Are gravity and space-time emergent?

Attempts to develop a quantum theory of gravity continue to falter and stagnate. Given this, it is worth considering approaches that start with what we know about gravity at the macroscale and investigate whether it provides any hints about some underlying more microscopic theory. One such approach was taken by Thanu Padmanabhan and is elegantly described and summarised in a book chapter.

Gravity and Spacetime: An Emergent Perspective

Insights about microphysics from macrophysics 

Padmanabhan emphasises Boltzmann's insight: "matter can only store and transfer heat because of internal degrees of freedom". In other words, if something has a temperature and entropy then it must have a microstructure.

The approach of trying to surmise something about microphysics from macrophysics has a long and fruitful history, albeit probably with many false starts that we do not hear about. Kepler's snowflakes may have been the first example. Before people were completely convinced about the existence of atoms, the study of crystal facets and of Brownian motion provided hints of the atomic structure of matter. Planck deduced the existence of the quantum from the thermodynamics of black-body radiation.

Arguably, the first definitive determination of Avogadro's number was from Perrin's experiments on Brownian motion which involved macroscopic measurements.

Comparing classical statistical mechanics to bulk thermodynamic properties gave hints of an underlying quantum structure to reality. The Sackur-Tetrode equation for the entropy of an ideal gas hints at the quantisation of phase space. The Gibbs paradox hints that fundamental particles are indistinguishable. The third law of thermodynamics hints at the idea of quantum degeneracy.

Puzzles in classical General Relativity

Padmanabhan reviews aspects of the theory that he considers some consider to be "algebraic accidents" but he suggests that they may be hints to something deeper. These include the role of boundary terms in variational principles and he suggests hint at a classical holography (bulk behaviour is determined by the boundary). He also argues that the metric of space-time should not be viewed as a field, contrary to most attempts to develop a quantum field theory for gravity.

Thermodynamics of horizons

The key idea that is exploited to find the microstructure is that can define a temperature and an entropy for null surfaces (event horizons). These have been calculated for specific systems (metrics) including the following:

For accelerating frames of reference (Rindler) there is an event horizon which exhibits Unruh radiation with a temperature that was calculated by Fulling, Davies and Unruh.

The black hole horizon in the Schwarschild metric has the temperature of Hawking radiation.

The cosmological horizon in deSitter space is associated with a temperature proportional to the Hubble constant H. [This was discussed in detail by Gibbons and Hawking in 1977].

Estimating Avogadro's number for space-time

Consider the number of degrees of freedom on the boundary, N_s, and in the bulk, N_b. 

On the boundary surface, there is one degree of freedom associated with every Planck area (L_p^2) where L_p is the Planck length, i.e,  N_s = A/ L_p^2, where A is the surface area, which is related to the entropy of the horizon (cf. Bekenstein and Hawking).

In the bulk equipartition of energy is assumed so the bulk energy E = N_b k T/2 where

An alternative perspective on cosmology 

He presents a novel derivation of the dynamic equations for the scale factor R(t) in the Friedmann-Robertson-Walker metric of the universe in General Relativity. His starting point is a simple argument leading to 

V is the Hubble volume, 4pi/3H^3, where H is the Hubble constant, and L_P is the Planck length.

The right-hand side is zero for the deSitter universe, which is predicted to be the asymptotic state of our current universe.

Possible insights about the cosmological constant

One of the biggest problems in theoretical physics is to explain why the cosmological constant has the value that it does.

There are two aspects to the problem.
1. The measured value is so small, 120 orders of magnitude smaller than what one estimates based on the quantum vacuum energy!

2. The measured value seems to be finely tuned (to 120 significant figures!) to the value of the mass energy.

He presents an argument that the cosmological constant is related to the Planck length 

where mu is of order unity.

Details of his proposed solution are also discussed here.

I am not technically qualified to comment on the possible validity or usefulness of Padmanabhan's perspective and results. However, I think it provides a nice example of a modest and conventional scientific alternative to radical approaches, such as the multiverse, or ideas that seem to be going nowhere such as AdS/CFT that are too often invoked or clung onto to address these big questions. 

