What is Scalar: Definition and 828 Discussions

In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

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  1. adoion

    Scalar field 2 dimensional discontinuous but differentiable

    Hi, f(X)=\frac{xy^2}{x^2+y^4} is the function in question, this is the value of the function at ##X=(x,y)## when ##x\neq0##, and ##f(X)=0## when ##X=(0,y)## for any ##y## even ##y=0##. Now, along any vector or line from the origin the directional derivative ##f'(Y,0)## (where ##Y=(a,b)## is...
  2. P

    Current of Complex scalar field

    I was trying to derive current for Complex Scalar Field and I ran into the following:So we know that the Lagrangian is: $$L = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^* \phi$$ The Lagrangian is invariant under the transformation: $$\phi \rightarrow e^{-i\Lambda} \phi $$ and $$\phi^*...
  3. T

    MTW Ch7: Choosing Lagrangian for Scalar Potential

    I have managed to get to the end of Chapter 6 and have done almost all of the exercises (I didn't get anywhere with exercise 5.6 (d) and have seen the Cesarth/TSny exchange and still don't feel I have a satisfactory solution...) but I have hit a bit of a wall with Exercise 7.1 (a).First question...
  4. B

    Scalar propagator for lightlike separation

    Hello everybody. I have a free scalar in two dimensions. I know that its propagator will diverge for lightlike separations, that is t= ±x. I have to find the prefactor for this delta function, and I don't know how to do this. How do I see from, for example, \int \frac{dk}{\sqrt{k^2+m^2}} e^{i k...
  5. L

    Cosine question. Scalar product.

    Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...
  6. M

    Difference between Scalar and Vectorial fields ?

    Dear all! I think the main difference between scalar and vector fields is that vectorial fields are composed of vector elements that varies among them. Scalar fields are fields that have large regions of equal magnitude, variations are just presented in different regions. Please bring me help...
  7. Xenosum

    Time Evolution of the Complex Scalar Field

    Homework Statement Consider the Lagrangian, L, given by L = \partial_{\mu}\phi^{*}(x)\partial^{\mu}\phi(x) - m^2\phi^{*}(x)\phi(x) . The conjugate momenta to \phi(x) and \phi^{*}(x) are denoted, respectively, by \pi(x) and \pi^{*}(x) . Thus, \pi(x) = \frac{\partial...
  8. S

    Lorentz-Invariant Scalar Products

    Hi, Are ALL scalar products of four-vectors Lorentz-invariant (as opposed to just the scalar product of a four-vector with itself)? And, if yes, what is the proof?
  9. Xenosum

    Real Scalar Field Fourier Transform

    Homework Statement Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field \phi(x) is written as \phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik...
  10. ChrisVer

    Introduction of the connection in Lagrangian for complex scalar field

    I am having some problem with this attached question. I also attached my answer... My problem is the appearence of the term: 2 e (A \cdot \partial C) |\phi|^2 which shouldn't appear...but comes from cross terms of the: A \cdot A \rightarrow ( A + \partial C) \cdot (A + \partial C) In my...
  11. T

    Primodial power spectrum of scalar perturbations

    For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum: P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2, where: c = G = ħ = 1, k is the...
  12. S

    Is Average Speed a Scalar quantity?

    Is Average Speed a Scalar quantity?
  13. V

    Expansion of scalar function times laplace's equation

    Apparently, f \nabla^2 f = \nabla \cdot f \nabla f - \nabla f \cdot \nabla f where f is a scalar function. Can someone please show me why this is step by step. Feel free to use suffix notation. Thanks in advance.
  14. S

    Why is charge a scalar in physics?

    Why is a charge from say an electron a scalar. It has a constant magnitude, and it has a direction.
  15. H

    Necessity for action to be a Lorentz scalar

    Hello, On p.573 of Jackson 2nd Ed. (section 12.1), he says, "From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extremum condition, \delta A=0." I agree that if the action is a Lorentz scalar, then...
  16. 2

    Is the inner product of two rank n tensors a scalar?

    Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...
  17. ChrisVer

    Scalar Propagator form with width

    I would like to ask when can someone add the width in a scalar particle's propagator. In general the scalar propagator can be: \frac{1}{k^{2}-m^{2}+i \epsilon} (\epsilon \rightarrow 0) However I read somewhere that if necessary one can include a width for the propagator...
  18. I

    Phyiscal consequences of a neutral massive scalar particle

    Let's add a simple uncharged massive particle to, say, the standard model of particle physics. The question is about the physical, observable consequences of this modification. The first question is if the following considerations are correct, and if not what is wrong. From a classical...
  19. Greg Bernhardt

    Exploring Scalar and Vector Flux in Electromagnetism

    Definition/Summary Flux sometimes means total flow through a surface (a scalar), and sometimes means flow per unit area (a vector). In electromagnetism, flux always means total flow through a surface (a scalar), and is measured in webers (magnetic flux) or volt-metres (electric flux)...
  20. A

