Infinite Charged Electron in QFT

[rogerl]

I'm reading this book "Deep Down Things" by Bruce A. Schumm about Quantum Field Theory. It says that the charge of the bare electron is infinite. Since the virtual particles calculations produced infinities too. Renormalization means substracting the infinite bare electron by infinite virtual particle calculations to come up with the small values of the charge. Do you actually believe this is true, that is, can the bare electron be really infinite in charge? How can this be?? QFT actually says this. But how can a bare particle be infinite in charge? Is this figurative or literal?

 

[weejee]

Afaik, the justification goes like this:

Although the expression for the perturbative correction is formally infinite, it diverges only logarithmically (~$$\alpha \ln \frac{\Lambda}{m_{e}}$$), where $$\Lambda$$ is the cutoff (some energy scale over which the theory isn't valid), and we know that the cutoff shouldn't be too large since QED is only the low-energy limit of the electroweak theory. Since $$\alpha$$ is a small number, the perturbative correction is still small in practice.

 

[rogerl]

Afaik, the justification goes like this:

Although the expression for the perturbative correction is formally infinite, it diverges only logarithmically (~$$\alpha \ln \frac{\Lambda}{m_{e}}$$), where $$\Lambda$$ is the cutoff (some energy scale over which the theory isn't valid), and we know that the cutoff shouldn't be too large since QED is only the low-energy limit of the electroweak theory. Since $$\alpha$$ is a small number, the perturbative correction is still small in practice.

What? Pls. translate it into language we newbies can understand. Or rather, pls. translate it to the actions of the virtual particles in the context mentioned by Bruce:

"When the virtual photon fluctuates into the virtual electron-positron pair, its energy must be shared, part going to the electron and part to the positron. But there are a large number - in fact, an infinite number - of different ways the energy can divide itself between the positron and electron."

<snip>

"In fact, because there's an infinite number of ways for the process to take place, the calculated interaction probability is infinite"


Pls. explain what you mean by "the perturbative correction is still small in practice" in the language of virtual particles such as mentioned above. Are you saying that the energy of the photon limits the energy of the virtual electron-positron pair such that it never goes up a certain value? If not. What do you mean as far as virtual particle and their fluctuations is concerned?

 

[A. Neumaier]

I'm reading this book "Deep Down Things" by Bruce A. Schumm about Quantum Field Theory. It says that the charge of the bare electron is infinite. Since the virtual particles calculations produced infinities too. Renormalization means substracting the infinite bare electron by infinite virtual particle calculations to come up with the small values of the charge. Do you actually believe this is true, that is, can the bare electron be really infinite in charge? How can this be?? QFT actually says this. But how can a bare particle be infinite in charge? Is this figurative or literal?

Everything labelled ''bare'' and ''virtual'', or ''infinite'' is purely figurative, and has no physical contents. it just tells something about formula manipulation in pseudo-intuitive terms. The real things are labelled ''dressed'' or ''renormalized''.

In a more precise language: The infinities (bare charges, integrals, etc.) are expressions that are huge when the integrals in the theory are cutoff at some large energy scale Lambda, and would diverge if one were to take the limit Lambda --> inf. But the limit is taken only after all calculations have been done, and then renormalized, finite results appear when everything was done right.

This is no magic, but something analogous to having two infinities u=x/(1-x) and v=x^2/(1-x) expressed in terms of a paramater x=1+O(Lambda^{-1}). As you remove the cutoff (Lambda --> inf), u and v become infinite, but the difference u-v of the two infinities can be rewritten as u-v=x/(1-x)-x^2/(1-x) =(x-x^2)/(1-x)=x, which has the respectable limit 1.

 

[weejee]

Enities (bare charges, integrals, etc.) are expressions that are huge when the integrals in the theory are cutoff at some large energy scale Lambda, and would diverge if one were to take the limit Lambda --> inf. But the limit is taken only after all calculations have been done, and then renormalized, finite results appear when everything was done right.

I have been confused about this. For example, when we talk about mass/wavefunction renormalization, we use the following relation to extract the "self energy" from the perturbation expansion of the propagator.

$$\frac{1}{X-Y} = \frac{1}{X} + \frac{1}{X}Y\frac{1}{X} + \frac{1}{X}Y\frac{1}{X}Y\frac{1}{X} + \cdots$$

The above relation only holds when $$|X|>|Y|$$. Shouldn't this mean that $$Y$$ can't really be an infinity?

How I understood this is that in QED, since "$$Y$$" is only logarithmically divergent and $$\alpha$$ is multiplied to the log, we can make it small enough as long as the cutoff isn't ridiculously big (QED breaks down at some reasonable energy scale after all). For $$\phi^{4}$$, for the same reason, we are required to have a bare mass of the order of the cutoff.

Am I missing something here? Maybe if we understand renormalization like this, no QFT can be UV complete?

