Photon emission, power output (and black holes)

[asimov42]

I recently re-read an article by Muller (https://arxiv.org/pdf/1606.07975.pdf) about the flow of time, and the possibility of time reversal given sufficient energy dissipation (basically during black hole evaporation, he concludes). Although the paper is on arXiv and not peer reviewed, Muller has written a book on the topic.

This got me thinking: Muller's point is that a sufficient amount of energy must be dissipated rapidly (if his theory is correct) - but in a much more mundane sense, let's consider the emission of a single photon from an excited hydrogen atom...

If photon emission is instantaneous, then one could say that the power output (energy dissipation rate) is infinite (finite energy photon emitted in zero time). I don't think this is what Muller is getting at... but in any quantized system (e.g., a blackbody radiator), the energy emission (power output) will vary, just by virtue of energy coming out in 'chunks'.

So, is photon emission truly instantaneous? Or is there a quantum effect of some type that actually brings time into play in some way? (those are my real questions)

In the former case, it would seem that Muller's idea could be tested without looking at black holes, but he specifically states that small evaporating black holes are the only objects that radiate sufficiently.

I'm totally confused about how to square the idea of continuous radiation output with the nature of a quantized system.

Thanks all.

 

[PeterDonis]

Muller's model appears to be classical, not quantum (he mentions quantum field theory, but the only actual math in his paper--which isn't much at all, and that makes me skeptical, but that's a whole other discussion--is classical), so I don't see any useful comparison between his model and a quantum model of emission of radiation.

 

[asimov42]

Thanks @PeterDonis - let's put Muller completely aside for now and just focus on photon emission absorption.

I've read in several places, including here when @vanhees71 noted that "the emission of a photon is a continuous process, described by unitary time evolution" that neither emission or absorption is instantaneous, which resolves the issue of infinite instantaneous power. However, there a good (peer reviewed) paper here:

http://www3.uji.es/~planelle/APUNTS/ESPECTROS/jce/JCEphoto.html

that explain in some detail how the process occurs. Now, making a measurement will force the system into a stationary state ... which it appears could occur in an arbitrarily short period of time. So, is there some minimum bound on the amount of time for a photon to be released / absorbed?

Going back to Muller briefly - his classical treatment demands a certain energy flux per unit time... but it seems that, if the release of a photon can take a arbitrarily short time (before it is measured), then Muller's conditions would be satisfied for plain ol' atoms... no black hole needed. So what am I missing?

Thanks all - any insights would be really helpful.

 

[vanhees71]

Is there a readable version of this paper somewhere?

 

[asimov42]

Is there a readable version of this paper somewhere?

Hi @vanhees71 - unfortunately, it's from (J. Chem. Educ. 1979 56 631-635, and I haven't been able to find a better version than the link above... not easy to get papers that old from the journal. But at least all the text and figures are in the web version (although the formatting is poor).

 

[vanhees71]

Thanks for the citation. Of course, there are good pdf's for this paper at the journal's website:

https://pubs.acs.org/toc/jceda8/56/10

Now I can read the paper :-)).

 

[asimov42]

Sorry @vanhees71 - I don't have access through the journal website (paywall). Thank you so much for having a look!

 

[asimov42]

Muller's model appears to be classical, not quantum (he mentions quantum field theory, but the only actual math in his paper--which isn't much at all, and that makes me skeptical, but that's a whole other discussion--is classical), so I don't see any useful comparison between his model and a quantum model of emission of radiation.

@PeterDonis , can you elaborate on the quantum model of emission? How does one measure power, in a quantum framework, when the flow of energy is not continuous?

 

[PeterDonis]

How does one measure power, in a quantum framework, when the flow of energy is not continuous?

Observables in QM are operators; so there is an operator for power. An operator can apply to any quantum state; the state tells you the probabilities of getting different measured results.

You appear to have a mental model where, in order to measure power, we somehow have to measure a continuous rate of energy flow. That would be true in classical physics, but there's no reason why it needs to be true in quantum physics; you just apply the power operator to the quantum state, as described above.

Alternatively, one can view "power" as an averaged quantity over time, i.e., over many individual emission events; this is probably more common in actual practice. In the QM formalism, that corresponds to computing a time averaged expectation value for the operator corresponding to power.

