Mirrors| Mirrors |
In Classical Physics > s.a. [optics]; sound [time
reversal].
* Status: Mirrors typically
absorb a few percent of the light; "Dielectric
mirrors" are highly reflective, but only for a band of wavelengths;
1998, The best mirror so far was built by MIT researchers [@ Fink et al Sci(98)nov].
In Quantum Field Theory > s.a. vacuum [focusing of fluctuations].
* Idea: The effects produced
by mirrors in quantum field theory are due to the boundary conditions they
impose on the fields.
Effect of a Mirror on a Thermal State
* Result: It is interesting
to notice that, if a thermal distribution of particles at a certain energy
is incident on a partially reflecting mirror,
the transmitted and reflected distributions are still thermal, but at a different
temperature; Notice that we are not talking of a thermal spectrum, with different
energies, but just of a thermal probability distribution, within
one mode, of finding a certain number of particles; Only black bodies in
equilibrium with the surrounding will emit a thermal spectrum of the same
temperature
as the incoming one [From a meeting with R Sorkin, 20.sep.1985].
* Proof of the above claim:
(a) Suppose we have ingoing modes from the left and the right with annihilation
operators a and b, respectively; Then, by unitarity, the
outgoing modes will be
A = 
a + 
b and B = * a +
* b ,
for some and such
that ||2 +
||2 =
1; If we now send in n particles
in a and 0 in b, we get m particles
in A and k in B, with m + k = n;
This ingoing state is given by
|
=
(n!)–1/2 a*n |0 ,
and the probability of getting m particles in mode A is
P(m ← n) = | 
0
| (m!)–1/2 Am
(n–m)!–1/2 B n–m | |2
=
[n!m!(n–m)!]–1 |
0| Am B n–m a*n
|0 |2 =
[n!m!(n–m)!]–1 |
0 | (a)m (–*b)n–m a*n |
0 |2
= n!/[m!(n–m)!]
||2m ||2(n–m) =
n!/[m!(n–m)!] T m Rn–m
,
if we call T:= ||2 and R:=
||2 (strange!).
(b) Now suppose we send in a thermal distribution in mode a,
Pin(n) = exp{–
n}/Z = xn/Z ,
where x:= exp{};
Then, from (a), the outcoming distribution
is
Pout(m)
= 
k=0infty Pin(m+k) T mR k
(m+k)!/(m!k!) = Z–1 xm T m k=0infty
xk Rk (m+k)!/(m!k!)
,
which is obviously again a thermal distribution; The summation in the last
expression gives something like (1–xR)–m–1;
Check.
* What to do afterwards:
We should also check that mout
=
T nin,
although it can't really fail; One could also see for
which T one gets ' =
.
References > s.a. time in
quantum mechanics.
@ Particles and detectors: Walker PRD(85); Beige et al PRA(02)qp [atom
in
front of mirror]; Galley et al qp/04-in.
@ Moving mirrors: Carlitz & Willey PRD(87)
[and black hole radiation]; Gjurchinovski AJP(04)
[light reflection and Lorentz contraction].
@ Accelerated mirrors: Jaekel & Reynaud QO(92)qp/01 [radiation
pressure],
QSO(95)qp/97,
RPP(97)qp;
Saa & Schiffer PRD(97)gq/96 [bound
states for massive scalars]; Van
Meter et al AJP(01)
[plane wave reflection]; Obadia & Parentani
PRD(01)
[massless fields], PRD(03)gq/02,
PRD(03)gq/02 [radiation];
Saharian CQG(02)ht/01 [vacuum
polarization];
Calogeracos JPA(02)gq/01,
JPA(02)gq/01 [radiation];
Marolf & Sorkin PRD(02)ht [self-accelerating
box paradox]; Haro & Elizalde a0709 [and
dynamical Casimir effect].
@ And thermodynamics: Cohadon et al PRL(99)qp [cooling
by radiation];
Helfer
PRD(01)ht/00;
Machado et al PRD(02)ht [radiation
pressure at finite T].
@ Related topics: Frolov & Singh CQG(99)gq [spherical
semitransparent]; Van
Den
Broeck
ht/00-wd [vacuum
forces from Tab].
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11 jul 2008