Relation between quantum fluctuations and vacuum energy?

[bhobba]

Is it purely due to the running of the coupling with the energy scale then? (I know a very small amount about the renormalization group, but not much)

Read my introduction to renormalisation:
https://www.physicsforums.com/insights/renormalisation-made-easy/

Its a very very brief, but as far as it goes, correct introduction.

Then read:
http://arxiv.org/pdf/hep-th/0212049.pdf

What you wrote is basically correct, but its a lot more sophisticated than that - its do do with renormalisation group flow. What fooled physicists for a long time was many of the constants that appear in QFT equations like charge, mass and (as you point out) the coupling constant depends on the cutoff.

But it can't be pushed too far because of what's called the Landau pole:
https://en.wikipedia.org/wiki/Landau_pole

In fact it's an indication that QED is sick, and indeed we know long before the Landau pole is reached another theory takes over - the electroweak theory. My understanding is that also has a Landau pole and exactly how that is handled my knowledge comes up short - maybe some of the more knowledgeable posters here can comment.

Thanks
Bill

 

[A. Neumaier]

whats called the Landau pole [...] In fact it's an indication that QED is sick,

The Landau pole is a perturbative phenomenon, and it is unknown whether it has any nonperturbative (i.e., physical) impact. It is speculation only (though widely thought to be credible) that this indicates a sickness of QED or electroweak theory. This is called the triviality problem - it is a completely open problem, even for $\phi^4$ theory! See here for my view of this problem.

 

[Frank Castle]

Read my introduction to renormalisation:
https://www.physicsforums.com/insights/renormalisation-made-easy/

Its a very very brief, but as far as it goes, correct introduction.

Then read:
http://arxiv.org/pdf/hep-th/0212049.pdf

What you wrote is basically correct, but its a lot more sophisticated than that - its do do with renormalisation group flow. What fooled physicists for a long time was many of the constants that appear in QFT equations like charge, mass and (as you point out) the coupling constant depends on the cutoff.

But it can't be pushed too far because of what's called the Landau pole:
https://en.wikipedia.org/wiki/Landau_pole

In fact it's an indication that QED is sick, and indeed we know long before the Landau pole is reached another theory takes over - the electroweak theory. My understanding is that also has a Ladau pole and exactly how that is handled my knowledge comes up short - maybe some of the more knowledgeable posters here can comment.

Thanks
Bill

Thanks for the links and info! So is the Landau pole of a theory the limit (as the energy scale increases) at which the coupling becomes infinite and thus the theory breaks down? If a theory doesn't have a Landau pole then as the energy scale increases the coupling becomes weaker and weaker, and in this limit the theory is said to be asymptotically free, and is thus stable at all energy scales (as I understand it QCD is such an example)?!

In terms of fluctuations of quantum fields, is any of what I wrote in post #27 correct? I have to admit I still have some confusion over it as there have been I few slightly conflicting posts (possibly some details lost in translation).

 

[bhobba]

//

Thanks for the links and info! So is the Landau pole of a theory the limit (as the energy scale increases) at which the coupling becomes infinite and thus the theory breaks down?

Well the usual perturbation methods break down - what that means, from my understanding, is still unclear.

In terms of fluctuations of quantum fields, is any of what I wrote in post #27 correct? I have to admit I still have some confusion over it as there have been I few slightly conflicting posts (possibly some details lost in translation).

Lets go back to basics. QM doesn't say what's going on when not observed so what does that imply about quantum fluctuations when not observed?

In QFT things are so far removed from usual experience all sorts of things are grasped at to have some of intuition, but its not really true.

The energy of a quantum field actually turns out to be infinite which is silly. Even sillier is, since the energy zero point is arbitrary, subtracting infinity from it and setting that to zero. The way its handled lies in something called normal ordering which you can look into.

It's important to understand that one can do QED without these infinities:
https://www.amazon.com/dp/0486492737/?tag=pfamazon01-20

It is pretty clear these days its an artifact of the mathematical methods used.

Thanks
Bill

 

[Frank Castle]

Lets go back to basics. QM doesn't say what's going on when not observed so what does that imply about quantum fluctuations when not observed?

