Tag Archives: GFT

Group field theory as the 2nd quantization of Loop Quantum Gravity by Daniele Oriti

This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference  paper – Group field theory: A quantum field theory for the atoms of space  and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:

Introduction

We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,
there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus
define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be  the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical
dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators  and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is  obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization  and Effective quantum cosmology from GFT condensates.

Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise

gfteqtfig1

The group elements Gij are assigned to each link of the graph, with Gij=Gij-1. Assume  gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

Spin networks in 2nd quantization

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum –  the natural background independent, diffeo-invariant vacuum state.

The  2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible
with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field  quantum is the spin network vertex or tetrahedron – the so called building block of space.

gftfig4

A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

gftfig5The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties  from results in quantum simplicial geometry.

 Related articles

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Quantum cosmology of loop quantum gravity condensates: An example by Gielen

This week I have mainly been studying the work done during the Google Summer of Code workshops, in particular that on sagemath knot theory at:

This work looks great and I’ll be using the results in some of my calculations later in the summer.

Another topic I’ve been reviewing is the idea of spacetime as a Bose -Einstein condensate. This together with emergent, entropic and  thermodynamic gravitation seem to be an area into which the quantum tetrahedron approach could naturally fit via statistical mechanics.

In the paper, Quantum cosmology of loop quantum gravity condensates, the author reviews the idea that spatially homogeneous universes can be described in loop quantum gravity as condensates of elementary excitations of space. Their treatment by second-quantised group field theory formalism allows the adaptation of techniques from the description of Bose–Einstein condensates in condensed matter physics. Dynamical equations for the states can be derived directly from the underlying quantum gravity dynamics. The analogue of the Gross–Pitaevskii equation defines an anisotropic  quantum cosmology model, in which the condensate wavefunction becomes a quantum cosmology wavefunction on minisuperspace.

Introduction
The spacetimes relevant for cosmology are to a very good approximation spatially homogeneous. One can use this fact and perform a symmetry reduction of the classical theory – general relativity coupled to a scalar field or other matter – assuming spatial
homogeneity, followed by a quantisation of the reduced system. Inhomogeneities are usually added perturbatively. This leads to models of quantum cosmology which can be studied  without the need for a full theory of quantum gravity.

Loop quantum gravity (LQG) has some of the structures one would expect in a full theory of quantum gravity: kinematical states corresponding to functionals of the Ashtekar–Barbero connection can be rigorously defined, and geometric observables such
as areas and volumes exist as well-defined operators, typically with discrete spectrum. The use of the LQG formalism in quantising symmetry-reduced gravity leads to loop quantum cosmology (LQC).

Because of the well-defined structures of LQG, LQC allows a rigorous analysis of issues that could not be addressed within the Wheeler– DeWitt quantisation used in conventional quantum cosmology, such as a definition of the physical inner product. More recently, LQC has made closer contact with CMB observations, and the usual inflationary scenario is now discussed within LQC.

A new approach towards addressing the issue of how to describe cosmologically relevant universes in loop quantum gravity uses the group field theory (GFT) formalism, itself a second quantisation formulation of the kinematics and dynamics of LQG: one has a Fock space of LQG spin network vertices or tetrahedra, as building blocks of a simplicial complex, annihilated and created by the field operator ϕ and its Hermitian conjugate ϕ†, respectively. The advantage of using this reformulation is that field-theoretic techniques are available, as a GFT is a standard quantum field theory on a curved group manifold. In particular, one can define coherent or squeezed states for the GFT field, analogous to states used in the physics of Bose– Einstein condensates or in quantum optics; these represent quantum gravity condensates. They describe a large number of degrees of freedom of quantum geometry in the same microscopic quantum state, which is the analogue of homogeneity for a differentiable metric geometry. After embedding a condensate of tetrahedra into a smooth manifold representing a spatial hypersurface, one shows that the spatial metric in a fixed frame reconstructed from the quantum state is compatible with spatial homogeneity. As the number of tetrahedra is taken to infinity, a continuum homogeneous metric can be approximated to a better and better degree.

At this stage, the condensate states defined in this way are kinematical. They are gauge-invariant by construction, and represent geometric data invariant under spatial diffeomorphisms. The strategy followed for extracting information about the dynamics of these states is the use of Schwinger–Dyson equations of a given GFT model. These give constraints on the n point functions of the theory evaluated in a given condensate state – approximating a non-perturbative vacuum, which can be translated into differential equations for the condensate wavefunction used in the definition of the state. This is analogous to condensate states in many-body quantum physics, where such an expectation value gives, in the simplest case, the Gross–Pitaevskii equation for the condensate
wavefunction. The truncation of the infinite tower of such equations to the simplest ones is part of the approximations made. The effective dynamical equations obtained can be viewed as defining a quantum cosmology model, with the condensate wavefunction interpreted as a quantum cosmology wavefunction. This provides a general procedure for deriving an effective cosmological dynamics directly from the underlying theory of quantum gravity. It canbe shown that  a particular quantum cosmology equation of this type, in a semiclassical WKB limit and for isotropic universes, reduces to the classical Friedmann equation of homogeneous,
isotropic universes in general relativity.

