149. Towers of vacua in SUSY field theories
ASTRO, HEP-TH/PH — By Dmitry Podolsky on December 21, 2008 at 10:10 pm
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Reading the previous post about dynamical RG treatment of the fractal surface growth problem some of you may have recalled that several months ago we (me, Niko Jokela and Jaydeep Majumder) have used dynamical RG methods to describe behavior of eternal inflation on a nearly continuous landscape. Later, I have also applied these methods to study eternal stochastic inflation in a random potential.
What does it mean that the landscape of vacua is nearly continuous? Imagine first that it only contains a large number of degenerate vacua 
. The rates of tunneling between them are equal, as well as the probabilities 
for an observer to find herself living in a one particular vacuum among them. If you write down the corresponding Fokker-Planck equation describing the evolution of , you will see that the late time asymptotics of its solution is given by a Bloch wave (indeed, consider a counterpart of the Fokker-Planck equation – a Schroedinger equation in the periodic potential; it is well known that the ground state is represented as a Bloch wave).
However, one can ask two questions – (1) can very large subsets of degenerate vacua appear on a realistic landscape and (2) if they can, how important are they for the eternal inflation? For example, dynamics of eternal inflation on the Bousso-Polchinski landscape is essentially defined by transitions from higher enery vacua to lower energy vacua, although large subsets of degenerate vacua also exist there.
The answer to the first question is provided by the latest paper by Dienes and Thomas, who give an explicit realistic example of the large landscape of degenerate vacua. Let us consider an N-site orbifold Abelian moose consisting of N 
gauge groups with a common coupling 
and N+1 chiral superfields 
. We will also introduce a kinetic mixing among the different factors of the theory such that the mixing occurs only between nearest neighbors of the moose and is determined by a single parameter 
.
(To get more technical, we first introduce a general superpotential for our theory in the form
.
Then, the gauge kinetic part of the Lagrangian is given by
,
where 
is the mixing matrix. We say that the only non-zero elements of it are diagonal and next-to-diagonal.)
There are N-1 vacua in the system separated by saddle-point solutions in the field space. The vacuum energies of the system are found to be
.
As we see, the case 
is special - all vacua are degenerate. If we now build an inflationary model in such a setup, we will find behavior of the type discussed above in the regime of eternal inflation.
The answer to the question (2) for this particular setup is that there are simply no other vacua, so Bloch waves give complete description of the vacuum dynamics. Of course, one can say that the kinetic mixing is non-symmetric or other non-diagonal elements in the mixing matrix are important. I guess, in the latter case the basis of Bloch waves is still sufficiently nice to describe the system in the first approximation (we can say that higher order non-diagonal kinetic mixing is small), and how order non-diagonal mixing should introduce some kind of scattering for the Bloch waves. In particular, randomly placed non-zero mixing matrix elements would correspond to a disordered potential for the Bloch waves to scatter on – the situation resembling the one discussed in our paper with Niko and Jaydeep.
18 Comments
I am pretty much convinced that the Bloch waves or linear superpositions of different vacua is not a legitimate notion, and if it is, the dynamics of such superpositions are vastly different from the naive Bloch example.
Dienes et al. – or someone like that – have also argued that our vacuum is a superposition of states from many similar superselection sectors. That’s even more unlikely because any single measurement – of a particle’s mass or the cosmological constant (or all of them) – should “collapse” the wave function to a single superselection sector.
In cosmology with finite volumes, there are really no superselection sectors because the orthogonality of states in different sectors is only obtained when the volume is infinite. However, the same finiteness of the volume also prevents us from the normal ways to calculate the transitions and even from discussing the inner product on the whole “cosmological” Hilbert space.
I am completely confused by another simple aspect of the Dienes-Thomas paper: they seem to intend to say something about cosmology and tunneling to be used in inflation, or at least you do
, but they discuss non-gravitational field theories. Are we supposed to believe that some classes of vacua in the landscape almost “decouple” not only from gravity but also from all vacua with different values of the C.C. or from any other nonperturbative signs of gravity? OK, I don’t believe it and I don’t think that they have offered reasons to believe this assumption.
