Canonical commutation relationships, in Space *and Time*, and Understanding the "Energy-Time" Uncertainty
Jay R. Yablon
been able to progress from the principle of general coordinate
invariance and gravitational theory, to electrodynamic gauge theory, to
Planck's quantization of energy, and now, to the *canonical commutation
relationships*, and to the *proper covariant extension* of these to
include the time component of these commutation relationships.
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-6.pdf
If you have ever wondered why:
[x^i,p_k]=i hbar delta^i_k (1)
is only a three-dimensional relationship and what how to cleanly
understand energy-time uncertainty and what is the "proper*
four-dimensional version of (1), then read section 8 and 9.
I have been working on this problem for about a year, starting back
during all the velocity operator and Foldy-Woutheysen discussions with
our again missing friend the Princess. ;-) Today, I have found the
solution to these conceptually-theory questions, and written them up in
sections 8 and 9. It requires bringing the potential four-vector A^u,
with its perturbations, into the canonical commutation relationships,
and then "trading" the time components in a rather nifty sort of way.
Next up will be the explicit construction of the quantum operators for
all of this, just to tie the whole bundle together. From general
relativity to electrodynamics to energy quantization to canonical
commutation and full scale quantum theory, all because geometric
coordinates x^u are not vectors under general coordinate transformations
and yet we seek to describe nature geometrically and in a fashion that
is invariant with respect to all coordinate transformations. Everything
follows from that. Wheeler's geometrodynamics is alive and well and
predicting the quantum mechanics we actually observe. General
relativity and quantum theory are now well down the aisle to their
marriage!
Thanks,
Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
Jay R. Yablon
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-7.pdf
Added quantum gravitation. This comes full circle, and more or less
completes the paper aside from tuneups.
Jay.
Peter
Hi Jay,
Looks great and marvellous, congratulation!
Methodologically, your manner of conclusion (eg, how non-commutativity
is introduced) is similar to that in the approach by Suisky & myself,
so I have no problem with it :-)
Of course, one has to go step by step through the details to identify
the prepositions (IMHO necessary for a publication), to check the
compatibility of the final result(s) (necessary for a publication),
and to check the rigour of each single step (IMHO not necessary for
the 1st publication)...
Looking forward,
Peter
Juan R.
> I have added several new sections to my work in progress, and have now
> been able to progress from the principle of general coordinate
> invariance and gravitational theory, to electrodynamic gauge theory, to
> Planck's quantization of energy, and now, to the *canonical commutation
> relationships*, and to the *proper covariant extension* of these to
> include the time component of these commutation relationships.
>
> http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-6.pdf
>
> If you have ever wondered why:
>
> [x^i,p_k]=i hbar delta^i_k (1)
>
> is only a three-dimensional relationship
This is explained in QM textbook. Time is an evolution parameter in QM
(just as in Newtonian mechanics) and not operator can be associated to it.
> and what how to cleanly
> understand energy-time uncertainty and what is the "proper*
> four-dimensional version of (1), then read section 8 and 9.
>
> I have been working on this problem for about a year, starting back
> during all the velocity operator and Foldy-Woutheysen discussions with
> our again missing friend the Princess. ;-) Today, I have found the
> solution to these conceptually-theory questions, and written them up in
> sections 8 and 9. It requires bringing the potential four-vector A^u,
> with its perturbations, into the canonical commutation relationships,
> and then "trading" the time components in a rather nifty sort of way.
The *consistent* four-dimensional version of (1) is very well-known, it is
[x^a,p^b] = i hbar \eta^ab [33]
where x^0 is the Stuckelberg operator corresponding to quantizing
Poincaré-Einstein concept of 'time'. This x^0 is *not* the time in QM; the
correct concept of time (as evolution parameter) in QM is still maintained
in Stuckelberg theory and often denoted by symbol "tau" (to differentiate
from Poincaré-Einstein "t") [#]
An intro to the literature is here
http://en.wikipedia.org/wiki/Relativistic_dynamics
References given there will explain you the x^0 operator in [33], its
correct time parametrization, its spectrum, and the derivation of the
Perls & Landau inequality of QFT from [33].
(...)
Regards
[#] Curioulsy, I like precisely the inverse notation in my own works. I
use "t" for time (as in QM and Newtonian literature) and use "tau" for
denoting the reading of a clock. But above I am using relativistic
dynamics literature usual convention "t" for clocks rate "tau" for
time.
--
http://www.canonicalscience.org/
Usenet Guidelines:
http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html
======================================= MODERATOR'S COMMENT:
Please avoid formulations like "The *consistent* four-dimensional version of (1) is very well-known", because it sounds like 'this is the only one' (cf the meaning of this group ;-)
Juan R.
(...)
> ======================================= MODERATOR'S COMMENT:
> Please avoid formulations like "The *consistent* four-dimensional
> version of (1) is very well-known", because it sounds like 'this is the
> only one' (cf the meaning of this group ;-)
Ok my mistake.
I did really mean [33] is the only expression I know was both consistent
and well-known.
If moderator can point to another four-dimensional expression was both
consistent and well-known, please provide references.
If moderator is pointing to his own version, then it may be not well-
known and moderator would still prove his own version is consistent.
--
http://www.canonicalscience.org/
Usenet Guidelines:
http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html
======================================= MODERATOR'S COMMENT:
My foregoing comment was of general nature; 'the only I know/I'm aware of' is fine
Ken S. Tucker
Hi Jay, I hit a bump at Eq.(9.14), (no need to post this,
ignore if it's ok).
Ken
======================================= MODERATOR'S COMMENT:
'e' is missing
Ken S. Tucker
Yeah, that's what I figured, once Jay gives the proof-read
order I'll check it more closely too, and if I do find anything
I'll leave it up to Jay if he wants it posted, jeesh maybe I'll
find a typo or two, of no interest to our group.
I'm certainly on standby to help a fellow theoretician with
refinement prior to submission.
Of course comments or criticism of a paper should be posted,
for all to review.
Regards
Ken
======================================= MODERATOR'S COMMENT:
Indeed, your comments may help other readers, too, go ahead!
Jay R. Yablon
"Peter" <end...@dekasges.de> wrote in message
news:b2b900a6-4a15-4840...@21g2000vbk.googlegroups.com...
Your kind remarks are very much appreciated, and even more so, thank you
for your helpful comments along the way.
I am now doing a very pedantic, careful proofread and revision of the
whole document, and beefing up references. Once I have completed that,
I will post it, and will be ready to submit this for refereed
considerations.
Thanks,
Jay
Jay R. Yablon
I have now cleaned everything up, and have a final draft which is posted
at the link below:
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-8.pdf
I have submitted this for publication, so now I will hold my breath for
awhile and hope that this finally flies.
If anyone notes anything of substance that needs to be changed, please
let me know, as I can always update the submitted manuscript. Of course
it is best to update with important changes and not just typos (which
should more or less be gone already anyway).
Thanks again!
Jay
"Peter" <end...@dekasges.de> wrote in message
news:b2b900a6-4a15-4840...@21g2000vbk.googlegroups.com...
Peter
>
> - Zitierten Text anzeigen -
Hello Jay,
You should at least partially follow my hints, in particular, what
concerns the motivation.
For instance, U is not an operator (there is nothing defined to act
upon), and you are *not* compelled to consider
exp(-i B_mu x^mu)
This makes your text looking like 'there should be some phase factor,
thus, we move one there', I'm afraid to have to say.
In the title, I would omit 'gravitation, because your exploitation of
general coordinate invariance in independent of gravitation, thus,
Inferring Electrodynamics and Quantum Theory from General
Coordinate Invariance
would be more exact (why there is a comma in your title?)
(3.7) and (3.8) are postulates rather than definitions
I doubt that it is justified to exploit Dirac's quantization condition
here, because you rest on standard cem, where is no monopole - or what
exactly do you mean with this condition? (since this is a crucial
step, you should provide more details)
One can say nothing about (4.11), because it's a consequence of a
"definition" (!!!) [viz. of (4.7)]. You obtain Planck's energy
quantization for fermions => something is wrong, perhaps, the origin
of this conclusion, eq.(4.7)
Your thoughts about energy-time uncertainty are most interesting, you
should relate it to Rost's approach, where time is imposed by the
environment (more details later - or see my book)
Looking forward,
Peter
Ken S. Tucker
Thank you Mr. Moderator.
Jay reports his paper is submitted to a refereed journal.
Suppose the referees read SPF, then, my/our comments
might influence their evaluation, and though that may be
unlikely, it's IMO best to await the findings of the referees,
and then work from there.
Ken
Jay R. Yablon
gotten lost in cyberspace. So, there is a possibility that in the end
it will show up twice. Jay.
"Peter" <end...@dekasges.de> wrote in message
news:059ca55d-de11-4ce1...@r34g2000vba.googlegroups.com...
Hi Peter,
I have changed the foregoing as you suggested.
>
> I doubt that it is justified to exploit Dirac's quantization condition
> here, because you rest on standard cem, where is no monopole - or what
> exactly do you mean with this condition? (since this is a crucial
> step, you should provide more details)
>
> One can say nothing about (4.11), because it's a consequence of a
> "definition" (!!!) [viz. of (4.7)]. You obtain Planck's energy
> quantization for fermions => something is wrong, perhaps, the origin
> of this conclusion, eq.(4.7)
I am not so sure re: fermions. Set aside the monopole issue where I
agree that we do not observe a monopole (at least for Abelian gauge
theory, as you know, I remain firmly convinced that baryons are the
magnetic monopoles of non-Abelian gauge theory but that is another
issue).
Keep in mind my section 4 derivation is based on Maxwell's magnetic
equation represented by Gauss' law, and that there are two aspects to
this: charge enclosed within the surface, and related field flux across
the surface. Any enclosed charge is a fermion, but the form A describes
the flow of photons across the surface.
So, what this says is two things: a) charge must be quantized (if there
are monopoles). We know charge is quantized, but we see no monopoles,
and this enigma remains as its always has been. b) the flow of photons
/ light energy cross the surface must be quantized. That is exactly
what I am applying here, that that is what we observe.
>
> Your thoughts about energy-time uncertainty are most interesting, you
> should relate it to Rost's approach, where time is imposed by the
> environment (more details later - or see my book)
I look forward to this, please also give me a link you your book. I
looked up Rost inline and he is as confused as everyone else about the
real nature of energy / time uncertainty.
Best regards,
Jay
>
> Looking forward,
> Peter
>
Jay R. Yablon
Hi Peter,
I have changed the foregoing as you suggested.
>
> I doubt that it is justified to exploit Dirac's quantization condition
> here, because you rest on standard cem, where is no monopole - or what
> exactly do you mean with this condition? (since this is a crucial
> step, you should provide more details)
>
> One can say nothing about (4.11), because it's a consequence of a
> "definition" (!!!) [viz. of (4.7)]. You obtain Planck's energy
> quantization for fermions => something is wrong, perhaps, the origin
> of this conclusion, eq.(4.7)
I am not so sure re: fermions. Set aside the monopole issue where I
agree that we do not observe a monopole (at least for Abelian gauge
theory, as you know, I remain firmly convinced that baryons are the
magnetic monopoles of non-Abelian gauge theory but that is another
issue).
Keep in mind my section 4 derivation is based on Maxwell's magnetic
equation represented by Gauss' law, and that there are two aspects to
this: charge enclosed within the surface, and related field flux across
the surface. Any enclosed charge is a fermion, but the form A describes
the flow of photons across the surface.
So, what this says is two things: a) charge must be quantized (if there
are monopoles). We know charge is quantized, but we see no monopoles,
and this enigma remains as its always has been. b) the flow of photons
/ light energy cross the surface must be quantized. That is exactly
what I am applying here, that that is what we observe.
>
> Your thoughts about energy-time uncertainty are most interesting, you
> should relate it to Rost's approach, where time is imposed by the
> environment (more details later - or see my book)
I look forward to this, please also give me a link you your book. I
Jay R. Yablon
I made all the changes you suggested, and especially worked to really
strengthen section 4 because as you point out, this is a pivotal
section. Resubmitted to the journal, and uploaded below:
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-9.pdf
Talk with you soon.
Jay
Peter
Hi Jay,
Good improvements :-)
You still claim to have derived something, where you have married 2
things; eg, you cannot exploit a wavefunction for deriving the
commutator between x and p, and then claim you have derived QM (this
is the most complicated part of editing: to find and eliminate all
these gaps and flaws in the conclusion chains!)
Thus, if you have not yet finished this work, it's safer to publish
without the unsure claims (eg, there must not be any postulate - as
a_p=a_G - within one chain!)
The more you claim, the less people will believe you ;-)
Looking forward,
Peter
Peter
...
> > Hello Jay,
>
> > You should at least partially follow my hints, in particular, what
> > concerns the motivation.
>
> > For instance, U is not an operator (there is nothing defined to act
> > upon), and you are *not* compelled to consider
>
> > � exp(-i B_mu x^mu)
>
> > This makes your text looking like 'there should be some phase factor,
> > thus, we move one there', I'm afraid to have to say.
>
> > In the title, I would omit 'gravitation, because your exploitation of
> > general coordinate invariance in independent of gravitation, thus,
>
> > � Inferring Electrodynamics and Quantum Theory from General
> > Coordinate Invariance
>
> > would be more exact (why there is a comma in your title?)
>
> > (3.7) and (3.8) are postulates rather than definitions
> Hi Peter,
>
> I have changed the foregoing as you suggested.
> > I doubt that it is justified to exploit Dirac's quantization condition
> > here, because you rest on standard cem, where is no monopole - or what
> > exactly do you mean with this condition? (since this is a crucial
> > step, you should provide more details)
>
> > One can say nothing about (4.11), because it's a consequence of a
> > "definition" (!!!) [viz. of (4.7)]. You obtain Planck's energy
> > quantization for fermions => something is wrong, perhaps, the origin
> > of this conclusion, eq.(4.7)
> I am not so sure re: fermions. �
Are there bosons obeying the Dirac eq.?