See posts:

Let’s  analyse more carefully the quantum cosmological models derived from quantum gravity condensate states in GFT. In particular, the formalism identifies the gauge-invariant configuration space of a tetrahedron with the minisuperspace of homogeneous generally anisotropic geometries.

Using a convenient set of variables the gauge-invariant geometric data, can be mapped to the variables of a general anisotropic Bianchi model it is possible to  find simple solutions to the full quantum equation, corresponding to isotropic universes.

They can only satisfy the condition of rapid oscillation of the WKB approximation for large positive values of the coupling μ in the GFT model. For μ < 0, states are sharply peaked on small values for the curvature, describing a condensate of near-flat building blocks, but these do not oscillate. This supports the view that rather than requiring semiclassical behaviour at the Planck scale, semiclassicality should be imposed only on large-scale observables.

 From quantum gravity condensates to quantum cosmology

Review the relevant steps in the construction of effective quantum cosmology equations for quantum gravity condensates. Use group field theory (GFT) formalism, which is a second quantisation formulation of loop quantum gravity spin networks of fixed valency, or their dual interpretation as simplicial geometries.

The basic structures of the GFT formalism in four dimensions are a complex-valued field ϕ : G⁴ → C, satisfying a gauge invariance property

gft1equ1

and the basic non-relativistic commutation relations imposed in the quantum theory

gft1equ2

These relations  are analogous to those of non-relativistic scalar field theory, where the mode expansion of the field operator defines annihilation operators.

In GFT, the domain of the field is four copies of a Lie group G, interpreted as the local gauge group of gravity, which can be taken to be G = Spin(4) for Riemannian and G = SL(2,C) for Lorentzian models. In loop quantum gravity, the gauge group is the one given by the classical Ashtekar–Barbero formulation, G = SU(2). This property encodes invariance under gauge transformations acting on spin network vertices.

The Fock vacuum |Ø〉 is analogous to the diffeomorphism-invariant Ashtekar–Lewandowski vacuum of LQG, with zero expectation value for all area or volume operators. The conjugate  ϕ acting on the Fock vacuum |Ø〉  creates a GFT particle, interpreted as a 4-valent spin network vertex or a dual tetrahedron:

gft1equ4

The geometric data attached to this tetrahedron, four group elements gI ∈ G, is interpreted as parallel transports of a gravitational connection along links dual to the four faces. The LQG interpretation of this is that of a state that fixes the parallel transports of the Ashtekar–Barbero connection to be gI along the four links given by the spin network, while they are undetermined everywhere else.

In the canonical formalism of Ashtekar and Barbero, the canonically conjugate variable to the connection is a densitised inverse triad, with dimensions of area, that encodes the spatial metric. The GFT formalism can be translated into this momentum space formulation by use of a non-commutative Fourier transform

gft1equ5

The geometric interpretation of the variables B ∈ g is as geometric bivectors associated to a spatial triad e, defined by the integral triadover a face △ of the tetrahedron. Hence, the one-particle state

gft1equ6

Defines a tetrahedron with minimal uncertainty in the fluxes, that is the oriented area elementstriad given by B . In the LQG interpretation this state completely determines the metric variables for one tetrahedron, while being independent of all other degrees of freedom of geometry in a spatial hypersurface.

The idea of quantum gravity condensates is to use many excitations over the Fock space vacuum all in the same microscopic configuration, to better and better approximate a smooth homogeneous metric or connection, as a many-particle state can contain information about the connection and the metric at many different points in space. Choosing this information such that it is compatible with a spatially homogeneous metric while leaving the particle number N free, the limit N → ∞ corresponds to a continuum limit in which a homogeneous metric geometry is recovered.

In the simplest case, the definition for GFT condensate states is

gft1equ7

where N(σ) is a normalisation factor. The exponential creates a coherent configuration of many building blocks of geometry. At fixed particle number N, a state of the form σⁿ|Ø〉 would be interpreted as defining a metric (or connection) that looks spatially homogeneous when measured at the N positions of the tetrahedra, given an embedding into space usually there is a sum over all possible particle numbers. The condensate picture does not use a fixed graph or discretisation of space.

