****
This page is the output of ChatGPT o1-pro with Deep Research option (on 03/11/25), on the topic of discrete quantum amplitudes.
I wanted to get a quick overview of what has been done on this question. The resulting report might be of interest to others.
*****
Quantum mechanics traditionally relies on the continuum of complex numbers to represent probability amplitudes. However, some researchers have explored whether these complex amplitudes might be discrete (drawn from a structured set of values) rather than truly continuous. Motivations for this idea include the observation that real/complex numbers carry infinite information, which may be unphysical in a finite universe (Indeterminist physics for an open world | ScienceDaily) (Geometry of discrete quantum computing). In this report, we survey theoretical frameworks that propose a discrete structure for quantum amplitudes, connections to number theory and algebraic structures, and the implications for quantum computing, quantum field theory, and quantum foundations. We also discuss any experimental attempts to detect such discreteness, and highlight key studies and contributors in this niche field.
Computable and Finite-Field Amplitude Hypotheses: One line of thought argues that physical laws might only involve computable numbers or elements of a finite field, avoiding the unphysical “infinite precision” of real numbers (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). Alan Turing’s seminal work on computable numbers and Rolf Landauer’s insights on finite information suggest that continuum mathematics (with uncountable real numbers) may not be necessary for physics (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). Building on this idea, A. J. Hanson and colleagues proposed Discrete Quantum Computing (DQC), replacing the field of complex numbers with a finite Galois field in quantum theory (Geometry of discrete quantum computing). They hypothesized that because the universe’s resources are finite, quantum mechanics could be formulated over computable numbers or finite fields without loss of physical validity (Geometry of discrete quantum computing). This framework uses complexified finite fields (finite fields with an imaginary unit) to construct Hilbert spaces with discrete amplitude values. Hanson et al. developed the geometry of discrete Hilbert spaces (complex projective spaces over $GF(p^2)$) and showed how traditional concepts like the Bloch sphere and entanglement can be translated into the discrete setting (Geometry of discrete quantum computing) (Geometry of discrete quantum computing).
Quantum Mechanics over Finite Fields: Independently, Lay Nam Chang, Djordje Minic, Tatsu Takeuchi, and colleagues at Virginia Tech constructed quantum mechanics on a vector space over a finite Galois field $GF(q)$ ([1208.5544] Some Mutant Forms of Quantum Mechanics). In their “Galois Field Quantum Mechanics,” states have probability amplitudes in a finite field rather than $\mathbb{C}$. Remarkably, they found that such a discrete model can reproduce many quantum features but with subtle differences in correlations. For example, a two-qubit system over $GF(q)$ cannot violate the standard Bell inequality (CHSH form), despite not being classically reproducible by any local hidden-variable model ([1208.5544] Some Mutant Forms of Quantum Mechanics). In other words, the discrete-amplitude theory remains non-classical (no hidden variable model can exactly mimic it) yet does not allow the full strength of quantum correlations that real/complex quantum mechanics predicts. This result hints that using a discrete number field for amplitudes might impose “superquantum” constraints or otherwise modify entanglement. The same team explored other “mutant” forms of quantum mechanics on finite fields and identified alternative constructions that preserve some quantum advantages while limiting others ([1208.5544] Some Mutant Forms of Quantum Mechanics). These studies demonstrate self-consistent quantum frameworks without a continuum of amplitudes, laying groundwork for discrete amplitude quantum theory.
Modal Quantum Mechanics (GF(2) Toy Model): Earlier, Benjamin Schumacher and Michael Westmoreland had introduced a simplified modal quantum mechanics using the field $GF(2)$ (binary field) instead of complex numbers. This toy model replaces amplitudes with elements of ${0,1}$ and reinterprets state addition as a logical exclusive-or operation. It lacks full quantum structure (no continuous phases), but it isolates the “quantum-esque” feature of interference via set exclusivity (Geometry of discrete quantum computing). Hanson et al. found that this extreme discretization led to unphysical “super-power” in computation (e.g. solving certain SAT problems in constant time) because $GF(2)$ lacks an analog of unitary rotations (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). This motivated them to move to larger finite fields (e.g. primes of form $p\equiv 3 \pmod 4$) which admit a richer structure (“complex” phases modulo $p$) and a notion of orthonormality (Geometry of discrete quantum computing). By using complexified Galois fields, they recovered the ability to define inner products and unitary transformations, eliminating the unphysical computational advantages (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). This progression illustrates how theoretical frameworks can trade off arithmetic complexity (field size) for physical fidelity, aiming to replicate standard quantum behavior with only discrete amplitude values.
Number-Theoretic Quantum Mechanics (p-adics and Adèles): Another class of proposals replaces the field of real or complex numbers with number systems from number theory. p-Adic quantum mechanics is an umbrella term for efforts to formulate quantum theory using p-adic numbers (extensions of the rationals with a non-Archimedean metric) instead of reals (p-adic quantum mechanics - Wikipedia). This idea was originally inspired by string theory: in 1987, Volovich noted that the famous Veneziano amplitude (an integral in bosonic string theory) could be reformulated over p-adic numbers (p-adic quantum mechanics - Wikipedia). This led to p-adic string theory and the conjecture that spacetime or physics at the Planck scale might be discrete in a number-theoretic sense (p-adic quantum mechanics - Wikipedia). In p-adic quantum mechanics, spatial coordinates or even wavefunctions take values in a p-adic domain. For instance, Vladimirov and Volovich (1989) studied p-adic analogues of the Schrödinger equation. One approach is to consider particles moving on a discrete ultrametric space (the p-adic number line) while maintaining complex-valued wavefunctions ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics). Another approach allows the wavefunction itself to be p-adic-valued, raising interpretational challenges since probabilities would then be p-adic. In practice, many formulations use p-adic path integrals or limit processes, because a direct p-adic Schrödinger evolution requires redefining concepts of normalization and probability. Adelic quantum mechanics combines the real and p-adic descriptions by using the mathematical adèle (an entity with components in each $p$-adic field and the reals). Researchers like Branko Dragovich have shown that one can formulate adelic quantum theory, which is invariant under interchange of real and p-adic fields (P-adic and adelic quantum mechanics - Inspire HEP). The adelic approach suggests that standard quantum mechanics might emerge as a special Archimedean case of a more general arithmetic quantum framework.
