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<\/p>\nWritten by: Atharv Kanchi<\/p>\n
The concept of duality can be defined in two different ways. Formally, duality is defined as relating mathematical concepts in a one-to-one relationship through something called involution<\/i>: if A is B, then B is A. In standard terms, duality is when two ideas are deeply related to each other, having concepts that are interconnected even across subjects.<\/p>\n
What is truly fascinating about duality, though, is how it so often reveals itself in nature, connecting two properties that may seem completely unrelated. In fact, most major fields of mathematics contain several examples of intertwined duality between disciplines, with one concept affecting another, creating a giant mathematical web. But where do these occur, why do they happen, and why does it matter?<\/p>\n
Dot Product Duality in Linear Algebra<\/b><\/p>\n
The first example of duality is within linear algebra \u2013 the study of vectors, matrices, and space transformations. One specific transformation you can apply between two vectors<\/i> (think about a vector as an arrow with a direction and magnitude) is the dot product<\/i>. It takes in two vectors and returns a scalar<\/i> (normal number), representing how much the vectors \u201calign.\u201dSpecifically, when you draw a line from the tip from one vector ,u,<\/b> to be perpendicular to some other vector, v<\/b>, the distance from the origin to the point on v<\/b> is the dot product. To compute the dot product, say you have two vectors <a, b> and <c, d>. The dot product is a * d + b * c.<\/p>\n
This operation is one of the most fundamental types of combining vectors in all of mathematics \u2014 it provides information about the angles and lengths of vectors, instructions on multiplying matrices, and even applications in machine learning to create algorithms that are used every day. Verbally, it is a bit hard to visualize, though, and here\u2019s where duality comes into play!<\/p>\n
Figure 1<\/p>\n
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Geometric interpretation of the dot product and its projection onto other vectors.<\/p>\n
Source: Dot products and duality. (2016). 3blue1brown.com. https:\/\/www.3blue1brown.com\/lessons\/dot-products<\/i><\/p>\n
\u00a0\u00a0\u00a0 For a second, imagine compressing the x and y axes onto a single horizontal number line, creating a 1D space where each vector corresponds to a single number. You just created the number line, the one you used to learn how to count in second grade! What\u2019s more important, however, is that this transformation can be represented by a 1×2 matrix <\/i>that scales the x and y components. Each column in the matrix roughly represents where the two axes ended up on the number line. For example, if your matrix was [-1 2], your vector for the x-axis ended up at -1, while your vector for the y-axis ended up at 2. But why does this matter? Well, it\u2019s where the duality pops up.<\/p>\n
The duality here comes from one key observation: if you take a vector on the normal 2D space and transform it using this matrix onto the 1D number line, you would multiply the two components of the vector with the two components of the matrix. In specific, if you had a vector <a, b> on the 2D space and multiplied it with the components of the matrix <c, d>, you would end up with some number that represents where the vector lies on this number line after the transformation.<\/p>\n
Doesn\u2019t that computation seem a bit familiar?<\/p>\n
It should be because that is exactly the computation for the dot product! These are two completely different ideas \u2014 transforming all of the 2D space onto a 1D number line and measuring how much two 2D vectors align. This highlights a fundamental form of duality in linear algebra: vectors can be both geometric objects and linear functions acting on other vectors.<\/p>\n
Figure 2<\/p>\n
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Basis vectors of a 2D space being transformed into a 1D number line<\/p>\n
Source: Dot products and duality. (2016). 3blue1brown.com. https:\/\/www.3blue1brown.com\/lessons\/dot-products<\/i><\/p>\n
\u00a0\u00a0 On a more advanced level, this phenomenon is explained by a duality between the V and V* spaces, where the vector space V <\/i>consists of vectors, while V* consists of linear functions that map vectors to scalars (like the dot product!). In this case, vectors in the V space are paired through a function in the V* space to return a scalar, which is essential to understanding vector space transformations in more advanced linear algebra, but that\u2019s for another time.<\/p>\n
Quantum Wave Particle Duality<\/b><\/p>\n
The next example of duality is that of quantum waves and particles, famously detailed in the double-slit experiment<\/i>. Imagine firing tiny particles at a barrier with two slits. Behind the slits is a screen that records where the particles land. When only one slit is open, the particles behave normally; they cluster together on the screen. But when both slits are open, something changes. Instead of forming two distinct clusters, the particles create an interference pattern of alternating light and dark bands \u2014 something that usually only happens with waves! It\u2019s almost like each particle travels through both slits at the same time and interferes with itself.<\/p>\n
However, if you place a detector to observe which slit each particle goes through, something changes again, and the interference pattern disappears. The particles act normally again, forming two distinct clusters behind the slits. This reveals a fundamental property of quantum mechanics, which is how \u201cmeasuring forces\u201d can impact particles to behave like, well, particles rather than waves.<\/p>\n
On a deeper level, the duality of particles and waves can be understood through two frameworks: the standard basis <\/i>and the Fourier basis<\/i>. When describing a particle in the standard, positional basis, it is represented by a wavefunction \u03c8(x), which describes the probability amplitude of finding a particle at a given position. This aligns with our intuition of a particle having a precise location at an instant in time.<\/p>\n
On the other hand, in the Fourier basis, we still describe the particle in terms of a wavefunction, however it uses the wavefunction \u03d5(p), which is the result of a fourier transform on \u03c8(x), that is expressed in terms of momentum. This results in a superposition<\/i> of many possible positions that the particle could be in, leading to interferences<\/i>. The wave function can interfere with itself and is spread out, which could create patterns and disturbances, leading to the bands created in the double-slit experiment.<\/p>\n
The duality is shown by keeping into account the observer effect<\/i>, where particles show interference patterns when not observed but behave like particles when they are observed. This duality is at the heart of quantum mechanics. It changes our perspective on matter, suggesting that, fundamentally, quantum objects don\u2019t have specific classifications. You can\u2019t have a quantum particle or a quantum wave, but you could have both depending on how you observe and measure it.<\/p>\n
Figure 3<\/p>\n
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Double slit experiment visualized, with the pattern representing the relative abundance of electrons in a certain location<\/p>\n
Source: Black, P. E., Kuhn, D. R., & Williams, C. J. (2002). Quantum Computing and Communication. Advances in Computers, 189\u2013244. https:\/\/doi.org\/10.1016\/s0065-2458(02)80007-9<\/i><\/p>\n
Why is Duality important?<\/b><\/p>\n
To top it all off, why is duality important? Well, in short, it reveals deep connections between topics that are seemingly unrelated. It provides deeper insight into the workings behind them and simplifies complicated problems. This principle underlies many key areas in math, from linear algebra to geometry and optimization. Understanding duality as a concept not only helps your problem-solving, but also helps you understand the structures within mathematics better and realize how everything is connected to each other through the universality of numbers.<\/p>\n
Sources<\/b><\/p>\n
1.13: Quantum Mechanics and the Fourier Transform<\/i>. (2019, March 21). Chemistry LibreTexts.<\/p>\n
https:\/\/chem.libretexts.org\/Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Quantum_Tutorials_%28Rioux%29\/01%3A_Quantum_Fundamentals\/1.13%3A_Quantum_Mechanics_and_the_Fourier_Transform<\/p>\n