So is the notion of a quantum fluctuation (in the ground state) deeply connected to the Heisenberg uncertainty principle, in that the values of the field, $\hat{\phi}(x)$ and its conjugate momentum, $\hat{\pi}(x)$ are not well-defined, and so we analogise this to our classical way of thinking in describing this as the quantum field fluctuating?!
I get that a quantum field fluctuates in the sense that obeys an equation of motion (some sort of wave equation) and hence quantised fluctuations propagating through the field correspond to quanta of that field - measurements of such quantisation being quantified by correlation functions of the field at different points. But what about in vacuum state of the quantum field? Is it simply that the field is in a fixed state, but that the uncertainties of its value (due to the uncertainty principle), we visualise this as the field itself fluctuating (so-called vacuum fluctuations of the field)?!

 

[bhobba]

So is the notion of a quantum fluctuation (in the ground state) deeply connected to the Heisenberg uncertainty principle, in that the values of the field, $\hat{\phi}(x)$ and its conjugate momentum, $\hat{\pi}(x)$ are not well-defined, and so we analogise this to our classical way of thinking in describing this as the quantum field fluctuating?!

A quantum field is an operator at a certain point in space time. Do position operators fluctuate, do momentum operators fluctuate? Are operators not well defined? Think about it. What fluctuates with quantum operators? Hint - its not called an observable for nothing.

Now it turns out if you chug through the math (do Fourier transforms and stuff) its exactly the same as one of the equivalent ways of doing QM:
http://www.colorado.edu/physics/phys5260/phys5260_sp16/lectureNotes/NineFormulations.pdf

See the second quastisation formulation. That's how particles enter into it but now the number of particles is not fixed.

I think you will benefit from studying the following book where a lot of these issues are examined in detail:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Thanks
Bill

 

[A. Neumaier]

Is it simply that the field is in a fixed state, but that the uncertainties of its value (due to the uncertainty principle), we visualise this as the field itself fluctuating (so-called vacuum fluctuations of the field)?

Yes. If the field is in the vacuum state, this is the complete, objective description of the field, and nothing at all goes on. Otherwise it is in a state with a (fixed or indefinite) particle content, only then something can happen. Again, the state describes everything that can objectively be said about the system.

Whatever is visualized in quantum field theory is added heuristics to aid human limitations in working with the formal content. Don't put too much into the visualization! It is just the way physicists familiarize themselves (and different physicists do it differently) with the formal machinery. To do anything in quantum field theory one needs to work with the latter, and the intuitive picture may be a help (in the earliest stages) or an obstacle (in an intermediate stage) for using this machinery. Ultimately the intuitive picture no longer matters - then one can say one has understood the formalism.

 

[bhobba]

Whatever is visualized in quantum field theory is added heuristics to aid human limitations in working with the formal content. Don't put too much into the visualization! It is just the way physicists familiarize themselves (and different physicists do it differently) with the formal machinery. To do anything in quantum field theory one needs to work with the latter, and the intuitive picture may be a help (in the earliest stages) or an obstacle (in an intermediate stage) for using this machinery. Ultimately the intuitive picture no longer matters - then one can say one has understood the formalism.

Thanks
Bill

 

[Frank Castle]

A quantum field is an operator at a certain point in space time. Do position operators fluctuate, do momentum operators fluctuate? Are operators not well defined? Think about it. What fluctuates with quantum operators? Hint - its not called an observable for nothing.

Now it turns out if you chug through the math (do Fourier transforms and stuff) its exactly the same as one of the equivalent ways of doing QM:
http://www.colorado.edu/physics/phys5260/phys5260_sp16/lectureNotes/NineFormulations.pdf

See the second quastisation formulation. That's how particles enter into it but now the number of particles is not fixed.

I think you will benefit from studying the following book where a lot of these issues are examined in detail:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Thanks
Bill

I get that the field operators themselves do not fluctuate, but what about their vacuum expectation values, won't they in general be of the form $\langle\hat{\phi}(x)\rangle =f(x)$?!

I've been reading through Quantum Field Theory for the gifted amateur and I haven't been able to find anything on quantum fluctuations yet (I'm up to page 140 so maybe I haven't gotten far enough into the text yet).
I get that in the context of QFT (where special relativity applies) that the vacuum energy contribution is physically meaningless since only energy differences are physically observable, hence we can legitimately apply a normal ordering procedure to obtain a renormalizable physical theory. At least I think that's right?!