Aside. In the same book, there is also a short and helpful chapter, Quantum Spacetime on loop quantum gravity by Carlo Rovelli. He explicitly identifies the "atoms" of space-time as the elements of "spin foam".

Emergent gauge fields in spin ices

Spin ices are magnetic materials in which geometrically frustrated magnetic interactions between the spins prevent long-range magnetic order and lead to a residual entropy similar to in ice (solid water).

Spin ices provide a beautiful example of many aspects of emergence, including how surprising new entities can emerge at the mesoscale. I think the combined experimental and theoretical work on spin ice was one of the major achievements of condensed matter physics in the first decade of this century.

Novelty

Spin ices are composed of individual spins on a lattice. The system exhibits properties that the individual spins and the high-temperature state do not have. The novel properties can be understood in terms of an emergent gauge field. Novel entities include spin defects reminiscent of magnetic monopoles and Dirac strings.

State of matter

Spin ices exhibit a novel state of matter, the magnetic Coulomb phase. There is no long-range spin order, but there are power-law (dipolar) correlations that fall off as the inverse cube of distance.

Toy models

Classical models such as the Ising or Heisenberg models with antiferromagnetic nearest-neighbour interactions on the pyrochlore lattice exhibit the emergent physics associated with spin ices: absence of long-range order, residual entropy, ice type rules for local order, and long-range dipolar spin correlations exhibiting pinch points. These toy models can be used to derive the gauge theories that describe emergent properties such as monopoles and Dirac strings.

Actual materials that exhibit spin ice physics such as dysprosium titanate (Dy₂Ti₂O₇) and holmium titanate (Ho₂Ti₂O₇) are more complicated. They involve quantum spins, ferromagnetic interactions, spin-orbit coupling, crystal fields, complex crystal structure and dipolar magnetic interactions. Chris Henley says these materials

"are well approximated as having nothing but (long-ranged) dipolar spin interactions, rather than nearest-neighbor ones. Although this model is clearly related to the “Coulomb phase,” I feel it is largely an independent paradigm with its own concepts that are different from the (entropic) Coulomb phase..."

Effective theory

Gauge fields described by equations analogous to electrostatics and magnetostatics in Maxwell’s electromagnetism are emergent in coarse-grained descriptions of spin ices. 

Consider a bipartite lattice where on each site we locate a tetrahedron. The "ice rules" require that two spins on each tetrahedron point in and two out. We can define a field L(i) on each lattice site i which is the sum of all the spins on the tetrahedron. The magnetic field B(r) is a coarse-graining of the field L(i). The ice rules and local conservation of flux require that 

The classical ground state of this model is infinitely degenerate. The emergent “magnetic” field [which it should be stressed is not a physical magnetic field] allows the presence of monopoles [magnetic charges]. These correspond to defects that do not satisfy the local ice rules in the spin system.

It can be shown that the total free energy of the system is

K is the "stiffness" or "magnetic permeability" associated with the gauge field. It is entirely of entropic origin, just like the elasticity of rubber.

[Aside: I would be curious to see a calculation of K from a microscopic model and an estimate from experiment. I have not stumbled upon one yet. Do you know of one? Henley points out that in water ice the entropic elasticity makes a contribution to the dielectric constant and this "has been long known."]

  A local spin flip produces a pair of oppositely charged monopoles. The monopoles are deconfined in that they can move freely through the lattice. They are joined together by a Dirac string.

This contrasts with real magnetism where there are no magnetic charges, only magnetic dipoles; one can view magnetic charges as confined within dipoles.

There is an effective interaction between the two monopoles [charges] that has the same form as Coulomb’s law.  There are only short-range (nearest neighbour) direct interactions between the spins. However, these act together to produce a long-range interaction between the monopoles (which are deviations from local spin order).

Universality

The novel properties of spin ice occur for both quantum and classical systems, Ising and Heisenberg spins, and for a range of lattices. The same physics occurs with water ice, magnetism, and charge order.