> Set aside the monopole issue where I
> agree that we do not observe a monopole (at least for Abelian gauge
> theory, as you know, I remain firmly convinced that baryons are the
> magnetic monopoles of non-Abelian gauge theory but that is another
> issue).
It's not the matter of existence of (magnetic) monopoles. I assume
that in extended microscopic Maxwell eqs., ie, with monopoles, the
relations between potentials and fields looks differently, hence,
gauge is different, or is it still U(1), as you presuppose?
> Keep in mind my section 4 derivation is based on Maxwell's magnetic
> equation represented by Gauss' law,
As I wrote, give more details for the 1st paragraph of Section 4!
Will study both your [12] (just printing ;-) and [9] Zee (Zee looks
grandiose like the Pauli lectures, but I see seldom, where he
concludes and where he only assumes)
> and that there are two aspects to
> this: charge enclosed within the surface, and related field flux across
> the surface. �Any enclosed charge is a fermion, but the form A describes
> the flow of photons across the surface.
The He-4 nucleus is a charged boson
Btw, what is the benefit of suddenly switching to forms?
> So, what this says is two things: a) charge must be quantized (if there
> are monopoles).
Please remove the logical flaw, that your conclusion is derived from a
*definition*
>�We know charge is quantized,
No, we only know that electric charge is *discretized* - whether this
is classical discretization like the harmonics of a string (frequency/
wavelength discretization), or non-classical discretization like the
energy of the undamped harmonic oscillator (energy quantization), is
not known, isn't it? (Will open a new thread about it :-)
> but we see no monopoles,
> and this enigma remains as its always has been. �b) the flow of photons
> / light energy cross the surface must be quantized. �That is exactly
> what I am applying here, that that is what we observe.
odal verbs tend to introduce anthropomorphic elements detoriating the
rigor ;-)
> > Your thoughts about energy-time uncertainty are most interesting, you
> > should relate it to Rost's approach, where time is imposed by the
> > environment (more details later - or see my book)
>
> I look forward to this, please also give me a link you your book.
> I looked up Rost inline and he is as confused as everyone else about the
> real nature of energy / time uncertainty.
I couldn't find any address to approach him - could you?
Looking forward,
Peter
neur...@yahoo.com.au
Princess Florence of Nightingale here. I'm the one who
waits with bandages and morphine in the Houses of Healing
for brave knights returning from battle with their legs cut
off - which seems imminent in your case.
I only skimmed your 40-page paper at warp speed, so I'll
just mention the few obvious things which jumped
out at me...
1) The way you spend many pages in geometrical arguments to
figure out the (3D) CCRs seems unnecessarily tedious. Are
you familiar with the standard geometrical derivation?
Eg: C.J. Isham, "Lectures on Quantum Theory - Mathematical
and Structural Foundations", ISBN 1-86094-000-5, section
7.2.2, p115. (This is just for flat space, but it's quite
quick and lucid.) [Prince Charles: is a derivation like that
somewhere on your website?]
2) When talking about problems associated with 4D CCRs (ie,
including a CR like [T,H] = ih), I only noticed you
mentioning the mass commutation issue, but not the more
serious issue: a CR like [T,H] = ih means there's no ground
state! Ie, if you postulate a Hilbert space which has a
ground state |vac> (ie of minimum energy), then if such a T
operator acts on that ground state, (eg, in the form of an
operator exp^{i eps T} with infinitesimal eps), it produces a
state of lower energy, contradicting the assumption that
|vac> was a ground state. To see this, evaluate
H' := (1 + i eps T) H (1 - i eps T) and apply it to |vac>.
3) A diagonal CCR such as [X_mu, H] ~ delta_{0\mu} ignores
the fact that [X_i, H] is a velocity operator v_i in quantum
theory. Restricting to [X_mu, H] ~ delta_{0\mu} is pretty
much equivalent to restricting to v_i = 0.
----
LoL from the Princess with the Lamp.
P.S: It's truly amazing how long some knights can continue
fighting around here, even with many limbs cut off. Some
seem to be having a contest like the black knight in Monty
Python, and continue battling even when reduced to a mere
head. Even two fully severed heads continue abusing and
biting each other.
Jay R. Yablon
news:8524cc94-a462-4e10...@b7g2000pre.googlegroups.com...
> Dear Brave Sir Jay,
>
> Princess Florence of Nightingale here. I'm the one who
> waits with bandages and morphine in the Houses of Healing
> for brave knights returning from battle with their legs cut
> off - which seems imminent in your case.
>
> I only skimmed your 40-page paper at warp speed, so I'll
> just mention the few obvious things which jumped
> out at me...
>
> 1) The way you spend many pages in geometrical arguments to
> figure out the (3D) CCRs seems unnecessarily tedious. Are
> you familiar with the standard geometrical derivation?
> Eg: C.J. Isham, "Lectures on Quantum Theory - Mathematical
> and Structural Foundations", ISBN 1-86094-000-5, section
> 7.2.2, p115. (This is just for flat space, but it's quite
> quick and lucid.) [Prince Charles: is a derivation like that
> somewhere on your website?]
Dear Princess,
Welcome back! I will get the book take a look at Isham's derivation to
see how it contrasts with what I am doing.
>
> 2) When talking about problems associated with 4D CCRs (ie,
> including a CR like [T,H] = ih), I only noticed you
> mentioning the mass commutation issue, but not the more
> serious issue: a CR like [T,H] = ih means there's no ground
> state! Ie, if you postulate a Hilbert space which has a
> ground state |vac> (ie of minimum energy), then if such a T
> operator acts on that ground state, (eg, in the form of an
> operator exp^{i eps T} with infinitesimal eps), it produces a
> state of lower energy, contradicting the assumption that
> |vac> was a ground state. To see this, evaluate
> H' := (1 + i eps T) H (1 - i eps T) and apply it to |vac>.
>
> 3) A diagonal CCR such as [X_mu, H] ~ delta_{0\mu} ignores
> the fact that [X_i, H] is a velocity operator v_i in quantum
> theory. Restricting to [X_mu, H] ~ delta_{0\mu} is pretty
> much equivalent to restricting to v_i = 0.
I do not believe that what I derive in the end conflicts on either of
these points. The p^0 component of the energy momentum remains
commuting with time, and the non-commutativity is between time and
perturbation, which replaces the energy / time uncertainty with energy
perturbation uncertainty (9.10) and maintains the Robertson link between
CCRs and uncertainty relationships.. The velocity operator that we
explored at length a year ago is not at all impacted. Sections 8 and 9
make this clear. Please take a look with slower than warp speed when
you have a moment.
Jay
Oh No
>Dear Brave Sir Jay,
>
>Princess Florence of Nightingale here. I'm the one who
>waits with bandages and morphine in the Houses of Healing
>for brave knights returning from battle with their legs cut
>off - which seems imminent in your case.
>
>I only skimmed your 40-page paper at warp speed, so I'll
>just mention the few obvious things which jumped
>out at me...
>
>1) The way you spend many pages in geometrical arguments to
>figure out the (3D) CCRs seems unnecessarily tedious. Are
>you familiar with the standard geometrical derivation?
>Eg: C.J. Isham, "Lectures on Quantum Theory - Mathematical
>and Structural Foundations", ISBN 1-86094-000-5, section
>7.2.2, p115. (This is just for flat space, but it's quite
>quick and lucid.) [Prince Charles: is a derivation like that
>somewhere on your website?]
How can I say? I have not read Isham. I have given Sir Jay the links
where the CCR's are derived. The method is standard, but I don't think
of it as geometrical so much as algebraic.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
Juan R.
> <neur...@yahoo.com.au> wrote in message
> news:8524cc94-a462-4e10...@b7g2000pre.googlegroups.com...
>> Dear Brave Sir Jay,
(...)
>> 2) When talking about problems associated with 4D CCRs (ie, including a
>> CR like [T,H] = ih), I only noticed you mentioning the mass commutation
>> issue, but not the more serious issue: a CR like [T,H] = ih means
>> there's no ground state! Ie, if you postulate a Hilbert space which has
>> a ground state |vac> (ie of minimum energy), then if such a T operator
>> acts on that ground state, (eg, in the form of an operator exp^{i eps
>> T} with infinitesimal eps), it produces a state of lower energy,
>> contradicting the assumption that |vac> was a ground state. To see
>> this, evaluate H' := (1 + i eps T) H (1 - i eps T) and apply it to
>> |vac>.
>>
>> 3) A diagonal CCR such as [X_mu, H] ~ delta_{0\mu} ignores the fact
>> that [X_i, H] is a velocity operator v_i in quantum theory. Restricting
>> to [X_mu, H] ~ delta_{0\mu} is pretty much equivalent to restricting to
>> v_i = 0.
>
> I do not believe that what I derive in the end conflicts on either of
> these points. The p^0 component of the energy momentum remains
> commuting with time,
>From (8.2)
[t, H] = ihbar
> and the non-commutativity is between time and perturbation, which
> replaces the energy / time uncertainty with energy perturbation
> uncertainty (9.10) and maintains the Robertson link between CCRs and
> uncertainty relationships..
>From (9.10)
[t, E] = 0
but H = E... in fact Dirac equation verifies
H^2 Psi = E^2 Psi
Moreover in the case when there is not perturbations your (9.3) reduces to
[t, E] = ihbar
instead your (9.4)...
What is the sense of (9.12), (9.13)?
> The velocity operator that we
> explored at length a year ago is not at all impacted. Sections 8 and 9
> make this clear. Please take a look with slower than warp speed when
> you have a moment.
I do not find any definition of "velocity operator" in those sections, and
as reported above by neuropulp "[X_mu, H] ~ delta_{0\mu}" is not correct.
It fails even for non-relativistic theory.
As said the Monday in another message, the only known consistent four
version of canonical conmmutations is given in Stuckelberg theory.
Regards
Juan R.
> <neur...@yahoo.com.au> wrote in message
> news:8524cc94-a462-4e10...@b7g2000pre.googlegroups.com...
>> Dear Brave Sir Jay,
(...)
>> 2) When talking about problems associated with 4D CCRs (ie, including a
>> CR like [T,H] = ih), I only noticed you mentioning the mass commutation
>> issue, but not the more serious issue: a CR like [T,H] = ih means
>> there's no ground state! Ie, if you postulate a Hilbert space which has
>> a ground state |vac> (ie of minimum energy), then if such a T operator
>> acts on that ground state, (eg, in the form of an operator exp^{i eps
>> T} with infinitesimal eps), it produces a state of lower energy,
>> contradicting the assumption that |vac> was a ground state. To see
>> this, evaluate H' := (1 + i eps T) H (1 - i eps T) and apply it to
>> |vac>.
>>
>> 3) A diagonal CCR such as [X_mu, H] ~ delta_{0\mu} ignores the fact
>> that [X_i, H] is a velocity operator v_i in quantum theory. Restricting
>> to [X_mu, H] ~ delta_{0\mu} is pretty much equivalent to restricting to
>> v_i = 0.
>
> I do not believe that what I derive in the end conflicts on either of
> these points. The p^0 component of the energy momentum remains
> commuting with time,
>From (8.2)
[t, H] = ihbar
> and the non-commutativity is between time and perturbation, which
> replaces the energy / time uncertainty with energy perturbation
> uncertainty (9.10) and maintains the Robertson link between CCRs and
> uncertainty relationships..
>From (9.10)
[t, E] = 0
but H = E... in fact Dirac wave equation verifies
H^2 Psi = E^2 Psi
Moreover in the case when there is not perturbations your (9.3) reduces to
[t, E] = ihbar
instead your (9.4)...
What is the sense of (9.12), (9.13)?
> The velocity operator that we
> explored at length a year ago is not at all impacted. Sections 8 and 9
> make this clear. Please take a look with slower than warp speed when
> you have a moment.
I do not find any definition of "velocity operator" in those sections, and
as reported above by neuropulp your "[X_mu, H] ~ delta_{0\mu}" is wrong.
It is even wrong in non-relativistic theory and also for classical
mechanics where [?] --> {?}/ihbar.
This pdf seems to me an inconsistent mixture of general relativity, with
field theory, and with relativistic quantum mechanics (not quantum field
theory!). It is worth to remind that equation 7.6 is not a Hamiltonian
operator in relativistic quantum field theory.
As said the Monday in another message, the only consistent four version of
Juan R.
> I do not find any definition of "velocity operator" in those sections,
> and as reported above by neuropulp your "[X_mu, H] ~ delta_{0\mu}" is
> wrong. It is even wrong in non-relativistic theory and also for
> classical mechanics where [?] --> {?}/ihbar.
Juan you are plain wrong :-D
The correspondence between quantum and classical brackets is
[?]/ihbar --> {?}
Jay R. Yablon
You raised some good questions, thank you!
These questions do, however, NOT affect the validity of what I have
done, but they do warrant further explanation which I have added to the
paper at an updated draft linked below:
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-101.pdf
I will point you inline below, to the specific portions of this new
draft which I believe should address your questions (and at least the
one princess raised about the velocity operator).
Thanks,
Jay
"Juan R. González-Álvarez" <juanR...@canonicalscience.com> wrote in
message news:pan.2009.05...@canonicalscience.com...
See the footnote on page 29. Also (9.17) and (9.18).
>
> What is the sense of (9.12), (9.13)?
If you did not understand this then you do not understand what I have
done. Take a close look again. These are now (9.14) and (9.15). They
provide a covaraiant four-version of the commutation relationships.
>
>> The velocity operator that we
>> explored at length a year ago is not at all impacted. Sections 8 and
>> 9
>> make this clear. Please take a look with slower than warp speed when
>> you have a moment.
>
> I do not find any definition of "velocity operator" in those sections,
> and
> as reported above by neuropulp your "[X_mu, H] ~ delta_{0\mu}" is
> wrong.
