Mathematical physics without partial derivatives

Question

Remark: All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.

Motivation:

I recently had an interesting exchange with several computational neuroscientists on whether organisms with spatiotemporal sensory input can simulate physics without computing partial derivatives. As far as I know, partial derivatives offer the most quantitatively precise description of spatiotemporal variations. Regarding feasibility, it is worth noting that a number of computational neuroscientists are seriously considering the question that human brains might do reverse-mode automatic differentiation, or what some call backpropagation [7].

Having said this, a large number of computational neuroscientists (even those that have math PhDs) believe that complex systems such as brains may simulate classical mechanical phenomena without computing approximations to partial derivatives. Hence my decision to share this question.

Problem definition:

Might there be an alternative formulation for mathematical physics which doesn't employ the use of partial derivatives? I think that this may be a problem in reverse mathematics [6]. But, in order to define equivalence a couple definitions are required:

Partial Derivative as a linear map:

If the derivative of a differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ at $x_o \in \mathbb{R}^n$ is given by the Jacobian $\frac{\partial f}{\partial x} \Bigr\rvert_{x=x_o} \in \mathbb{R}^{m \times n}$, the partial derivative with respect to $i \in [n]$ is the $i$th column of $\frac{\partial f}{\partial x} \Bigr\rvert_{x=x_o}$ and may be computed using the $i$th standard basis vector $e_i$:

\begin{equation} \frac{\partial{f}}{\partial{x_i}} \Bigr\rvert_{x=x_o} = \lim_{n \to \infty} n \cdot \big(f(x+\frac{1}{n}\cdot e_i)-f(x)\big) \Bigr\rvert_{x=x_o}. \tag{1} \end{equation}

This is the general setting of numerical differentiation [3].

Partial Derivative as an operator:

Within the setting of automatic differentiation [4], computer scientists construct algorithms $\nabla$ for computing the dual program $\nabla f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ which corresponds to an operator definition for the partial derivative with respect to the $i$th coordinate:

\begin{equation} \nabla_i = e_i \frac{\partial}{\partial x_i} \tag{2} \end{equation}

\begin{equation} \nabla = \sum_{i=1}^n \nabla_i = \sum_{i=1}^n e_i \frac{\partial}{\partial x_i}. \tag{3} \end{equation}

Given these definitions, a constructive test would involve creating an open-source library for simulating classical and quantum systems that doesn’t contain a method for numerical or automatic differentiation.

The special case of classical mechanics:

For concreteness, we may consider classical mechanics as this is the general setting of animal locomotion, and the vector, Hamiltonian, and Lagrangian formulations of classical mechanics have concise descriptions. In all of these formulations the partial derivative plays a central role. But, at the present moment I don't have a proof that rules out alternative formulations. Has this particular question already been addressed by a mathematical physicist?

Perhaps a reasonable option might be to use a probabilistic framework such as Gaussian Processes that are provably universal function approximators [5]?

Koopman Von Neumann Classical Mechanics as a candidate solution:

After reflecting upon the answers of Ben Crowell and gmvh, it appears that we require a formulation of classical mechanics where:

1. Everything is formulated in terms of linear operators.
2. All problems can then be recast in an algebraic language.

After doing a literature search it appears that Koopman Von Neumann Classical Mechanics might be a suitable candidate as we have an operator theory in Hilbert space similar to Quantum Mechanics [8,9,10]. That said, I just recently came across this formulation so there may be important subtleties I ignore.

Related problems:

Furthermore, I think it may be worth considering the following related questions:

1. What would be left of mathematical physics if we could not compute partial derivatives?
2. Is it possible to accurately simulate any non-trivial physics without computing partial derivatives?
3. Are the operations of multivariable calculus necessary and sufficient for modelling classical mechanical phenomena?

A historical note:

It is worth noting that more than 1000 years ago as a result of his profound studies on optics the mathematician and physicist Ibn al-Haytham(aka Alhazen) reached the following insight:

Nothing of what is visible, apart from light and color, can be perceived by pure sensation, but only by discernment, inference, and recognition, in addition to sensation.-Alhazen

Today it is known that even color is a construction of the mind as photons are the only physical objects that reach the retina. However, broadly speaking neuroscience is just beginning to catch up with Alhazen’s understanding that the physics of everyday experience is simulated by our minds. In particular, most motor-control scientists agree that to a first-order approximation the key purpose of animal brains is to generate movements and consider their implications. This implicitly specifies a large class of continuous control problems which includes animal locomotion.

Evidence accumulated from several decades of neuroimaging studies implicates the role of the cerebellum in such internal modelling. This isolates a rather uniform brain region whose processes at the circuit-level may be identified with efficient and reliable methods for simulating classical mechanical phenomena [11, 12].

