This article is based on my research project¹ in mathematical physics. I outline the central ideas in broader terms. The project examines the possibility that gravity may be an emergent phenomenon rather than a fundamental interaction. It begins with black hole thermodynamics and entropy bounds, including Bousso’s bound, and discusses the holographic principle and its realization in the AdS/CFT correspondence. Finally, it presents selected approaches to emergent gravity, most notably Jacobson’s thermodynamic derivation of Einstein’s equations and Verlinde’s entropic gravity.

Why consider gravity as emergent?

One of the central open problems in modern theoretical physics is the formulation of a consistent theory of quantum gravity. General relativity describes gravity as curvature of spacetime, governed by the Einstein equations

At low energies this classical theory works remarkably well. However, attempts to quantize gravity directly face serious difficulties. From the point of view of quantum field theory, general relativity is non-renormalizable, even though it can be treated as an effective field theory at low energies.

These obstacles motivate a different question. Instead of quantizing gravity in the same way as other interactions, could gravity itself be an emergent phenomenon, analogous to hydrodynamics or elasticity?

An important hint in this direction appeared already in the 1970s through an unexpected connection between black holes and thermodynamics.

Black hole thermodynamics

In 1973, Bardeen, Carter and Hawking formulated the four laws of black hole dynamics².
These are theorems of classical general relativity, yet they closely resemble the laws of thermodynamics.

For a stationary black hole:

1. The surface gravity \(\kappa\) is constant over the event horizon.

2. For stationary perturbations one has

   \[\delta M = \frac{\kappa}{8\pi} \, \delta A + \omega \, \delta J + \varphi \, \delta Q ,\]

   where \(M\) is the mass, \(A\) the horizon area, \(J\) the angular momentum and \(Q\) the charge.

3. The area of the event horizon never decreases.

4. It is impossible to reduce \(\kappa\) to zero by a finite sequence of physical operations.

If one formally identifies surface gravity with temperature and area with entropy, the analogy with thermodynamics becomes apparent. Initially, this was widely regarded as just that, an analogy. In classical general relativity, black holes are perfect absorbers and do not radiate. It was therefore unclear in what sense they could have a temperature.

Jacob Bekenstein argued that black holes should carry entropy proportional to their horizon area. Otherwise, when matter with entropy falls into a black hole, there would be no way to verify that the second law of thermodynamics continues to hold. His proposal was controversial at the time.

Stephen Hawking set out to show that Bekenstein was wrong. Instead, in his semiclassical calculation of particle creation in a black hole background³, he discovered that black holes radiate as black bodies with temperature

The analogy became physical. Combining this result with the first law fixes the entropy of a black hole to be

in Planck units.

This so called Bekenstein–Hawking formula shows that black hole entropy is proportional to horizon area rather than volume. This fact will play a central role in what follows.

It suggests that gravity may impose a fundamental limitation on the number of independent degrees of freedom that can be associated with a spatial region. We now turn to entropy bounds and the holographic principle, where this idea is developed in a more general and covariant form.

Entropy bounds and holography

The proportionality \(S_{BH} \propto A\) raises a natural question. Is the area scaling a special feature of black holes, or does it reflect a more general principle about gravity?

In ordinary local quantum field theory without gravity, entropy typically scales with volume. One expects the number of degrees of freedom in a region to be proportional to its volume. The Bekenstein–Hawking formula suggests something radically different, the maximal entropy contained in a region may scale with the area of its boundary.

This idea is sharpened by entropy bounds.

Among them, the most robust and covariant formulation is the Bousso’s covariant entropy bound⁴:

Here \(B\) is a \(d-2\) dimensional spacelike surface and \(L(B)\) is a light-sheet constructed from it.

A light-sheet is defined as a null hypersurface generated by light rays orthogonal to \(B\) whose expansion is initially non-positive. The null geodesics are followed until they reach a caustic or their expansion becomes positive. Intuitively, one selects those light rays that are locally contracting.

To illustrate this notion, consider for example \(B\) to be a circle in 3D Minkowski space as in the image above. There are four null congruences orthogonal to it: future-outgoing \(\mathcal{C}_1\), past-ingoing \(\mathcal{C}_2\), future-ingoing \(\mathcal{C}_3\) and past-outgoing \(\mathcal{C}_4\). In the figure, the congruences that qualify as light-sheets are the contracting cones \(\mathcal{C}_2\) and \(\mathcal{C}_3\).

The focusing of null geodesics by matter, encoded in the Raychaudhuri equation, ensures that entropy flux through such contracting congruences is constrained by the initial area.

The von Neumann entropy of a quantum system is bounded above by the logarithm of the dimension of its Hilbert space. The covariant bound therefore suggests a limitation on the dimension of the Hilbert space describing a region.

We may formulate this idea precisely as follows.

Conjecture (Holographic principle):
In some possibly emergent spacetime, let \(B\) be a \(d-2\) dimensional spacelike surface and let \(L(B)\) denote one of its light-sheets. If \(\mathcal{H}[L(B)]\) is the Hilbert space describing the degrees of freedom on \(L(B)\) in the final theory, then

\[\dim \mathcal{H}[L(B)] \le e^{A(B)/4}.\]

If such a bound holds universally, the fundamental number of degrees of freedom associated with a region is controlled by boundary area rather than bulk volume. This statement is known as the holographic principle.

Its most concrete realization to date is the AdS/CFT correspondence.

AdS/CFT correspondence

A particularly explicit realization of the holographic principle is provided by the AdS/CFT correspondence, originally conjectured by Maldacena⁵. It is widely regarded as one of the most concrete and far-reaching insights in the search for a consistent theory of quantum gravity.

In its most studied example, the duality states that a theory of quantum gravity, namely type IIB superstring theory in an asymptotically \(\mathrm{AdS}_5 \times S^5\) spacetime with \(N\) units of five-form flux on \(S^5\), is exactly equivalent to a quantum field theory without gravity — the 3+1 dimensional \(\mathcal{N}=4\) supersymmetric Yang–Mills theory with gauge group \(U(N)\).

The bulk theory is ten-dimensional and contains dynamical gravity. The boundary theory is four-dimensional and contains no graviton at all. Nevertheless, the two descriptions are conjectured to be fully equivalent. Every physical observable in the gravitational theory has a precise counterpart in the field theory. In this sense, the correspondence realizes the idea that spacetime and gravity may emerge from more fundamental, non-gravitational degrees of freedom.