> It is even wrong in non-relativistic theory and also for classical
> mechanics where [?] --> {?}/ihbar.
See new (8.14) and (8.15). I would look at those first, then go to the
section 9 changes noted above. You cannot set operators to zero; if the
observable is zero, that comes out of the operator operating, not by
setting the operator to zero.
> This pdf seems to me an inconsistent mixture of general relativity,
> with
> field theory, and with relativistic quantum mechanics (not quantum
> field
> theory!). It is worth to remind that equation 7.6 is not a Hamiltonian
> operator in relativistic quantum field theory.
That is fair enough, and I will make sure that I am not referring to QFT
anywhere that might be in inappropriate.
>
> As said the Monday in another message, the only consistent four
> version of
> canonical conmmutations is given in Stuckelberg theory.
I respectfully believe that this works as well, and does not contradict
anything that should not be contradicted.
Jay
Jay R. Yablon
>
> Princess Florence of Nightingale here. . . .
>
> 3) A diagonal CCR such as [X_mu, H] ~ delta_{0\mu} ignores
> the fact that [X_i, H] is a velocity operator v_i in quantum
> theory. Restricting to [X_mu, H] ~ delta_{0\mu} is pretty
> much equivalent to restricting to v_i = 0.
Dear princess Florence: ;-)
This is not inconsistent with the velocity operator. After all of our
discussion last year I made sure of that. I added two equations at the
end of section 8, in the update below, to make this clear.
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-101.pdf
I did not specifically reference the velocity operator in the earlier
draft and should have. Now I have.
The whole purpose of going to great lengths to maintain a commuting
mass -- in part -- is to maintain the velocity operator as well, since
that is one byproduct of a commuting mass.
Jay.
neur...@yahoo.com.au
> I do not believe that what I derive in the end conflicts on
> either of [my previous points 2 & 3].
I'll try below to explain how/where the problem lies at a
deeper foundational level than what you're focusing on.
> The p^0 component of the energy momentum remains commuting
> with time, and the non-commutativity is between time and
> perturbation, which replaces the energy / time uncertainty
> with energy perturbation uncertainty (9.10) and maintains
> the Robertson link between CCRs and uncertainty
> relationships.. The velocity operator that we explored at
> length a year ago is not at all impacted. Sections 8 and 9
> make this clear. Please take a look with slower than warp
> speed when you have a moment.
It takes more than a "moment" to disentangle everything.
I've now spent more than a moment, but my warp-speed
impressions remain unchanged...
IIUC, you're making a distinction between the free
Hamiltonian (which you seem to regard as the "energy") and
"perturbation" which is the interaction potential, ie the
interaction part of the full Hamiltonian. Then you try to
treat the interaction potential as distinct, and say that
the time operator commutes with the free Hamiltonian
([t,E]=0), but not with the interaction part of the
Hamiltonian ([t,V] = -ih). In that case, the time operator
still satisfies [t,H] = -ih (where H is the full
Hamiltonian). Therefore your quantum Hilbert space doesn't
have a state of lowest total energy. The observable
corresponding to what we measure experimentally as "energy"
of an interacting system is the full Hamiltonian, not the
free Hamiltonian.
Similarly, when one writes [X^i,H] = v^i, the H is also the
full Hamiltonian (if the X^i are to be interpreted as the
position configuration observables).
> This is not inconsistent with the velocity operator. After
> all of our discussion last year I made sure of that. ...
A lot of what we discussed before about Dirac velocity
operators, Newton-Wigner position operators, and
Foldy-Wouthuysen transformation was strictly in the context
of the _free_ Dirac theory. Textbooks typically go further and
discuss what happens in an interacting theory (showing that
the transformation cannot be derived in closed form, but
only perturbatively). But the important point here is that
for the operators in "[X^i,H] = v^i" to have their usual
intuitive physical meanings, H must be the full Hamiltonian.
Aside: various related lines of research were blown out of
the water a long time ago by the "no-interaction" theorem
from this paper:
Currie, Jordan, Sudarshan: "Relativistic Invariance and
Hamiltonian Theories of Interacting Particles",
Rev Mod Phys, vol 35, No 2, (1963), p350.
The essence of the theorem is that the combined assumptions
of Lorentz symmetry and Lorentz transformation of particle
positions rule out any interaction. I.e., in any theory
that tries to combine these notions, there cannot be any
interaction between particles.
---
LoL from Princess Florence as she moves on to other patients,
having bandaged the current one as much as she reasonably can
in the time available.
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:6c4cadea-f88d-4d2f...@f28g2000pra.googlegroups.com...
Before I pick up my sword and dash back to the battle, two questions:
1) are you saying that
[t,H] = 0 (1)
is the only acceptable commutator between a time operator and the
Hamiltonian which avoid the lowest energy problem? If not, what are the
constraints on [t,H] required to not have this problem?
2)
Do you agree that for the rest mass:
[t,m] = 0 (2)
and more generally:
[x^u,m] = 0 (3)
is an absolute requirement?
Thanks,
Jay.
Juan R.
> Juan,
>
> You raised some good questions, thank you!
>
> These questions do, however, NOT affect the validity of what I have
> done, but they do warrant further explanation which I have added to the
> paper at an updated draft linked below:
>
> http://jayryablon.files.wordpress.com/2009/05/covariance-and-
gauge-101.pdf
>
> I will point you inline below, to the specific portions of this new
> draft which I believe should address your questions (and at least the
> one princess raised about the velocity operator).
>
> Thanks,
>
> Jay
>
>
> "Juan R. González-Álvarez" <juanR...@canonicalscience.com> wrote in
> message news:pan.2009.05...@canonicalscience.com...
>> Jay R. Yablon wrote on Thu, 14 May 2009 22:55:06 -0600:
>>
>>> <neur...@yahoo.com.au> wrote in message news:8524cc94-
a462-4e10-909...@b7g2000pre.googlegroups.com...
>>>> Dear Brave Sir Jay,
(...)
>> Moreover in the case when there is not perturbations your (9.3) reduces
>> to
>>
>> [t, E] = ihbar
>>
>> instead your (9.4)...
>
> See the footnote on page 29. Also (9.17) and (9.18).
The footnote is incorrect because for free particles the perturbation
operator is given by the zero *operator*: \hat{A} = \hat{0}.
In fact if there was not zero operators, the space of operators (needed
to give mathematical meaning to algebraic operations with operators as b
c = d) would be not defined. Moreover, you are assuming existence of zero
operators in your own theory. This is why instead writting the
Hamiltonian operator
H = H_0 + V_EM + V_Gravity + V_anyother
you can write just
H = H_0 + V_EM
and ignore other interactions.
>From (9.3), for free particles, follows [t, E] = ihbar instead your (9.4)
and from Dirac Klein constraint (H^2 Psi = E^2 Psi) follows (8.2) and the
inconsistency in your work between both conmmutators is solved.
Moreover, that [t, E] cannot be zero can be also seen from writing the
definitions for the two operators and computing the conmmutator.
>> What is the sense of (9.12), (9.13)?
>
> If you did not understand this then you do not understand what I have
> done.
100% agree ;-)
(...)
> See new (8.14) and (8.15). I would look at those first, then go to the
> section 9 changes noted above.
First, your (8.15) is incompatible with your (8.7) plus (8.9).
Second, even if you correct that and give (8.2) as the velocity operator
for Dirac theory, still your theory is inconsistent because that operator
is not observable (components even do not conmute!) which means (8.15)
defines an unphysical velocity.
Integrating (8.15), it can be showed that the your x^i in the quantum
part is not compatible with the classical x^i in the element of line ds^2
in electrodynamics, for instance.
This inconsistency is also well-known in literature and was the reason
which x^i is not an Dirac operator in quantum electrodynamics.
Your claims that Dirac electron at rest would give [t, m] =/= 0 are not
valid, because by the Zbw effect the Dirac electron is *never* at rest...
eugene_st...@usa.net
> Aside: various related lines of research were blown out of
> the water a long time ago by the "no-interaction" theorem
> from this paper:
>
> Currie, Jordan, Sudarshan: "Relativistic Invariance and
> Hamiltonian Theories of Interacting Particles",
> Rev Mod Phys, vol 35, No 2, (1963), p350.
>
> The essence of the theorem is that the combined assumptions
> of Lorentz symmetry and Lorentz transformation of particle
> positions rule out any interaction. I.e., in any theory
> that tries to combine these notions, there cannot be any
> interaction between particles.
You are absolutely right that interaction is not compatible with
Lorentz transformations of particle positions. However, it is
important to understand which conclusion you make from this theorem.
My thinking is this: It is impossible to deny that particles do
interact. Then it follows that their positions do not transform by
Lorentz. Is there anything wrong with this logic?
Oh No
Not sure what we mean by Lorentz transforming particle positions anyway.
Obviously in gtr position is not a vector, but disregarding that
particle position is not, in general, a well defined quantity in quantum
theory. At best you might hope to Lorentz transform the wave function,
but then, when you do a measurement of position, the wave function
collapses on a synchronous slice.
maxwell
At least, you got it, Eugene.
Juan R.
Currie, Jordan, and Sudarshan only showed what happen when one confounds
the concept of time in quantum mechanics (evolution parameter) with the
concept of time in relativistic *classical* physics (dimension). Thus
their Hamiltonian was the generator of time translations for x^0 which is
nonsensical.
Of course, it is possible to build a Hamiltonian theory with positions
transforming by the LT. Example Stuckelberg theory.
maxwell
>
> I'll try below to explain how/where the problem lies at a
> deeper foundational level than what you're focusing on.
> Aside: various related lines of research were blown out of
> the water a long time ago by the "no-interaction" theorem
> from this paper:
>
> Currie, Jordan, Sudarshan: "Relativistic Invariance and
> Hamiltonian Theories of Interacting Particles",
> Rev Mod Phys, vol 35, No 2, (1963), p350.
>
> The essence of the theorem is that the combined assumptions
> of Lorentz symmetry and Lorentz transformation of particle
> positions rule out any interaction. I.e., in any theory
> that tries to combine these notions, there cannot be any
> interaction between particles.
>
> having bandaged the current one as much as she reasonably can
> in the time available.
Well said, Princess. Most theorists are still hung up on Hamiltonian
physics, which like Lagrangian physics, is based on single time
theories at a point in space. This involves both absorbing all the
dynamics into a potential & beginning with the idea of 'free'
particles (or fields, if you are a QFT enthusiast). Timeless
potentials imply instantaneous interactions across unlimited spatial
separations & this is incompatible with reality (asynchronous delays
between electrons) & all derivations of the Lorentz Transformations
(heavens above!). There NO such objects as free particles - a purely
mathematical conception with no correspondence in reality.
Dear Princess, please stop on by whenever you can. Your wisdom &
compassion are much appreciated (at least by some).
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> Currie, Jordan, and Sudarshan only showed what happen when one confounds
> the concept of time in quantum mechanics (evolution parameter) with the
> concept of time in relativistic *classical* physics (dimension). Thus
> their Hamiltonian was the generator of time translations for x^0 which is
> nonsensical.
>
> Of course, it is possible to build a Hamiltonian theory with positions
> transforming by the LT. Example Stuckelberg theory.
>
> --http://www.canonicalscience.org/
So, you propose to have two different notions of time in physical
theory. Which one of these two notions corresponds to the "time"
measured by experimentalists (by looking at their laboratory clocks)?
And what is the reason to keep the other notion? It is suspicious when
the theory has two different quantities while the experiment has only
one.
neur...@yahoo.com.au
> Of course, it is possible to build a Hamiltonian theory with positions
> transforming by the LT. Example Stuckelberg theory.
Have you got an online review of Stuckelberg theory somewhere?
(I'm pretty sure I don't know enough about it, unless it's also known
under some other name.)
---
LoL from the Princess!
neur...@yahoo.com.au
> Well said, Princess. [....] please stop on by whenever you
> can. Your wisdom & compassion are much appreciated [...]
Thank you indeed for your kind words, dear sir.
(Though let us steer well clear of the slippery slope of
unkind gossip about our intelligent and learned colleagues.)
[And BTW, don't let my sister Princess Spank-a-Lot overhear
such compliments. She gets jealous. She's not nice when
those green eyes are lit up.]
> There NO such objects as free particles - a purely
> mathematical conception with no correspondence in reality.
You mean in the sense that we never actually measure a
non-interacting particle, right?
eugene_st...@usa.net
> Not sure what we mean by Lorentz transforming particle positions anyway.
> Obviously in gtr position is not a vector, but disregarding that
> particle position is not, in general, a well defined quantity in quantum
> theory. At best you might hope to Lorentz transform the wave function,
> but then, when you do a measurement of position, the wave function
> collapses on a synchronous slice.
What would you say about the classical limit of quantum theory in the
absence of gravity? Do you think that 4 components of position-time
transform as a 4-vector in this limit? Then, according to the Currie-
Jordan-Sudarshan theorem, you cannot have interactions.
Eugene.
neur...@yahoo.com.au
> 1) are you saying that
>
> [t,H] = 0 (1)
>
> is the only acceptable commutator between a time operator
> and the Hamiltonian which avoid the lowest energy problem?
The "only" acceptable commutator? Who knows? There might be
other Lie algebras (involving more observables) which somehow
solve the problem, but if so I don't know what they are.
> If not, what are the constraints on [t,H] required to not
> have this problem?
I don't have an answer to this in closed form...
There might be many ways to approach the problem, such as
introducing other observables in the algebra (but then the
Casimirs would probably be different, and "mass" might need
to be generalized somehow).
Or there might be alternative formulations of quantum-like
theories which only produce eigenvalue _differences_ (and
that might help because we only measure energy differences
anyway).
But this is all speculation. They say dragons roam beyond
the edge of the known world.
> 2) Do you agree that for the rest mass:
> [t,m] = 0 (2)
> and more generally:
> [x^u,m] = 0 (3)
> is an absolute requirement?