On the other hand, if the paper studies purely non-gravitational field theories only, I have no idea how the results can be relevant for self-confident applications such as the stringy landscape of the C.C. problem.
Dear Lubos
True, the measurement of the cosmological constant will lead to the collapse of the wave function. Not sure about electron though and that’s why… you have a tunneling Hamiltonian, and in order to find correct degrees of freedom you need to diagonalize it. The spectrum of the diagonalized Hamiltonian will be interpreted in terms of quasiparticles. Are you sure your electrons are fundamental excitations and not quasiparticles?
Ok, there is collapse in Dienes-Thomas setup, so what if the probability to measure a given value of the cosmological constant is determined by a Bloch wave-like wave function? (Well, not exactly – if you take instantons into account, the degeneracy of vacua will be removed.)
Anyway, in the setup I always keep in mind there is even no wave function collapse. I was talking about Fokker-Planck eq. describing the evolution of eternal inflation in the setup with multiple vacua. If vacua are degenerate, then the general solution of the F.-P. can be surely represented in terms of Bloch waves.
I don’t think that is what they want to say. Let us send M_P to infinity in the very first approximation. You find that the Dienes-Thomas system has a large number of degenerate vacua with positive vacuum energies. What is going to happen if I make M_P finite but very large? Inflation would start, I guess. Isn’t that in a sense what KKLT do? (Note that all they show is that the vacuum energy is uplifted to a positive value, but no actual 4d inflation in the sense of exponentially changing metrics is seen.)
So, what exactly makes you so unconfortable with this setup, corrections to the potential?
Cheers,
Dmitry.
Dear Dmitry, I don’t think that there is any physical difference between what you call “fundamental particles” and “quasiparticles”. Both of them can appear as physical particles and poles in scattering amplitudes of others. And in many cases, both of them may even occur as elementary fields in a description. In different descriptions related by dualities, there are different “fundamental particles”, so this distinction only refers to a particular weakly coupled limit.
At any rate, I am not getting your point of raising this distinction because the “collapse” of the wave function occurs even if you measure properties of “quasiparticles” (or any other objects). Again, I am not arguing with the measurement of the C.C. itself. I am saying that it is an inconsistent picture for the measurement of anything else. The particle masses are given by the second derivative of the potential, so to say, so they (and 99.9999% of physics) will depend on the point in the landscape and you simply can’t make any macroscopic superpositions (unless the weight of almost anyone except for one vacuum is virtually zero) because that would be just like the mixed Schrodinger cat (worse, in fact, because astronomical in size).
Concerning your “surely” comment, I am not disputing that in some artificial, mathematically masturbational toy model, XY is expressed as a superposition, whatever XY is. Instead, I am giving you a simple yet indisputable proof that such a picture cannot have anything to do with a Universe that at least remotely resembles ours. We’re measuring things like particle masses all the time which is why, assuming that the masses of a “particular” particle species has many eigenvalues (which itself sounds very bizarre because particle species in different Universes are usually not mapped to one another across the Universe in any canonical fashion), we must live in an eigenstate of a particle mass which means that the superposition to start with is now gone, anyway.
Your last paragraph: I am also uncomfortable with the attempt to talk about tunneling, yet ignoring gravity and differences in the C.C. The papers starting from KKLT surely don’t do this mistake. In fact, my criticism against the Dienes et al. picture is exactly that it disagrees with KKLT-like features of the tunneling. In KKLT, the typical tunneling dramatically changes the value of the C.C. In the eternal inflation, the C.C. jumps frantically. You (almost) never tunnel into something that has the same C.C. That’s why you cannot neglect gravity because it dramatically changes during the tunneling, and you need the gravitational Coleman-DeLuccia instantons instead of the field-theoretical Coleman tunneling only.
Dear Lubos
Well, the fundamental difference between, say, electrons and Cooper pairs is their characteristic size
For a weakly coupled superconductor characteristic size of a Cooper pair is much larger than the typical distance between electrons (mean free path).
Let me take this analogy with superconductor further in order to understand what you want to say
You want to focus on physics in the vicinity of a single atom (that is your vacuum, well, at least – potential well), namely, you seat directly on the nuclei 
and claim that my correct elementary excitations are related to the energy levels of the atom.
