance, in GR space is neither homogeneous nor isotropic [#], but
if one performed the same LOCAL experiment here or in Andromeda, one
would obtain the same results (given identical local conditions, of
course). The laws of physics are invariant under spatial and temporal
translations, but space itself is not.
[#] nor is spacetime.
It is important to realize that "derivations" of Lorentz
transformations (quoted as references [127-130] in my book) do not
use the above definition of the "isotropy and inhomogenity of space".
Of course not! Nobody uses that definition for "isotropy and homogeneity
of space", because it applies to the laws of physics, not to space.
[... a repeat, confusing space with interactions]
Once you avoid confusing space with interactions, you can then explore
what "isotropy and homogeneity of space" really mean. Then you'll
understand why SR is based on these concepts, but GR does not -- before
GR space and time were a "stage" on which the physics played, but did
not participate in any dynamics; since GR they are no longer passive
entities but have become active participants in the dynamics of physical
systems.
[See another recent post of mine in this thread for the
definitions of these terms.]
Tom Roberts
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| User: "Androcles" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 03:03:14 PM |
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"Tom Roberts" <tjroberts137@sbcglobal.net> wrote in message
news:4MGyi.50259$YL5.49252@newssvr29.news.prodigy.net...
: wrote:
: > I think it is important to distinguish two very different meanings of
: > the expression "isotropy and homogeneity of space".
:
: I think
LIAR!
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
21 Aug 2007 03:46:07 PM |
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On Aug 21, 12:09 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
I think first it is important to resolve your wildly unconventional
usage of he word "space".
The question we were discussing was about derivations of Lorentz
transformations for (e.g.) the position of the particle A in a two-
particle system (A+B). Are we allowed to assume that all direction
around particle A are equal? I think that we are not allowed to assume
that, because there is particle B around and A interacts with B. So,
it seems to me, there IS a preferred direction. Apparently, you are
willing to ignore this circumstance and assume that this direction
doesn't matter. Do you have a justification for that?
You see, I haven't even used the words "space" or "spacetime" here.
So, whatever my misunderstanding of these notions is, it is
irrelevant.
Eugene.
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| User: "Shubee" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
22 Aug 2007 07:20:16 AM |
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On Aug 20, 11:47 pm, wrote:
On Aug 19, 9:31 am, Shubee <e.Shu...@gmail.com> wrote:
Why should the isotropy and homogeneity of space be considered
troublesome? It's just a postulate. Can Eugene prove that this
postulate is inconsistent? Eugene might call it a naive postulate but
to the best of my understanding and belief, there is, at the moment,
no conclusive argument or experimental evidence that contradicts it.
I think it is important to distinguish two very different meanings of
the expression "isotropy and homogeneity of space".
The first meaning is just the basic principle of relativity: Identical
experiments performed in different regions of space (here in the Solar
system or in the Andromeda galaxy) or at different times (today or
tomorrow) yield identical results. I fully agree with this postulate.
Although there are theories which challenge these assumptions (e.g.,
claiming that physical constants change with time), I don't think
these theories have any credible support.
It is important to realize that "derivations" of Lorentz
transformations (quoted as references [127-130] in my book) do not
use the above definition of the "isotropy and inhomogenity of space".
They use the same words, but apply them in a completely different
situation. To show the absurdity of these "derivations",
Eugene, aren't you saying that not only are those wrongheaded
derivations wrong but that their ultimate conclusion (the Lorentz
transformation itself) is wrong also in a universe where relativistic
non-instantaneous action-at-a-distance forces between particles
operate?
let us take a
system of two classical particles A and B that interact with each
other. (For example, these could be two point charges) Suppose that
observer at rest O measures particle A position as r_A. Suppose that
we want to find what is the position of A from the point of view of
the moving oberver O' (r_A'). In the above references it is (wrongly)
assumed that when calculating r_A' from r_A we can think that all
directions from the point A are physically equivalent.
Are you criticizing a well-established physics textbook? Which one?
The rules of established theories of physics are extraordinarily
clear. In this instance the name of the game is called Lorentz
invariance. That means that events transform according to the Lorentz
transformation.
Shubee
This would be
correct if particles A and B didn't interact with each other. However,
they do interact, and this interaction certainly makes the direction
from A to B somewhat different from other directions.
