Exploring Alternatives to QFT: A Critique of Non-Interacting Quantum Fields

[strangerep]

I'm trying to learn what Haag's theorem is, and googling brings up articles by Fraser, and Earman and Fraser. It looks as if Haag's theorem only needs Euclidean invariance, so it would seem to apply to non-relativistic QFT. Does Haag's theorem apply in the non-relativistic QFT used in condensed matter? If Haag's theorem doesn't apply, is it because Euclidean invariance is broken by the lattice?

I looked at some of the Fraser/Earman papers several years ago and got the impression that they're more philosophers than physicists (being in the Dept. of History and Philosophy of Science at Pittsburgh). They seemed to be most interested in exploring the fact that, in infinite dimensions, there can exist unitarily inequivalent representations of the CCRs -- and one certainly doesn't need full Poincare relativity to explore that. The textbooks of Umezawa et al ("Thermofield Dynamics & Condensed States" and "Advanced Field Theory") contain useful introductions to inequivalent reps.

For Haag's theorem in a relativistic context, there's always Streater & Wightman's "PCT, Spin, Statistics, and all that". But the first exposition of Haag's theorem that I could actually follow (including the proof) was in Barton's little-known book:

G. Barton,
Introduction to Advanced Field Theory,
Interscience, 1963.

He also has a chapter near the end with some interesting remarks and speculations about the role of unitarily inequivalent representations in full QFT.

 

[atyy]

@waterfall, supersymmetry isn't meant to solve the problem of infinities, it's meant to solve the "naturalness" problem. It's not primarily a mathematical problem, more a problem of explaining why some parameters in the standard model have to specified so precisely to match experimental observations. There's a discussion of this on p6 of the Zinn-Justin chapter. The Wilson-Kadanoff viewpoint that non-renormalizable theories are acceptable effective field theories, and that renormalization is just a way to see how they look like at low energies, underlies two different approaches to quantum gravity: string theory and asymptotic safety. A further argument for the Wilson-Kadanoff viewpoint is the gauge/gravity conjecture in which the renormalization flow is transformed into a spatial dimension.

@strangerep, thanks for the references! I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser, but also further indicates that actual meaning should be given by Euclidean field theory, checking if the Osterwalder-Schrader axioms are satisfied, and analytically continuing to Minkowski space. I think one of the problems in LQG is choosing between unitarily inequivalent representations due to Haag's theorem. Apparently Thiemann's master constraint programme tries to use dynamics to choose the appropriate representation. There seems to be an analogy with a particle on a circle.

 

[strangerep]

I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser,

If you're not familiar with the Epstein-Glaser method, try Scharf's book:

G. Scharf,
Finite Quantum Electrodynamics -- The Causal Approach,
Springer, 2nd Ed., 1995. ISBN 3-540-60142-2

(Make sure you get the 2nd edition -- it has a lot more stuff than the first.)

But the basic idea of Epstein-Glaser-Scharf is that QFT infinities arise from multiplying distributions by $\Theta(t)$ (step-function) in the time-ordered products. The discontinuity in the step function means that the product is no longer a tempered distribution. The method then revolves around inserting correction terms perturbatively to fix it -- using causality as a guide. But it's quite a few years since I went through the 1st edition of Scharf's book, back when I knew far less QFT and math than now. I really should read the 2nd edition thoroughly some time. :-(

[...] use dynamics to choose the appropriate representation.

Haag also makes a brief mention in his book about how choosing the representation is a "dynamical problem". I guess that means choosing an appropriate time-dependent Bogoliubov transformation, but I don't understand that stuff very well -- and modern LQG even less. :-(

 

[waterfall]

@waterfall, supersymmetry isn't meant to solve the problem of infinities, it's meant to solve the "naturalness" problem. It's not primarily a mathematical problem, more a problem of explaining why some parameters in the standard model have to specified so precisely to match experimental observations. There's a discussion of this on p6 of the Zinn-Justin chapter. The Wilson-Kadanoff viewpoint that non-renormalizable theories are acceptable effective field theories, and that renormalization is just a way to see how they look like at low energies, underlies two different approaches to quantum gravity: string theory and asymptotic safety. A further argument for the Wilson-Kadanoff viewpoint is the gauge/gravity conjecture in which the renormalization flow is transformed into a spatial dimension.

@strangerep, thanks for the references! I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser, but also further indicates that actual meaning should be given by Euclidean field theory, checking if the Osterwalder-Schrader axioms are satisfied, and analytically continuing to Minkowski space. I think one of the problems in LQG is choosing between unitarily inequivalent representations due to Haag's theorem. Apparently Thiemann's master constraint programme tries to use dynamics to choose the appropriate representation. There seems to be an analogy with a particle on a circle.

From all these infinities and renormalization thing. It looks like our physics is mainly about interactions between particles. So I think it's true they are just lower limit or classical limit of a completely radical theory. Remember Zinn-Justin's last sentence in the book you shared where it is quoted "One uses the terminology of effective QFT, approximations of an as yet unknown more fundamental theory of a radically different nature."

