NEQNET: non-equilibrium phenomena in physics, social dynamics, markets and much more. No nonsense, just science.

Correlator of Wilson and t’Hooft loops at strong coupling in N=4 SYM theory

Posted by Andrew Zayakin

at May 22, 2009

Save This Post as PDF

Andrew Zayakin works at LMU, Munich and ITEP, Moscow. His interests include non-perturbative physics of QCD, string theory and AdS/CFT correspondence. Dmitry.

This post is about my recent paper with Alexander Gorsky and Alexander Monin about a correlator of a Wilson and a ‘t Hooft loop. Before I proceed, I should explain what these objects are and why they are important to be studied. QCD possesses a consistent description in terms of “dual variables” – charges and monopoles. Reader familiar with the systematics of particle-like solutions in different theories would stop me at this very moment by pointing out there are no monopoles in QCD. True, there are no monopoles in the sense of e.g. Georgi-Glashow model. However, effectively there is such a thing as monopole, which is widely observed on lattice as a non-zero Abelian flux through a closed lattice surface. A lot is known on “thermodynamics” and “phenomenology” of these quasiparticles. They do not exist in the sense of theory spectrum. Still, they are an important tool of describing QCD. The QCD phase transition, which is an element of common lore, can easily be understood in terms of monopoles (Fig.1).

QCD phase transition

This diagram distinguishes between a phase of condensed, strongly correlated or dominating monopoles in each particular phase of QCD. In the transition area both monopoles and charges are determining the dynamics of the theory. In general, most interesting thermodynamic properties of the theory are supposed to take place in the region of the phase transition. Therefore, after we know something about monopoles and charges per se, one would ask himself what the dynamics of their interaction is.

A typical measure of charge or monopole mutual properties are Wilson lines. In particular, they allow one to introduce quark-quark potential by studying a correlator of two straight lines, and many other useful things non-perturbatively. A reasonable guess then would be to use Wilson lines as a measure of quark-monopole interaction. Therefore, a natural idea for someone interested in QCD properties next to phase transition point would be to consider a correlator of corresponding Wilson loops. actually the loop carrying magnetic charge is also known quite well in the literature as ‘t Hooft loop.

One could actually have done this on a lattice (actually, this was how this project started due to a discussion with our lattice-oriented colleagues). This however hasn’t been accomplished so far. Perturbative calculation is out of question, for a small coupling of the charge will automatically make monopole charge large. A resummation technique a la Erickson-Semenoff-Szabo-Zarembo may apply, but that would be a long story in its own turn. So the only way to calculate something well-defined would be to use duality approach.

There have already been some posts discussing some aspects of AdS/CFT here at NEQNET, as well as more particular models of AdS/QCD, so I will just briefly remind the reader a simplest formulation of duality: maximally supersymmetric gauge theory on a 4-dimensional boundary is equivalent to IIB string theory in the $AdS_5\times S^5$ bulk, the equivalence requiring identification of currents in the boundary with bulk SUGRA fields’ limit towards boundary. When we calculate a Wilson or ‘t Hooft loop on the boundary, its AdS counterpart is a string two-dimensional surface in bulk, whose one-dimensional boundary is the contour of the loop. This recipe will be used by us as well. To find a correlator of a Wilson and a ‘t Hooft loop we must consider a configuration in AdS with two different boundaries, carrying electric and magnetic charges correspondingly.

Charge conservation would not allow to connect directly an electric string to a magnetic one. Thus an intermediate dyonic string world-surface emerges. Happily, there exists a vertex “charge, monopole, dyon”, or, more generically, $\{(p,q),(p-1,q),(1,0)\}$ vertex, which relates more generic dyons with each other. This vertex will allow the connection between electric, magnetic and dyonic string surfaces.

The question on whether monopole and charge degrees of freedom are mutually strongly correlated or not acquires a simple and beautiful explanation. Dependent on conditions which I discuss below, either configuration shown in Fig. 2

Charge and monopole correlated

may be realized. The configuration on the left takes the advantage of the dyon vertex. The configuration on the left consists of free, non-correlated charge and monopole. Actually they are not absolutely non-interacting, since they are capable of exchanging SUGRA modes. Still, this is already at PT level and is not of foremost interest to us.