To make it even more clear, let us apply the same idea of "isotropy"
to time translations (dynamics) rather than to boosts. Suppose that we
want to find a relationship between the position of A in the reference
frame O and the position of A in the reference frame O'' displaced in
time with respect to O. If we apply the same logic about "isotropic
space" we would reach an absurd conclusion that for the particle A
there is no preferred direction of movement. This is wrong, because
there is a vector of force acting on A from the direction of B. So,
for calculations of trajectories of interacting particles we cannot
use the "isotropy" argument. If this argument doesn't work for time
translations, why should it work for boosts?
Eugene.
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
22 Aug 2007 01:00:14 PM |
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On Aug 22, 5:20 am, Shubee <e.Shu...@gmail.com> wrote:
Eugene, aren't you saying that not only are those wrongheaded
derivations wrong but that their ultimate conclusion (the Lorentz
transformation itself) is wrong also in a universe where relativistic
non-instantaneous action-at-a-distance forces between particles
operate?
I think that Lorentz transformations should be modified in the
presence of ANY interaction, even the retarded one. The reason is very
simple. Interaction means a deviation of the Poincare group
representation from its non-interacting form. This modification must
preserve the commutation relations of the Poincare Lie algebra in
order to be consistent with the principle of relativity. This means
that both generators of time translations (the Hamiltonian) and boosts
should be interaction-dependent. Interaction-dependent boost
generators imply interaction-dependent boost transformations of
observables.
Are you criticizing a well-established physics textbook? Which one?
You can pick any special relativity textbook you like, and I will show
you the exact place where the author makes an unsubstiated assumption
which makes his/her "derivation" invalid, or, at least, inaccurate.
Eugene.
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
22 Aug 2007 08:32:11 PM |
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wrote:
I think that Lorentz transformations should be modified in the
presence of ANY interaction, even the retarded one.
First you should understand what is being modeled, and what is NOT being
modeled. In SR, spacetime is a model of the spatial and temporal
relationships among objects, generalized to points of a manifold.
Interactions are NOT modeled. This is in the pre-GR approach that space
and time are "stages" on which the physical phenomena play out their
drama (dynamics). Because interactions do not affect the underlying
spacetime manifold, one can add to SR whatever dynamical theory one
likes, as long as it is consistent with SR (i.e. is Lorentz invariant).
The reason is very
simple. Interaction means a deviation of the Poincare group
representation from its non-interacting form.
No. In the model of SR, the underlying spacetime is not affected by any
interactions, and neither are its symmetries, such as the Poincare'
group. There is no such thing as "interacting Poincare' group" and
"non-interacting Poincare' group", there is just the Poincare' group.
This modification must
preserve the commutation relations of the Poincare Lie algebra in
order to be consistent with the principle of relativity.
But that is the whole group! (well, up to isomorphism...) This is
inconsistent with your first statement quoted above.
It seems to me you are trying to "graft" a portion of GR onto SR -- that
is hopeless. In GR, the presence of interactions does indeed modify the
structure of the spacetime manifold, and the manifold and its metric
participate in the dynamics. This has a PROFOUND effect on the theory.
But in GR there is no Lorentz symmetry, there is only a LOCAL Lorentz
symmetry that is APPROXIMATE in the neighborhood of a specified point
(the region of its validity depends on the local curvature of the
manifold and on the accuracy to which it is supposed to hold).
So I suggest you learn about GR, rather than trying to "modify" SR. A
good start is:
Geroch, _General_Relativity_from_A_to_B_.
Tom Roberts
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| User: "Androcles" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
22 Aug 2007 08:36:01 PM |
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"Tom Roberts" <tjroberts137@sbcglobal.net> wrote in message
news:Ts5zi.854$YQ.848@nlpi061.nbdc.sbc.com...
: wrote:
: > I think that Lorentz transformations should be modified in the
: > presence of ANY interaction, even the retarded one.
:
: First you should understand
Second you should shut the ***** up and STUDY, Roberts.
'we establish by definition that the "time" required by
light to travel from A to B equals the "time" it requires
to travel from B to A' because I SAY SO and you have to
agree because I'm the great genius, STOOOPID, don't you
dare question it. -- Albert Einstein,
who in 1895 failed an examination that would have allowed
him to study for a diploma as an electrical engineer at
the Eidgenössische Technische Hochschule in Zurich
(couldn't even pass the SATs).