My original confusion over the subject arose from reading a few texts on trying to solve the cosmological constant problem. Many of the author's note that vacuum energy is not necessarily ignorable in the realm of general relativity. If we assume that the gravity sector is classical, i.e. described by GR and the mate sector is quantum, i.e. described by QFT, then if the vacuum energy is physical it should be a source of curvature. The problem is, they attribute the source of this vacuum energy as being from the "quantum fluctuations in the matter fields" , which I'm assuming they mean fluctuations in their vacuum expectation values (since the source is from the vev of their energy-momentum tensor). This is primarily what has caused confusion for me.

 

[bhobba]

I get that the field operators themselves do not fluctuate, but what about their vacuum expectation values, won't they in general be of the form $\langle\hat{\phi}(x)\rangle =f(x)$?!

Expectations of what? What needs to be done in QM before you get an outcome? I basically told you the answer in a previous post. I can easily tell you, and you will likely kick yourself if I did, but its a point so basic, and so often ignored in these discussions you need to see if for yourself. Its one of those things Dr Neumaier mentioned as part of its heuristics - but it doesn't make the statement the vacuum is a broiling sea with constantly changing energy from the Heisenberg uncertainty principle as per the link below correct:
https://en.wikipedia.org/wiki/Quantum_fluctuation

It a common, no very common, heuristic but you should immediately see its problem - it goes right to the foundations of QM.

Its even used, incorrectly, to explain spontaneous emission:
http://www.calphysics.org/articles/Fain1982.pdf

It is wrong - the real answer is its because an electron is not an isolated system and is entangled with the vacuum so is not in a stationary state. But vacuum fluctuations is very commonly used as a heuristic such as in the above. It's still wrong and I am hoping you can see why it wrong. Then you can see why using it as a heuristic usually helps, but its crucial to understand its a heuristic - that it most definitely is NOT what is going on.

At least I think that's right?!

Basically - yes. That book explains normal ordering in detail.

My original confusion over the subject arose from reading a few texts on trying to solve the cosmological constant problem. Many of the author's note that vacuum energy is not necessarily ignorable in the realm of general relativity. If we assume that the gravity sector is classical, i.e. described by GR and the mate sector is quantum, i.e. described by QFT, then if the vacuum energy is physical it should be a source of curvature. The problem is, they attribute the source of this vacuum energy as being from the "quantum fluctuations in the matter fields" , which I'm assuming they mean fluctuations in their vacuum expectation values (since the source is from the vev of their energy-momentum tensor). This is primarily what has caused confusion for me.

That is not the source of energy in QFT. The ground state has infinite energy which is obviously an absurdity. A more careful look has it as zero.

Thanks
Bill

 

[vanhees71]

Although A. Neumair states otherwise, I still think that indeed there is no way to fix the absolute value of energy in special relativity. You have to define the Hamiltonian, and already for free fields you get a divergent result, but the trouble is due to the problem of multiplying field operators at the same space-time point, because the canonical commutators shows that they are distribution valued operators. For a field component $\hat{\phi}$ and its canonical conjugate momentum $\hat{\Pi}$ you have the canonical equal-time commutator
$$[\hat{\phi}(t,\vec{x}),\hat{\Pi}(t,\vec{y})]=\mathrm{i} \delta^{(3)}(\vec{x}-\vec{y}).$$
Using the mode decomposition of free fields in terms of annihilation and creation operators show, however, that you can introduce the norma-ordering procedure to define a finite Hamiltonian up to an additive c-number constant, but this additive constant is irrelevant for the dynamics and thus can be neglected. Then you define the absolute value of the energy of the vacuum state, which is the ground state of the free-field system, to 0.

 

[A. Neumaier]

I still think that indeed there is no way to fix the absolute value of energy in special relativity. You have to define the Hamiltonian, and already for free fields you get a divergent result

The infinities and the resulting ambiguity appear only in a naive Lagrangian approach to field theory, where already the Lagrangian is ill-defined, i.e., mathematically meaningless.

In axiomatic quantum field theory, which is consistent with all our knowledge about quantum fields (though interacting models in 4D haven't yet been constructed rigorously), a field theory is defined by a family of vacuum N-point functions satisfying certain axioms, among them translation invariance. Given these N-point functions there is a canonical way to construct a Hilbert space with a representation of the Poincare group, satisfying the traditional commutation relations. It is these commutation relations that do not allow one to shift the Hamiltonian, defined as the generator of the time translations.