    Poynting vector and intensity in scalar diffraction theory

    The power density of an electromagnetic wave is proportional to the absolute square of the electric field |E|^2 (assuming a plane wave). Here, E is a vector so the absolute square involves all three of Ex, Ey, and Ez. In homogeneous, linear media, it's easy to show that each component of E...
  21. V

    Derivative of scalar triple product

    Homework Statement If u(t) = σ(t) . [σ'(t) x σ''(t)], show that u'(t) = σ(t) . [σ'(t) x σ'''(t)]. Homework Equations The rules for differentiating dot products and cross products, respectively, are: d/dt f(t) . g(t) = f'(t) . g(t) + f(t) . g'(t) d/dt f(t) x g(t) = f'(t) x g(t) +...
  22. ChrisVer

    Scalar field in Expanding Universe EOM

    I would like to ask something. How is the solution of EOM for the action (for FRW metric): S= \int d^{4}x \sqrt{-g} [ (\partial _{\mu} \phi)^{2} - V(\phi) ] give solution of: \ddot{\phi} + 3H \dot{\phi} + V'(\phi) =0 I don't in fact understand how the 2nd term appears... it...
  23. B

    Finding the electric and magnetic fields from a scalar potential?

    E = - grad*phi - 1/c (dA/dt). phi is the scalar potential, and is given. How do I calculate the vector potential = A ? Is it A = (v/c) * phi ? If it is, then where is this equation coming from? Thank you.
  24. J

    Understanding Scalar Fields: Tools for Studying Vector Field Behavior

    After watch this video , I understood that for study the behavior of the vector field, just use 2 tools, the line integral and the surface integral, and actually too, the divergence and the curl. In accordance with this, the maxwell's equations are justly the line integral, the surface integral...
  25. J

    'Constant' vector field is equivalent to some scalar field

    To every scalar field s(x,y) there corresponds a 'constant' vector field x = A s(x,y) and y = B s(x,y), where A,B are direction cosines. The vector field is only partially constant since only the directions, and not the magnitudes, which are equal to |f(x,y)|, of the field vectors are constant...
  26. xortdsc

    How to compute the energy of scalar wave equation

    Hi, considering the scalar wave equation $$ { \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u $$ (where ∇^2 is the (spatial) Laplacian and where c is a fixed constant) how can I derive the potential and kinetic energy for a given state u and u' ? Thanks and cheers
  27. I

    Sketching the Level Surface of a Scalar Field

    Question: For the scalar field \Phi = x^{2} + y^{2} - z^{2} -1, sketch the level surface \Phi = 0 . (It's advised that in order to sketch the surface, \Phi should be written in cylindrical polar coordinates, and then to use \Phi = 0 to find z as a function of the radial coordinate \rho)...
  28. I

    Vector Calculus - Laplacian on Scalar Field

    A scalar field \psi is dependent only on the distance r = \sqrt{x^{2} + y^{2} + z^{2}} from the origin. Show: \partial_{x}^{2}\psi = \left(\frac{1}{r} - \frac{x^{2}}{r^{3}}\right)\frac{d\psi}{dr} + \frac{x^{2}}{r^{2}}\frac{d^{2}\psi}{dr^{2}} I've used the chain and product rules so...
  29. S

    Find Value of α for Scalar Product a\cdotb = 0 & Explain Phys. Significance

    Vectors a and b correspond to the vectors from the origin to the points A with co-ordinates (3,4,0) and B with co-ordinates (α,4, 2) respectively. Find a value of α that makes the scalar product a\cdotb equal to zero, and explain the physical significance. My attempt: The scalar product...
  30. A

    Calculating Beta Function for Scalar QCD Theory

    Hi all, I am currently trying to calculate the beta function for scalar QCD theory (one loop for general su(n)). I therefore need to calculate the Feynman rules in order to apply them to the one loop diagrams. Unfortunately I am getting very confused with what the Lagrangian for scalar QCD...
  31. J

    Decompostion of scalar field

    If a vector field can be decomposed how a sum of a conservative + solenoidal + harmonic field... so, BTW, a scalar field can be decomposed in anothers scalar fields too?
  32. J

    Genaral PDE for scalar and vectors

    I realized that a PDE of 2nd order can written like: A:Hf+\vec{b}\cdot\vec{\nabla}f+cf=0 \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}f+\begin{bmatrix} b_1\\ b_2\\...
  33. J

    Curl in 2D is a vector or a scalar?