The duality is encoded in the GKPW prescription (Gubser–Klebanov–Polyakov–Witten), which states that the generating functional of the conformal field theory equals the partition function of the bulk theory with specified boundary conditions:

Here \(J(x)\) is a source coupled to some local operator \(\mathcal{O}(x)\) in the boundary theory, and \(\phi\) is the corresponding bulk field whose boundary value acts as that source. In this way, bulk fields and boundary operators are put into one-to-one correspondence.

In general this relation involves the full string theory path integral. However, in the limit of large \(N\) and large ‘t Hooft coupling, the bulk theory simplifies and is well approximated by classical supergravity. In appropriate low-energy truncations, supergravity reduces to Einstein’s gravity (with a negative cosmological constant and additional matter fields), so the dynamics is governed by classical general relativity in AdS spacetime. In this regime the partition function is dominated by its classical saddle:

Here \(S_{\mathrm{AdS}}\) is the classical supergravity action evaluated on-shell in the AdS background, and \(\phi_{\mathrm{cl}}\) denotes the classical solution of the bulk equations of motion subject to the boundary condition

The factor of \(N^2\) reflects the scaling of the number of degrees of freedom in the gauge theory. In the large-\(N\) limit the bulk path integral reduces to a stationary phase approximation, so that quantum gravity in the bulk becomes classical gravity, while the boundary theory remains fully quantum.

This equivalence provides a concrete realization of holography: a gravitational theory in five bulk dimensions is completely encoded in a non-gravitational theory living on its four-dimensional boundary. The scaling of degrees of freedom as \(N^2\) matches the area scaling suggested by entropy bounds, giving a precise framework in which the holographic principle is realized.

Black holes in AdS correspond to thermal states in the boundary CFT. In the classical limit, black hole thermodynamics maps directly to ordinary thermodynamics of the gauge theory. In this way, horizon entropy and other gravitational phenomena acquire a microscopic interpretation in terms of quantum field theory degrees of freedom.

Below is an artistic illustration of the correspondence (drawing by my sister):

The cylindrical geometry represents anti–de Sitter spacetime. The circular section at the top can be viewed as it’s spatial slice depicted here as a hyperbolic Poincaré disc. The vertical direction suggests time evolution, while the radial direction into the bulk encodes the energy scale of the boundary theory. The structure inside the cylinder is meant to evoke bulk string dynamics whose effects are fully captured at the boundary, reflecting the central idea that quantum gravity in the bulk is encoded in a lower-dimensional quantum field theory.

Jacobson’s emergent gravity

One of the most conceptually precise emergent-gravity proposals is due to Ted Jacobson⁶.

Instead of assuming Einstein’s equations, Jacobson assumes:

• proportionality between entropy and horizon area,

• validity of the Clausius relation

  \[\delta Q = T \, \delta S,\]
• Unruh temperature for local Rindler horizons.

The key idea is local. Around any spacetime point one introduces a local Rindler horizon and associates to it an entropy proportional to its area and a temperature given by the Unruh temperature of an accelerated observer. Requiring that the Clausius relation holds for all such local horizons leads directly to the Einstein equations.

In this approach, the Einstein equations resemble equations of state, similar to how the Navier–Stokes equations emerge from statistical mechanics. Gravity is not introduced as a fundamental interaction but appears as a thermodynamic consistency condition imposed on local causal horizons.

There is also a natural connection to entropy bounds discussed earlier. The assumption \(\delta S \propto \delta A\) can be viewed as a local saturation of an area-entropy relation. In particular, Jacobson’s reasoning is closely aligned with the covariant spirit of Bousso’s bound, since the entropy is associated with local light-like horizons rather than global, static screens. In this sense, the framework is conceptually closer to the covariant entropy bound than to the original Bekenstein bound, which applies to isolated systems in more restricted settings.

Verlinde’s entropic gravity

Erik Verlinde, influenced by Jacobson, proposed understanding gravity as an entropic force⁷. The basic idea comes from statistical mechanics. Consider, for example, a polymer immersed in a thermal bath. When stretched, the number of available microscopic configurations decreases, lowering the entropy. The system responds with an effective macroscopic force that tends to restore the polymer to a configuration of higher entropy. In such situations the force is not fundamental but emergent, and is given by

Here the entropy plays a role analogous to a potential, while the temperature sets the proportionality scale.

Verlinde applies this intuition to gravity. He considers holographic screens, surfaces that encode information about the bulk degrees of freedom. When a particle moves relative to such a screen, the entropy associated with the microscopic degrees of freedom changes. This change in entropy induces an effective force on the particle.

Using Bekenstein’s entropy bound together with holographic arguments and identifying the temperature with the Unruh temperature experienced by an accelerated observer, one can recover Newton’s law of gravitation in this framework.

While this approach provides appealing intuition and connects naturally to holography and AdS/CFT, it is structurally more restrictive. It relies directly on Bekenstein’s bound, which is less covariant than Bousso’s entropy bound and applies most cleanly in quasi-static situations. Extending the argument beyond the Newtonian regime requires working in special settings of static spacetimes and has not been fully carried out in a general relativistic formulation.

In the project, I compare this framework with Jacobson’s and analyse conceptual limitations that arise when attempting a modern relativistic formulation.

What this project aimed to do

This research project is primarily conceptual and structural rather than model-building. The goal was to:

• review black hole thermodynamics rigorously,
• analyse entropy bounds and their geometric content,
• explain the holographic principle in modern language,
• clarify the relation between Jacobson’s and Verlinde’s approaches,
• discuss how generalized entropy functionals may modify cosmological equations.

The aim of this work was not to claim that gravity is definitively emergent. Rather, the recurring appearance of thermodynamic, holographic and entanglement-based structures across different approaches suggests that they play a central role in any attempt to understand gravity at a more microscopic level. Whether these structures are fundamental or effective remains open, but they consistently point toward a deep relation between spacetime dynamics and quantum degrees of freedom.

1: J. Masák, Emergent gravity, Research Project, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague (2025), PDF.

2: J. M. Bardeen, B. Carter, S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161–170, doi:10.1007/BF01645742.

3: S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199–220, doi:10.1007/BF02345020.

4: R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825–874, doi:10.1103/RevModPhys.74.825.

5: J. M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133, doi:10.1023/A:1026654312961.

