Friday, April 28, 2017

Transition times through the black hole bounce

Tuesday, Apr 4th
Parampreet Singh, LSU
Title: Transition times through the black hole bounce 
PDF of the talk (2M)
Audio+Slides [.mp4 18MB]


by Gaurav Khanna, University of Massachusetts Dartmouth


Loop quantum cosmology (LQC) is an application of loop quantum gravity theory in the context of spacetimes with a high degree of symmetry (e.g. homogeneity, isotropy). One of the main successes of LQC is the resolution of "singularities" that generically appear in the classical theory. An example of this is the "big bang" singularity that causes a complete breakdown of general relativity (GR) in the very early universe. Models studied within the framework of LQC replace this "big bang" with a "big bounce" and do not suffer a singular breakdown like in the classical theory.


It is, therefore, natural to consider applying similar techniques to study black holes; after all, these solutions of GR are also plagued with a central singularity. In addition, it is plausible that a LQC model may shed some light on long-standing issues in black hole physics, i.e., information loss, Hawking evaporation, firewalls, etc.


Now, if one restricts the model only to the Schwarzschild black hole interior region, the spacetime can actually be considered as a homogeneous, anisotropic cosmology (the Kantowski-Sachs spacetime). This allows techniques of LQC to be readily applied to the black hole case. In fact, a good deal of study has been performed in this direction by Ashtekar, Bojowald, Modesto and many others for over a decade. While these models are able to resolve the central black hole singularity and include important improvements over previous versions, they still have a number of issues.


Recently, Singh and Corichi (2016), proposed a new LQC model for the black hole interior that attempts to address these issues. In this talk, Singh describes some of the resulting phenomenology that emerges from that improved model.

The main emphasis of this talk is on the following questions:
 

(1) Is the "bounce" in the context of a black hole LQC model, i.e., transition from a black hole to a white hole, symmetric? Isotropic and homogeneous models in LQC have generally exhibited symmetric bounces. But, that is not expected to hold in the context of more general models.
(2) Does quantum gravity play a role only once during the bounce process?
(3) What quantitative statements can be made about the time-scales of this process; and what are the full implications of those details?
 (4) Do all black holes, independent of size, exhibit very similar characteristics?


Based on detailed numerical calculations that Singh reviews in his presentation, he uncovers the following features from this model:


(1) The bounce is indeed not symmetric; for example, the sizes of the parent black hole and the offspring white hole are widely different. Other details on this asymmetry appear below.
(2) Two distinct quantum regimes appear in this model, with very different associated time-scales.
(3) In terms of the proper time of an observer, the time spent in the quantum white hole geometry is much larger than in the quantum black hole. And, in particular, the time for the observer to reach the white hole horizon is exceedingly large. This also implies that the formation of the white hole interior geometry happens a lot quicker than the formation of its horizon.
(4) The relation of the bounce time with the black hole mass, does depend on whether the black hole is large or small.


On the potential implications of such details on some of the important open questions in black hole physics, Singh speculates:


(1) For large black holes, the time to develop a white hole (horizon) is much larger than the Hawking evaporation time. This may suggest that for an external observer, a black hole would disappear long before the white hole appears!
(2) For small black holes, the time to form a white hole is smaller than Hawking time, i.e., small black holes explode before they can evaporate!


These could have some interesting implications for the various proposed black hole evaporation paradigms. Given the concreteness of the results Singh presents, they are also likely to be relevant to the many previous phenomenological studies on black hole to white hole transitions including Planck stars.


The two main limitations of Singh's results are: (1) the current model ignores the black hole exterior entirely; and (2) the conclusions rely on effective dynamics, and not the full quantum evolution. These may be addressed in future work.


