 

[asimov42]

@PeterDonis - thanks, yes, this is in fact exactly what I was thinking - I think

My question then, is going back to the Muller result (right or wrong) - his assertion regarding time reversal is that it requires a specific energy outflow rate, or power... but in the framework of GR, this seems to need to be a continuous flow. I don't know how to square this with the quantum model - Muller's classical approach (even though it's applied to black holes, and quantum effects) requires continuity, but QM is not.

So if I require energy dissipation at a certain continuous rate, but the QM operator gives me 'on' and 'off' amounts, what then? How does one reconcile the two?

Or, is it the case that the power operator is continuous in time? That is, for any quantum state (and transition between states) does the power operator change smoothly? All that I've read above (related to photon emission and absorption etc.) says that is should... in particular, here's a quote from the J. Chem. Edu. article:

A common interpretation is that an instantaneous quantum jump in the energy occurred at some unpredictable time during the transition period. This interpretation may in turn suggest that the superposition function is merely a mathematical formalism; if one could observe the evolution of the state, it would also exhibit an abrupt discontinuity or change from

i to

j or vice versa... These experiments confirm that there are no "quantum jumps" in the non-stationary state; rather there are smooth, continuous periodic changes in the magnetic and electric properties of a system undergoing a transition.

 

[PeterDonis]

if I require energy dissipation at a certain continuous rate, but the QM operator gives me 'on' and 'off' amounts, what then?

Since the classical model is an approximation to the quantum model, the usual procedure is to take appropriate time averages and expectation values of the quantum operators and treat them as the classical continuous amounts. In other words, the underlying quantum process is not continuous, but at the level of the classical approximation it looks continuous--you can't see the individual quantum events, just their averages over time.

is it the case that the power operator is continuous in time?

Many quantum operators have a continuous dependence on time, but the quantum operators and states only give you probabilities for events to occur; so basically the probabilities change continuously with time. But that doesn't mean the actual quantum events are continuous in time.

 

[asimov42]

You appear to have a mental model where, in order to measure power, we somehow have to measure a continuous rate of energy flow. That would be true in classical physics, but there's no reason why it needs to be true in quantum physics; you just apply the power operator to the quantum state, as described above.

A related question - what would the power operator give you as a result when applied to say, the transition of an electron from one orbital to another in a hydrogen atom?

 

[PeterDonis]

what would the power operator give you as a result when applied to say, the transition of an electron from one orbital to another in a hydrogen atom?

I don't know because I don't know the specific form of the power operator. I'm also not sure the concept of "power" even makes sense when applied to a single transition. If it didn't, then there wouldn't actually be a power operator; "power" would be something that only made sense, in a quantum model, as a time average of expectation values of other operators. I don't know enough about how power is treated in quantum models to know for sure.

 

[asimov42]

Is there anywhere I could look for more information on the power operator and single transitions? (@vanhees71 may know) I've had a look around, but I can't seem to find much information... if at all.

 

[vanhees71]

I've never heard about a "power operator". What should that be?

If it comes to radiation you can get very far with the semiclassical approximation, i.e., you treat the radiation field as classical electromagnetic field and only the matter quantum theoretical (in atomic physics of light atoms you get also a long way using non-realtivistic quantum theory). The only thing you are missing is spontaneous emission.

A full picture of radiation is only QED, and there it is quite well defined what's observable, namely transition probabilities (aka cross sections) between asymptotic free states and correlation functions between gauge-invariant local observables, like an intensity of the em. field (i.e., expectation values of components and correlation functions of the energy-momentum tensor).

 

[PeterDonis]

I've never heard about a "power operator". What should that be?

I'm not sure; as I said in my previous post, it's possible that there isn't one, i.e., that emitted power for radiation is not a direct observable. It might only be definable as some kind of average of expectation values.

 

[asimov42]

Ok, so just going back to Muller's paper (as an example) - when he gives an example of a black hole that must radiate at a given rate, what does that mean in a quantum system?