6: T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State, Phys. Rev. Lett. 75 (1995) 1260–1263, doi:10.1103/PhysRevLett.75.1260.

7: E. Verlinde, On the origin of gravity and the laws of Newton, JHEP 2011 (2011) 029, doi:10.1007/JHEP04(2011)029.

This article is based on my seminar report¹ on topological insulators, prepared at Leiden University. The work gives a pedagogical overview of the topological description of band insulators, with a primary focus on the Kane–Mele model of graphene and the emergence of the ℤ₂ topological phase. Along the way, I introduce the necessary mathematical tools like vector bundles and their topological invariants and illustrate them on the Haldane and Kane–Mele models.

Introduction

Topological insulators are phases of matter which cannot be distinguished from ordinary insulators by the usual Landau picture based on spontaneous symmetry breaking and local order parameters. Instead, what matters is the global structure of the electronic bands. Roughly speaking, the valence bands of an insulator can carry non-trivial topology, and this topology cannot be changed continuously unless the energy gap closes.

One of the main reasons this topic became so important is that non-trivial bulk topology is accompanied by robust conducting states at the boundary. These edge or surface states are not just a mathematical curiosity. They are probably the most interesting feature of topological phases from the point of view of applications, because they can be protected against certain kinds of disorder and backscattering. The subject is therefore interesting not only conceptually, but also technologically. In the broader family of topological phases, related ideas also appear in the search for Majorana quasiparticles and in proposals for topological quantum computing, which promises error protection already at the hardware level. Microsoft is one of the companies actively pursuing this direction, as illustrated for example by the recent Nature paper⁶. At the same time, I think it is important to keep some scepticism here. There is a lot of excitement around these applications, but it is still not clear how close we are to fully practical devices built on these principles.

The modern study of topological insulators was shaped in large part by the two seminal papers of Kane and Mele²³, which showed that graphene with spin–orbit coupling provides a natural setting for a new kind of time-reversal invariant topological phase. Kane also later wrote a very nice pedagogical introduction to the topic, which I found especially useful for building intuition.⁴ Although the original Kane–Mele model is not realized in actual graphene because the spin–orbit coupling there is too weak, closely related physics was later realized experimentally in HgTe quantum wells.⁵

From bands to topology

To understand where topology enters band theory, it is useful to think of the eigenstates of the Bloch Hamiltonian as varying over momentum space. In a crystal, Bloch’s theorem tells us that single-particle states can be labelled by crystal momentum \(\mathbf{k}\), which lives in the Brillouin zone. In two dimensions, the Brillouin zone is topologically a torus \(\mathbb{T}^2\).

At each momentum \(\mathbf{k}\), the Bloch Hamiltonian has a finite-dimensional Hilbert space of eigenstates. If we focus on the states below the Fermi energy, that is the valence bands, these states form what is called the valence Bloch bundle over the Brillouin zone. This is the object whose topology we study.

One can think of a vector bundle very roughly as assigning a vector space to each point of some base space. Here the base space is the Brillouin zone and the fibres are the spaces of valence states. A trivial bundle is just a simple product, where one can choose the states continuously everywhere in momentum space without running into any obstruction. A non-trivial bundle is twisted in a global way.

An intuitive picture is the difference between a cylinder and a Möbius strip. Locally they both look simple, but globally the Möbius strip contains a twist that cannot be removed continuously.

This picture should not be taken too literally, but it captures the essential idea. In band theory, non-trivial topology means that the valence states cannot be globally chosen in a completely smooth and symmetry-compatible way over the whole Brillouin zone. As long as the gap between valence and conduction bands stays open, this topology cannot change under adiabatic deformations of the Hamiltonian.

Graphene

The basic arena for both the Haldane and Kane–Mele models is graphene. Graphene has a honeycomb lattice, which is not itself a Bravais lattice but can be viewed as two interpenetrating triangular sublattices, usually denoted \(A\) and \(B\).

Because of this two-sublattice structure, the low-energy electronic states naturally come with a two-component structure. Near the special points \(\pm\mathbf{K}\) in the Brillouin zone one may write the momentum as \(\mathbf{k} = \pm \mathbf{K} + \mathbf{q}\), where \(\mathbf{q}\) measures the deviation from the Dirac points. In this regime the effective Hamiltonian takes the form of a two-dimensional Dirac Hamiltonian

In this low-energy description the topological properties are controlled precisely by how this gap parameter behaves at the two valleys \(\pm\mathbf{K}\). This is one of the reasons the Dirac picture is so useful: it makes the origin of the topology very transparent. What matters is not every microscopic detail of the lattice model, but the structure of the gaps in the effective theory near the points where the bands would otherwise touch.

Chern insulators and the Haldane model

Before turning to the Kane–Mele model, it is useful to look at the simpler case of a Chern insulator. Here the relevant topological invariant is the Chern number.

To see where it comes from, recall that the valence states of an insulator form the valence Bloch bundle over the Brillouin zone \(\mathbb{T}^2\). A natural gauge freedom appears because the overall phase of a single-particle wave function is not observable. This means that the bundle carries a \(U(1)\) gauge symmetry, very much like in ordinary gauge theory.

Once such a gauge freedom is present, it is natural to introduce a connection on the bundle. In condensed matter physics one usually works with the Berry connection, which is the connection associated with the adiabatic variation of the Bloch Hamiltonian with momentum \(k\). But since we are computing a topological invariant one may choose completely arbitrary connection.

Every connection \(A\) has an associated curvature two-form

The topology of the bundle can then be characterized using Chern–Weil theory, which associates topological invariants to curvature forms. For a single valence band the resulting invariant is the first Chern number

and the remarkable fact is that this quantity is always an integer, which ultimately follows from the underlying algebraic topology of the bundle.

Physically, the Chern number is important because of the so called TKNN relation, which connects it directly to Hall conductivity:

So in this case topology is not just abstract mathematics. It becomes directly observable as a quantized transport coefficient.

A famous theoretical model realizing this physics on the graphene lattice is Haldane’s model. It describes a system with no net magnetic flux through the unit cell, but with a pattern of complex next-nearest-neighbour hopping terms that still breaks time-reversal symmetry. In this way one obtains a quantized Hall effect without an overall external magnetic field, the so-called anomalous quantum Hall effect.

In the low-energy description discussed in the previous section, the Hamiltonian still has the Dirac form (see the equation above). The effect of the Haldane terms is simply to generate a valley-dependent gap parameter \(\Delta = \lambda_v \pm 3\sqrt{3}\,t_2 \sin\phi\), where the sign again depends on around which Dirac point one expands.