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
22 Aug 2007 11:41:35 PM |
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On Aug 22, 6:32 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Interaction means a deviation of the Poincare group
representation from its non-interacting form.
No. In the model of SR, the underlying spacetime is not affected by any
interactions, and neither are its symmetries, such as the Poincare'
group. There is no such thing as "interacting Poincare' group" and
"non-interacting Poincare' group", there is just the Poincare' group.
Yes, there is just one Poincare group which is the group of
transformations between different inertial observers. And you might
notice that I didn't say "interacting Poincare' group" or "non-
interacting Poincare' group". I was talking about different
*representations* of the same Poincare group. These representations
can be either non-interacting or interacting. This notion of Poincare
group representations is the most fundamental notion in relativistic
physics. It is absolutely necessary in order to understand how
relativistic interacting theories can be formulated. The idea is most
easily explained for quantum theories.
It begins with Wigner's work
E. P. Wigner "On unitary representations of the inhomogeneous Lorentz
group" Ann. Math. 40 (1939), 149
where he showed that any relativistic quantum system is described by a
Hilbert space in which a unitary representation of the Poincare group
is defined. This means (in a somewhat simplified language) that for
each element g of the Poincare group a unitary operator U_g
corresponds, so that the multiplication law of operators U_g is the
same as the multiplication law in the Poincare group. Moreover, if F
is operator of an observable in the reference frame O, and reference
frame O' is related to O by a transformation g, then the operator of
observable in the reference frame O' is
F' = U_g F U_g^{-1}................(1)
The significance of this result is that as soon as we know the
representation of the Poincare group in the Hilbert space of the
system, we know everything we need to do physics. For example, if g is
time translation, then eq. (1) tells us everything about dynamics. If
g is boost, then eq. (1) gives us the boost transformation law for any
observable. In principle, we should be able to derive Lorentz
transformations from (1).
Wigner provided a classification of all possible (irreducible) unitary
representations of the Poincare group for 1-particle systems. The
choice of the representation U_g for multiparticle systems is the most
challenging problem in relativistic quantum physics. For a given
multiparticle system the representation U_g can be defined in an
infinite number of ways. The simplest representation U_g0 corresponds
to non-interacting particles. It can be shown that with this
representation (in the classical limit) all particles move with
constant velocities along straight lines and boost transformations of
particle observables are given by Lorentz formulas.
Other possible representations correspond to different
relativistically-invariant ways to introduce interaction between
particles. These different "forms" of interaction were classified in
another important paper
P. A. M. Dirac, "Forms of relativistic dynamics" Rev. Mod. Phys. 21
(1949), 392.
From comparison with observations it follows that interactions in
nature belong to the so-called "instant form" of Dirac's dynamics. In
this case, representatives of generators of space translations and
rotations in the Hilbert space keep their non-interacting form, while
representatives of generators of time translations (the Hamiltonian)
and boosts acquire additional (interaction) terms that make them
different from non-interacting representatives of these
transformations. It is important to note that the Poincare group and
algebra of its generators remains the same in the presence of
interactions. However representatives (unitary operators U_g in the
Hilbert space of the system) of group elements g may be different.
The presence of interaction terms in the Hamiltonian is a well-known
fact. As a result of these terms particles no longer move along
straight lines. Their trajectories become curved due to their mutual
interactions.
It is less known that this relativistic theory demands that
representatives of boost transformations in the Hilbert space also
contain interaction terms. The immediate consequence of this fact is
that boost transformations of observables of interacting particles are
no longer given by simple linear universal Lorentz formulas. Boost
transformations become interaction-dependent.
This modification must
preserve the commutation relations of the Poincare Lie algebra in
order to be consistent with the principle of relativity.
But that is the whole group! (well, up to isomorphism...) This is
inconsistent with your first statement quoted above.
As I wrote above, the structure of the Poincare group is not affected
by interaction. Unitary representation preserves this structure as
well, and generators of the representation have the same commutation
relations as in the Lie algebra of the Poincare group itself.
Interaction is revealed in the difference between generators of this
representation and generators of the non-interacting representation.