The vacuum is defined as a translation invariant state (not only up to phase), hence the Hamiltonian has the eigenvalue zero in this state. For a free field, the Hamiltonian turns out uniquely to be $H=\int Dp a^*(p)pa(p)$, where $Dp$ is the invariant measure integrating over momenta on the appropriate mass shell. This is the unique manner of proceeding in a manifestly covariant way; there is no freedom except for choosing mass and spin. Nowhere is an infinity or a vacuum energy encountered, no field operators are multiplied at the same position. The canonical commutation relations follow from the representation theory of the Poincare group and the connection between spin and statistics. The details are in Weinberg's QFT book, surely a standard work. Causal perturbation theory extends this perturbatively to the interacting case. Again, nowhere is an infinity or a vacuum energy encountered, no field operators are multiplied at the same position.

If you introduce a nontrivial zero-point energy, Lorentz covariance implies in another frame nontrivial zero-point momenta. But there is no covariant way to do this consistently., quite apart from the question what such a zero-point momentum should mean. The vacuum doesn't move! This shows that in a relativistic context the zero-point energy is a spurious concept.

 

[Frank Castle]

Expectations of what? What needs to be done in QM before you get an outcome? I basically told you the answer in a previous post. I can easily tell you, and you will likely kick yourself if I did, but its a point so basic, and so often ignored in these discussions you need to see if for yourself. Its one of those things Dr Neumaier mentioned as part of its heuristics - but it doesn't make the statement the vacuum is a broiling sea with constantly changing energy from the Heisenberg uncertainty principle as per the link below correct:
https://en.wikipedia.org/wiki/Quantum_fluctuation

Is the point that you have to interact with the system in order to make a measurement of it therefore entangling the state of the system you are measuring with whatever measurement "probe" you are using?

t a common, no very common, heuristic but you should immediately see its problem - it goes right to the foundations of QM.

Its even used, incorrectly, to explain spontaneous emission:
http://www.calphysics.org/articles/Fain1982.pdf

It is wrong - the real answer is its because an electron is not an isolated system and is entangled with the vacuum so is not in a stationary state. But vacuum fluctuations is very commonly used as a heuristic such as in the above. It's still wrong and I am hoping you can see why it wrong. Then you can see why using it as a heuristic usually helps, but its crucial to understand its a heuristic - that it most definitely is NOT what is going on.

I wish lecturers would be more upfront about this - at least in my experience, not nearly enough emphasis is made that this notion of quantum fluctuations is purely a heuristic aid to understand calculations.

Is it simply that higher-order corrections in perturbation theory, to correlation functions such as $\langle 0\lvert\hat{\phi}(x)\hat{\phi}(y)\rvert 0\rangle$, are referred to as "quantum fluctuations" since they are represented by loops in the corresponding Feynman diagrams and so are heuristically thought of as virtual particles being created and annihilated in vacuum?

 

[A. Neumaier]

Is it simply that higher-order corrections in perturbation theory, to correlation functions such as $\langle 0\lvert\hat{\phi}(x)\hat{\phi}(y)\rvert 0\rangle$, are referred to as "quantum fluctuations" since they are represented by loops in the corresponding Feynman diagrams and so are heuristically thought of as virtual particles being created and annihilated in vacuum?

Conventional Feynman diagrams represent time-ordered expectation values, not Wightman correlation functions. (The latter need the CTP formalism.) The term fluctuations derives from the informal generalization of the fact that fluctuating time series also have correlation functions. Identifying these fluctuations with virtual particles popping in and out of existence for a short time is additional but unrelated imagery for these fluctuations, as described here.

If you really want to understand quantum physics, concentrate on the formulas, and view the talk about it only as a very loose and fallible guide.

 

[bhobba]

Is the point that you have to interact with the system in order to make a measurement of it therefore entangling the state of the system you are measuring with whatever measurement "probe" you are using?

Bingo.

QM is a theory about observations. Whats going on when not observed it says nothing about. The uncertainty relations only applies to observations. Applying it to the vacuum ground state means if you observe it then the results you get follows from the uncertainty relations - but only if you observe it. If you don't it says a big fat nothing. The vacuum state isn't in a constant fluctuations etc etc.

What going on and why its a valuable heuristic to look at it that way is from the Von Neumann regress where collapse can be basically placed anywhere so if you think of it at the vacuum then generally no harm is done and it a good way of getting an intuitive grip. But that is NOT what is happening.

I wish lecturers would be more upfront about this - at least in my experience, not nearly enough emphasis is made that this notion of quantum fluctuations is purely a heuristic aid to understand calculations.