As also already mentioned the topology is determined by the behaviour of \(\Delta\) at the Dirac points, specifically the Chern number can be expressed as

One sees that, if the gaps at the two valleys have opposite signs, the system is in a topological phase with \(c_1=\pm1\). If the signs are the same, the contributions cancel and the phase is topologically trivial.

This model is conceptually very important even though it is not naturally realized in graphene itself. It provides the cleanest prototype of a Chern insulator and, more importantly for us, it serves as the building block for understanding the Kane–Mele model²³.

Kane–Mele model and the quantum spin Hall effect

The Kane–Mele model extends the graphene Dirac picture by including the electron spin. The Hamiltonian can be written as two blocks corresponding to the two spin sectors plus a spin mixing term

where \(H_H^{\uparrow}\) and \(H_H^{\downarrow}\) are Haldane Hamiltonians with opposite effective fluxes. In particular they correspond to the Haldane model with parameters \(t_2 = \lambda_{so}\) and \(\phi = \pm \frac{\pi}{2}\).

For sufficiently strong intrinsic spin–orbit coupling \(\lambda_{so}\) these two Haldane models therefore lie in topological phases with Chern numbers \(c_1^{\uparrow}=+1\) and \(c_1^{\downarrow}=-1\).

This structure originates from spin–orbit coupling, which couples the electron spin to its motion in the lattice potential. In graphene the intrinsic spin–orbit interaction effectively generates opposite Haldane-type terms for the two spin sectors.

In the low-energy theory near the Dirac points the intrinsic spin–orbit interaction takes the simple form

The Kane–Mele model can also contain a Rashba spin–orbit term, which mixes the spin sectors. In the effective theory it reads

Unlike the intrinsic spin–orbit term, this interaction does not preserve the spin \(z\)-component and therefore prevents us from viewing the model simply as two independent Haldane systems. In the regime where the Rashba coupling is small, however, the two spin sectors remain approximately decoupled.

In that simplified regime the model effectively splits into two copies of Haldane’s model: one for spin up and one for spin down. These two copies are related by time-reversal symmetry, so their Chern numbers are opposite \(c_1^\uparrow = -c_1^\downarrow\). As a result the total charge Hall conductivity vanishes, which is exactly what one expects in a time-reversal invariant system. However, one can still define a spin Hall conductivity. If we denote it by

then using the TKNN relation one finds

where \(n_s = \frac{c_1^{\uparrow}-c_1^{\downarrow}}{2}\).

Thus the spin Hall response is quantized. This is the quantum spin Hall (QSH) phase discussed in the original Kane–Mele paper.²

From this point of view the \(\mathbb{Z}_2\) invariant admits a very intuitive interpretation. One may define \(z_2 \equiv n_s \pmod{2}\).

If the two spin sectors carry opposite Chern numbers, then \(z_2=1\) and the phase is topologically non-trivial. If they are both trivial, then \(z_2=0\). In this regime one may therefore think of the quantum spin Hall phase simply as two time-reversed Haldane models glued together.

This is not yet the full story. In the more general Kane–Mele model the Rashba interaction mixes the spin sectors and the separate Chern numbers are no longer well defined. Nevertheless the non-trivial \(\mathbb{Z}_2\) phase survives beyond this simplified regime. For the purposes of this post I kept the discussion at the level of the quantum spin Hall regime, but more about the treatment of the general case can be found in the work itself.¹

Edge states and why they matter

The most striking consequence of non-trivial bulk topology is the existence of boundary modes. In a Chern insulator, these are chiral edge states: they move only in one direction along the boundary. In the quantum spin Hall regime of the Kane–Mele model, they become helical edge states: opposite directions of motion are tied to opposite spin sectors.

This edge physics is in many ways the most important part of the story. At the interface between a topological and a trivial phase, the bulk gap must close, and this produces gapless states localized near the boundary. In the topological regime these edge states cross the bulk gap and connect valence and conduction bands. In the trivial regime they do not.

In the helical case, a right-moving edge mode and a left-moving edge mode form a time-reversal pair. Because of this, elastic backscattering from non-magnetic disorder is strongly suppressed. To scatter backwards, the electron would have to flip into its time-reversed partner, and time-reversal symmetry prevents generic perturbations of this kind from opening a gap. This robustness is the main reason edge states are so interesting for transport and for applications.

Of course, one should not oversell this. Real materials are never ideal, interactions can matter, disorder is not always harmless, and practical device design is much harder than the clean theoretical picture suggests. Still, the general mechanism is extremely robust conceptually: bulk topology enforces protected boundary states, and those boundary states can carry current in an unusually stable way.

This is one of the central lessons of topological condensed matter. The important physics is not only in the local form of the Hamiltonian, but in its global topological structure. And once that topology is non-trivial, the boundary has no choice but to respond.

Why I find this topic interesting

What I find especially attractive about this subject is how naturally it connects abstract mathematics with concrete physics. Objects like vector bundles, Berry curvature and topological invariants are not added artificially. They arise quite directly once one starts taking the global structure of Bloch states seriously.

This topic is a nice example of a broader theme that appears again and again in theoretical physics: ideas that originally look abstract or formal often end up describing very real phenomena. In this case, concepts from geometry and topology became part of the language used to understand actual quantum materials, and possibly future technologies built from them.

1: J. Masák, Topological insulators, Seminar Report, Science faculty, Leiden university (2025), PDF.

2: C. L. Kane, E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95 (2005) 226801, doi:10.1103/PhysRevLett.95.226801.

3: C. L. Kane, E. J. Mele, $\mathbb{Z}_2$ Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95 (2005) 146802, doi:10.1103/PhysRevLett.95.146802.

4: C. L. Kane, Topological Band Theory and the $\mathbb{Z}_2$ Invariant, in Topological Insulators, Contemporary Concepts of Condensed Matter Science 6, Elsevier (2013) 3–34, doi:10.1016/B978-0-444-63314-9.00001-9.

5: M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, S.-C. Zhang, Quantum Spin Hall Insulator State in HgTe Quantum Wells, Science 318 (2007) 766–770, doi:10.1126/science.1148047.

6: M. Aghaee, A. Alcaraz Ramirez, Z. Alam, et al., Interferometric single-shot parity measurement in InAs–Al hybrid devices, Nature 638 (2025), doi:10.1038/s41586-024-08445-2.

