Duality Definition and 282 Threads

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. I

    Wave-Particle Duality and Particle Specific Identification

    I have been struggling with one simple question. How can one measure the momentum of a particle within a field without disrupting the entire field, all together? If the particle is under observation at at t_0, how is it verifiable that at t_1 the same particle is being observed? Obviously spin...
  2. M

    Question on Montonen-Olive duality

    Hi folks, I've been reading about Montonen-Olive duality and understand that two different classical theories can give rise to the same QFT. In particular, we can have a classical theory of electrically charged particles giving rise to a magnetic monopole, and a classical theory of magnetic...
  3. I

    Simplex Method, Duality Problem

    Hello everyone, I have the following question: Show without using the simplex method that x1=5/26, x2=5/2, x3=27/26 is an optimal solution to the following LPP. Maximize z=9x1+14x2+7x3 subject to 2x1+x2+3x3<= 6 5x1+4x2+x3<= 12 12x2 <= 5 x1,x2,x3 unrestricted...
  4. ShayanJ

    Are Phonons Particles? Exploring the Wave-Particle Duality

    Phonons are said to be the result of the quantization of crystal waves.Let the function u=u_0 e^{i(\vec{k}\cdot\vec{r}-\omega t)} describe such a wave. Is it right to say that if we assume u to be the quantum mechanical wave function of a particle, that particle is a phonon? Thanks
  5. T

    MHB Linear programming duality

    I have an hour to solve this problem. I have calculated the duality of linear problem. My question is,whats the range i can change some values,so the optimal solution stay as it is? Is there an easy way to calculate it? Thanks to all!
  6. M

    Three questions on Montonen-Olive duality

    Hi folks -- I have three questions on Montonen-Olive duality. This duality switches the electric and magnetic couplings, where these couplings are related by the Dirac quantization condition eg = 2\hbar\pin Here e is the electric charge and g the magnetic (monopole) charge. 1. First of...
  7. N

    Exploring Light & Electron Duality in QED

    I was curious about two things. I just started reading Feynman's QED book this week. He starts by using an analogy of a clock spinning and that determines the direction (which I think is being analogous to the amplitude of the frequency?) My first question is, as a layperson, what causes...
  8. P

    Possible explanation for the wave-particle duality ?

    Hi everyone, today I had a thought coming across my mind when I woke up, and I think it might be an explanation for the particle-wave duality. Now, when we are talking about a particle, one thing that has to be mentioned is the uncertainty principle. If you divide space into equal volumes for...
  9. C

    Why is wave/particle duality so hard to imagine?

    Why is it difficult (counter intuitive) to imagine an entity (like an electron) that is a particle and a wave? I have seen in many texts/lectures that it is counter intuitive to imagine this sort of thing but I don't understand why. I have this image in my head that seems to make pretty good...
  10. Islam Hassan

    QFT: Point-Like Particles vs Wave-Particle Duality

    Is there a (theoretical) partial or total inconsistency in QFT's postulate/premise/description of elementary particles as dimensionless, point-like objects with respect to the wave-particle duality nature of QM? This is in the sense that such description is *only* particle-like. Clearly, even...
  11. F

    Reason for wave-particle duality?

    Recently been thinking about the double slit experiment, and Schrodinger's cat thought experiment. And realized that quantum mechanics only applies to the atomic, and subatomic level of the universe as far as we know. And what's more that I realized that gravity does not play any significant...
  12. B

    Vector Calculus Theorems - Duality Question

    I'm trying to go over some vector analysis using forms & kind of noticed what look like random vector identities are more appropriately thought of, to me at least, as differential analogues of the classical integral theorems in the way Maxwell's equations can be cast in differential & integral...
  13. D

    Particle-Wave Duality of Elementary Particles

    Hi, I am new to this forum, and to physics in general. I have been reading the basics of General Relativity and Quantum Mechanics but am yet to learn the mathematical side. I am just trying to wrap my head around particle-wave duality and specifically, the wave quantifying of elementary...
  14. S

    Wave-particle duality in the Standard Model

    I posted this question in an another forum but I didn't receive any answers, so I'll post it here again: Do all the fundamental particles in the Standard Model (61 fundamental particles) exhibit wave-particle duality? From my understanding, a photon acts more like a wave than it does a...
  15. I

    Understanding wave particle duality

    I haven't studied physics, but I've had this thought in my head lately and I want to get it out. I'm hoping someone here can either help me understand this or set me straight if I'm on the wrong track. I've been mulling over wave particle duality in my head, and I realized that Einstein sort of...
  16. N

    Wave Particle Duality: Masses & Light Speed

    Do all masses show wave particle duality at near about speed of light?
  17. P

    Question about SO(32) heterotic - Type I duality, M-theory perspective

    I'm having trouble understanding the arguments in http://arxiv.org/abs/hep-th/9510209 They propose that M-theory on the orbifold R^9 \times S^1 \times S^1 / Z_2(1) yields Type I' on R^9 \times S^1, wheras M-theory on R^9 \times S^1 / Z_2 \times S^1(2) is E_8 \times E_8 on R^9 \times S^1. What...
  18. M

    Wave-Particle Duality: Double & Single Slit Experiments

    My initial understanding of the double-slit experiment was that the resulting interference pattern demonstrated that particles such as electrons behaved like a wave. However, if one of the slits was covered, then they reverted to their particle-like nature. However, the single-slit experiment...
  19. W

    Duality Principle | Understand the Concept

    Have i done it correctly?
  20. D

    What quantity shows the wave particle duality of light?

