# The Theory of Causal Fermion Systems

## Comments on Various Quantum Gravity Wishlists

### Prerequisites

In absence of experimental guidance for the development of a Unified Theory / Quantum Gravity, scientists in the field often argue with mathematical beauty, general guiding principles or conceptual issues in current theories which they consider to be particularly relevant. (These wishlists usually correlate strongly with the set of problems which the author’s preferred theory is most likely to address.)

In what follows, we give a collection of Unified Theory / Quantum Gravity wishlists by different people. To each problem, we comment on what there is to say in the context of causal fermion systems. We also give our opinion to what degree we think that the Theory of Causal Fermion Systems solves or will be able to solve that particular problem.

If you have further questions or suggestions, please do not hesitate to contact us (see our community page). We will get back to you as soon as we find time.

For the answers below, it is assumed that you read through the introduction to physical aspects and at least the more detailed explanations listed in the prerequisites in the quick-navigation sidebar.

# Lee Smolin – The Trouble with Physics

Problem 1: Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature. This is called the problem of quantum gravity.

Both General Relativity and Quantum Field Theory (including the gauge group of the Standard Model) are obtained from the Theory of Causal Fermion Systems in an appropriate limiting case (for clarity, we point out that the QFT limit is a relatively recent development which has not yet been worked out in full detail). In this sense, causal fermion systems are a theory of quantum gravity. However, it is unclear whether it includes an actual “quantization” of the gravitational interaction itself (including the graviton as interaction particle). On the other hand, there is no apparent need for a graviton for a complete picture of our physical reality in the context of causal fermion systems. The reason is that general relativity emerges as a next to next to leading order effect, and hence gravity only plays a role in long range interactions.

Conclusion: The theory of Causal Fermion Systems seems to solve this problem within its scope.

Problem 2: Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory as it stands, or by inventing a new theory that does make sense.

The Theory of Causal Fermion Systems by itself does not directly say anything about quantum mechanics. However, there is hope that the theory is compatible with Fröhlich’s Events, Trees, Histories (ETH) approach to quantum mechanics. You can find a rough outline of this project here.

Conclusion: It is unclear whether the CFS theory solves or addresses this problem.

Problem 3: Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestation of a single, fundamental entity.

In the Theory of Causal Fermion Systems, every property of the physical system of interest is encoded in a single mathematical object, namely the universal measure $\rho$. All fields and the geometry of spacetime emerge as effective descriptions in the continuum limit.

Conclusion: The CFS theory resolves this problem completely.

Problem 4: Explain how the values of the free constants in the standard model of particle physics are chosen in nature.

At the moment, the framework of causal fermion systems seems unable to shed any light on the values of the constants in the effective theories. However, a more detailed analysis of the continuum limit might eventually change that. But there is one notable exception: the fact that gravitation comes in as a next-to-next-to-leading-order effect explains why it is so much weaker than the interactions in the standard model, as the coupling constant picks up a factor $\delta^2$, where $\delta$ is related to the regularization length and comparable to the Planck length. (Note: This is one incarnation of the hierarchy problem)

Conclusion: The CFS theory resolves one aspect of this problem, but leaves it largely open.

Problem 5: Explain dark matter and dark energy. Or, if they don’t exist, determine how and why gravity is modified on large scales. More generally, explain why the constants of the standard model of cosmology, including dark energy, have the values they do.

The theory of Causal Fermion systems can to reproduce the Standard Model and General Relativity without postulating any additional particles. However, in principle it is possible to add more fermionic sectors and thus describe physical models with additional particles and potentially additional interactions. However it is unclear how flexible the framework is with that respect.

Regarding the cosmological constant, it is worth to note that the cosmological constant problem might be resolved along the ideas of Padmanabhan. In this context, one should keep in mind that when evaluating the Euler-Lagrange equations of the causal action principle in the continuum limit, one gets the Einstein equations only in the form where both tensor indices contracted with a lightlike vector $\xi$, i.e.

$\Big( R_{jk} – \frac{1}{2}\: R\: g_{jk} – \kappa \,T_{jk} \Big) \xi^j \xi^k = 0$

(for details see [cfs16, Section 4.9]). Therefore, the cosmological term drops out. The cosmological constant enters as a free constant to be determined by observation.

Conclusion: It remains unclear whether the CFS theory can resolve this problem.