This article is based on a computational physics project¹ carried out at Leiden University. In the project, the two-dimensional XY model was simulated on a square lattice using the Metropolis Monte Carlo algorithm with periodic boundary conditions. The goal was to explore the thermodynamic behaviour of the model and to reproduce the Kosterlitz–Thouless phase transition. Here I give a short conceptual overview of the work. The simulation code used in the project is available on GitHub.

Introduction

The XY model imagines a two-dimensional lattice \(\Lambda\) of sites \(i \in \Lambda\), at each of which there is a vector \(\mathbf{s}_i \in \mathbb{S}^1\) of unit length. This can be viewed as a generalisation of the Ising model because \(\mathbb{S}^0 = \{-1,1\}\). Each configuration is then represented by a set of spins \(\{\mathbf{s}_i\}_{i \in \Lambda}\) or, equivalently, through the parametrisation \(\mathbf{s}_i = (\cos \theta_i, \sin \theta_i)\), with \(\theta_i \in [-\pi,\pi)\), by a set of angles \(\Theta = \{\theta_i\}_{i \in \Lambda}\), which is the representation I will use most often.

The Hamiltonian used in the simulations can be written as a sum of two parts,

The first term

describes the interaction between nearest neighbours and favours alignment of neighbouring spins. The second term

represents the coupling to an external magnetic field of magnitude \(h = \lVert \mathbf{h} \rVert\) pointing in the \(x\) direction.

The key property of the model lies in the symmetry of the interaction term \(H_0\). Because it depends only on angle differences, it has the symmetry

which is a global \(U(1)\) symmetry. This is probably the most important difference from the Ising model, which also has a global symmetry, but only a discrete \(\mathbb Z_2\) symmetry. Because of this continuous symmetry, we have the Mermin–Wagner theorem², which states that in a system with such local interactions, continuous symmetries cannot be spontaneously broken at finite temperature when the spatial dimension is \(\leq 2\). This means that in the thermodynamic limit the system cannot develop a non-zero mean magnetisation serving as an ordinary order parameter.

The physical mechanism behind this is the existence of long-wavelength spin waves. Deviations from order can then be quantified by the momentum-space integral

If the lattice is \(N\times N\), this integral scales as \(\ln N\) and therefore diverges in the thermodynamic limit. Since in the simulation the lattice is finite, the divergence is only logarithmic, and as we will see, the system can still look magnetised at low temperatures. This is one of the reasons finite-size effects are important in the XY model.

The absence of spontaneous magnetisation also means that the system does not fit the standard Landau picture of phase transitions based on a local order parameter. Nevertheless, the two-dimensional XY model still exhibits a phase transition, first described by Kosterlitz and Thouless³. It is a topological phase transition, associated with the behaviour of vortices. In our work, we also studied the effect of an external magnetic field. Since the term \(H_h\) explicitly breaks the global \(U(1)\) symmetry, one expects the Kosterlitz–Thouless transition to disappear once the field is switched on. This is indeed what the simulation suggests.

Monte Carlo integration and the Metropolis algorithm

The Monte Carlo method is a powerful approach for studying the properties of a system at equilibrium when we are not interested in its dynamics and need to calculate high-dimensional integrals. In statistical physics, we typically have to evaluate integrals of the form

with \(\mathcal{D}\Theta = \prod_{i\in\Lambda} d\theta_i\) and \(M = [-\pi,\pi)^{N^2}\), which can be thought of as the configuration space of the system. We therefore have a very high-dimensional integral, and additional complexity arises from the fact that the function \(A[\Theta]e^{-\beta H[\Theta]}\) is usually very sharply peaked. This requires that we do not sample over the configuration space with the uniform distribution, as the simplest Monte Carlo method does, but instead use importance sampling.

Importance sampling can be implemented by means of the Metropolis algorithm⁴. This algorithm is also convenient because it assumes that we know the target distribution only up to a normalisation factor. This is exactly the case with the Boltzmann distribution

because we do not know the partition function \(Z\).

The Metropolis algorithm can be understood using the concept of a Markov chain: a set of random states \(\Theta(t)\), where the probability of \(\Theta(t+1)\) depends only on \(\Theta(t)\). A Markov chain is fully described by transition probabilities \(T(\Theta\rightarrow\Theta')\) such that

In the Metropolis algorithm, one divides these probabilities as

The first factor \(w(\Theta' \mid \Theta)\) denotes the probability to propose a new state \(\Theta'\) given a state \(\Theta\), and the second factor \(A(\Theta' \mid \Theta)\) denotes the probability to accept the proposed new state \(\Theta'\) if the system was before in the state \(\Theta\).

To ensure that a Markov process has a unique stationary distribution, two conditions must be met. First, there has to exist a stationary distribution. A sufficient, though not necessary, condition is the detailed balance condition

Second, the stationary distribution should be unique. This is guaranteed by ergodicity of the Markov chain, a property commonly assumed in the Metropolis algorithm. A key component of ergodicity is irreducibility, which ensures that the Markov chain can explore the entire configuration space over time. As will be important later, in our system this assumption may break down at low temperatures because of finite-size effects.

If we assume that \(w\) is symmetric, that is,

then the detailed balance condition can be written as

which can be satisfied with

The Metropolis algorithm thus consists of the following steps:

1. Start with a random initial state \(\Theta(0)\).
2. Generate a state \(\Theta'\) from \(\Theta(t)\).
3. If \(\pi[\Theta']>\pi[\Theta(t)]\), then \(A(\Theta'\mid\Theta)=1\) and set \(\Theta(t+1)=\Theta'\). Otherwise, accept the move with probability \(p=\pi[\Theta']/\pi[\Theta(t)]\).
4. Continue with step 2.

In our case, \(\pi[\Theta(t)]\) is the Boltzmann distribution. Thus, a trial move that lowers the energy is always accepted, while a move that increases the energy by an amount

is accepted with probability \(e^{-\beta \Delta E}\). This means that the system tries to move towards lower total energy, but the higher the temperature, the higher is the probability to go to a state with more energy.

For practical implementation, we chose the trial-step probabilities to be

This means that when the spins are in state \(\Theta\), we create a trial state \(\Theta'\) by picking one spin at random and changing its angle by a random increment from an interval \([-\Delta,\Delta]\subset[-\pi,\pi]\). In the simulation, this increment was chosen so that the average acceptance probability stayed close to \(0.5\), which allows the configuration space to be explored efficiently. After that, one computes the energy difference \(\Delta E\) by calculating the change in the interaction energy of the chosen spin with its four neighbours.

