# What is Duality: Definition and 282 Discussions

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.

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1. ### Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1

This is problem 28 from chapter 3F "Duality" of Axler's Linear Algebra Done Right, third edition. I spent quite a long time on this problem, like a few hours. Since there is no available solution, I am wondering if my solution is correct. One assumption in this problem is that...
2. ### Prove that range ##T'## = ##(\text{null}\ T)^0##

My question is about item (b). (b) Here is what I drew up to try to visualize the result to be proved The general idea, I think, is that 1) ##(\text{null}\ T)^0## and ##\text{range}\ T'## are both subspaces of ##V'=L(V,\mathbb{F})##. 2) We can show that they have the same dimension. 3)We...
3. ### I Exploring Particle/Wave Duality in Photons

(Beginner) - W.o. going into particle/wave duality, we know the resultant image came ONLY through something going through this nano opening we left uncovered. We also see that the resultant image still neatly shows diff. colors. But ALL the objects reflecting are sending different colors...
4. ### B Quantum field theory and wave particle duality

I recently watched this lecture "Quantum Fields: The Real Building Blocks of the Universe" by David Tong where the professor provides a succinct explanation of QFT in about 6 minutes around the midway mark. The main point being that there are fields for particles and fields for forces and the...
5. ### B Wave/Particle Duality: Helical Paths & Planar Projections

With respect to wave/particle duality, is it correct to think of a particle traveling a wave path? And if so, given that no orientation with respect to an observer of that wave is generally considered, is the "wave" that is referred to then more accurately the planar projection of a helical path?
6. ### B Wave-particle duality of atoms and molecules

Hello ladies and gentlemen, On the website of Encyclopedia Brittanicca I read the article about wave - particle duality. The article says that the wave - particle duality is experimentally established for light, electrons and protons. However, i found other internet sources which say that the...
7. F

### Fourier transform: duality property and convolution

Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...
8. ### I Is the Continuum Limit of Spin Foam Dynamics Equivalent to Massless Gravitons?

The spin-foam approach to quantum gravity is part of the class of approaches, that also include loop quantum gravity and a variety of other methods, that sets out to quantize space-time rather than the gravitational force itself. But, according to a new paper, it turns out that "the continuum...
9. ### B Wave-Particle Duality -- When is it a wave and when is it a particle?

In order to trigger this "interaction at a point as a particle" does an entity need to meet a certain criteria? Why doesn't any other entity on its way force this transition? Can the properties of this wave be altered? Thank you.
10. ### What is the relationship between particles and waves in QED?

"Everything is a particle whose position is predicted by a mathematical wave. Light is not a wave but is packet of energy whose position is predicted by the wave.The same goes for an electron. Interference pattern is a probability distribution of where we are likely to find an electron. When...
11. ### I A quasi-analogy of wave-particle duality?

One Major question I have about wave-particle duality of say a photon... Could we describe it like a rock falling vertically into a still pond. Around this point of contact we establish a circular wall which detects the contact of the wave. Two things are evident here: the rock keeps on moving...
12. ### A Is wave-matter duality a proven theory?

The observed diffraction patterns in slit experiments are held up as proof of wave-particle duality. But wave theory diffraction (borrowed from optics - Kirchoff's Laws, Fresenel & Fraunhofer diffraction) don't quite fit the experimental results. There is always some tinkering to get theory to...

15. ### Showing two groups are equal/Pontryagin duality

I am confused because ##H## is a subgroup of ##G## and ##H^{\perp\perp}## is a set of homomorphisms. Are we trying to show ##f(H) = H^{\perp\perp}## where ##f## is the isomorphism defined in the definitions above? Proof: We want to show ##H = (H^\perp)^\perp##. ##(\subset):## Let ##h \in H##...
16. ### I Article on photon wave-particle duality

We all know that Feynman declared wave-particle duality as the central/only problem of quantum physics. Not sure how to evaluate a recent publication summary on this topic: https://sciencex.com/news/2020-11-wave-particle-duality-entanglement-customary-pitfalls.html Would like others take on...
17. ### B Mach-Zehnder Experiment: Wave-Particle Duality Demonstrated

Can the essence o the Mach-Zehnder experiment be resumed as shown, to emphasize the analogy with the double-slit experiment? When the beams are brought together on a screen they form an interference pattern (no which-path inormation; wave behaviour). If the screen is removed and replaced by two...
18. ### B Wave-Particle Duality Controlled by Single-Photon Self-Entanglement

I stumbled over this[1] paper recently, in which they report of experimental evidence for the claim made in earlier work[2] that the amount of "waveness" V and "particleness" P of a photon is related to the amount of self-entanglement C via the equation V^2 + P^2 + C^2 = 1. I found it...
19. ### B How does the observed version of the double slit get by without math?

