Re: h bar and Lorentz transformation

From: Non Ame (noname_at_nospam.net)
Date: 03/26/05 

Date: Sat, 26 Mar 2005 03:21:51 +0000 (UTC)

"Ken S. Tucker" <dynamics@vianet.on.ca> writes:

>Non Ame wrote:
>> "Ken S. Tucker" <dynamics@vianet.on.ca> writes:
>>
>> >To Bilge & Non Ame...
>>
>> >Bilge wrote:
>> >> Ken S. Tucker:
>> >> >Non Ame wrote:
>> >>
>> E(r,t) is the electric field at position r and time t. B(r',t) is
>the
>> magnetic field at position r' and time t. B(r',t') is the magnetic
>field
>> at position r' and time t'.

>That notation is clear now.

>> E_x(r,t) B_y(r',t') - B_y(r',t') E_x(r,t)

>>is the commutator of E_x(r,t) and B_y(r',t').

>That makes no sense to me, does he have a
>tensor? I don't have the book you recommended,
>likely I'm missing something.

Perhaps you need an introductory book on quantum mechanics. There was no
reason for us to expect you to know the correct formula for

        E_x(r,t) B_y(r',t') - B_y(r',t') E_x(r,t),

which was why I gave the explicit formula. But it is surprising that you
are surprised that E_x(r,t) B_y(r't') and B_y(r',t') E_x(r,t) may not be
equal in the case of the quantum EM field. If you had any understanding
of quantum mechanics, you would have known that that was a possibility.
E_x(r,t) and B_y(r',t') are real q-numbers, to use the parlance of Dirac,
so, from that information alone, it is possible to conclude that

        E_x(r,t) B_y(r',t') - B_y(r',t') E_x(r,t)

is an imaginary q-number, but no more can be determined. Specifically,
the imaginary q-number need not be equal to zero.

>>E_x(r,t) B_y(r',t) - B_y(r',t) E_x(r,t)

>>is a special case when E_x and B_y are
>>evaluated at exactly the same time.

>> Schiff gives the formulae
>>
>> E_s(r,t) E_{s'}(r',t') - E_{s'}(r',t') E_s(r,t)
>>
>> = B_s(r,t) B_{s'}(r',t') - B_{s'}(r',t') B_s(r,t)
>>
>> = 4 pi i hbar ((delta_{ss'}/c^2) d/dt d/dt' - d/dr_s d/dr'_{s'})
>>
>> D(r - r', t - t'),
>>
>> E_s(r,t) B_{s'}(r',t') - B_{s'}(r',t') E_s(r,t)
>>
>> = - 4 pi i hbar epsilon_{ss's"} d/dr_{s"} d/dt' D(r - r', t - t'),
>>
>> for s, s' = x, y, z, where delta_{ss'} is the Kronecker delta,
>> epsilon_{ss's"} is the Levi-Civita symbol determined by epsilon_{xyz}
>= 1,
>> summation is taken over all values of s" in the last line above, and
>>
>> D(rho, tau) = (4 pi |rho|)^{-1} [delta(|rho|+c tau) - delta(|rho|-c
>tau),
>>
>> where delta(r) is the Dirac delta function on R^3.
>>
>> When t' = t, these equations reduce to
>>
>> E_s(r,t) E_{s'}(r',t) - E_{s'}(r',t) E_s(r,t)
>>
>> = B_s(r,t) B_{s'}(r',t) - B_{s'}(r',t) B_s(r,t)
>>
>> = 0,

>That's what I thought in the first place.

But you thought it for the wrong reason.

>> E_s(r,t) B_{s'}(r',t) - B_{s'}(r',t) E_s(r,t)
>>
>> = - 4 pi i hbar c epsilon_{ss's"} d/dr_{s"} delta(r - r')
>>
>> = 4 pi i hbar c epsilon_{ss's"} d/dr'_{s"} delta(r - r'),
>>
>> for s, s' = x, y, z.
>>
>> Note that E_x(r,t) and B_x(r',t') commute for all values of r, r', t,
>t'.

>I'd be careful there.

Not at all. It is a fact that, for the quantum radiation EM field in
Minkowski space, these two fields must commute for all pairs of events.

>Suppose Non Ame (K) and I (K') are sitting in a
>tree K.I.S.S.I.N.G. we can agree r = -r' in our
>relative spatial relation, between kisses.
> But the time t = t' holds during the passion.

Cut that out, you sleazeoid. You really revolt me. We are supposed to be
discussing the quantum electromagnetic field in special relativity here,
not inter-personal relationships and groping. You are obviously obsessed
with sex, and seem to think that all women are your inferiors. We are not
inferior to men at all. I would love to see you suffer the same degree of
extreme pain that a woman does during childbirth, because then you might
give women the respect we deserve. Get your mind out of the gutter and
turn your thoughts to what is really important here.

>> All components of the field strengths must commute if (r,t) and
>(r',t')
>> are separated by a space-like interval, otherwise causality would be
>> violated.

>Ah, set up the experiment using *radar ranging*
>between the bodies and determine (r,t) and (r't').

And what is the relevance of this comment to the discussion?

>> Schiff comments that because all components of the field strengths
>commute
>> if the interval between (r,t) and (r',t') is not light-like,

>What is "not light-like" where measuring intervals
>(like ds) are concerned?

In Minkowski space, the interval between (r,t) and (r',t'), where (r,t)
and (r't') are not equal, is called:

        light-like if |r-r'|^2 = c^2 (t-t')^2,

        space-like if |r-r'|^2 > c^2 (t-t')^2,

        time-like if |r-r'|^2 < c^2 (t-t')^2.

This classification of intervals is Lorentz invariant (moreover, the
classification is Poincare invariant). Two events are separated by a
space-like interval iff there exists a reference frame in which the two
events are simultaneous (but at different places). Two events are
separated by a time-like interval iff there is a reference frame in which
the two events are in the same place (but at different times). Two events
are separated by a light-like interval iff it is possible for light in a
vaccuum to travel from the earlier event to the later event.

Light-like motion means that ds = 0, and in the case of General
Relativity, light-like motion is described by the formula ds = 0.

>>the quantized
>> electromagnetic field is propagated at the classical speed of light,
>c.

>There are too many double negatives to sort
>threw in this post.

That's "through", not "threw". You really should get a dictonary. Your
mistake is also quite unforgivable, as I have already gone to considerable
trouble to correct you about this error on your part. As I have already
informed you, the word 'threw' is the past tense of 'throw'.

Schiff's statement was to the effect that all components of the quantized
EM field at one event commute with all components of the quantized EM
field at another event, if the interval separating the events is either
space-like or time-like. He also stated that as a consequence of this,
the quantized EM field propagates at speed c.

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