The radical theory would make possible for example the holodeck in Star Trek where one can manifest any object or make them physical. This is engineering beyond the vacuum. It seems our present physics just focus on the interactions of particles, they don't even know how spacetime is connected to quantum particles. So spacetime could be just a temporary construction, and if we can have access to the more fundamental theory, then we can reprogram spacetime and matter to make possible the idea of Holodeck in Star Trek. This is possible isn't it? You can't make arguments about our mere physics of interactions to judge the limit of what is possible. Do you accept this (and others too)?

 

[strangerep]

[...] the holodeck in Star Trek [...] This is possible isn't it?

You just crossed over into the twilight zone of crackpot speculation.

(Moderators: maybe it's time to close this thread?)

 

[waterfall]

You just crossed over into the twilight zone of crackpot speculation.

(Moderators: maybe it's time to close this thread?)

I'm just asking if our physics is the final.. but I noticed they are mostly based on interactions... on non-interacting quantum fields and renormalization group that is ad hoc. Don't worry. I'm not a star trek fan. But it's just asking if our physics is really the final.. or just the beginning to another chapter like from Newtonian to einsteinian or quantum...

 

[martinbn]

I'm just asking if our physics is the final.. but I noticed they are mostly based on interactions... on non-interacting quantum fields and renormalization group that is ad hoc. Don't worry. I'm not a star trek fan. But it's just asking if our physics is really the final.. or just the beginning to another chapter like from Newtonian to einsteinian or quantum...

What if it is not the final theory? In fact, I don't think that anybody thinks that it is.

 

[waterfall]

What if it is not the final theory? In fact, I don't think that anybody thinks that it is.

It's reported in many news and magazines that when the Higgs will be found found. Physics will be almost complete. But it may be just the beginning.. perhaps we are like starting in Newton stage comparatively and physics would continue to develope the next 400 years...

With non-positive results in Loop quantum gravity and Superstrings, we may be on a wrong foundation and quantum gravity may be more than a century away... you think we can solve it before year 2100?

 

[tom.stoer]

It's reported in many news and magazines that when the Higgs will be found found. Physics will be almost complete.

Standard model (including Higgs) + classical gravity (general relativity) is by no means complete

1) the perturbation series of standard model QFTs does not converge (here I do not mean the infinities in each term but the series as a whole
2) there are problems in the UV, especially for the Higgs
3) gravity is not quantized, but we know that QFT + classical gravity is incomplete
4) gravity itself is incomplete (singularities)

Of course there are additional physical issues like unification, reason for SU(3)*SU(2)*U(1), coupling constants, particles, fermion generations etc.; but even w/o taking these questions into account, the mathematical structure "standard model + classical gravity" is ill-defined.

 

[martinbn]

tom.stoer, a side question about your 1), 4).

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

4) why do singularities mean that gravity is incomplete?

waterfall,

if I understand you correctly you afraid that physicists think that physics is almost complete, and you disagree, but i don't think that is the case, dispite of what some news and magazines may say. Also I get the feeling that you believe that about 80 years ago physics took a wrong turn with QFT and now it is in a dead end street, so they should all stop what they are doing and go back to the begining. That is misunderstanding what physics is and what it has done. Of course I may be completely missing you point.

 

[waterfall]

Standard model (including Higgs) + classical gravity (general relativity) is by no means complete

1) the perturbation series of standard model QFTs does not converge (here I do not mean the infinities in each term but the series as a whole
2) there are problems in the UV, especially for the Higgs
3) gravity is not quantized, but we know that QFT + classical gravity is incomplete
4) gravity itself is incomplete (singularities)

Of course there are additional physical issues like unification, reason for SU(3)*SU(2)*U(1), coupling constants, particles, fermion generations etc.; but even w/o taking these questions into account, the mathematical structure "standard model + classical gravity" is ill-defined.

After a week of understanding the essence of QFT, I just realized how badly is our situation. It's like we were back in the days of Newton. When you don't know QFT. You think i'ts very impressive and we are near to the solution of everything. Isn't it that Steven Weinberg wrote how we soon would have a theory of everything. See:

http://www.math.vt.edu/people/gao/physics/weinberg.html

What would it take to create interacting fields. Maybe we need to find alternatives for the fock space which doesn't even interact. It's quiet bad. We have quantum field theory, but the fields don't interact and we have to use artificial means and ad hoc pertubation series.

I think it's time I should reread Lee Smolin Not Even Wrong and Peter Woit Physics Wrong Turn.. because there is a possibility they may be right and String theory and even Loop Quantum Gravity are just "Recreational Mathematical Theology". I forgot all their arguments.

 

[waterfall]

tom.stoer, a side question about your 1), 4).

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

4) why do singularities mean that gravity is incomplete?

waterfall,

if I understand you correctly you afraid that physicists think that physics is almost complete, and you disagree, but i don't think that is the case, dispite of what some news and magazines may say. Also I get the feeling that you believe that about 80 years ago physics took a wrong turn with QFT and now it is in a dead end street, so they should all stop what they are doing and go back to the begining. That is misunderstanding what physics is and what it has done. Of course I may be completely missing you point.

