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. C

    I Wave particle duality and phonons

    I was reading the thread about wave particle duality linked from the newsletter, and I noticed it said (to use my own words) that the conflict between wave and particle dynamics can be avoided by using operator dynamics instead. Unfortunately, in the case of phonons, I've never seen a...
  2. F

    B Is wave particle duality really wrong?

    Can all seasoned physicists please confirm if it is true that wave particle duality is really wrong as when Bill stated in the decoherence branches thread "This is a little different than what you will find in a beginner text becauise it doesn't use the wave particle duality which is wrong - but...
  3. eow

    I Are Superposition, the Uncertainty Principle, and Duality the same?

    Are quantum superposition, Heisenberg uncertainty principle, wave/particle duality the same thing? Do they imply each other? They all seem the same to me like the wave-like nature of photons -> superposition, and superposition means there's not definite position -> uncertainty principle. Sorry...
  4. N

    B Any good lay explanation of the Schrödinger cat duality?

    Once again, I came across the notion (in a lay physics book by an academic) that two mutually exclusive states simultaneously exist until observed (based on the Schrödinger living/dead cat thought experiment), which, on its face, seems to me to belong to theology and not to science. That both...
  5. ShayanJ

    A Problem with fields and operators in holographic duality

    I'm reading McGreevy's lecture notes on holographic duality but I have two problems now: (See here!) 1) The author considers a matrix field theory for large N expansion. At first I thought its just a theory considered as a simple example and has nothing to do with the ## \mathcal N=4 ## SYM...
  6. R

    B Is wave / particle duality linked to superposition?

    I was watching a MIT opencourseware video on quantum mechanics. The first video is on superposition. The link is here: http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/lecture-1/ In short the lecture shows a series of experiments done on electron spin in the...
  7. S

    A Concept of duality for projective spaces and manifolds

    I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...
  8. F

    I Does the particle wave duality create unique signatures...

    I know how the particle duality works, but does the duality create its unique pattern. I want to argue no because one can obtain the same momentum with different velocities or mass.
  9. Peter99

    I Wave-particle duality: nature of wave-particle as it travels

    Hi, I'm trying to conceptualize the life of a particle as it travels through free space. I wish to start simple and then build from there. Speaking about the wave-particle duality that we observe in fundamental particles. Let's start with electromagnetic radiation (then move on to...
  10. Z

    I Legitimacy of Particle-Wave Duality

    I'm asking this question not because of my own misgivings but someone else on this forum some time ago made a comment in a different context which, when talking about teaching Quantum Mechanics, was something like, "Do they still teach things like the particle-wave duality, too?" as a criticism...
  11. amjad-sh

    I Wave-particle duality and localization

    I have read recently that the motion of an electron of momentum p must be described by the means of a plane waves :\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r -wt)}=Ae^{i(\vec p\cdot \vec r -Et)/\hbar} de Broglie hypothesis states that every particle of momentum p has a wavelength lamda. I will...
  12. J

    Fourier Transform using duality property?

    Homework Statement Find the Fourier transform of x(t) = 4 / (4 - i*t)^2 where i is imaginary Homework Equations Duality Property F(t) ↔ 2πf(-ω) when f(t) ↔ F(ω) The Attempt at a Solution I am not sure if duality property is the way to solve this. I look at a list of properties and this...
  13. M

    Can the concept of wave-particle duality be applied to subatomic systems?

    hi, this could be nonsens, I just had to ask :-) If my girlfriend shouts something at me, the sound travels through the room before i hear it. The sound is a wave, I know that. But the message seems to me to be the particlepart. Can this in any way be what happens in subatomic systems as well...
  14. domainwhale

    Insights High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree - Comments

    domainwhale submitted a new PF Insights post High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree Continue reading the Original PF Insights Post.
  15. G

    Wave Particle Duality: Understanding Single Slit Experiment

    In the single slit experiment, individual photons may be diffracted. The electric field of the em wave should accelerate charged particles in the screen (if it were a capacitor for example) but only one packet of energy exists. The photon may be absorbed after many wavelengths of light have...
  16. Y

    Wave particle duality: accept a new entity?

    Hi, I have always been presented with a description of photons (and electrons more recently) as being entities that at times behave like waves and at times like particles (and I understant that typically particle is a substitude word for small little solid object of mass). So I would like to...
  17. ShayanJ

    Gauge/Gravity duality and evidence for superstring theory

    I don't know much about gauge/gravity duality but I know that its a conjecture coming from superstring theory and one of its examples is AdS/CFT duality. Now I have two questions: 1) As I said, I know that gauge/gravity duality and superstring theory are related but I don't know how much. Is it...
  18. M

    What Shows the Wave-Particle Duality of Light in the Compton Effect Experiment?

    Homework Statement What quantity measured in the Compton effect experiment show the wave-particle duality of light? Homework EquationsThe Attempt at a Solution the question is out of 2 marks and all I can think about is momentum is there anything I am missing or am i completely wrong?
  19. KiNGGeexD

    Exploring the Nature of Light: Is Light Made of Particles or Waves?

