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.
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.
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
and the basic non-relativistic commutation relations imposed in the quantum theory
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:
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
The geometric interpretation of the variables B ∈ g is as geometric bivectors associated to a spatial triad e, defined by the integral over a face △ of the tetrahedron. Hence, the one-particle state
Defines a tetrahedron with minimal uncertainty in the fluxes, that is the oriented area elements 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
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
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, 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
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.
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.