As for the question of whether the mind/brain may actually be modelled by Turing machines, I believe this was precisely Alan Turing’s motivation in conceiving the Turing machine [13]. For a concrete example of neural computation, it may be worth looking at recent research that a single dendritic compartment may compute the xor function: [14], Reddit discussion.

References:

1. William W. Symes. Partial Differential Equations of Mathematical Physics. 2012.
2. L.D. Landau & E.M. Lifshitz. Mechanics (Volume 1 of A Course of Theoretical Physics). Pergamon Press 1969.
3. Lyness, J. N.; Moler, C. B. (1967). "Numerical differentiation of analytic functions". SIAM J. Numer. Anal. 4: 202–210. doi:10.1137/0704019.
4. Naumann, Uwe (2012). The Art of Differentiating Computer Programs. Software-Environments-tools. SIAM. ISBN 978-1-611972-06-1.
5. Michael Osborne. Gaussian Processes for Prediction. Robotics Research Group Department of Engineering Science University of Oxford. 2007.
6. Connie Fan. REVERSE MATHEMATICS. University of Chicago. 2010.
7. Richards, B.A., Lillicrap, T.P., Beaudoin, P. et al. A deep learning framework for neuroscience. Nat Neurosci 22, 1761–1770 (2019). doi:10.1038/s41593-019-0520-2.
8. Wikipedia contributors. "Koopman–von Neumann classical mechanics." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 19 Feb. 2020. Web. 7 Mar. 2020.
9. Koopman, B. O. (1931). "Hamiltonian Systems and Transformations in Hilbert Space". Proceedings of the National Academy of Sciences. 17 (5): 315–318. Bibcode:1931PNAS...17..315K. doi:10.1073/pnas.17.5.315. PMC 1076052. PMID 16577368.
10. Frank Wilczek. Notes on Koopman von Neumann Mechanics, and a Step Beyond. 2015.
11. Daniel McNamee and Daniel M. Wolpert. Internal Models in Biological Control. Annual Review of Control, Robotics, and Autonomous Systems. 2019.
12. Jörn Diedrichsen, Maedbh King, Carlos Hernandez-Castillo, Marty Sereno, and Richard B. Ivry. Universal Transform or Multiple Functionality? Understanding the Contribution of the Human Cerebellum across Task Domains. Neuron review. 2019.
13. Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. 2 (published 1937). 42: 230–265. doi:10.1112/plms/s2-42.1.230. (and Turing, A.M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction". Proceedings of the London Mathematical Society.
14. Albert Gidon, Timothy Adam Zolnik, Pawel Fidzinski, Felix Bolduan, Athanasia Papoutsi, Panayiota Poirazi, Martin Holtkamp, Imre Vida, Matthew Evan Larkum. Dendritic action potentials and computation in human layer 2/3 cortical neurons. Science. 2020.

Answer 1

As to question 2, there are certainly plenty of non-trivial discrete models in statistical physics, such as the Ising or Potts models, or lattice gauge theories with discrete gauge groups, that require no partial derivatives (or indeed any operations of differential calculus) at all to formulate and simulate.

Similarly, quantum mechanics can be formulated entirely in the operator formalism, and an entity incapable of considering derivatives could still contemplate the time-independent Schrödinger equation and solve it algebraically for the harmonic oscillator (using the number operator) or the hydrogen atom (using the Laplace-Runge-Lentz-Pauli vector operator).

So an answer to question 1 might be "at least anything that can be written as a discrete-time Markov chain with a discrete state space, as well as anything that can be recast as an eigenvalue problem", and other problems that can be recast in purely probabilistic or algebraic language should also be safe (although it might be hard to come up with their formulations without using derivatives at some intermediate step).

As to question 3, I personally don't believe that an approach to classical mechanics or field theory can be correct if it isn't equivalent (at least at a sufficiently high level of abstraction) to formulating and solving differential equations. But the level of abstraction could conceivably be quite high -- for an attempt to formulate classical mechanics without explicitly referring to numbers (!) cf. Hartry Field's philosophical treatise "Science without Numbers".

Answer 2

Is it possible to accurately simulate any non-trivial physics without computing partial derivatives?

Yes. An example is the nuclear shell model as formulated by Maria Goeppert Mayer in the 1950's. (The same would also apply to, for example, the interacting boson model.) The way this type of shell model works is that you take a nucleus that is close to a closed shell in both neutrons and protons, and you treat it as an inert core with some number of particles and holes, e.g., $^{41}\text{K}$ (potassium-41) would be treated as one proton hole coupled to two neutrons. There is some vector space of possible states for these three particles, and there is a Hamiltonian that has to be diagonalized. When you diagonalize the Hamiltonian, you have a prediction of the energy levels of the nucleus.