Tuesday, March 28, 2017

Holographic signatures of resolved cosmological singularities

Tuesday, March 21st
Norbert Bodendorfer, LMU Munich
Title: Holographic signatures of resolved cosmological singularities 
PDF of the talk (2M)
Audio+Slides [.mp4 10MB]
By Jorge Pullin, Louisiana State University

One of the most important results in string theory is the so called “Maldacena conjecture” or “AdS/CFT correspondence” proposed by Juan Maldacena. This conjecture states that given a space-time with cosmological constant (known as anti De Sitter space-time or AdS) the behavior of gravity in it is equivalent to the behavior of a field theory living on the boundary of the space-time. These field theories are of a special type known as “conformal field theories”. Hence the AdS/CFT name. Conformal field theories are considerably better understood than quantum gravity so to make the latter equivalent to them opens several new possibilities. The discussion of AdS/CFT has mostly taken place in the context of string theory which has general relativity as a classical limit. This opens the question of what kind of imprint the singularities that are known to exist in general relativity leave in the conformal field theory.

On the other hand, loop quantum gravity is known for eliminating the singularities that arise in general relativity. They get replaced by regions of high curvature and fluctuations of it that are not well described by a semiclassical geometry. However, nothing is singular, physical variables may take large –but finite-values. If AdS/CFT were to hold in the context of loop quantum gravity the question arises of what imprint would the elimination of the singularity leave on the conformal field theory. The seminar dealt with this point by considering certain functions known as correlation functions in the conformal field theory that characterize its behavior. In particular how the singularities of general relativity get encoded in these correlation functions and how their elimination in loop quantum gravity changes them. The work is at the moment only a model in five dimensions of a particular space-time known as the Kasner space-time.

Future work will consist in expanding the results to other space-times. Of particular interest would be the extension to black hole spacetimes, which loop quantum gravity also rids of singularities. As is well known, black hole space-times have the problem of the “information paradox” stemming from the fact that black holes evaporate through the radiation that Hawking predicted leaving in their wake only thermal radiation no matter what process led to the formation of the black hole. It is expected that when the evaporation is viewed in terms of the conformal field theory, this loss of information about what formed the black hole will be better understood.

In addition to the specific results, the fact that this work suggests points of contact between loop quantum gravity and string theory makes it uniquely exciting since both fields have developed separately over the years and could potentially benefit from cross pollination of ideas.

Wednesday, February 22, 2017

Gravity as the dimensional reduction of a theory of forms in six or seven dimensions

Tuesday, February 21st
Kirill Krasnov, University of Nottingham
Title: 3D/4D gravity as the dimensional reduction of a theory of differential forms in 6D/7D 
PDF of the talk (5M)
Audio+Slides [.mp4 16MB]


by Jorge Pullin, Louisiana State University


Ordinary field theories, like Maxwell’s electromagnetism, are physical systems with infinitely many degrees of freedom. Essentially the values of the fields at all the points of space are the degrees of freedom. There exist a class of field theories that are formulated as ordinary ones in terms of fields that take different values at different points in space,  but that whose equations of motion imply that the number of degrees of freedom are finite. This makes some of them particularly easy to quantize. A good example of this is general relativity in two space and one time dimensions (known as 2+1 dimensions). Unlike general relativity in four-dimensional space-time, it only has a finite number of degrees of freedom that depend on the topology of the space-time considered. This type of behavior tends to be generic for these types of theories and as a consequence they are labeled Topological Field Theories (TFT). These types of theories have encountered application in mathematics to explore geometry and topology issues, like the construction of knot invariants, using quantum field theory techniques. These theories have the property of not requiring any background geometric structure to define them unlike, for instance, Maxwell theory, that requires a given metric of space-time in order to formulate it.

Remarkably, it was shown some time ago by Plebanski, in 1977 and later further studied by Capovilla-Dell-Jacobson and Mason in 1991 that certain four dimensional TFTs, if supplemented by additional constraints among their variables, were equivalent to general relativity. The additional constraints had the counterintuitive effect of adding degrees of freedom to the theory because they modify the fields in terms of which the theory is formulated. Formulating general relativity in this fashion leads to new perspectives on the theory. In particular it suggests certain generalizations of general relativity, which the talk refers to as deformations of GR.

