The 23th DAE-BRNS High Energy Physics Symposium was held at IIT Madras from Dec 10 - Dec 14. It was an interesting event. I met lots of very smart people. My abstract based on my paper (Vaid 2017) had been selected for a talk in the “Formal Theory” parallel session on Dec 11. Interestingly the talk preceding mine was delivered by Suresh Govindarajan (INSPIRE) who is a faculty at IITM and a hardcore string theorist. Also in attendance was Prof G. Rajasekaran, who is an emeritus faculty at the Institute of Mathematical Sciences, Chennai and himself a distinguished high energy physicist. Naturally, following Govindarajan’s highly mathematical talk on the existence of BKM superalgebras - of which, I understood perhaps the first three slides - I felt a little trepidatious, especially since the number of mathematical formulae in his talk was several orders of magnitude greater than in mine! Naveen, a PhD student also from NITK, was kind enough to record my talk on his phone 1 . The result is viewable on YouTube.
Connecting String Theory and LQG
Couple of days later I had a nice conversation with Prof Govindarajan 2 where he conveyed to me that the general feeling among many in the strings community was that loops and strings would ultimately have to come together. He mentioned the following questions as his main concerns:
Matter Degrees of Freedom
Where is the matter in LQG? In String Theory matter arises “naturally” from compactification of \(n>4\) dimensions. The compactified dimensions behave like scalar and gauge fields in the non-compactified geometry. I mentioned to him that LQG does have candidates for matter in the form of topological degrees of freedom known as “preons” (Bilson-Thompson 2005; Vaid 2010). Of course, much work is still to be done to understand how the entire spectrum of the standard model can arise from these topological defects. I made some early efforts trying to connect quantum computation gates and elementary particles in LQG in (Vaid 2013a) and to show how non-abelian gauge fields - such as those in the Standard Model - can arise naturally from defects in spin-networks in (Vaid 2013b).
Degenerate geometry in LQG
Suresh’s next concern was about the existence of geometries in LQG where the tetrad \(e_\mu^I\)(which determines the metric geometry via the relation\(g_{\mu\nu} = e_\mu^I e_\nu^J \eta_{IJ}\)) is allowed to be degenerate - \(\ie~\text{det}(e_\mu^I) = 0\). In such cases the resulting metric exists, however its inverse \(g^{\mu\nu}\) does not (because matrices with zero determinant do not have well-defined inverses). His concern might have been motivated by Kaul and Sengupta’s recent work on degenerate spacetimes in the connection formulation of gravity. I explained that there is nothing non-physical about having degenerate spacetimes. One can do all the usual physics with scalars, vectors and spinors in such geometries. However, one also has new physics in such regimes which cannot be captured by the metric formulation. In particular with degenerate tetrads one can have geometries with non-zero torsion even without any spinning matter present (Kaul and Sengupta 2016a, 2016b).
Background dependence of string theory
String theory as it is usually defined, is a manifestly background dependent theory. Now, presumably a theory of quantum gravity should be background independent. One should be able to extract physical information such as correlation functions, scattering amplitudes and such without having to worry about the background geometry the given processes take place in. Moreover, since in the strong quantum gravity regime, even the gravitational field will be involved in scattering processes, our quantum gravity theory should be able to handle cases which involve transitions between geometries which cannot be treated as perturbations of a given background. Suresh recognizes this shortcoming of String Theory and mentioned it as such to me. It is in this respect that LQG trumps String Theory. Background independence is manifest and non-negotiable in LQG. We need to be able to incorporate background independence in a meaningful way in the String framework whether it is via String Field Theory or some other approach. LQG can provide pointers on how this might be accomplished.
Extra dimensions or Lack Thereof
Extra dimensions are a given in String Theory. The requirement of conformal invariance of the string worldsheet enforces that the spacetime dimensions much be \(D = 26\)for the bosonic string and\(D = 10\)for the fermionic (or supersymmetric) string. In order to obtain our familiar four dimensional spacetime, these extra dimensions have to be gotten “rid off” in some way. The most method is compactification (Kawai, Lewellen, and Tye 1986; Kawai, Lewellen, and Henry Tye 1987) , wherein six (in the case of\(D = 10\) superstring theory) dimensions are “rolled up” and only four “large” space+time dimensions remain. The compactified dimensions still manifest physically as effective scalar or gauge fields propagating in the background of the four large dimensions. These extra dimensions are also the source of much criticism of String Theory. It turns out that there is a huge (\(\sim 10^{500}\) ways in which the extra dimensions can be compactified and which one of these, if any, compactifications can give rise to our Universe with the Standard Model and all its interactions is a notoriously intractable problem. Extra dimensions are not present, and in fact are not needed, in Loop Quantum Gravity. This is considered a net plus for LQG. However the optimism might be short-lived. All those extra dimensions in String Theory, which seem like so much clutter, can actually be used to produce experimentally verifiable predictions about QCD scattering processes! Sakai and Sugimoto (Sakai and Sugimoto 2005) first initiated this approach by constructing a holographic dual of large-N QCD. Following this work many authors, including Aalok Misra and his brilliant student Vikas Yadav, whom I had the pleasure of meeting at this conference, have managed to use the Sakai-Sugimoto framework to predict decay widths of glueballs (Yadav and Misra 2018) (bound states of gluons) which appear to match very nicely with lattice QCD calculations. LQG is yet to deliver on any such quantitive particle physics related predictions, though it does have several predictions on the astrophysical front (Rovelli and Vidotto 2014; Barrau, Rovelli, and Vidotto 2014; Barrau et al. 2017), which if confirmed would be a stunning success.