I'm not saying it's a dead end street. I think it's similar to what happened in General Relativity. Had Einstein not discover certain math technique (I forget if it's differential geometry or tensor calculus), he won't be able to perfect GR and make things lorentz covariant. So I think we have missed the right mathematical tool or language for true QFT instead of the sporatic Fock space that may be just child's play. Now I wonder what is the right mathematicals for fields that indeed interact. It means we have to replace or rather enchance Hilbert Space too with a more superior mathematics. Anyone has any idea what kind of math algorithm for it and if I make sense at all?

 

[tom.stoer]

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

I agree; this is perhaps not a fundamental issue.

It's not true that there are no non-perturbative tools, but one cannot say that there is a fully developed non-perturbative approach applicable to all questions in QFT; it strongly depeds on the use case.

4) why do singularities mean that gravity is incomplete?

b/c GR is formulated for smooth manifolds w/o boundary and w/o defects; at singularities the theory is no longer predictive; you cannot formulate boundary or initial conditions; you don't know where all the matter goes in a black hole (the Schwarzschild metric is a vacuum solution with a point-like singularity); b/c when combined with QFT a black hole it violates unitarity; ...

 

[waterfall]

I was rereading Lee Smolin "Trouble with Physics". He was saying in the following in page 249 that Loop Quantum Gravity was trying to reinvent QFT?? I thought LQG is all about gravity. How come I don't hear about QFT being redone in LQG formulation?

"This work was made possible by Ashtekar's great discovery that general relativity could be expressed in language like that of a gauge field. The metric of spacetime, then, turns out to be something like an electric field. When we tried to treat the corresponding field lines quantum-mechanically, we were forced to treat them without a background because there was none - the field lines already described the geometry of space. Once we made them quantum-mechanical, there was no classical geometry left. So we had to reinvent quantum field theory in order to work without a background metric. To make a long story short, it took the input of many people, with a variety of skills from physics and mathematics, but we succeeded. The result is loop quantum gravity."

Do you agree we have to reinvent quantum field theory in order to work without a background metric? Btw.. why hasn't anyone told me the answer to the "Alternative to QFT " in my thread question is nothing but Loop Quantum Gravity as Smolin mentioned?

 

[suprised]

I think it's time I should reread Lee Smolin Not Even Wrong and Peter Woit Physics Wrong Turn..

It's amazing how one can mess up every little detail. Sorry, don't be surprised about not gettting answers, it's just too far off.

 

[waterfall]

It's amazing how one can mess up every little detail. Sorry, don't be surprised about not gettting answers, it's just too far off.

What? It's not my idea but Smolin's. Anyway. I converted all texts of Lee Smolin "Trouble With Physics" to speech and I'll listen to it all day and night in my ipod. Here's Smolin main theme or punchline:

"This is the story of a quest to understand nature at its deepest level. Its protagonists are the scientists who are laboring to extend our knowledge of the basic laws of physics. The period of time I will address - roughly since 1975 - is the span of my own professional career as a theoretical physicist. It may also be the strangest and most frustrating period in the history of physics since Kepler and Galileo began the practice of our craft four hundred years ago. The story I will tell could be read by some as a tragedy. To put it bluntly - and to give away the punch line - we have failed. We inherited a science, physics, that had been progressing so fast for so long that it was often taken as the model for how other kinds of science should be done. For more than two centuries, until the present period, our understanding of the laws of nature expanded rapidly. But today, despite our best efforts, what we know for certain about these laws is no more than what we knew back in the 1970s. How unusual is it for three decades to pass without major progress in fundamental physics? Even if we look back more than two
hundred years, to a time when science was the concern mostly of wealthy amateurs, it is unprecedented. Since at least the late eighteenth century, significant progress has been made on crucial questions every quarter century"

 

[Physics Monkey]

What I find amazing is the unbelievable hubris required to equate a few esoteric questions in high energy physics with the progress of all physics.

 

[friend]

Btw.. why hasn't anyone told me the answer to the "Alternative to QFT " in my thread question is nothing but Loop Quantum Gravity as Smolin mentioned?

Does your concern about the right math in physics come from a perception that present math may not be able to give us a complete theory of everything? Or does it seem that the mathematical origins of QFT seem arbitrary? What would prove that we are using the correct math? We would have to be able to derive QFT from deductive logic in order to show that there is even any chance of proving the completeness of physics. Otherwise, our theories will always be contingent on the next experiment confirming their predictions. We can never measure everything, so their always remains the possibility that our theory can be proven wrong by some experiment.

 

[tom.stoer]

LQG is certainly not the alternative
- it's not a complete theory but work in progress (neither mathematically nor physically)
- it's a theory about gravity only; full inclusion of matter is still missing
- it's by no means a theory aiming for unification
- the definition of obervables is not fully understood
- nobody knows how to do simple low-energy scattering calculations
-...