When seeking improved theories of physics, "absolute" is a
tricky word, best avoided. It's better to try and understand
the structure of existing theories more deeply - to
distinguish between what's truly physics, what's truly
mathematics, and what's an unjustified extrapolation between
the two.
For example, if one takes seriously the idea that (spatial)
position should be an operator satisfying the CCRs, then
expressions like [x^i,m] need to correspond to something
like dm/dp_i (partial derivatives) in a classical context
(if we tacitly assume that our commutators came from
classical Poisson brackets somehow, as is often the case in
quantization). But that's a bit silly in a relativistic
context where m^2 = E^2 - (p_i)^2, since then dm/dp_i is not
zero, so it wouldn't make much sense for [x^i,m] to be zero
either.
neur...@yahoo.com.au
after an absence! Sound the trumpets! What news from
the wilderness Master Eugene?
On May 17, 7:28�am, eugene_stefanov...@usa.net wrote:
> it is important to understand which conclusion you make
> from [the CJS] theorem.
Wise words.
> My thinking is this: It is impossible to deny that particles
> do interact. Then it follows that their positions do not
> transform by Lorentz. Is there anything wrong with this
> logic?
That's what I got initially from the CJS no-interaction
theorem.
But CJS also try to explain the distinction between
the abstract Poincare algebra and its representation(s) on
Minkowski spacetime, which is a caution for us all.
And heh, Minkowski spacetime was always just a
figment of our imaginations, right? :-)
---
LoL from the Princess!
("Neuropulp": n, an inevitable by-product from studying QFT.)
neur...@yahoo.com.au
> Not sure what [Master Eugene means] by Lorentz transforming
> particle positions anyway. ...
The Currie/Jordan/Sudarshan paper I mentioned deals with
trajectories of particles in Minkowski spacetime. When
they talk about "Lorentz transforming particle positions"
they mean that trajectories should transform as you would
naively expect if the spacetime points therein transformed
according to the usual relativistic formulae. (And yeah, I know
that entire way of looking at physics is wide open to plenty
of objections.)
But their point is this: if one constructs a relativistic
interacting theory, where by "relativistic" they mean a
representation of the Poincare algebra, where Minkowski
spacetime points just parameterize the representation, then
particle trajectories in Minkowski space don't
Lorentz-transform in the way one might naively expect.
Like many "no-go" theorems, it highlights that something
is probably wrong with our current mix of physical prejudices
and mathematical axioms.
[BTW, if anyone wants to study the CJP paper (which is
probably essential to discuss these points more deeply),
try Google Scholar and then web search. You might get
lucky and find an easily accessible copy. (Sssh!)]
eugene_st...@usa.net
> Master Stefanovich returns unexpectedly from the wilderness
> after an absence! Sound the trumpets! What news from
> the wilderness Master Eugene?
CJS theorem is my favorite subject, so I couldn't stay silent any
longer. You may stop the trumpets now. Thank you.
> But CJS also try to explain the distinction between
> the abstract Poincare algebra and its representation(s) on
> Minkowski spacetime, which is a caution for us all.
> And heh, Minkowski spacetime was always just a
> figment of our imaginations, right? :-)
In quantum mechanics the major role is played by representations of
the Poincare algebra in Hilbert spaces. In the classical limit they
transform to representations on the phase space. These representations
are sufficient to build a complete physical theory. So, "Minkowski
spacetime" representations are pretty much useless. They are simple,
cute and popular, but physics can (and should) be done without them,
especially when interacting systems are involved. That's the
conclusion I drew from the CJS theorem.
Oh No
>Prince Charles wrote:
>
> > Not sure what [Master Eugene means] by Lorentz transforming
> > particle positions anyway. ...
>
>The Currie/Jordan/Sudarshan paper I mentioned deals with
>trajectories of particles in Minkowski spacetime. When
>they talk about "Lorentz transforming particle positions"
>they mean that trajectories should transform as you would
>naively expect if the spacetime points therein transformed
>according to the usual relativistic formulae.
Pretty meaningless then huh.
>(And yeah, I know
>that entire way of looking at physics is wide open to plenty
>of objections.)
You disappoint me. There was I thinking we could have our first row.
>
>But their point is this: if one constructs a relativistic
>interacting theory, where by "relativistic" they mean a
>representation of the Poincare algebra, where Minkowski
>spacetime points just parameterize the representation, then
>particle trajectories in Minkowski space don't
>Lorentz-transform in the way one might naively expect.
>Like many "no-go" theorems, it highlights that something
>is probably wrong with our current mix of physical prejudices
>and mathematical axioms.
A little bit like Haag's no go theorem, but less applicable?
>
>[BTW, if anyone wants to study the CJP paper (which is
>probably essential to discuss these points more deeply),
>try Google Scholar and then web search. You might get
>lucky and find an easily accessible copy. (Sssh!)]
>
Can we cut off its arms and legs, and see if it will still jump about
shouting challenges?
Oh No
>
>("Neuropulp": n, an inevitable by-product from studying QFT.)
>
to carry out research? I remain of the view that one should first cut
off its arms and legs, and reduce it to operators on Fock space.
Hendrik van Hees
> You are absolutely right that interaction is not compatible with
> Lorentz transformations of particle positions. However, it is
> important to understand which conclusion you make from this theorem.
> My thinking is this: It is impossible to deny that particles do
> interact. Then it follows that their positions do not transform by
> Lorentz. Is there anything wrong with this logic?
There's everything wrong with this logic since it contradicts every-day
experience: There are interacting particles, and these interacting
particles can be described to high accuracy by Poincare-covariant QFT.
Thus Lorentz invariance (which is part of Poincare invariance) is in no
way contradicted by the existence of interactions (or vice versa).
Oh No
>On May 16, 3:08�pm, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>
>> Not sure what we mean by Lorentz transforming particle positions anyway.
>> Obviously in gtr position is not a vector, but disregarding that
>> particle position is not, in general, a well defined quantity in quantum
>> theory. At best you might hope to Lorentz transform the wave function,
>> but then, when you do a measurement of position, the wave function
>> collapses on a synchronous slice.
>
>What would you say about the classical limit of quantum theory in the
>absence of gravity?
Ultimately I think I would say that it is meaningless.
>Do you think that 4 components of position-time
>transform as a 4-vector in this limit?
I think I just said that I don't even know what this is.
>Then, according to the Currie-
>Jordan-Sudarshan theorem, you cannot have interactions.
>
It does not appear meaningful to treat interactions between particles in
a classical approximation.
eugene_st...@usa.net
giessen.de> wrote:
> eugene_stefanov...@usa.net schrieb:
When talking about "Lorentz invariance" people often mix together two
quite different things. One thing is the "Lorentz group", which is a
part (that includes rotations and boosts of inertial reference frames)
of the Poincare group. The symmetry of observations with respect to
elements of this group is an exact symmetry of nature. This is also
called "relativistic invariance", and the CJS theorem talks explicitly
about relativistically invariant interacting Hamiltonian theories.
The other notion named after Lorentz is the notion of space-time
"Lorentz transformations". These are specific linear formulas that
(supposedly) connect positions and times of events measured in
different frames of reference. The CJS theorem tells that trajectories
(or worldlines) of interacting particles do not transform between
different frames by those linear Lorentz formulas.
In other words, "Lorentz transformations" provide a specific (4-
dimensional) representation of the "Lorentz group" in the Minkowski
space. However one group may have many different representations. The
CJS theorem basically says that time-positions of events with
interacting particles transform by some other representation of the
Lorentz group. This representation may be unrelated to the Minkowski
space-time, it may be even non-linear. The important thing is that it
cannot be the usual representation by "Lorentz transformations".
Yes, you are right that QFT is invariant with respect to the Lorentz
(even Poincare) group. However, as far as I know, nobody has ever
calculated time-dependent trajectories of interacting particles within
QFT (or its classical limit). If such a calculation was done, I am
sure it would find that these trajectories do not transform by Lorentz
formulas between different reference frames. In accordance with the
CJS theorem.
======================================= MODERATOR'S COMMENT:
CJS theorem = ? For the convenience of a broader readership please write not so widely known abbreviations once in detail
Oh No
>Yes, you are right that QFT is invariant with respect to the Lorentz
>(even Poincare) group. However, as far as I know, nobody has ever
>calculated time-dependent trajectories of interacting particles within
>QFT (or its classical limit).
There is a good reason for this. These trajectories are meaningless in
quantum theory.
Princess Florence, may I present you with this arm, as a token of my
esteem. I am sure you can find a dismembered head which has use for it.
Of course, Sir Eugene, if you really do mean the classical limit there
is nothing to stop you describing the trajectories of electrons in a
loop of wire, propelled by a changing magnetic field. The locus of the
trajectories transforms perfectly well.
Princess Florence, please take another arm for your collection.
Of course, the classical approximation is, by definition, approximation
to something else, and I am sure that is not what CJS was about.
Princess Florence, a leg for you, perchance.
>If such a calculation was done, I am sure it would find that these
>trajectories do not transform by Lorentz formulas between different
>reference frames. In accordance with the CJS theorem.
There is no way to anticipate what will be found by someone carrying out
an incorrect calculation. As Russell remarked, any result can be
obtained from a wrong postulate.
Ah, my dear Princess, I believe that makes a complete set.
Juan R.
I reproduce some basic formulae of the classical version on
http://canonicalscience.blogspot.com/2007/08/relativistic-lagrangian-and-
limitations_20.html
The quantum version is given by directly quantizing the classical theory
by usual methods. The difference with non-relativistic QM is that now
exists a new operator associated to Poincaré-Einstein time
(x^0 = ct --> c\hat{t})
The quantum equation of motion is
ihbar {\partial\Psi \over \partial\tau} = \hat{K} \Psi
where tau is the evolution parameter (Newtonian concept of time) of the
overall system.
Unlike classical and quantum field theory, this equation describes the
motion of many-body systems for arbitrary regimes, reduces exactly to non-
relativistic limit, and is free of several problems of usual relativistic
theory. E.g. there is no problem of time as in others quantum theories.
Regards
Juan R.
> On May 16, 4:56 pm, "Juan R." González-Álvarez
> <juanREM...@canonicalscience.com> wrote:
>> Currie, Jordan, and Sudarshan only showed what happen when one
>> confounds the concept of time in quantum mechanics (evolution
>> parameter) with the concept of time in relativistic *classical* physics
>> (dimension). Thus their Hamiltonian was the generator of time
>> translations for x^0 which is nonsensical.
>>
>> Of course, it is possible to build a Hamiltonian theory with positions
>> transforming by the LT. Example Stuckelberg theory.
>>
>> --http://www.canonicalscience.org/
>
>
> So, you propose to have two different notions of time in physical
> theory.
Actually, there exist two different concepts of time in physical theory.
Time as evolution parameter: non-relativistic both classical and quantum
mechanics, Coulomb electrodynamics, Newtonian gravity, thermodynamics and
non-relativistic statistical mechanics.
Time as dimension: special and general relativity, field electrodynamics,
quantum field theory, field theory of gravity, and several approaches to
quantum general relativity or field theory of gravity.
> Which one of these two notions corresponds to the "time" measured by
> experimentalists (by looking at their laboratory clocks)?
Both
> And what is the reason to keep the other notion? It is suspicious when
> the theory has two different quantities while the experiment has only
> one.
It is ironic that you are suspicious about this when your own "dressed
theory" is built over an inconsistent mixture of two concepts of time. You
take the 't' from quantum mechanics and mix it with the 't' from quantum
field theory giving a final theory is neither one nor other.
At least the theory worked by Stueckelberg, Feynman, Horwitz, Piron,
Schieve, and others is internally consistent because not mix the
different concepts of time.
Regards
Juan R.
Nothing more wrong! That "x" you find in RQFT textbooks is not related to
particle positions. That Lorentz invariance you find in RQFT textbooks
refers to *unobservables* "x" and "t" which are parametric arguments for
quantum fields.
RQFT was explicitely build to avoid the no-interaction theorem given
above by rejecting to follow the details of the quantum dynamics of
particles.
RQFT is enough for particle physicists, only interested in scattering
amplitudes. But rest of us need a relativistic *dynamics*. When I study
chemical process in a wessel or a biological process in a cell I can
assume (as particle physicists do) that particles are infinitely
separated at infinite past, collide and go to infinite future again
infinitely separated.
Juan R.
> RQFT is enough for particle physicists, only interested in scattering
> amplitudes. But rest of us need a relativistic *dynamics*. When I study
> chemical process in a wessel or a biological process in a cell I can
> assume (as particle physicists do) that particles are infinitely
> separated at infinite past, collide and go to infinite future again
> infinitely separated.
Strong typo
RQFT is enough for particle physicists, only interested in scattering
amplitudes. But rest of us need a relativistic *dynamics*. When I study
chemical process in a wessel or a biological process in a cell I *cannot*
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:81a6f91c-68fc-47c5...@n7g2000prc.googlegroups.com...
If I apply Poisson bracket to the time/energy relationship, I get:
{t,E}=1 (1)
from the term (dt/dt)(dE/dE).
This would mean going back to quantum that:
[t,E]=ihbar (2)
Isn't this a problem too?
LoL, Jay
maxwell
> Thus spake neurop...@yahoo.com.au
Charles, the Princess has just chopped off your (theoretical) arms,
legs & head and the best you can do is to attempt some weak humor.
Surely, your own theory (RQG) deserves a better defense than a laugh.
maxwell
> Sir Maxwell wrote most gallantly:
>
> �> Well said, Princess. [....] please stop on by whenever you
> �> can. Your wisdom & compassion are much appreciated [...]
>
> Thank you indeed for your kind words, dear sir.
> (Though let us steer well clear of the slippery slope of
> unkind gossip about our intelligent and learned colleagues.)
>
> [And BTW, don't let my sister Princess Spank-a-Lot overhear
> such compliments. She gets jealous. She's not nice when
> those green eyes are lit up.]