The GFT condensate is defined in terms of a wavefunction on G⁴
invariant under separate left and right actions of G on G⁴ . The strategy is then to demand that the condensate solves the GFT quantum dynamics, expressed in terms of the Schwinger–Dyson equations which relate different n-point functions for the condensate. An important approximation is to only consider the simplest Schwinger– Dyson equations, which will give equations of the form

gft1equ10

This is analogous to the case of the Bose–Einstein condensate where the simplest equation of this typegives the Gross–Pitaevskii equation.

In the case of a real condensate, the condensate wavefunction Ψ (x), corresponding to a nonzero expectation value of the field operator, has a direct physical interpretation: expressing it in terms of amplitude and phase, psi one can rewrite the
Gross–Pitaevskii equation to discover that ρ(x) and v(x) = ∇θ(x) satisfy hydrodynamic equations in which they correspond to the density and the velocity of the quantum fluid defined by the condensate. Microscopic quantum variables and macroscopic classical variables are directly related.

The wavefunction σ or ξ of the GFT condensate should play a similar role. It is not just a function of the geometric data for a single tetrahedron, but equivalently a function on a minisuperspace of spatially homogeneous universes. The effective dynamics for it, extracted from the fundamental quantum gravity dynamics given by a GFT model, can then be interpreted as a quantum cosmology model.

Minisuperspace – gauge-invariant configuration space of a tetrahedron

Condensate states are determined by a wavefunction σ, which is
a complex-valued function on the space of four group elements for given gauge group G which is invariant under

gft1equ11

is a function on G\G⁴/G. This quotient space is a smooth manifold
with boundary, without a group structure. It is the gauge-invariant configuration space of the geometric data associated to a tetrahedron. When the effective quantum dynamics of GFT condensate states is reinterpreted as quantum cosmology equations, G\G⁴/G becomes a minisuperspace of spatially homogeneous geometries.

Conclusion
Condensate states in group field theory can be used to derive effective quantum cosmology models directly from the dynamics of a quantum theory of discrete geometries. This can be illustrated by the interpretation of the configuration space of gauge-invariant geometric data of a tetrahedron, the domain of the condensate
wavefunction, as a minisuperspace of spatially homogeneous 3-metrics.

I’ll also looking at the calculations behind this paper in more detail in a later post.

 

 

 

 

 

 

 

 

 

 

Disappearance and emergence of space and time in quantum gravity by D. Oriti

This week I’ve been studying GFT condensates, Bose-Einstein condensates and the Gross-Pitaevskii Equation and the emergence of spacetime. I have posted on spacetime as a GFT condensate and numerical work with python on Bose-Einstein condensates. This post is based on D. Oriti’s paper, ‘Disappearance and emergence of space and time in quantum gravity’

In this paper he looks at the disappearance of continuum space and time at a microscopic scale. These include arguments for the discrete spacetimes and non-locality in a quantum theory of gravity. He also discusses how these ideas are realized in specific quantum gravity approaches. He then considers the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space such as quantum tetrahedra, and for understanding spacetime as a kind of condensate and presents the case for this emergence process being the result of a phase transition, called ‘geometrogenesis’. Oriti then discusses some conceptual issues of this scenario and of the idea of emergent spacetime in general. A concrete example is given in the form of the GFT framework for quantum gravity, and he illustrates a procedure for the
emergence of spacetime in this framework.
GEOMETROGENESIS
An example of emergent spacetime in the context of the GFT framework is given in this paper. It aims at extracting cosmological dynamics directly from microscopic GFT models, using the idea of continuum spacetime as a condensate, possibly emerging from a big bang phase transition.

GFTs are defined usually in perturbative expansion around the Fock vacuum. In this approximation, they describe the interaction of quantized simplices and spin networks, in terms of spin foam models and simplicial gravity. The true ground state of the system, however, for non-zero couplings and for generic choices of the macroscopic parameters, will not be the Fock vacuum. The interacting system will organize itself around a new, non-trivial state, as in the case of standard Bose condensates. The relevant ground states for different values of the parameters will correspond to the different macroscopic, continuum phases of the theory, with the dynamical transitions
from one to the other being phase transitions of the physical system called spacetime.
The fact that the relevant ground state for a proper continuum geometric phase would probably not be the GFT Fock vacuum can be argued also on the basis of the pregeomet ”meaning of it: it is a quantum state in which no pregeometric excitations at all are present, no simplices, no spin networks. It is a no space state, the absolute void. It can be the full non-perturbative, diffeo-invariant quantum state around which one defines the theory – in fact, it is analogous to the diffeoinvariant vacuum state of loop quantum gravity, but it is not where to look for effective continuum physics. Hence the need to change vacuum and study the effective geometry and dynamics of a different
one.

As described in my last post it is possible to define an approximation procedure that associates an approximate continuum geometry to the set of data encoded in a generic GFT state. This applies to GFT models whose group and Lie algebra variables admit an interpretation in terms of discrete geometries, i.e. in which the group chosen is SO(3, 1) in the Lorentzian setting or SO(4) in the Riemannian setting and additional simplicity conditions are imposed, in the model, to reduce generic group and Lie algebra elements to discrete counterparts of a discrete tetrad and a discrete gravity connection.