One example of this difference is the "potential energy" operator in
the Hamiltonian. Another example is the "potential boost" operator in
the Hilbert space representative of the boost transformation between
observers.
It seems to me you are trying to "graft" a portion of GR onto SR -- that
is hopeless. In GR, the presence of interactions does indeed modify the
structure of the spacetime manifold, and the manifold and its metric
participate in the dynamics. This has a PROFOUND effect on the theory.
But in GR there is no Lorentz symmetry, there is only a LOCAL Lorentz
symmetry that is APPROXIMATE in the neighborhood of a specified point
(the region of its validity depends on the local curvature of the
manifold and on the accuracy to which it is supposed to hold).
In the above description I haven't used words "metric" or "spacetime
manifold" or "curvature". I can repeat that Lorentz (or Poincare)
group is an exact group of transformations between inertial observers.
The structure of this group does not depend on whether the observed
physical system is interacting or not. The Poincare group structure is
preserved when transformations between inertial observers are
represented by unitary operators in the Hilbert space of the system.
The only thing that depends on interaction is how generators of this
representation are related to generators of the non-interacting
representation, and what are commutators between interacting
generators and one-particle observables. These commutators ultimately
determine how particle observables transform with respect to boosts.
It can be proven that in an interacting theory these transformations
are different from Lorentz formulas.
Eugene.
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
23 Aug 2007 10:40:33 AM |
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wrote:
I was talking about different
*representations* of the same Poincare group. These representations
can be either non-interacting or interacting.
Hmmm. They are all related by isomorphisms....
This notion of Poincare
group representations is the most fundamental notion in relativistic
physics. It is absolutely necessary in order to understand how
relativistic interacting theories can be formulated. The idea is most
easily explained for quantum theories.
Certainly. The different representations of the Poincare' group apply to
different fundamental particles, distinguished by their intrinsic
spin/parity. But for all representations, the transforms of spacetime
coordinates are the same (they differ in the transform properties of
internal variables related to spin).
This has nothing whatsoever to do with your claims of non-isotropy due
to interactions. Indeed, in the theories you discuss spacetime is
Minkowski spacetime, and as I said before it is unaffected by the
presence of interactions. The overall physics depends on those
interactions, but the metric and other properties of spacetime do not.
It begins with Wigner's work
[...]
Yes, of course.
It is less known that this relativistic theory demands that
representatives of boost transformations in the Hilbert space also
contain interaction terms. The immediate consequence of this fact is
that boost transformations of observables of interacting particles are
no longer given by simple linear universal Lorentz formulas. Boost
transformations become interaction-dependent.
Sure, transforms IN THE HILBERT SPACE depend on which representation is
used; transforms IN SPACETIME do not. The underlying spacetime is
unaffected by the interactions.
A more modern viewpoint is that the interactions occur in a fiber bundle
over spacetime -- a completely geometric approach. The fibers consist of
a tensor product of the various representations of the Poincare' group
for the different spins of the elementary particles of the theory. The
structure of the fibers does not affect the base manifold (Minkowski
spacetime), and its isometry group remains the Poincaré group with its
usual Lorentz transforms.
It seems to me you are trying to "graft" a portion of GR onto SR -- that
is hopeless. [...]
In the above description I haven't used words "metric" or "spacetime
manifold" or "curvature".
No. The aspect of GR you seem to be trying to apply to SR is that
interactions affect the underlying spacetime manifold. In SR they simply
do not do so, and nothing you have said about group representations in a
Hilbert space does so.
I can repeat that Lorentz (or Poincare)
group is an exact group of transformations between inertial observers.
The structure of this group does not depend on whether the observed
physical system is interacting or not.
I agree with what you say here. But this seems to be considerably
different from what you claimed earlier.
The Poincare group structure is
preserved when transformations between inertial observers are
represented by unitary operators in the Hilbert space of the system.
The only thing that depends on interaction is how generators of this
representation are related to generators of the non-interacting
representation, and what are commutators between interacting
generators and one-particle observables.
All that is in the Hilbert space, not in spacetime.
These commutators ultimately
determine how particle observables transform with respect to boosts.
It can be proven that in an interacting theory these transformations
are different from Lorentz formulas.