Alternative Algebraic Structures: Beyond finite fields and p-adics, physicists have also considered other replacements for the complex number system in quantum theory. Quaternionic quantum mechanics (studied by Stephen Adler and others) uses quaternions instead of complex numbers. Quaternions form a division algebra that is still continuous, not discrete, but they extend the number system in a different direction (non-commutative coefficients). While quaternionic quantum mechanics has interesting mathematical properties, it does not make the amplitude space discrete, so we mention it only for completeness. More pertinently, finite-dimensional division rings or rings of algebraic integers could provide discrete subsets of $\mathbb{C}$. For example, one might imagine restricting amplitudes to the Gaussian integers or to rational numbers. In practice, any such restriction must still allow a dense set of phases to reproduce interference phenomena. (If only a coarse discrete phase set is allowed, many quantum interference patterns would be lost or visibly altered.) Nonetheless, rational quantum mechanics has been contemplated: if all amplitudes were rational numbers, the theory might evade certain no-go theorems at the expense of exact unitarity. In fact, a result by Meyer (1999) showed that if one only demands agreement with quantum predictions up to finite precision, one can construct a hidden-variable model using a dense subset of quantum states (with rational-valued coordinates) that avoids the Kochen–Specker contradiction (Finite Precision Measurement Nullifies the Kochen-Specker Theorem | Phys. Rev. Lett.). This indicates that the continuum of values in quantum theory is intimately connected to its contextuality; a discrete approximation to the complex plane might allow a classical description that is otherwise forbidden by the full continuum theory. (We return to this point in the Foundations section.)
In summary, a variety of theoretical frameworks replace the continuum complex field with a discrete or different arithmetic structure: finite Galois fields, p-adic number fields, or other algebraic constructs. These frameworks must redefine core quantum concepts (state spaces, inner products, unitary evolution, etc.) in the new arithmetic setting. The works of Hanson et al. (Geometry of discrete quantum computing) and Chang et al. ([1208.5544] Some Mutant Forms of Quantum Mechanics) demonstrate that one can, at least in finite-dimensional systems, formulate consistent “quantum-like” theories without continuous amplitudes. Meanwhile, p-adic approaches probe what quantum mechanics might look like in a fundamentally discrete spacetime or with number-theoretic coordinates, an area that remains more speculative but rich in mathematical structure.
The exploration of discrete amplitude spaces connects with broader themes of discrete quantum mechanics and number theory in physics. In discrete quantum mechanics (as often discussed in literature), one usually means quantum models where time, space, or other variables take on discrete values. Examples include difference-equation versions of the Schrödinger equation or quantum systems on lattices. These models typically retain the usual complex amplitudes but assume a discrete spectrum or lattice in physical variables. By contrast, here we are focused on discreteness of the amplitude values themselves. Nevertheless, both lines of inquiry share a common spirit: questioning the continuum assumptions in quantum theory. For instance, efforts to test spatial discreteness via neutron interferometry (e.g. C. Wolf’s proposal) involve deriving phase shifts under a hypothesis of spatial lattice and seeing if experiment deviates from continuum predictions. If space or time were discrete at a very fine scale, the interference pattern of neutrons or other particles might show slight anomalies (C. Wolf, Testing discrete quantum mechanics using neutron ...). Similarly, if amplitude values were discrete or rational, there could be subtle deviations in interference visibility or phase that accumulate in complex experiments.
Number-theoretic approaches to quantum physics often originate from the idea that nature may be number-theoretically structured. Besides p-adic physics, there are models invoking modular arithmetic, lattice of integers, or algebraic number fields for state space. One intriguing angle is the idea that quantum phases could be quantized. Normally, a qubit’s relative phase can be any real number modulo $2\pi$. But one could imagine a theory where only certain phase angles are allowed (for example, phases must be multiples of some fundamental angle $\Delta\phi$). This would effectively discretize the complex unit circle. Early quantum logic researchers like Birkhoff and von Neumann proved that aside from real, complex, or quaternionic fields, no other complete division algebra can serve as the scalars of Hilbert space if one insists on standard axioms (like Gleason’s theorem assumptions). However, if one relaxes completeness or other assumptions, finite fields become possible scalar candidates, as demonstrated by the finite-field quantum mechanics constructions. The price one pays is that the usual Hilbert space properties (such as an ordering for probabilities or a continuum of orthonormal bases) need modification. In number-theoretic quantum frameworks, probability may need redefinition – for example, one might map finite-field amplitudes to real probabilities via a norm or embedding. Chang et al. sidestepped the issue of defining probabilities in $GF(q)$ by focusing on correlation predictions (like Bell-test statistics) which can be compared to real-world outcomes ([1208.5544] Some Mutant Forms of Quantum Mechanics). In p-adic quantum mechanics, one often ends up with ordinary real probabilities at the end, even if the intermediate calculation uses p-adic valued wavefunctions ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics). This is seen in recent work by Zúñiga-Galindo (2024), who constructed a p-adic Schrödinger equation and analyzed the two-slit interference: the wavefunction lives on a p-adic spatial lattice, but only $|\psi|^2$ (a real number) has physical meaning, yielding an interference pattern in the detector counts ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics). Interestingly, in that model each particle can be thought of as going through one slit at a time (no simultaneous path superposition in the p-adic sense), yet an interference pattern still emerges in the probability distribution ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics). This counter-intuitive result underscores how deeply probability amplitudes and their continuity are tied to our usual interpretation of quantum phenomena.