Modularity at the mesoscale

The system can be understood as a set of weakly interacting modular units. These include the tetrahedra of spins, the magnetic monopoles, and the Dirac strings. The measured temperature dependence of the specific heat of Dy₂Ti₂O₇  is consistent with that calculated from Debye-Huckel theory for deconfined charges interacting by Coulomb's law, and shown as the blue curve below. The figure is taken from here.

Pinch points.

The gauge theory predicts that the spin correlation function (in momentum space) has a particular singular form exhibiting pinch points [also known as bow ties], which are seen experimentally.

Unpredictability

Most new states of matter are not predicted theoretically. They are discovered by experimentalists, often by serendipity. Spin ice and the magnetic Coulomb phase seems to be an exception. Please correct me if I am wrong.

Sexy magnetic monopoles or boring old electrical charges?

I am hoping a reader than clarify this issue. What is wrong with the following point of view. In the discussion above the "magnetic field" B(r) could equally well be replaced with an "electric field" E(r). Then the spin defects are just analogous to electrical charges and the "Dirac strings" become like a polymer chain with opposite electrical charges at its two ends. This is not as sexy. 

Note that Chris Henley says Dirac strings are "a nebulous and not very helpful notion when applied to the Coulomb phase proper (with its smallish polarisation), for the string's path is not well defined... It is only in an ordered phase... that the Dirac string has a clear meaning."

Or is the emergent field actually "magnetic"? It describes spin defects and these are associated with a local magnetic moment. Furthermore, the long-range dipolar correlations (with associated pinch points) of the gauge field are detected by magnetic neutron scattering and so the gauge field should be viewed as "magnetic" and not "electric".

Emergent gauge fields in quantum many-body systems?

In spin ice, the emergent gauge field is classical and arises in a spin system that can be described classically. This does raise two questions that have been investigated extensively by Xiao-Gang Wen. First, he has shown how certain mean-field treatments of frustrated antiferromagnetic (with quantum spin liquid ground states) and doped Mott insulators lead to emergent gauge fields. As fascinating as his work is, it needs to be stressed that there is no definitive evidence for these emergent gauge fields. They just provide appealing theoretical descriptions. This is in contrast to the emergent gauge fields for spin ice.

Second, based on Wen's success at constructing these emergent gauge fields he has pushed provocative (and highly creative) ideas that the gauge fields and fermions that are considered "fundamental" in the standard model of particle physics may be emergent entities. This is the origin of the subtitle of his 2004 book, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons.

To prepare this post I found the articles below helpful.

Emergent particles and gauge fields in quantum matter

Ben J. Powell

Maxwell electromagnetism as an emergent phenomenon in condensed matter

J. Rehn and R. Moessner

The “Coulomb Phase” in Frustrated Systems

Chris Henley

Emergence and protein folding

Proteins are a distinct state of matter. Globular proteins are tightly packed with a density comparable to a crystal but without the spatial regularity found in crystals. The native state is thermodynamically stable, in contrast to the globule state of synthetic polymers which is often glassy and metastable, with a structure that depends on the preparation history.

For a given amino acid sequence the native folded state of the protein is emergent. It has a structure, properties, and function that the individual amino acids do not, nor does the unfolded polymer chain. For example, the enzyme catalase has an active site whose function is as a catalyst to make hydrogen peroxide (which is toxic) decay rapidly.

Protein folding is an example of self-organisation. A key question is how the order of the folded state arises from the disorder (random configuration) of the unfolded state.

There are hierarchies of structures, length scales, and time scales associated with the folding.

The hierarchy of structures are primary, secondary, tertiary, and ternary structures. The primary structure is the amino acid sequence in the heteropolymer. Secondary structures include alpha-helices and beta-sheets, shown in the figure above in orange and blue, respectively. The tertiary structure is the native folded state. An example of a ternary structure is in hemoglobin which consists of four myoglobin units in a particular geometric arrangement.

The hierarchy of time scales varies over more than fourteen orders of magnitude, including folding (msec to sec), helix-coil transitions (microsec), hinge motion (nanosec), and bond vibrations (10 fsecs).