    What quantity measured in the Compton effect experiment shows the wave- particle duality of light? the momentum shows the wave particle duality of light. is this true? The Compton Effect is An increase in wavelength of X-rays or gamma rays that occurs when they are scattered.
  21. M

    Question on Olive-Montonen Duality

    Hi there -- I've just started looking at the topic of dualities in quantum field theory, and I've hit the wall right at the beginning! Looking at the classic Olive-Montonen paper (from 1977), they show how we get elementary quanta of photon, higgs and heavy gauge boson from the Georgi-Glashow...
  22. T

    How does Particle Wave Duality Work?

    I really don't understand particle wave duality whatsoever. Please explain it to me, as I am very interested. Thanks, Joe
  23. G

    Wave Particle Duality: Is It Mass-Energy Equivalence?

    I was thinking about wave particle duality, and it's always made out to seem so mysterious and paradoxical. But then it occurred to me that it might just be the same thing as mass energy equivalence, as waves are just energy and particles are 'mass'. Is this right or am I missing something?
  24. Hercuflea

    Quick beginner question on wave-particle duality

    For a given entity, what is the convention for determining whether it behaves as a wave or as a particle? I know that we generally treat neutrons as waves when they travel faster than .2c, but is there an "absolute" way of determining this for a general particle, or is there a wave-threshold...
  25. O

    MHB Least upper bound - greatest lower bound duality

    Hello everyone! There's a point I didn't get in Rudin's theorem 1.11 that says: Suppose S is an ordered set with the LUB property, and B $\subset$ S, B is not empty and B is bounded below. Let L be the set of lower bounds of B. Then a = sup L exists in S, and a - inf B. In particular inf B...
  26. A

    Duality of Boolean Expressions: Can They Be Compared?

    Can somebody check if I answer them right? Write the dual of the following boolean expressions: 1.x’(y+ z’)+z = x'+(yz')+z 2.x(y+ z)’y’ = x+(y'z')+y' 3.xy+ y’z’+xz = (x+y)(y+z)'(x+z)
  27. S

    Can wave particle duality be viewed at macroscopic scale ?

    can wave particle duality be viewed at macroscopic scale...?? ok so far we have discussed the wave particle duality in case of electrons of-course at microscopic level through the 1) interference phenomenon showing its wave behavior 2)photoelectric effect discussing its particle...
  28. C

    Questions about wave-particle duality

    Please bear with me as I have been reading about quantum physics and, not being a physicist, I have some questions regarding the nature of wave-particle duality. First of all, according to the special theory of relativity a particle with a rest mass would need infinite energy to accelerate to...
  29. L

    Is this person right about particle wave duality?

    I was having a discussion on some other forums about this. Here is the conversation Person 1: Here is my favorite hit off a quick google search for an equation intensive look at QM: http://www.physics.sfsu.edu/~greensit/book.pdf Starting with De Broglie wavelength since it's the most fun...
  30. O

    Wave-particle duality is a correct theory of reality

    Do share your thoughts on this statement.
  31. J

    Weak Duality Theorem (according to my course notes)

    Homework Statement Weak Duality Theorem. If x ε ℝn is feasible for P and x ε ℝm is feasible for D, then cTx ≤ yTAx ≤ bTy. Thus, if P is unbounded, then D is necessarily infeasible, and if D is unbounded, then P is necessarily infeasible. Moreover, if cTx = bTy with x feasible for P and y...
  32. L

    Strong duality and coordniate transform

    Hello all, Assume we have an optimization problem and that strong duality holds. Will it also hold for another optimization problem obtained from the initial by a bijective coordinates transform? Thank you in advance.
  33. R

    Explain Duality Theory: Relativity & Davisson-Germer Exp.

    explanation of duality theory ? relativity phenomenon is largely based on the duality principle . the davisson and germer experiment proved the wave nature of electrons . however i wish to understand the cause of it .
  34. G

    Wave Particle Duality Question

    Okay, so after a little research I think I have a fairly good understanding of wave particle duality. A wave function is just a wave of probability. You don't know where the particle will be until you measure it. Until then, the particle will be in superposition. While it is in superposition, it...
  35. O

    String Theory & Particle-Wave Duality: Can Oscillating Strings Explain?