 

[rogerl]

Everything labelled ''bare'' and ''virtual'', or ''infinite'' is purely figurative, and has no physical contents. it just tells something about formula manipulation in pseudo-intuitive terms. The real things are labelled ''dressed'' or ''renormalized''.
QUOTE]

Do all mainstream physicists believe that the "bare", "virtual", "infinite" is purely figurative? Or are they divided on this just like some physicists believe the wave function is real (as in Bohmian Mechanics), while the other believe they are just mathematical tool (Copenhagen)? I think you are the latter. Do you also believe virtual particles don't have literal existence but just mathematical stuff? But then ever heard of the SLAC experiment where virtual particles are elevated into actual particles? The following is the details:

http://www.slac.stanford.edu/exp/e144/science1202.html

"Melissinos views the result as the first direct demonstration of "sparking the vacuum," a long-predicted phenomenon. In it, the energy of a very strong electromagnetic field promotes some of the fleeting, "virtual" particles that inhabit the vacuum, according to QED, to become pairs of real particles."

So you see. Virtual particles are not just mathematical entities. They can be real too. And if they are real, and virtual particles dressed the bare particle. Then "bare", "virtual", "infinite" are real entities! Is it only you who believe they are just figurative due to your own unique quantum interpretation?

 

[A. Neumaier]

$$\frac{1}{X-Y} = \frac{1}{X} + \frac{1}{X}Y\frac{1}{X} + \frac{1}{X}Y\frac{1}{X}Y\frac{1}{X} + \cdots$$
The above relation only holds when $$|X|>|Y|$$. Shouldn't this mean that $$Y$$ can't really be an infinity?

Strictly speaking, yes. But the formula can also be viewed as an identity in the field of all Laurent series. Then it is true for indeterminates x, y. This allows you to do lots of formal manipulations safely. Then, upon finding a relation that can be finitely evaluated, one pretends (and in some cases can prove) that if everything would be analytical in the stuff you expand then any formally derived formula between finite quantities would translate into a true formula. This bears out correctly in many cases, and physicists get a feel for when they can trust this sort of argument - with later correction from experiments of peers, if necessary.

 

[weejee]

Strictly speaking, yes. But the formula can also be viewed as an identity in the field of all Laurent series. Then it is true for indeterminates x, y. This allows you to do lots of formal manipulations safely. Then, upon finding a relation that can be finitely evaluated, one pretends (and in some cases can prove) that if everything would be analytical in the stuff you expand then any formally derived formula between finite quantities would translate into a true formula. This bears out correctly in many cases, and physicists get a feel for when they can trust this sort of argument - with later correction from experiments of peers, if necessary.

Could you elaborate a little bit? For example, Laurent expansion of which function around which point? Shouldn't we still have some (usually finite) region of convergence? Or are you saying that we don't need to associate X and Y with definite numbers and consider the formula as some formal identity, and shouldn't care too much about the convergence at intermediate steps? If so, it sounds quite confusing to me.

Furthermore, I'd really appreciate if you can tell me how you think about the way I understand the problem of infinity (or point out something wrong in the argument.)

Thank you.p.s. I think that at least the mass renormalization of phi^4 and the charge renormalization of QED can be understood in my way, even in the limit where the bare quantities and perturbative corrections tend to infinity. Although |Y| is an infinity (or a very big number), we can make |X| even bigger such that X-Y becomes some finite number. Here, |Y/X| is still smaller than 1 and the series converges).

 

[weejee]

What? Pls. translate it into language we newbies can understand. Or rather, pls. translate it to the actions of the virtual particles in the context mentioned by Bruce:

"When the virtual photon fluctuates into the virtual electron-positron pair, its energy must be shared, part going to the electron and part to the positron. But there are a large number - in fact, an infinite number - of different ways the energy can divide itself between the positron and electron."

<snip>

"In fact, because there's an infinite number of ways for the process to take place, the calculated interaction probability is infinite"Pls. explain what you mean by "the perturbative correction is still small in practice" in the language of virtual particles such as mentioned above. Are you saying that the energy of the photon limits the energy of the virtual electron-positron pair such that it never goes up a certain value? If not. What do you mean as far as virtual particle and their fluctuations is concerned?

I actually think that Schumm's line of argument can be quite misleading. Still, since I'm not sure how to explain it any better in terms of ordinary language, I'll just add things to his explanation.

1. As Schumm said, the frequency and also the momentum of the virtual photon divide themselves into a virtual pair, but it isn't the whole story. Actually, it is unfavorable for the virtual pair to have momentums or frequencies that are too big compared to the rest energy of the electron. Then, we have two competing effects, and the net result is dependent on the details(e.g. dimensionality of the space-time). Sometimes it can be even finite. For the case of QED in four space-time dimensions, it is still divergent, but the divergence is not very severe. What do I mean by saying that the divergence is severe or not? I'll explain it in the following paragraph.