Send a classical particle into a three dimensional potential well. 0 Probability of it tunneling. It's a classical trajectory with wobble from uncertainty. It's not a wave, but gets wobble from the quantum field influencing it. https://farside.ph.utexas.edu/teaching/315/Waveshtml/node95.html...
20. ### A Regarding Coleman's Sine-Gordon Duality

The link to the paper is here; http://www.sbfisica.org.br/~evjaspc/xviii/images/Nunez/Jan26/AdditionalMaterial/Coleman_paper/Coleman_paper.pdf Im confused over what exactly he is calculating on page 5 (2092) equation 4.11. Specifically what is \mathcal{T} \langle 0 | \prod_i A_+ (x_i)...
21. ### I Wave-particle duality revisited: Neither wave nor particle

Jan Sperling et al., Wave-particle duality revisited: Neither wave nor particle, arXiv:1907.09836 From the abstract:
22. ### B Bohr's duality paradox 100 years later?

Bohr declared it the duality paradox. It works. But a paradox in science is an unresolved problem. Are we any closer to resolving that problem 100 years after Bohr's declaration, or has the physics community just grown numb to it?
23. ### B So can I use "wave particle duality?"

Some sources say wave particle duality is very much a principal, others say it's oversimplified and not exactly true but still okay to use in order to explain things, and many say it's flat out wrong... so who's right? Can I say "wave particle duality" when talking quantum mechanics or no?
24. ### B Understanding Wave-Particle Duality with Water Analogy

I'm thinking about the wave–particle duality and I like to understand it through analogy. So I'm thinking about how it relates to for example water. Water can be seen as both having wave characteristics, but also as particles (water molecules). Do you think this is a good analogy to the...
25. ### A Does it make sense to break T duality at fundamental level?

https://arxiv.org/abs/hep-th/0605250 Here is an attempt to make bohmian mechanics compatible with string theory. It posits that T duality breaks at the fundamental level, and that the is no minimal length in the theory. Does this proposal make sense?
26. ### I Light : Wave - particle duality

Interference pattern made by light shows the wave nature of light and photoelectric effect shows particle nature of light. So, what is light? According to the photoelectric effect, light consists of photons with energy E and momentum ## \vec p##. According to the interference pattern, we...
27. ### B Is wave-particle duality a result of the relativity theory?

Have thought about this for sometime but couldn't get deeper. I have speculated that wave-particle duality is a direct result of the relativity theory. Especially it could arise from the factor of length contraction (dimensionality reduction). So far, the particle reality is based on various...
28. ### Wave / particle duality of light

Homework Statement I've just read that light can behave like a wave at times, and a particle at other times. How does the light from the Sun traveling towards Earth behave? A wave? A particle? Or both? And is it in any sense something that actually starts at the Sun, travels across space and...
29. ### I Understanding particle/wave duality and the Wave Function

Im just starting to try to break into and understand quantum physics and so this question may be a completely absurd but I am curious as to whether or not its been proven that a particle really does act like a wave until observed or if the "spin" of two entangled atoms actually changes opposite...
30. ### Hamilton-Jacobi equation and particle-wave motion

I've seen somwhere a claim that Hamilton-Jacobi euqation is the only formulation of classical mechanics which can treat motion of particle as wave motion. There was something about hamilton prinicpal function, hamilton characteristic function and one of these change in time like wavefront or...
31. ### A question about wave/particle duality

Subatomic particles can take the form of a wave or a particle. While in wave form, it is not like a physical wave, but rather a probability wave, (i.e. a wave of information about where the particle is probably located etc.) And while in particle form, a photon, for example, can knock electrons...
32. ### B Double slit and wave particle duality

Just for my knowledge, not to confuse the OP, why would you say, for example, the double slit experiment does not show both of these properties in one measurement method? [Mentors' more: this thread was forked off from another thread]
33. ### B What, in essence, distinguishes light from matter?

Before quantum mechanics, light was generally seen as a wave and matter as particles (biliards). From e.g. the discovery of the photoelectric effect, one saw that light can also be seen as a particle. From e.g. the double slit experiment, one makes the interpretation that matter can also be seen...
34. ### Wave Particle Duality For Electrons and Photons

Homework Statement Discuss the concept of the wave-particle duality for electrons and photons and include an equation which connects the wave like and particle like properties. Homework EquationsThe Attempt at a Solution So I am having trouble with how to word this question and generally...
35. ### Question regarding Fourier Transform duality

Homework Statement Given the Fourier transformation pair ##f(t) \implies F(jw)## where ##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions: a) ##g(t)=\frac{2}{t^2+1}## b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)## Homework...
36. ### B Light, Photons, Waves, Particles: Wave-Particle Duality....