You do have to determine the matrix elements of the Hamiltonian in whatever basis you've chosen. There are various methods for estimating these. (They cannot be determined purely from the theory of quarks and gluons, at least not with the present state of the art.) In many cases, I think these estimates are actually done by some combination of theoretical estimation and empirical fitting of parameters to observed data. If you look at how practitioners have actually estimated them, I'm sure their notebooks do contain lots of calculus, including partial derivatives, or else they are recycling other people's results that were certainly not done in a world where nobody knew about partial derivatives. But that doesn't mean that they really require partial derivatives in order to find them.

As an example, people often use a basis consisting of solutions to the position-space Schrodinger equation for the harmonic oscillator. This is a partial differential equation because it contains the kinetic energy operator, which is basically the Laplacian. But the reality is that the matrix elements of this operator can probably be found without ever explicitly writing down a wavefunction in the position basis and calculating a Laplacian. E.g., there are algebraic methods. And in any case many of the matrix elements in such models are simply fitted to the data.

The interacting boson model (IBM) is probably an even purer example of this, although I know less about it. It's a purely algebraic model. Although its advocates claim that it is in some sense derivable as an approximation to a more fundamental model, I don't think anyone ever actually has succeeded in determining the IBM's parameters for a specific nucleus from first principles. The parameters are simply fitted to the data.

Looking at this from a broader perspective, here is what I think is going on. If you ask a physicist how the laws of physics work, they will probably say that the laws of physics are all wave equations. Wave equations are partial differential equations. However, all of our physical theories except for general relativity fall under the umbrella of quantum mechanics, and quantum mechanics is perfectly linear. There is a no-go theorem by Gisin that says you basically can't get a sensible theory by adding a nonlinearity to quantum mechanics. Because of the perfect linearity, our physical theories can also just be described as exercises in linear algebra, and we can forget about a specific basis, such as the basis consisting of Dirac delta functions in position space.

In terms of linear algebra, there is the problem of determining what is the Hamiltonian. If we don't have any systematic way of determining what is an appropriate Hamiltonian, then we get a theory that lacks predictive power. Even for a finite-dimensional space (such as the shell model), an $n$-dimensional space has $O(n^2)$ unknown matrix elements in its Hamiltonian. Determining these purely by fitting to experimental data would be a vacuous exercise, since typically the number of observations we have available is $O(n)$. One way to determine all these matrix elements is to require that the theory consist of solutions to some differential equation. But there is no edict from God that says this is the only way to do so. There are other methods, such as algebraic methods that exploit symmetries. This is the kind of thing that the models described above do, either partially or exclusively.

References

Gisin, "Weinberg's non-linear quantum mechanics and supraluminal communications," http://dx.doi.org/10.1016/0375-9601(90)90786-N , Physics Letters A 143(1-2):1-2

Answer 3

Well if you take out partial derivatives, at least quantum field theory and in particular conformal field theory will survive the massacre. The reason is explained in my MO answer: $p$-adic numbers in physics

One can use random/quantum fields $\phi:\mathbb{Q}_{p}^{d}\rightarrow \mathbb{R}$ as toy models of fields $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$. In this $p$-adic or hierarchical setting, Laplacians and all that are nonlocal and not given by partial derivatives.

Most equations in physics are local and therefore need partial derivatives in order to be formulated. What should remain, in the very hypothetical scenario proposed in the question, is everything pertaining to nonlocal phenomena.

Answer 4

I'd query the contention that organisms or even inorganic matter compute in the sense described.

For example, if I drop a stone on the surface of the earth, it falls in a straight line. To call this as 'computing' a straight line seems rather a stretch of the word computation; to my thinking, to compute, means that one ought to be conscious that one is carrying out a computation. That is the person who dropped it is computing the straight line - and not the stone itself. It merely moves in a straight line. We know it moves in a straight line, and hence by dropping it, are describing a straight line.

Answer 5

One way for reformulating all of (classical) mechanics is Peridynamics, which does away with derivatives. It is essentially a non-local reformulation.

Javili, Ali, et al. "Peridynamics review." Mathematics and Mechanics of Solids 24.11 (2019): 3714-3739.

Answer 6

For one example of non-trivial physics without partial derivatives, one can look into Volume 1 of the Feynman lectures. In Chapter 28, Feynman starts to develop electrodynamics without partial derivatives — they only appear in Volume 2.

Instead of the Maxwell equation, Feynman uses a somewhat complex formula for the field that is generated by a single moving charge. The formula only has ordinary derivatives — the first and second time derivative of the speed of the charged particle — but becomes a bit unusual in that uses retarded time: The field at a distant point is determined by the movement of the particle some time ago, to correct for the finite speed of light.