The choice between the two configurations is done dynamically, dependent on which of the two actions is the lower. Contribution by the other configuration to the partition function is then exponentially suppressed. The parameters on which the action depends are temperature, internal radius and external radius. I show in Fig. 3 the phase transition between the connected and disconnected phases dependent on radii ratio. Denoting them in the Figure as “interacting” and “inert” I mean the absence of non-perturbative correlation between the particles; they may still interact via exchange of supergravity modes.

Phase transition

The meaning of this transition is simple and physically quite clear: only when monopoles and charges are sufficiently close, they are non-perturbatively mutually strongly correlated. At low temperatures, this limiting ratio of radii is approximately 0.6. When R_1/R_2<.6 , the loops are already uncorrelated. However, at high temperatures the critical ratio falls like 1/T , therefore, monopoles and charges may become correlated.

We have shown some temperature results, although we never said where the temperature came from. For that purpose the pure AdS space is generalized towards either a t-direction compactified version of the latter or a black-hole-in-AdS background. The transition between the two is known as Hawking-Page transition. Although we are far away from normal QCD with fundamental fermions, this phase transition maybe in some or other way related to QCD phase transition at zero chemical potential.

The results for the phase transition may lead us to a speculation on thermal behaviour of QCD, remembering again the caution that should be taken when extrapolating SUSY results upon a non-supersymmetric theory. We however remember here the standard lore that at finite temperature QCD and SUSY are closer in their dynamics, thus we hope that these results are actually more useful in a non-SUSY context than what one might think from the first sight.

The other issue we discuss in our paper, related to the existence of the vertex and different configurations of monopole/charge/dyon worldsheets is the generalization of the entanglement entropy.

This notion has been introduced long time ago but attracted a lot of attention recently because of its effective derivation in the holographic picture. Roughly speaking if we have a set of regions
divided by boundaries than the entanglement entropy is defined as the entropy seen by an observer in a region who does not communicate with the other regions. In the simplest case one has two regions A and B and introduce the vacuum density matrix $\rho_0=|0><0|$ . Then the reduced density matrix $\rho_A=Tr_{B} \rho_0$ defines the entanglement entropy $S_A=- Tr_{A} \rho_{A} log\rho_a$ . The entanglement entropy is generically UV divergent but the UV divergent part of the entropy does not depend on the size of the region hence the finite -dependent contribution to the entanglement entropy can be, for instance, safely defined as the difference of the entropies at two different L_1 and L_2 .

The multicomponent regions has been investigated as well and the following generalization has been suggested, inspired by the one-dimensional case

$S(X_1\cup X_2.. X_p)$ $=\sum_{i,j} S_{(|a_i -b_j|)}-\sum_{i<j} S_{(|a_i -a_j|)}-\sum_{i<j} S_{(|b_i-b_j|)},$

where is the entropy of the single component and a_i and b_i are the right and left boundaries of the -th component. An important question concerns the property of the strong subadditivity $S_A + S_B \ge S_{A\bigcup B} +S_{A\bigcap B},$ which has been proven in holographic picture by Hirata. Another interesting feature of the system to study is the extensive mutual information

$I(A,B\cup C)=I(A,C) +I(A,C),$

where

$I(A,B)=S(A) +S(B) -S(A\cup B)$ .

It was argued in literature that the extensivity does not generically hold which is triggered by nonvanishing tripartite information function

$I(A,B,C)=I(A,B) +I(A,C) -I(A,B\cup C).$

The holographic calculation of the entanglement entropy is practically identical to the Wilson loop calculation hence our mixed correlators suggest the natural generalization of the entanglement entropy when the charges (p_i,q_i) are attributed to each boundary. That is, the entropy function for each interval takes values in $SL(2,Z)\otimes SL(2,Z)$ lattice and has the following structure

$S_i=S_{(p_i,q_i)}^{(p_{i+1}, q_{i+1})}$

for the -th interval. In the conformal case the calculation of the generalized entropy corresponds to the calculation of the partition function with nontrivial boundary conditions. One can define the generalized entropy by summing over the all boundary charges or introducing a kind of boundary chemical potentials for different charges.