Measured quantities

The following is an overview of the quantities measured in the simulation. We denote the total magnetisation by \(\mathbf M = \sum_{i\in\Lambda}\mathbf s_i\). The total magnetisation per spin is then \(\mathbf m = \mathbf M / N^2\). Vector norms are denoted simply by \(M = \lVert \mathbf{M} \rVert\) and \(m = \lVert \mathbf{m} \rVert\). The time averages that should converge to ensemble averages are denoted as

One of the outputs of the simulation is the average magnitude of total magnetisation per spin, \(\langle m \rangle = \langle M \rangle / N^2\). Another possibility would be to measure \(\langle \mathbf m \rangle\), which should be zero if \(h=0\), but in practice it is unstable at low temperatures. Below the critical temperature, the system acquires a magnetisation and it would take a very long time for it to average back to zero. Because of this, \(\langle \mathbf m \rangle\) is not a stable quantity and it is more useful to measure \(\langle m \rangle\). Analogously, the energy per spin is measured as \(e = E/N^2\), where \(E\) is the total energy of the system.

The standard deviation of magnetisation, and analogously of energy, is calculated as

where \(\tau\) is the autocorrelation time, that is the time it takes for the system to forget its previous state. The autocorrelation time is calculated from the autocorrelation function

The autocorrelation function should decay as \(\chi(t) \sim e^{-t/\tau}\). In the simulation, \(\tau\) is approximated as

where the last approximation holds for large \(t_{\max}\). So in practice the autocorrelation time is estimated using this sum, with \(t_{\max}\) chosen so that the summation stops once \(\chi(t)<0,\) which signals the onset of poor statistics.

Additional quantities measured in the simulation are the magnetic susceptibility per spin,

and the specific heat per spin,

An interesting feature that appears in the XY model at low temperatures is the presence of vortex–antivortex pairs. To quantify this phenomenon, the simulation measures the average vortex density and plots the distribution of vortex-pair member distances. The presence of bound vortex pairs in the low-temperature phase, and their unbinding in the high-temperature phase, is one of the central aspects of the Kosterlitz–Thouless picture⁵.

Vortices and the Kosterlitz–Thouless transition

The Kosterlitz–Thouless transition is not best understood through an ordinary local order parameter. Its central objects are vortices, that is topological defects of the spin field. At low temperatures vortices and antivortices tend to appear as tightly bound pairs. At higher temperatures these pairs unbind, and this unbinding is precisely the mechanism underlying the topological phase transition.

At low temperature one still sees large correlated regions, even though in the thermodynamic limit true long-range order is absent. Because the lattice in the simulation is finite, the system can effectively select one global orientation and fluctuate only weakly around it. This behaviour is related to the finite-size effects discussed earlier: although long-wavelength spin waves prevent true spontaneous symmetry breaking in the thermodynamic limit, on a finite lattice the system may still appear magnetised for long periods of time.

The same finite-size effects also affect the Monte Carlo sampling. As mentioned earlier in the discussion of the Metropolis algorithm, the assumption that the Markov chain efficiently explores the whole configuration space can become problematic at low temperatures. In practice the simulation may remain trapped near one orientation of the spins, which is why the low-temperature configuration above shows relatively small angular fluctuations.

At high temperature the configuration is visibly more disordered.

The video illustrates the same physics more directly: vortices and antivortices appear as localized defects in the spin field, and at low temperatures they tend to remain pairwise bound.

Results and discussion

A particularly clear quantity is the autocorrelation time. The two plots below show the autocorrelation function at \(T=0.7\) and \(T=1.1\) for \(h=0\).

If the temperature is high, the autocorrelation function follows the exponential approximation quite accurately, at least in the initial regime where \(\chi(t)>0\). At low temperatures the decay is slower and the difference between the exponential fit and the actual values becomes significant. This supports the concern already mentioned above: at low temperatures the finite lattice may fail to explore configuration space properly and can get stuck around a non-zero magnetisation.

The corresponding autocorrelation times are shown below.

Without an external magnetic field, one can clearly see the existence of a critical point around \(T_C \approx 0.9\). The autocorrelation time grows strongly there, as expected from the Kosterlitz–Thouless transition theory. At low temperatures the autocorrelation time is not properly defined and its value differs between runs. This again arises from the finite-size effects already discussed.

The same figure also displays the effect of external magnetic fields. Even in the case of a weak field \(h=0.05\), the autocorrelation time no longer diverges at the critical point. Although the values are still somewhat larger at low temperatures, they remain in a reasonable range. A stronger magnetic field \(h=0.4\) decreases the autocorrelation time dramatically. This supports the main message of the simulation: the external magnetic field effectively removes the topological phase transition, although some critical behaviour remains.

The vortex-pair distance distributions tell a similar story.

At low temperature there are relatively few vortices and they are mostly pairwise bound, with distances typically of one lattice spacing. At high temperature the distribution becomes much broader, signalling the unbinding of vortex pairs. This is one of the clearest geometric manifestations of the Kosterlitz–Thouless transition.

The simulation therefore reproduces the basic theoretical picture rather well. Without external magnetic field the model undergoes a topological phase transition associated with vortex unbinding and critical slowing down. When a magnetic field is applied, the transition disappears because the global \(U(1)\) symmetry is broken explicitly, but the system still exhibits some residual critical-looking behaviour.

Conclusion

What I find especially attractive about this project is that it makes a rather subtle phase transition visible in a very concrete way. The two-dimensional XY model does not fit the usual symmetry-breaking paradigm, yet it still exhibits non-trivial thermodynamic behaviour governed by topological defects. The simulation captures this picture quite well: one sees critical slowing down, vortex proliferation above the critical region, and the qualitative disappearance of the transition once an external magnetic field is applied.

1: J. Masák, E. Tõkke, Computational Physics Project 2: Monte Carlo simulation of the two-dimensional XY model, Course Project Report, Science faculty, Leiden university (2025), PDF.

2: N. D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133, doi:10.1103/PhysRevLett.17.1133.

3: J. M. Kosterlitz, D. J. Thouless, Ordering, Metastability and Phase Transitions in Two-Dimensional Systems, J. Phys. C: Solid State Phys. 6 (1973) 1181, doi:10.1088/0022-3719/6/7/010.

