Finance as a Field Concept

I love math because it's so weird. Math at its highest levels becomes a curious amalgamation of logic, philosophy, perspective, imagination, and the practice of unbending your mind to peer at the bizarre underbelly of the universe. Einstein put it best, "Pure Mathematics is, in its way, the poetry of logical ideas." I can appreciate that math itself, as a meta concept, is a dynamical system, which I suspect has chaotic attractors. Mathematics, even in its most abstruse fields, is grounded in simple ideas. Yet, these simple ideas, these bare building blocks, take on lives of their own. ones and zeros get compiled from machine code into a vastly powerful emergent form, far different from their base behavior (Boeing 2016). There is a sense that even the most unapproachable subjects of math have understandable building blocks, waiting to be decomposed and understood. Today, let's explore a few more of the building blocks of our quantum financial future using simple intuition generated by path integrals, the structural ingenuity of tensors, and the power of memory uncovered by Hurst Exponents.  Hopefully we can eventually arrive at a more elegant understanding of our material world.

The Role of Tensors in Modern Computing

Tensors are at the heart of modern computing. Tensors provide maps between concepts; in essence, mathematical dynamic lookups over several multidimensional sheets. One way to understand tensors intuitively is to think about them as higher-order representations of relationships in vector spaces  (Vasilescu, 2009). Formally, a simple definition of a tensor was beautifully described on stackoverflow: A 𝑘-tensor is a multilinear function from 𝑉×𝑉×⋯×𝑉 to the reals, where 𝑉 is a vector space and 𝑘 is the number of 𝑉's in the above Cartesian product. (Spivak, 1965). Applying this definition to the real world, you could think of scalars, vectors, and matrices as varying ranks of a tensor. A rank 3 tensor could be thought of as a three-dimensional array; however, it's limiting to think of these in terms of structures. It is far more useful to understand how to transform vectors across vector spaces. In truth, the rank of a tensor cannot be described with just one input. For example, vectors are rank (0,1) tensors, and a "3D tensor" is written as follows, where m is a rank 3 tensor: m=(((1,2),(2,4)),((2,3),(5,6))), essentially a nested list of matrices.

For computer science, the structure of tensors allows GPUs and the processing of large models to work more efficiently at scale. As a data structure, tensors are the innovation that has enabled computer science to take deep learning to new heights. The tensor cores created by Nvidia reportedly lean on these same principles to ride the AI boom to new heights. Beyond Nvidia, you must have heard of TensorFlow, which Google also leans on the ingenuity of tensors to power complex learning models. Tensors will be the cement, the reference point, the map, and the guide between our other base building blocks.

Tensors in Finance

Now that we've given some context to what a tensor is generally, let's narrow our focus to the financial world. Tensors offer elegant solutions to the disaster of matrix dimensionality. Without getting into a linear algebra lecture, a lot of our matrix math relies on having square matrices. For example, portfolio optimization uses matrices to represent covariance; covariance matrices are always symmetric and square(matricies with equal number of columns and rows)--but data in real life doesn't fit neatly into those patterns. In order to fit standard axioms, m by n matrices are sheared into n by n matrices, and square matrices are forced into semi-positive definite shapes in order to be invertible. The beauty of tensors is that they are multilinear maps that can be defined beyond rank 2; we could potentially give 2D representations to higher-order concepts (Quadratic Optimization Problems, UPENN).

Take, for example, the allure of not having to engage in complicated smoothing to fix the volatility smile in options pricing; tensors could offer a novel solution to expressing the nonlinearity driving the extrinsic value of options prices without having to approximate quite so much (Zeron & Ruiz, 2022). Unlike a set of matrices that are cobbled together or a stochastic process, a tensor itself does not change but still gives us the right directions and magnitude to understand complex relationships. We could create a multilinear map of all the relevant first- and second-order Greeks with as much precision as we can afford.

Challenges and Speculations

Side note: I haven't done this sort of calculation, wouldn't know where to start, nor would I imagine that it would make much sense to think about tensors when decomposition algorithms can probably avoid much of this messy math, or if it makes sense to care about refining covariance optimization math when values seem to converge to the same central tendencies at sufficiently large n; but, like a lot of my theories, I am being forward-looking (Central Limit Theorem, Boston University). As Math and programming grow asymptotically more advanced we will eventually reach a time when tensors are a realistic way to model complex economic interlinkages.