 

[waterfall]

Does your concern about the right math in physics come from a perception that present math may not be able to give us a complete theory of everything? Or does it seem that the mathematical origins of QFT seem arbitrary? What would prove that we are using the correct math? We would have to be able to derive QFT from deductive logic in order to show that there is even any chance of proving the completeness of physics. Otherwise, our theories will always be contingent on the next experiment confirming their predictions. We can never measure everything, so their always remains the possibility that our theory can be proven wrong by some experiment.

I have to study deeper the math of QFT to be able to answer that. But I heard from Fredrik that even those with Ph.D. in Physics doesn't mean they are already expert in QFT. So it's kinda heartbreaking. From graduate in college of BS in Physics to Ph.D. I think takes 4 to 5 years more for total of 8 years. And yet they are not yet master in it. Are you a physicist? I'm thinking whether to go back to school and become one. Because there is a possibility our physicists may just miss it all and won't see the light even after 20 years or year 2032. This is because they are doing it blind. They don't have any guiding principle much like when Einstein got the insight about the Equivalence Principle and spent 10 years to perfect it to produce GR. I think I have a guiding principle insight too and just need to find the right math. Actually some have the same guiding principle insight but they are just not trained to math to develope it fully. And physics is just so important to leave it to physicists. Important choices to make maybe not just me.. but also you. So is your course related to physics? What do you make of Smolin and Woit book. Woit book is more mathematical and I think I'll try to understand it deeper after learning here that QFT is good for only free fields and Fock space is none-interacting proving Smolin and Woit is not smoking pots but are partly if not more right in their critique of modern physics.

 

[ofirg]

Why is QFT treated here as its definition only makes sense with perturbation theory?
I understood that the path integral definition ( and the canonical formalism also, at least of you can find an appropriate fock space) doesn't rely on perturbation theory, and we use it simply because we aren't able solve the full theory.

 

[marcus]

Hi Waterfall, your thread about a possible wrong turn reminds me of a different discussion we had here a few years back. A mentor named "SelfAdjoint" took part in the discussion.

https://www.physicsforums.com/showthread.php?t=124999

It began with a poll asking when people thought a wrong turn was made.

 

[juanrga]

Why is QFT treated here as its definition only makes sense with perturbation theory?
I understood that the path integral definition ( and the canonical formalism also, at least of you can find an appropriate fock space) doesn't rely on perturbation theory, and we use it simply because we aren't able solve the full theory.

The «full theory» does not exist as QFT... because only free fields are well-defined.

 

[atyy]

after learning here that QFT is good for only free fields and Fock space is none-interacting

First, the renormalization group shows that theories do not have to be defined at all energies to yield great predictions at low energies. QED is such a theory.

Second, it is not true that only free fields are rigourously defined at all energies. Some nonlinear self-interacting quantum fields have been rigourously constructed in 2 and 3 dimensional spacetimes. The rigourous construction of Yang-Mills theory in 4 dimensional spacetime is thought possible because of asymptotic freedom, but it is still being researched. In fact, the Clay Institute is offering a prize of $1 million for a rigourous construction of Yang Mills theory and a demonstration that it has a mass gap.

http://www.claymath.org/millennium/Yang-Mills_Theory/
http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf

 

[marcus]

LQG is certainly not the alternative
- it's not a complete theory but work in progress (neither mathematically nor physically)
- it's a theory about gravity only; full inclusion of matter is still missing
- it's by no means a theory aiming for unification
- the definition of obervables is not fully understood
- nobody knows how to do simple low-energy scattering calculations
-...

I have no reason to object to any of these 5 points and don't want to argue about any of this.

But I want to mention what my perspective is on your "certainly not the alternative" phrase.

LQG could well be on the correct path to the alternative even though it is itself not the last step nor does try to be.

The LQG program may be *on the path to unification" because it strives for a new (no-prior-geometry) representation of spacetime. One which takes into account how geometry responds to measurement and interacts with matter.

On the path, because it may happen to be necessary to settle on a quantum theory of geometry (interacting with matter) before one can build a new representation of the whole.

And in particular it may be necessary to arrive first at a testable QG model of early universe cosmology, like that of LQC--something which explains how the big bang occurred, resolves some problems with dark matter and inflation, and predicts various features to observe in the background of ancient light.

 

[friend]

Hi Waterfall, your thread about a possible wrong turn reminds me of a different discussion we had here a few years back. A mentor named "SelfAdjoint" took part in the discussion.

https://www.physicsforums.com/showthread.php?t=124999

It began with a poll asking when people thought a wrong turn was made.

That thread ended with

riginally Posted by Mike2
"We don't really know WHY the math is the way it is. "

To which the reply was,
"We never knew that, and we will never know that."

My opinion is that we will never really be satisfied until we can derive physics from the first principles of logic. For that's really the only means of really "knowing' what is true beyond any argument. Otherwise, theories developed by just guessing the math can be falsified by some observation in the future, and we can never be totally sure about them because we can not measure everything to actually prove that the theory predicts all measurements.

So, my efforts have been towards a derivation of physics from logic for the past 5-10 years. I seem to have made progress (at least no one is showing me any blatant errors), and I think I am getting very close. However, my efforts are not on the arXiv yet. So it might still be considered speculative here. But if you really want to examine my work in progress, give me a Private Message.

