Group field theory

Quantum Gravity

Modern physics rests on two extraordinarily successful descriptions of nature. General relativity explains gravity as the dynamics of spacetime geometry and accurately describes phenomena ranging from planetary motion to black holes, gravitational waves and the evolution of the Universe. Quantum mechanics, with equally remarkable precision, governs the microscopic world. In its modern formulation, particles are understood as excitations of quantum fields: dynamical entities that can create and annihilate particles and mediate their interactions. Quantum-field-theoretic methods can also be applied to gravity and provide a successful description at sufficiently low energies. The outstanding problem is to construct a fundamental quantum theory that remains valid when quantum fluctuations of spacetime itself can no longer be neglected. Achieving this is one of the deepest unresolved problems in theoretical physics.

There are several indications that such a theory is necessary. The singularity theorems of general relativity show that, under broad physical assumptions, the classical description of spacetime becomes incomplete in situations such as the interior of black holes and the beginning of the Universe. In these extreme regimes, general relativity reaches the boundary of its applicability and can no longer determine the continuation of spacetime evolution. Moreover, since matter and all other known fundamental interactions possess quantum fluctuations, consistency strongly suggests that the gravitational field they source must be capable of fluctuating quantum mechanically as well. A further clue comes from the close relationship between gravity and thermodynamics. Black holes possess entropy and temperature, and Einstein’s equations themselves can be interpreted as an equation of state associated with the thermodynamics of spacetime. Much as the thermodynamic behaviour of a fluid reflects the collective motion of microscopic molecules, this suggests that spacetime geometry may arise from more fundamental microscopic constituents.

Group Field Theories

There are many approaches to quantum gravity, emphasizing different aspects of the problem and proposing different candidates for its fundamental degrees of freedom. Group Field Theories, or GFTs, occupy a particularly interesting position within this landscape because they are closely connected with several of these approaches. They provide a field-theoretic formulation of the quantum-geometric structures appearing in loop quantum gravity, generate spin-foam models through their Feynman expansion, and encode discrete gravitational path integrals related to lattice approaches. At the same time, they generalize matrix and tensor models of random geometry to include gravitational data, and share with string field theory the idea of defining a quantum field theory whose elementary excitations are extended or geometric objects rather than point particles.

GFTs develop this idea by extending the logic of quantum field theory to spacetime geometry itself. Their fundamental excitations are not particles propagating through a pre-existing space, but elementary building blocks of quantum geometry. Depending on the model, an individual quantum may be pictured as a quantum tetrahedron or, more generally, as a small polyhedral region carrying discrete geometric information such as areas and volumes. Many such quanta can be combined to form extended quantum geometries, just as many atoms combine to form a macroscopic material.

In this sense, GFTs can be viewed as a discrete realization of the “third quantization” of gravity. Ordinary quantum field theory allows particles to be created and annihilated, so that their number is not fixed. Third quantization applies the same field-theoretic logic to geometry. In GFT, field operators create and annihilate elementary quanta of space, while their interactions describe how these quanta are joined, rearranged and separated. The number of geometric building blocks and the relations among them are therefore dynamical rather than specified in advance.

The gravitational content of the theory is encoded in its quantum amplitudes. As in any quantum field theory, physical processes can be organized in terms of Feynman diagrams. In an ordinary particle theory, such a diagram represents a possible history of particles interacting within an already existing spacetime. In GFT, by contrast, the diagram itself represents a possible discrete spacetime. Its lines and vertices record how elementary quanta of geometry are assembled, while the corresponding Feynman amplitude takes the form of a gravitational path integral for the resulting discrete geometry. The quantum dynamics of GFT therefore generates both possible spacetime structures and the amplitudes associated with them.

Spacetime is consequently not introduced as a fundamental background on which the theory lives. Instead, it is expected to emerge from the collective behaviour of a large number of GFT quanta. The situation is analogous to a fluid: concepts such as density, pressure and continuous flow are not properties of a single molecule, but become meaningful only for a large collective system. Similarly, a smooth spacetime geometry—and potentially its cosmological dynamics—should arise only in an appropriate many-body regime of the underlying quantum theory. From this perspective, the central task is to understand how the microscopic quantum dynamics encoded by a GFT gives rise to the continuum spacetime described by general relativity, and whether traces of this microscopic structure can survive at observable scales.

Current Research

My research on GFTs addresses this challenge from several complementary directions. I study how cosmological physics can be extracted from their underlying quantum dynamics, with the aim of deriving effective models for the early and late Universe and identifying possible observational signatures (see here for more). At the same time, I investigate the mechanisms through which continuum spacetime and its dynamics emerge from many interacting quanta of geometry, seeking greater theoretical and mathematical control over the approximations involved in this process. A third central aspect of my work is the definition of physical observables within these theories—quantities that can be given an operational meaning and ultimately connected with cosmological observations (see here for more). Together, these directions aim to build a concrete bridge from the microscopic dynamics of quantum geometry to testable physics.

Selected work

Related publications