Me and my friend have recently (half a year ago) had a huge debate, between ourselves, about the wave-particle duality. We took sides in light being a particle or a wave. I was for particle he was for waves. At the end of a hot-filled week of arguing, the debate ended up with the acceptance of...
37. ### I DeBroglie wavelength & particle-wave duality

We are told to exorcise wave-particle duality because it belongs to the past, in fact prior to 1927. So it’s acceptable to believe matter is neither wave nor particle. But in exorcising wave-particle duality. Why does the deBroglie relationship of momentum being inversely proportional to...
38. ### B Is Decoherence Enough to Solve the Measurement Problem in Wave-Particle Duality?

After reading on the basics of this & watching videos, here are some questions I can't help asking. When the wave is traveling in the room that is filled with air, will the photon interact with all the air molecules continuously collapsing the wave aspect? As I understand...
39. ### I Is Wave-Particle Duality Still Relevant in Modern Physics?

https://www.nature.com/search?journal=nphys&q=wave%20particle%20duality&page=1 When people come to this forum enquiring about the concept of wave particle duality the usual advice seems to be based on the idea that the concept is outdated and has historical interest only. The problem is that...
40. ### I Wave-Particle Duality of Electrons

The wave-particle duality of light was demonstrated first with Thomas Young's 1801 Interference Experiment...and then more clearly with the Double Slit Experiment. Both of these were done with light (so photons). My question is -- How did we come to understand the same of electrons? Did we...
41. ### I Wave-particle duality general question

I just read the following paper that was written in 2014: https://arxiv.org/abs/1403.4687 It proposes that wave-particle duality and the momentum-position uncertainty (entropic) principle are physically the same phenomena. The paper was discussed at some length in this forum in the following...
42. ### I Why does "wave particle duality" not exist anymore?

I think I've asked this question here before but I'm still not clear about the answers. To reiterate, some experts on this forum seem to be of the opinion that wave particle duality doesn't exist anymore and this has been expressed in different threads and in different ways with certain...
43. ### I My stupid question about particle-wave duality

Again, my knowledge of quantum physics is so poor, I have no reference to give me any idea about the measure of stupidity of this question, so please bear with me.: Could it be that the nature of a ‘particle’ doesn’t change? That when decoherence occurs the wave function collapses into a new...
44. ### B Wave particle duality - my visual/mental image

I'm trying to get my head around some ideas in Quantum Mechanics, but I'm trying to find a visual/mental image of the ideas, so that I may, in some cases, understand a little better, if that's even possible. I have a mental image of wave/particle duality and I'd like to know if it's a valid...
45. ### I Doesn't wave duality for massive objects favor pilot-wave?

It is well known that wave particle duality was also observed with large particles up to 10000 amu: Abstract of Paper : https://arxiv.org/abs/1310.8343 Doesn't that make pilot wave theory a more logical interpretation? For example it is easier than the whole material becoming a wave...
46. ### B Does "wave–particle duality" exist in low frequency waves?

Hello, I'm new to the forum, and Physics is not my area of study, but I have a doubt that I couldn't find a satisfactory answer on Google, so I created this account in this forum just to ask this, because I'm really curious to know the answer to this doubt. Does "wave–particle duality" exist in...
47. ### I Can particles exist without duality?

Does Duality serve any other purpose than preventing investigators from fully understanding the framework of our reality?
48. ### B Photon-Wave Duality: Why can't they be separated?

Hi! So the whole Photon-Wave Duality phenomenon really bothers me. I found this little video earlier today: I'm far more comfortable with reality *being* reality. You're either here or there. Every instant contains a single reality. You cannot be in two places at once, nor can you follow two...
49. ### B Quantum Mechanics: Wave-Particle Duality Explained

Hi, i am doing an introductory course in quantum mechanics (that would be equal to first two chapters in griffith's quantum mechanics). I have the doubt that what exactly do we consider in quantum mechanics. Let me say like the electron is a particle and when we will observe it will have a...
50. ### B How does electron microscopy exploit wave/particle duality?

I have been researching wave/particle duality, and I have trouble comprehending how electron microscopy actually exploits wave/particle duality to operate. From Wikipedia, " Wave–particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can...