# Is it possible to combine string theory and LQG?

1. Apr 26, 2012

### CarlosLara

Good evening. I am wondering if string theory and loop quantum gravity could be combined into a single theory. I have been trying to decide which of the two I should choose as my "religion," but I feel that both are correct. Could string theory and loop quantum gravity be different manifestations of a deeper theory?

2. Apr 26, 2012

### Nabeshin

Well, just as a bare bones issue string theory is a potential 'theory of everything' whereas LQG is not. By this I mean that string theory at least has the hope of describing ALL the fundamental forces and particles of nature, whereas a canonical QG theory such as LQG seeks only to quantize general relativity. So the possible relationship could be that the QG of LQG is the same as the QG obtained by string theory, but string theory is the broader theory. I don't know too much about either, so I don't know if there's some simple reason that their approaches could be incompatible. Marcus, our resident LQG guy, might be able to shed some light on this.

Also, it's probably too early for you to 'choose sides' if you will. But a more detailed discussion along those lines is probably better suited for elsewhere.

3. Apr 26, 2012

### atyy

4. Apr 27, 2012

### MathematicalPhysicist

The only way to see this is if you specialize in both of them. Good luck with that. :-)

5. Apr 27, 2012

### tom.stoer

It is too early for a reliable assessment.

String theory is - as far as I can see - still no single, well-defined theory with unchanging foundation. There is no single Lagrangian from which the full theory can be derived. Instead it's a web of equations, relations and dualities which point towards some underlying trooth still to be uncovered.

String theory seems to be - afaik - a framework for constructing (consistent) theories of supergravity + (supersymmetric) gauge theories; just like "gauge theory" is such a framework w/o gravity.

LQG as of today is not a candidate ToE, but string theory is.

LQG is mostly formulated for 4-dim spacetime and symmetry group SO(3) ~ SU(2) but I don't see any reason not to start with an arbitrary graph plus labels of some (arbitrary) group G - which can very well be different from SO(3,1). LQG can be extended with additional labels for gauge and matter fields. There are attempts to understand the LQG quantization method in more than 4 dimensions and possibly for supergravity as well. So LQG seems to be a quantization method applicable to different theories, not only a single theory (the quantization method is still not fully understood, neither in the PI / spin foam, nor in the canonical approach)

The difference to string theory is that you start with such a "standard" theory and apply the LQG quantization scheme, whereas in string theory you write down a string theory, calculate it's vacuum plus its spectrum and derive the low-energy theory). The whole approach is different.

I would say they are different paradigms, not only different theories.

6. Apr 27, 2012

### QGravity

There are some partial results and some part of one of these theories can be obtained from the other but there is no singe theory (as far as I Know) that both of these theories can be derived :

For example the following papers may be useful:

String Field Theory from Quantum Gravity by Louis Crane

Abstract:
Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group $E_6$, as explained by the quantum Mckay correspondence of Frenkel Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space by Thomas Thiemann

Abstract:
We combine I. background independent Loop Quantum Gravity (LQG) quantization techniques, II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space. While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincar\'e invariance and 8. without picking up UV divergences. The existence of this stable solution is exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string. Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem. The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces.

Topological M-theory as Unification of Form Theories of Gravity by Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke, Cumrun Vafa

Abstract:
We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory one can classically obtain 6-dimensional topological A and B models, the self-dual sector of loop quantum gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective we find hints of an intriguing holographic link between non-supersymmetric Yang-Mills in 4 dimensions and A model topological strings on twistor space.

The cubic matrix model and a duality between strings and loops by Lee Smolin

Abstract:
We find evidence for a duality between the standard matrix formulations of M theory and a background independent theory which extends loop quantum gravity by replacing SU(2) with a supersymmetric and quantum group extension of SU(16). This is deduced from the recently proposed cubic matrix model for M theory which has been argued to have compactifications which reduce to the IKKT and dWHN-BFSS matrix models. Here we find new compactifications of this theory whose Hilbert spaces consist of SU(16) conformal blocks on compact two-surfaces. These compactifications break the SU(N) symmetry of the standard M theory compactifications, while preserving SU(16), while the BFSS model preserve the SU(N) but break SU(16) to the SO(9) symmetry of the 11 dimensional light cone coordinates. These results suggest that the supersymmetric and quantum deformed SU(16) extension of loop quantum gravity provides a dual, background independent description of the degrees of freedom and dynamics of the M theory matrix models.

7. Apr 28, 2012

### tom.stoer

I tried to understand these ideas a couple of years ago but I see two major problems:
1) the individual theories are by no means complete (so to unify strings and loops one would have to be clear about what strings and loops are - this is still not the case)
2) there seems to be no progress at all regarding unification of strings and loops over the last 10 years or so; the two fields are nearly disjoint

8. Jun 5, 2012

### julian

Interesting paper, "Simplicial Gravity and Strings" arXiv:1110.5088:

"String theory, as a theory containing quantum gravity, is usually thought to require more dimensions of spacetime than the usual 3+1. Here I argue on physical grounds that needing extra dimensions for strings may well be an artefact of forcing a fixed flat background space. I also show that discrete simplicial approaches to gravity in 3+1 dimensions have natural string-like degrees of freedom which are inextricably tied to the dynamical space in which they evolve. In other words, if simplicial approaches to 3+1 dimensional quantum gravity do indeed give consistent theories, they may essentially contain consistent background-independent string theories."