Folding exhibits a hierarchy of processes, summarised in the figure below which is taken from

Cooperativity and modularity in protein folding

Masaki Sasai, George Chikenji, Tomoki P. Terada

Modularity 

"Protein foldons are segments of a protein that can fold into stable structures independently. They are a key part of the protein folding process, which is the stepwise assembly of a protein's native structure." (from Google AI)

See for example.

Discontinuities

The folding-unfolding transition [denaturation] is a sharp transition, similar to a first-order phase transition. This sharpness reflects the cooperative nature of the transition. There is a well-defined enthalpy and entropy change associated with this transition.

Universality

Proteins exhibit "mutational plasticity", i.e., native structures tolerant to many mutations (changes in individual amino acids). Aspects of the folding process such as its speed, reliability, reversibility, and modularity appear to be universal, i.e., hold for all proteins.

Diversity with limitations

On the one hand, there are a multitude of distinct native structures and associated biological functions. On the other hand, this diversity is much smaller than the configuration space, presumably because thermodynamic stability vasts reduces the options.

Effective interactions

These are subtle. Some of the weak ones matter as the stabilisation energy of the native state is of order 40 kJ per mole, which is quite small as there are about 1000 amino acids in the polymer chain. Important interactions include hydrogen bonding, hydrophobic, and volume exclusion. In the folded state monomers interact with other monomers that are far apart on the chain. The subtle interplay of these competing interactions produces complex structures with small energy differences, as is often the case with emergent phenomena.

Toy models

1. Wako-Saito-Munoz-Eaton model

This is an Ising-like model on a chain. A short and helpful review is

A simple theoretical model goes a long way in explaining complex behavior in protein folding

Victor Muñoz 

Note that the interactions are not pairwise but involves strings of "spins" between native contacts.

2. Dill's HP polymer on a lattice

This consists of a polymer which has only two types of monomer units and undergoes a self-avoiding walk on a lattice. H and P denote hydrophobic and polar amino acid units, denoted by red and blue circles, respectively, in the figure below. The relative simplicity of the model allows complete enumeration of all possible confirmations for short chains. The model is simpler in two dimensions, yet still captures essential features of the folding problem.  

As the H-H attraction increases the chain undergoes a relatively sharp transition to just a few conformations that are compact and have hydrophobic cores. The model exhibits much of the universality of protein folding. Although there are 20 different amino acids in real proteins, the model divides them into two classes and still captures much of the phenomena of folding, including mutational plasticity.

co-operativity - helical order-disorder transition is sharp

Organising principles

Certain novel concepts such as the rugged energy landscape and the folding funnel apply at a particular scale.

This post drew on several nice papers written by Ken Dill and collaborators including

The Protein Folding Problem, H.S. Chan and K.A. Dill, Physics Today, 1993

The Protein Folding Problem: The Role of Theory

Roy Nassar, Gregory L. Dignon, Rostam M. Razban, Ken A. Dill, Journal of Molecular Biology, 2021

Interestingly, in the 2021 article, Dill claims that the protein folding problem [which is not the prediction problem] has now essentially been solved.

A very effective Hamiltonian in nuclear physics

Atomic nuclei are complex quantum many-body systems. Effective theories have helped provide a better understanding of them. The best-known are the shell model, the (Aage) Bohr-Mottelson theory of non-spherical nuclei, and the liquid drop model. Here I introduce the Interacting Boson Model (IBM), which provides somewhat of a microscopic basis for the Bohr-Mottelson theory. Other effective theories in nuclear physics are chiral perturbation theory, Weinberg's theory for nucleon-pion interactions, and Wigner's random matrix theory.

The shell model has similarities to microscopic models in atomic physics. A major achievement is it explains the origins of magic numbers, i.e., nuclei with atomic numbers 2, 8, 20, 28, 50, 82, and 126 are particularly stable because they have closed shells. Other nuclei can then be described theoretically as an inert closed shell plus valence nucleons that interact with a mean-field potential due to the core nuclei and then with one another via effective interactions.

For medium to heavy nuclei the Bohr-Mottelson model describes collective excitations including transitions in the shape of nuclei.