You want to measure a) a momentum of a single electron or b) its coordinate (we cannot measure both simultaneously, it is clear). You say, after the measurement the wave function of electron will collapse, and the corresponding Cooper pair the electron belonged to will be broken (we loose coherence, at least, at astronomical scales!). The question is – how don’t these Cooper pairs get constantly broken – they scatter on each other, on phonons etc.?
Wait, don’t leave yet, let us masturbate together for a bit. Let us consider a very abstract model probably having nothing to do with objective reality – a QM of 1-dim particle in double well potential (two degenerate vacua). Let us measure the energy levels of the particle. Are you saying that my particle has no idea about the second minimum when I measure its mass near the first minimum?
From your comment above I do not see how the picture will dramatically change when you switch the gravity on. The tunneling rates will change – fine. Without gravity you had a system with multiple degenerate vacua. Taking instantons into account slightly broke the degeneracy.
What is going to happen if you turn the gravity on?
Clearly, you’ll get the same – multiple slightly non-degenerate vacua.
Cheers,
Dmitry.
Sorry, Dmitry, but completely everything you write above seems to be completely wrong. First of all, I was talking about the measurement of the mass. For example, measure the momentum and the energy and calculate E squared minus p squared, to get m squared. Or calculate the energy in the rest frame, directly.
I was never talking about any measurement of position, so the relevance of all your comments about the size are exactly zero. But even if I did need any measurements of positions, I have absolutely no idea why the slightly larger size of Cooper pairs relatively to electrons would play any role. That could only play role for some detailed quantitative questions – like the speed of decoherence – but certainly not for the very qualitative fact that observables that are sufficiently accurately measured collapse into an eigenstate.
It really seems that you believe that there is some God-given qualitative difference between “elementary particles” and “quasiparticles”, that they come in classes. In my example, they don’t, but in the rest of physics, they don’t either. Everything you say about the qualitatively different behavior of these two types of particles is a deep misunderstanding. Physics doesn’t work like that. All particles in physics work on qualitatively identical footing and all differences are quantitative only.
Your breaking of Cooper pairs is yet another unphysical addition to this story. What does “breaking of Cooper pairs” have to do with the measurement of a particle’s mass, energy, momentum (or even position)? If it is a legitimate approximation to say that a particle exists, and electrons surely do, then it is equally necessary to say that they don’t break throughout their existence. Sure, the more unstable a particle is, the greater the width, and the less the particle “exists” as a real, lasting object.
For some low-energy questions, atoms or Cooper pairs are better degrees of freedom, but for high-energy physics questions, it is the electrons and quarks that matter and whether they choose to become a part of a bound state is up to their freedom. But all these objects – atoms, Cooper pairs, electrons – are particles, so at very low energies, they behave like particles.
The stability is why I chose electrons that are completely stable. But I could have done comparably well with decaying particles, too. Their masses are still sharp enough for lasting superpositions of universes with different values to be unsustainable. You know, by measuring electron’s mass to be 511 keV, we are not measuring some central value of a distribution that goes from 200 to 800 keV. Instead, we are showing that all electrons in the Universe have 511 keV plus minus a supertiny error. All vacua where the number would differ substantially are eliminated, their amplitude drops to zero (and dropped many times because we have measured the e-mass zillion times directly or indirectly).
Concerning the masturbation paragraph. Of course. If you only measure the counterpart of “mass” in your QM model, which is the force gradient (one number!) acting on the particle, a particle near the first minimum can have no clue about the second minimum. In fact, much stronger statement holds. The whole perturbative expansion of your QM model – whatever you quantity you ask about – around the first minimum is completely unaffected by the second minimum! This is a well-known thing that new minima away from the chosen vacuum only have nonperturbative influence on physics near the first vacuum. That holds even in QM when there are no superselection sectors. So yes, you seem to be deeply misled about the effect of distant minima, too. It is very tiny.