If the radiative output is quantum in nature, then you'll never have a continuous output - so when someone says "this effect occurs when radiation is dissipated at a rate of 1,000,000,000 Watts" (and let's say that's a tight, exact bound - for argument's sake), then how is it at all possible to measure this?

You could take a time average, but over what interval? And if you did, because everything comes in discretized quanta (and no output occurs at precisely the same time), you'd end up with periods of zero output (over short intervals, in between emission).

@vanhees71 I take your point about QED, but i may not be far enough along to fully grasp everything.

I'm just using Muller's paper as an example (but I'm more curious in general) - he's saying that in a GR sense, energy dissipation must reach a given rate (for time reversal - very controversial idea, but anyway)... now this is given as a continuous rate - how does one reconcile that with a quantum system, where the output will not be continuous?

 

[PeterDonis]

and let's say that's a tight, exact bound - for argument's sake

Since this assumption obviously contradicts your previous assumption, that quantum radiation isn't continuous, then the problem is that you are making two contradictory assumptions and then wondering why the result doesn't make sense. The correct response is "don't do that".

You could take a time average, but over what interval?

Long enough to have a reasonably large number of quantum events; small enough that the probability of a quantum event per unit time doesn't change appreciably over the interval.

 

[asimov42]

Since this assumption obviously contradicts your previous assumption, that quantum radiation isn't continuous, then the problem is that you are making two contradictory assumptions and then wondering why the result doesn't make sense. The correct response is "don't do that".

Hehe Ok @PeterDonis, very good point - what would the quantum equivalent of the statement of "Dissipate radiation at a rate of 1 billion Watts" be? (you may have already answered for the most part in the previous post).

This is really what confuses me about the Muller paper... I presume (any help here much appreciated - sorry for the all questions) that he's getting at the fact that to change the GR metric tensor significantly, you have to move a lot of mass around, and (in his case) do it in a short time?

Thanks @PeterDonis and all for the help!

 

[PeterDonis]

what would the quantum equivalent of the statement of "Dissipate radiation at a rate of 1 billion Watts" be?

An appropriate number of quantum emission events, at some average energy per event, in an appropriate interval of time.

This is not really anything particular to quantum mechanics; it's just basic math. If you have some number of events in some interval of time, you divide the two to get a rate. I don't understand what's difficult to grasp about that.

I presume (any help here much appreciated - sorry for the all questions) that he's getting at the fact that to change the GR metric tensor significantly, you have to move a lot of mass around, and (in his case) do it in a short time?

I can't say about Muller's hypothesis specifically, since I'm highly skeptical about the whole "time reversal" thing. But in GR generally, a power (or more precisely a power density, like watts per cubic meter) is a rate of change of a component of the stress-energy tensor. So it tells you how fast the stress-energy tensor (and hence the Einstein tensor, which describes spacetime curvature) is changing, yes.

 

[asimov42]

An appropriate number of quantum emission events, at some average energy per event, in an appropriate interval of time.

This is not really anything particular to quantum mechanics; it's just basic math. If you have some number of events in some interval of time, you divide the two to get a rate. I don't understand what's difficult to grasp about that.

I can't say about Muller's hypothesis specifically, since I'm highly skeptical about the whole "time reversal" thing. But in GR generally, a power (or more precisely a power density, like watts per cubic meter) is a rate of change of a component of the stress-energy tensor. So it tells you how fast the stress-energy tensor (and hence the Einstein tensor, which describes spacetime curvature) is changing, yes.

Nothing difficult to grasp about that - in the GR case, if one requires a *continuous* power density (for Muller's case) - how does one reconcile that with the quantum nature of the system? In GR, things change continuously, in QM they don't - the rate of change of the stress energy tensor is a derivative in GR, but it's definitely not a smooth function in QM... so the instantaneous change that Muller seems to be suggesting is needed in the GR case, will never happen in the QM case.

There's got to be something about QM and QFT here that makes approximations such as Muller's (a continuum limit of a discrete process) valid, I just don't know what they are. @vanhees71 any thoughts?