 

[ohwilleke]

"- it's a theory about gravity only; full inclusion of matter is still missing
- it's by no means a theory aiming for unification
. . .

But I want to mention what my perspective is on your "certainly not the alternative" phrase.

LQG could well be on the correct path to the alternative even though it is itself not the last step nor does try to be.

The LQG program may be *on the path to unification" because it strives for a new (no-prior-geometry) representation of spacetime. One which takes into account how geometry responds to measurement and interacts with matter."

To restate the point Marcus is making in a slightly different way, there is active research underway regarding how existing Standard Model interactions can be formulated in the context of a LQG space-time that seem to be making more progress than efforts express Standard Model interactions in a manner consistent with General Relativity rather than just Special Relativity. LQG efforts are proving more fruitful because one of the main the difficulties in integrating Standard Model interaction with General Relativity flow directly from the simultaneous assumptions that space-time is continuous (from GR) and that fundamental fermions and bosons are point-like (from the Standard Model) which naturally implies that every single particle is a singularity and isn't easily fixed. Point-like Standard Model fundamental fermions and bosons cause far less mischief in LQG because the discreteness of space-time discourages singularities from arising.

Thus, LQG is a promising possible way to formulate the Standard Model and GR in ways that are at least theoretically capable of being consistent with each other at the scale of individual particles.

Moreover, if you can formulate Standard Model interactions in an LQG space-time, then a "Theory of Everything" in which all four fundamental forces are different manifestations of a single deeper form of boson mediated particle interaction becomes a category error. If space-time has the character that LQG tries to model it as then gravity since it flows from the nature of space-time itself, is fundamentally different in kind than the other three fundamental forces. Hence, if you can formulate a mere GUT within an LQG space-time, you have taken the reductionist agenda for fundamental physics as far as it can go.

The genius of this research program is that string theory needs more than four dimensions of space-time almost entirely to accommodate the need to make gravity weak relative to the other three fundamental forces which by themselves can be formulated quite workably in a four dimensional space time. Indeed, the whole braneworld concept, in which string theorists conceive of a world where everything but gravity is confined to a four dimensional brane which is embedded in a larger dimensional space, illustrates how unnatural it is to try to treat gravity and the other three fundamental forces of the Standard Model as different versions of the same thing. In contrast, in LQG, an emergently four dimensional space-time arises naturally, and since you should need extra dimensions to formulate Standard Model equations designed for four dimensional Minkowski space into emergently four dimensional LQG space, you have eliminated the need to deal with unnatural extra dimensions from the get go.

A deep hate for point-like particles is deeply embedded in the very DNA of GR equations formulated in continous, perfectly local, space-time, and the Standard Model is equations are very deeply wedded to having point-like particles, which is what has made the marriage of the two so difficult to navigate. But, neither GR nor the Standard Model are all that deeply wedded to a continuous space-time. Continuity is an assumption thrown into the mix mostly for mathematical convenience and the heuristic that drives calculus itself is the notion that it is possible to exquisitely accurately (and indeed beyond accuracy to the point of a Platonic ideal) model sums of infinitessimal quantities as continuous quantities. LQG simply steps back off this assumption used for mathematical convenience to go to the source of the heuristic and computes quantities the hard way rather than analytically. Indeed, we routinely use discrete numerical methods to approximate Standard Model interactions by themselves, and to approximate GR by itself.

Yes, LQG is a work in progress. The task of formulating the Standard Model interactions in an LQG geometry is not ready for prime time yet. And, even if your ultimate goal is to formulate a GUT within an LQG geometry, there is not point in even trying until you accomplish this prerequisite task. You have to hope that the progress you make in formulating Standard Model interactions in an LQG geometry will point you in the right direction regarding what to do next.

But, once you can accomplish the more doable seeming task of formulating Standard Model interations in an LQG geometry, even the goal of then integrating those interactions into a GUT is less pressing. One of the reasons a Theory of Everything is such an alluring Holy Grail is that this would solve the current mess in which are two most perfect and wonderful theories of fundamental physics (the Standard Model and GR) are theoretically inconsistent with each other in obvious ways, despite the fact that both in their own domains repeatedly manage to describe experimental data with exquisite precision. But, if you formulate Standard Model interactions in an LQG geometry, that isn't a problem that needs to be solved by devising a Theory of Everything any more.

With that problem already solved, the only real problem left to tackle by taking the reformulated Standard Model interactions and integrating them into a GUT is to develop better insight about the physics of extremely high energy systems approaching Big Bang conditions. And, even those problems may not be as acute for the Standard Model interactions embedded in LQG, because the LQG geometry is already going to give rise by its very nature to some subtle high energy system modifications to Standard Model predictions because it integrates GR effects that have only been considered in an ad hoc non-rigorous way to date, for example, in asymptotic gravity driven predictions about Higgs mass and clever suppositions formulated just so necessary to predict Hawking radiation. For example, Standard model interactions embedded in LQG are very likely to have a lower unification energy scale than SUSY or SUSY inspired theories, and the natural UV bounds in LQG also tame a lot of the rigor concerns associated with commonly used renormalization methods.