    I have been reading Richard Feynmans Quantum Electrodyamics and quite early in the first chapter he asserts that Photons are particles. His reasoning that as you decrease the intensity of light incident on a photomultiplier the clicks which the multiplier make become less frequent but equally...
  20. E

    Color Kinematics Duality and Asymptotic Safety

    So, you'll have to forgive me, but I am an engineer, not a physicist, but I take an interest in quantum gravity. My understanding is primarily conceptual, not mathematical. So if my question is dumb, I apologize. Ok, here goes: I'm aware of Zwi Bern's conjectured color-kinematics duality...
  21. J

    Question regarding wave/particle duality

    Regarding to wave/particle duality. Are there places in the universe without magnetic fields or where magnetic fields are too weak for light to be able to behave as a wave (since they travel in the magnetic field as a wave right?). Thanks in advance!
  22. P

    Hodge duality and differential forms

    If we have,$$A=d[(\bar{\alpha}-\alpha)(dt+\lambda)]$$ where $$\alpha$$ is a complex function and $$\lambda$$ is a 1-form. t here represents the time coordinate. If we want to calculate $$d\star A=0$$ where $$\star$$ is hodge star, we get if I did my calculations correctly...
  23. I

    Wave/Particle duality in Blackhole paradox?

    Alrighty, this is my second attempt at this thread, as my first was removed due to speculating theories so i will try to keep this short and clear. Is it possible to apply the concept of Wave/Particle duality to explain the event horizon paradox in which a particle can cross the horizon in a...
  24. F

    Wave-particle duality explanation?

    Hi, Is there a mathematical equation that for itself can explain, or be deducted from it the wave-particle duality? I mean something that connects that 2 concepts in physics equations? sorry if my english isn't very good.. PD: If you can recommend me a paper or a book that describes and...
  25. Ranvir

    EPFL's claim about wave particle duality

    Recent claims by scientists at EPFL suggesting that they somehow manged to photograph light simultaneously as a particle and as a wave raises more misconceptions than ever. The experiment on it's own is an excellent example of recent improvements in technological and experimental techniques, but...
  26. P

    Hodge duality and some properties

    So we know that [Hodge duality](http://en.wikipedia.org/wiki/Hodge_dual) works this way $$⋆(dx^i_1 \wedge ... \wedge dx^i_p)= \frac{1}{(n-p)!} \epsilon^{i_1..i_p}_{i_{p+1}..i_n} dx^{i_{p+1} } \wedge dx^{i_n}$$ where p represents the p in p-form and n is the dimensional number. My question is...
  27. P

    Are observables like position emergent properties?

    Title basically says it all. I'm a physics undergrad trying to wrap my head around quantum physics, and I was hoping people here could help. My question comes from something in one of my textbooks. It tries to explain particle-wave duality through a piece of string, which I'll quickly go over as...
  28. sunrah

    Is the concept of wave-particle duality outdated in modern quantum theory?

    For example with photons in a double slit, knowing the path of the photon leads to particle-like measurements and not knowing leads to interference (wave-like). In the interference case, is it fair to suppose the photon travels through more than one slit at the same time? Is this accepted...
  29. G

    Wave-particle duality and my non-physicist friend

    Hi, I'm a vet with an amateur interest in physics. In discussion with a friend about the usefulness of physics he stated that physicists had not even decided whether light was a wave or a particle. I said the following: 'The question of whether light is a wave or particle is not one which...
  30. C

    Physical meaning of a wave packet w/ respect to HUP&duality

    I'm a QM noob/newb trying to understand the physical implication of a wave packet, in my mind it is something like this: On the x-axis there is displacement (vibration), probability on the y. I Imagine stretching and compressing the wave packet. When I stretch it out, the amplitude must...
  31. G

    Is wave-particle duality a myth?

    Hi, This paper- http://arxiv.org/abs/quant-ph/0609163 - suggests (on p4) that the term 'wave-particle duality' is an incorrect description of the phenomenon, but then goes into a bit of heavy maths to describe the realities of it, so I'm left a little confused. Is my understanding correct when...
  32. J

    Does wave/particle duality apply to the full EM spectrum?

    My question relates to wave/particle duality. If light is regarded as possessing both wave like and particle like properties, and visible light is just part of the whole EM spectrum of radiation, is it correct to say that radio waves for instance, ( being themselves part of the whole spectrum...
  33. Ahmed Abdullah

    Equivalence of waveparticle duality to uncertainty principle

    http://www.sciencedaily.com/releases/2014/12/141219085153.htm It says "Here's a nice surprise: quantum physics is less complicated than we thought. An international team of researchers has demonstrated that two peculiar features of the quantum world previously considered distinct are different...
  34. S

    ? connection between uncertainty and wave-particle duality

    After reading these recent articles on proof of the theory that merges the "duality" of .. " The connection between uncertainty and wave-particle duality comes out very naturally when you consider them as questions about what information you can gain about a system" Can someone point me to a...
  35. P

