Mirrors in Quantum Field Theory |
In General > s.a. vacuum [focusing of fluctuations];
{for mirrors in optics, see optical technology}.
* 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, 1985-09-20].
* 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 = α a + β b and B = α* a + β* b ,
for some α and β such that |α|2 + |β|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
|ψ\(\rangle\) = (n!)−1/2 a*n |0\(\rangle\) ,
and the probability of getting m particles in mode A is
P(m ← n) = |\(\langle\)0 | (m!)−1/2 Am (n−m)!−1/2 B n−m | ψ\(\rangle\)|2
= \(1\over n!\,m!\,(n-m)!\)|\(\langle\)0| Am B n−m a*n |0\(\rangle\)|2 = \(1\over n!\,m!\,(n-m)!\)|\(\langle\)0 | (αa)m (−β*b)n−m a*n | 0\(\rangle\)|2
= \(n!\over m!\,(n-m)!\) |α|2m |β|2(n−m) = \(n!\over m!\,(n-m)!\) T m Rn−m ,
if we call T:= |α|2
and R:= |β|2 (strange!).
(b) Now suppose we send in a thermal distribution in mode a,
Pin(n) = exp{−βωn}/Z = xn/Z ,
where x:= exp{βω}; Then, from (a), the outcoming distribution is
Pout(m) = ∑k=0∞ Pin(m+k) T mR k (m+k)!/(m!k!) = Z−1 xm T m ∑k=0∞ 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 \(\langle\)mout\(\rangle\)
= T\(\langle\)nin\(\rangle\),
although it can't really fail; One could also see for which
T one gets ω' = ω.
References
@ Particles and detectors:
Walker PRD(85);
Beige et al PRA(02)qp [atom front of mirror];
Galley et al qp/04-proc.
@ Moving mirrors: Carlitz & Willey PRD(87) [and black-hole radiation];
Gjurchinovski AJP(04)oct [light reflection and Lorentz contraction],
EJP(13) [light reflection];
Castaños & Weder PS-a1410 [electromagnetic field].
@ 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)jul [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 JPA(08)-a0709,
Fosco et al PLB(08)-a0807 [and dynamical Casimir effect];
Fulling & Wilson a1805-fs [stationary mirror in Rindler space];
Wan Mokhtar a1806 [radiation, fermions vs bosons].
@ 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 \(\langle\)Tab\(\rangle\)];
news sn(13)jul [debut of the perfect mirror].
> Related topics: see
propagation of gravitational waves;
time in quantum mechanics.
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 2 jul 2018