2. Since we can't directly deal with infinity, we adopt something called 'cutoff' when we calculate this "interaction probability" (Actually it isn't much of a probability, let's suppose it is just some measure of the strength of the interaction). Roughly speaking, the cutoff is the maximal allowed energy of a virtual electron or a virtual positron. Then, the result is given as a function of the cutoff and we look at how it behaves as the cutoff tends to infinity. For QED in 4-dim. space-time, the result goes like alpha*log(cutoff/M), where 'alpha' is a small number called the fine-structure constant(~1/137) and 'M' is roughly the electron mass. The log function increases extremely slowly as we raise the cutoff, although it eventually becomes infinity.

3. One thing we know is that QED is only a low energy limit of a bigger theory, which is called the electroweak theory. So, all descriptions based on electron-positron pairs and photons become invalid at some high energy scale (electroweak scale), and we can consider this energy scale as some sort of 'cutoff' for QED. This energy scale is high, but not quite enormous such that it makes log(cutoff/M) a big number.

Hence, we can say that the "interaction probability", which goes like alpha*log(cutoff/M), is not too big, although it diverges with an arbitrarily high cutoff.

I hope I didn't confuse you further.

 

[rogerl]

I actually think that Schumm's line of argument can be quite misleading. Still, since I'm not sure how to explain it any better in terms of ordinary language, I'll just add things to his explanation.

1. As Schumm said, the frequency and also the momentum of the virtual photon divide themselves into a virtual pair, but it isn't the whole story. Actually, it is unfavorable for the virtual pair to have momentums or frequencies that are too big compared to the rest energy of the electron. Then, we have two competing effects, and the net result is dependent on the details(e.g. dimensionality of the space-time). Sometimes it can be even finite. For the case of QED in four space-time dimensions, it is still divergent, but the divergence is not very severe. What do I mean by saying that the divergence is severe or not? I'll explain it in the following paragraph.

2. Since we can't directly deal with infinity, we adopt something called 'cutoff' when we calculate this "interaction probability" (Actually it isn't much of a probability, let's suppose it is just some measure of the strength of the interaction). Roughly speaking, the cutoff is the maximal allowed energy of a virtual electron or a virtual positron. Then, the result is given as a function of the cutoff and we look at how it behaves as the cutoff tends to infinity. For QED in 4-dim. space-time, the result goes like alpha*log(cutoff/M), where 'alpha' is a small number called the fine-structure constant(~1/137) and 'M' is roughly the electron mass. The log function increases extremely slowly as we raise the cutoff, although it eventually becomes infinity.

3. One thing we know is that QED is only a low energy limit of a bigger theory, which is called the electroweak theory. So, all descriptions based on electron-positron pairs and photons become invalid at some high energy scale (electroweak scale), and we can consider this energy scale as some sort of 'cutoff' for QED. This energy scale is high, but not quite enormous such that it makes log(cutoff/M) a big number.

Hence, we can say that the "interaction probability", which goes like alpha*log(cutoff/M), is not too big, although it diverges with an arbitrarily high cutoff.

I hope I didn't confuse you further.

Do you believe virtual particles are real or just imagined entities? how about those descriptions about them being "bare", "dressed", "infinite".. Are they real processes going on?


Do all mainstream physicists believe that the "bare", "virtual", "infinite" is purely figurative? Or are they divided on this just like some physicists believe the wave function is real (as in Bohmian Mechanics), while the other believe they are just mathematical tool (Copenhagen)? I think you are the latter. Do you also believe virtual particles don't have literal existence but just mathematical stuff? But then ever heard of the SLAC experiment where virtual particles are elevated into actual particles? The following is the details:

http://www.slac.stanford.edu/exp/e144/science1202.html

"Melissinos views the result as the first direct demonstration of "sparking the vacuum," a long-predicted phenomenon. In it, the energy of a very strong electromagnetic field promotes some of the fleeting, "virtual" particles that inhabit the vacuum, according to QED, to become pairs of real particles."

So you see. Virtual particles are not just mathematical entities. They can be real too. And if they are real, and virtual particles dressed the bare particle. Then "bare", "virtual", "infinite" are real entities! Is it only you who believe they are just figurative due to your own unique quantum interpretation?

 

[A. Neumaier]

Virtual particles are not just mathematical entities. They can be real too.

You misunderstand the situation. A ''virtual particle becoming real'' is not a process in time - it is impossible to describe this as a time-dependent process in terms of quantum mechanics! The virtual particles there are still only metaphors for multivariate integrals, whereas the real particles are the observable things - having an associated wave function and a computable probability of observation.

 

[rogerl]

You misunderstand the situation. A ''virtual particle becoming real'' is not a process in time - it is impossible to describe this as a time-dependent process in terms of quantum mechanics! The virtual particles there are still only metaphors for multivariate integrals, whereas the real particles are the observable things - having an associated wave function and a computable probability of observation.

You attribute "real" as those that can be described in a time-dependent process in terms of QM as you stated. But what if the time processes of virtual particles as dictated by the Heidenberg Uncertainty Principle even though it has no real time processes is also physical. That is. A timeless process doesn't occur in time is also physical?