Our recipe for the holographic calculation of the generalized entanglement entropy is very transparent. One has just to calculate of the area of the composite minimal surface as the function of the geometrical characteristics. Since all boundaries generically have (p,q) electric and magnetic charges, the corresponding boundary contour has to be a boundary of the (p,q) string worldsheet. Such connected composite surfaces may exist or not depending on the geometry of the boundary regions. Similar to canonical entanglement entropy, charged entanglement entropy is UV divergent but the UV divergent part is independent of the geometrical factors.

A natural question concerns the properties of the generalized entropy. The first one to be mentioned is the strong subadditivity which can be simply tested in the holographic picture. A comparison of the corresponding area indicates that for the simplest (0,1)-(1,0) correlator this property is satisfied, however, the analysis of the multiple (p_i,q_i) loop correlators deserves a special consideration. The most interesting question related to the generalized entanglement entropy concerns its modular properties. Indeed, when we have a correlator of multiple dyonic loops, it takes values in $SL(2,Z)^{\otimes k}$ with some integer and it would be very interesting to investigate the action of the -duality group on it, which could be related to the deconfinement phase transition.

Certainly there are other questions to come after this work. It would be interesting to recognize the phase transition in terms of the summation of the perturbative series in the spirit of Zarembo. However in the case under consideration the perturbation analysis is more involved since the interactions between electric and magnetic objects have to be summed up.

One of the most interesting questions concerns the action of the S-duality group on the generic correlators of the dyonic (p,q) loops. A generic correlator of (p_1,q_1) , (p_2,q_2) dyonic loops has to possess interesting properties under the action of $SL(2,Z)\otimes SL(2,Z)$ group. In particular, it would be interesting to investigate the modular properties of phase transition points of dyonic loop correlator.

Calculation of a correlator of several nonlocal observables has a lot in common with the calculation of the entanglement entropy. Our calculation suggests the natural generalization of the entanglement entropy notion to the case when the boundaries of the regions are charged under the -duality group. That is, generically the generalized entanglement entropy for the region with boundaries takes values in the group tensor product $SL(2,Z)^{\otimes k}$ . Since the entanglement entropy at strong coupling is similar to the Bekenstein-Hawking black hole entropy the generalized entanglement entropy can be considered as an analogue of charged black hole entropy. We plan to discuss these issues elsewhere.

Journal club, Master Postdoc, Quantum field theory, Quark confinement, String theory

106. Criteria for confinement. Wilson loop – getting more technical

85. Hard thermal loops: what is it?

126. From quarks to strings. Migdal-Makeenko equation and AdS-CFT correspondence

155. Witten explains how to quantize gauge theory

String theory and the diffusion equation

Posted by Gianluca Calcagni

at May 21, 2009

Print This Post

Save This Post as PDF

Gianluca Calcagni is a postdoc at Penn State working in the group of Martin Bojowald. His interests include string theory, string field theory and cosmology. Dmitry.

This post is based on arXiv:0904.3744, in collaboration with Giuseppe Nardelli. Check the links for references and introductory reviews on the subject.

A question. The prototype of instanton in local scalar theories is the classical Euclidean solution (a hyperbolic tangent) for a double-well potential, $-V(\Phi)\sim \Phi^2-\Phi^4$ . The study of instantonic solutions is an essential tool to understand the vacuum structure of the corresponding Lorentzian theory with the potential upside down. The same problem is neither trivial nor of mere academic interest as far as nonlocal theories (i.e., with an infinite number of derivatives) are concerned.

A simple example of a nonlocal scalar with a static double-well potential is provided by the tachyon of open string field theory (OSFT). Nice reviews on OSFT were written by Ohmori, Sen, and Fuchs & Kroyter. The effective lowest-level Euclidean equation of motion for the tachyon of the polynomial superstring field theory is

$(\partial_t^2-m^2)e^{-r\partial_t^2}\Phi=\sigma\Phi\,e^{r\partial_t^2}\Phi^2\,.\hspace{5.3cm} (1)$

This equation is highly nonlocal and it is not obvious how to solve it. For instance, an expansion of the operator $e^{r\partial_t^2}\approx 1+r\partial_t^2+\dots$ would not do, since theories with higher-order derivatives are physically inequivalent to nonlocal theories (on general grounds, the former have ghosts, the latter have not). These “perturbative” solutions have a limited range of validity and by no means include all possible solutions. For this reason, equation (1) has never been solved nonperturbatively, although it admits an oscillatory solution corresponding to a brane with marginal deformations. This class of solutions plays a major role in string field theory, as they describe the initial or final stage of tachyon condensation.