The talk considered a series of field theories in six and seven dimensions. The theories do not require background structures for their definition but unlike the topological theories we mentioned before, they do have infinitely many degrees of freedom. Then the dimensional reduction to four dimensional of these theories was considered. Dimensional reduction is a procedure in which one “takes a lower dimensional slice” of a higher dimensional theory, usually by imposing some symmetry (for instance assuming that the fields do not depend on certain coordinates). One of the first such proposals was considered in 1919 by Kaluza and further considered later by Klein so it is known as Kaluza-Klein theory. They considered general relativity in five dimensions and by assuming the metric does not depend on the fifth coordinate, were able to show that the theory behaved like four-dimensional general relativity coupled to Maxwell’s electromagnetism and a scalar field. In the talk it was shown that the seven dimensional theory considered, when reduced to four dimensions, was equivalent to general relativity coupled to a scalar field. The talk also showed that certain topological theories in four dimensions known as BF theories (because the two variables of the theory are fields named B and F) can be viewed as dimensional reductions from topological theories in seven dimensions and finally that general relativity in 2+1 dimensions can be viewed as a reduction of a six dimensional topological theory.


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At the moment is not clear whether these theories can be considered as describing nature, because it is not clear whether the additional scalar field that is predicted is compatible with the known constraints on scalar-tensor theories. However, these theories are useful in illuminating the structures and dynamics of general relativity and connections to other theories.


Tuesday, February 7, 2017

Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy

Tuesday, February 7th
Muxin Han, Florida Atlantic University
Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy 
PDF of the talk (2M)
Audio+Slides [.mp4 18MB]
by Jorge Pullin, Louisiana State University


The cosmological constant is an extra term that was introduced into the equations of General Relativity by Einstein himself. At the time he was trying to show that if one applied the equations to the universe as a whole, they had static solutions. People did not know in those days that the universe expanded. Some say that Einstein called the introduction of this extra term his “biggest blunder” since it prevented him from predicting the expansion of the universe which was observed experimentally by Hubble a few years later. In spite of its origin, the term is allowed in the equations and the space-times that arise when one includes the term are known as de Sitter space-times in honor of the Dutch physicist who first found some of these solutions. Depending on the sign of the cosmological constant chosen, one could have de Sitter or anti-de Sitter (AdS) space-times.


It was observed in the context of string theory that if one considered quantum gravity in anti-de Sitter space-times, the theory was equivalent to a certain class of field theories known as conformal field theories (CFT) living on the boundary of the space-time. The result is not a theorem but a conjecture, known as AdS/CFT or Maldacena conjecture. It has been verified in a variety of examples. It is a remarkable result. Gravity and conformal field theories are very different in many aspects and the fact that they could be mapped to each other opens many possibilities for new insights. For instance, an important open problem in gravity is the evaporation of black holes. Although nothing can escape a black hole classically, Hawking showed that if quantum effects are taken into account, black holes radiate particles like a black body at a given temperature. The particles take away energy and the black hole shrinks, eventually evaporating completely. This raises the question of what happened to matter that went into the black hole. Quantum mechanics has a property named unitarity that states that ordinary matter cannot turn into incoherent radiation, so this raises the question of how it could happen in an evaporating black hole. In the AdS/CFT picture, since the evaporating black hole would be mapped to a conformal field theory that is unitary, that would provide a way to study quantum mechanically how matter turns into incoherent radiation.


Several authors have connected the AdS/CFT conjecture to a mathematical construction known as tensor networks that is commonly used in quantum information theory. Tensor networks have several points in common with the spin networks that are the quantum states of gravity in loop quantum gravity. This talk spells out in detail how to make a correspondence between the states of loop quantum gravity and the tensor networks, basically corresponding to a coarse graining or averaging at certain scales of the states of quantum gravity. This opens the possibility of connecting results from AdS/CFT with results in loop quantum gravity. In particular the so-called Ryu-Takahashi formula for the entropy of a region can be arrived from in the context of loop quantum gravity.


