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 |
(*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}
*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];
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 \(\langle\)*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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