# Could wavefunction values be quantized?

According to standard quantum mechanics, Hilbert space is defined over the complex numbers, and amplitudes in a superposition can take on values with arbitrarily small magnitude. This probably does not trouble non-realist interpretations of the wavefunction, or explicit wave function collapse interpretations, but does come into play in serious considerations of realist interjections that reject explicit collapse (e.g. many worlds, and the quantum suicide paradox).

Are there, or could there be, models of "quantum mechanics" on Hilbert-like spaces where amplitudes belong to a discretized subset of the complex numbers like the Gaussian integers -- in effect, a module generalization of a Hilbert space -- and where quantum mechanics emerges as a continuum limit? (Clearly ray equivalence would still be an important consideration for this space, since almost all states with Gaussian-integer valued amplitudes will have $$\left< \psi | \psi \right> > 1$$.

Granted, a lot of machinery would need to change, with the normal formalism emerging as the continuous limit. An immediate project with discretized amplitudes could be the preferred basis problem, as long as there are allowed bases that themselves differ by arbitrarily small rotations.

Note: the original wording the question was a bit misleading and a few responders thought I was requiring all amplitudes to have a magnitude strictly greater than zero; I've reworded much of the question.

• Tip: Avoid writing 'edit' or 'edited' in the post. This is already mentioned in the edit history. Commented Aug 15, 2021 at 14:36
• Do you want the discretize-complex-numbers version of the theory to be indistiguishable from the original, as far as realistic experiments can tell? Commented Aug 15, 2021 at 17:26
• are you asking about "third quantization"? Commented Aug 16, 2021 at 14:51

## 2 Answers

One problem with this idea is normalization:

$$\int_{\mathbb R} \psi^* (x) \psi(x)~ dx = 1$$

You are integrating over infinite space. If $$\psi$$ has a minimum non-zero value, $$\psi$$ must be $$0$$ everywhere except a finite volume.

Now switch to the momentum basis. Because $$\psi$$ has bounded support, the Fourier Transform of it cannot have. To be normalizable, the tails would have to have infinitesimal values. So you cannot have discrete values in momentum space. Does this fit your theory?

Another problem is that wave functions are continuous. If there are only a discrete set of values, you would have discontinuous functions.

Unless you are talking about a space with holes in it? Constant values in distinct regions?

Given

$$\hat p \psi(x) = -i\hbar \frac{\partial\psi}{\partial x}$$

a $$\psi$$ that was constant, except where interrupted by discontinuities would correspond to $$\hat p = 0$$ except where it is undefined or perhaps has infinite spikes.

Likewise

$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} = E \psi$$

would lead to $$E = 0$$ except perhaps at the discontinuities.



• Oh, I realize now my wording was maybe a bit ambiguous. By "minimum non-zero magnitude for any non-zero amplitude" I mean a model where amplitudes are either zero or have a minimum magnitude. Maybe the first "non-zero" in that sentence was redundant and misleading. What I had in mind were amplitudes that were proportional to the Gaussian integers. Commented Jul 30, 2021 at 1:48
• I think the fact that wave functions are continuous in the position basis may, mathematically, be more a consequence of initial conditions that a necessary property, since there are certainly other bases where they are not continuous (and the position basis shouldn't be preferred). In any case, discontinuity in any basis is an intentional (if somehow fatal) property of this hypothetical. Commented Jul 30, 2021 at 1:52
• It took me a couple reads but I belatedly see your point. I see it as a more dramatic example of the preferred basis problem I alluded to in the question -- clearly any such model would have to deem some bases (e.g. some position bas(ie)s) as acceptables, and others (plane wave momentum states) as forbidden, or vice versa. Commented Aug 16, 2021 at 15:48
• When you write "$\psi$ is bounded", do you mean "$\psi$ has bounded support"? Commented Aug 16, 2021 at 18:24
• @Noiralef - Thank you. I added a correction. Commented Aug 16, 2021 at 22:47

Couple of papers found through google search:

Abstract

We present a discrete model theory similar in structure to ordinary quantum mechanics, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the “modal” concepts of possibility and necessity rather than quantitative probability measures. Despite its simplicity, our model theory includes entangled states and has versions of both Bell’s theorem and the no cloning theorem.

Further details on MQT could be found in arXiv:1204.0701.

Another paper:

Abstract

We explore finite-field frameworks for quantum theory and quantum computation. The simplest theory, defined over unrestricted finite fields, is unnaturally strong. A second framework employs only finite fields with no solution to $$x^2 + 1 = 0$$, and thus permits an elegant complex representation of the extended field by adjoining $${\rm i}=\sqrt{-1}$$. Quantum theories over these fields recover much of the structure of conventional quantum theory except for the condition that vanishing inner products arise only from null states; unnaturally strong computational power may still occur. Finally, we are led to consider one more framework, with further restrictions on the finite fields, that recovers a local transitive order and a locally-consistent notion of inner product with a new notion of cardinal probability. In this framework, conventional quantum mechanics and quantum computation emerge locally (though not globally) as the size of the underlying field increases. Interestingly, the framework allows one to choose separate finite fields for system description and for measurement: the size of the first field quantifies the resources needed to describe the system and the size of the second quantifies the resources used by the observer. This resource-based perspective potentially provides insights into quantitative measures for actual computational power, the complexity of quantum system definition and evolution, and the independent question of the cost of the measurement process.

As far as I can tell those papers do not particularly care about “continuous limits”.

Also, there are some interesting links on MathOverflow question “Toy Models of Quantum Mechanics”.

• In case anybody's not familiar with finite fields: this description is relatively concise and clear. Note that a finite field is not a subset of the field of complex numbers: the rules of arithmetic are different (still commutative and associative, etc, but with different addition/multiplication tables). Commented Aug 17, 2021 at 0:37