Wednesday, January 25, 2017

Symmetries and representations in Group Field Theory

Tuesday, January 24th
Alexander Kegeles, Albert Einstein Institute
Title: Field theoretical aspects of GFT: symmetries and representations 
PDF of the talk (1M)
Audio+Slides [.mp4 11MB]


by Jorge Pullin, Louisiana State University


In loop quantum gravity the quantum states are labeled by loops, more precisely by graphs formed by lines that intersect at vertices and that are “colored”, meaning each line is associated with an integer. They are known as "spin networks". As the states evolve in time these graphs "sweep" surfaces in four dimensional space-time constituting what is known as a “spin foam”. This is a representation of a quantum space-time in loop quantum gravity. The spin foams connect an initial spin network with a final one and the formalism gives a probability for such “transition” from a given spatial geometry to a future spatial geometry to occur. The picture that emerges has some parallel with ordinary particle physics in which particles transition from initial to final states, but also some differences.

However, it was found that one could construct ordinary quantum field theories such that the transition probabilities of them coincided with those stemming from spin foams connecting initial to final spatial geometries in loop quantum gravity. This talk concerns itself with such quantum field theories, known generically as Group Field Theories (GFTs). The talk covered two main aspects of them: symmetries and representations.

Symmetries are important in that they may provide mathematical tools to solve the equations of the theory and identify conserved quantities in it. There is a lot of experience with symmetries in local field theories, but GFT’s are non-local, which adds challenges. Ordinary quantum field theories are formulated starting by a quantity known as the action, which is an integral on a domain. A symmetry is defined as a map of the points of such domain and of the fields that leaves the integral invariant. In GFTs the action is a sum of integrals on different domains. A symmetry is defined as a collections of maps acting on the domains and fields that leave invariant each integral in the sum. An important theorem of great generality stretching from classical mechanics to quantum field theory is Noether’s theorem, that connects symmetries with conserved quantities. The above notion of symmetry for GFTs allows to introduce a Noether’s theorem for them. The theorem could find applicability in a variety of situations, in particular certain relations that were noted between GFTs and recoupling theory and better understand various models based on GFTs.

In a quantum theory like GFTs the quantum states structure themselves into a mathematical set known as Hilbert space. The observable quantities of the theory are represented as operators acting on such space. Hilbert spaces are generically infinite dimensional and this introduces a series of technicalities both in their own definition and in the definition of observables for quantum theories. In particular one can find different families of inequivalent operators related to the same physical observables. This is what is known as different representations of the algebra of observables. Algebra in this context means that one can compose observables to form either new observables or linear combinations of known observables. An important type of representation in quantum field theory is known as Fock representation. It is the representation on which ordinary particles are based. Another type of representations is the condensate representation which, instead of particles, describes their collective (excitations) behaviour and is very convenient for systems with large (infinite) number of particles. A discussion of Fock and condensate like representations in the context of GFTs was presented and the issue of when representations are equivalent or not was also addressed.

Future work looks at generalizing the notion of symmetries presented to find further non-standard symmetries of GFTs. Also investigating “anomalies”. This is when one has a symmetry in the classical theory that may not survive upon quantization. The notion of symmetry can also be used to define an idea of “ground state” or fundamental state of the theory. In ordinary quantum field theory in flat space-time this is done by seeking the state with lower energy. In the context of GFTs one will invoke more complicated notions of symmetries to define the ground state. Several other results of ordinary field theories, like the spin statistics theorem, may be generalizable to the GFT context using the ideas presented in this talk.


Friday, March 11, 2016

Symmetry reductions in loop quantum gravity






Tuesday, Dec. 8th
Norbert Bodendorfer, Univ. Warsaw 
Title: Quantum symmetry reductions based on classical gauge fixings 
PDF of the talk (1.4MB)
Audio [.wav 35MB]
YouTube.

Tuesday, Nov. 10th
Jedrzej Swiezewski, Univ. Warsaw 
Title: Developments on the radial gauge 
PDF of the talk (4MB)
Audio [.mp3 40MB]

by Steffen Gielen, Imperial College


A few months ago, physicists around the world celebrated the centenary of the field equations of general relativity, presented by Einstein to the Prussian Academy of Sciences in November 1915. Arriving at the correct equations was the culmination of an incredible intellectual effort by Einstein, driven largely by mathematical requirements that the new theory of gravitation (superseding Newton's theory of gravitation, which proved ultimately incomplete) should satisfy. In particular, Einstein realized that its field equations should be generally covariant – they should take the same general form in any coordinate system that one chooses to use for the calculation, say whether one uses Cartesian, cylindrical, or spherical coordinates. This property sets the equations of general relativity apart from Newton's laws of motion, where changing coordinate system can lead to the appearance of additional “forces” such as centripetal or Coriolis forces.

