In absence of experimental guidance for the development of a Unified Theory/ Quantum Gravity, people resort to guidance in beauty and principles as well as certain conceptional issues in current theories they consider particularly relevant. (These wish lists usually correlate strongly with the set of problems, the author’s pet theory is most likely to address.)
In the following I made a collection of Unified Theory/Quantum Gravity wish lists by different people.
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.
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
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.
Problem 4: Explain how the values of the free constants in the standard model of particle physics are chosen in nature
Problem 5: Explain dark matter and dark energy. Or, if they don’t eists, 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 challenges we face can be separated, in degree of difficulty, in the following three:
- 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.
- 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.
- 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
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
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
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:
1. 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)
2. conservation of probability for the evolution if anything like the
Born Rule is fundamental (not necessarily unitarity, but it would
be nice to have)
It would also be wonderful if the theory of quantum gravity resolved
what I see as the deepest conceptual problem, respectively, of each
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 manifestation of an underlying unified entity.
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.
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.
- 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.
- 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.
- 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.
The Hierarchy problem: why is Gravitation so much weaker than the other forces.