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, 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 = 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(mn) = |\(\langle\)0 | (m!)–1/2 Am (nm)!–1/2 B nm |ψ\(\rangle\)|2

= \(1\over n!\,m!\,(n-m)!\)|\(\langle\)0| Am B nm a*n |0\(\rangle\)|2 = \(1\over n!\,m!\,(n-m)!\)|\(\langle\)0 | (αa)m (–β*b)nm a*n | 0\(\rangle\)|2

= \(n!\over m!\,(n-m)!\) |α|2m |β|2(nm) = \(n!\over m!\,(n-m)!\) T m Rnm ,

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 mk=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].
@ 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.


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