Sure, for elements of the Hilbert space. But not for spacetime -- the
projection onto the spacetime manifold is the same for all
representations of the Poincaré group. This is so fundamental that most
textbooks do not even mention it, and concentrate on the
representations' behavior in the Hilbert space.
Tom Roberts
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| User: "Peter Christensen" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
25 Aug 2007 01:48:45 AM |
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On Aug 23, 5:40 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
eugene_stefanov...@usa.net wrote:
I was talking about different
*representations* of the same Poincare group. These representations
can be either non-interacting or interacting.
Hmmm. They are all related by isomorphisms....
This notion of Poincare
group representations is the most fundamental notion in relativistic
physics. It is absolutely necessary in order to understand how
relativistic interacting theories can be formulated. The idea is most
easily explained for quantum theories.
Certainly. The different representations of the Poincare' group apply to
different fundamental particles, distinguished by their intrinsic
spin/parity. But for all representations, the transforms of spacetime
coordinates are the same (they differ in the transform properties of
internal variables related to spin).
This has nothing whatsoever to do with your claims of non-isotropy due
to interactions. Indeed, in the theories you discuss spacetime is
Minkowski spacetime, and as I said before it is unaffected by the
presence of interactions. The overall physics depends on those
interactions, but the metric and other properties of spacetime do not.
It begins with Wigner's work
[...]
Yes, of course.
It is less known that this relativistic theory demands that
representatives of boost transformations in the Hilbert space also
contain interaction terms. The immediate consequence of this fact is
that boost transformations of observables of interacting particles are
no longer given by simple linear universal Lorentz formulas. Boost
transformations become interaction-dependent.
Sure, transforms IN THE HILBERT SPACE depend on which representation is
used; transforms IN SPACETIME do not. The underlying spacetime is
unaffected by the interactions.
A more modern viewpoint is that the interactions occur in a fiber bundle
over spacetime -- a completely geometric approach. The fibers consist of
a tensor product of the various representations of the Poincare' group
for the different spins of the elementary particles of the theory. The
structure of the fibers does not affect the base manifold (Minkowski
spacetime), and its isometry group remains the Poincar=E9 group with its
usual Lorentz transforms.
Round fibers - That must just be what we call 'flat spacetime'-
HeHe :-)
Or usual Minkowski space, I mean...
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| User: "Juan R." |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
23 Aug 2007 01:17:01 PM |
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On Aug 23, 5:40 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
eugene_stefanov...@usa.net wrote:
Indeed, in the theories you discuss spacetime is
Minkowski spacetime, and as I said before it is unaffected by the
presence of interactions. The overall physics depends on those
interactions, but the metric and other properties of spacetime do not.
Well, that is the standard asumption. Could you substantiate it, for
instance citing some two-body FULL relativistic Lagrangian (e.g. EM
interactions) and proving that effectively spacetime (LI, etc) remains
intact when one adds the interactions?
E=2Eg. in QED the metric and other properties of spacetime do not change
by introducting interactions over an original system of free particles
but the prize is too high, there is not time! No dinamics!
Sure, transforms IN THE HILBERT SPACE depend on which representation is
used; transforms IN SPACETIME do not. The underlying spacetime is
unaffected by the interactions.
The same comments than above.
A more modern viewpoint is that the interactions occur in a fiber bundle
over spacetime -- a completely geometric approach.
A more modern view is that geometric formulations of interactions are
limited to one-body solutions or certain dinamical limits for N-body
systems: scattering, c^2.
I would be glad to see one of your "fiber bundle over spacetime
approach" for next system:
Two charges (e.g. two electrons) switching between two stationary
regimes by an intermediate regime at constant v. Please formulate the
"geometric approach" and proves it is complete. Explicitely prove
continuity and locality during the switchs. By comodity you can work
with classical interactions: LW potentials.
The fibers consist of
a tensor product of the various representations of the Poincare' group
for the different spins of the elementary particles of the theory.
What elementary particles? Free, bound, bare, dressed? Define!
The
structure of the fibers does not affect the base manifold (Minkowski
spacetime), and its isometry group remains the Poincar=E9 group with its
usual Lorentz transforms.
Also for interacting particles? What is the quantum or classical state
then?
No. The aspect of GR you seem to be trying to apply to SR is that
interactions affect the underlying spacetime manifold. In SR they simply
do not do so, and nothing you have said about group representations in a
Hilbert space does so.
