Re: h bar and Lorentz transformation

From: Non Ame (noname_at_nospam.net)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 02:08:19 +0000 (UTC)

Non Ame <noname@nospam.net> writes:

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

>>To Bilge & Non Ame...

>>Bilge wrote:
>>> Ken S. Tucker:
>>> >Non Ame wrote:
>>>
>>> >> One thing that I have long suspected is that Ken expects quantum
>>> >> observables to commute, and I recently got some evidence in
>>favour of
>>> >> that hypothesis after I recently explicitly wrote one expression
>>in
>>> >> Quantum Electrodynamics as
>>> >>
>>> >> E_x(r,t) B_y(r',t) - B_y(r',t) E_x(r,t),
>>> >>
>>> >> and his response included the indication that he believed the
>>above
>>> >> expression to be equal to zero.
>>> >
>>> >Non ames expression above has poor notation, for
>>> >example, if both sides are equal (unless he forgot
>>> >a cross product or some other specification), between
>>> >B and E then it's zero. Rewrite as,
>>>
>>> Ken, his notation is fine. What he wrote is the commutator of
>>> B and E, each measured in different frame. If it were identically
>>equal
>>> to zero, a radio wouldn't work, since a zero commutator implies
>>> the interval between measurements of E(r,t) and B(r',t') is
>>spacelike.

>>See that Bilges B(r',t') differs from Non Ame's
>>B_y(r',t), substantially, where prime's are
>>concerned. Which of you is correct?
>>Need clarification.

>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'.

>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'). 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.

That is not quite right. delta(r) is the Dirac delta function on R, here.

>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,

> 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.

Here, delta(r) is the Dirac delta function on R^3.

>Note that E_x(r,t) and B_x(r',t') commute for all values of r, r', t, t'.

>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.

>Schiff comments that because all components of the field strengths commute
>if the interval between (r,t) and (r',t') is not light-like, the quantized
>electromagnetic field is propagated at the classical speed of light, c.

Relevant Pages

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