An answer. Eventually, we have been able to find an approximate solution of equation (1), namely, the error function

$\textrm{erf}\left(\frac{t}{\sqrt{4r}}\right) \equiv \frac{2}{\sqrt{\pi}} \int_0^{t/\sqrt{4r}} d \tau\, e^{-\tau^2}\,.$

The global accuracy of this solution is between 0.9% and 1.5%, depending on some details, and it can be estimated via two duplication formulae for incomplete gamma functions. To show that $\textrm{erf}$ is a solution, we used a method developed in a series of papers. The idea is to promote to an auxiliary direction and impose $\Phi$ to obey the diffusion equation

$\left(\partial_r-\partial_t^2\right)\,\Phi(r,t)=0\,.$

Then equation (1) is localized, since nonlocal operators act as translations along the extra direction:

$e^{s\partial_t^2}\Phi(r,t)=\Phi(r+s,t)\,.$

(In general, powers of the scalar field do not obey the diffusion equation, but in this case they approximately do with very good accuracy.) In particular, solutions can be explicitly constructed. This type of theories is ghost-free and characterized by a well-defined Cauchy problem. The error function is a kink:

Error function

For comparison we show also the usual kink $\textrm{tanh}$ of the local theory.

One can compute the probability of the “quantum mechanical” instanton to tunneling between the two vacua (minima of the effective potential) and the result is very close to the corresponding local system ( r=0 ), despite the fact that the local equation and its solution are radically different.

Note that the error function is also solution of the equation

$(\partial_t^2-m^2)e^{-r\partial_t^2}\Phi=\sigma\Phi^3\,,\hspace{6cm} (2)$

with different values of the parameters. This equation has been often used in the literature as a simpler substitute of equation (1).

Inverse problem. A different way to recast the above results is to start with the following inverse problem:

What is the simplest nonlocal system which generalizes the double-well instanton and has the error function as a soluton?

The answer is:

The tachyonic effective action of open string field theory at lowest truncation level!

The mass and nonlocal exponent appear as separate inputs in the effective equation of the OSFT tachyon, although both are determined by conformal invariance. Obviously, a time rescaling can change their ratio, which is precisely the job done by the parameter in our model. However, it turns out that their product m^2r is fixed once is chosen. The remarkable fact is that the value of the parameter in the simplified equation of motion is very close to the one dictated by string theory, once the mass is fixed to the OSFT tachyon mass m^2=-1/2 . In particular,

$r\approx 0.515\,,\qquad r_\textrm{string}=2 \ln (3^{3/2}/4)\approx 0.523\,.$

Branes. Regarding the above equations of motion as living on Minkowski and changing $t\to x$ to a spatial coordinate, the solution becomes a spatial Minkowski kink, that is to say, a soliton. In fact, the energy of this configuration is peaked around $x\sim 0$ , and the latter can be interpreted as a lower-dimensional brane according to Sen and Horava. More precisely, this solution represents a unstable (non-BPS) -brane in a (p+1) -dimensional target spacetime decaying into a stable (BPS) D(p-1) -brane.