You probably would still need to make a few leaps of insight to get a GUT embedded within an LQG geometry rather than merely the Standard Model embedded within an LQG geometry, but there is very good reason to think that those leaps of insight would have to be more modest in the LQG context than in the continuous space context, because there are fewer problems less to resolve. Once you've got the Standard Model embedded within LQG, pretty much all you need to do is to come up with a way to describe each of the twelve fundamental fermions (three generations each of two kinds of quarks, one kind of charged lepton and one kind of neutrino; if one does not count variations in the color charge/matter-antimatter/parity directions as different fermions), and each of the twelve Standard Model bosons (photon, 2 Ws and a Z, and eight gluons, ignoring any other possible variations on these bosons) as manifestations of one more fundamental thing. Turning one thing into twenty-four things with variations that have the right properties is tough, but not nearly as Herculean as the task currently facing people trying to devise a TOE via string theory.

I wouldn't be at all surprised to see insight developed in one or another prior GUT (perhaps even the original SU(5) GUTs) that failed and developed pathologies when formulated in Minkowski space, apply quite directly to a Standard Model embedded in LQG geometry that would somehow miraculously resolve the pathologies that arose in previous attempts to apply those insights.

 

[waterfall]

First, the renormalization group shows that theories do not have to be defined at all energies to yield great predictions at low energies. QED is such a theory.

Second, it is not true that only free fields are rigourously defined at all energies. Some nonlinear self-interacting quantum fields have been rigourously constructed in 2 and 3 dimensional spacetimes. The rigourous construction of Yang-Mills theory in 4 dimensional spacetime is thought possible because of asymptotic freedom, but it is still being researched. In fact, the Clay Institute is offering a prize of $1 million for a rigourous construction of Yang Mills theory and a demonstration that it has a mass gap.

http://www.claymath.org/millennium/Yang-Mills_Theory/
http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf

Fock space is derived from Hilbert space which is derived from the Schrodeinger Equations. We know there are other candidate equations or formulations like Matrix Mechanics and Path Integral Approach for example (although I know they are identical in essence). If we were to return to the early 1900s. What kind of math must happen or approach for the quantum fields to be completely interacting? Or is there none at all? Why? Or maybe Fock Space/Hilbert space is just too coarse for it. Also I wonder if this has to do with quantum interpretations. If we can somehow distinguish the right interpretation, would it make the fields become naturally interacting?

 

[friend]

Yes, LQG is a work in progress. The task of formulating the Standard Model interactions in an LQG geometry is not ready for prime time yet. And, even if your ultimate goal is to formulate a GUT within an LQG geometry, there is not point in even trying until you accomplish this prerequisite task. You have to hope that the progress you make in formulating Standard Model interactions in an LQG geometry will point you in the right direction regarding what to do next.

I have to wonder how the symmetries of the standard model would fit into the LQG scheme of things. How does the U(1)XSU(2)XSU(3) symmetry fit into the descrete spacetime of LQG? I thought these symmetries were continuous. Would discrete spacetime destroy them?

 

[waterfall]

"- it's a theory about gravity only; full inclusion of matter is still missing
- it's by no means a theory aiming for unification
. . .

But I want to mention what my perspective is on your "certainly not the alternative" phrase.

LQG could well be on the correct path to the alternative even though it is itself not the last step nor does try to be.

The LQG program may be *on the path to unification" because it strives for a new (no-prior-geometry) representation of spacetime. One which takes into account how geometry responds to measurement and interacts with matter."

To restate the point Marcus is making in a slightly different way, there is active research underway regarding how existing Standard Model interactions can be formulated in the context of a LQG space-time that seem to be making more progress than efforts express Standard Model interactions in a manner consistent with General Relativity rather than just Special Relativity. LQG efforts are proving more fruitful because one of the main the difficulties in integrating Standard Model interaction with General Relativity flow directly from the simultaneous assumptions that space-time is continuous (from GR) and that fundamental fermions and bosons are point-like (from the Standard Model) which naturally implies that every single particle is a singularity and isn't easily fixed. Point-like Standard Model fundamental fermions and bosons cause far less mischief in LQG because the discreteness of space-time discourages singularities from arising.

Thus, LQG is a promising possible way to formulate the Standard Model and GR in ways that are at least theoretically capable of being consistent with each other at the scale of individual particles.

Moreover, if you can formulate Standard Model interactions in an LQG space-time, then a "Theory of Everything" in which all four fundamental forces are different manifestations of a single deeper form of boson mediated particle interaction becomes a category error. If space-time has the character that LQG tries to model it as then gravity since it flows from the nature of space-time itself, is fundamentally different in kind than the other three fundamental forces. Hence, if you can formulate a mere GUT within an LQG space-time, you have taken the reductionist agenda for fundamental physics as far as it can go.