>
> �> mathematical conception with no correspondence in reality.
>
> You mean in the sense that we never actually measure a
> non-interacting particle, right?
No, my lovely Lady, I was simply referring to the present view that
the EM interaction exhibited by charged particles, such as electrons,
acts continuously throughout all of time it never 'shuts off' & never
behaves sporadically. The false impression given by naive
interpretations of Feynman diagrams, is that at first order, there is
only ONE interaction. Students should be reminded that this is only
the first term in an infinite expansion of the Lagrangian argument in
the exponential function under a time integral; IOW, it has to
integrated across ALL time, implying 'forever' interactions between
two electrons, EVEN at the first order . This also implies that the
'photon' is NOT a particle (a concept implying localization in real
space) but a mathematical description of the INTERACTION across all
time between any two electrons.
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> > And what is the reason to keep the other notion? It is suspicious when
> > the theory has two different quantities while the experiment has only
> > one.
>
> It is ironic that you are suspicious about this when your own "dressed
> theory" is built over an inconsistent mixture of two concepts of time. You
> take the 't' from quantum mechanics and mix it with the 't' from quantum
> field theory giving a final theory is neither one nor other.
You possibly misunderstood. The parameter 't' in quantum field theory
has nothing to do with real observable time. It is simply a dummy
integration variable. Similarly, three other arguments (x,y,z) of
quantum fields have nothing to do with the physical observable of
position. I write about that in my book.
maxwell
> Thus spake eugene_stefanov...@usa.net
>
> >On May 16, 3:08�pm, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>
> >> Not sure what we mean by Lorentz transforming particle positions anyway.
> >> Obviously in gtr position is not a vector, but disregarding that
> >> particle position is not, in general, a well defined quantity in quantum
> >> theory. At best you might hope to Lorentz transform the wave function,
> >> but then, when you do a measurement of position, the wave function
> >> collapses on a synchronous slice.
>
> >What would you say about the classical limit of quantum theory in the
> >absence of gravity?
>
> Ultimately I think I would say that it is meaningless.
>
> >Do you think that 4 components of position-time
> >transform as a 4-vector in this limit?
>
> I think I just said that I don't even know what this is.
>
> >Then, according to the Currie-
> >Jordan-Sudarshan theorem, you cannot have interactions.
>
> It does not appear meaningful to treat interactions between particles in
> a classical approximation.
>
Come, come, Charles. All we're asking is let your math go to the
limit where the relative velocity is much less than 'light-speed'.
Your reply is inadequate. Please try again.
Oh No
>On May 17, 3:04�am, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>> Thus spake neurop...@yahoo.com.au
>>
>
>legs & head
As she did not come close, I wouldn't insult her by suggesting she even
attempted such a thing.
>and the best you can do is to attempt some weak humor.
>Surely, your own theory (RQG) deserves a better defense than a laugh.
Why would it need a defence? It is not under attack. You may not have
noticed but we are talking of a paper by Currie/Jordan/Sudarshan, which
has no bearing at all on rqg.
Oh No
quantum particles as though they have a classical trajectory. It is
plain nonsense, and not surprising that CJS have proved it impossible.
Peter
> Thus spake maxwell <s...@shaw.ca>
> >Charles, the Princess has just chopped off your (theoretical) arms,
> >legs & head
> As she did not come close, I wouldn't insult her by suggesting she even
> attempted such a thing.
> >and the best you can do is to attempt some weak humor.
> >Surely, your own theory (RQG) deserves a better defense than a laugh.
> Why would it need a defence? It is not under attack. You may not have
> noticed but we are talking of a paper by Currie/Jordan/Sudarshan, which
> has no bearing at all on rqg.
Mr. Mehdorn was for many years the CEO of the public German railway
company Deutsche Bahn AG (btw, AG = shareholder company, and this is
part of the story), and he was quite successful in several areas.
However, as he has announced in an radio interview I have listened, he
considers himself as perfect and as making only minor faults which are
quickly corrected by the team. In the leading German talk show, he
criticised the request of trade unions for higher salaries. The
question, how he compares this with the fact, that his own salary has
been multiplied by a factor of 10 (!) during the last 10 years, he
replied, that "this is something completely different".
Mr. Mehdorn has recently resigned under the pressure of a scandal (in
which he was as little involved as the CEO of Siemens in the Siemens
scandal - of course ;-)
"Similarities with living persons are quite by chance and not
intended"
Hope this helps,
Peter
Juan R.
eugene_stefanovich wrote on Sun, 17 May 2009 14:11:42 -0600:
> On May 17, 7:15 am, "Juan R." González-Álvarez
Of course, in relativistic quantum field theory the parameters (x,y,z,t)
are not observable. However, the "t" in quantum fields is associated to
quantum field theory concept of time ("t" has units of time). It is also
that "t" which enters in the propagator for QED: U = exp(-i H_QED t).
Moreover, those parameter are related to the original (x,y,z,t) of the
classical field theory and special relativity, which *are observable*.
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> Of course, in relativistic quantum field theory the parameters (x,y,z,t)
> are not observable. However, the "t" in quantum fields is associated to
> quantum field theory concept of time ("t" has units of time). It is also
> that "t" which enters in the propagator for QED: U = exp(-i H_QED t).
>
> Moreover, those parameter are related to the original (x,y,z,t) of the
> classical field theory and special relativity, which *are observable*.
I don't quite understand what you are saying. In the first sentence
you state that the field parameter t is not observable. Then you seem
to claim the opposite.
My position is this: The only physical quantity calculated by QFT is
the S-matrix. (I am talking here about relativistic QFT, such as QED.
Quantum field theories in solid state physics is a different matter)
In these calculations all parameters t in quantum fields get
integrated out. So, these parameters should not be associated with the
time measured in laboratory. QFT in its traditional renormalized form
has nothing to say about the time evolution of states and observables
(renormalized QFT even does not have a well-defined Hamiltonian - the
generator of time translations), so the "physical" time parameter is
simply not needed in QFT.
The "physical" time parameter can be introduced only in theories
having well-defined and finite Hamiltonian. Then you can define the
time evolution operator exp(i H t) and you can study the time
dependence of states and observables. This can be achieved, for
example, in the "dressed particle" approach to QFT.
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:e5dcaeb9-1978-405f...@i28g2000prd.googlegroups.com...
Jay.
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:6c4cadea-f88d-4d2f...@f28g2000pra.googlegroups.com...
> On May 16, 12:05 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>
> > I do not believe that what I derive in the end conflicts on
> > either of [my previous points 2 & 3].
>
> I'll try below to explain how/where the problem lies at a
> deeper foundational level than what you're focusing on.
>
> > The p^0 component of the energy momentum remains commuting
> > with time, and the non-commutativity is between time and
> > perturbation, which replaces the energy / time uncertainty
> > with energy perturbation uncertainty (9.10) and maintains
> > the Robertson link between CCRs and uncertainty
> > relationships.. The velocity operator that we explored at
> > length a year ago is not at all impacted. Sections 8 and 9
> > make this clear. Please take a look with slower than warp
> > speed when you have a moment.
>
> It takes more than a "moment" to disentangle everything.
> I've now spent more than a moment, but my warp-speed
> impressions remain unchanged...
>
> IIUC, you're making a distinction between the free
> Hamiltonian (which you seem to regard as the "energy") and
> "perturbation" which is the interaction potential, ie the
> interaction part of the full Hamiltonian. Then you try to
> treat the interaction potential as distinct, and say that
> the time operator commutes with the free Hamiltonian
> ([t,E]=0), but not with the interaction part of the
> Hamiltonian ([t,V] = -ih). In that case, the time operator
> still satisfies [t,H] = -ih (where H is the full
> Hamiltonian). Therefore your quantum Hilbert space doesn't
> have a state of lowest total energy. The observable
> corresponding to what we measure experimentally as "energy"
> of an interacting system is the full Hamiltonian, not the
> free Hamiltonian.
>
> Similarly, when one writes [X^i,H] = v^i, the H is also the
> full Hamiltonian (if the X^i are to be interpreted as the
> position configuration observables).
>
> > This is not inconsistent with the velocity operator. After
> > all of our discussion last year I made sure of that. ...
>
> A lot of what we discussed before about Dirac velocity
> operators, Newton-Wigner position operators, and
> Foldy-Wouthuysen transformation was strictly in the context
> of the _free_ Dirac theory. Textbooks typically go further and
> discuss what happens in an interacting theory (showing that
> the transformation cannot be derived in closed form, but
> only perturbatively). But the important point here is that
> for the operators in "[X^i,H] = v^i" to have their usual
> intuitive physical meanings, H must be the full Hamiltonian.
>
> Aside: various related lines of research were blown out of
> the water a long time ago by the "no-interaction" theorem
> from this paper:
>
> Currie, Jordan, Sudarshan: "Relativistic Invariance and
> Hamiltonian Theories of Interacting Particles",
> Rev Mod Phys, vol 35, No 2, (1963), p350.
>
> The essence of the theorem is that the combined assumptions
> of Lorentz symmetry and Lorentz transformation of particle
> positions rule out any interaction. I.e., in any theory
> that tries to combine these notions, there cannot be any
> interaction between particles.
>
>
> ---
> LoL from Princess Florence as she moves on to other patients,
> having bandaged the current one as much as she reasonably can
> in the time available.
>
Princess and others:
I am taking everything you and Juan and others have said under
advisement.
Assuming for the sake of discussion that section 6 onward of
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-101.pdf
may need some revision, I would ask you or somebody to please take a
look at sections 1 through 5 alone, and let me know if you have any
critique.
What everyone seems to be missing is that we would not even be
discussing what we are discussing if I had not been able to derive at
least the three dimensional commutators:
[x^i,p_k]=i hbar delta^i_k
on the basis of general covariance. In the process we connected
together the gravitational gauge with the electrodynamic gauge, and
obtained energy quantization.
So, please take your noses for the moment out of anything later than
section 5, and let me please know if I have gotten to at least that
point with a few limbs left intact. ;-)
Thanks,
Jay.
eugene_st...@usa.net
try this one
http://wildcard.ph.utexas.edu/~sudarshan/pub/1963_005.pdf
======================================= MODERATOR'S COMMENT:
Thank you, links to open sources are very much welcome (PE)
neur...@yahoo.com.au
> What everyone seems to be missing is that we would not even be
> discussing what we are discussing if I had not been able to derive at
> least the three dimensional commutators:
>
> [x^i,p_k]=i hbar delta^i_k
>
> on the basis of general covariance.
(One of) the points of my earlier reference to Isham's treatment
is that you don't need general covariance to get that far.
P.S: Let me know if Master Eugene's link doesn't work for you
and I'll try to re-find the one I remembered.
---
LoL from a Princess in haste. (The many other followups in
this thread need more careful thought before I reply.
I certainly never tried to cut anyone legs off. :-)
Juan R.
> On May 17, 8:06 pm, "Juan R." González-Álvarez
> <juanREM...@canonicalscience.com> wrote:
>
>> Of course, in relativistic quantum field theory the parameters
>> (x,y,z,t) are not observable. However, the "t" in quantum fields is
>> associated to quantum field theory concept of time ("t" has units of
>> time). It is also that "t" which enters in the propagator for QED: U =
>> exp(-i H_QED t).
>>
>> Moreover, those parameter are related to the original (x,y,z,t) of the
>> classical field theory and special relativity, which *are observable*.
>
> I don't quite understand what you are saying. In the first sentence you
> state that the field parameter t is not observable. Then you seem to
> claim the opposite.
This is your misreading. In the first sentence I make an statement *about
RQFT*. In the second I make a statement *about special relativity and
classical field theory*.
> My position is this: The only physical quantity calculated by QFT is the
> S-matrix.
Untrue.
> (I am talking here about relativistic QFT, such as QED. Quantum field
> theories in solid state physics is a different matter) In these
> calculations all parameters t in quantum fields get integrated out.
There exists a well-defined reason for that.
> So, these parameters should not be associated with the time measured in
> laboratory.
Correct, as is stated in many textbooks in papers.
> QFT in its traditional renormalized form has nothing to say about the
> time evolution of states and observables
Untrue. QFT deals with scattering evolution of free states between
infinite past and infinite future.
> The "physical" time parameter can be introduced only in theories having
> well-defined and finite Hamiltonian. Then you can define the time
> evolution operator exp(i H t) and you can study the time dependence of
> states and observables. This can be achieved, for example, in the
> "dressed particle" approach to QFT.
The "dressed particle" approach is an attempt to substitute the unphysical
concept of bare particles in QRFT by the physical "dressed" particles.
Your so called dressed particle 'theory', is a different beast. This is a
completely inconsistent approach, which mixes elements from quantum
mechanics and quantum field theory. Already Dirac warned about being two
inconsistent theories.
The many flaws of your 'theory' have been extensively discussed in
sci.physics.research for years, as moderator Igor Kavhkine pointed in a
recent thread.
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:be6b4d9c-8966-4b64...@f28g2000pra.googlegroups.com...
> Sir Jay wrote:
>
>> What everyone seems to be missing is that we would not even be
>> discussing what we are discussing if I had not been able to derive at
>> least the three dimensional commutators:
>>
>> [x^i,p_k]=i hbar delta^i_k
>>
>> on the basis of general covariance.
>
> (One of) the points of my earlier reference to Isham's treatment
> is that you don't need general covariance to get that far.
Princess haste:
I just ordered Isham, so we will pick up this discussion downstream.
> P.S: Let me know if Master Eugene's link doesn't work for you
> and I'll try to re-find the one I remembered.
Yes it did. Thanks.
Jay.
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> Your so called dressed particle 'theory', is a different beast. This is a
> completely inconsistent approach, which mixes elements from quantum
> mechanics and quantum field theory. Already Dirac warned about being two
> inconsistent theories.