To support this claim, we must check that the ratio of the brane tensions is the one prescribed by Sen’s descent relations. At its local maximum the effective tachyon potential equals the tension of the non-BPS -brane, which is

${\cal T}_p=\frac{1}{2\pi^2g_o^2}\,,$

where g_o is the open string coupling. When p=9 , the brane coincides with the target spacetime of Type I/IIA theory. On the other hand, the tension of the stable D(p-1) -brane is

$\tilde{\cal T}_{p-1}=\sqrt{2}\pi{\cal T}_p.$

The prefactor takes into account reduction of dimensionality of the brane ( ${\cal T}_{p-1}=2\pi{\cal T}_p$ ) and the fact that the tension of an unstable -brane is $\sqrt{2}$ times the tension of a BPS -brane. To proceed, we first revert to the original effective action S_* of string field theory and then fix the normalization of the solution. The truncation level of the action affects the value the non-BPS brane tension ${\cal T}_p$ and possibly the ratio

$-\frac{S_*[\Phi]}{{\cal T}_p} \stackrel{?}{=} \sqrt{2}\pi\,.$

For the approximate potential in equation (2),

$-\frac{S_*[\Phi]}{{\cal T}_p}\approx 4.435=0.998\times (\sqrt{2}\pi)\,.$

Considering that $\textrm{erf}$ was regarded as the approximate solution of the lowest-level approximate effective action, the agreement is impressive. We can conclude that the error function is a nonperturbative OSFT tachyonic profile. This does not correspond to a marginal deformation, so it is not clear, at least to me, how to obtain a similar result with the modern techniques recently developed in the full theory.

The parameters of the system with the nonlocal potential, equation (1), are not so close to OSFT as those of the simplified system or, if they are, the global accuracy of the solution is lower (around 3 to 4%). Nonetheless, the brane tension ratio is about the same, if not better. The evaluation of the action on the solution cannot be done numerically unless one implements a careful numerical procedure which takes into account the nonlocal operators (a truncation of the latter would not be reliable). However, we can use the duplication formulae again. One gets

$-\frac{S_*[\Phi]}{{\cal T}_p}\approx 4.517=1.017\times (\sqrt{2}\pi)\,.$

We conjecture that the exact numerical result is extremely close to the theoretical value.

A puzzle. To summarize, the effective equation of the string tachyon with similar values of the coupling constants, as well as the brane descent relation in Sen’s tachyon condensation, have been obtained starting from an apparently different framework. It would be desirable to explain this open problem. The fact that string field theory may be viewed as a diffusing system was already pointed out in arXiv:0708.0366 and arXiv:0802.4395, where tachyon solutions of OSFT and boundary string field theory were mapped onto each other. In a forthcoming study we will argue that the diffusion equation naturally implements some large gauge symmetries of OSFT at the level of the effective dynamics. In the meanwhile, we can discuss together on NEQNET.

Journal club, Master Postdoc, Quantum field theory, String theory

43. Schwinger-Keldysh: Martin-Siggia-Rose diagrammatics (non-equilibrium diagrammatic methods 2)

285. Dephasing and diffusion of quantum particles

42. Recalling a couple of facts about 2D and 3D Ising models

290. Last two weeks of February on NEQNET

Vorticity generation in cosmological perturbation theory

Posted by Adam Christopherson

at May 20, 2009

Print This Post

Save This Post as PDF

Adam Christopherson is a PhD student at Queen Mary, U. of London working with Karim Malik on cosmological perturbation theory. Dmitry.

In this blog post, I will summarize recent work on vorticity generation in cosmological perturbation theory, undertaken by Karim Malik, David Matravers and myself. The main result of the paper this is based on, arxiv:0904.0940, is that at second order in perturbation theory, vorticity generation is sourced by entropy gradients.

Vorticity is a common phenomenon in nature and is defined, in classical fluid dynamics, as the curl of the fluid velocity. It will arise in most situations involving real fluids. However, despite its prevalence, vorticity has rarely been studied in the early universe and in cosmology.

The ’standard cosmological model’ is an expanding homogeneous and isotropic model, described by the Friedmann-Robertson-Walker (FRW) solution of General Relativity. But this is an approximation, and in reality things are not so simple.

Observational evidence indicates that the universe is not exactly homogeneous and isotropic. Since General Relativity is highly nonlinear, finding an exact inhomogeneous solution is extremely difficult, so cosmologists resort to using a powerful method called Cosmological Perturbation Theory. Starting with a FRW universe as the background spacetime, small inhomogeneous perturbations are added, order by order. In this work we consider only scalar and vector perturbations, since tensor perturbations are not crucial to the result. After gauge choice, the perturbed metric is

$ds^2=a^2(\eta)\left[-(1+2A)d\eta^2+B_id\eta dx^i+\delta_{ij}dx^idx^j\right] \,,$

where is the lapse function, and B_i is the shear. Since Einstein’s equations connect the geometry of the universe to its matter content, perturbations in the metric imply perturbations in the energy-momentum tensor.