Don't worry about it. It happens in physics all the time. You start out with stuff of dubious validity that gets corrected later, either explicitly or you are supposed to cotton onto it yourself. But what you have done is developed intuition which actually is more important. The only issue is not realizing what's going on and even then it generally causes issues only when discussing foundational issues rather than actually solving problems. Guess what most discuss here .

The other group it causes issues with is those that actually think - so give yourself a pat on the back. I first came across this when studying how transistors work. They work via holes which was explained to be the absence of electrons. It was a big fact crock of the proverbial. It annoyed the bejesus out of me - it was so obviously wrong. Then as a footnote in a textbook I read the alert reader will recognize it as nonsense but rest assured a QM analysis shows quasi particles exist that act like that.

Thanks
Bill

 

[Frank Castle]

Conventional Feynman diagrams represent time-ordered expectation values, not Wightman correlation functions. (The latter need the CTP formalism.)

Sorry, I meant it to be a time-ordered correlator, but just got lazy when I was typing and didn't include the time-ordering symbol.

If you really want to understand quantum physics, concentrate on the formulas, and view the talk about it only as a very loose and fallible guide.

I try to do that, but then I feel like I'm following the math blindly without having any physical intuition and this in itself feels uncomfortable to me.

Don't worry about it. It happens in physics all the time. You start out with stuff of dubious validity that gets corrected later, either explicitly or you are supposed to cotton onto it yourself. But what you have done is developed intuition which actually is more important. The only issue is not realizing what's going on and even then it generally causes issues only when discussing foundational issues rather than actually solving problems. Guess what most discuss here .

Thank you for being patient with me, I really appreciate it!

I think a big part of my problem is, when I first was taught QFT I didn't give these notions much thought as I was already completely bogged down with trying to get a grip with all the formalism, and it wasn't really expounded upon by my lecturer. Now that I've had a little more experience with QFT I'm questioning all these aspects a lot more, and I am in a constant state of worry and confusion that I don't really understand at all what is going on

 

[bhobba]

Now that I've had a little more experience with QFT I'm questioning all these aspects a lot more, and I am in a constant state of worry and confusion that I don't really understand at all what is going on

Then you have reached the second later phase I talked about

Thanks
Bill

 

[A. Neumaier]

I meant it to be a time-ordered correlator, but just got lazy when I was typing and didn't include the time-ordering symbol.

is this your way to excuse yourself for having talked nonsense? You repeatedly wrote what you later claimed you didn't mean - this makes it very difficult to communicate. It is much better to check if what you wrote is what you meant before sending things off.

As for the talk accompanying the formulas, they don't mean much, and if one can't make sense of them it is often because they don't make sense - except by very superficial analogy. The only thing to make sense of is the formulas, and how they relate to (possibly) experimentally accessible information. So you have to look for the latter in the words.

 

[Frank Castle]

is this your way to excuse yourself for having talked nonsense?

No, definitely not. I put down what I'm thinking at the time and I'm fully aware that sometimes (hopefully not always) it can be nonsense, but it's just me trying to make sense of things. Often I think a bit more about it afterwards and take into account what you and others have written in response and then try and formulate my updated thoughts. I apologise that I'm sometimes incoherent, I will try to improve my posts in future.

Then you have reached the second later phase I talked about

I guess at least it's reassuring that I'm not the only one that has struggled/is struggling with these things.

 

[aries0179]

Sorry, I meant it to be a time-ordered correlator, but just got lazy when I was typing and didn't include the time-ordering symbol.
I try to do that, but then I feel like I'm following the math blindly without having any physical intuition and this in itself feels uncomfortable to me.
Thank you for being patient with me, I really appreciate it!

I think a big part of my problem is, when I first was taught QFT I didn't give these notions much thought as I was already completely bogged down with trying to get a grip with all the formalism, and it wasn't really expounded upon by my lecturer. Now that I've had a little more experience with QFT I'm questioning all these aspects a lot more, and I am in a constant state of worry and confusion that I don't really understand at all what is going on