Noncommutative Geometry and Preons
Another very interesting talk was delivered by Prof. Rajiv Gavai from TIFR on work (Ghoderao, Gavai, and Ramadevi 2018) done with Pulkit Ghoderao and P. Ramadevi, on the possible detection of non-commutative effects by measuring the Lamb shift of the hydrogen atom or in accelerator experiments. Two mathematical quantities \(A, B\)are said to be “non-commutative” (or “non-commuting” or “do not commute”), when they dont’ satisfy the following relation:\([A,B] = 0\)where\([A,B] = AB - BA\) is referred to as the “commutator”. Experts may skip the following subsection.
Non-commutativity for Non-Experts
A trivial example is any pair of complex numbers \(z_1, z_2\). Using the rules of complex multiplication one can easily see that \([z_1, z_2] = 0\)for all\(z_1, z_2 \in \mathbb{C}\). This is also obviously also true for all real numbers which are a subset of the complex numbers. However, this is not true for quaternions and octonions, which along with the real and complex numbers, constitute the only four normed division algebras (see for e.g. (Baez 2001)) possible mathematically. It is also not true, in general, for matrices. One can see this by taking any two \(2 \times 2\)matrices with random elements and calculating the commutator. Another example is given by operators in quantum theory. Position\(x\)and momentum\(p\)are represented by operators\(\hat x\)and\(\hat p\), respectively. While the classical variables commute: \({ x, p } = 0\)3, their operator versions don’t:\([ \hat x, \hat p] = i\hbar \ne 0\). In this sense, the phase space of a quantum mechanical system is an example of a non-commutative geometry. The non-commutativity Ghoderao et al.’s work is concerned with is between different spatial co-ordinates: \([x_i, x_j] = i \theta_{ij}\)where\(\theta_{ij}\)measures the amount of non-commutativity. Now this is very interesting. For instance, if you consider a particle in a plane, the operation\(x y\)correspond to walking one unit in the\(y\)direction, followed by one unit in the\(x\)direction.\(y x\)is defined in the same way. Now, normally we expect that both operations should get us to the same point,\(\ie\) \(xy = yx\). However, if we were living on a non-commutative plane then this would no longer be true. In a sense, non-commutativity measures the presence of “non-abelian defects” in geometry. Both String Theory and LQG generically predict non-commutative effects arising from quantum geometry. Thus the existence of such an effect would provide very strong support for both theories and also allow us to differentiate between various models.
Composite Particles and Non-commutativity
Ghoderao et al’s result can be summarized in one sentence
in a non-commutative geometry, quarks can form composite particles such as protons and neutrons, if and only if, they (quarks) have substructure.
Now this is a stunning result which also applies to leptons such as electrons, muons and neutrinos. The reason I found this work particularly exciting is because it provides very strong circumstantial evidence for the preon model of elementary particles developed by my good friend and collaborator Sundance Bilson-Thompson (Bilson-Thompson 2005) 4. This model predicts precisely such a substructure for the leptons and quarks. It would be very interesting to try to understand the relationship between the Bilson-Thompson model and non-commutative geometry.
Fermi Arcs and AdS/CFT
Finally, there was very interesting work presented by Wadbor Wahlang who is a graduate student working under Sayan Chakrabarti at IIT Guwahati. This work was about understanding the origin of Fermi arcs in Weyl semi-metals 5 from a holographic perspective. As is well understood by now (Hartnoll 2009; Hartnoll et al. 2008; Hartman and Hartnoll 2010) the AdS/CFT correspondence can be used to explore the phase diagram of condensed matter systems. Essentially what Wahlang and Chakrabarti do is to couple free fermion fields to the usual scalar field living in the bulk AdS spacetime and use that to calculate the spectral function of the boundary field theory. In the event that the fermionic fields they use are Weyl fermions, the spectral function exhibits Fermi arcs. I am looking forward to seeing this work on the arXiv.
- It might seem a bit narcissistic to some to record one’s own conference talks. However, for scientists, talks are the best way to communicate our ideas to our own community and to the general public. They also serve to help us improve our presentation skills. There’s really no downside to recording your own talks, except perhaps the realization that you’re not quite as slim as you’d like to imagine.
- This version is only my recollection of my conversation with Prof Govindarajan and has not been endorsed or approved by him. Any errors or omissions are solely mine.
- We are using curly braces \({,}\)because in classical mechanics the commutator is given by the Poisson bracket which is written in this way, whereas square braces\([,]\) are typically used to represent commutators of quantum mechanical quantities.
- Just to clarify, I met Sundance long after he had discovered the “Bilson-Thompson” mode and I had no role in its discovery. I did try to explain how it could be embedded into LQG in (Vaid 2010)
- See, for \(\eg\) (Rao 2016) for an introduction to the concept of Weyl fermions, Weyl semi-metals and Fermi arcs.