Remember that in Casimir plates, virtual particles can affect physical matter. Therefore rather than saying they are just mathematical fiction.. why not say they are real processes that occur in special timeless mode that is also physical? Unless you only attribute physical to something that has time dependent process? And attribute timeless process as only metaphors for multivariate integrals as you stated? But since the math has real world consequences. Then we can say that multivariate integrals occur physically but in a pseudo-time that is outside our Time definition but still in a physical timeless world. Well?

 

[A. Neumaier]

You attribute "real" as those that can be described in a time-dependent process in terms of QM as you stated. But what if the time processes of virtual particles as dictated by the Heidenberg Uncertainty Principle even though it has no real time processes is also physical.

There are no ''time processes of virtual particles''. Nobody ever has written down an equation for the time evolution of virtual particles. While a case can be made that virtual particles exist at least as lines on paper, no such case can be made for their time evolution.

Remember that in Casimir plates, virtual particles can affect physical matter.

No. The plates exert a Casimir force upon each other. Virtual particles are irrlevant for this force. See the entry '''Does the Casimir effect prove the existence of virtual particles?' in Chapter A7 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#casimir

 

[rogerl]

There are no ''time processes of virtual particles''. Nobody ever has written down an equation for the time evolution of virtual particles. While a case can be made that virtual particles exist at least as lines on paper, no such case can be made for their time evolution.

But couldn't timeless processes of virtual particles be considered part of "physical" too? You seem to be saying that only if something has time component is it physical. But a timeless state could also have physical contents. No? You seem to be saying that anything that has no time component is purely mathematical. But remember mathematical processes can affect the physical.. like QED calculations of gymagnetic ratio of the electron which has experimental results of 1.00115965219 by taking into account Feynman diagrams with up to seven minimal interaction vertices of the virtual fluctuations. Now since virtual particles can affect experimental result as shown... then virtual particles although they occur in timeless states is also part of physical reality. Well?

 

[A. Neumaier]

a timeless state could also have physical contents. No?

No. How should it? it would have to be describable by physics, and this means by a dynamical law telling how the state changes and interacts.

But remember mathematical processes can affect the physical.

This is an illusion. Mathematics may _describe_ what is real but does not _affect_ it.

like QED calculations of gymagnetic ratio of the electron which has experimental results of 1.00115965219

The electrons in the universe behaved according to this value even before the first mathematician thought of a way to do QED calculations. Even before the first person thought about counting or measuring distance.

 

[rogerl]

No. How should it? it would have to be describable by physics, and this means by a dynamical law telling how the state changes and interacts.

This is an illusion. Mathematics may _describe_ what is real but does not _affect_ it.

The electrons in the universe behaved according to this value even before the first mathematician thought of a way to do QED calculations. Even before the first person thought about counting or measuring distance.

Are you saying that our Quantum Field Theory is so because of the unique mathematical
formalisms of the past and in another planet, the development of their QFT can be described without the concept of virtual particles??

If you state instead that in all the planets in the universe, their QFT is the same as that on Earth with virtual particles and renormalization, then there is something unique about the math. As if the math is more primary, which produced the following scenerio.

A few years back, there was a movie The Matrix staring Keanu Reeves. It's about humans living in computer generated virtual world (inside a computer) with the real bodies in sleeping chambers. Is it possible that our world is like a Matrix or computer generated. Here the mathematics are part of the programming algorithm. The output of the program on the screen are analogous to measurements in our physics. So when we measure certain attributes of the electron. We are interacting with the screen dynamics. But what goes behind it is complex mathematics which can't be displayed on the screen. But the mathematics have consequences in that it affects what is measured in the screen. And the avatars in the screen developed their own maths to describe the dynamics of their virtual world. Here it support your statement that "Mathematics may _describe_ what is real but does not _affect_ it". This is because the real mathematics occur behind the scene in the programming algorithm while the math of the virtual characters in the virtual world is unique to their own, for their understanding.

This scenerio is possible if you are saying that virtual particles can affect the lamb shift although virtual particles are not real and just mathematical artefacts. Here the mathematics of the virtual particles are part of the computer programming algorithm that creates the computer world.

If you don't believe Matrix reality is possible (do you?) then virtual particles are part of physical world. Here we can extend the meaning of physical world even to timeless entities. Without this adjustment, everything doesn't make any sense and you are limiting yourself to simply "shut up and calculate" without any regards to thinking of what goes behind the scene or calculations. What is the case of your thought process? Pls explain so I can get an idea how you think about all this.

 

[weejee]

Do all mainstream physicists believe that the "bare", "virtual", "infinite" is purely figurative? Or are they divided on this just like some physicists believe the wave function is real (as in Bohmian Mechanics), while the other believe they are just mathematical tool (Copenhagen)? I think you are the latter. Do you also believe virtual particles don't have literal existence but just mathematical stuff? But then ever heard of the SLAC experiment where virtual particles are elevated into actual particles? The following is the details:

http://www.slac.stanford.edu/exp/e144/science1202.html

"Melissinos views the result as the first direct demonstration of "sparking the vacuum," a long-predicted phenomenon. In it, the energy of a very strong electromagnetic field promotes some of the fleeting, "virtual" particles that inhabit the vacuum, according to QED, to become pairs of real particles."