Path Integrals: Connecting Building Blocks

If tensors were the directions between our building blocks, path integrals would serve as the roads. The path integral is not new. Dirac started work on precursors to the path integral formulation as early as 1928, but in his work on the Lagrangian of Quantum Mechanics in 1932, he essentially showed that quantum mechanics or quantum mechanical properties could be expressed with classical calculus (Dirac, 1932). Richard Feynman crystallized the initial path integral formulation laid out by Dirac; it can most simply be explained as: "...very much like the partition function of a 1-d classical statistical system except for one crucial difference; the integrand of the path-integrals is a complex phase..."(Dass,2020). In more simplistic terms, a path integral can loosely be defined as a functional integral that finds the sum over an infinity of quantum mechanical paths (Hall 2013). Imagine a Riemann Sum from Calculus 1 but for the quantum world, taking the sum of infinitely many probabilities over a partitioned area; being able to describe the partition function of our economy could elegantly represent complex financial relationships(Hall 2013).

Implications for Financial Systems

I think this approach could be revolutionary for financial assets because globalization has made all financial systems have a great degree of topological mixing. Ripple effects flow, and small perturbations have disproportionate global effects (Jha, 2023). Take, for example, how indeterminate pricing for something on the internet has become. While we don't have perfect knowledge, an online consumer could be changing my quoted price for a good or service every few minutes based on a probability density function grounded in petabytes of data about my consumer preferences. As Dirac would probably put it, I think the economic world around us has a strong quantum analogue. It might not be proper to consider financial economics to truly exist in physical terms. It would be asinine to try and model entanglement with quarks, spins, and other nonsense like the physical world; rather, I think it's a robust framework for us economists to dissect and apply.

Potential Future Applications

I can foresee a world where we could use tensors as maps to project forward rates onto manifolds, in which path integrals would be extremely useful as a probabilistic measure of yield; imagine knowing the weighted sum of the infinitely different possibilities of a forward curve instead of just one prediction or even a few million predictions if you wanted to use a Monte Carlo simulation (Bayraktar et al., 2005). I recognize that my claims might seem a bit grandiose, especially because while I am talking about novel research, I don't seem to be doing much of my own calculations; however, I would rather speak confidently about what I know with certainty than speak beyond my scope of knowledge. I will rely on folks who have spent considerably more time researching these subjects to give you some more substantive proof. Researchers around the world have already started doing groundbreaking work in these areas.

Examples from Academia and Industry

Take, for example, researchers at Princeton who used tensor networks as a structure to implement path integrals with great success, noting that the flexibility of their framework shows great promise to model non-equilibrium quantum dynamics (Bose and Walters, 2021). Or take the example of an institute in Canada that successfully developed a quantum-based portfolio optimizer using novel hardware that simulates quantum gates and tensor networks (Mugel et al., 2020) . Finally, look at these people in industry who have developed a pragmatic framework to apply path integrals to great effect on short-rate models and bond pricing (Kakushadze, 2015). Who knows when or if these insights will make it to mainstream thinking--they are definitely brewing under our academic surface; with the development of quantum computing impending, I can only think these ideas will become increasingly important.

Connecting Philosophy and Quantum Mechanics

Casting aside pure math and computing for a second, I want to lean on philosophy and the idea of a quantum consciousness to connect the importance of the fractal dimension and the Hurst exponent to my quantum world. I will not get too far into the weeds with fractals in this already bloated article, but think of the Hurst Exponent as a measure of the memory of a time series, defined between 0 and 1. A time series that has a Hurst exponent value between 0 and 0.5 could be defined as indicative of antipersistent behavior, and 0.5 to 1 could be defined as having persistent behavior. I was gobsmacked when I heard about such a concept because to me it would vastly clarify a lot of market strategies in our material world; we could potentially put a magnitude on how likely something is to remember its true value or not.

Quantum Consciousness and the Hurst Exponent

As I reflected further, I realized that the deeper implications could be even bigger. I think the idea of autocorrelation/memory of a time series is a strong qualitative analogy to quantum consciousness (Qian and Rasheed). Not a direct analogue, as we aren't seeing where our wave function would collapse upon observation, but rather we could use a similar concept to express our material understanding of interference and bias. The memory of our data, its tendency to revert back to a base state, really how it scales differently due to the fractal dimension, is a strong analogue to a chaotic attractor, something that puts a nice bow on our complex quantum approximation because it introduces the idea of bias, interference, and can account for how feedback loops affect complex relationships (Ferrara, 2024).

The Future of Quantum Building Blocks

My quantum building blocks are still only for the future--far in the future. There are very good reasons why people have let Dirac's ideas stew for almost 100 years: our conventional wisdom is good enough, and quantum mechanics, no matter how robust, is a theory, not a fact of law. Furthermore, even if we could, like I have mentioned countless times, we wouldn't bother as of right now. Path integrals are hard to solve and definitely not worth the effort, money, and time with our current toolset. Researchers, economists, and forecasters are not looking for an all-encompassing forecasting tool; math doesn't neatly fit into the theory-of-everything paradigm. Rather, they need the tool that is most efficient at explaining their problem simply. Something to keep in mind, however, is that our progress will not be static as we continue to grow; there will be a time when the conventional building blocks of our material world become lackluster to represent our reality--that is when we can look to my ideas to fuel a quantum world.