In number theory, concepts like Galois fields, rings of integers mod $n$, and p-adics introduce discretization at a fundamental level. These structures have algebraic symmetries and properties that could, in principle, correspond to physical symmetries. For example, using $GF(p)$ imposes an inherent periodicity (mod $p$ arithmetic), which might be connected to phase periodicity or other quantum cyclic phenomena. Replacing the continuum with a large finite field might be viewed as imposing a colossal but finite ``resolution'' on amplitude values – somewhat akin to digital sampling of an analog signal. If $p$ is astronomically large, the discrete model would be empirically indistinguishable from the continuum in all current experiments, yet it would satisfy a principle of finite information content. This interplay between discrete mathematics and physical theory is a developing area of quantum foundations that bridges physics with theoretical computer science and logic.
If the amplitudes in quantum mechanics were discrete rather than continuous, the implications for quantum computing would be profound. Quantum algorithms often rely on delicate interference between amplitudes that are tuned to exact real values (such as the specific rotation angles in the Quantum Fourier Transform or the continuous parameter in Shor’s algorithm). In a discrete-amplitude theory, only certain amplitude values or phase angles might be allowed, potentially limiting the operations that a quantum computer could natively perform.
On the positive side, a discrete amplitude theory aligns with the notion of a digital quantum computer that ultimately processes finite information. If nature only permitted computable amplitudes, then any quantum computer is effectively simulating within a rational approximation of quantum mechanics. This could mean that quantum computers wouldn’t have unbounded analog precision power, reassuring us that the Church-Turing thesis (no physical device can compute non-computable functions) holds even in quantum theory. In fact, one motivation behind Hanson et al.’s DQC program was to remove “uncomputable” resources from quantum computing (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). They showed that using finite fields avoids certain unphysical algorithmic advantages. For instance, their early toy model (modal quantum computing over $GF(2)$) strangely could solve the UNIQUE-SAT problem with exponential speed-up by exploiting a loophole (Geometry of discrete quantum computing). Once they moved to larger fields and a proper discrete Hilbert space structure, that “supernatural” power disappeared (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). This suggests that a properly constructed discrete amplitude theory might naturally enforce the same computational complexity limits as standard quantum mechanics, whereas ad hoc discrete models could either cripple quantum algorithms or give unrealistic power.
Quantum error correction and fault tolerance could also be impacted. Discrete amplitude values might provide inherent noise resilience if small deviations are simply not allowed (similar to how digital signals resist small analog noise). However, if an error pushes an amplitude to the nearest allowed value, it might lead to a different type of error syndrome. The design of quantum gates would likely be restricted to those that map allowed amplitudes to allowed amplitudes. In finite field quantum computing, one must define unitary operations carefully – for example, Chang et al. and Hanson et al. construct analogs of Pauli matrices and Hadamards within $GF(q)$ arithmetic (Geometry of discrete quantum computing). Some gates familiar in continuous quantum computing might not exist in a truly discrete framework, or they might exist only approximately.
Another implication is on simulation: A quantum computer with inherently discrete amplitudes might be efficiently simulable by a classical computer, depending on the field size. If the field is huge (e.g. 160-bit prime), classical simulation is still difficult, but if the field is small (like $GF(2)$ or $GF(3)$), one could simulate the quantum evolution by brute force over the finite state space. Researchers have noted that using finite fields sometimes allows exhaustive simulation to check properties, since the state space, though exponential in qubit number, is finite () (). This was one reason finite-field quantum computing models were attractive to study – one can search for patterns or validate identities by iterating over all field elements for small systems ().
From an algorithmic perspective, discrete amplitude quantum computing might restrict certain algorithms that rely on intermediate amplitude values interfering precisely. But it could also open new algorithmic ideas. Perhaps arithmetic over a finite field could be directly embedded into quantum computations, enabling new forms of quantum cryptography or computations on finite rings. Indeed, one paper titled “Geometry of Discrete Quantum Computing” explores how entanglement properties depend on the size of the underlying field, and how one might count entangled states in a discrete Hilbert space (Geometry of discrete quantum computing) (Geometry of discrete quantum computing). These insights could inform alternate quantum protocol designs that are optimized for discrete amplitude systems.
In summary, if quantum amplitudes are discrete, quantum computers would essentially operate on “digital” quantum states. This could limit their power (preventing them from solving certain problems exponentially faster than classical computers in unintended ways) but also ensure they are physically realizable. The work by Hanson, Chang, and others indicates that careful choice of the discrete structure is needed to preserve quantum computing’s known capabilities (like entanglement and computational speedups for some problems) while respecting new limits. It remains an open question how much of the quantum algorithm repertoire (Shor, Grover, etc.) could be ported to a discrete-amplitude quantum computer without significant loss of efficiency.
