Field Definition and 1000 Threads

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

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  1. M

    A Any relation between Quantum Chaos and Quantum Field Theory?

    A few years ago I took a course on QC from the mathematics department. But I didn't see a direct connection to QFT, it seems QC is more connected to QM than to QFT. But I may be wrong I am just a novice here. Maybe a good title for a PhD dissertation... (now just to fill the details) so much...
  2. L

    Electric field of an infinitely long wire with radius R

    Hi, I don't know if I have calculated the electric field correctly in task a, because I get different values for the Poisson equation from task b The flow of the electric field only passes through the lateral surface, so ##A=2\pi \varrho L## I calculated the enclosed charge as follows...
  3. H

    A Physical interpretation of hopping between orthogonal orbitals

    Suppose two orthogonal neighbouring orbitals ##|\phi _1 \rangle## and ##|\phi _2 \rangle## so that ##\langle \phi_1|\phi _2 \rangle =0##. Applying an electric field adds a new term ##u (c_1^{\dagger}c_1-c_2^{\dagger}c_2)## to the Hamiltonian which u is a constant potential. Obviously, we still...
  4. athenad07

    Force on a point charge near a uniformly charged rod

  5. Paul Colby

    B In QFT, do Fermion fields belonging to distinguishable particles commute or anti commute?

    In the standard model fermion field components collect quite a few labels. The basic fermion field has 4 components that obey anti commutation relations. If one has two types of fermions, say electrons and muons. Do these commute or anti commute? Same question for other labels like gauge group...
  6. L

    Surface integrals of vector fields, normal - does scaling matter?

    Source: https://tutorial.math.lamar.edu/Problems/CalcIII/SurfIntVectorField.aspx Alright so my confusion lies in the following step: Consider the side x = 0. Okay, from the formula (I am just going to insert an image here) Okay, so gradient of f would just be <1, 0, 0>, which is simply i...
  7. M

    I Electric field inside & outside of a spherical shell

    Hi, I am reading Griffiths Introduction to electrodynamics. Currently I am solving problem 2.11 which asks to find an electric field inside and outside a spherical shell of radius R. Inside: $$\int{E \cdot da} = \frac{Q}{e_0} = |E|4\pi r^2 = \frac{Q}{e_0} = 0$$ The result is $$0$$ because we...
  8. pinball1970

    Giveaway QFT for the gifted amateur

  9. Hill

    Giveaway Student Friendly QFT by Klauber

    https://www.amazon.com/Student-Friendly-Quantum-Field-Theory/dp/0984513957?tag=pfamazon01-20
  10. Hill

    Giveaway Quantum Field Theory and the Standard Model

    It is a giveaway. https://www.amazon.com/Quantum-Field-Theory-Standard-Model/dp/1107034736/?tag=pfamazon01-20
  11. BvU

    Sell Itzykson/Zuber QFT

    Cost me a fortune when the dollar was so high in the eighties :rolleyes: ##\ ##
  12. P

    I Gravitational field in Galilean relativity

    My question is why the direction of gravitational field doesn't change relative to moving observer take for example gravitational field in the y direction relative to stationary observer but relative to an observer moving with velocity v in the x direction the field should have x component
  13. Q

    A How does QFT treat Young’s Double-Slit Experiment?

    How does QFT treat the Young’s DSE? Is there a wave function (wave packet) attached (and created at the moment of launching of the photon) or the modes of the EM quantum field are pre-existing due to experimental configuration (including the screen) and do they play the role the wave function is...
  14. BuggyWungos

    Why does electric potential energy increase if you move against the field?

    My understanding of this question is that, if you have a proton standing against a positive electric field, and move it in the opposite direction of the field, you're putting in work and therefore should have greater electric potential energy. But that idea breaks down when you consider a...
  15. phys9928

    Magnetic field in and around a conductive hollow cylinder

    My attempt at a solution: Cylindrical coordinate system with ##r##, ##\theta##, ##z##. Conductivity ##\sigma## and permeability ##\mu_0##. Inner radius ##a## and outer radius ##b##. (##b>a##) The external field is spatially uniform and driven at sinusoidally at frequency ##f##. The external...
  16. physics_rocky

    A What Is the Expression for Settling Velocity in a Rotating Field?

    Hi there, The settling velocity is terminal velocity at which drag force is equal to the gravity. When the sphere is in rotating field the terminal settling velocity is reduced. What will be the expression for it in rotating field?
  17. kirito

    Find the electric field between 2 finite discs

    I did make the problem simpler by looking at the the part from d/2 down the upper plate here are my initial parameters I am making my size step be h since lowering it may make calculating harder I am especially getting weird results for the field and capacitance R = 0.1; % Radius of the...
  18. PeterDonis

    A A possible quantum field theory of tachyons?