The genius of this research program is that string theory needs more than four dimensions of space-time almost entirely to accommodate the need to make gravity weak relative to the other three fundamental forces which by themselves can be formulated quite workably in a four dimensional space time. Indeed, the whole braneworld concept, in which string theorists conceive of a world where everything but gravity is confined to a four dimensional brane which is embedded in a larger dimensional space, illustrates how unnatural it is to try to treat gravity and the other three fundamental forces of the Standard Model as different versions of the same thing. In contrast, in LQG, an emergently four dimensional space-time arises naturally, and since you should need extra dimensions to formulate Standard Model equations designed for four dimensional Minkowski space into emergently four dimensional LQG space, you have eliminated the need to deal with unnatural extra dimensions from the get go.

A deep hate for point-like particles is deeply embedded in the very DNA of GR equations formulated in continous, perfectly local, space-time, and the Standard Model is equations are very deeply wedded to having point-like particles, which is what has made the marriage of the two so difficult to navigate. But, neither GR nor the Standard Model are all that deeply wedded to a continuous space-time. Continuity is an assumption thrown into the mix mostly for mathematical convenience and the heuristic that drives calculus itself is the notion that it is possible to exquisitely accurately (and indeed beyond accuracy to the point of a Platonic ideal) model sums of infinitessimal quantities as continuous quantities. LQG simply steps back off this assumption used for mathematical convenience to go to the source of the heuristic and computes quantities the hard way rather than analytically. Indeed, we routinely use discrete numerical methods to approximate Standard Model interactions by themselves, and to approximate GR by itself.

Yes, LQG is a work in progress. The task of formulating the Standard Model interactions in an LQG geometry is not ready for prime time yet. And, even if your ultimate goal is to formulate a GUT within an LQG geometry, there is not point in even trying until you accomplish this prerequisite task. You have to hope that the progress you make in formulating Standard Model interactions in an LQG geometry will point you in the right direction regarding what to do next.

But, once you can accomplish the more doable seeming task of formulating Standard Model interations in an LQG geometry, even the goal of then integrating those interactions into a GUT is less pressing. One of the reasons a Theory of Everything is such an alluring Holy Grail is that this would solve the current mess in which are two most perfect and wonderful theories of fundamental physics (the Standard Model and GR) are theoretically inconsistent with each other in obvious ways, despite the fact that both in their own domains repeatedly manage to describe experimental data with exquisite precision. But, if you formulate Standard Model interactions in an LQG geometry, that isn't a problem that needs to be solved by devising a Theory of Everything any more.

With that problem already solved, the only real problem left to tackle by taking the reformulated Standard Model interactions and integrating them into a GUT is to develop better insight about the physics of extremely high energy systems approaching Big Bang conditions. And, even those problems may not be as acute for the Standard Model interactions embedded in LQG, because the LQG geometry is already going to give rise by its very nature to some subtle high energy system modifications to Standard Model predictions because it integrates GR effects that have only been considered in an ad hoc non-rigorous way to date, for example, in asymptotic gravity driven predictions about Higgs mass and clever suppositions formulated just so necessary to predict Hawking radiation. For example, Standard model interactions embedded in LQG are very likely to have a lower unification energy scale than SUSY or SUSY inspired theories, and the natural UV bounds in LQG also tame a lot of the rigor concerns associated with commonly used renormalization methods.

You probably would still need to make a few leaps of insight to get a GUT embedded within an LQG geometry rather than merely the Standard Model embedded within an LQG geometry, but there is very good reason to think that those leaps of insight would have to be more modest in the LQG context than in the continuous space context, because there are fewer problems less to resolve. Once you've got the Standard Model embedded within LQG, pretty much all you need to do is to come up with a way to describe each of the twelve fundamental fermions (three generations each of two kinds of quarks, one kind of charged lepton and one kind of neutrino; if one does not count variations in the color charge/matter-antimatter/parity directions as different fermions), and each of the twelve Standard Model bosons (photon, 2 Ws and a Z, and eight gluons, ignoring any other possible variations on these bosons) as manifestations of one more fundamental thing. Turning one thing into twenty-four things with variations that have the right properties is tough, but not nearly as Herculean as the task currently facing people trying to devise a TOE via string theory.

I wouldn't be at all surprised to see insight developed in one or another prior GUT (perhaps even the original SU(5) GUTs) that failed and developed pathologies when formulated in Minkowski space, apply quite directly to a Standard Model embedded in LQG geometry that would somehow miraculously resolve the pathologies that arose in previous attempts to apply those insights.

I didn't focus on LQG years ago because the spin networks couldn't even approximate General Relativity in the classical limit. I wonder if it is more optimistic now or still the same problem. If anyone has references concerning updates of how to make LQG approximate the manifold of GR. Let me know.

 

[atyy]

Fock space is derived from Hilbert space which is derived from the Schrodeinger Equations. We know there are other candidate equations or formulations like Matrix Mechanics and Path Integral Approach for example (although I know they are identical in essence). If we were to return to the early 1900s. What kind of math must happen or approach for the quantum fields to be completely interacting? Or is there none at all? Why? Or maybe Fock Space/Hilbert space is just too coarse for it. Also I wonder if this has to do with quantum interpretations. If we can somehow distinguish the right interpretation, would it make the fields become naturally interacting?