At the level of quantum field theory (QFT), assuming a discrete amplitude parameter space leads to intriguing possibilities. Quantum field theories involve not only quantum amplitudes but also typically an infinity of degrees of freedom (field modes at each point in space). A fundamental discreteness in amplitude might serve as a form of built-in regularization: if fields can only take values from a discrete set (or if field operators only have discrete spectra of coefficients), then ultraviolet divergences might be tamed. Some researchers have speculated that a discretization of the number system underlying QFT could cure certain infinities or make the theory manifestly finite. For instance, p-adic string theory provided finite results for scattering amplitudes that in real-number string theory required renormalization (p-adic quantum mechanics - Wikipedia). The p-adic formulation essentially replaces continuous integration measures with sums over a discrete (p-adic) domain, eliminating continuum ultraviolet behavior. This raises the question: could the physical world be such that at Planck scales, the continuum is replaced by a discrete number structure, making all physical amplitudes effectively finite? Ivan Volovich’s 1987 conjecture directly stated that at the Planck length, the spacetime geometry and the values of physical fields might be p-adic rather than real (p-adic quantum mechanics - Wikipedia). If true, our continuum QFT would be a low-energy effective description, and deviations might become evident only at extreme energies or tiny distances.
An implication for QFT is also on the violation of certain symmetries or axioms. For example, standard QFT heavily relies on principles like locality, Lorentz invariance, and unitarity in a complex Hilbert space. A discrete amplitude space might subtly violate locality (as finite fields often lack an order or topology needed to define continuity in space). In the p-adic Schrödinger model mentioned earlier, the resulting Hamiltonian was nonlocal ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics) – essentially because p-adic space is ultrametric (every jump is either zero or large, no notion of arbitrarily small localized interaction). Thus, a QFT built on a discrete or p-adic number system might inherently be nonlocal at the fundamental scale, potentially offering a different route to quantum gravity (since many quantum gravity approaches also predict some form of fundamental nonlocality or discreteness).
In practice, there have been concrete attempts to merge these ideas: p-adic quantum field theory and adelic field theory. For example, Freund and Witten (1987) famously computed p-adic string amplitudes and found a relation between p-adic and ordinary string amplitudes via an adelic product formula. This suggested that the ordinary string amplitude could be factored into contributions from all $p$-adic sectors and the Archimedean (real) sector, hinting at a hidden number-theoretic structure in the theory of strings. Researchers like Vladimir Dragovich and collaborators extended this to field theory, studying p-adic analogues of scalar field dynamics and instantons. These models often yield insights into nonperturbative aspects of QFT by using arithmetic analytic continuation.
For foundational interpretations of quantum field theory, a discrete amplitude space could support super-deterministic or hidden-variable formulations that are otherwise disallowed. In a continuous complex field, Bell’s theorem and related results lean on the continuum of measurement settings and outcomes. In a discrete setting, some of those assumptions are modified. We already saw an example in the finite field Bell test result: no CHSH violation ([1208.5544] Some Mutant Forms of Quantum Mechanics). If quantum fields were defined over finite or algebraic number domains, one might similarly find that certain quantum correlations have different limits. This could affect our understanding of topics like entanglement entropy (which in a continuum field is formally infinite for a sharp boundary, but with discrete amplitude values might be naturally capped). Also, the path integral formulation of QFT might be reconceived as a sum over discrete-field histories rather than an integral over continuous field configurations. This is somewhat analogous to how lattice gauge theory defines the path integral as a finite (though large) sum. Typically in lattice gauge theory, spacetime is discretized but fields (group variables) are continuous. A fully discrete amplitude approach might discretize the field values as well, potentially yielding a double discretization (lattice spacetime and finite field values). Some computational evidence shows that even coarse discretizations of field values can flow to the correct continuum limit under renormalization group, which is encouraging if one hopes that the true theory has a built-in discreteness at microscopic scales.
Lastly, we note an interesting bridge to algebraic number theory in QFT: There are efforts to understand symmetries and dualities via number fields. For example, some authors have investigated whether the appearance of special numbers (like Dedekind eta function values, modular forms, etc.) in quantum amplitudes hints at an underlying arithmetic structure. If complex amplitudes were discrete or algebraic, many of these special functions would take values in algebraic number fields, perhaps simplifying or explaining why certain cancellations occur. Although this is speculative, it resonates with the broader theme that maybe quantum mechanics with continuous complex numbers is an approximation to a deeper theory with discrete/finite arithmetic that naturally encodes what we observe.
The question of discrete versus continuous amplitude touches on the foundations of quantum mechanics. Foundational interpretations – such as whether quantum probabilities are objective or epistemic, whether hidden variables exist, or how measurement works – could all be affected if the underlying math is discrete.