So you see. Virtual particles are not just mathematical entities. They can be real too. And if they are real, and virtual particles dressed the bare particle. Then "bare", "virtual", "infinite" are real entities! Is it only you who believe they are just figurative due to your own unique quantum interpretation?

Do you have some background in quantum mechanics? There, people talk about "real" or "virtual" transitions between different energy levels. Being real or virtual in quantum field theory mean pretty much the same thing, although renormalization complicates the problem.

Anyway, the example you mentioned is a "real" process. It's conceptually the same as an atom in the ground state getting excited by absorbing a photon with the matching energy. If the energy doesn't match, real transition doesn't happen, but excited states(minus one photon) still mix in slightly as a quantum supersition, and we call this "virtual transition".

 

[rogerl]

Do you have some background in quantum mechanics? There, people talk about "real" or "virtual" transitions between different energy levels. Being real or virtual in quantum field theory mean pretty much the same thing, although renormalization complicates the problem.

Anyway, the example you mentioned is a "real" process. It's conceptually the same as an atom in the ground state getting excited by absorbing a photon with the matching energy. If the energy doesn't match, it can't make a transition, but excited states still mix in slightly, and we call this "virtual transition".

Do you also believe like Neumaier that "The virtual particles there are still only metaphors for multivariate integrals"? But in the SLAC experiment as mentioned, electron-positron are elevated to real world. So how can metaphors become reality unless the virtual particles are there in dormant form awaiting a chance to precipitate into the real world. If you will state that multivariate integrals can become real electron-positron then it's like saying math can affect real world. Or better yet, what is the physical correlate of multivariate integrals that can actually produce electron-positron in the SLAC experiment?

 

[A. Neumaier]

Are you saying that our Quantum Field Theory is so because of the unique mathematical formalisms of the past and in another planet, the development of their QFT can be described without the concept of virtual particles??

If you state instead that in all the planets in the universe, their QFT is the same as that on Earth with virtual particles and renormalization

I am saying neither of the two. i am saying that QFT is mathematically the same in all civilizations of our universe (apart from language and symbols used, and equivalent rewritings of the theory), without virtual particles but with renormalization.

Virtual particles are an inconsequential aid to helping visualize complicated integrals, and can be dispensed with in QFT without any loss of predictivity.

 

[weejee]

Do you also believe like Neumaier that "The virtual particles there are still only metaphors for multivariate integrals"? But in the SLAC experiment as mentioned, electron-positron are elevated to real world. So how can metaphors become reality unless the virtual particles are there in dormant form awaiting a chance to precipitate into the real world. If you will state that multivariate integrals can become real electron-positron then it's like saying math can affect real world. Or better yet, what is the physical correlate of multivariate integrals that can actually produce electron-positron in the SLAC experiment?

Considering my condensed matter background, it doesn't sound natural to me to regard only the renormalized(or physical) quantities as real and consider bare quantities and the renormalization process as purely figurative or mathematical things.

I believe that I allow virtual particles some more reality than 'mere metaphors for multivariate integrals'. Still, I think that the 'reality' must exactly match with what the multivariate integrals denote. So, I'm very reluctant to say yes to questions like "are virtual particles real?", since such statement may mean million times more than what it should.

For this laser experiment, virtual particles being promoted to real particles just means that the coefficient of the state with an e-p pair was tiny at first (virtual) and then became dominant (real). No mystery in it.

 

[rogerl]

I am saying neither of the two. i am saying that QFT is mathematically the same in all civilizations of our universe (apart from language and symbols used, and equivalent rewritings of the theory), without virtual particles but with renormalization.

Virtual particles are an inconsequential aid to helping visualize complicated integrals, and can be dispensed with in QFT without any loss of predictivity.

Is this also the belief of mainstream physicists, or only selected physicists like you?

In the following SLAC creation of electron-positron:

http://www.slac.stanford.edu/exp/e144/science1202.html

"Melissinos views the result as the first direct demonstration of "sparking the vacuum," a long-predicted phenomenon. In it, the energy of a very strong electromagnetic field promotes some of the fleeting, "virtual" particles that inhabit the vacuum, according to QED, to become pairs of real particles."

So in your language of multivariate integrals. Are you saying that very strong electromagnetic field can promote some of the fleeting, "multivariate integrals" to become pairs of real particles? But since you don't believe mathematics may affect physical processes. Then what is the physical correlate or causal mechanism of the multivariate integrals in the vacuum that can create actual pairs of electron-positron?? There must be something that bind the two somehow.

 

[A. Neumaier]

Is this also the belief of mainstream physicists, or only selected physicists like you?