If Dirac ever said that, then I would like to disagree with him. In my
opinion, quantum field theory is simply quantum mechanics of systems
with variable number of particles. QFT is just a particular case of
QM. It seems that Weinberg would agree with me, as he wrote on page
49 of his vol. 1: "First some good news: quantum field theory is
based on the same quantum mechanics that was invented by
Schroedinger..."
Cheers.
Eugene.
Oh No
>On May 18, 8:15�am, "Juan R." Gonz�lez-�lvarez
><juanREM...@canonicalscience.com> wrote:
>
>> Your so called dressed particle 'theory', is a different beast. This is a
>> completely inconsistent approach, which mixes elements from quantum
>> mechanics and quantum field theory. Already Dirac warned about being two
>> inconsistent theories.
>
>If Dirac ever said that, then I would like to disagree with him. In my
>opinion, quantum field theory is simply quantum mechanics of systems
>with variable number of particles.
While I agree with this in so far as my own development is concerned,
please recognise that in other treatments it is simply not true, and
certainly not in yours. The undefinability of quantum fields is a major
unsolved problem. Simply claiming that qft reduces to qm is unjustified
unless you also present an explicit construction of qft from qm, and
explain in detail how the mathematical problems are overcome. Waving
your hands and dressing particles simply does not do this in any
mathematically acceptable manner.
>QFT is just a particular case of
>QM. It seems that Weinberg would agree with me, as he wrote on page
>49 of his vol. 1: "First some good news: quantum field theory is
>based on the same quantum mechanics that was invented by
>Schroedinger..."
>
Weinberg would certainly not agree with you. Like other quantum field
theorists, by "based on", he means using argument by analogy. This is
very different from a construction of qft from the quantum mechanics of
many particles. There is a large literature on this, which you should
familiarise yourself with before making unjustified claims.
eugene_st...@usa.net
> The undefinability of quantum fields is a major
> unsolved problem. Simply claiming that qft reduces to qm is unjustified
> unless you also present an explicit construction of qft from qm, and
> explain in detail how the mathematical problems are overcome. Waving
> your hands and dressing particles simply does not do this in any
> mathematically acceptable manner.
I am not quite sure what you mean by "undefinability of quantum
fields". Free quantum fields are well-defined linear combinations of
creation and annihilation operators of particles. So, there is no
problem. Possibly, you are talking about "interacting" quantum fields.
But, in my opinion, they are useless, and entire theory can be
formulated without them.
In quantum field theory states of the system are represented as
vectors in the Hilbert space, observables are represented as Hermitian
operators in the same Hilbert space. Dynamics and other inertial
transformations are given by an unitary representation of the Poincare
group in the Hilbert space. So, all features of the standard quantum
mechanics are there. The only non-standard thing is that interactions
can change the number of particles.
You are right that traditional QFT has a number of nasty problems,
e.g., divergences. However, they are fixed in the "dressed particle"
approach. So, in this approach, QFT is not different from
"relativistic quantum mechanics with variable number of particles".
Oh No
>On May 18, 2:09�pm, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>
>> The undefinability of quantum fields is a major
>> unsolved problem. Simply claiming that qft reduces to qm is unjustified
>> unless you also present an explicit construction of qft from qm, and
>> explain in detail how the mathematical problems are overcome. Waving
>> your hands and dressing particles simply does not do this in any
>> mathematically acceptable manner.
>
>I am not quite sure what you mean by "undefinability of quantum
>fields". Free quantum fields are well-defined linear combinations of
>creation and annihilation operators of particles.
No they are not. The definition you use leads to inconsistency in the
form of divergences. Writing down a formal expression for the free
fields is very different from being able to say that that formal
expression is mathematically well defined.
> So, there is no
>problem. Possibly, you are talking about "interacting" quantum fields.
>But, in my opinion, they are useless, and entire theory can be
>formulated without them.
>
and in mine.
>In quantum field theory states of the system are represented as
>vectors in the Hilbert space, observables are represented as Hermitian
>operators in the same Hilbert space. Dynamics and other inertial
>transformations are given by an unitary representation of the Poincare
>group in the Hilbert space. So, all features of the standard quantum
>mechanics are there. The only non-standard thing is that interactions
>can change the number of particles.
plus the inconvenient fact that the field operators are not well
defined.
>
>You are right that traditional QFT has a number of nasty problems,
>e.g., divergences. However, they are fixed in the "dressed particle"
>approach. So, in this approach, QFT is not different from
>"relativistic quantum mechanics with variable number of particles".
The dressed particle approach merely covers things which are not well
defined with other things which are not well defined. This is no help.
You cannot fix physical theory by making it sufficiently complicated
that you no longer see its failings, or by sticking your head in the
sand and refusing to acknowledge what those failings are.
eugene_st...@usa.net
> > So, all features of the standard quantum
> >mechanics are there. The only non-standard thing is that interactions
> >can change the number of particles.
>
> plus the inconvenient fact that the field operators are not well
> defined.
In its pure form the "dressed particle" approach does not need field
operators at all. The Hamiltonian of the "dressed particle" QED
contains a sum of inter-particle potentials (just as in ordinary
quantum mechanics). All loop integrals in S-matrix calculations are
convergent. These statements have been proved in www.arxiv.org/abs/physics/0504062
The only reason I use field operators in the book is to make
connection with the standard formulation of QED.
Oh No
>On May 18, 4:14�pm, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote:
>
>> > So, all features of the standard quantum
>> >mechanics are there. The only non-standard thing is that interactions
>> >can change the number of particles.
>>
>> plus the inconvenient fact that the field operators are not well
>> defined.
>
>In its pure form the "dressed particle" approach does not need field
>operators at all. The Hamiltonian of the "dressed particle" QED
>contains a sum of inter-particle potentials (just as in ordinary
>quantum mechanics). All loop integrals in S-matrix calculations are
>convergent. These statements have been proved in www.arxiv.org/abs/phys
>ics/0504062
I have read your book and much of it is good, but it does not do to make
extravagant claims which are not born out in the text. You have not even
touched on the issues which your claims require you to have resolved.
Your Hamiltonian is not well defined. When last I looked you did not
even have a formal expression for the Hamiltonian, let alone a proof of
convergence. As with other treatments, your manipulations in S-matrix
calculations use further undefined expressions. When we have discussed
it you have not even seen that it is necessary to establish that an
integral or an infinite sum is well defined. This being the case, it
would be better if you left out the extravagant claims and said only
that it is a heuristic account, which it is, and not a bad one at that..
neur...@yahoo.com.au
> Can we cut off [the arms and legs of the CJS paper], and see
> if it will still jump about shouting challenges?
It wasn't clear to me from what you've written in followup
postings whether you're familiar with the detail of the CJS
paper, or whether you're just responding to the brief snippets
mentioned by Master Eugene and this damsel. (It might be
helpful for me if you could clarify this.)
The CJS "no-interaction" theorem is just that -- a theorem.
As such, the paper is a useful contribution to science, imho.
People can then ponder what inputs to the theorem should
be relaxed or discarded - followed by inevitable punchups
when opinions on the latter disagree. :-(
---
LoL from an overworked Princess Florence.
(BTW, there's no point sending me arms and legs.
I'm not a micro-surgeon.)
"Welcome to S.P.F. - Serious.Punchups.Frequently"
neur...@yahoo.com.au
> If I apply Poisson bracket to the time/energy relationship,
> I get:
>
> {t,E}=1 (1)
>
> from the term (dt/dt)(dE/dE).
>
> This would mean going back to quantum that:
>
> [t,E]=ihbar (2)
>
> Isn't this a problem too?
This question needs a much longer answer (to be helpful) than
I can realistically offer on a newsgroup. But I'll try to
summarize a few things.
"Poisson brackets" usually means we're working in the
context of non-relativistic classical mechanics. (Do you
have any textbooks on classical mechanics, eg Goldstein?) In
that context, it's hard to make sense of "{t,E}" (if by that
you mean "{t,H}", with H being the Hamiltonian). Since H is
a function of p,q (spatial position and momentum variables),
you'd have to evaluate the Poisson bracket "{t,H}" very
carefully, and somehow invert q(t),p(t) in order to
differentiate time wrt q,p. The result (assuming it's even
well-defined) depends on the particular system.
Extending the Hamiltonian phase-space approach to a
relativistic context is problematic because of the
distinction between proper time and coordinate time.
(Goldstein discusses some of the difficulties.)
Even if one could construct a satisfactory relativistic
phase space formulation, there's the issue of constraints to
plague you. By "constraints", I mean ones like "m^2 = p^2",
where m=constant is imposed as a constant. (Also another one
for total spin.) Constrained quantization is a tricky
subject - you can't just pass from a Poisson bracket to a
commutator like in ordinary QM. There's a technique called
"Dirac-Bergman" quantization which must be used in this
case. The Wiki page on Dirac Bracket explains some of the
basics:
http://en.wikipedia.org/wiki/Dirac_bracket
The essence of the problem is that Poisson brackets are the
Lie product in a Lie algebra consisting of phase space
functions. But such an algebra must be "closed" else it's
not well-defined. (By "closed", I mean the Poisson bracket
of two such functions f,g from the algebra must yield
another function h which is also in that algebra.) But if
there's constraints on the functions (such as p^2=constant),
there's no guarantee that the Poisson bracket respects that
constraint, so h might not satisfy the constraint - which is
bad. Dirac-Bergman quantization is the careful way of
dealing with this, resulting in a modified quantum
commutator in general.
---
LoL from the Princess.
eugene_st...@usa.net
> Your Hamiltonian is not well defined. When last I looked you did not
> even have a formal expression for the Hamiltonian, let alone a proof of
> convergence. As with other treatments, your manipulations in S-matrix
> calculations use further undefined expressions. �When we have discussed
> it you have not even seen that it is necessary to establish that an
> integral or an infinite sum is well defined. This being the case, it
> would be better if you left out the extravagant claims and said only
> that it is a heuristic account, which it is, and not a bad one at that..
I accept a good portion of this criticism. Yes, a closed expression
for the "dressed particle" QED Hamiltonian does not exist at the
moment. What I have is only a prescription of how to get interaction
terms in each perturbation order by "dressing" the Hamiltonian of
traditional renormalized QED. I was able to follow this prescription
and obtain explicit interaction terms only in the lowest 2nd
perturbation order. Even the next 4th order presents significant
difficulties related to infrared divergencies. Currently, I have no
idea how to get around this problem. More work is needed. I have even
less idea about the convergence of the perturbation expansion for the
Hamiltonian. If you think these obvious drawbacks make my approach
"heuristic", so be it.
Oh No
>Prince Charles wrote:
>
>> Can we cut off [the arms and legs of the CJS paper], and see
>> if it will still jump about shouting challenges?
>
>It wasn't clear to me from what you've written in followup
>postings whether you're familiar with the detail of the CJS
>paper, or whether you're just responding to the brief snippets
>mentioned by Master Eugene and this damsel. (It might be
>helpful for me if you could clarify this.)
In all honesty, I am just responding to snippets mentioned by yourself
and Eugene. My impression is that it is a "no go" to somewhere I would
never have been going in the first place. I am happy to leave it at
that.
>
>The CJS "no-interaction" theorem is just that -- a theorem.
>As such, the paper is a useful contribution to science, imho.
>People can then ponder what inputs to the theorem should
>be relaxed or discarded - followed by inevitable punchups
>when opinions on the latter disagree. :-(
Perhaps we can leave it with its arms and legs then.
>LoL from an overworked Princess Florence.
>(BTW, there's no point sending me arms and legs.
>I'm not a micro-surgeon.)
As you are clearly dripping with magic, I hardly thought microsurgery
would be required.
neur...@yahoo.com.au
> I just ordered Isham, so we will pick up this discussion downstream.
I didn't realize you needed to buy it. I thought you'd just look at
it in a library somewhere. If you're into buying such books,
you might also consider Ballentine's "modern" introduction to
(non-relativistic) QM, which seems to be widely well-regarded
(not just by yours truly). He treats some of the foundations in
a more easily readable manner than Isham (although I don't seem
to recall Ballentine giving the particular derivation I had in mind
from Isham).
neur...@yahoo.com.au
> The many flaws of [Master Eugene's] 'theory' have been extensively
> discussed in sci.physics.research for years, as moderator Igor Kavhkine
> pointed in a recent thread.
I remember when Master Eugene had a Great Battle with Iggy-babes
and the fearsome Sage of Vienna, but I didn't know enough back
then to be confident of saying anything. Your remark above made me
go back into the spr archives and re-read the battle. Now that I
know a little more about both QFT and the dressed-particle
unitary-transformation approaches, I'm pretty sure that much of
the Great Battle arose from misunderstands on both sides.
BTW, I had a look around your website. (The link you gave me
earlier brings up one of your "micro-thoughts", but I was hoping
something more like a peer-reviewed article such as one might
find in Rev Mod Phys.) As a constructive criticism, your website
seems a bit messy to a newcomer like me. I couldn't find concise
info about peer-reviewed publications (or even just arxiv papers),
nor info and CVs for people in your center. I'm such info must be
there somewhere, but if so maybe it could be made easier to find?)
neur...@yahoo.com.au
>> I am not quite sure what you mean by "undefinability of quantum
>> fields". Free quantum fields are well-defined linear combinations of
>> creation and annihilation operators of particles.
Prince Charles puzzled this Princess by replying:
> No they are not. [...]
In the _free_ theory, I was under the impression that Gel'fand
triples and the nuclear spectral theorem provide an
adequately rigorous framework.
(The interacting case is of course quite another matter.)
--
LoL from the Princess.
neur...@yahoo.com.au
> In all honesty, I am just responding to snippets [of the CJS paper]
> mentioned by yourself and Eugene. My impression is that it is a
> "no go" �to somewhere I would never have been going in the
> first place. I am happy to leave it at that.
> [...]
> Perhaps we can leave [The CJS "no-interaction" theorem] with its
> arms and legs then.