One key implication is on the Bell and Kochen-Specker no-go theorems, as hinted earlier. Meyer’s result in 1999 showed that if one assumes only finite precision in measurements (effectively replacing the continuum of directions by a dense discrete subset), the Kochen-Specker theorem (which rules out non-contextual hidden variables in standard QM) can be circumvented (Finite Precision Measurement Nullifies the Kochen-Specker Theorem | Phys. Rev. Lett.). Following Meyer, Adrian Kent and Rob Clifton also argued along similar lines that the apparent contextuality of quantum mechanics might be an artifact of assuming exact real number values for measurement settings. If nature only allows a dense discrete set of states or measurement orientations, a non-contextual hidden-variable model becomes possible that agrees with all practical experiments (since experiments have finite precision). This line of reasoning suggests that a discrete-amplitude (or discrete-state) theory might permit local-realistic or non-contextual explanations that standard quantum mechanics forbids. In other words, the “weirdness” of quantum mechanics (contextuality, nonlocality) could be softened or removed in a theory where the continuum is replaced by a discrete structure – but only if that discrete structure is carefully chosen to be just dense enough to mimic the quantum predictions within experimental bounds. Chang et al.’s finite field model is a concrete example: it is local realistic enough not to violate a Bell inequality, yet still not exactly classical ([1208.5544] Some Mutant Forms of Quantum Mechanics). This places it in an intriguing intermediate position from a foundations perspective, something like a superdeterministic theory or a post-quantum theory with restricted contextuality. Foundationally, such models challenge the assumption that we have already experimentally verified the full continuum of quantum prediction. If our actual experiments only ever sample rational approximations of amplitudes, one could posit a hidden variable model operating on those rational points consistent with everything seen so far.
Another implication is for the Born rule and probability interpretation. In standard quantum theory, the Born rule (probability = $|\psi|^2$) is a separate postulate that works hand-in-hand with the field of complex numbers to ensure continuous probability distributions. If amplitudes were discrete, one must ensure that probabilities still come out as meaningful real numbers in [0,1]. Finite field or p-adic amplitudes are not ordered in the usual sense, so one typically defines an embedding or uses a squared-norm in a conventional real sense. The foundational question arises: is probability itself quantized? Could it be that there is a smallest nonzero probability that can occur in nature? Most approaches say no – even discrete amplitude models yield effectively continuous probabilities, since experimental frequencies are rational anyway (N occurrences out of M trials). However, it’s worth noting: if amplitude space is discrete, there might be subtle deviations from the exact Born rule. Perhaps interference is not perfectly destructive in cases where it should cancel out, if the cancellation cannot be represented exactly in the discrete arithmetic. Experiments have tested the Born rule’s exactness by looking for third-order interference (beyond the pairwise interference of the double-slit). The landmark triple-slit experiment by Sinha et al. (2010) looked for any sign of an extra term when three paths interfere simultaneously. They bounded any such deviation to less than 1 part in 100 of the expected intensity ([1007.4193] Ruling Out Multi-Order Interference in Quantum Mechanics). This strongly supports the absence of weird discreteness or nonlinearity at that level – effectively confirming that standard quantum interference (assuming continuous amplitudes) holds to high precision. Such results put constraints on any discrete-amplitude theory: it must mimic continuous quantum mechanics extremely well for superpositions of up to at least three or more paths. If the amplitude space were coarse or had a minimal granularity that was too large, we would likely have seen these interference anomalies. So far, no experiment has detected a finite-step size in probability amplitudes; quantum predictions appear smooth even at very low probabilities (experiments have verified interference with single-photon sources where probabilities like $10^{-6}$ still behave as expected).
Interpretations like QBism or relational quantum mechanics emphasize the role of information and might be amenable to a discrete amplitude reinterpretation, since they downplay any physical reality of the wavefunction itself. In QBism, for example, the wavefunction is just an agent’s information; if that information is ultimately stored in a finite brain, one could argue it’s naturally discrete (rational probabilities). On the other hand, objective collapse theories or many-worlds interpretations lean on the amplitude’s ontological role – altering the nature of amplitude could have major effects there. A discrete amplitude could act like a built-in regularization that causes effective collapse when amplitudes fall below a certain granularity (since they might not be representable if too small). This is speculative, but one could imagine a collapse mechanism that “kicks in” when the amplitude ratio between branches becomes smaller than one quantum of amplitude. No such theory has been fleshed out in detail, but it’s an interesting avenue: it might connect to ideas of gravity-induced collapse (Penrose) if one thinks the gravitational field cannot distinguish changes below a certain amplitude, effectively collapsing the state.
Finally, information theoretic interpretations (like Wheeler’s “it from bit”) resonate strongly with discrete amplitudes. If information is fundamental, one might expect everything, including amplitude values, to be quantized in information units. John Wheeler speculated that the physical world is at bottom made of binary information. If we take that seriously, perhaps the complex phase of a qubit is not a continuum but composed of bits of information. Recent philosophical works by Nicolas Gisin have argued that the use of real numbers in physics is an unjustified idealization: “Real numbers are, de facto, the hidden variables of classical physics” (Items where Author is "Gisin, Nicolas" - PhilSci-Archive), carrying infinite information that leads to determinism. Gisin suggests reformulating physics in terms of processes that produce one decimal (or bit) at a time (Indeterminist physics for an open world | ScienceDaily). In a quantum context, this could mean that as time evolves, the amplitude’s precision grows digit by digit, rather than assuming an exact real value. Such an interpretation is deeply radical but aligns with an intuitionistic mathematics view: the universe doesn’t “know” an infinite decimal expansion, it only realizes as much detail as needed. While these ideas remain philosophical, they do provide a conceptual foundation for why amplitude might be discrete or gradually defined, rather than a static continuum.
Detecting whether quantum amplitudes are discrete is extremely challenging, because any discretization – if it exists – is likely at a scale far beyond current experimental resolution. Essentially, as long as the discrete step size in amplitudes (or phases) is tiny compared to experimental precision, all experiments will match the continuous theory. That said, physicists have performed experiments that, while not targeting “amplitude discretization” per se, do test the linearity and continuity of quantum mechanics at high precision.