Fluids in the early universe can be well modeled as perfect fluids. Like any thermodynamical system, this can be characterized by three variables, two of which are independent. A natural choice for the independent variables are the energy density, $\rho$ , and the entropy, , with the pressure being $P\equiv P(\rho,S)$ . The pressure perturbation can then be expanded in a Taylor series as

$\delta P=\frac{\partial P}{\partial S}\delta S + \frac{\partial P}{\partial \rho}\delta\rho\equiv \delta P_{\rm{nad}}+c_{\rm{s}}^2\delta\rho\,,$

where we have defined the non-adiabatic pressure (or entropy) perturbation, as $\delta P_{\rm{nad}}\equiv\left.\frac{\partial P}{\partial S}\right|_{\rho}\delta S$ . This can be readily extended to second order by simply not truncating the Taylor series after linear order terms.

Vorticity Evolution

The vorticity tensor is defined as the projected, anti-symmetrised covariant derivative of the fluid four velocity:

$\omega_{\mu\nu}=\mathcal{P}_\mu^{~\alpha}\mathcal{P}_\nu^{~\beta}u_{[\alpha;\beta]},\,$

where $\mathcal{P}_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$ is the projection tensor into the instantaneous fluid rest space. This definition is equivalent to $\vec{\omega}=\vec{\nabla}\times \vec{u}$ in standard fluid mechanics. We can then calculate the evolution of the vorticity, making use of energy momentum conservation equations and constraint equations. At first order, we reproduce the well known result that vorticity decays in the absence of anisotropic stress, and that if it is initially zero, it will remain zero. Our novel result comes at second order. We find the evolution equation for the second order vorticity to be

$\omega_{2ij}^\prime -3\mathcal{H}c_{\rm{s}}^2\omega_{2ij}=\frac{2a}{\rho_0+P_0}\left\{3\mathcal{H} V_{1[i}\delta P_{\rm{nad}{1,j]}} +\frac{\delta\rho_{1,[j} \delta P_{\rm{nad}1,i]}}{\rho_0+P_0}\right\}\,,$

even in the case of vanishing vorticity. Thus, at second order in cosmological perturbation theory, vorticity is sourced by entropy gradients. We note that for barotropic fluids, where the entropy perturbation is zero, we recover the result of Lu et al in arxiv:0812.1349.

It is important to consider the possible observational signatures that vorticity in the early universe will have. As we know, the Cosmic Microwave Background (CMB) is polarized, and the polarization can be classified into E-modes (curl-free) and B-modes (divergence-free). The latter is produced (at linear order) only by vector and tensor perturbations. Tensor perturbations correspond to gravitational waves, and are predicted by inflation, whereas inflation does not produce vector perturbations. However, vector perturbations at second order generated by density perturbations, as discussed in this work, exhibit B-mode polarization as an observational signature. This will prove important for current and future experiments, such as Planck or CMBPol.

Furthermore our result, that vorticity is non-zero at second order in the presence of non-adiabatic or entropy perturbations, has immediate implications on the study of magnetic fields in the early universe, since Biermann showed that the generation of magnetic fields is intimately related to vorticity.

Further Reading:

References to relevant literature can be found in our paper, arxiv:0904.0940. For a recent review on Cosmological Perturbation Theory, see Malik and Wands, arxiv:0809.4944.

Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Hydrodynamics, Journal club, Quantum field theory

357. Vortex line representation. Coulomb interaction of vortex lines

111. Talk in Munich. One interesting infrared scale in inflationary cosmology

347. Numerical simulation of vortices: video of the day

353. Vortex line representation. Cauchy invariant

Correcting the initial vacuum state in quantum gravity

Posted by Emre Kahya

at May 18, 2009

Print This Post

Save This Post as PDF

Emre Kahya is a postdoc at Koc University, Turkey (he is a former graduate student of Richard Woodard). Dmitry.