 

[PeterDonis]

In GR, things change continuously, in QM they don't

So obviously one of them is just an approximation. Since we know the quantum field theory of radiation is more fundamental than the classical theory, obviously the GR analysis which assumes radiation is continuous is the approximation. As I already said back in post #11. Again, I don't see what's so hard to grasp about that.

the instantaneous change that Muller seems to be suggesting is needed in the GR case

I don't know why Muller's analysis would require continuity to be exact, any more than any other analysis in GR. As long as continuity is a good enough approximation for the case under discussion, everything works just fine.

There's got to be something about QM and QFT here that makes approximations such as Muller's (a continuum limit of a discrete process) valid

What normally makes a continuum approximation of a discrete process valid? Obviously, as I've already said, that you have a large enough number of events in a short enough time interval for averages and rates to be meaningful. Again, there's nothing particular about quantum physics here; the same thing would apply if you were trying to approximate coin flips or die rolls as a continuous process.

 

[asimov42]

So obviously one of them is just an approximation. Since we know the quantum field theory of radiation is more fundamental than the classical theory, obviously the GR analysis which assumes radiation is continuous is the approximation. As I already said back in post #11. Again, I don't see what's so hard to grasp about that.

I don't know why Muller's analysis would require continuity to be exact, any more than any other analysis in GR. As long as continuity is a good enough approximation for the case under discussion, everything works just fine.

What normally makes a continuum approximation of a discrete process valid? Obviously, as I've already said, that you have a large enough number of events in a short enough time interval for averages and rates to be meaningful. Again, there's nothing particular about quantum physics here; the same thing would apply if you were trying to approximate coin flips or die rolls as a continuous process.

Thanks for your patience @PeterDonis - I think my confusion with continuum approximations is the following, as as example (rephrased from above): You need 1,000,000 Watts output... so, if you were to measure the emission of a single 1 joule photon (just putting simple numbers in) over 1e-6 seconds, you'd end up with a power of 1,000,000 Watts (of course you would not end up with *1,000,000 joules*). But, the GR stress energy tensor in the region of, say, a neutron star or a black hole, would change completely negligibly with the release of this photon, correct?

So it seems that Muller's argument requires a specific amount of energy to be dissipated, to change the stress energy tensor significantly. This is where my confusion comes in (otherwise, if his theory was correct, we'd see these time reversal events all over the place).

 

[asimov42]

Or, alternatively, it seems like every single photon emission should satisfy Muller's criterion (finite energy emitted over near zero time), and so why do we need a black hole at all?

 

[PeterDonis]

if you were to measure the emission of a single 1 joule photon (just putting simple numbers in) over 1e-6 seconds, you'd end up with a power of 1,000,000 Watts (of course you would not end up with *1,000,000 joules*).

Not really, because you can't compute a useful average from one photon. Also, 1 Joule is an extremely large energy for a photon; try comparing it to, for example, the energy of the highest energy gamma ray ever detected. A more realistic scenario would be, say, 1 billion photons of average energy 1 nanojoule being emitted in an interval of a microsecond.

the GR stress energy tensor in the region of, say, a neutron star or a black hole, would change completely negligibly with the release of this photon, correct?

The units of the stress-energy tensor are energy density, not energy, so you would have to know from what volume the photon was being emitted. If it was being emitted by a neutron star, so the effective emission volume was some appreciable fraction of the neutron star's volume, since the mass equivalent of 1 Joule is going to be many orders of magnitude smaller than the neutron star's mass, then yes, the effect of emitting 1 1 Joule photon would be negligible.

it seems that Muller's argument requires a specific amount of energy to be dissipated, to change the stress energy tensor significantly

Again, I'm highly skeptical of his whole thesis, so I would not be surprised if there isn't really a consistent criterion in his paper.

 

[asimov42]

The units of the stress-energy tensor are energy density, not energy, so you would have to know from what volume the photon was being emitted. If it was being emitted by a neutron star, so the effective emission volume was some appreciable fraction of the neutron star's volume, since the mass equivalent of 1 Joule is going to be many orders of magnitude smaller than the neutron star's mass, then yes, the effect of emitting 1 1 Joule photon would be negligible.