Finally. No, gravity and the C.C. change the eternal inflation dramatically because most tunneling is to other vacua with the C.C. differing by a huge amount of order Planck density, and it is only a matter of coincidences that one ends up with a C.C. near zero, like in our Universe. This is the whole point of “randomness” in their anthropic reasoning. There is huge traffic between highly dS vacua and the nearly flat ones and slightly AdS ones. It’s just completely wrong to imagine that most of the traffic in a sector of the landscape is between nearly flat vacua. The vacua with a small C.C. are very “rare” and they are very distant from each other on the configuration space.
The C.C. gap between two connected vacua can be slightly smaller than the Planck scale if there are some parameters but there is certainly not sublandscape of KKLT’s just “slightly nondegenerate vacua” that you fantasize about. Everything you write about these issues is wrong, too.
Dear Lubos
The fact that you used the word “completely” twice in the same sentence shows that you are deeply distracted
Yes, you were talking about measurement of position – you want (and only can) to measure electron mass in a given Hubble patch, the one you live in.
“Slightly” larger?? Lubos, you are clearly in need to reread something basic about superconductivity
Also, compare the size of Cooper pair with the lattice scale (distance between atoms – for the landscape counterpart picture this is the distance between vacua).
My example was supposed to show that correct eigenstates of the low energy physics are Cooper pairs and not electrons, so wave function collapse in a superconductor due to scattering, interactions etc. happens also into egeinstates-Cooper pairs. Even simpler – if I only probe Cooper pairs with other Cooper pairs, then ultimate outcome of my measurements is nothing but… right, Cooper pairs
, states with coherence length much larger than the characteristic distance between the atoms.
If I go back to the landscape picture, correct excitations can be very well those with wavefunction that knows about other vacua, too (its width is much larger than the distance between different vacua – as in the case of Cooper pairs).
The point where you make mistake is when you suppose that you measure electron mass, spin etc. etc. with some ultimate tool. However, every tool you can use for such measurement is made of the same electrons, protons, etc.
And what if the field theory at 1 TeV is also an effective field theory? Your appealing to the width of the distribution of electron mass is misleading. If you measure the mass of the Cooper pair, you’ll get 2m_e, but it will hardly make its wavefunction localized near some particular atom (vacuum of the effective potential that Cooper pairs are moving in).
Yeah, sure
Well, anyway, in conclusion, the spectrum knows about other vacua, too. Although the splitting of energy levels due to tunneling is extremely small (as you indicate), the number of vacua is extremely large. In particular, this fact makes the spectrum of excitations in solid state media nearly continuous.
If you allow me a little hint to bring discussion to the original topic – and what if only degenerate vacua are present in the setup like in Dienes-Thomas’ one?
If you repeat it 10^250 times more, you will feel much easier
Sorry, have to leave now.
Cheers,
Dmitry.
Dear Lubos
It looks like Dienes/Thomas are not going to come although I invited them. Hoping that it will bring more people into the discussion, I tried to defend their point of view as I imagined they would do it, but ultimately it is a weak argumentation and a flowed setup, so I don’t see why I should continue defending it
Indeed, any single measurement of any single quantity in our Hubble patch should lead to the wave function collapse and breakdown of interesting superpositions even if they were initially there. Note that this argument is perfectly also for the setup used by Henry Tye or by Mersini-Houghton with collaborators (the latter studies WdW in a non-trivial potential with large number of minima).
Somehow, you preferred to ignore the only physically interesting statement I made – that I am actually dealing with inflationary Fokker-Planck and not with WdW. The latter described the probability P to measure given values of interesting scalars (inflaton, effective cosmological constant, moduli) in a given Hubble patch. The probability P is time dependent, and due to constant CdL or Hawking-Moss tunneling between the vacua it remembers about the overall vacuum structure and initial conditions for eternal inflation on the landscape for a very long time.
Although time independent asymptotics of P is given by the expression for Hartle-Hawking wavefunction, it does not have much to do with “wavefunction” (and with collapse) – simply because the expectation value of the inflaton \phi is classical but stochastic quantity.
As for “dramatic change of C.C.” due to quantum tunneling, note that the tunneling downwards in the Bousso-Polchinski landscape is more probable than tunneling upwards, so the system very quickly reaches the subset of low energy minima, and in principle there may be many of them with very close energies, so the system lives for a very long time in this subset of vacua.
Cheers,
Dmitry.