    I have come across the following paper (arxiv preprint link, it looks like it is published in Phys Rev D) that claims to have found a consistent quantum field theory of tachyons, overcoming three issues which have been said in previous literature to make such a theory impossible...
  19. tellmesomething

    Charge conservation for 3 parallel charged plates

    I took a point P on the conductor with charge Q. We know that the field inside a conductor is zero in electrostatic equilibrium Therefore I took induced charge on the neutral conductor to be x and gave the respective charges to the others. On the side facing conductor with charge Q the neutral...
  20. tellmesomething

    Gauss' law problem -- Should the field due to both the inside and outside charges be taken into account?

    My teacher said that gauss law may not accounts for the field due to the outside charges in the LHS in this expression ##\int E.ds## = ##\frac{q}{\epsilon}## as field lines coming in the surface leave it as well. Hence the total flux is 0 However i dont think thats very consistent with the...
  21. ShivamM

    A Where Can I Find Detailed Derivation of Teleparallel Gravity Field Equations?

    Where can I find detailed derivation of Field Equation of Teleparallel Gravity from variation of Action ?
  22. D

    I Why can't we photograph magnetic lines of force emitted by planetary and beyond objects (like we can for the Sun)?

    First astrophotography was black and white. So then we colored it in. Then the light got filtered so people can see the universe better. When I ask why are there no photographs of magnetic field lines in space, or anywhere; I read that it's because our eyes cannot see them, but we can't see...
  23. S

    Electric field of point along the central vertical axis of a triangle

    here is my attempted solution. ## d^2 = z^2 + \frac {L^2} {3} ## ## C ## is coulomb constant since the point is symmetric, only the vertical component of the electric field remains. So, $$ E = 3 E_y =3 \frac {C Q cos \theta} {d^2} $$ $$ E= 3 \frac {C Q z} {d^3} $$ thus part (a) is done ( i...
  24. Zayan

    Electric Field of a quarter ring

  25. DaveC426913

    B Is the Sun's Magnetic Field About to Flip?

    https://www.space.com/sun-magnetic-field-flip-solar-maximum-2024 I know theyre not the same physics. I just wondered if the the Suns 11 year polarity flip has any analogy to the flipping T-wrench effect.
  26. Bling Fizikst

    Magnetic field outside a semi-infinite solenoid

    The solenoid can be assumed to be a stack of rings of width ##dz## each where ##z:0\to-\infty## . $$i=nIdz$$ $$\sin\theta=\frac{R-r}{l}$$ $$\cos\theta=\frac{z+x}{l}$$ $$dB=\frac{\mu_{\circ}nIR d\varphi dz}{4\pi l^2}$$ Call ##\frac{\mu_{\circ}nIR}{4\pi}=\alpha## $$ B_y =\int dB\cos\theta ...
  27. aliinuur

    B E-field of a constant velocity charged particle

  28. kirito

    Why are the electrostatic field lines normal to a charged conductor?

    I understand the following .a conductor is made of atoms and atoms always strive to be at equilibrium and that's why the electric field inside a conductor is zero because the electros distribute themselves in such a way so that they are in equilibrium , yet they do produce an electric field...
  29. kirito

    Electric field of a sphere and a shell

    q encolsed =0 Second case q enclosed q by gauss law At 2r
  30. K

    I Are gravitons necessary?

    If gravity is not a force, but rather curvature of spacetime, are gravitons necessary to couple with gravitational field?
  31. hongseok

    B How is a magnetic field formed in a space surrounded by one pole?

    There is a magnet in the shape of a hollow sphere. And the inner surfaces are all n-poles, and the outer surfaces are all s-poles. At this time, how is the magnetic field formed in the empty space of the sphere? There is no s pole for the magnetic force lines from the n pole to enter, so can...
  32. J

    I How to get metric field using a path dependent parallel propagator

    If a vector ##V(x)## being transported down a path ##l##, The vector field is described with equation: $$\partial_\mu V(x)=\Gamma_\mu V(x)$$ The solution of the equation can be described with parallel propagator ##P(x, x_0)##(in mathematics it is also called product integration): $$V(x)=P(x...
  33. tellmesomething

    Find the electric field at a point in 3 dimensional space

    I tried resolving the semi infinite rods into arcs of 90 degree each placed on the three axes but that doesnt take me anywhere.... Alternatively I tried finding out the field at the point due to each rod but im unable to find the perpendicular distance from the point to the rod...I dont think...
  34. tellmesomething

    Prove that the field due to a circular arc is same as its bounded tangent at the centre of curvature

    Lambda = charge density I tried first taking out the field due to the circular arc and I got $$ (lambda / 4π (epsilon knot) ) (2 sin (theta)) $$ For reference this is the arc that was provided in the question of angle 2(theta) and the tangent What I dont understand is how can the fields be...
  35. ergospherical

    I Killing fields commute if they are asymptotically coordinate?