@PeterDonis sorry to beat a dead horse. Again putting Muller aside, let's envision a situation: i require an outflow of energy of 10 kilowatts continuously to affect the metric tensor of GR in the manner i desire (whatever that may be). But quanta are release over very short intervals... so does the 1 nanojoule photon in 1e--12 seconds satisfy this requirement? Or do I need to wait? GR wants the metric tensor to change continuously, but it doesn't - so I don't understand how a rate change in the metric tensor can possibly be correlated with discrete events. Does one nanojoule/1e-12 seconds trigger the change I want, or not?

With changes in the metric tensor being continuous, but QM being discrete, it seems that i'd come up with completely different answers solely based on my sampling period, and that just seems wrong. Said differently - do I care about the total amount of energy dissipated (as Muller would suggest), or the rate of individual quantum events... if the latter, then it seem like these 'high power' events should happen all the time (just depends on the timescale over which you look)

 

[PeterDonis]

i require an outflow of energy of 10 kilowatts continuously to affect the metric tensor of GR in the manner i desire

Why? Remember the units: the stress-energy tensor has units of energy density. So changes in stress-energy are changes in energy density. So any requirement for a particular change in the stress-energy tensor must be a requirement for a particular change in stress-energy. It can't be a requirement for a particular power.

do I care about the total amount of energy dissipated

Why would you care about anything else, since the total energy change (or more precisely the total energy density change) is the change in the stress-energy tensor?

 

[asimov42]

Why? Remember the units: the stress-energy tensor has units of energy density. So changes in stress-energy are changes in energy density. So any requirement for a particular change in the stress-energy tensor must be a requirement for a particular change in stress-energy. It can't be a requirement for a particular power.

Muller's requirement is that there be a specific change in the stress-energy tensor in a specific time - but this does not jive with discrete quantum events. How is it any different to say that I require an energy dissipation of 1 million watts (in one second) for hundreds of billions of photons, versus 1 million watts in 1e-12 seconds for 10,000 photons. The ultimate change in the stress energy tensor is very different, but the rate is the same? So in the Muller case (or for any similar problem) what would lead to time reversal? Here, again, all we've done is change the period over which we average.

 

[PeterDonis]

Muller's requirement is that there be a specific change in the stress-energy tensor in a specific time - but this does not jive with discrete quantum events.

Sure it does; it just means you whatever events happen within that time interval need to involve a sufficient aggregate energy change.

How is it any different to say that I require an energy dissipation of 1 million watts (in one second) for hundreds of billions of photons, versus 1 million watts in 1e-12 seconds for 10,000 photons. The ultimate change in the stress energy tensor is very different, but the rate is the same?

Well, you just said that the requirement is that there be a specific change in the stress-energy tensor in a specific time. It's obvious that the two examples you give here give very different results for that. And it's straightforward to compare either one with a given specific requirement: for example, say we need a change of 1 million Joules in a time interval of 1 second. The first example obviously meets that requirement; the second obviously falls way short. So what's the problem?

in the Muller case (or for any similar problem) what would lead to time reversal?

I have no idea since, as I've already said a couple of times now, I'm skeptical of Muller's whole proposal. Please limit this discussion to the basic question of GR continuity vs. QM discreteness. (Discussion of Muller's proposal is really off topic for this forum anyway, it should go in Beyond the Standard Model if it is going to be discussed at all.)

 

[asimov42]

@PeterDonis - I've grasped all you've said above, and i'll give a go at articulating what's still bugging me (thick head below):

For the sake of argument (Muller's or any other) let's say I require a black hole to radiate away energy at the rate of 1 million watts (a rate)... in order to cause space to contract, i.e., the local metric-tensor to change, or for whatever other condition you choose to impose to occur - this is similar in many ways to the expansion rate of the universe, again, it's a rate). Now, 1 million watts is 1 joule per microsecond, yes?

Then, further say that, most of the time, we don't achieve this energy dissipation rate (and it is a rate). But, for some short period, we do manage to dissipate 1 joule within 1 microsecond. So, for that short period, although we haven't perhaps changed the metric tensor very much, we did meet the required rate of energy dissipation. So did we achieve our goal over that short period, or not? Is it a matter (no pun intended) of meeting a total quantity of energy dissipated over a longer time, or simply a rate of dissipation?