# Some aspects from a cosmological angle by Padmanahban

1) The cosmos, at very large scales, provides us with an absolute frame of rest (in which the CMB is isotropic) and an absolute time coordinate (in the form of the temperature of the CMB). You can measure your absolute motion using the dipole anisotropy of the CMB, which acts like a cosmic aether. However, we see no experimental trace of the existence of such an absolute standard of rest at subcosmic scales (if we do not use the CMB). In other words, sub-cosmic scale physics appears to be invariant under a much larger group (viz., the general coordinate transformation group) than the very large scales which deﬁne a cosmic aether and a preferred coordinate system. This enhancement of symmetry at small scales vis-a-vis the largest scales is intriguing.

In the Theory of Causal Fermion Systems, the regularization breaks the Lorentz symmetry. If the regularization length is sufficiently small, it is plausible that local measurements can not detect the regularization. Nevertheless, the regularization distinguishes a globally preferred coordinate system in the effective spacetime description.

Conclusion: The problem remains open in the context of the CFS theory but there exist some preliminary ideas to address it.

2) Cosmic evolution introduces an arrow of time even though Einstein’s equations, like the rest of physics, are invariant under time reversal. (This arrow of time arises through dynamics rather than through statistical coarse graining etc.) While the unstable mode, corresponding to the expansion factor a(t), might have something to do with this, the details are still uncertain. In particular, this explanation requires the effective Hamiltonian describing the cosmos to be unbounded; it is not clear how such a feature could be embedded in a quantum gravitational model without affecting small scale physics. Once again, we see a conceptual tension between the description of sub-cosmic scale physics and the dynamics of the large scale universe.

It is unclear whether the Theory of Causal Fermion Systems can say anything with regard to this question. One remark though: The regularization is allowed to change over time, but at present it is unclear how that would affect the time evolution (for a first step in this direction see the preprint [dgc16]). An interesting observation is that the large scale dynamics tends to to dilute/remove the material degrees of freedom from the universe, because the gravitational collapse eventually hides them away behind an event horizon and leaves us with a vacuum space time. Cosmological expansion simply disperses the matter degrees of freedom driving the spactime towards a vacuum configuration as well.

Conclusion: At present, the Theory of Causal Fermion System does not really say anything about this problem.

3) The universe is probably the only system known to us which made a spontaneous transition from quantum mechanical behaviour to classical behaviour — i.e, treated as a dynamical system, it became more and more classical as time went on. Systems with bounded Hamiltonians cannot do this. This suggests a relation between spontaneous transition to classicality and the arrow of time, possibly again through the “wrong sign” for the kinetic energy associated with the expansion factor. While this could very well be the reason, the details remain to be worked out.

It is unclear whether the framework of causal fermion systems can shed any light on this question. It might be worth remarking, however, that in Fröhlich’s ETH formulation of quantum theory, decoherence of pure states occurs automatically. This might present another route towards a quantum to classical transition than the one envisioned by Padmanabhan.

Conclusion: At present, the Theory of Causal Fermion systems does not address this question.

# Juan Maldacena’s Wishlist

The challenges we face can be separated, in degree of difficulty, in the following three:

1) Formulate an internally consistent theory of quantum gravity. By this we mean a theory which reduces at low energies, E<<10^19 GeV, to general relativity but in which we can perform quantum calculations to any order we wish. This theory should solve some fundamental problems with quantum gravity such as explaining the origin of black hole entropy, etc. These are questions about gravity which do not involve directly the fact that we also have the particle physics we see in nature.

The Theory of Causal Fermion Systems is able to reproduce General Relativity in the continuum limit as a next-to-next-to-leading-order effect in the neighborhood of Minkowski space. Black hole related questions can not be addressed yet. There exist local correlation maps to describe globally hyperbolic spacetimes as causal fermion systems (see [nlstg17]). But the  causal action principle has not yet been analyzed in such spacetimes.

The connection to area and matter flux has been made for general causal fermion systems (see [cfi19]). Nevertheless, it is unclear whether black hole entropy is an important question in the context of the Theory of Causal Fermion Systems for the following reason: If we manage to merge the Theory of Causal Fermion Systems with Fröhlich’s Events Trees Histories (ETH) approach to quantum theory, then loss to access of information is in fact assumed for all real physical systems. (To be transparent: yes this means that the effective time evolution of a physical system in Fröhlichs ETH approach to quantum mechanics is NOT unitary)

Conclusion: The Theory of Causal Fermion Systems resolves some aspects of the problem, while for other aspects, it is unclear whether they are at all relevant in this context.

