Configuration space is a critical element of standard quantum mechanics, in most of its various formulations, including the classical Hilbert space formulation evolving according to the Schrodinger wave equation, and also the pilot-wave framework. It is the space defined by the multiparticle configuration of all the elements of relevance to a given experimental setup being analyzed. Critically, the size of this space increases exponentially as a function of the number of such elements, because it is effectively a tensor product space, i.e., the outer product across all elements.
For example, the basic wave equation for a single “particle” (e.g., using the Klein-Gordon or Schrödinger equation) can be represented using the cellular automaton (CA) framework with two real-valued numbers per each state (e.g., for position and velocity, or the two parts of the complex number in the Schrödinger version), using a 3D state space discretized at a specific resolution. If we use R discrete cells per spatial dimension, the total size of this state space is:
\[ N = 2R^{3} \]
The exponential character of this space becomes apparent when we see that each particle or element requires its own 3 additional dimensions in this space, where M represents the total number of such elements:
\[ N = 2R^{3M} \]
What this means is that you effectively have different discrete CA cells for each possible combination of individual dimensions for each element. For example, there is a unique cell representing the wave function for the specific combination of position 23 on the X axis of particle A and position 12 of the Z axis of particle B. This is what an outer product means: all possible combinations are represented with their own distinct state value.
It should be clear that this space is entirely non-local across particles. Each possible spatial location in one particle is “crossed” with each possible spatial location in the other particle. This is why the standard formalism has no problem dealing with non-local entanglement across spatially-separated particles, as explored in the Bell tests (must-read for concrete example of how this all works).
From a purely computational point of view, the outer-product representation is maximally expressive, and maximally expensive. It is essentially computationally universal, allowing the representation of any and all interactions in any way that might be necessary. Thus, it is not unreasonable to suggest that this representation is a calculational tool, consistent with various arguments in the literature (Wallace, 2021; Myrvold, 2015; Bell, 1975; Peres & Terno, 2004), but not without some debate (Carroll, 2021; North, 2012; Ney & Albert, 2013).
Specifically, there really is no single privileged version of a configuration space that could be used as a universal basis space, and, as a clear signature of a calculational tool, the specific version of the space that is used depends entirely on the type of problem being solved, and the approach being used. It is not plausible to imagine that there is one single such space for the entire universe, because for example the basic particle elements of the universe are not persistent. This is the reason that the Fock space is used in field theory, to support the use of creation and annihilation operators, but that space is in Fourier space, not 3D space.
In the end, there is a compelling argument that conventional 3D space provides the only standard reference frame for physics that is universally applicable (Wallace, 2021), which is consistent with the privileged status of the position basis in the pilot wave framework.
Necessarily exponential?
The exponential size of configuration space represents a fatal barrier to any kind of plausible CA model of physics, in the same way that non-local computation does. Indeed, as shown in the Bell test discussion, the behavior of configuration space is generally incompatible with special relativity, even though the measurement results that can be computed from it are not (Bell, 1975; Aharonov & Albert, 1981; Peres, 2000; Peres & Terno, 2004; Norsen, 2011). Therefore, we are strongly motivated to adopt virtually any other kind of physical mechanism that might avoid the need for such a thing.
Interestingly, there are various lines of evidence that the exponential size is indeed excessive, exponentially so. First, Poulin et al. (2011) concluded that the exponential size of configuration space was unnecessary, because the actual physical time-dependent Hamiltonians don’t end up using the vast majority of the space. This appears to contradict a proof from Montina (2008) showing that the exponential space is necessary, but they made much stronger assumptions in this proof, including the ability to transition from any state to any other via a finite number of quantum gates. One important difference is that the first paper used a weaker constraint (polynomial vs. finite). Furthermore, the second paper requires a Markovian state evolution mechanism, which they argue is the first assumption to question in challenging the proof. Thus, the full resolution at this abstract mathematical level appears to be inconclusive.
Norsen et al. (2015) analyzed the contributions of configuration space to a pilot-wave based framework. They concluded that indeed the configuration space contains a large amount of “redundant” information, and that even the simplest approximation for the inter-particle interaction terms does a reasonable (yet imperfect) job of capturing the behavior of the full configuration-space model. Exploration of higher-order terms in this approximation is ongoing (Norsen, 2022).
Many practical applications in quantum chemistry take advantage of the low-dimensional Hartree-Fock space, which models all the interaction terms using a single averaged mean field value, instead of the standard exponential configuration space. This mean field space specifically fails to capture the detailed particle-level correlations, and thus also fails the Bell’s inequality tests to measure non-locality (Wang & Kais, 2007). However, a considerable recent literature has explored the use of neural network models to more efficiently capture the structure present in the full exponential configuration space, with one result showing that \(N^2\) parameters are sufficient to capture most of the relevant structure (Geier et al., 2025). This provides an additional suggestive indication that the full exponential space may be significantly overpowered.
Another major constraint on both non-locality and configuration space comes from considerations of integrating gravity with quantum mechanics, which is famously difficult, and seemingly impossibly contradictory in many respects. At a very basic level, general relativity, like special relativity, is fundamentally local in spacetime, and it is unclear how the apparent non-locality of the quantum world could be reconciled. The ultraviolet divergence of the standard model, reflected in the renormalization procedures, also makes it unclear how much energy might be in a given localized region of space.
A further contradiction is that according to general relativity, the entropy inside a given region of spacetime is strictly limited, as a function of the surface area of the region, known as the Bekenstein-Hawking bound (Bekenstein, 1981). This implies a corresponding limit in the number of degrees of freedom in the corresponding quantum state, which would be well below that corresponding to any kind of exponential configuration space. The interplay between this constraint and considerations from quantum field theory remains an active topic of investigation (Bousso et al., 2020; Witten, 2022; Giddings, 2015; Witten, 2018; Casini, 2008).
Overall, it is difficult to draw strong conclusions from these existing analyses, but it does not seem certain that the exponential size of configuration space is an absolute irrevocable requirement to capture the phenomenology of quantum physics.
Ultimately, the most relevant data will come from quantum computing, which is pushing the envelope on this issue, as the computational power derives directly from the exponential size of configuration space.