    Vector, by definition, have 2 or 3 scalar components (generally), but the curl of a vector field f(x,y) in 2D have only one scalar component: \left ( \frac{\partial f_y}{\partial x} -\frac{\partial f_x}{\partial y} \right )dxdy So, the Curl of a vector field in 2D is a vector or a scalar?
  34. V

    Energy scale of Hubble constant for dark energy scalar field

    Hello All, In Carroll's there is a brief introduction to a dynamical dark energy in which the equation of motion for slowly rolling scalar field is discussed. Then to give an idea about the mass scale of this field it is compared to the Hubble constant, saying that it has an energy of...
  35. P

    Chapter 21 Ray D'Inverno Scalar Optics, congruence of null geodesics

    First of all this is my first thread, so I apologize for any mistake. Perhaps this is a stupid question, but i need some help in exercise 21.10 of D'Inverno, to write down geodesic equation for l^a, which is a vector tangent to a congruence of null geodesics and then by a rescaling of l^a...
  36. S

    Finding the instantaneous velocity of a scalar function?

    Homework Statement Find the instantaneous velocity of f(x, y, z) = xyz, at (1, 2, 1) Homework Equations The Attempt at a Solution I think this problem our proffesor gave us wasn't formulated correctly. The only time when we calculated instantaneous velocity was when we had a...
  37. B

    Can dyons be broken down into scalar and pseudoscalar components?

    If an electrically charged particle is a scalar, and a magnetically charged particle is a pseudoscalar, then what is a dyon (having both types of charge), a "mixed-type" scalar? Are there "scalar-like" quantities that can be decomposed into a scalar part and a pseudoscalar part?
  38. N

    Closed under addition and closed under scalar

    Could someone explain to me about what closed under addition and closed under scalar multiplication means? I have a patchy idea of what it is but how does it relates to A = {(x,y) | x^2 + y^2 <= 1}? What does A stands for? What does the language implies? Edit: My interpretation: Let's...
  39. S

    Scalar Triple Product Derivative

    Homework Statement Find an expression equivalent for the derivative of the scalar triple product a(t) . (b(t) x c(t))The Attempt at a Solution Initially I figured since whatever comes out of B X C is being dotted with A, I can use the derivative rules of a dot product: (a(t)' . (b(t) x...
  40. T

    Force can be represented by a scalar?

    Homework Statement True or.false Force can be represented by a scalar? Homework Equations The Attempt at a Solution I say false is this correct?
  41. T

    Is Scalar Addition Equivalent to Adding Parallel Vectors?

    Homework Statement T/F: The addition of two scalars is equivalent to the addition of parallel vectors. Select one: a. False b. True Homework Equations The Attempt at a Solution i said true reason being the magnitude of vectors can be added if they are in the same direction by...
  42. S

    Fortran FORTRAN error array bound is not scalar integer

    FORTRAN error "array bound is not scalar integer" I'd like to know if a loop can be created, inside which I can call a subroutine in which there are arrays to be defined whose size varies as a function of loop variable. I tried as following, but got error "array bound is not scalar integer"...
  43. T

    Non-canonical terms of scalar fields

    Hello! Well, I guess it's all in the title, really. I was reading about k-essence, and it was described as a scalar field having a non-canonical kinetic term. I did a bit of browsing and couldn't find a clear explanation of what, exactly, a non-canonical kinetic term is. Any help would be...
  44. E

    Can the Vector Scalar Product Equation be Solved for x Using Cross Product?

    I have been asked to solve the equation for x - All letters bar λ are vectors. λx + (a cross x) = b I have worked it down as far as x.(λb + (b cross a)) = |b|^2 by taking the dot product with b of both sides. But is there any way I can now solve this equation for x?
  45. M

    Scalar in adjoint representation

    Hello, people. I'm studying (as an exercise) the breaking of an SU(3) gauge group to SU(2) x U(1) via a Higgs mechanism. The scalar responsible for the breaking is \Phi, who transforms under the adjoint representation of SU(3) (an octet). First of all I want to construct the most general...
  46. andrewkirk

    Intuitive description of what the Ricci tensor & scalar represent?

    Is there a simple intuitive description of what the Ricci tensor and scalar represent? I have what seems to me a straightforward understanding of what the Riemann tensor Rabcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined...
  47. J

    Scalar, vector and tensor calculus

    I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...
  48. junfan02

    Transformations that are scalar invariant

    I am a bit confused about something! Exactly under what kind of transformations are scalars invariant in the domain of classical mechanics? The fact which is disturbing me is, say we have a moving body of certain kinetic energy in a certain inertial frame of ref, and then we choose to.observe...
  49. shounakbhatta

    Is the Lagrangian Invariant Under Coordinate Transformations?

    Can you please tell me whether I am right or wrong? Lagrangians are scalars. They are NOT invariant under coordinate transformations[ the simplest example is when you have a gravitational potential(V=mgz) and you translate z by "a"(some number)...
  50. I

    Correct my simple error (property of scalar multiplication)

    say i have some vector ##\vec{v}## multiplied by a scalar k. the norm of ##\vec{v}## would be just ##||\vec{v}||## and the norm of ##k\vec{v}## is claimed to be ##|k|||\vec{v}||## ie, ##||k\vec{v}||=|k|||\vec{v}||## ie the sign of the constant is irrelevant. when i work it out...
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