Hi,Dmitry!
I was read your paper. Very interestingly. I have interest in a picture of a landscape as a graph of connected metastable vacua(weighted CY for example) too. Especially in a dynamics on such graphs. I think that this dynamics must be non equilibrium and non stationary. Therefore the linearity is a “instantaneous phenomenon” and they do not control a long time evolution. Roughly speaking: Eigenvalues are in the moduli space.
I think key factor here is a memory. This mean non Markovian dynamics(non Gamiltonian as a consequence or v.v.). Infinitesimally(ODE) this may appear as a high derivative extention of Lagrangians(Very popular in arxiv in recent). But this lead to an initial condition problem
(higer dimensional branes).
I think that the memory is entropic phenomenon.
Roughly speaking schematically: the mesure consentration-entropy decreasing-conformal factor increasing-phase transition-mesure relaxation- and so on.
Some my exploration
I interest in your expert opinion and in constructive discussion. Theme is very interesting in more context not only in a string landscape.
Thanks.
Maxim.
Hi Max
Dynamics of what? If you are talking about eternal inflation, then the distribution function achieves equilibrium if the graph is finite (and ultimately it is always finite – as is the total number of vacua).
Again, are you talking about eternal inflation? Ultimately, instead of the inflationary Fokker-Planck you should have some integro-differential equation for the probability measure, associated memory effects etc. etc. since the dynamics of the inflaton is unitary. After introducing coarse graining (CG scale of the order of the single Hubble patch size) you are left with Fokker-Planck, information loss etc.
There should me some time scale t_m associated with memory effects (probably of the order of the inverse tunneling rate or smaller) such that memory effects are important for t < t_m.
Dynamics of constructive measure and dynamics of entropy and dynamics of metrics in configuration space(conformal evolution).
I do not speak about eternal inflation in an conventional sense. Though in my paper there is an interpretation of the attractor destruction phase as an inflation(superinflation)-blow up tunneling.
I have the discrete stochastic dynamical system (Langeven type)on coarse grained space. This is vector(velocity) -covector(gradient of the environment potential) relation by means of the renormalized metrics(potential as a sequence). This metrics is a entropy functional(non decreasing function of entropy). Details in my paper.
Were is no fixed time scale in this system because of the time inhomogeneity.
In paper I estimate a time of correlation which is renormalized too.
The Main advantages are independense
“topological convergence” to flatness, independense from initial conditions and so on.
As an Algorithm this system search the most flat plases in an arbitrary landscapes(not necessary a stringy landscape). The Error functional Curvature of the neuronet for example.
Hi Max
If you respect your readers, let us go one step at a time.
What is the physical system you want to describe? If there is no particular system in your mind, what you are talking about sounds to me as Langevin in a random potential. Many people discussed this setup starting from early eighties. What is new in your work compared to, say, work of Fisher (1984)?
Cheers,
Dmitry.
Hi Dmitry
Excuse me! My respect!
Ok!
I have no less than three particular system in my mind.
//What is new in your work compared to, say, work of Fisher (1984)?//
First of all this is entropic memory.
Formal reflector mechanism of such memory is a dynamic metrics-the drift factor(multiplier of the environment potential gradient )wich depends on entropy(on all history of the trajectory).
Non formal illustration:
Let trajectory that genrtates measure in some configuration space equipped by some partition (moduli space for example)trapped by an attractor(vacuum) wich contains in a partition’s cell(or set of cells). This leads to the measure concentration and as result to growth of drift factor(decreasing of the effective attractor’s curvature)as result the destruction of attractor and transition to another one.
This seems like a renormalization and ,after more detail analysis, remind of standart renormgroup behaviour(because of the entropy’s logarithmical nature as you may see)
In my paper where are estimation of the attractors lifetime as a function of his curvature and entropy.
But more interesting in more global view-the interattractors dynamics. The system remembered the curvature of the self past exitation(in entropic form)and interprets the high energy part of spectrum as a single destroyed attractor-topological factorization(asymptotic freedom?). Here the memory emerges as the interest: System
do not stay too long there where it was recently(excuse me for my English).
This looks like the anomalous diffusion.