Many conferences were held honoring the anniversary of Einstein's achievement. What was discussed at those conferences was partially the historical context, the beauty of the form of the equations, or the precise mathematical and conceptual significance of general covariance. However, the most important legacy of general relativity and the main inspiration for modern research have been the new physical phenomena that appear in general relativity but not in Newtonian gravity: black holes are regions of spacetime where gravity becomes so strong that not even light can escape; the strong gravitational field outside a black hole leads to a time dilation so strong that an hour nearby a black hole can correspond to years on Earth, as used recently in the film Interstellar; and we now believe that the universe as a whole is expanding, and has been since the Big Bang which is thought of as the beginning of space and time.

In order to understand these dramatic consequences of the Einstein equations, physicists had to find solutions to these equations. This is rather challenging in general: the Einstein equations are complicated differential equations for ten functions, depending on one time and three space dimensions, that encode the gravitational field of spacetime. Furthermore, the conceptually appealing property of general covariance means that apparently different solutions of the equations can be simply the same physical configuration looked at in different coordinates. Indeed, both issues – finding solutions to the equations at all and understanding their meaning – were challenges in the early days of the theory, when physicists tried to make sense of Einstein's equations.

Despite this formidable challenge, the Prussian lieutenant of the artillery Karl Schwarzschild, while serving on the Eastern front in World War I, was able to derive an exact solution of Einstein's equations in vacuum within weeks of their publication, much to the surprise of Einstein himself. This solution, now called the Schwarzschild solution, describes a black hole, and is one of the most important solutions of general relativity. What Schwarzschild did in order to solve the equations was to assume a symmetry of the solution: he assumed that the configuration of the gravitational field should be spherically symmetric. In spherical coordinates, where each point in space is specified by one radial and two angular coordinates, it should be independent of any change in the angular directions. This means that one describes space as a collection of regular, concentric spheres. What Schwarzschild found was that the spheres did not have to be glued together to simply give normal flat space, but one could form a curved geometry out of them, with curvature increasing as one heads towards the centre (eventually forming a black hole), while still solving Einstein's equations. To be able to do the calculation, Schwarzschild had to choose a particularly suitable coordinate system, hence exploiting the property of general covariance in his favor.

This strategy of finding solutions is typical for practitioners of general relativity: cosmological solutions could similarly be found by assuming that the universe looks exactly the same at each point and in each direction in space (in mathematical terms, it is homogeneous and isotropic), and only changes in time. This reduces the problem of solving Einstein equations to a much simpler task, and explicit solutions could be written down, again in a suitable coordinate system. These simplest solutions already exhibit the main features of our universe (overall expansion and an initial Big Bang singularity) and are fairly realistic – indeed our Universe seems to display only small variations between different large-scale regions, and at the very largest scales is, within an approximation, well described by a geometry that simply looks the same everywhere in space.

Loop quantum gravity is an approach at a quantization of general relativity, aiming to extend general relativity by making it compatible with quantum mechanics. What distinguishes it from other approaches is that the main property of general relativity, general covariance, is taken as a central guiding principle towards the construction of a quantum theory. In some respects, the status of loop quantum gravity can be compared to the early days of general relativity: while it is now known that a quantum theory compatible with general covariance can be constructed, and its mathematical structure is well understood, one now needs to understand the new physical phenomena implied by the quantization, beyond general relativity. Just as in the time after November 1915, today's physicists should find explicit solutions to the equations of loop quantum gravity that can be used to study the physical implications of the (relatively) new framework.