It is the mainstream view of those who care about a precise language. In my FAQ, I gave lots of references, but I also wrote:

Physicists talk about virtual particles as illustrative language for
internal lines in so-called Feynman diagrams. A Feynman diagram is a
mnemonic graphical representation of a multiple integral contributing
to a scattering amplitude in a collision process between real (i.e.,
measurable) particles. The collision process is characterized by
ingoing and outgoing real particles, represented in the Feynman diagram
as external lines, which suggests an interpretation of the remaining,
internal, lines as sort of short living intermediated products of the
collision, made up of virtual (i.e., nonmeasurable) particles.

While this gives the Feynman diagrams an intuitive interpretation,
it is impossible to give this intuition a deeper foundation in terms
of processes happening in space and time. The attempt to do so leads
to a phantastic view of the microscopic world.

This _is_ the mainstream view. The language is used a lot, to illustrate the otherwise abstract findings on a figurative level, but the talk is not taken serious as being about something real.

In the following SLAC creation of electron-positron:
http://www.slac.stanford.edu/exp/e144/science1202.html
"Melissinos views the result as the first direct demonstration of "sparking the vacuum," a long-predicted phenomenon. In it, the energy of a very strong electromagnetic field promotes some of the fleeting, "virtual" particles that inhabit the vacuum, according to QED, to become pairs of real particles."

The article you quote from is a report about the research paper
Positron Production in Multiphoton Light-by-Light Scattering
Phys. Rev. Lett. 79, 1626 (1997).
(the only paper fitting the description ''Melissinos, a spokesperson for the group, [...] In the 1 September Physical Review Letters'') There you can read in the abstract the scientific version of the above:

. The positrons are interpreted as arising from a two-step process in which laser photons are backscattered to GeV energies by the electron beam followed by a collision between the high-energy photon and several laser photons to produce an electron-positron pair. These results are the first laboratory evidence for inelastic light-by-light scattering involving only real photons.

What you quoted is a rephrasing of this precise scientific statement, in terms that generate a more vivid impression but are useful _only_ for impressive announcements, not for doing real science.

The research version of the paper mentions the word ''virtual'' precisely three times - in each case it is about a virtual photon, and describes a particular Feynman diagram (without having to draw it).

Not the slightest mention is made of virtual particles becoming real. The real particles produced are electrons and positrons. and they were created from real photons.

However, on the figurative level (i..e, with figures on paper or in the mind), talking of a virtual electron-positron pair becoming real invokes some loose sort of understanding why the experimental findings make sense - and this is the sole purpose for talking like this to colleques: The Feynman diagram representing the experimentally observed process (two gamma in, e^- and e^+ out) and the Feynman diagram representing the process responsible for detecting the positrons (e^- and e^+ in, two gamma out) put together (assuming that the two electrons are the same - which they aren't in practice) form a typical 1-loop diagram for photon-photon scattering via a virtual e^-/e^+ pair.

If you still believe in the reality of virtual photons, you can't be helped...

 

[weejee]

Everything labelled ''bare'' and ''virtual'', or ''infinite'' is purely figurative, and has no physical contents. it just tells something about formula manipulation in pseudo-intuitive terms. The real things are labelled ''dressed'' or ''renormalized''.

According to your viewpoint, bare quantities and perturbative corrections to them are purely formal things, and they just give us some rules to obtain renormalized quantities? Is it that you don't even think about the Hilbert space before renormalization?

I wonder. Maybe to have a UV complete theory, a construction like this is crucial? Then, if we are dealing with effective field theories with some physical cutoff, do we still need to stick to such a viewpoint for some reason?

 

[A. Neumaier]

According to your viewpoint, bare quantities and perturbative corrections to them are purely formal things, and they just give us some rules to obtain renormalized quantities? Is it that you don't even think about the Hilbert space before renormalization?

In QFT, the Hilbert space is _constructed_ through renormalization; the bare stuff is only the scaffolding to construct an appropriate limit containing the physics. See the thread
https://www.physicsforums.com/showthread.php?t=476412 , and post #100 in
https://www.physicsforums.com/showthread.php?t=474666

if we are dealing with effective field theories with some physical cutoff, do we still need to stick to such a viewpoint for some reason?

Well, one still needs renormalization to get the physical states...

 

[weejee]

Well, one still needs renormalization to get the physical states...

That's right. Still, when we have a physical cutoff, can't we relax our view on the bare quantities that they are purely figurative, and think about some Hilbert space even at the bare level?

Maybe the conclusion is dependent on what kind of cutoff we have?

 

[A. Neumaier]

That's right. Still, can't we relax our view on the bare quantities that they are purely figurative, and think about some Hilbert space even at the bare level?

One can think about many things, but no amount of thinking changes the fact that the bare Hilbert space is completely irrelevant for physics. To do physics, you need a mathematical representations for the states that can be prepared and measured, and once you have that you have everything needed.

To keep the bare stuff around is like living in a house where all the raw stuff and machinery needed to build it still lies around. It is in the way, it is ugly, and it turns what could be a beautiful place into a permanent construction site.

