Re: h bar and Lorentz transformation
From: Non Ame (noname_at_nospam.net)
Date: 03/26/05
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Date: Sat, 26 Mar 2005 02:35:34 +0000 (UTC)
Non Ame <noname@nospam.net> writes:
>Non Ame <noname@nospam.net> writes:
>>"Ken S. Tucker" <dynamics@vianet.on.ca> writes:
>>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.
These formulae can be put into Lorentz covariant form.
F_{uv}({x^u}) F_{u'v'}({x'^u}) - F_{u'v'}({x'^u}) F_{uv}({x^u})
= 4 pi i hbar (g_{uu'} d/dx^v d/dx^{v'} - g_{uv'} d/dx^v d/dx^{u'}
- g_{vu'} d/dx^u d/dx^{v'} + g_{vv'} d/dx^u d/dx^{u'}) D(x^u - x'^u),
where F_{uv}({x^u}) is the value of the appropriate component of the
quantum electromagnetic field at the event whose coordinates are
(x^0,x^1,x^2,x^3), F_{u'v'}({x'^u}) is the value of the appropriate
component of the quantum electromagnetic field at the event whose
coordinates are (x'^0,x'^1,x'^2,x'^3), D is as above, and the metric has
signature (+---), and c has been set equal to 1.
The formula above is actually Poincare covariant.
Some may object that the above formula is not completely in Lorentz
covariant form since D is not in a Lorentz invariant form. The fact that
D(rho, tau) is Lorentz invariant (i.e. it is invariant under special
orthochronous Lorentz transformations, which preserve the direction of
time) is very simple to prove directly. Alternatively, one can prove that
D(x^1, x^2, x^3, x^0) = - (1/(2 pi)) sign(x^0) delta(g_{uv} x^u x^v).
since we are restricting our attention to Lorentz transformations which
are in the connected component containing the identity, the sign of x^0
is invariant for events on the light cone under all such trasformations.
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