From:  Steve Fulling <fulling@math.tamu.edu>
Date:  Fri, 5 Jun 2020 20:15:00 -0500

ANNOUNCEMENT:  Along with my department's IT head, I am still 
looking for a system to automate the compilation of your messages 
into daily mailings.  Please enquire whether your institution 
has a contract with a service called Slack.  (See article "Slack 
(software)" in Wikipedia.)  If the answer is "yes" for most 
people, it will be cheaper for us to move to that platform.  And 
that will save me an hour or more of cutting and pasting every 
evening!  I understand that Slack also will allow us to put TeX 
expressions into the messages via MathJax or something similar!

===================================================================

From: George Matsas <george.matsas@unesp.br>
Date: Fri, 5 Jun 2020 17:58:25 -0300

> Dear Steve, Eduardo and Jose,
>
> Thank you for your comments. Let me try to answer you guys in a 
> single message since some of your comments can be traced back 
> to the same common origin.

EDITOR'S NOTE:  For the full text of the two comments in 
question, please see yesterday's mailing (matsas1_comments604), 
the first and last items.  

> But before discussing the specific points raised on the UD 
> detectors, let me reemphasize that my main aim in this "talk" 
> was to call attention to Ref. [1]. Any discussions about /"in 
> what sense UD detectors with Delta E=0 can (or cannot?) be seen 
> as structureless scalar sources" /do not affect [1].In 
> addition, as said before, for those who (for some reason?) do 
> not like the introduction of the dipole in [1], please see Ref. 
> [8]. In Ref. [8] we analyze in the inertial and uniformly 
> accelerated frames a single electric charge uniformly 
> accelerated in the z-direction and rotating along the XY plane 
> and find explicitly that the emission of a Minkowski photon can 
> be interpreted as the absorption or emission ofRindler photon 
> in the Unruh thermal bath -- quite expected. Now, Ref. [1] 
> results can be straightforwardly recovered by vanishing the 
> charge rotation in the XY plane. In this sense, Ref. [1] is a 
> particular case of Ref. [8] where no UD detectors, no dipole, 
> only a plain electric charge is used. We believed that this 
> should put to rest any doubts on Ref.[1] (namely that the 
> emission of a usual photon from a uniformly accelerated charge 
> in the Minkowski vacuum can be seen to correspond to either the 
> absorption from or the emission to the Unruh thermal bath of a 
> zero-energy Rindler photon.)
>
> Now back to the UD detector questions.
>
> 1.Let me begin with Steve's second question (not related to 
> Eduardo and Jose's ones)": ... doesn't that mean we should 
> AVERAGE over initial states (or take a linear combination of 
> them (as density matrices) with positive coefficients (a,b) 
> that add to 1), rather than adding them?
>
> I do not think so. The reason the combined absorption and 
> emission rates should not be averaged in this case is that I 
> need to use Unruh and Wald's (1984) result [4] to link the 
> Rindler calculation with the radiation seen by inertial 
> observers [see paragraph (28)]. According to [4], it is the 
> combined, absorption + emission, rates of Rindler particles 
> which corresponds to the emission rate of Minkowski particles 
> as seen by inertial observers.
>
> 2.Now, let me discuss Steve's 3rdpoint: "That is not the end of 
> the story, however.If we take the energy splitting to 0 at the 
> beginning, then the argument in par. 23 is no longer valid: all 
> 4 terms contribute equally, not just 2."
>
> Yes. That's right. I am aware of it. But by starting with 
> \Delta E = 0 from the beginning it would jeopardize from 
> scratch my first aim (par 5):to render the original argument 
> given in [1] in terms of Unruh-DeWitt detectors which are more 
> familiar to most of the people. If I began the calculation 
> assuming a structureless source from scratch, then it had 
> better not to talk of UD detector and reproduce Ren and 
> Weinberg [5] which is basically Ref. [1] with our electric 
> charge replaced by a structureless scalar source.
>
> 3.As I see it, Steve's point 2 above is related with Eduardo 
> and Jose's file complain at (26-end): "Note that now, when 
> taking the limit \Omega -> 0, tanh \beta\Omega 2 -> 0, thus the 
> response vanishes."
>
> Let me point out that this is already codified in my file and 
> why this does not spoil the result - on the contrary. In par. 
> 23 we have
>
> Gamma^L  \approx   Gamma^deexc_em + \Gamma^exc_abs
>
> Because of Unruh+Wald result [5], this is the emission rate of 
> Minkowski particles seen by inertial observers. This can be 
> recast as
>
> Gamma^L  \approx  \int dk_\bot d\omega_R  (|A^deexc_em|^2 * 
> (1+thermal factor) + |A^exc_abs|^2 * (thermal factor))
>
> Using that|A^deexc_em|=|A^exc_abs|, this is simplified:
>
> Gamma^L \approx \int dk_\bot d\omega_R
> |A^deexc_em|^2/ tanh (\pi \omega/a)
>
> where 1 / tanh (\pi \omega/a) comes from the Unruh thermal bath.
>
>
> Now, it is easy to see that in the particular case, where Delta 
> E =0, indeed A^deexc_em=A^exc_abs =0 -- as Eduardo and Jose 
> pointed out. Can we conclude from this that inertial observers 
> will see no radiation coming out from the source? No. Why not? 
> Because the factor 1/ tanh (\pi \Delta /a) (which multiplies 
> |A^deexc_em|^2) diverges with the same power. [Physically 
> speaking -- and this connected with Don's question -- the 
> plethora of zero-energy particles in the bath stimulates 
> infinitely the absorption and emission of zero-energy particles 
> rendering the response nonzero (and finite).] (At this point, I 
> would like Eduardo and Jose to reconsider when they say that 
> they accept that the acceleration radiation may be associated 
> with the zero frequency Rindler modes but /that they are 
> suspicious of the relation between this and the Unruh effect. 
> -- The 1 / tanh (\pi \omega/a)which saves the response from 
> vanishing comes from the Unruh the bath.)
>
>
> Concerning the indeterminacy "0 x Infty" which is the problem 
> crux, it must be regularized somehow. In [1] we regularized the 
> result using an oscillating dipole. In [5] Ren and Weinberg 
> regularized the result using an oscillating scalar source. In 
> [8] we began with a plain swirling charge, vanishing the 
> swirling at the end to recover a uniformly accelerated charge. 
> Here I tried a variation of Ren and Weinberg approach, 
> replacing their oscillating source with UD detectors. All 
> results agree with each other and, most importantly, they agree 
> with independent inertial frame calculations which assume plain 
> uniformly accelerated sources/charges from scratch.
>
> Thank you all for the comments.
>
> Best wishes, George
>