In Muller's paper (sorry to go back to it, but it only serves to illustrate the points I'm trying to make about continuous vs. discrete), spacetime expands at a certain continuous rate, and to counter that, you must dissipate energy at a given rate (drips and drabs or continuously, I'm not sure).

Since QM consists of discrete events, the only way I can make sense of all of the discussion above (i.e., sufficient QM events within a given time) interval is if the metric tensor changes smoothly and continuously as quanta are dissipated - and I think that it should, as energy is radiated away from a source. Without this, there seems to be no link between the different outputs above, and achieving 1 joule per microsecond now and again should be just as good as achieving 1 million Watts.

But I really don't think that's what Muller's paper is getting at by requiring a small black hole evaporating rapidly (as i said, there wouldn't be any need for a hole if you could dissipate little bits of energy here and there without ever getting close to the total output from evaporation).

I don't know if this makes anything more clear - my whole confusion is really how you connect the continuity of GR (and continuous rates of change) with the discrete world of QM.

If anyone else wants to chime in and set me straight, I'd be most grateful.

 

[PeterDonis]

So, for that short period, although we haven't perhaps changed the metric tensor very much, we did meet the required rate of energy dissipation. So did we achieve our goal over that short period, or not?

I have no idea, since at this point you're just making up a criterion and yet professing to not understand what it says. You can make the answer be anything you want by defining the criterion appropriately, and you're the one that made up the criterion, so why are you asking me what it is telling you?

my whole confusion is really how you connect the continuity of GR (and continuous rates of change) with the discrete world of QM.

And I've already repeatedly explained how you do that: continuity is an approximation. I don't know what more I can say. It looks to me like you are insisting on beating yourself over the head with something that's irrelevant.

It might be that the question you really want to ask is, if continuity is only an approximation, how can you apply a criterion that requires continuity to be exact? And the answer to that should be obvious: you can't. So either you're misunderstanding the criterion (maybe Muller didn't intend for his rate to be an exact continuous rate valid at any single instant), or the person who made up the criterion didn't think it through (maybe Muller didn't stop to consider that the criterion he was proposing can't actually be applied). But no amount of discussion here will address that issue; you'd have to ask Muller what he meant.

 

[asimov42]

@PeterDonis - thanks for the assistance. I've clearly annoyed you, so my apologies. I'm not doing a good job of explaining the issue I'm wondering about, which is my fault. The intent above was not to make up arbitrary criteria and then ask what they're telling me...

I'm sure Muller has at least a reasonable idea of what he's doing - he would not mention black hole evaporation and the corresponding energy dissipation for nothing.

Re-reading Muller's paper, and just thinking about GR, it seems that the key is that the metric tensor for a region of spacetime changes continuously, and it must (as mass/energy moves around) - this is what links the quantum nature of black hole evaporation with his 'rate of creation' of new space/time' in Muller's eyes... there has to be a tie in between the continuous and the discrete (until we get to quantum gravity perhaps). So, you're right - "either you're misunderstanding the criterion (maybe Muller didn't intend for his rate to be an exact continuous rate valid at any single instant)." This is exactly the case.

I'd welcome any further discussion of Muller's work, as I find it quite interesting - but I'll be sure to move any further questions over to Beyond the Standard Model.

 

[PeterDonis]

I've clearly annoyed you, so my apologies.

No need to apologize, I'm not annoyed. It just seemed like you were struggling to squeeze out the window when the door was wide open, so to speak.

(maybe Muller didn't intend for his rate to be an exact continuous rate valid at any single instant)." This is exactly the case.

That would be my interpretation of Muller's intent, yes.

 

[asimov42]

Thanks @PeterDonis and all! Just as a last, short question - am I correct in understanding that the metric tensor for a region of spacetime changes continuously as mass and energy move around? If so, then the tensor (curvature) in a region will continue to change as energy radiates away.

(Just making sure I understand the smooth manifold structure of spacetime).

 

[PeterDonis]

am I correct in understanding that the metric tensor for a region of spacetime changes continuously as mass and energy move around?

A better way of putting it is that the metric tensor can have different values at different events in spacetime. Remember that spacetime already includes time; it doesn't "change", it just is, but the "is" includes descriptions of everything that an observer following a particular worldline would observe to "change" with respect to his own proper time.