4: N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21 (1953) 1087, doi:10.1063/1.1699114.

5: J. Tobochnik, G. V. Chester, Monte Carlo Study of the Planar Spin Model, Phys. Rev. B 20 (1979) 3761, doi:10.1103/PhysRevB.20.3761.

This article is based on a computational physics project¹ carried out at Leiden University. In the project, the motion of a relatively small number of Argon atoms was simulated using simple Newtonian dynamics with the Lennard–Jones potential, integrated using the Verlet algorithm with periodic boundary conditions. The aim was to see how this microscopic model reproduces the three phases of matter and to study their basic properties. Here I give a short conceptual exposition of the work. The simulation code used in the project is available on GitHub.

Molecular dynamics and Argon

Molecular dynamics aims to follow the motion of many interacting particles by numerically solving their equations of motion. Early simulations of this kind already demonstrated phase behaviour in simple interacting systems². In this approach each atom is treated as a classical particle and its trajectory is obtained by integrating Newton’s equations. Even a relatively simple interaction model can already reproduce many properties of real materials.

In the simulation each particle obeys Newton’s second law

If the interaction potential is a sum of pair interactions, the force on particle \(\alpha\) can be written as

Once the potential is specified, the task of the simulation is therefore to repeatedly compute the forces and update the particle positions.

Lennard–Jones potential

For Argon atoms the interaction can be modeled by the Lennard–Jones two-particle potential

Here \(r\) is the distance between the two particles, while \(\varepsilon\) and \(\sigma\) set the energy and length scales of the interaction.

The \(r^{-12}\) term produces a strong short-range repulsion, preventing atoms from occupying the same position. The \(r^{-6}\) term represents an attractive van der Waals interaction. Together they create a preferred separation between atoms, corresponding to the minimum of the potential at \(r_{\min} = 2^{1/6}\sigma\).

This preferred distance later appears as the first peak in the pair-correlation function.

Numerical integration

To evolve the system in time we integrate the equations of motion numerically. In this project the velocity Verlet algorithm⁴ was used because it is simple and stable for long simulations.

The positions are updated according to

The velocity update uses the forces at the beginning and end of the step

In the continuous equations of motion the total energy is conserved. This follows from the invariance of the physical laws under time translations, as expressed by Noether’s theorem. When the equations are discretised this symmetry is generally broken. For example, a simple Euler integration scheme treats the forward and backward directions in time differently and typically leads to a gradual drift of the energy.

The Verlet algorithm improves this behaviour by making the update rule explicitly symmetric under the transformation \(t \to -t\). Although the energy is not exactly conserved, it typically oscillates around the correct value instead of drifting systematically. This makes the method particularly well suited for long molecular dynamics simulations.

Periodic boundary conditions and the minimum image convention

The number of particles in a simulation is necessarily limited. To mimic a much larger system we use periodic boundary conditions.

Particles move inside a cubic box of side length \(L\). Whenever a particle leaves the box through one side it re-enters through the opposite side. One can think of the simulation box as being surrounded by infinitely many copies of itself. This removes artificial boundary effects and makes the system behave more like a small piece of bulk matter.

Under periodic boundary conditions each particle has infinitely many periodic images. When computing forces we therefore need a rule that determines which image interacts with a given particle. The minimum image convention states that particles interact only with the nearest periodic image as illustrated in the image below.

Formally the relative position between two particles can be written as

Because the simulation box is orthogonal, the shortest displacement can be obtained by shifting each coordinate difference into the interval \((-L/2, L/2)\). In practice this can be implemented using the modulo operation

and similarly for \(\Delta y\) and \(\Delta z\). The displacement vector is then \(\mathbf{r}_{\alpha\beta} = (\Delta x, \Delta y, \Delta z)\).

Initial configuration and equilibration

To avoid extremely large forces the atoms are initially placed on a face-centered cubic lattice, which is the crystal structure into which Argon solidifies. The initial separation between atoms is chosen to correspond to the observed lattice spacing of solid Argon. In practice such structural parameters can be obtained from experiment or from electronic structure calculations. The unit cell of the face-centered cubic lattice is illustrated in the image below.

Initial velocities are chosen from a Maxwell distribution corresponding to the desired temperature. The system is then evolved for a number of steps while rescaling the velocities so that the kinetic energy approaches the expected value

This equilibration stage allows the system to settle into a representative state before measurements are taken.

Pair-correlation function

One of the most informative observables is the pair-correlation function, which was famously analysed in Rahman’s early molecular dynamics simulation of liquid Argon³. In the project some of the results reported in that work were also reproduced numerically, allowing for a direct comparison with the classical simulation of liquid Argon.

In the simulation the pair-correlation function is defined as

Here \(n(r)\) denotes the number of particles found at a distance between \(r\) and \(r+\Delta r\) from a chosen reference particle.

Roughly speaking, \(g(r)\) measures how likely it is to find another atom at distance \(r\) compared to a completely uniform distribution. The structure of this function therefore reveals the spatial organization of the system.

Gas phase

In the gas phase the particles move freely and there is little spatial structure. The trajectories in the animation below show that the atoms travel through the simulation box with frequent collisions but without forming any persistent structure.

The corresponding pair-correlation function is shown in the plot below.

It is nearly flat with \(g(r) \approx 1\) except at very small distances where the strong repulsive core of the Lennard–Jones potential suppresses close approaches between atoms.

Liquid phase

In the liquid phase the atoms still move throughout the system but remain correlated with their neighbours. This can already be seen in the trajectories below, where particles continuously change their neighbours while remaining part of a relatively dense local environment.

The pair-correlation function shown below reflects this short-range order.

A pronounced first peak appears near the preferred interatomic separation, together with weaker oscillations at larger distances. These peaks indicate that atoms tend to arrange themselves in shells around each other, even though the long-range order of a crystal is absent.

Solid phase

In the solid phase atoms remain localized near lattice sites and only oscillate around their equilibrium positions. This behaviour can be seen clearly in the trajectories below.

The corresponding pair-correlation function shows a series of sharp peaks.

These peaks correspond to the well-defined distances between atoms in the underlying crystal lattice. In contrast to the liquid case, the peaks remain pronounced even at larger distances, reflecting the long-range order characteristic of the solid phase.