An example of the trends in the low-lying excitation spectrum  to explain is shown in the figure below. The left spectrum is for nucleus with close to a magic number of nuclei and the right one for an almost half-filled shell. R_4/2 is the ratio of the energies of the J=4+ state to that of the 2+ state, relative to the ground state. B(E2) is the strength of the quadrupole transition between the 2+ state and the ground state.

The Interacting Boson Model (IBM) is surprisingly simple and successful. It illustrates the importance of quasi-particles, builds on the stability of closed shells, and neglects many degrees of freedom. It describes even-even nuclei, i.e., nuclei with an even number of protons and an even number of neutrons. The basic entities in the theory are pairs of nucleons. These are taken to be either an s-wave state or a d-wave state. There are five d-wave states (corresponding to the 2J+1 possible states of total angular momentum with J=2). Each state is represented by a boson creation operator and so the Hilbert space is six-dimensional. If the states are degenerate [which they are not] the model has U(6) symmetry.

The IBM Hamiltonian is written in terms of the most general possible combinations of the boson operators. This has a surprisingly simple form.

Note that it involves only four parameters. For a given nucleus these parameters can be fixed from experiment, and in principle calculated from the shell model. The Hamiltonian can be written in a form that gives physical insight, connects to the Bohr-Mottelson model and is amenable to a group theoretical analysis that makes calculation and understanding of the energy spectrum relatively simple.

Central to the group theoretical analysis is considering subalgebra chains as shown below

 

An example of an energy spectrum is shown below.

The fuzzy figures are taken from a helpful Physics Today article by Casten and Feng from 1984 (Aside: the article discusses an extension of the IBM involving supersymmetry, but I don't think that has been particularly fruitful).

The figure below connects the different parameter regimes of the model to the different subalgebra chains.

The nucleotide chart below has entries that have colour shading corresponding to their parameter values for the IBM model according to the symmetry triangle above.

The different vertices of the triangle correspond to different nuclear geometries and allow a connection to Aage Bohr's model for the surface excitations. 

This is discussed in a nice review article, which includes the figure above.

Quantum phase transitions in shape of nuclei

Pavel Cejnar, Jan Jolie, and Richard F. Casten

Aside: one thing that is not clear to me from the article concerns questions that arise because the nucleus has a finite number of degrees of freedom. Are the symmetries actually broken or is there tunneling between degenerate ground states?   

Colloquium on 2024 Nobel Prizes

This friday I am giving a colloquium for the UQ Physics department.

2024 Nobel Prizes in Physics and Chemistry: from biological physics to artificial intelligence and back

The 2024 Nobel Prize in Physics was awarded to John Hopfield and Geoffrey Hinton “for foundational discoveries and inventions that enable machine learning with artificial neural networks.” Half of the 2024 Chemistry prize was awarded to Dennis Hassabis and John Jumper for “protein structure prediction” using artificial intelligence. I will describe the physics background needed to appreciate the significance of the awardees work. 

Hopfield proposed a simple theoretical model for how networks of neurons in a brain can store and recall memories. Hopfield drew on his background in and ideas from condensed matter physics, including the theory of spin glasses, the subject of the 2021 Physics Nobel Prize.

Hinton, a computer scientist, generalised Hopfield’s model, using ideas from statistical physics to propose a “Boltzmann machine” that used an artificial neural network to learn to identify patterns in data, by being trained on a finite set of examples. 

For fifty years scientists have struggled with the following challenge in biochemistry: given the unique sequence of amino acids that make up a particular protein can the native structure of the protein be predicted? Hassabis, a computer scientist, and Jumper, a theoretical chemist, used AI methods to solve this problem, highlighting the power of AI in scientific research. 

I will briefly consider some issues these awards raise, including the blurring of boundaries between scientific disciplines, tensions between public and corporate interests, research driven by curiosity versus technological advance, and the limits of AI in scientific research.

Here is my current draft of the slides.

John Hopfield on what physics is

A decade ago John Hopfield reflected on his scientific life in Annual Reviews in Condensed Matter Physics, Whatever Happened to Solid State Physics?