Cosmology is becoming the most active area of research in theoretical physics for the last 10 years. We now understand that initial quantum fluctuations are reasons of our existence with in the context of Inflation. This brings the following question: Can we make quantum gravity calculations and expect to test them by some means? Naively one would say no. One reason is the smallness of the coupling constant:
quantum gravity effects have the order of magnitude $0^{\rm th}{\rm Order} [1+\alpha_1 G + \alpha_2G^2 + ...]$

Since goes like $\frac{\rm length^3}{\rm time}^2 {\rm mass}}$ , the coupling constant $\alpha_1$ should go like E^2/(h c^5)} , therefore $\alpha_1 G$ would approximately be $(\frac{E}{10^{19} GeV})^2$ . Working with energy scales of LHC, 1 TeV, the biggest quantum gravitational correction is at of the order $10^{-32}$ . And the fact that this series diverge at all, puzzles some of us. In any way, one might loose hope of getting any experimentally testable quantum gravity effect due to this smallness problem.

But cosmology is a unique area where the quantum gravitational effects might add up to overcome this problem. This make cosmology a very interesting area of research for theoreticians from string theorists to astrophysicists. I have been interested in quantum gravitational loop effects during the de Sitter phase of Inflation for the last five years or so. In a recent paper, arXiv:0904.4811 V. K. Onemli, R. P. Woodard and myself worked on a problem of correcting the initial vacuum state of a massless minimally coupled scalar with $\lamda \varphi^4$ on non-dynamical de Sitter.

In flat space QFT one calculates the expectation value of matrix elements between true in and out vacuum states in the infinite past and future. This is very useful in scattering problems in flat space QFT but not very relevant for cosmology where the universe began with an initial singularity and particle production precludes the in vacuum from evolving to the out vacuum. The relevant thing to do would be to release the universe from a prepared state at finite time and let it evolve as it will. This can be done by using the In-In formalism or the so called Schwinger-Keldysh formalism.

Since we are doing perturbation theory, the lowest order correction can be obtained by using the free vacuum. It occasionally is the case that one obtains secular growth from higher order corrections and correcting the initial state does not affect this. For some cases corrections to the initial state might be as big as 4-volume effects. But for both cases it would certainly cause some problems related to initial value divergences. In this recent work, we showed that the divergent terms that appear in the expectation value of the stress energy tensor at two loops order in the presence of free Bunch-Davies vacuum, can be absorbed by correcting the initial vacuum state.

In short, cosmology is a natural locale for quantum gravity since gravity is a long range force and it knows about the scale factor a(t) . In-in formalism fits much better in cosmology and one would prefer to use that instead of the in-out formalism that we are used to in flat space QFT calculations. Taking the free vacuum is easier but it might result into terms which would diverge in initial value surface, and might even result into changes at the order of 4-volume effects. And this problem can be avoided by correcting the initial vacuum state.

Further detail and references can be found in our paper:

E. O. Kahya, V. K. Onemli and R. P. Woodard “A Completely Regular Quantum Stress Tensor with w < -1″” arXiv:0904.4811 [gr-qc].

Cosmology, Entered Apprentice, Journal club, Quantum field theory

Related posts:

69. Happy Halloween!

Real-time gauge/gravity duality

31. Schwinger-Keldysh: brief review (Nonequilibrium diagrammatic methods 1)

45. Quantization of cosmological perturbations. Mukhanov-Sasaki variable (Inflationary perturbations 5)

373. Some musings about Unruh effect

One week to spend in US

Posted by Dmitry

at May 17, 2009

Print This Post

Save This Post as PDF

Cleveland

At this very moment – when you are reading this – I am flying to US, where the final destination of my trip is Cleveland. The plan is to take part in the 3-day Workshop on tests of gravity and gravitational physics at Case Western Reserve U.

The schedule there is very dense and workshop promises to be interesting – Avi Loeb, Samir Mathur, France Pretorius, Bill Unruh and Nima Arkani-Hamed are among the participants. I really hope that a) I’ll have internet access there to update NEQNET and b) I’ll find some time for blogging If both conditions (a) and (b) will hold, you’ll get a coverage of the workshop from the first hands. If not, I’m back on May 24th.