    I'm not sure if this question is something that has been asked before or if i thought about it myself or if it's completely ridiculous! But I was thinking, if, according to String Theory, particles are 1-dimensional strings that go through modes of oscillation, could they describe how a...
  36. D

    LaTeX Observing Wave-Particle Duality in a Classical Experiment

    This is what's happening: (space after "-demonstrates wave particle duality") This is the surrounding code: \begin{multicols}{2} \section*{\centerline{Classical Experiment}} \begin{singlespace*} \begin{itemize*} \item demonstrates particle/wave duality \end{itemize*}...
  37. C

    Understanding Electromagnetic Duality in Free Space without Sources

    What is electromagnetic duality for Maxwell's equations in free space without sources? Is this symmetry valid when there are conventional sources? If not, what hypothetical additional sources are needed to restore duality symmetry?
  38. B

    Wave/Particle Duality: Observing Reality Differently?

    If a photon can be observed as a wave or a particle, and that observation decides whether it is in fact a wave or a particle, what happens when to observers observe the same particle differently? Is it both? Is reality different for each observer?
  39. M

    Strong Duality Theorem in Linear Programming

    If a linear Program (P) has a feasible solution x_{o}, ( x_{o} not necessarily optimal),does it follow that there exists a feasible solution to the dual problem (D) as well? If yes, why? I know that the Strong Duality Theorem guarantees an optimal finite solution to the dual problem if the...
  40. Z

    Confused about Duality in Projective Geometry

    So I'm studying projective geometry and I'm confused about duality. In particular, I'm confused about drawing dual pictures. If you look at Menelaus's Theorem and Ceva's Theorem, they are supposedly dual diagrams. http://en.wikipedia.org/wiki/Menelaus'_theorem...
  41. L

    Wave-particle duality and special relativity

    hi, i am in year 13 of school, so i would appreciate it if any responses to this question weren't too far over my head. saying that i have a fairly decent back ground knowledge of some of these ideas. so, we have learned about the whole lead up to deciding photons etc. are not waves or...
  42. C

    There is a high concentration of introductory material on wave-particle duality

    (A disclaimer: I'm not an expert, having genuine confusion) There is a high concentration of introductory material on wave-particle duality that claims that the uncertainty principle manifests itself identically to the behavior of waves in the macro world. This however appears to be in direct...
  43. F

    Could wave/particle duality have to do with the speed limit of light?

    This has got to be theorized before, and google is failing me. So someone point me in the direction of the Theory I am looking for. I am not over speculating, I am trying to find the name of a similar theory. When I doing some Feynman reading I started getting some off topic ideas, so I took a...
  44. bohm2

    Wave-particle duality at Macro scale?

    No-one is suggesting that this is exactly the same as the wave-particle duality that exists at the quantum scale (e.g. non-locality) but I thought these papers looking at the behaviour of “walking droplets” that can be seen at the macroscale were very interesting: Quantum mechanics writ...
  45. C

    How Does Duality in Fourier Transform Help Solve F(exp(iax)f(x))(k)=X(k-a)?

    Homework Statement How do I use the (FF(f))(x)=2\pi f(-x) where F is the Fourier transform. and F(f(x-a))(k)=\exp(-ika) X(k) where X(k)=F(f(x)) to get F(\exp(iax)f(x))(k)=X(k-a) Homework Equations Please see above. The Attempt at a Solution The variables confuse me. I can do it by brute...
  46. E

    Light Particle & Wave: 4D Space Reality?

    Is there a possibility that light can be a particle and wave at the same time? If so then there is the proof that there exists a physical manifestation of 4D space.
  47. P

    Bohr Model and wave-particle duality

    I am currently studying wave-particle duality. However, at the moment I do not see how the Bohr Model is connected to this topic. I know that the Bohr Model failed to recognise various aspects of the atom and that it works only for single atoms e.g. Hydrogen, but what did it fail to do when it...
  48. H

    How Does a Particle's Kinetic Energy Affect Its Wave-Particle Duality?

    Homework Statement At what Kinetic energy will a particle's debroglie wavelength equal its Compton Wavelength Homework Equations DeBroglie λ = h/mv Compton λ = h/mc The Attempt at a Solution Setting the two equations equal to each other, I got v = c, then said KE = (1/2)mc^2...
  49. J

    Is Ads/CFT correspondance the same as gauge /gravity duality?

    Is Ads/CFT correspondance the same as gauge /gravity duality?
  50. F

    Is a Bounded Dual Linear Program Indicator of a Feasible Primal?

    Say I have a primal LOP K and a dual LOP D If D is bounded, which means it is feasible, does that mean P is also bounded? Because if P is unbounded, then D is infeasible Likewise if D is unbounded, P is infeasible So D is bounded, P is feasible? Make sense?
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