One of the main successes of loop quantum gravity has been its application to cosmology. Homogeneous solutions of the Einstein equations that approximately describe our universe have been shown to receive modifications once loop quantum gravity techniques are used, leading to a resolution of the Big Bang singularity by a Big Bounce, and potentially observable quantum effects. However, the resulting models of the universe are not solutions of the full theory of loop quantum gravity: rather, they arise from quantization of a reduced set of solutions of classical general relativity with loop quantum gravity techniques. There is no reason, in general, to expect that these are exact solutions of loop quantum gravity. Quantum mechanics is funny: quantization can lead to many inequivalent theories, depending on how one decides to do it. By assuming that the universe is homogeneous from the outset, one obtains a quantum theory of only a finite, rather than an infinite number of “degrees of freedom”. It is well known that quantum theories can behave differently depending on whether they have a finite or infinite number of degrees of freedom.

In their ILQGS seminars, Jedrzej and Norbert presented work towards resolving this tension. Namely, they presented an approach in which, similar to how Schwarzschild and contemporaries proceeded 100 years ago, one identifies a suitable coordinate system in which the spacetime metric, representing the gravitational field, is represented. In a quantum theory where general covariance is implemented fundamentally, this means one has to perform a “gauge-fixing”; the freedom of changing the coordinate system must be “fixed” consistently in the quantum theory. Gauge-fixings mean that one works with fewer variables, and has to worry less about different but physically equivalent solutions that are only related by changes in the coordinate system. Achieving them is often quite hard technically. Together with collaborators in Warsaw, Jedrzej and Norbert have made progress on this issue in recent years.

The second step, after a convenient coordinate system (think of spherical coordinates for treating the Schwarzschild black hole) has been chosen, is to do a “symmetry reduction” in the full quantum theory: rather than on the most general quantum universes, one now focusses on those that have a certain symmetry property. Norbert showed a detailed strategy for how to do this. One identifies an equation satisfied by all classical solutions with the desired symmetry, such as isotropy (i.e. looking the same in all directions). The quantum version of this equation is then imposed in loop quantum gravity, leading to a full quantum definition of symmetries like “isotropy” or “spherical symmetry” in loop quantum gravity. The obvious applications of the mechanism, which are being explored at the moment, are identifying cosmological and black hole solutions in loop quantum gravity, studying their dynamics, and verifying whether the resulting effects are in accord with what has been found in the simpler finite-dimensional quantum models described above. In particular, one would like to know whether singularities inside black holes and at the Big Bang, where Einstein's theory simply breaks down, can be resolved by quantum mechanics, as is hoped.


Jedrzej also showed how the methods developed in different “gauge-fixings” for classical general relativity could be used to resolve a disputed issue in the context of the AdS/CFT correspondence in string theory, where one faces a similar problem of fixing the huge freedom under changes in the coordinate system in order to identify the invariant physical properties of spacetime. In particular, a certain choice of gauge-fixing has been discussed in AdS/CFT, which leads to unfamiliar consequences such as non-locality in the gauge-fixed version of the theory. The tools developed by Jedrzej and collaborators could be used to clarify precisely how this non-locality occurs. They hence provide a somewhat unusual example of the application of methods developed for loop quantum gravity in a string theory-motivated context, clearly a positive example that can inspire more work on closer connections between methods used in these different communities. 

Monday, May 25, 2015

Separability and quantum mechanics


Tuesday, Apr 21st
Fernando Barbero, CSIC, Madrid 
Title: Separability and quantum mechanics 
PDF of the talk (758k)
Audio [.wav 20MB]

by Juan Margalef-Bentabol, UC3M-CSIC, Madrid

Classical vs Quantum: Two views of the world

In classical mechanics it is relatively straightforward to get information from a system. For instance, if we have a bunch of particles moving around, we can ask ourselves: where is its center of mass? What is the average speed of the particles? What is the distance between two of them? In order to ask and answer such questions in a precise mathematical way, we need to know all the positions and velocities of the system at every moment; in the usual jargon, we need to know the dynamics over the state space (also called configuration space for positions and velocities, or phase space when we consider positions and momenta). For example, the appropriate way to ask for the center of mass, is given by the function that for a specific state of the system, gives the weighted mean of the positions of all the particles. Also, the total momentum of the system is given by the function consisting of the sum of the momenta of the individual particles. Such functions are called observables of the theory, therefore an observable is defined as a function that takes all the positions and momenta, and returns a real number. Among all the observables there are some ones that can be considered as fundamental. A familiar example is provided by the generalized position and momenta denoted as and .