look, its kinda a major undertaking to even be in a quantum class, and i know cause i went through 12 years of study and working towards my own answers that i wasnt satisfied with in a text or class. the problems begin when all the calculations and measures take over your ability to process the scenario and be confident in your own knowledge and experiences. the qualified academics in any form of quantum sciences generally do not have the true qualifications to lead upcoming brains in the extremities of for example quantum mechanics and quantum theory, those happen to be the interests that i pursued based on the entire quantum spectrum of sub choices to focus my own personal talents and intellect towards those portions of a greater entity then i think its possible for anyone person to have even a good grasp upon... its too much information and a lot of the various scientists are teaching or saying the same things, they just rather have the exclusive rights on the ideas and formulas but they dont. so do not get down from the lack of understanding that someone else may have, and i will say this with high certainty, the knowledge that i have achieved has been mainly from my own ideas and knowledge based on exploring the accepted science and being able to use my brain to get beyond what is accepted by the masses. one more thing it does make a difference who you listen and learn from and the only way to get the best and most respected minds to use as a guide is to do some background on a few people like Michio Kaku, or university ofberkeley professor alex filippenko. from my own experience it was so liberating and thoughtful for me when I've gone to a weekend of lectures to be able to get first hand guidance and help from the people that i respect and look at as mentors, sorry i kinda am lengthy with my answers i guess that may come with the territory, but i wanted to give you several aspects of helpful ideas to think over and go with your comfort level.

 

[Frank Castle]

look, its kinda a major undertaking to even be in a quantum class, and i know cause i went through 12 years of study and working towards my own answers that i wasnt satisfied with in a text or class. the problems begin when all the calculations and measures take over your ability to process the scenario and be confident in your own knowledge and experiences. the qualified academics in any form of quantum sciences generally do not have the true qualifications to lead upcoming brains in the extremities of for example quantum mechanics and quantum theory, those happen to be the interests that i pursued based on the entire quantum spectrum of sub choices to focus my own personal talents and intellect towards those portions of a greater entity then i think its possible for anyone person to have even a good grasp upon... its too much information and a lot of the various scientists are teaching or saying the same things, they just rather have the exclusive rights on the ideas and formulas but they dont. so do not get down from the lack of understanding that someone else may have, and i will say this with high certainty, the knowledge that i have achieved has been mainly from my own ideas and knowledge based on exploring the accepted science and being able to use my brain to get beyond what is accepted by the masses. one more thing it does make a difference who you listen and learn from and the only way to get the best and most respected minds to use as a guide is to do some background on a few people like Michio Kaku, or university ofberkeley professor alex filippenko. from my own experience it was so liberating and thoughtful for me when I've gone to a weekend of lectures to be able to get first hand guidance and help from the people that i respect and look at as mentors, sorry i kinda am lengthy with my answers i guess that may come with the territory, but i wanted to give you several aspects of helpful ideas to think over and go with your comfort level.

Thanks, I appreciate your comments.

Are there any good notes by Michio Kaku, or Alex Filppenko (or others) on the subject that you would recommend reading?

 

[Frank Castle]

Then you have reached the second later phase I talked about

Thanks
Bill

In a set of notes I've read in the last couple of days the author refers to quantum fluctuations of a quantum field in terms of the variance of its Fourier modes $\delta\phi( t, \mathbf{k}) =\sqrt{\langle\phi^{2}\rangle}$ about its expectation value $\langle\phi\rangle =0$. I'm not saying I place any trust in this description, but I was just wondering what your thoughts are on it?

 

[aries0179]

from the looks of it it seems incomplete or maybe to not fully cover the full amount of required information to get a correct field variance, i only say that because this may be a calculation based on a set of info that in those settings the calculation always reads as such. but i have to say i can't find any real issue with the values in relation to the sum, i know that may not help but i would need to do a little digging to verify for sure. for instance i need to know where the equation came from who ran the numbers and how they came to the solution and base that on the values accounted for. make sense?

 

[Frank Castle]

from the looks of it it seems incomplete or maybe to not fully cover the full amount of required information to get a correct field variance, i only say that because this may be a calculation based on a set of info that in those settings the calculation always reads as such. but i have to say i can't find any real issue with the values in relation to the sum, i know that may not help but i would need to do a little digging to verify for sure. for instance i need to know where the equation came from who ran the numbers and how they came to the solution and base that on the values accounted for. make sense?

I think their reasoning is that although the expectation value (average) of the field will be zero, the variance in general won't be and this quantifies how the value of the field fluctuates around the expectation value (but I'm aware this may be nonsense).

 

[A. Neumaier]

lthough the expectation value (average) of the field will be zero, the variance in general won't be

The variance is infinite hence meaningless, as it also is for olassical stochastic processes. The fluctuations are descibed by the correlations (covariance functions) or their Fourier transform, which give an equivalent description.