As long as a theory is a construction site, it is messy and prone to misunderstanding. Once tidied up, it becomes clear and efficient for use. But some never get around clearing their mind from all the construction work, keeping only the finished products. Those who do are much better off.

 

[weejee]

One can think about many things, but no amount of thinking changes the fact that the bare Hilbert space is completely irrelevant for physics. To do physics, you need a mathematical representations for the states that can be prepared and measured, and once you have that you have everything needed.

To keep the bare stuff around is like living in a house where all the raw stuff and machinery needed to build it still lies around. It is in the way, it is ugly, and it turns what could be a beautiful place into a permanent construction site.

As long as a theory is a construction site, it is messy and prone to misunderstanding. Once tidied up, it becomes clear and efficient for use. But some never get around clearing their mind from all the construction work, keeping only the finished products. Those who do are much better off.

I kind of see where my confusion comes from.

In condensed matter, we just define the "bare" theory in the Fock space. (The contents in the bare theory are already renormalized in the high-energy sense, but if there is something like the Fermi sea, we can expect further renormalization of the low-energy excitations.)

However, if we really want to quantize some wave equation with non-linear equation of motion, there is no way around this rigorous stuff, whether there is a physical cutoff or not.

Am I right?

 

[A. Neumaier]

In condensed matter, we just define the "bare" theory in the Fock space. (The contents in the bare theory are already renormalized in the high-energy sense, but if there is something like the Fermi sea, we can expect further renormalization of the low-energy excitations.)

However, if we really want to quantize some wave equation with non-linear equation of motion, there is no way around this rigorous stuff, whether there is a physical cutoff or not.

Am I right?

Not yet but in the right direction. In condensed matter, the bare theory makes sense in principle, since (due to lack of particle creation and annihilation) it is the theory of the asymptotic particles at zero temperature. But to do calculations, one wants to think of the effective degrees of freedom of the solid, which are the ground state excitations. One wants to consider the ground state to be the vacuum, and the collective excitations to be the (quasi-)particles. To get this view, one needs to renormalize the theory - this is like in particle physics, except that, because the bare theory is physically meaningful, all renormalizations are finite.

In relativistic quantum field theory, there is no substance (aether) that would fill the ground state (vacuum), upon which one could build the theory. The bare particles pretend to be such a substance but is found inadequate, as seen by the divergences. But by taking careful limits and adjusting the parameters of the bare theory to diverge while taking the limit - which is possible only since the bare stuff is unphysical - one can still arrive at a renormalized theory in which the vacuum is a Poincare invariant state and the Hilbert space has the required property of carrying covariant and causal field operators.

This has nothing to do with being or not being an effective theory - the latter only need an infinite number of renormalization parameters for their construction.

It has also nothing to do with nonlinearities - the field equations underlying condensed matter are also nonlinear. The difference to condensed matter theory comes from the requirement of causality, which necessitates processes that change particle number.

 

[weejee]

It has also nothing to do with nonlinearities - the field equations underlying condensed matter are also nonlinear. The difference to condensed matter theory comes from the requirement of causality, which necessitates processes that change particle number.

Well, I was wondering if we can tell whether the bare theory is unphysical, simiply by looking at the field equation. Are you saying that it is the causality requirement which makes certain relativistic field theories unphysical at the bare level?

As for field equations of interacting condensed matter systems(ex: fermions interacting via Coulomb potential), I definitely can see that they are non-linear. However, all the condensed matter books I've seen so far, simply start from the Fock space and consider how the Coulomb potential acts on the space, but never care to quantize the original field equation from the beginning. Doesn't this cause any problem theoretically?

Thank you very much for your careful and patient answers. :)

 

[rogerl]

It is the mainstream view of those who care about a precise language. In my FAQ, I gave lots of references, but I also wrote:

This _is_ the mainstream view. The language is used a lot, to illustrate the otherwise abstract findings on a figurative level, but the talk is not taken serious as being about something real.

The article you quote from is a report about the research paper
Positron Production in Multiphoton Light-by-Light Scattering
Phys. Rev. Lett. 79, 1626 (1997).
(the only paper fitting the description ''Melissinos, a spokesperson for the group, [...] In the 1 September Physical Review Letters'') There you can read in the abstract the scientific version of the above:

What you quoted is a rephrasing of this precise scientific statement, in terms that generate a more vivid impression but are useful _only_ for impressive announcements, not for doing real science.

The research version of the paper mentions the word ''virtual'' precisely three times - in each case it is about a virtual photon, and describes a particular Feynman diagram (without having to draw it).

Not the slightest mention is made of virtual particles becoming real. The real particles produced are electrons and positrons. and they were created from real photons.

However, on the figurative level (i..e, with figures on paper or in the mind), talking of a virtual electron-positron pair becoming real invokes some loose sort of understanding why the experimental findings make sense - and this is the sole purpose for talking like this to colleques: The Feynman diagram representing the experimentally observed process (two gamma in, e^- and e^+ out) and the Feynman diagram representing the process responsible for detecting the positrons (e^- and e^+ in, two gamma out) put together (assuming that the two electrons are the same - which they aren't in practice) form a typical 1-loop diagram for photon-photon scattering via a virtual e^-/e^+ pair.