2) Be capable of incorporating the Standard Model. So the theory should be such that at low energies it can contain chiral gauge fields, fermions, etc.

The Theory of Causal Fermion Systems can reproduce the particle and field configuration of the Standard Model including the gauge interactions (without any need for postulating any additional particles). It also provides a motivation for the existence of three generations of fermions in the Standard Model.

Conclusion: This problem is completely resolved in the Theory of Causal Fermion Systems.

3) Explain the Big-Bang and the parameters of the Standard Model. We should understand the resolution of the initial singularity in cosmology and we should understand why we have the Standard Model. We should understand how the Standard Model parameters arise, which parameters are related, and which parameters (if any) arise as a ”historical” accident

It is unclear what the framework of causal fermion systems can say about the observed parameters, neither in cosmology nor in the standard model. There are two notable exceptions:

1. The fact that gravitation comes in as a next-to-next-to-leading-order effect explains why it is so much weaker than the interactions in the standard model, because the coupling constant picks up a factor $\delta^2$, where $\delta$ is related to the regularization length and comparable to the Planck length. (Note again: This is one incarnation of the hierarchy problem).
2. Regarding the cosmological constant, it is worth to note that the cosmological constant problem might be resolved along the ideas of Padmanabhan. In the framework of causal fermion systems, the causal action principle only determines the equations up to the cosmological term. The cosmological constant then enters as an integration constant to be determined by observation.

Conclusion: The Theory of Causal Fermion Systems resolves some aspects of the problem. But, at present, it also leaves many aspects of the problem open.

# Erik Curiel’s Wishlist

A theory of quantum gravity should combine general relativity and quantum theory in a way that holds on to (as much as possible of) what is most characteristic of each as an account of the way that the relevant part of the physical world works—the deepest lessons each has taught us about the nature of the world.  For general relativity, I think this is:

1) The geometric nature of “gravity”—its inextricability from  “spacetime”, that it is not a separate force or interaction, as  electromagnetism is; ultimately this means that, no matter exactly what the field equations may be (Einstein’s or otherwise), above some spatiotemporal scale and below some energy scale spacetime and gravity are jointly well modeled using the machinery of Lorentzian geometry

Certain causal fermion systems reproduce Lorentzian manifolds in the continuum limit. At present, an explicit realization of this limit as a minimizer exists only for Minkowski space. In the Theory of Causal Fermion Systems, the emerging spacetime structure in the continuum limes has exactly one time like direction, and the maximal extension for the Clifford Algebra is 5 dimensional (for details see [lqg11, Section 3]. This is exactly the world we observe, with 3+1 spacetime dimension plus a helicity degree of freedom.

Conclusion: The Theory of Causal Fermion Systems resolves this problem completely.

2) Its causal structure—not only the constraints on dynamics and possible symmetries of physical systems that the structure of a Lorentz metric imposes, but also, contrarily, the fact that the causal structure of a spacetime alone already gives one essentially 9/10 of the metric.

The framework of causal fermion systems allows us to define a generalized causal structure on the space of all possible events. One can argue that the causal action principle does nothing but select the optimal weighted causal structure from all possible weighted causal structures. These causal structures  agree with Lorentzian causal structures except on very short scales.

Conclusion: This problem is resolved completely by the Theory of Causal Fermion Systems.

For quantum theory, I think this is what is encoded purely in the structure of the dynamics—what you can tell me about the nature and dynamics of a system if I give you its algebra of observables and its possible states (essentially bounded self-adjoint operators on a Hilbert space).  This includes:

3) entanglement (*not* necessarily superposition, which is a basis-dependent notion—a “privileged basis” makes no sense to me—though it may be that there can be no entanglement in the relevant sense without basis-dependent superposition)

In the limiting case of Quantum Field Theory (see the mathematics section → Connection to Quantum Field Theory), entanglement and superpositions arise in the Fock space description in the usual manner. However, It is still unclear how the notion of entanglement arises in the CFS framework on a fundamental level. This is due to the fact that the connection between the universal measure $\rho$ and the usual notion of a “physical state” still needs to be understood better.

Conclusion: This problem is partially addressed by the Theory of Causal Fermion Systems.