A generic GFT state with a fixed number N of GFT quanta will be associated to a set of 4N Lie algebra elements: BI(m) , with m = 1, …,N running over the set of tetrahedra/vertices, I = 1, …, 4 indicating the four triangles of each tetrahedron. In turn, the geometricity conditions imply that only three elements are independent for each tetrehadron.

 

Emergence - generic quantum state

If the tetrahedra are embedded in a spatial 3-manifold M with a transitive
group action. The embedding is defined by specifying a location of the tetrahedra, i.e. associating for example one of their vertices with a point on the manifold, and three tangent vectors defining a local frame and specifying the directions of the three edges incident at that vertex. The vectors eA canbe interpreted  as continuum tetrad vectors integrated over paths in M corresponding to the edges of the tetrahedron. Then, the variables gij(m) can be used to define the coefficients of continuum metric at a finite number N of points, as: gij(m) = g(xm)(vi(m), vj(m)), invariant under the action of the group SO(4).

The reconstruction of metric coefficients gij(m), at a finite number of points, from the variables associated to a state of N GFT quanta – such as quantum tetrahedra, depends only on the topology of the assumed symmetric manifold M and on the choice of group action H. The approximate metric will be homogenous if it has the same coefficients gij(m) at any m. This captures the notion of the metric being the same at every point. It also implies that the same metric would be also isotropic if H = R3 or H = SU(2).

The quantum GFT states obtained using the above procedure can be interpreted as continuum homogeneous quantum geometries. In such a second quantized setting, the definition of states involving varying and even infinite numbers of discrete degrees of freedom is straightforward, and field theory formalism is well adapted to dealing with their dynamics.

The crucial point, from the point of view of emergent spacetime and of the idea of spacetime as a condensate of quantum pregeometric and not spatio-temporal building blocks is that quantum states corresponding to homogeneous continuum geometries are exactly GFT condensate states. The hypothesis of spacetime as a condensate, as a quantum fluid, is realized in a literal way. The simplest state of this type -one-particle GFT condensate, for which we assume a bosonic quantum statistics, is

Emergence - one particle equ

This describes a coherent superposition of quantum states of arbitrary number of GFT quanta, all of them described by the same distribution ϕo of pregeometric variables. The function ϕo is a collective variable characterizing such continuum geometry, and it depends only on invariant homogeneous geometric data.  It is a second quantized state characterized by the fact that the mean value of the fundamental quantum operator φ is non-zero:

<ϕo|φ(gi)|ϕo> = ϕo(gi),

contrary to what happens in the Fock vacuum.

The effective dynamical equations for the condensate can be extracted directly from the fundamental GFT quantum dynamics.The generic form of the dynamics for the condensate ϕo is, schematically:

Emergence - GPE equ

where Keff and Veff are modified versions of the kinetic and interaction kernels entering the fundamental GFT dynamics, reflecting the approximations needed to interpret ϕo as a cosmological condensate.This is a non-linear and non-local, Gross-Pitaevskii-like equation for the spacetime condensate function ϕo. In the simple case in which:

Emergence - laplace op

and we assume that the function ϕo depends on four such SU(2) variables. The final equation one gets for non-degenerate geometries) is:

Emergence - FW equ

that is the Friedmann equation for a homogeneous universe with constant curvature k.

Numerical work using python 1: Bose-Einsten condensation

In my last post ‘Constructing spacetime from the quantum tetrahedron: Spacetime as a Bose-Einstein Condensate’ I started to explore how spacetime could be formed as a Bose-Einstein condensate of quantum tetrahedra within the framework of Group Field Theory (GFT).

This week I have been adapting some of the algorithms found in Statistical Mechanics: Algorithms and Computations by Werner Krauth  to allow me to begin exploring Spacetime as a Group Field Theory Bose-Einstein Condensate.

First I used a python program to look at how bosons occupy energy levels as the temperature decreases: Basically they all end up occuping the lowest energy level.

BE figure 0

The data obtained is plotted in a graph of occupancy number against Energy – notice how the lower energy levels are more highly occupied that the higher ones.

BE figure_1

Following this I used a python program with a more sophisticated algorithm to explore how the degree of condensation varies with temperature as used this to produce an animation showing the formation of the Bose-Einstein condensate.

BE fig 4

The animation of the data obtained from the python program shows how the Bose-Einstein condensate forms as the temperature is lowered.

Spacetime as Bose-Einstein condensate v2

(Click picture to see animation)

Given the crudeness of the algorithms I am using so far I think this models an actual Bose-Einstein condensate formation quite well.

Actual BE condensate formation