You appear confused about how extract the spacetime information, Tom.
Eugene is arguing that the concept of spacetime is not applicable to
fully interacting systems. Which is *very* different from GR lessons.
Similar thoughts are explained in [1] in a classical context. The two-
body classical relativistic potential in [1, 2] is NOT based in an
underlying spacetime manifold.
That potential would be useless for a N-body theory. At least so
useless like standard relativist approaches have been for the N-body
system during a century.
Tom Roberts
[1] Classical Relativistic Many-Body Dynamics. Springer; 1999. Trump,
Matthew A; Schieve, William C.
[2] http://canonicalscience.blogspot.com/2007/08/relativistic-lagrangian-an=
d-limitations_20.html
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D
Note for readers]
Because some past episodes of flamming in sci.physics.relativity, both
comments in my blog and my newsgroup e-mail are disabled.
Note for 'experts' and 'professionals']
Avoid to reply this message if you are in my blacklist below. The
capacity of any human for correcting your endless conceptual nonsenses
and foolish mathematical mistakes is, unfortunately, just finite. Also
my occupations do not include to teach you to read others, not to
teach you dimensional analysis or even pre-university physics.
Since you will be sanely ignored here in thereafter you are open to
misread, misquote, or misinterpret me in any way you want, specially
when that adds some light to your grey existence. You are open to
write any triviality; to invent any mistake I did not really did. You
can cite discredited, outdated, or wrong references. You can
manipulate or misread references. You are also open to address any
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extend your insults to any poster, institution, colleague, friend,
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You can also try to falsify ratings, voting against me dozens or
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same dishonest tactic for increasing the rating of your akins.
{BLACKLIST
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Shuba; }}
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
23 Aug 2007 02:13:34 PM |
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On Aug 23, 8:40 am, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
I was talking about different
*representations* of the same Poincare group. These representations
can be either non-interacting or interacting.
Hmmm. They are all related by isomorphisms....
No, this is not true. Some of them are isomorphic to each other,
others are non-isomorphic and non-equivalent.
This notion of Poincare
group representations is the most fundamental notion in relativistic
physics. It is absolutely necessary in order to understand how
relativistic interacting theories can be formulated. The idea is most
easily explained for quantum theories.
Certainly. The different representations of the Poincare' group apply to
different fundamental particles, distinguished by their intrinsic
spin/parity. But for all representations, the transforms of spacetime
coordinates are the same (they differ in the transform properties of
internal variables related to spin).
I can agree with you if you are talking about (irreducible)
representations which describe single-particle systems. I can also
agree with you if you are talking about non-interacting
representations for multiparticle systems. However, the universality
of boost transformations of spacetime coordinates is no longer valid
in the case of interacting representations of the Poincare group in
Hilbert spaces of multiparticle systems. The commutators between the
interacting boost operator and operators of particle positions are non-
trivial and interaction-dependent.
This has nothing whatsoever to do with your claims of non-isotropy due
to interactions. Indeed, in the theories you discuss spacetime is
Minkowski spacetime, and as I said before it is unaffected by the
presence of interactions. The overall physics depends on those
interactions, but the metric and other properties of spacetime do not.
In the theories I discuss there is no such notions as "spacetime" and
"metric". These are entirely algebraic theories in which physical
observables (such as particle positions) are represented by Hermitian
operators in the Hilbert space. The relationships between their
observables are provided by commutators of operators rather than by
mapping to the hypothetical "spacetime".
It is less known that this relativistic theory demands that
representatives of boost transformations in the Hilbert space also
contain interaction terms. The immediate consequence of this fact is
that boost transformations of observables of interacting particles are
no longer given by simple linear universal Lorentz formulas. Boost
transformations become interaction-dependent.
Sure, transforms IN THE HILBERT SPACE depend on which representation is
used; transforms IN SPACETIME do not. The underlying spacetime is
unaffected by the interactions.
I happen to believe that in a consistent theory each "element of
reality" should be represented by a single mathematical object. If
there are two conflicting representatives, one of them should be
eliminated. Following you logic, there are two mathematical
representatives for the notion of "particle position". One is the
operator of particle position in the Hilbert space. Boost
transformations of this operator are interaction-dependent (I am glad
that we agree on this point). The other representative is the space
coordinate in the Minkowski space-time whose boost transformations are
universal and interaction-independent. Apparently, there is a
contradiction. Which of these two mathematical objects is the true
reflection of the physical property - position?