4) conservation of probability for the evolution if anything like the Born Rule is fundamental (not necessarily unitarity, but it would be nice to have)

In the limiting case of Quantum Field Theory (see the mathematics section → Connection to Quantum Field Theory), the Fock space norm is conserved under time evolution. However, if it is possible to merge the CFS framework with Fröhlich’s ETH approach to quantum mechanics, then unitarity would survive for probabilistic predictions, however the actual evolution of the physical system would turn out to be non unitary.

Conclusion: This requirement is satisfied for predictions but not necessarily for the effective evolution.

It would also be wonderful if the theory of quantum gravity resolved what I see as the deepest conceptual problem, respectively, of each theory.

5) General Relativity: I think the deepest problem, given the way that we tend to try to think about the world, is that there is no clear distinction between “matter” and “geometry/gravity”.  The “obvious” way to distinguish them—matter is Ricci curvature, gravity is Weyl curvature—doesn’t really work.  When one takes quantum effects into account, “Weyl curvature can transform into matter” (e.g., the Hawking effect); in other contexts, such as gravitational collapse to a singularity, “matter transforms into Weyl curvature”.  This may suggest that, at some deeper level, “matter” and “curvature/gravity” are just different manifestations of an underlying unified entity.

On the fundamental level the answer is simple: the universal measure encodes all properties of the universe and hence the distinction between matter degrees of freedom and gravitational degrees of freedom does not exist.

On the level of the effective theories, however, we can only give a tentative answer. The key to address this question lies in a reversal of Wheeler’s interpretation of Einstein’s Field equations: “Matter tells spacetime how to curve and curvature tells matter how to move”.
In fact, the causal action principle determines the entire causal structure of our physical system. Spacetime geometry and matter are effective descriptions. Indeed, one can take the point of view that matter is nothing else than the failure of the effective geometric description of the causal structure to be Ricci flat. Therefore, a transition between matter degrees of freedom and gravitational degrees of freedom is not a problem because those are only effective descriptions in the first place. This might be a crucial point in determining the origin of matter in the universe. Furthermore, a macroscopically Ricci flat effective description will deviate on microscopic levels from the fundamental causal structure. This could give rise to an effective description involving vacuum fluctuations of the matter fields.

Conclusion: The problem is resolved in the Theorem of Causal Fermion Systems on the fundamental level. Moreover, there are tentative ideas for how the problem could be resolved on the level of the effective description.

6) Quantum Theory: I think the deepest problem, again given the way we tend to like to try to think about the world, is that there is no clear distinction between, on the one hand, interaction between separate systems and, on the other, dynamical evolution of a single system.  In classical physics (Newtonian mechanics, Lagrangian mechanics), a dynamical evolution is represented by a vector field on the space of states; an interaction with another system consists of an externally imposed force (a “generalized force” in Lagrangian mechanics), which is represented by *a different kind* of vector field on the space of states.  Dynamical evolution and interaction are mathematically, physically and conceptually distinct; correlatively, one can always, without ambiguity, separate the degrees of freedom of a given system from those of any other systems it may be interacting with.  In quantum theory, dynamical evolutions are generated by exponentiating a self-adjoint operator (the “Hamiltonian”)—but interactions are *exactly the same thing*, just adding on the exponentiation of another Hamiltonian.  They are mathematically, physically and conceptually *not* distinct, but, again, seem to be different aspects of some underlying unified conceptual structure that we have not adequately articulated and grasped.  Correlatively, it is not always possible to cleanly separate the degrees of freedom of a given system from those of others it is interacting.  This, I think, is the heart of the Measurement Problem.

Here it is unclear how the answer in the framework of causal fermion systems might look like. This is mostly due to the fact that the causal action principle determines a measure that describes the entire universe. Therefore, a separation into “system” and “interaction” is possible only in the effective description. Nevertheless, a connection with Fröhlich’s ETH approach to quantum theory might address the measurement problem.

Conclusion: It is unclear whether this problem can be resolved in the framework of causal fermion systems.

7) Best of all would be for a theory of quantum gravity to solve both problems in a way that shows them to be related to each other, for it seems to me an evocative and compelling idea that the lack of a clear distinction between interaction and evolution in quantum theory is intimately tied up with the lack of a clear distinction between matter and geometry in general relativity.

There might indeed be a way to stitch together the above two answers, however this is still in such a speculative stage that any comments would be premature.

Conclusion: This problem is wide open in the Theory of Causal Fermion Systems.

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