I decided that I needed to re-study the CJS paper carefully before
saying
anything else about it.
This will take while to do properly, but already I've noticed a couple
of
technical points that actually seem incorrect to me. I'll start a
fresh
thread later if I can't overcome these suspicions.
> As you are clearly dripping with magic,[...]
You must be thinking of one my sisters.
---
LoL from Princess Florence.
Jay R. Yablon
<neur...@yahoo.com.au> wrote in message
news:3fb97938-af1d-422c...@g3g2000pra.googlegroups.com...
Let me get your (and anyone else's) thoughts on the following:
I just took a very brief look at CJS, so let me speak informally, with
the understanding that I am not looking for a detailed dissertation.
Let's think about physics in terms of symmetries, and in terms of
multiple symmetries "overlaid" on one another and having to peacefully
coexist with one another without conflict. (Not like SPF ;-) So, each
time you add a symmetry, you are further constraining what is permitted,
and sooner or later in principle you provide enough symmetry constraints
that nature can only behave in one unique fashion, which is what we
physically observe in all respects.
Also, think about starting with some symmetry, then breaking that
symmetry.
Finally, think about gravitational theory in a "pre-metric" form, and
then, adding a metric as one symmetry constraint.
What CJS seems to be saying is that once you add together several
symmetry constraints at once, you get to a point that they so crowd each
other out, that you have ended up with a physics which permits no
interaction, and so in some sense "over constrains" nature because taken
together the result is something that is not physical, i.e., no
interaction. So, something has to give way, and the discussion of what,
of course, is where the fun and games begin from this theorem.
I am now starting to see the paper with which I started this thread, at:
http://jayryablon.files.wordpress.com/2009/05/covariance-and-gauge-9.pdf
in this sort of context. Specifically, I am simply exploring one
symmetry -- general coordinate invariance in a pre-metric geometry --
and seeing where that leads. And, where it leads is to electrodynamics
gauge transformations, energy quantization, and four-dimensional
canonical commutation, and even quantum gravitation in the final
section.
As soon as one starts to impose added symmetry constraints, things start
to tighten up. So, for example, the four-dimensional commutation needs
to give way, or the symmetry of this needs to be "broken" in some way
not understood. In particular, as soon as I first make use of:
m^2=p^u p_u (1)
or Dirac's:
m psi = gamma^u p_u psi (2)
I am now taking a pre-metric geometry and adding metricty, since (1) is
parallel equation to the metric equation (and has a gravitation-like
equation of motion which is the covariant first derivative of (1) or
(2)), and (2) also imposes such things as how spin behaves, especially
once I add a gauge field with
p^u-->p^u+eA^u. (3)
In that light, I am inclined to back out any constraints other than
general coordinate invariance, show how far that can bring us (which is
all the way to the framework for quantum gravitation unconstrained by
anything else), and present this as just that. Obviously, this is then
not physical, but is "pre-physical" in the same sense as pre-metric
gravitation. That provides a good baseline to work from, because then
we have introduced no prejudice whatsoever, other than the requirement
for a generally coordinate-invariant description of whatever goes on in
nature, and can talk then about what to add and in what way and how that
changes the description, rather than trying to back thing out once we
are deeply committed on various further symmetry prejudices.
Does this make some sense as a way in which to view and present this?
Thanks,
Jay
Oh No
>Master Eugene wrote:
>
>>> I am not quite sure what you mean by "undefinability of quantum
>>> fields". Free quantum fields are well-defined linear combinations of
>>> creation and annihilation operators of particles.
>
>Prince Charles puzzled this Princess by replying:
>
>> No they are not. [...]
>
>In the _free_ theory, I was under the impression that Gel'fand
>triples and the nuclear spectral theorem provide an
>adequately rigorous framework.
>
such as rigged Hilbert space (another name Gel'fand triple for the
uninitiated), another to show that there is something in mathematics
which obeys that structure (yes, I know this is physics, but if there is
nothing in maths corresponding to what we think the physics is, then
there is a fundamental problem with mathematical modelling of the
physics).
For example, in the case of distribution theory it is not enough just to
define the Dirac delta as a distribution. To show existence one really
has to show convergence of the limits using a nascent delta function on
the given test space.
(again, for the uninitiated I give a link to explain terminology)
http://en.wikipedia.org/wiki/Dirac_delta_function#Representations_of_the
_delta_function
The central problem for the existence (in the formal mathematical sense)
of the field operators is the equal point multiplication. One can
attempt to define a field operator psi(x) as an operator valued
distribution, but then products psi_bar(x) psi(y) must also be defined.
But psi_bar(x) psi(x) is not defined as a distribution - integrals
containing it will diverge.
The problem can be solved by removing psi_bar(x) psi(x) from the theory.
This is done formally in causal perturbation theory (Scharf, Finite
Quantum Electrodynamics), by splitting the advanced and retarded
distributions using a continuous switching function which is zero at the
equal point multiplication. This doesn't actually give you a formal
construction of the field operators as distributions, but it does enable
the theory to be constructed and loop diagrams evaluated using the
advanced and retarded parts without the need for renormalisability.
In my view, the only real problem with Scharf's account is that causal
perturbation theory is an S-matrix approach, and no physical reason
is given for the inclusion of the switching function. There are those
who object to the way limits at infinity are taken (the interaction has
to be "switched off" to get a meaningful limit in S-matrix theory). I am
not actually uncomfortable with that. The real impact of the treatment
is not at infinity but at close range. Effectively the switching
function switches off the interaction at close range.
I came to all of this independently from a different angle, a little
after Epstein & Glaser (who did the fundamental work on which causal
perturbation theory is based), and I only showed the cause of the
divergence as the undefined integrals arising from the equal point
multiplication, whereas they developed a method for evaluating the
integrals which is now central to Finite QED.
The main formal difference between my treatment and theirs is that I use
a step function rather than a continuous switching function to remove
the equal point multiplication. This makes no difference to the formal
definition of the integrals. So, the result is the same, and I just cite
Scharf at that point of my account. In all of my work I have been much
more interested in setting up theory and correspondence to physics than
in calculation of results, so I am pleased that what I see as the
difficult bit was done by Epstein & Glaser & Scharf.
My account starts with interpretation of quantum theory. It is
relativistic from the outset. I remove the equal point multiplication on
physical grounds by interpreting quantum theory as a description of
interactions between pointlike particles in the absence of spacetime
background. Interactions are discrete, and the equal point
multiplication is removed on physical grounds, because only one
interaction is allowed for a particle at a time.
I don't know if any of this can catch your interest, but I am sure you
will appreciate that, for an account which properly makes sense, posts
on a NG are inadequate. One has to start properly from square one and
develop things very carefully, which is why I have done the website.
neur...@yahoo.com.au
> I decided that I needed to re-study the CJS paper
> carefully before saying anything else about it. [...]
In pursuit of this, I have a question for Master Eugene
(or anyone else who has _carefully_studied_ the
technical detail of the CJS paper)...
CJS use a formalism based on multi-particle phase space.
In the section leading up to eqs 3.5, 3.6 they show how
this formalism recovers Hamilton's equations of motion.
In particular eq 3.6:
d/dt p_i^n(t) = - d/dq_i^n H(q,p)
They don't explicit derive this, but it's similar to
the way they derive the other Hamilton equation 3.5.
Clearly, they need [P_i^n(q,p) , H(q,p] to be nonzero
for particle "n" to experience a nonzero force.
But over the page near eq 3.9 and the sequel, they
impose the Poincare [H,P_1]=0, then use this when
analyzing q'_j^n(t) to get the usual CCRs between
position and momentum.
It seems to me that this is tantamount to vetoing
any possibility of nonzero force being experienced
by any of the particles. Sure, one would expect the
composite system to satisfy the Poincare [H,P_1]=0,
but I don't see how they're justified in imposing
this on individual particles. (In a composite system,
two particles can be accelerating relative to
each other even though the total system experiences
zero force as a whole.)
Am I reading CJS correctly? (If so, it seems unsurprising
and trivial that they derive impossibility of interaction.)
---
LoL from Princess Perplexed.
Juan R.
> On May 18, 8:15Â am, "Juan R." González-Álvarez
> <juanREM...@canonicalscience.com> wrote:
>
>> Your so called dressed particle 'theory', is a different beast. This is
>> a completely inconsistent approach, which mixes elements from quantum
>> mechanics and quantum field theory. Already Dirac warned about being
>> two inconsistent theories.
>
> If Dirac ever said that, then I would like to disagree with him.
Dirac, one of founders of both quantum mechanics and quantum field theory
wrote in "Mathematical Foundations of Quantum Theory":
The appearance of this [Dirac] equation did not solve the general
problem of making quantum mechanics relativistic. [...] Most physicists
are very satisfied with this situation. They argue that if one has rules
for doing calculations and the results agree with observation, that is
all that one requires. But it is not all that one requires. One requires
a single comprehensive theory applying to all physical phenomena. Not
one theory for dealing with non-relativistic effects and a separate
disjoint theory for dealing with certain relativistic effects. [...] For
these reasons I find the present quantum electrodynamics quite
unsatisfactory.
> In my
> opinion, quantum field theory is simply quantum mechanics of systems
> with variable number of particles. QFT is just a particular case of QM.
> It seems that Weinberg would agree with me, as he wrote on page 49 of
> his vol. 1: "First some good news: quantum field theory is based on the
> same quantum mechanics that was invented by Schroedinger..."
As explained in virtually any textbook in relativistic quantum field
theory, this is a different theory from quantum mechanics.
For instance, Mandl and Shaw in the page 10 of their classic texbook
(revised version) in quantum field theory notice:
"an important difference between a quantized field theory and
non-relativistic quantum mechanics."
Dirac used the term "a separate disjoint theory" for emphasizing the
difference with quantum mechanics.
Best regards.
Juan R.
> Hi Don Juan,
>
>> The many flaws of [Master Eugene's] 'theory' have been extensively
>> discussed in sci.physics.research for years, as moderator Igor Kavhkine
>> pointed in a recent thread.
>
> I remember when Master Eugene had a Great Battle with Iggy-babes and the
> fearsome Sage of Vienna, but I didn't know enough back then to be
> confident of saying anything. Your remark above made me go back into the
> spr archives and re-read the battle. Now that I know a little more about
> both QFT and the dressed-particle unitary-transformation approaches, I'm
> pretty sure that much of the Great Battle arose from misunderstands on
> both sides.
And mutual mistakes? I agree.
But Eugene assertion that he has developed a consistent relativistic
quantum mechanics that solves the difficulties of QED does not hold upon
close inspection.
> BTW, I had a look around your website. (The link you gave me earlier
> brings up one of your "micro-thoughts", but I was hoping something more
> like a peer-reviewed article such as one might find in Rev Mod Phys.) As
> a constructive criticism, your website seems a bit messy to a newcomer
> like me. I couldn't find concise info about peer-reviewed publications
> (or even just arxiv papers), nor info and CVs for people in your center.
> I'm such info must be there somewhere, but if so maybe it could be made
> easier to find?)
I gave you a link to my personal blog (which I initially used for
canonical science issues, therein the old domain name). That personal blog
gives some basic expressions for the theory you asked and one monograph
in the issue.
I do not know any Rev. Mod. Phys. I have some papers PLA, JMP... on the
application of Stuckelberg theory to atomic problems and a brief review of
the applications to molecular systems in a recent Handbook dealing with
molecular relativistic problems.
Juan R.
> This Princess said:
>
>> I decided that I needed to re-study the CJS paper carefully before
>> saying anything else about it. [...]
>
> In pursuit of this, I have a question for Master Eugene (or anyone else
> who has _carefully_studied_ the technical detail of the CJS paper)...
>
> CJS use a formalism based on multi-particle phase space. In the section
> leading up to eqs 3.5, 3.6 they show how this formalism recovers
> Hamilton's equations of motion. In particular eq 3.6:
>
> d/dt p_i^n(t) = - d/dq_i^n H(q,p)
>
> They don't explicit derive this,
It is a standard result.
> but it's similar to the way they derive
> the other Hamilton equation 3.5. Clearly, they need [P_i^n(q,p) , H(q,p]
> to be nonzero for particle "n" to experience a nonzero force.
>
> But over the page near eq 3.9 and the sequel, they impose the Poincare
> [H,P_1]=0, then use this when analyzing q'_j^n(t) to get the usual CCRs
> between position and momentum.
>
> It seems to me that this is tantamount to vetoing any possibility of
> nonzero force being experienced by any of the particles. Sure, one would
> expect the composite system to satisfy the Poincare [H,P_1]=0, but I
> don't see how they're justified in imposing this on individual
> particles.
They work with detail an interacting two-particle system, and prove that
Lorentz invariance applied to particle trajectories only holds when
particles are free. Thus I fail to understand you.
> (In a composite system, two particles can be accelerating
> relative to each other even though the total system experiences zero
> force as a whole.)
>
> Am I reading CJS correctly? (If so, it seems unsurprising and trivial
> that they derive impossibility of interaction.)
>
> ---
> LoL from Princess Perplexed.
--
eugene_st...@usa.net
> But over the page near eq 3.9 and the sequel, they
> impose the Poincare [H,P_1]=0, then use this when
> analyzing q'_j^n(t) to get the usual CCRs between
> position and momentum.
>
> It seems to me that this is tantamount to vetoing
> any possibility of nonzero force being experienced
> by any of the particles. Sure, one would expect the
> composite system to satisfy the Poincare [H,P_1]=0,
> but I don't see how they're justified in imposing
> this on individual particles. (In a composite system,
> two particles can be accelerating relative to
> each other even though the total system experiences
> zero force as a whole.)
In their notation P_1 is the generator of displacement along the x-
axis. In the quantum case this would be the operator of the total
momentum (= the vector sum of momenta of all particles, as in eq.
(4.2)). So, the condition [H, P_1] = 0 is simply the conservation law
for the total momentum of the system. This condition does not imply
zero forces on all particles. It says that the sum of all forces is
zero, i.e., the third Newton's law.