 

[bhobba]

In a set of notes I've read in the last couple of days the author refers to quantum fluctuations of a quantum field in terms of the variance of its Fourier modes $\delta\phi( t, \mathbf{k}) =\sqrt{\langle\phi^{2}\rangle}$ about its expectation value $\langle\phi\rangle =0$. I'm not saying I place any trust in this description, but I was just wondering what your thoughts are on it?

How can an operator vary? Obviously they are in some way referring to the variance of an observation of some sort.

Thanks
Bill

 

[A. Neumaier]

How can an operator vary?

In general, an operator is the noncommutative version of a random variable. (Probability theory is equivalent to the theory of operators acting by multiplication on a function space. The multiplication operators all commute.)

 

[Brage]

I think it is not quite right to say that vacuum fluctuations are not physically real (which is what it seems to me you are saying), as a lot of effects are very precisely described by them. Take for instance Hawking radiation and the lambda shift (which was not predicted by the Dirac equation). I believe I saw you (A. Neumaier) comment earlier on how (and I paraphrase) "Hawking radiation is produced from the gravitational field" (or so I gathered), and therefore not due to virtual particles. Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

 

[bhobba]

the lambda shift

Have you heard of Lattice Gauge Theory:
https://en.wikipedia.org/wiki/Lattice_gauge_theory

That predicts the Lamb shift and has no vacuum fluctuations or virtual particles. Nor is it the explanation for Hawking radiation as has been discussed many times on this forum.

All such are is intuitive cruxes for the math to aid intuition. That is not to belittle their importance - developing intuition is vital They get repeated in populist press and we get people who, for reasons that perplex me, won't accept what those that have studied the theory in detail tell them - namely they are fictions.

This isn't the only area it happens - it occurs all the time in advanced areas. I mentioned previously about the 'hole' issue in how transistors work. You will find electronics textbook after textbook that promulgates the same load of rubbish. But one must start somewhere. Feynman commented on it. He would prefer to tell the truth, the whole truth, and nothing but the truth, from the start. But being the great educator he was and having thought hard about it realized it would be very unwise doing that. So as you proceed from beginner to advanced you need to unlearn things. Its nothing to really worry about except when those at the beginner level do not believe what those at the more advanced level tell them. That is unbelievably frustrating as I know only too well.

Its one of the many myths of QM:
http://arxiv.org/abs/quant-ph/0609163

I have posted the above many times. It should really be the end of the matter but it isn't - people have so many ingrained ideas that are hard to shift.

Thanks
Bill

 

[A. Neumaier]

"Hawking radiation is produced from the gravitational field" (or so I gathered), and therefore not due to virtual particles. Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

If you insist on waiting for a consistent quantum theory of gravity one cannot discuss Hawking radiation at all - any statement about it would be ill-founded according to this overly stringent criterion!

The whole concept of Hawking radiation depends (at present) on semiclassical reasoning. And in semiclassical reasoning, Hawking radiation is produced from the gravitational field.

 

[bhobba]

Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

Validated by experiment probably has some validity (although I would like an experimental type to comment) but its a, admittedly very common, misconception we don't have a quantum theory of gravity.

We indeed have one, and it elegantly explains Hawking radiation:
http://blogs.umass.edu/grqft/effective-field-theory/

What we don't have is one valid to beyond about the Plank scale - but gravity is not alone in that - the standard model is not trusted in that region either.

As I am won't to say it doesn't make developing a full quantum theory of gravity any easier, but is an interesting take that keeps what really is going on in perspective. Its not quite the mess some sources make it out to be.

Thanks
Bill

 

[vanhees71]

If you insist on waiting for a consistent quantum theory of gravity one cannot discuss Hawking radiation at all - any statement about it would be ill-founded according to this overly stringent criterion!

The whole concept of Hawking radiation depends (at present) on semiclassical reasoning. And in semiclassical reasoning, Hawking radiation is produced from the gravitational field.

Well, what's even more interesting is, whether there is any empirical evidence for Hawking radiation. I'm not aware of any...

 

[A. Neumaier]

Well, what's even more interesting is, whether there is any empirical evidence for Hawking radiation. I'm not aware of any...

We can be lucky, it would require a very big black hole close to us...

 

[vanhees71]

Well, whether this is a lucky situation, is another question ;-)).

 

[George Jones]

We can be lucky, it would require a very big black hole close to us...

Big?