In a quantum setting answering, and even asking, such questions is however much trickier. It can be properly justified that the needed classical ingredients have to be significantly changed:
  1. The state space is now much more complicated, instead of positions and velocities/momenta we need a (usually infinite dimensional) complex vector space with an inner product that is complete. Such vector space is called a Hilbert space and the vectors of are called states (up to a complex multiplication).
  2. The observables are functions from to itself that "behave well" with respect to the inner product (these are called self-adjoint operators). Notice in particular that the outputs of the quantum observables are complex vectors and not numbers anymore!
  3. In a physical experiment we do obtain real numbers, so somehow we need to retrieve them from the observable associated with the experiment. The way to do this is by looking at the spectrum of , which consists of a set of real numbers called eigenvalues associated with some vectors called eigenvectors (actually the number that we obtain is a probability amplitude whose absolute value squared is the probability of obtaining as an output a specific eigenvector).
The questions that arise naturally are: how do we choose the Hilbert space? how do we introduce fundamental observables analogous to the ones of classical mechanics? In order to answer these questions we need to take a small detour and talk a little bit about the algebra of observables.

Algebra of Observables

Given two classical observables, we can construct another one by means of different methods. Some important ones are:
  • By adding them (they are real functions)
  • By multiplying them
  • By a more sophisticated procedure called the Poisson bracket
The last one turns out to be fundamental in classical mechanics and plays an important role within the Hamiltonian form of the dynamics of the system. A basic fact is that the set of observables endowed with the Poisson bracket forms a Lie algebra (a vector space with a rule to obtain an element out of two other ones satisfying some natural properties). The fundamental observables behave really well with respect to the Poisson bracket, namely they satisfy simple commutation relations i.e. if we consider the - position observable and "Poisson-multiply" it by the - momentum observable, we obtain the constant function if , or the constant function if .

One of the best approaches to construct a quantum theory associated with a classical one, is to reproduce at the quantum level some features of its classical formulation. One way to do this is to define a Lie algebra for the quantum observables such that some of such observables mimic the behavior of the Poisson bracket of some classical fundamental observables. This procedure (modulo some technicalities) is known as finding a representation of this algebra. In order to do this, one has to choose:
  1. A Hilbert space .
  2. Some fundamental observables that reproduce the canonical commutation relations when we consider the commutator of operators.
In standard Quantum Mechanics the fundamental observables are positions and momenta. It may seem that there is a great ambiguity in this procedure, however there is a central theorem due to Stone and von Neumann that states that, under some reasonable hypothesis, all the representations are essentially the same.

Separability

One of the hypotheses of the Stone-von Neumann theorem is that the Hilbert space must be separable. This means that it is possible to find a countable set of orthonormal vectors in (called Hilbert basis) such that any state -vector- of can be written as an appropriate countable sum of them. A separable Hilbert space, despite being infinite dimensional, is not "too big", in the sense that there are Hilbert spaces with uncountable bases that are genuinely larger. The separability assumption seems natural for standard quantum mechanics, but in the case of quantum field theory -with infinitely many degrees of freedom- one might expect to need much larger Hilbert spaces i.e. non separable ones. Somewhat surprisingly, most of the quantum field theories can be handled with our beloved and "simple" separable Hilbert spaces with the remarkable exception of LQG (and its derivative LQC) where non separability plays a significant role. Henceforth it seems interesting to understand what happens when one considers non separable Hilbert spaces [3] in the realm of the quantum world. A natural and obvious way to acquire the necessary intuition is by first considering quantum mechanics on a non-separable Hilbert space.

The Polymeric Harmonic Oscillator

The authors of [2,3] discuss two inequivalent (among the infinitely many) representations of the algebra of fundamental observables which share a non familiar feature, namely, in one of them (called the position representation) the position observable is well defined but the momentum observable does not even exist; in the momentum representation the roles of positions and momenta are exchanged. Notice that in this setting, some familiar features of quantum mechanics are lost for good. For instance, the position-momentum Heisenberg uncertainty formula makes no sense at all as both position and momentum observables need to be defined.