    Basic question in electromagnetic duality

    We originally have $$\overrightarrow{\nabla}\cdot\overrightarrow{B} = 0$$ $$\overrightarrow{\nabla}\times\overrightarrow{E} = -\frac{\partial \overrightarrow B}{\partial t}$$ When electromagnetic duality is concerned this rank 2 tensor kicks in:$$G^{\mu\nu}$$ And most of books and sites...
  36. phinds

    History of wave/particle duality

    [Moderator's note - this thread was split out from another thread as an interesting but independent discussion] I really don't want to sidetrack this thread with something that will further confuse the OP, but I must point out to you that the concept of "wave particle duality" was dumped some...
  37. P

    Exploring Duality in $$S_{F,\Lambda}$$

    (This question is only long because I tried to give all the details (necessary and not) so that you don't refer to the paper every now and then) In this paper http://arxiv.org/pdf/hep-th/9705122.pdf We have $$S_A = \frac{1}{4g^2} \int{d^4x F_{\mu\nu}(A)F^{\mu\nu}(A)}$$ where $$F_{\mu\nu}(A) =...
  38. R

    Wave Particle Duality & Uncertainty Principle: Which Came First?

    Which of them comes first,(I mean the concept or the theory)? Did Heisenberg come up with the idea of uncertainty principle because of the discover of the wave particle duality? Thanks for paying attention~
  39. G

    Understanding Electromagnetic Duality: From B_i to E_i Using Tensor Notation

    So, I'm trying to show that by duality B_{i}\rightarrow E_{i}, using tensor notation. I've done it in a different way, and it works (starting from \overline{F}_{ij}, the dual of F_ij), but I would like to get it from B_i directly. Where am I going wrong? This is what I did...
  40. J

    Duality and Orthogonality: What's the Difference?

    1. I cannot understand the difference between orthogonality and duality? Of course orthogonal vectors have dot product zero but isn't this the condition of duality as well? Swinging my head around it my cannot find the answer on the internet as well. 2.Relating to same concept is orthogonality...
  41. E

    Solving structure optimization with Lagrange duality

    Homework Statement I need to optimize the following structure with respect to compliance P\cdot u_y. the constraint is that the volume of the truss must not exceed V_0. The design variables are the bar cross sections A_1,A_2,A_3 Homework Equations The mathematical programming problem I got is...
  42. B

    De Broglie Wavelength Calculation for an Electron with 120 eV Energy

    Homework Statement Calculate the de Broglie wavelengths of an electron with energy 120 eV ...Homework Equations lambda = h\p where p = sqrt(2*Me*E) The Attempt at a Solution E=1.6E-19*120ev.. Then sub into equation and I get 1.1E-10m for the wavelength, which is the answer quoted. The...
  43. Krasi_Gavrilov

    Has the double-slit experiment been conducted with multiple observers?

    Hello All, I hope that this question has not yet been asked. I'm interested if the double-slit experiment has been conducted with more than one observer. I mean one (or more) "light-measuring devices" that have been set on every slit for the sake of verifying if the results come out...
  44. aditya ver.2.0

    The Wave-Particle Duality of light and the laws of reflection

    I wonder how the wave particle duality property of light persists with the laws of reflection? How exactly is a photon directed in the same angle (i.e. the angle of incidence) by a particle as reflection while abiding through the quantum theory and the Raman Effect? Such as we throw an elastic...
  45. aditya ver.2.0

    On the Wave-Particle Duality property of light

    I just want to confirm a statement the - Light travels in the form of electromagnetic waves in open space, not particles, but converts to a particle while encountering an obstacle deserting its wave form. So is the statement correct or not? And does it persists any anomaly or exception while...
  46. L

    What waves in wave-particle duality according to de Broglie

    NON-LINEAR WAVE MECHANICS A CAUSAL INTERPRETATION by LOUIS DE BROGLIE "Since 1954, when this passage was written, I have come to support wholeheartedly an hypothesis proposed by Bohm and Vigier. According to this hypothesis, the random perturbations to which the particle would be constantly...
  47. StevieTNZ

    Duality principle is 'safe and sound'

    http://phys.org/news/2014-08-duality-principle-safe-apparent-violation.html It seems an experiment was performed back in 2012 that measured which-way information, but also found interference. (I may have come across it at the time, but cannot recall reading about it from memory.) How could this...
  48. Greg Bernhardt

    How Does the Double-Slit Experiment Illustrate Wave-Particle Duality?

    Definition/Summary The double-slit (or 'two-slit') experiment clearly demonstrates that individual particles exhibit a wave-like behavior, in that an interference pattern can be shown to build up over time, despite the presence of only one particle in the experimental apparatus at any given...
  49. H

    Wave-particle duality confusion

    Hi, I am a high-school student who recently finished the chapter on QM. I thought I completely understood it, but when I started to look back at what I’ve learnt, then suddenly, nothing really makes sense. And one of the things I find really really hard is the nature of electrons. My...
  50. B

    Hypothesis of Quantum Entanglement acting up particle/wave duality

    Pardon my wording as I do not have a solid background in physics, while pondering The effects of observation on a particle changing its state upon observation, and the "theory" that the particle in question went back in time to change its state, My idea or hypothesis on this was that it might...
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