Therefore system drifts(non monotone) to the low energy (low curvature) part of attractors spectrum. And oscillates around the flatness.
Slogan:
No singularity
No gaussianity
Pre ergodicity
Algorithmicity:)
Cheers,
Maxim.
Hi Max
What are these systems?
Sorry for maybe too many questions, but I really want to understand what are you talking about.
That’s where you already loose me
From my point of view, memory is existence of certain correlations between snapshots of physics taken at different times, i.e., it is preservation of information. On the other hand, entropy is measure of disorder in a physical system. Integro-differential Schwinger-Dyson equation with build-in memory effects is unitary, it describes evolution of a pure state with associated entropy = 0. On the other hand, when you do coarse graining, Schwinger-Dyson eq. reduces to kinetic (or even Fokker-Planck 
) equation, which are dissipative, information is lost, and entropy is growing with time.
So, what is exactly “entropic memory”?
Cheers,
Dmitry.
Hi Dmitry
Thanks for your questions!
1. I have a computer programm: neurunet simulator “Ariadna” in wich I realized whis system as a learning algorithm.
“Ariadna” visualized the evolution of the entropy and the error(mean square variance). The entropy time series is not monotone.
The Neuronet may be considered as a universal dynamical partition, by the way! This fact excites my fantasy! My neurons:)
2. The abstact attractors may be interpreted in the more contexts. As a discrete energy spectrum in the quantum system for example.
In may paper the estimation of attractor’s lifetime had the form of the uncertainty relation(renormalized).
3. Non perturbative dynamics in the moduli space.
1 in the permanent digital experiment.
2 and 3 in progress. I think about it.
//So, what is exactly ?entropic memory?? //
You are right. The Memory may be various : associative, Boltzmann brains false memory, RAM, bad:)
I want to add the cognitive memory or the entropic memory.
The cognitivity manifests from the bifurcation relation in my paper. The relation between the attractor’s curvature and the entropy.
The trajectory to measure the attractor.
The entropic memory is a nonstationarity or the long time correlations,
when the entropy not only the effect of the dynamics but also its cause:)
My equation for the trajectory but not for probability density. The self-consistency appears from the fact that the trajectory generates the measure (pre ergodicity).
The Coarse grained partition is necessary for the definition the space measure from the time(trajectory) measure.
May be it is cause of your question.
It is very difficult to exactly explain the memory effects without the formulas:)
Cheers,
Max.
Some my exploration:
http://arxiv.org/abs/0801.3996
Hi Dmitry,
Hi Lubos
About the landscape topology.
//In particular, this fact makes the spectrum of excitations in solid state media nearly continuous.//
//There is huge traffic between highly dS vacua and the nearly flat ones and slightly AdS ones//
//…because most tunneling is to other vacua with the C.C. differing by a huge amount of order Planck density, and it is only a matter of coincidences that one ends up with a C.C. near zero, like in our Universe.//
//The vacua with a small C.C. are very ?rare? and they are very distant from each other on the configuration space.//
I think what metrical topology of the landscape must vary according to the gravitational backreaction.
This may be realized as the renormalization of the landscape(superpotential) or C.C.
In the system the trajectory-environment(TE) there is an effect of the dynamical factorization.
In TE the high energy part of the landscape(dynamical level is determined by the entropy) is considered as an one bug vacua with some C.C(relatively big).
The Topology of this island may be complicated.
The evolution of this island may be condidered as a tpajectory in homological lattice or in cohomological ring (singelarities sequence).
The trajectory dynamics on this island very fast and near random (pre ergodicity).
This “makes nearer” the vacua that are “very distant from each other on the configuration space”.
I think that this effect may by interpreted also as the asymptotic freedom.
This is the manifestation of the entropic memory.
Cheers,
Max.
P.S. What about inet? Problems with .net, .com, .org(arXiv)! Do not download. Frozen:( New Year!:)
Have you a such problem?
runet(.ru)-no problem!
iron curtain or western crisis?;)
Hi Max
It seems that West is doing fine, we don’t have any problems with internet around
Cheers and Happy New Year,
Dmitry.
I hope that this attractor has a big curvature.
Cheers and Happy New Year,
Maxim.
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