I know that Simplical gravity is not the same as LQG but it is also background-independent and the two may be related in some way.

Smolin also fould string like objects in certain forms of LQG..."Three roads to quantum gravity".

Last edited: Jun 5, 2012
9. Jun 5, 2012

### julian

Smolin, who has also done research in string theory as well, wrote the review: "How far are we from the quantum theory of gravity?" where he compares the status of string theory and LQG - arXiv:hep-th/0303185.

Last edited: Jun 5, 2012
10. Jun 25, 2012

### isidrago

I must disagree. Loop Quantum Gravity could be a theory of everything!
Moreover it can be unified with String Theory. Read this attachment... the unification is been already done. It's the Arrangement Field Theory (AFT).

#### Attached Files:

• ###### AFT-PART2.pdf
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11. Jun 27, 2012

### isidrago

I would like to bring to your vision the following papers.

The arrangement field theory (AFT). Part 2
http://arxiv.org/abs/1206.5665
Abstract << In this work we apply the formalism developed in the previous paper ("The arrangement field theory") to describe the content of standard model plus gravity. We discover a triality between Arrangement Field Theory, String Theory and Loop Quantum Gravity which appear as different manifestations of the same theory. Finally we show as three families of fields arise naturally and we discover a new road toward unification of gravity with gauge and matter fields. >>

The arrangement field theory (AFT)
http://arxiv.org/abs/1206.3663
Abstract << We introduce the concept of "non-ordered space-time" and formulate a quaternionic field theory over such generalized non-ordered space. The imposition of an order over a non-ordered space appears to spontaneously generate gravity, which is revealed as a fictitious force. The same process gives rise to gauge fields that are compatible with those of Standard Model. We suggest a common origin for gravity and gauge fields from a unique entity called "arrangement matrix" (M) and propose to quantize all fields by quantizing $M$. Finally we give a proposal for the explanation of black hole entropy and area law inside this paradigm. >>

12. Jun 28, 2012

### julian

Ashtekar in a recent review (http://arxiv.org/pdf/1201.4598.pdf - page 26) makes interesting comments on unification and LQG and how it could provide a bridge to string theory:

"Unification. Finally, there is the issue of unification. At a kinematical level, there is
already an unification because the quantum configuration space of general relativity is the
same as in gauge theories which govern the strong and electro-weak interactions. But the
non-trivial issue is that of dynamics. To conclude, let us consider a speculation. One possibility
is to use the ‘emergent phenomena’ scenario where new degrees of freedom or particles,
which were not present in the initial Lagrangian, emerge when one considers excitations of
a non-trivial vacuum. For example, one can begin with solids and arrive at phonons; start
with superfluids and find rotons; consider superconductors and discover cooper pairs. In
loop quantum gravity, the micro-state representing Minkowski space-time will have a highly
non-trivial Planck-scale structure. The basic entities will be 1-dimensional and polymerlike.
one can argue that, even in absence of a detailed theory, the fluctuations of these
1-dimensional entities should correspond not only to gravitons but also to other particles,
including a spin-1 particle, a scalar and an anti-symmetric tensor. These ‘emergent states’
are likely to play an important role in Minkowskian physics derived from loop quantum
gravity. A detailed study of these excitations may well lead to interesting dynamics that
includes not only gravity but also a select family of non-gravitational fields. It may also
serve as a bridge between loop quantum gravity and string theory. For, string theory has
two a priori elements: unexcited strings which carry no quantum numbers and a background
space-time. Loop quantum gravity suggests that both could arise from the quantum state
of geometry, peaked at Minkowski (or, de Sitter) space. The polymer-like quantum threads
which must be woven to create the classical ground state geometries could be interpreted
as unexcited strings. Excitations of these strings, in turn, may provide interesting matter
couplings for loop quantum gravity."

13. Jul 9, 2012

### mitchell porter

If some outgrowth of loop gravity ever produces a type of string theory, it might be through the use of twistor variables. My logic: loop gravity is largely based on topological QFT, and the topological string can be defined on twistor space.

14. Jul 10, 2012

### tom.stoer

The more I think about it the more I come to the conclusion that string theory is a kind of effective theory and is not - at least not in its present formulation - based on fundamental d.o.f. Therefore I do not expect a combination but rather a certain limit (of LQG or some other theory) from which string theory could emerge.

15. Jul 10, 2012

### julian

Completely agree.

The big attraction of string theory is it's promise to provide a T.O.E. ...if LQG (or some other theory) had `string theory' in some limit then LQG (or some other theory) could also claim to promise a T.O.E.

Last edited: Jul 10, 2012
16. Jul 10, 2012

### julian

It was Joe Polchinski who said in 1999 "all good ides are part of string theory"...

...it may turn out to be the other way round and that string theory is part of some more fundamental approach.

Last edited: Jul 10, 2012
17. Jul 11, 2012