To improve the understanding of such systems and gain some insight for the application to LQG and LQC, the authors of [1] (re)study the -dimensional Harmonic Oscillator (PHO) in a non separable Hilbert space (known in this context as a polymeric Hilbert space). As the space is non separable, any Hilbert basis should be uncountable. This leads to some unexpected behaviors that can be used to obtain exotic representations of the algebra of fundamental observables.

The motivation to study the PHO is kind of the same as always: the HO, in addition to being an excellent toy model, is a good approximation to any 1-dimensional mechanical system close to its equilibrium points. Furthermore, free quantum field theories can be thought of as ensembles of infinitely many independent HO's. There are however many ways to generalize the HO to a non separable Hilbert space and also many equivalent ways to realize a concrete representation, for instance by using Hilbert spaces based on:
The eigenvalue equations in these different spaces take different forms: in some of them they are difference equations, whereas in others they have the form of the standard Schrödinger equation with a periodic potential. It is important to notice nonetheless that writing Hamiltonian observables in this framework turn out to be really difficult, as only one of the position or momentum observables can be strictly represented. This means that for the other one it is necessary to rely on some kind of approximation (that can be obtained by introducing an arbitrary scale) and choosing a periodic potential with minima corresponding to the one of the quadratic operator. The huge uncertainty in this procedure has been highlighted by Corichi, Zapata, Vukašinac and collaborators. The standard choice leads to an equation known as the Mathieu equation but other simple choices have been explored, as the one shown in the figure.










Energy eigenvalues (bands) of a polymerized harmonic oscillator. The horizontal axis shows the position (or the momentum depending on the chosen representation), the vertical axis is the energy and the red line represents the particular periodic extension of the potential used to approximate the usual quadratic potential of the HO. The other lines plotted in this graph correspond to auxiliary functions that can be used to locate the edges of the bands that define the point spectrum in the present example.

As we have already mentioned, the orthonormal bases in non separable Hilbert spaces are uncountable. A consequence of this is the fact that the orthonormal basis provided by the eigenstates of the Hamiltonian must be uncountable, i.e. the Hamiltonian must have an uncountable infinity worth of eigenvalues (counted with multiplicity). A somewhat unexpected result that can be proved by invoking classical theorems on functional analysis in non-separable Hilbert spaces is the fact that these eigenvalues are gathered in bands. It is important to point out here that only the lowest-lying part of the spectrum is expected to mimic reasonably well the one corresponding to the standard HO, however it is important to keep also in mind the huge difference that persists: even the narrowest bands contain a continuum of eigenvalues.

Some physical consequences

The fact that the spectrum of the polymerized harmonic oscillator displays this band structure is relevant for some applications of polymerized quantum mechanics. Two main issues were mentioned in the talk. On one hand the statistical mechanics of polymerized systems must be handled with due care. Owing to the features of the spectrum, the counting of energy eigenstates necessary to compute the entropy in the microcanonical ensemble is ill defined. A similar problem crops up when computing the partition function of the canonical ensemble. These problems can probably be circumvented by using an appropriate regularization and also by relying on some superselection rules that eliminate all but a countable subset of energy eigenstates of the system.

A setting where something similar can be done is in the polymer quantization of the scalar field (already considered by Husain, Pawłowski and collaborators). As this system can be thought of as an infinite ensemble of harmonic oscillators, the specific features of their (polymer) quantization will play a significant role. A way to avoid some difficulties here also relies on the elimination of unwanted energy eigenvalues by imposing superselection rules as long as they can be physically justified.

Bibliography

[1] J.F. Barbero G., J. Prieto and E.J.S. Villaseñor, Band structure in the polymer quantization of the harmonic oscillator, Class. Quantum Grav. 30 (2013) 165011.
[2] W. Chojnacki, Spectral analysis of Schrodinger operators in non-separable Hilbert spaces, Rend. Circ. Mat. Palermo (2), Suppl. 17 (1987) 135–51.
[3] H. Halvorson, Complementarity of representations in quantum mechanics, Stud. Hist. Phil. Mod. Phys. 35 (2004) 45-56.