Conclusion

Although the model is very simple, it already reproduces the qualitative behaviour of gases, liquids and solids. Even with a relatively small number of particles the characteristic differences between these phases become visible in the particle trajectories and in the pair-correlation function.

This illustrates a central idea of statistical physics: macroscopic properties of matter can often be understood as emerging from simple microscopic dynamics.

1: J. Masák, E. Tõkke, Computational Physics Project 1: Molecular Dynamics Simulation of Argon Atoms, Course Project Report, Science faculty, Leiden university (2025), PDF.

2: B. J. Alder, T. E. Wainwright, Phase Transition for a Hard Sphere System, J. Chem. Phys. 27 (1957) 1208, doi:10.1063/1.1743957.

3: A. Rahman, Correlations in the Motion of Atoms in Liquid Argon, Phys. Rev. 136 (1964) A405, doi:10.1103/PhysRev.136.A405.

4: L. Verlet, Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Phys. Rev. 159 (1967) 98, doi:10.1103/PhysRev.159.98.

This article is based on my Bachelor’s thesis¹ in mathematical physics. I outline the main ideas and motivations of the work in broader terms. In the thesis, I investigate how deformations of the standard uncertainty relations can be interpreted, at the classical level, as deformations of the Poisson algebra of observables. I then explore how this modified structure affects Hamiltonian dynamics, with applications to geodesic motion in Schwarzschild and Kerr spacetimes and to black hole thermodynamics.

Uncertainty relations

In quantum mechanics, physical quantities such as position or momentum are represented by operators. Given two observables \(a\) and \(b\), their commutator is defined as \([a,b] = ab - ba\). If \([a,b]=0\), the observables and said to commute. Physically, this can be understood as measuring \(a\) and then \(b\) giving the same result as measuring them in the opposite order. When the commutator does not vanish, the order of measurements matters.

The uncertainty relation follows directly from this non-commutativity. For two observables \(a\) and \(b\),

Here \(\Delta a\) denotes the standard deviation of the measurements of observable \(a\) and \(\langle \cdots \rangle\) is an expectation value (average of measurement values in this case of the commutator operator). In this work we assume vanishing expectation values of \(a\) and \(b\), so for instance \(\Delta a\) directly characterizes the typical size of fluctuations.

For position \(x\) and momentum \(p\), the canonical commutation relation \([x,p] = i\hbar\) leads to the familiar Heisenberg uncertainty relation

A basic physical intuition already follows from this formula: if one tries to localize a particle more and more precisely, the momentum uncertainty tends to grow.

Motivation for generalized uncertainty relations

Many theoretical frameworks suggest that the standard uncertainty relation may be modified at very short distances. These include for example numerous thought experiments, string theory, curved momentum space, noncommutative geometry, … Basis of many of these ideas is in one way or another existence of an effective minimal length scale.

A commonly studied deformation is a generalized uncertainty relation (GUP) of the form,

\(\beta\) being some constant of appropriate dimensions. The plot below illustrates the qualitative consequence of this relation for the case \(\beta = \ell_p^2/\hbar^2\), where \(\ell_p\) is the Planck length.

In this plot the coloured region represents the allowed values of \(\Delta x\) and \(\Delta p\). One can see that there is a minimum value of \(\Delta x\) (in this case precisely \(\ell_p\)) which can be interpreted as existence of a minimal length, below which localization is no longer possible. While the precise interpretation varies between frameworks, this qualitative feature appears very robust. For instance, similar behaviour arises in string theory due to the extended nature of strings.

Reverse quantisation and deformed algebra

In classical mechanics, observables are functions on phase space, and their algebraic structure is encoded in the Poisson bracket \(\{a,b\}\). The equations of motion then take the form

where dots denote time derivatives and \(H\) is the Hamiltonian. These equations determine the trajectories of classical particles.

When one wants to promote a classical theory to a quantum one follows procedure known as quantisation. The canonical quantisation procedure is to replace Poisson brackets as

In my thesis, I take the reverse perspective. Because commutators give us uncertainty relations, GUP motivates a deformed quantum algebra. In my work, I ask what classical structure corresponds to one such deformed algebra.

Particularly I look at the Snyder algebra², given in the quantum theory by

Where \(x^\mu\) denotes the spacetime coordinate operators and \(p^\mu\) four-momentum operators (four-momentum combines momentum and energy). Taking the classical limit leads to a deformed Poisson algebra with the same structure. This then leads to modified Hamiltonian equations, and therefore to modified classical trajectories.

Deformed solutions of general relativity

In general relativity, the geometry of spacetime is encoded in the metric tensor \(g_{\mu\nu}\). Free test particles then follow geodesics, which can be derived using the Hamiltonian formalism.

The Schwarzschild solution of general relativity for example describes the spacetime outside a spherically symmetric mass. It applies to systems like the Solar System as well as, for example non-rotating black holes. A common visualization of this spacetime provides the Flamm’s paraboloid, shown below, which represents a spatial slice in the radial direction.

The Kerr solution generalizes this to rotating black holes (though basically just to black holes) and features additional structures such as an ergosphere and an inner horizon.

In my work, I study particle motion in both Schwarzschild and Kerr spacetimes using the pseudo-Newtonian limit. In this approximation one assumes weak gravitational field and spatial components of canonical four-momenta much smaller than the time component. Implementing the Snyder-deformed Poisson algebra in this setting then leads to modified geodesic equations and hence to deformed trajectories.

What was done and what was left out

The main contribution of the thesis is a logically consistent implementation of a deformation motivated by GUP, together with its application to geodesic motion in curved spacetimes within a pseudo-Newtonian approximation. While similar deformations of Schwarzschild geodesics have been studied before³, the approach taken here differs both conceptually and technically.

Many aspects were left out of this short overview. The thesis¹ contains for example extended discussion on the motivations of such deformation and explicit calculations of the modified equations of motion. At the end applications of GUP to black hole thermodynamics are also discussed. Interested readers can find all details in the full text.

1: J. Masák, Generalized uncertainty relation and its use in cosmology, Bachelor’s thesis, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague (2023), dspace.cvut.cz/handle/10467/111416.

2: H. S. Snyder, Quantized Space-Time, Physical Review 71 (1947), 38–41, doi:10.1103/PhysRev.71.38.

3: S. Mignemi and R. Strajn, Geodesics in the Schwarzschild metric with Snyder dynamics, Phys. Rev. D 90, 044019 (2014), doi:10.1103/PhysRevD.90.044019.