In my approach the Hilbert space operator of position is directly
related to experimental measurements, and Minkowski space-time
coordinates are abstract unobservable things. Minkowski spacetime
picture is redundant. It is not needed for the mathematical
description of behavior of quantum systems of particles. The Hilbert
space and its operators is all we need for doing physics.
A more modern viewpoint is that the interactions occur in a fiber bundle
over spacetime -- a completely geometric approach. The fibers consist of
a tensor product of the various representations of the Poincare' group
for the different spins of the elementary particles of the theory. The
structure of the fibers does not affect the base manifold (Minkowski
spacetime), and its isometry group remains the Poincar=E9 group with its
usual Lorentz transforms.
Could you please be more specific, which "modern viewpoint" you have
in mind. I thought that the most modern physical theory is quantum
mechanics in which states of the system are represented by vectors in
the Hilbert space and observables are represented by Hermitian
operators. There is no place for "fibre bundles" and "base manifolds"
in this formalism.
Perhaps, you are talking about quantum field theory. It is true that
in QFT quantum fields are defined as operator distributions on the
Minkowski spacetime. However, I would dare to say that this spacetime
is an artificial mathematical object. Coordinates in this spacetime
have no relationship to experimentally measured particle positions.
Quantum fields are formal mathematical objects whose only purpose is
to assist in constructing relativistically invariant interaction
operators, which define the interacting unitary representation of the
Poincare group in the Hilbert (or Fock) space of the system. When the
S-matrix is calculated in QFT, the x and t arguments of quantum fields
serve as integration variables, so they do not appear in final
results. The best description of this viewpoint on QFT is given in
vol. 1 of S. Weinberg "The quantum theory of fields".
In the above description I haven't used words "metric" or "spacetime
manifold" or "curvature".
No. The aspect of GR you seem to be trying to apply to SR is that
interactions affect the underlying spacetime manifold. In
No, there is no such thing as "underlying spacetime manifold" in my
approach. There is a parameter "time" t which simply labels reference
frames time-displaced with respect to each other. There is also an
observable of "position" r which is represented by a Hermitian
operator in the Hilbert space. Each particle in a multiparticle system
has its own position operator; there is also an operator for the
center-of mass position. Transformations of these operators with
respect to time translations, boosts, and other intertial
transformations can be calculated from commutators of these
observables with generators of the Poincare group representation in
the Hilbert space.
I can repeat that Lorentz (or Poincare)
group is an exact group of transformations between inertial observers.
The structure of this group does not depend on whether the observed
physical system is interacting or not.
I agree with what you say here. But this seems to be considerably
different from what you claimed earlier.
Perhaps I was not entirely clear in what I wrote before. The correct
statement is that the Poincare group structure is unique and
universal. However, the representation of the Poincare group in the
Hilbert space of the system depends on interaction. Different
representations correspond to different (relativistically invariant)
ways to introduce interaction between particles.
The Poincare group structure is
preserved when transformations between inertial observers are
represented by unitary operators in the Hilbert space of the system.
The only thing that depends on interaction is how generators of this
representation are related to generators of the non-interacting
representation, and what are commutators between interacting
generators and one-particle observables.
All that is in the Hilbert space, not in spacetime.
If you want to have Minkowski spacetime in quantum theory, you must
ensure its compatibility with the Hilbert space description. In
particular, it is important that Lorentz transformations of Minkowski
space coordinates match with transformations of particle positions
described as Hermitian operators in the Hilbert space. The latter
transformations are provided by unitary operators of the Poincare
group representation in the Hilbert space. Unfortunately, this match
between the Minkowski spacetime and Hilbert space descriptions can be
achieved only in the case of non-interacting particles. This theorem
is proven in
D=2E G. Currie, T. F. Jordan, E. C. G. Sudarshan, "Relativistic
invariance and Hamiltonian theories of interacting particles", Rev.
Mod. Phys., 35 (1963), 350.
So, we need to choose which path to follow for the description of
interacting systems: either Hilbert space or Minkowski spacetime. We
cannot have both without contradictions. I choose the Hilbert space
description and ignore the Minkowski spacetime picture.