One class of tests involves interference experiments with very low probabilities. The triple-slit experiment mentioned above is one example, setting bounds on non-standard interference terms ([1007.4193] Ruling Out Multi-Order Interference in Quantum Mechanics). Another example is interferometry with single neutrons or single electrons over long baselines, which can test if the interference pattern is exactly as predicted by a continuous wavefunction. If amplitude were discrete at some scale, one might expect a slight diffraction or decoherence effect when path amplitudes become very small or very close in phase, analogous to pixelation in a digital image. Experiments with neutron interferometers (like the COW experiment testing gravitational phase shifts, or tests of split-beam phase stability) have generally found perfect agreement with quantum mechanics, constraining any “discreteness” effects. For instance, studies by Zeilinger et al. and others in the 1980s showed interference of single neutrons even when the beam intensity (hence amplitude in each path) was extremely low, with no anomalous loss of coherence beyond what’s expected from environmental decoherence. These results, while not usually phrased in terms of amplitude quantization, imply that if there is a minimum amplitude increment, it must be much smaller than $10^{-6}$ in typical units of wavefunction amplitude (since probabilities on the order of $10^{-12}$ have been indirectly verified by observing millions of particles).
Another route is to test quantum mechanics in regimes where a discrete model might differ. One suggestion in the literature is to examine entangled states for Bell-inequality saturation. Chang et al.’s model predicts no CHSH violation ([1208.5544] Some Mutant Forms of Quantum Mechanics); thus if one could prepare a state that in standard QM gives maximal violation of CHSH, but an experiment finds it does not violate beyond the classical bound, that could hint at underlying discretization. So far, experiments with photons, electrons, ions, etc., have violated CHSH up to the Tsirelson bound (≈2.828) as expected with continuous amplitudes. No observation has shown a systematic shortfall below the quantum prediction that would indicate a discrete amplitude cap. (All results are either in agreement with quantum theory or occasionally lower due to experimental inefficiencies, which are well-understood classical effects.)
Physicists have also looked at spectroscopic tests for any nonlinearity or discreteness in the Schrödinger equation. A famous set of experiments (e.g. by Weinberg’s suggestion) looked for energy level shifts or bifurcations that would indicate a slight departure from linear evolution. Although those tests were targeting nonlinear terms, a discrete amplitude could manifest effectively as a nonlinear perturbation (since rounding to a discrete grid could break the superposition principle subtly). The outcome has been that no evidence of nonlinear or discretized behavior was found down to very sensitive levels (e.g., in atomic hyperfine transitions).
If amplitude discretization is related to a fundamental length (like Planck length $~10^{-35}$ m), one might need to probe processes involving enormous energy or time precision to see effects. For example, some have speculated that cosmological observations or high-energy cosmic ray interference might show anomalies if spacetime or amplitudes are discrete. So far, cosmic interference (like diffraction patterns in gravitational lensing or double-slit analogs with astrophysical sources) has not revealed any discrepancies attributable to amplitude quantization.
Another indirect way to constrain amplitude discreteness is via quantum computing experiments themselves. If we build small quantum processors (with a handful of qubits) and compare their behavior to simulations using only rational amplitudes of limited precision, any deviation might be a sign that nature requires the continuum. So far, quantum gates and circuits perform as expected; any errors are attributed to noise, not a fundamental limit on amplitude values. As quantum processors scale up, if one ever observed that certain interference-based algorithms consistently underperform even after accounting for noise, one might question if the continuum theory is fully accurate. No such red flag has appeared — quantum gates can produce superposition states with amplitudes as arbitrarily fine as we can indirectly measure (through tomography), limited only by calibration accuracy.
Given the difficulty of direct detection, researchers often frame the search for amplitude discreteness in terms of setting upper bounds on possible granularity. For instance: “If amplitude is discrete, the step size must be smaller than $10^{-k}$ (in absolute amplitude) or else experiment X would have observed an anomaly.” With current technology, one could argue k is very large (perhaps on the order of 20-30 or more) meaning amplitudes could be discrete at the $10^{-30}$ level and we’d not yet know. In many discrete models (like finite fields), the “step size” is not linear but depends on mod p arithmetic. In such cases, constraints might be phrased as “the modulus p must be larger than about $10^N$” to be consistent with experiments. Some theoretical papers (e.g. by Janet Conrad and others investigating Planck-scale discreteness in neutrino oscillations) have derived that if neutrino flavor amplitudes were discrete at the Planck scale, it would induce tiny probability violations that next-generation experiments could in principle detect. So far, no such effect has been seen, but these remain interesting potential windows.
In summary, no experiment to date has found evidence of a discrete amplitude space – quantum mechanics with continuous complex amplitudes has passed every test. Experimental attempts that relate (testing interference linearity, higher-order interference, precise entanglement correlations) all support the continuity of the wavefunction. However, the door isn’t completely closed: it’s possible that amplitude discretization exists at scales far beyond current reach. The challenge for experimental physicists is to design clever tests that could amplify a tiny discreteness effect. One idea could be to use quantum amplification processes (like in a quantum chaotic system or a sensitive interferometer network) where a minute deviation grows nonlinearly to an observable level. Thus far, none of these ideas have borne fruit, but as quantum control improves, who knows – we might one day detect a small “graininess” in the fabric of the complex plane.