If you still believe in the reality of virtual photons, you can't be helped...

Ok. I can accept what you said that virtual particles are just mathematical artifacts or multivariate integrals. But virtual particles produce an *observable* effect.

You said mathematics only describe reality and they can't affect the physical. But how come the physical can be described perfectly by mathematics? Don't say they just do. You must explain why. This is physics. And physics should involve understanding why and not just describe measurements only.

 

[rogerl]

Not yet but in the right direction. In condensed matter, the bare theory makes sense in principle, since (due to lack of particle creation and annihilation) it is the theory of the asymptotic particles at zero temperature. But to do calculations, one wants to think of the effective degrees of freedom of the solid, which are the ground state excitations. One wants to consider the ground state to be the vacuum, and the collective excitations to be the (quasi-)particles. To get this view, one needs to renormalize the theory - this is like in particle physics, except that, because the bare theory is physically meaningful, all renormalizations are finite.

In relativistic quantum field theory, there is no substance (aether) that would fill the ground state (vacuum), upon which one could build the theory. The bare particles pretend to be such a substance but is found inadequate, as seen by the divergences. But by taking careful limits and adjusting the parameters of the bare theory to diverge while taking the limit - which is possible only since the bare stuff is unphysical - one can still arrive at a renormalized theory in which the vacuum is a Poincare invariant state and the Hilbert space has the required property of carrying covariant and causal field operators.

This has nothing to do with being or not being an effective theory - the latter only need an infinite number of renormalization parameters for their construction.

It has also nothing to do with nonlinearities - the field equations underlying condensed matter are also nonlinear. The difference to condensed matter theory comes from the requirement of causality, which necessitates processes that change particle number.

You mentioned that the vacuum is a poincare invariant state. So the quantum vacuum is another mathematical figment of imagination just like the virtual particles?? Or is quantum vacuum located in space? Or not? If not, then space has only empty contents that has no vacuum and virtual particles?

 

[A. Neumaier]

Well, I was wondering if we can tell whether the bare theory is unphysical, simiply by looking at the field equation.

Yes, one can tell from that: Trying to solve the field equations together with equal-time CCR produces the dreaded infinities. This was the major obstacle in using QED between 1930 and 1948.

all the condensed matter books I've seen so far, simply start from the Fock space and consider how the Coulomb potential acts on the space, but never care to quantize the original field equation from the beginning. Doesn't this cause any problem theoretically?

No, as long as one keeps the particle number fixed. The reason is that particle number N is conserved. Usually, the thermodynamic limit N --> inf is done at a late stage where this is also harmless.

 

[A. Neumaier]

Ok. I can accept what you said that virtual particles are just mathematical artifacts or multivariate integrals. But virtual particles produce an *observable* effect.

No. Interactions produce an observable effect. The interactions are represented pictorially by vertices in Feynman diagrams representing perturbative contributions to a scattering amplitude, and the lines between them are pictorially called virtual particles.
They cause nothing.

You said mathematics only describe reality and they can't affect the physical. But how come the physical can be described perfectly by mathematics?

This is one of the great insights of science throughout the centuries, starting with Pythagoras and a^2+b^2=c^2. But the description is not perfect. All our models are only approximations to reality.

That it works so well is a miracle whose splendedness increases the better we understand nature.
http://en.wikipedia.org/wiki/Unreasonable_Effectiveness
It is easiest explained by assuming that God created the world according to a well-planned overall design. Some explain it instead by assuming that it is the result of chance and necessity - without being able to say where the laws defining necessity come from.
Some explain it instead by saying it is our minds who impose order on the universe, - but where is the evidence that we have such a power when confronted with a mess of any kind?

 

[A. Neumaier]

You mentioned that the vacuum is a poincare invariant state. So the quantum vacuum is another mathematical figment of imagination just like the virtual particles?? Or is quantum vacuum located in space? Or not? If not, then space has only empty contents that has no vacuum and virtual particles?

Physical states describe the possible contents of space-time. One of the many possibilities (and one not realized in a world containing us) is that it is completely empty. Then there is no way to distinguish one point from another or one orientation from another, which is why this state is Poincare invariant. (In general relativity it is even much more invariant.)

 

[rogerl]

Physical states describe the possible contents of space-time. One of the many possibilities (and one not realized in a world containing us) is that it is completely empty. Then there is no way to distinguish one point from another or one orientation from another, which is why this state is Poincare invariant. (In general relativity it is even much more invariant.)

Are you saying that space is synonymous to the quantum vacuum? I was asking about the quantum vacuum and whether it is another mathematical figment of imagination like virtual particles and you answered that space could be empty. Are you saying that space is another mathematical figment of imagination? But we move in space, how could that be? Unless you mean space is real while the quantum vacuum is just a mathematical entity or only in the mathematics like virtual particles. Is this what you mean?