I don't know.

Maybe you could take a look at the known successful constructions of interacting quantum fields referenced in Jaffe and Witten's http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf.

The rigourous construction of quantum Yang-Mills theory remains http://www.claymath.org/millennium/Yang-Mills_Theory/. Douglas describes the http://www.claymath.org/millennium/Yang-Mills_Theory/ym2.pdf.

Gupta's Introduction to Lattice QCD could also be helpful. Gupta references a proof of Osterwalder-Schrader reflection positivity which is a condition for the rigourous construction of quantum fields, including an appropriate Hilbert space. A description of all the Osterwalder-Schrader conditions, and why they lead to a rigourous construction of quantum field theory, is found in Glimm and Jaffe's book.

 

[waterfall]

I don't know.

Maybe you could take a look at the known successful constructions of interacting quantum fields referenced in Jaffe and Witten's http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf.

The rigourous construction of quantum Yang-Mills theory remains http://www.claymath.org/millennium/Yang-Mills_Theory/. Douglas describes the http://www.claymath.org/millennium/Yang-Mills_Theory/ym2.pdf.

Gupta's Introduction to Lattice QCD could also be helpful. Gupta references a proof of Osterwalder-Schrader reflection positivity which is a condition for the rigourous construction of quantum fields, including an appropriate Hilbert space. A description of all the Osterwalder-Schrader conditions, and why they lead to a rigourous construction of quantum field theory, is found in Glimm and Jaffe's book.

Ok. I'll take a look at them. For an hour now. I kept looking for Peter Woit "Not Even Wrong" in my attic and I can't seem to find it. The book is deeply mathematical and when I read it years ago. It was hard to follow, but now I'll try it again.. it mentioned about S-matrix and other mathematical techniques that I'm sure others who haven't read with more mathematical background can appreciate it more.

Anyway. I think there are two ways of looking at it. If space is discrete, etc. then it is the physical description that is not complete, this means the math ultraviolet divergence for example is due to the physical problem and not the math. But if space is continuous, then it is the math limitation. I think people will agree to this catogorizing...

 

[marcus]

I didn't focus on LQG years ago because the spin networks couldn't even approximate General Relativity in the classical limit. I wonder if it is more optimistic now or still the same problem. If anyone has references concerning updates of how to make LQG approximate the manifold of GR. Let me know.

A recent paper:
http://arxiv.org/abs/1201.2187
A spin-foam vertex amplitude with the correct semiclassical limit
Jonathan Engle
(Submitted on 10 Jan 2012)
Spin-foam models are hoped to provide a dynamics for loop quantum gravity. All 4-d spin-foam models of gravity start from the Plebanski formulation, in which gravity is recovered from a topological field theory, BF theory, by the imposition of constraints, which, however, select not only the gravitational sector, but also unphysical sectors. We show that this is the root cause for terms beyond the required Feynman-prescribed exponential of i times the action in the semiclassical limit of the EPRL spin-foam vertex. By quantizing a condition isolating the gravitational sector, we modify the EPRL vertex, yielding what we call the proper EPRL vertex amplitude. This provides at last a vertex amplitude for loop quantum gravity with the correct semiclassical limit.
Comments: 4 pages

Some other recent papers:
https://www.physicsforums.com/showthread.php?p=3755045#post3755045

 

[atyy]

Ok. I'll take a look at them. For an hour now. I kept looking for Peter Woit "Not Even Wrong" in my attic and I can't seem to find it. The book is deeply mathematical and when I read it years ago. It was hard to follow, but now I'll try it again.. it mentioned about S-matrix and other mathematical techniques that I'm sure others who haven't read with more mathematical background can appreciate it more.

Anyway. I think there are two ways of looking at it. If space is discrete, etc. then it is the physical description that is not complete, this means the math ultraviolet divergence for example is due to the physical problem and not the math. But if space is continuous, then it is the math limitation. I think people will agree to this catogorizing...

The big problem is gravity which is perturbatively not UV renormalizable. The Wilson-Kadanoff picture of renormalization as a way of seeing how a theory looks like at low energies points to two different approaches. The first is that the theory is incomplete, and new degrees of freedom enter - this is the approach of string theory. The second is that the theory could be UV complete if the renormalization flow is non-perturbatively reversed to high energies - this approach is called Asymptotic Safety.

 

[marcus]

...- this approach is called Asymptotic Safety.

If the FGZ program is successful then LQG could turn out to be in effect a background-independent form of Asymptotic Safety.

The FGZ idea is to reformulate classical GR using graphs to truncate GR to finitely many degrees of freedom. So "loop classical gravity" LCG would simply be an alternative formulation of GR. This seems to capture the kinetics of GR. What remains to examine is dynamics.

For sure you know the paper Atyy, anyone not familiar with it can google "freidel geiller ziprick" and get http://adsabs.harvard.edu/abs/2011arXiv1110.4833F