The exploration of discrete amplitude spaces in quantum mechanics is a multidisciplinary endeavor spanning quantum foundations, mathematical physics, and quantum computing theory. Below we highlight some key contributors and works:
A. J. Hanson (Indiana University) and collaborators – “Geometry of Discrete Quantum Computing” (J. Phys. A, 2013) (Geometry of discrete quantum computing). This work lays out a rigorous framework for quantum states over finite Galois fields and studies entanglement and state geometry in that context. It builds on earlier ideas by Turing (Geometry of discrete quantum computing) and Landauer about computable physics, and references V. I. Arnold’s mathematical work on projective geometry over Galois fields. Hanson’s group demonstrated how conventional quantum geometric structures emerge in the limit of large finite fields, and how discrete analogues can be constructed for small fields.
Lay Nam Chang, Djordje Minic, Tatsu Takeuchi, Zachary Lewis (Virginia Tech) – “Galois Field Quantum Mechanics” (Mod. Phys. Lett. B, 2013) and “Some Mutant Forms of Quantum Mechanics” (AIP Conf. Proc. 2012) ([1208.5544] Some Mutant Forms of Quantum Mechanics). Their papers introduce quantum mechanics over $GF(q)$ and analyze fundamental features like Bell inequalities in this setting. They are notable for explicitly showing a model that is “quantum” yet does not violate CHSH, highlighting new possibilities for the foundations of QM. This team’s work has been influential in prompting questions about what features of quantum theory truly require the continuum, and which do not.
P-adic Quantum Physics Pioneers: Vladimir S. Vladimirov and Igor V. Volovich (Steklov Mathematical Institute, Russia) are credited with early foundational work (late 1980s) on p-adic quantum mechanics and quantum field theory. Volovich’s 1987 paper conjecturing p-adic spacetime (p-adic quantum mechanics - Wikipedia) initiated a field of study that includes p-adic string theory and cosmology. Branko Dragovich (University of Belgrade, Serbia) is known for extensive reviews and developments in p-adic and adelic quantum mechanics (e.g. arXiv:hep-th/0312046) ([hep-th/0312046] p-Adic and Adelic Quantum Mechanics - arXiv). W. Zúñiga-Galindo (Centro de Investigación en Matemáticas, Mexico) is a more recent contributor, with works on p-adic quantum equations and experiments (2023/24) analyzing how two-slit interference would work in a discrete p-adic space ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics).
Modal Quantum Mechanics: Benjamin Schumacher (Kenyon College) and Michael D. Westmoreland (Denison University) proposed modal quantum theory as a simplified discrete model (information-based approach). Their work (circa 2010) showed how a quantum-like theory over the boolean field could be constructed and analyzed. Although initially seeming too “toy-like”, it became a stepping stone for more advanced finite-field theories (as cited in Hanson et al.’s work) (Geometry of discrete quantum computing).
Kochen-Specker and Discreteness: David Meyer (UC San Diego) – “Finite Precision Measurement Nullifies the Kochen-Specker Theorem” (Phys. Rev. Lett. 1999) (Finite Precision Measurement Nullifies the Kochen-Specker Theorem | Phys. Rev. Lett.). Meyer’s argument that using only rational-valued directions avoids the KS paradox was a cornerstone in suggesting that maybe nature’s indeterminacy could stem from a hidden discreteness. Adrian Kent (University of Cambridge) and Rob Clifton (late, University of Pittsburgh) also wrote about this “finite precision” loophole in the late 1990s, provoking discussions in the quantum foundations community about whether continuity is an experimentally unsupported assumption. Although some disagreed with the implications, these works underscored the interplay between mathematical precision and physical reality.
Nicolas Gisin (University of Geneva) – While not proposing a new discrete model of quantum mechanics, Gisin has been a vocal proponent of revisiting the role of real numbers in physics. His essay “Are There Real Numbers in Physical Reality, or Could Quantum Mechanics be an Emergent Phenomenon?” and a Nature Physics article in 2020 questioned the necessity of uncountably infinite information in physical models (Indeterminist physics for an open world | ScienceDaily). By drawing attention in the broader physics community (through media like ScienceDaily (Indeterminist physics for an open world | ScienceDaily)), Gisin’s work has lent support to those pursuing discrete or algorithmic reformulations of quantum theory.
Gerard ’t Hooft (Utrecht University) – Nobel laureate ’t Hooft has proposed a deterministic, cellular-automaton interpretation of quantum mechanics, outlined in his book “The Cellular Automaton Interpretation of Quantum Mechanics” (2016). His model assumes an underlying discrete phase space and deterministically evolving “beables,” with quantum states as epistemic overlays. While his focus is more on determinism vs. quantum indeterminism, his approach is compatible with the idea that the continuum is emergent. He conjectures that quantum amplitudes and Hilbert space arise from averaging over discrete states. This is another influential voice suggesting that quantum theory’s continuity might not be fundamental.
John Archibald Wheeler (Princeton) – Through his famous phrase “it from bit”, Wheeler in the 1980s advocated that information theory might underlie physics. He speculated that space-time and quantum phenomena might be built from yes/no binary events at Planck scale. Wheeler’s legacy inspires modern researchers to think about discrete information units as the substrate of the wavefunction.