"What is physics? To me—growing up with a father and mother who were both physicists—physics was not subject matter. The atom, the troposphere, the nucleus, a piece of glass, the washing machine, my bicycle, the phonograph, a magnet—these were all incidentally the subject matter. The central idea was that the world is understandable, that you should be able to take anything apart, understand the relationships between its constituents, do experiments, and on that basis be able to develop a quantitative understanding of its behavior. 

Physics was a point of view that the world around us is, with effort, ingenuity, and adequate resources, understandable in a predictive and reasonably quantitative fashion. Being a physicist is a dedication to the quest for this kind of understanding."

He describes how this view was worked out in his work in solid state theory and moved into biological physics and the paper for which he was awarded the Nobel Prize. 

"Eventually, my knowledge of spin-glass lore (thanks to a lifetime of interaction with P.W. Anderson), Caltech chemistry computing facilities, and a little neurobiology led to the first paper in which I used the word neuron. It was to provide an entryway to working on neuroscience for many physicists..."

After he started working on biological physics in the late 1970s he got an offer from Chemistry and Biology at Caltech and Princeton Physics suggested he take it. 

"In 1997, I returned to Princeton—in the Molecular Biology Department, which was interested in expanding into neurobiology. Although no one in that department thought of me as anything but a physicist, there was a grudging realization that biology could use an infusion of physics attitudes and viewpoints. I had by then strayed too far from conventional physics to be courted for a position in any physics department. So I was quite astonished in 2003 to be asked by the American Physical Society to be a candidate for vice president. And, I was very happy to be elected and ultimately to serve as the APS president. I had consistently felt that the research I was doing was entirely in the spirit and paradigms of physics, even when disowned by university physics departments."

2024 Nobel Prize in Physics

 I was happy to see John Hopfield was awarded the Nobel Prize in Physics for his work on neural networks. The award is based on this paper from 1982

Neural networks and physical systems with emergent collective computational abilities

One thing I find beautiful about the paper is how Hopfield drew on ideas about spin glasses (many competing interactions lead to many ground states and a complex energy landscape).

A central insight is that an efficient way to store the information describing multiple objects (different collective spin states in an Ising model) is in terms of the inter-spin interaction constants (J_ij's) in the Ising model. These are the "weights" that are trained/learned in computer neural nets.

It should be noted that Hopfield's motivation was not at all to contribute to computer science. It was to understand a problem in biological physics: what is the physical basis for associative memory? 

I have mixed feelings about Geoffrey Hinton sharing the prize.  On the one hand, in his initial work, Hinton used physics ideas (Boltzmann weights) to extend Hopfields ideas so they were useful in computer science. Basically, Hopfield considered a spin glass model at zero temperature and Hinton considered it at non-zero temperature. [Note, the temperature is not physical it is just a parameter in a Boltzmann probability distribution for different states of the neural network]. Hinton certainly deserves lots of prizes, but I am not sure a physics one is appropriate. His work on AI has certainly been helpful for physics research. But so have lots of other advances in computer software and hardware, and those pioneers did not receive a prize.

I feel a bit like I did with Jack Kilby getting a physics prize for his work on integrated circuits. I feel that sometimes the Nobel Committee just wants to remind the world how physics is so relevant to modern technology.

Ten years ago Hopfield wrote a nice scientific autobiography for Annual Reviews in Condensed Matter Physics,

Whatever Happened to Solid State Physics?

After the 2021 Physics Nobel to Parisi, I reflected on the legacy of spin glasses, including the work of Hopfield.

Aside: I once pondered whether a chemist will ever win the Physics prize, given that many condensed matter physicists have won the chemistry prize. Well now, we have had an electronic engineer and a computer scientist winning the Physics prize.

Another side: I think calling Hinton's network a Boltzmann machine is a scientific misnomer. I should add this to my list of people getting credit for things that did not do. Boltzmann never considered networks, spin glasses or computer algorithms. Boltzmann was a genius, but I don't think we should be attaching his name to everything that involves a Boltzmann distribution. To me, this is a bit like calling the Metropolis algorithm for Monte Carlo simulations the Boltzmann algorithm.