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> As explained in virtually any textbook in relativistic quantum field
> theory, this is a different theory from quantum mechanics.
So bad, then I disagree with "virtually any textbook". I wrote my own
textbook to prove the point that QFT (properly formulated) is just
good old quantum mechanics with a twist (the possibility of processes
that change the number of particles). If you say that QFT and QM are
fundamentally different, then could you specify which postulates of QM
are violated in QFT?
Oh No
>This Princess said:
>
>Am I reading CJS correctly? (If so, it seems unsurprising
>and trivial that they derive impossibility of interaction.)
you mean you might cut off its arms and legs anyway?
>LoL from Princess Perplexed.
>
I too am perplexed. I do not understand how a paper using the
commutation relations quantum theory can also talk of trajectories, and
yet this is my clear impression from all you, Eugene and Don Juan have
said.
Oh No
>eugene_stefanovich wrote on Mon, 18 May 2009 12:41:55 -0600:
>
>>
>>> Your so called dressed particle 'theory', is a different beast. This is
>>> a completely inconsistent approach, which mixes elements from quantum
>>> mechanics and quantum field theory. Already Dirac warned about being
>>> two inconsistent theories.
>>
>> If Dirac ever said that, then I would like to disagree with him.
>
>Dirac, one of founders of both quantum mechanics and quantum field theory
>wrote in "Mathematical Foundations of Quantum Theory":
How very rude of him. :-) Mathematical Foundations of Quantum Theory
was written by Von Neumann.
>
> The appearance of this [Dirac] equation did not solve the general
> problem of making quantum mechanics relativistic. [...] Most physicists
> are very satisfied with this situation. They argue that if one has rules
> for doing calculations and the results agree with observation, that is
> all that one requires. But it is not all that one requires. One requires
> a single comprehensive theory applying to all physical phenomena. Not
> one theory for dealing with non-relativistic effects and a separate
> disjoint theory for dealing with certain relativistic effects. [...] For
> these reasons I find the present quantum electrodynamics quite
> unsatisfactory.
Nonetheless this remark is now out of date. It is not particularly
simple to show non-relativistic quantum theory from qed, but it can be
done. This is what the Fouldy-Wouthuyson transformation is for.
eugene_st...@usa.net
> I too am perplexed. I do not understand how a paper using the
> commutation relations quantum theory can also talk of trajectories, and
> yet this is my clear impression from all you, Eugene and Don Juan have
> said.
They work under the assumption that in the classical limit (hbar goes
to zero, states of particles are described by well-localized wave
packets) quantum Hamiltonian theory transforms into classical
Hamiltonian theory. In this approximation, particle trajectory should
be understood as the time dependence of the "center of mass" of the
wave packet. This assumption makes sense, because we do observe
particle trajectories in macroscopic world in spite of the fact that
this world is governed by quantum mechanics.
Oh No
to form a classical particle from quantum particles I need to put a
large number of quantum particles together, not let hbar go to 0. Then
the interactions of the conglomeration are not the same as the
interactions of a single elementary particle. There are no fundamental
interactions of the classical particle anyway - its component quantum
particles do the interacting (perhaps this is what the theorem proves).
We can't understand a particle trajectory as plotting the centre of mass
of the wave packet, because it isn't. It could be the centre of mass of
many wavepackets.
Juan R.
> Thus spake Juan R. González-Álvarez <juanR...@canonicalscience.com>
>>eugene_stefanovich wrote on Mon, 18 May 2009 12:41:55 -0600:
>>
>>> On May 18, 8:15Â am, "Juan R." González-Álvarez
>>> <juanREM...@canonicalscience.com> wrote:
>>>
>>>> Your so called dressed particle 'theory', is a different beast. This
>>>> is a completely inconsistent approach, which mixes elements from
>>>> quantum mechanics and quantum field theory. Already Dirac warned
>>>> about being two inconsistent theories.
>>>
>>> If Dirac ever said that, then I would like to disagree with him.
>>
>>Dirac, one of founders of both quantum mechanics and quantum field
>>theory wrote in "Mathematical Foundations of Quantum Theory":
>
> How very rude of him. :-) Mathematical Foundations of Quantum Theory
> was written by Von Neumann.
What is the point of this /ad hominem/ against a dead guy? And its
utility for this discussion?
>> The appearance of this [Dirac] equation did not solve the general
>> problem of making quantum mechanics relativistic. [...] Most
>> physicists are very satisfied with this situation. They argue that if
>> one has rules for doing calculations and the results agree with
>> observation, that is all that one requires. But it is not all that one
>> requires. One requires a single comprehensive theory applying to all
>> physical phenomena. Not one theory for dealing with non-relativistic
>> effects and a separate disjoint theory for dealing with certain
>> relativistic effects. [...] For these reasons I find the present
>> quantum electrodynamics quite unsatisfactory.
>
> Nonetheless this remark is now out of date. It is not particularly
> simple to show non-relativistic quantum theory from qed, but it can be
> done. This is what the Fouldy-Wouthuyson transformation is for.
It is not outdated. Non-relativistic quantum theory is not derivable from
QED. In particular the Fouldy-Wouthuyson transformation is only useful to
find certain non-relativistic limits of certain expressions for free
particle RQM.
Regards
eugene_st...@usa.net
> I am still perplexed. The assumptions here seem very rash. For a start,
> to form a classical particle from quantum particles I need to put a
> large number of quantum particles together, not let hbar go to 0. Then
> the interactions of the conglomeration are not the same as the
> interactions of a single elementary particle. There are no fundamental
> interactions of the classical particle anyway - its component quantum
> particles do the interacting (perhaps this is what the theorem proves).
> We can't understand a particle trajectory as plotting the centre of mass
> of the wave packet, because it isn't. It could be the centre of mass of
> many wavepackets.
It is not necessary to consider many-particle systems to reach the
classical limit. Heavy elementary particles have more or less
classical properties, i.e., their wave packets do not spread fast and
move along pretty well-defined trajectories. The same is true for
light particles with high momentum. Fast electrons have well-defined
trajectories and do not show significant interference effects. Even
photons can be reasonably well described in the corpuscular picture.
Oh No
interactions take place. A track left by an electron, fast moving or
not, is very different from saying that the electron has precise
position at each precise time.
Juan R.
> On May 19, 7:26 am, "Juan R." González-Álvarez
> <juanREM...@canonicalscience.com> wrote:
>
>> As explained in virtually any textbook in relativistic quantum field
>> theory, this is a different theory from quantum mechanics.
>
> So bad, then I disagree with "virtually any textbook". I wrote my own
> textbook to prove the point that QFT (properly formulated) is just good
> old quantum mechanics with a twist (the possibility of processes that
> change the number of particles).
Evidently you can disagree, but the problem is you avoid the issues
discussed in textbooks.
> If you say that QFT and QM are
> fundamentally different, then could you specify which postulates of QM
> are violated in QFT?
As stated both are different. This is explained in QFT textbooks and
papers. Some differences are treated in the reference I gave and you
deleted now :-D
I will not give a detailed discussion, but a brief review of some
fundamental differences between quantum mechanics of charged particles
(QM), relativistic quantum mechanics of charged particles (RQM), and
relativistic quantum field theory (RQFT) for electromagnetic field and
fermions.
QM is essentially a dynamical theory of particles with AAAD scalar
potentials Phi(R(t)) where x is an operator associated to the dynamical
variable that localizes the particle in space. There is not time operator.
Interactions are Galilean invariant (aka instantaneous).
RQM is essentially a dynamical theory of particles with field potentials
Phi(x,t) and A(x,t) where x is an operator that lacks the corresponding
dynamical variable that localizes the particle in space. There is not time
operator. Interactions are Lorentz invariant (aka non-instantaneous), but
this invariance is tricky (strict non-equivalence between space and time,
quantum version of the no-interaction theorem, etc.).
The difficulties within RQM to define an observable position x for
particles, the localization inconsistencies (non-analitical propagators,
light cone causality...), Klein paradox, negative probabilities, etc. did
that finally people abandoned the idea of unify special relativity with QM
(RQM) and developed RQFT, which is a different beast.
QFT is essentially a non-dynamical theory of fields with local potentials
Phi(x,t) and A(x,t) where x is not an operator because there is not
dynamical variable that localizes the particle. Of course, there is not
time operator. Again interactions are Lorentz invariant (aka non-
instantaneous).
For the sake of comparison the Stuckelberg Feynman Horwitz Piron
relativistic quantum dynamics (cited before) is build as a relativistic
generalization of QM. This is essentially a dynamical theory of particles
with AAAD scalar potentials Phi(rho(tau)) where both x and t are
operators associated to the dynamical variables that localizes the
particle in spacetime. Interactions are Lorentz invariant in x and t but
still instantaneous with regard to the evolution time *tau*.
It has been proved in literature that this relativistic theory rigorously
reduces to QM in the non-relativistic limit. E.g. the many-body
interaction Hamiltonian reduces to exact QM form
V = V(rho(tau)) ---> V(R(t))
However, quantum field theory fails to get the limit because in none limit
the interaction Phi(x,t) used in QED gives the Phi(R(t)). In fact, the
imposibility to obtain QM from QFT (remind also Dirac quote about being
disjoint theories) was one of the reasons for the development of this
theory Stuckelberg, Feynman, Horwitz, Piron, Schieve, and others.
Moreover, Eugene know that in my draft I have proved that classical
relativistic dynamics version of Stuckelberg et al theory reduces exactly
to classical mechanics with Coulomb/Newtonian interactions. I have
rigorously proved the classical limit of
V = V(rho(tau)) ---> V(R(t))
However, the classical field electrodynamics fails to reduce because in
none limit the interaction Phi(x,t) used in field electrodynamics gives
the Phi(R(t)) in the classical mechanics for charged particles (Phys.
Rev. E 1996, 53, 5373).
Cheers.
Oh No
>
>>>eugene_stefanovich wrote on Mon, 18 May 2009 12:41:55 -0600:
>>>
>>>>
>>>>> Your so called dressed particle 'theory', is a different beast. This
>>>>> is a completely inconsistent approach, which mixes elements from
>>>>> quantum mechanics and quantum field theory. Already Dirac warned
>>>>> about being two inconsistent theories.
>>>>
>>>> If Dirac ever said that, then I would like to disagree with him.
>>>
>>>Dirac, one of founders of both quantum mechanics and quantum field
>>>theory wrote in "Mathematical Foundations of Quantum Theory":
>>
>> How very rude of him. :-) Mathematical Foundations of Quantum Theory
>> was written by Von Neumann.
>
>What is the point of this /ad hominem/ against a dead guy? And its
>utility for this discussion?
Do you not understand jokes at all? While Dirac may well have written
this, Von Neumann's book would certainly be a strange place for him to
have written it.
>
>>> The appearance of this [Dirac] equation did not solve the general
>>> problem of making quantum mechanics relativistic. [...] Most
>>> physicists are very satisfied with this situation. They argue that if
>>> one has rules for doing calculations and the results agree with
>>> observation, that is all that one requires. But it is not all that one
>>> requires. One requires a single comprehensive theory applying to all
>>> physical phenomena. Not one theory for dealing with non-relativistic
>>> effects and a separate disjoint theory for dealing with certain
>>> relativistic effects. [...] For these reasons I find the present
>>> quantum electrodynamics quite unsatisfactory.
>>
>> Nonetheless this remark is now out of date. It is not particularly
>> simple to show non-relativistic quantum theory from qed, but it can be
>> done. This is what the Fouldy-Wouthuyson transformation is for.
>
>It is not outdated. Non-relativistic quantum theory is not derivable from
>QED. In particular the Fouldy-Wouthuyson transformation is only useful to
>find certain non-relativistic limits of certain expressions for free
>particle RQM.
>
In fact you are wrong. There is a derivation of the interacting Dirac
equation at http://rqgravity.net/CEM, and reduction to the non-
relativistic equation is simple from there.
eugene_st...@usa.net
> I would regard this as very approximate on the scale on which
> interactions take place. A track left by an electron, fast moving or
> not, �is very different from saying that the electron has precise
> position at each precise time.
Of course, the classical limit is just an approximation. But this
approximation has served us pretty well for many centuries. Moreover,
the time-position Lorentz transformations (which are the subject of
the CJS paper) can be discussed only in this approximation, because in
the quantum world the idea of localized events having exact positions
does not work, as you correctly pointed out.
eugene_st...@usa.net
<juanREM...@canonicalscience.com> wrote:
> I will not give a detailed discussion, but a brief review of some
> fundamental differences between quantum mechanics of charged particles
> (QM), relativistic quantum mechanics of charged particles (RQM), and
> relativistic quantum field theory (RQFT) for electromagnetic field and
> fermions.
Hi Juan,
thank you for a good review of fundamental problems associated with
traditional approaches to relativistic quantum theory. However, I am
afraid most of this criticism does not apply to my approach. I still
insist that by successive approximations one can go from the "dressed
particle" QED to relativistic quantum mechanics (which is different
from RQM described by you; for example, the RQM I am talking about is
based on the Breit Hamiltonian rather than on the Dirac equation) and
then to the usual non-relativistic quantum mechanics. Moreover, one
can also obtain consistent classical limits of these theories.
In all these theories interactions are described by instantaneous
(action-at-a-distance) potentials. However, this does not violate the
relativistic invariance. All necessary arguments and explanations are
in the book. If you have specific questions about it, I would be happy
to explain.
neur...@yahoo.com.au
> In that light, I am inclined to back out any constraints other than
> general coordinate invariance, [...]
> That provides a good baseline to work from, because then
> we have introduced no prejudice whatsoever, other than the requirement
> for a generally coordinate-invariant description of whatever goes on in
> nature, [...]
There still seems to be a rather large prejudice lurking in there: the
tacit
assumption of a common background spacetime manifold.