A strong constraint for understanding quantum physics can be derived from the attempt to define a universal basis space that can be used to represent and compute any arbitrary physics problem that can occur in nature. Logically, Nature must somehow operate within such a basis space, assuming that it is not dynamically reconfiguring itself on the fly for different situations at different points in space / time. The value of this universal basis constraint is that it is entirely abstract and functionally-defined, and has no dependencies on intuitions based on macroscopic physics, and yet it appears to be strongly constraining. Indeed, there are no existing frameworks that satisfy this constraint for the full scope of the Standard Model.
Aside from universal expressiveness, the only other constraint for the universal basis representation is:
If there are numbers that appear in a relevant physics equation, there must be an accounting for how Nature would have access to the values described by those numbers.
This is fundamentally an information processing level issue, which is independent of any given algorithm or detailed programming language choices: the issue is about the availability of the information needed to drive the forces that then update the state at the next moment in time. This kind of abstract information processing framework is what the cellular automaton (CA) provides: a way of organizing the data and the computation so that everything can happen locally, in parallel, with fully determined finite computation at each location and time step, and no open-ended search or long-range variable lookup routines required.
The universal basis constraint can be used to determine if a given framework is a calculational tool, most of which involve the use of bespoke representational spaces that extract the most relevant and convenient basis for calculation of a given problem. For example, the use of \(r\) in the \(1/r^2\) inverse-square-law form of the Coulomb electric force law (or Newton’s law of gravity) is an example of a non-universal representational variable. Do we imagine that Nature is somehow computing and storing these r values for each possible pair of charged particles in the universe? And updating them in real-time as they move? And what happens when such particles are destroyed or created?
In short, it is clear that this is not a universal basis representation, because the representation must be updated in highly non-local ways, with a required capacity that is both highly variable over time and exponential in size. The need to associate particles with their properties in a non-local way creates a significant logical problem: do we imagine there is a giant lookup table somewhere, cross-referenced against every other particle? How does all of this drive local forces experienced by each such particle, in an efficient, parallel manner?
By contrast with the Coulomb gauge calculational tool, wave equations provide a suitable universal basis representation for electromagnetism (EM) in the Lorenz gauge (Maxwell), and likewise for general relativity. All of the relevant state is available locally to drive the updating of the wave state over time, and a fixed set of state variables organized in 3D space can be used to represent any arbitrary EM configuration that can occur in Nature.
According to the above criteria, configuration space is another example of a bespoke calculational space, where the dimensionality and basis are chosen specifically for the problem at hand (often a simple 2D spin or qubit space), and the tensor product nature results in a lookup-table like expressive power to capture interactions or correlations. However, as discussed on that page, there is no plausible configuration space that provides a universal basis (Wallace, 2021), due to the same kinds of issues as the inverse-square law. Furthermore, the manifest non-local nature of configuration space is unambiguously incompatible with special relativity (Bell, 1975; Aharonov & Albert, 1981; Peres, 2000; Peres & Terno, 2004; Norsen, 2011).
The complementarity of variables in quantum physics (e.g., position vs. momentum) also poses a specific challenge for finding a universal basis. For example, the Fock space used in field theory adopts the momentum basis by using Fourier space, where the momentum of a given particle can be represented precisely. But, to keep things tractable, such particles are represented in a “pure” manner, as an infinite sine wave, so any information about a localized position is lost. Adding such information to the Fourier space requires adding many additional sine waves at different phases and frequencies, which then creates a significant burden for tracking all of these associated waves and calculating over them. Likewise, spin and other 2D basis spaces do not represent either momentum or position information.
By contrast, the wave equation in a CA naturally represents both the position and momentum aspects simultaneously, with the complementarity emerging due to the natural properties of waves, without the need to choose one exclusive basis or the other.
In summary, most of the primary tools used in the Standard Model are clearly not candidate universal basis representations, because they simply do not encode all of the relevant information. Furthermore, the information they do encode fails the “access to the numbers” test.
Ultimately, it may be that all attempts to discover a suitable universal basis representation for Nature fail, and we just have to accept that something inexplicable is going on relative to these basic constraints. However, the fact that so much of quantum physics can be captured with basic wave equations, which are entirely amenable to a very simple CA-like universal basis representation, suggests that such an effort is not obviously doomed to failure, and we ought to at least exhaust all alternatives before giving up.
Position as a universal basis
J. S. Bell persuasively argued that all measurement in quantum physics ultimately boils down to measuring positions of massive particles (fermions), thereby establishing the privileged status of the position basis (Bell, 1982, p. 996):
The second moral is that in physics the only observations we must consider are position observations, if only the positions of instrument pointers. It is a great merit of the de Broglie-Bohm picture to force us to consider this fact. If you make axioms, rather than definitions and theorems, about the “measurement” of anything else, then you commit redundancy and risk inconsistency.
This idea was further developed and formalized (Daumer et al., 1996; Dürr et al., 2004), all of which is based on the pilot-wave model, which provides an unambiguous framework for understanding what a measurement is actually doing, as compared to the standard Copenhagen framework which leaves this entirely underspecified.
Other proposals
There does not appear to be a significant literature on the goal of establishing universal basis space representations of the form defined above. Stoica (2025) appeals to the use of a basis space for the universe in deriving the Born probability rule. The requirement of a representation to make sense of a situated observer within the formalism is also related (Barrett, 2021), and is also used to support the pilot wave approach.