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} *A*^{m}
(*n*–*m*)!^{–1/2}
*B*^{ n–m} |*
ψ*\(\rangle\)|^{2}

=
\(1\over n!\,m!\,(n-m)!\)|\(\langle\)0| *A*^{m} *B*^{ n–m} *a**^{n}
|0\(\rangle\)|^{2} =
\(1\over n!\,m!\,(n-m)!\)|\(\langle\)0 | (*α**a*)^{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}* R*^{n–m}
,

if we call *T*:= |*α*|^{2} and *R*:=
|*β*|^{2} (strange!).

(b) Now suppose we send in a thermal distribution in mode *a*,

*P*_{in}(*n*) = exp{–*βωn*}/*Z* = *x*^{n}/*Z* ,

where *x*:= exp{*βω*};
Then, from (a), the outcoming distribution is

*P*_{out}(*m*)
= ∑_{k=0}^{∞} *P*_{in}(*m*+*k*) *T*^{ m}*R*^{ k}
(*m*+*k*)!/(*m*!*k*!) = *Z*^{–1} *x*^{m} *T*^{ m} ∑_{k=0}^{∞}
*x*^{k} *R*^{k} (*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 \(\langle\)*m*_{out}\(\rangle\)
= *T*\(\langle\)*n*_{in}\(\rangle\),
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].

@ __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 \(\langle\)*T*_{ab}\(\rangle\)]; news sn(13)jul [debut of the perfect mirror].

> __Related topics__: see
propagation of gravitational waves; time in
quantum mechanics.

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