    Some axially symmetric star has two independent KVFs, ##T## and ##\Phi##. We don't know the expressions for these at all points -- the only thing we know is that as ##r^2 + z^2 \rightarrow \infty##, that ##T \rightarrow \partial/\partial t## and ##\Phi \rightarrow \partial/\partial \phi##. The...
  36. kated

    How to find the potential of a field that has regions of non-zero curl

    We know that in electrostatics, there is path independency for line integral of E, so E is a conservative field and thus we have E=-gradV. Integrating this from ro(reference point of our choice) to the point r we are studying, along a random path, we get the solution of the above equation...
  37. M

    B Solar Wind Particles, - reaching into Earth's magnetic field?

    How many particles (approximately) from the solar wind reach deep into the Earth's magnetic field every second. Possibly measured in mass?
  38. aryan kumar pandey

    Why is the electric field inside a conducting hollow/filled sphere zero?

  39. Ben2

    Field due to an electric dipole (Halliday & Resnick, Ch. 27 problem 23)

    Using either H&R's Chapter 27 Example 3 or Problem 590 of the ##\mathbf{Physics Problem Solver}##, I've been unable to get the component ##E_x## or ##E_y##. There are now different angles at the charges. My thanks to berkeman for LaTeX advice, but any errors are of course my own. Thanks in...
  40. maxelcat

    Which way does this coil turn in a magnetic field?

    I use FLHR and the plan diagram. This suggests that the left hand side of the coil is going down. I have added annotations to this, which agree with the mark scheme which then goes onto suggest that the coil turns clockwise. To me though this would make it go anti clockwise. So... what am I...
  41. Rick16

    B What ever happened to Classical Unified Field Theory?

    Are there currently professional physicists who work on Einstein’s quest to unify the gravitational field and the electromagnetic field (as classical fields), or has this idea been completely abandoned? Is it simply a hopeless endeavour?
  42. L

    Direction of electric field vector on the surface of charged conductor

    Consider a negatively charged spherical conductor. On the surface of it, what is the direction of its electric field? Well, the definition of the direction of an electric field is the direction a positive test charge would go if placed at that point. But... it wouldn't move anywhere! So is the...
  43. B

    Planar Coil Rotating in a Magnetic Field

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  44. Hill

    A Vector field vs. four scalar fields ("QFT and the SM", Schwartz)

    This is the statement in question: But if they were scalar fields, they would not transform at all. How could they contribute differently if they didn't change?
  45. Hill

    I What does "transforms covariantly" mean here?

    The Lagrangian, $$\mathcal L(x)= \frac 1 2 \partial^{\mu} \phi (x) \partial_{\mu} \phi (x) - \frac 1 2 m^2 \phi (x)^2$$ for a scalar field ##\phi (x)## is said to be Lorentz invariant and to transform covariantly under translation. What does it mean that it transforms covariantly under translation?
  46. A

    Magnetic Field Due To Wire Of Semi-Circular Cross Section

    I proceeded as follows Current in sector ##d\theta=## is: $$dI=\int_{x=0}^{x=R}{\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x dx}$$ Field due to sector ##d\theta## is therefore $$dB=\int^{x=R}_{x=0}{\frac{\mu_○}{2\pi x}\times\frac{I}{\pi R^2/2}\times\frac{d\theta}{2\pi}\times 2\pi x...
  47. K

    Calculation of the magnetic field of four quarter circles

    The diagram is something like this, and I want to calculate the magnetic field at the center using the Biot-Savart law. In this case, do the magnetic fields formed by the quarter circles pairwise alternate with each other?
  48. tanmays1

    Does levitation distance increase as Fg increase???

    I'm trying to conduct an experiment where I calculate the magnetic field strength of a magnet, by comparing the levitation distances between two magnets. My experiment involves using different masses to anchor down magnetic repulsion between 2 magnets. Fg = Fm. The formula I am using for this...
  49. R

    B Help with science experiment magnets/force/acceleration/speed

    Sorry, I guess I should have remembered all of this from my school days, but right now I have forgotten so much that I need some help. I am developing some simple experiments for school children (age ca. 12). This one involving magnets. I am not asking for detailed calculations, that is way...
  50. Hill

    Quantum Is Quantum Field Theory for the Gifted Amateur Effective for Self-Study?

    I'd like to hear your professional opinion on and experience with using Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen J. Blundell as a self-study textbook. Thank you.
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