Institutions and Collaborations: This topic cuts across fields, so there isn’t a single institute solely dedicated to it. However, groups like the Institute for Quantum Computing (IQC) at University of Waterloo and Perimeter Institute have hosted talks and workshops on quantum foundations where such ideas are explored. The International Centre for Theory of Quantum Technologies (ICTQT) in Gdańsk, Poland (where some authors of the 2022 discretized observables paper (Discretized continuous quantum-mechanical observables that are neither continuous nor discrete | Phys. Rev. Research) are based) is also known for foundational research. On the number theory side, the Steklov Institute and mathematical departments like CINVESTAV in Mexico (Zúñiga-Galindo’s affiliation) contribute. The Quantum Theory and Symmetries conferences and the Växjö series of conferences on quantum foundations have seen presentations (e.g. Takeuchi’s 2012 talk ([1208.5544] Some Mutant Forms of Quantum Mechanics)) on these topics, indicating a sustained, if niche, interest.
Research into a discrete complex number parameter space for quantum amplitudes remains a blend of speculative theory, deep mathematical insight, and foundational soul-searching. Theoretical frameworks using finite fields and p-adic numbers show that quantum-like behavior can persist without a continuous spectrum of amplitudes (Geometry of discrete quantum computing) ([1208.5544] Some Mutant Forms of Quantum Mechanics), although subtle differences arise that could, in principle, be testable. Number-theoretic approaches like p-adic quantum mechanics tie into broader efforts to reconcile quantum theory with quantum gravity and the limits of computation. The implications of amplitude discreteness touch almost every aspect of quantum science – from the power of quantum computers to the interpretation of quantum reality.
So far, experiments have not demanded a departure from the continuum hypothesis; quantum mechanics over the complex numbers continues to match observations to high precision. But the possibility of an underlying discreteness is intellectually enticing: it promises a physical world that is ultimately digital at its core, perhaps resolving philosophical puzzles about infinite precision in physics (Indeterminist physics for an open world | ScienceDaily) (Geometry of discrete quantum computing). As technology advances, we may gain the tools to probe quantum mechanics at unprecedented precision or in regimes (energy scales, composite systems) where any small deviation could become apparent. Until then, the hypothesis of discrete quantum amplitudes serves as a fertile ground for theoretical exploration, pushing us to clarify why quantum theory has the form it does. In doing so, it enriches our understanding of quantum mechanics, whether or not nature ultimately deems the continuum to be an illusion carved out by our approximations.
References:
Hanson, A. J., et al. (2013). Geometry of Discrete Quantum Computing, J. Phys. A 46: 185301. [Replaces complex amplitudes with a finite Galois field, developing discrete analogues of quantum state space geometry] (Geometry of discrete quantum computing) (Geometry of discrete quantum computing).
Chang, L. N., Minic, D., Lewis, Z., & Takeuchi, T. (2013). Galois Field Quantum Mechanics, Mod. Phys. Lett. B 27: 1350064. [Finite-field Hilbert space formalism; notes no CHSH violation despite non-classical predictions] ([1208.5544] Some Mutant Forms of Quantum Mechanics).
Schumacher, B., & Westmoreland, M. D. (2012). Modal Quantum Theory. [Toy model using $GF(2)$; highlights the role of interference logic in quantum behavior] (Geometry of discrete quantum computing).
Vladimirov, V. S., Volovich, I. V., & Zelenov, E. I. (1994). p-Adic Analysis and Mathematical Physics. [Monograph on p-adic numbers in physics, including quantum mechanics and field theory] (p-adic quantum mechanics - Wikipedia).
Dragovich, B. (2006). Adelic Models of Quantum Mechanics, Proc. Steklov Inst. Math. 252: 64–73. [Review of adelic QM, combining real and p-adic descriptions; discusses path integrals invariant under number field interchange].
Meyer, D. A. (1999). Finite Precision Measurement Nullifies the Kochen-Specker Theorem, Phys. Rev. Lett. 83: 3751 (Finite Precision Measurement Nullifies the Kochen-Specker Theorem | Phys. Rev. Lett.). [Shows a dense discrete subset of directions permits a hidden-variable assignment, challenging the necessity of continuum assumptions for contextuality].
Sinha, U., et al. (2010). Ruling Out Multi-Order Interference in Quantum Mechanics, Science 329: 418 ([1007.4193] Ruling Out Multi-Order Interference in Quantum Mechanics). [Triple-slit experiment setting bounds on any extra interference terms; supports the exactness of Born’s rule and by extension the linear superposition with continuous amplitudes].
Gisin, N. (2019). Real Numbers are the Hidden Variables of Classical Mechanics, Quantum Stud.: Math. Found. 7: 197 (Items where Author is "Gisin, Nicolas" - PhilSci-Archive). [Philosophical and conceptual argument that continuum real numbers are unphysical idealizations; suggests reformulating physics without actual infinities] (Indeterminist physics for an open world | ScienceDaily).
Takeuchi, T. (2012). Some Mutant Forms of Quantum Mechanics, AIP Conf. Proc. 1508: 502 ([1208.5544] Some Mutant Forms of Quantum Mechanics). [Conference talk paper describing quantum mechanics over $GF(q)$ and an alternate “mutation”; provides an overview of finite-field QM results].
Zúñiga-Galindo, W. A. (2024). The p-Adic Schrödinger Equation and the Two-slit Experiment, arXiv:2308.01283 ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics) ([2308.01283] The p-Adic Schrödinger Equation and the Two-slit Experiment in Quantum Mechanics). [Develops a Schrödinger equation on p-adic space; finds that interference pattern arises even though particles don’t follow superposed paths in the usual sense].