These commutators ultimately
determine how particle observables transform with respect to boosts.
It can be proven that in an interacting theory these transformations
are different from Lorentz formulas.
Sure, for elements of the Hilbert space. But not for spacetime -- the
projection onto the spacetime manifold is the same for all
representations of the Poincar=E9 group. This is so fundamental that most
textbooks do not even mention it, and concentrate on the
representations' behavior in the Hilbert space.
I respectfully disagree. In my opinion, it is impossible to reconcile
the Hilbert space and the Minkowski spacetime descriptions in the case
of interacting systems. One of them should be abandoned. I wish
textbooks were more clear on this fundamental issue.
Eugene.
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
24 Aug 2007 03:09:13 PM |
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wrote:
[...]
It's clear we are talking past each other, without either of us
understanding the other's position.
Tom Roberts
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
24 Aug 2007 04:18:05 PM |
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On Aug 24, 1:09 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
eugene_stefanov...@usa.net wrote:
[...]
It's clear we are talking past each other, without either of us
understanding the other's position.
Tom Roberts
I think I understand your position very well. It is represented in
most books on relativistic quantum mechanics and QFT. However, I think
that this position is not easy to defend. Basically, it tries to
combine two conflicting descriptions of reality. One is the quantum-
mechanical description with Hilbert spaces, Hermitian operators, etc.
Another description involves Minkowski spacetime and quantum fields
defined as operator functions on this spacetime.
These two descriptions partially overlap. For example, each of them
has its own notion of position. In the Hilbert space description the
position is a Hermitian operator, while in the Minkowski spacetime
description the position is a label of spacetime points. In the
Hilbert space description, time is a parameter which cannot be
combined with position in one 4-vector quantity. In the Minkowski
spacetime picture, time and position are equal and interchangeable
coordinates in the manifold.
My major point is that these two descriptions cannot logically coexist
with each other. There are just too many contradictions (which,
unfortunately, are rarely mentioned in most textbooks). I offer the
following solution: accept that the primary description of reality
should be in terms of Hilbert spaces etc. Then Minkowski spacetime and
quantum fields are just formal mathematical objects which play a
subordinate role. For example, the only significance of quantum fields
is to provide "building blocks" for the construction of relativistic
interaction operators in the Hilbert (Fock) space of systems with
variable number of particles. This point of view is represented well
in vol. 1 of Weinberg's "The quantum theory of fields".
I understand that my views are very unconventional. Their complete
description is in http://www.arxiv.org/abs/physics/0504062. If you
found a hole in my logic, I would be glad to discuss it.
Eugene.
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| User: "Tom Roberts" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
24 Aug 2007 09:23:38 PM |
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wrote:
accept that the primary description of reality
should be in terms of Hilbert spaces etc. [...]
Ah. Rather than quantum fields being a fiber bundle over spacetime,
there is no spacetime manifold, and what we perceive as such a manifold
is really just observables of the underlying Hilbert space. Or rather,
the distance between objects is such an observable WHEN IT IS MEASURED,
and there is nothing at all like the abstract points of a manifold,
there are only observable operators which represent distance between
objects in the Hilbert space.
Interesting. Yes, that's a whole new approach to me.... I'll have to
think about that....
Tom Roberts
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| User: "" |
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| Title: Re: Shubert's Derivation of the Lorentz Transformations |
25 Aug 2007 12:49:47 AM |
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On Aug 24, 7:23 pm, Tom Roberts <tjroberts...@sbcglobal.net> wrote:
Ah. Rather than quantum fields being a fiber bundle over spacetime,
there is no spacetime manifold, and what we perceive as such a manifold
is really just observables of the underlying Hilbert space. Or rather,
the distance between objects is such an observable WHEN IT IS MEASURED,
and there is nothing at all like the abstract points of a manifold,
there are only observable operators which represent distance between
objects in the Hilbert space.
Interesting. Yes, that's a whole new approach to me.... I'll have to
think about that....
Yes, exactly! I want you to forget about the spacetime manifold. In
quantum mechanics, everything measurable should be expressed in terms
of Hermitian operators. Particle positions are represented by
corresponding operators. Distances between particles are differences
between eigenvalues of these operators. Think about it.
Eugene.
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