Scalar field Definition and 207 Threads

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. W

    Constant Scalar Field: Meaning & Relationship to Surface S

    What does it mean if a scalar field φ is said to be constant on a surface S? Does φ then have a particular mathematical relationship with S?
  2. U

    How do I differentiate this Scalar Field?

    Homework Statement (a) Find the christoffel symbols (Done). (b) Show that ##\phi## is a solution and find the relation between A and B.[/B] Homework EquationsThe Attempt at a Solution Part(b) \nabla_\mu \nabla^\mu \phi = 0 I suppose for a scalar field, this is simply the normal derivative...
  3. F

    How to prove some functions are scalar field or vector field

    Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?
  4. cvex

    How to get the laplacian of a scalar field?

    Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...
  5. N

    Evaluating Time-Ordered Product with Wick's Theorem

    Homework Statement [/B] Consider a real free scalar field Φ with mass m. Evaluate the following time-ordered product of field operators using Wick's theorem: ∫d^4x <0| T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4) |0> (T denotes time ordering) Homework Equations Wick's theorem: T((Φ(x1)...Φ(xn)) = ...
  6. G

    Feynman rules for this real scalar field in 2d

    Homework Statement Consider the following real scalar field in two dimensions: S = \int d^2 x ( \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 - g \phi^3) What are the Feynman rules for calculating < \Omega | T(\phi_1 ... \phi_n ) | \Omega > 2. Homework...
  7. R

    Finding a scalar field given two gauge fields

    Homework Statement Demonstrate the equivalence between the gauge fields A1=(0,bx,0) and A2=)-yB/2,xB/2,0) and find the scalar field Φ for which A1= A2 + ∇ΦHomework Equations B = ∇XA The Attempt at a Solution The first part is fine, you just plug it into the above relevant equation and you get...
  8. Q

    Energy of Scalar Field: Evaluating Rubakov's Expression

    Homework Statement My question is just about a small mathematical detail, but I'll give some context anyways. (From Rubakov Sec. 2.2) An expression for energy is given by E= \int{}d^3x\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}\dot{\phi}(\vec{x}) - L, where L is the Lagrangian...
  9. 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...
  10. 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^*...
  11. 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...
  12. K

    Direction of the maximum gradient (scalar fields)

    If a question asks for the direction of the maximum gradient of a scalar field, is it acceptable to just use del(x) as the answer or is the question asking for a unit vector? Thanks
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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)...
  19. 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...
  20. J

    Can Scalar Fields Be Decomposed into Symmetric and Antisymmetric Parts?

    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?
  21. 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...
  22. 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)...
  23. B

    Evaluating Scalar Field in Spherical Coordinates

    Homework Statement Evaluate the scalar field ##f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid## in spherical coords. Homework Equations Law of Cosines? ##\mid \vec{A} + \vec{B} \mid = \sqrt{A^2+B^2+2ABCos(\theta)}## The Attempt at a Solution I'm not sure the law of cosines...
  24. Ace10

    Complex scalar field propagator evaluation.

    Good afternoon fellow scientists,i have a small problem in evaluating the propagator for the complex Klein-Gordon field. Although the procedure is the one followed for the computation of the propagator of the real K-G field, a problem comes up: As known: <0|T\varphi^{+}(x)\varphi(y)|0> =...
  25. H

    Solutions to equations of motion for free scalar field

    I hope this fits this section. This doesn't all fit into the title, but this comes from a homework on conformal field theory, and I am slightly stumped on it. I just can't seem to get anything sensible out of it at the end, but it may be because I've just done something wrong (even though I've...
  26. E

    Lorentz Invariance of Propagator for Complex Scalar Field

    Homework Statement Show that [\hat{\phi}(x_1),\hat{\phi}^\dagger(x_2)] = 0 for (x_1 - x_2)^2 < 0 where \phi is a complex scalar field Homework Equations \hat{\phi}=\int\frac{d^3 \mathbf{k}}{(2\pi)^3 \sqrt{2\omega}}[\hat{a}(k)e^{-ik\cdot x} + b^\dagger(k)e^{ik\cdot x}]...
  27. G

    Total momentum operator for free scalar field

    Sorry for reopening a closed thread. But I have exactly the same doubt as this guy: https://www.physicsforums.com/showthread.php?t=346730 And the answer doesn't actually answer his question. I do get delta(p+p'), but they just help me in getting a_{p}a_{-p} and a_{p}^{\dagger}a_{-p}^{\dagger}...
  28. R

    Scalar field lagrangian in curved spacetime

    Homework Statement I am studying inflation theory for a scalar field \phi in curved spacetime. I want to obtain Euler-Lagrange equations for the action: I\left[\phi\right] = \int \left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + V\left(\phi\right) \right]\sqrt{-g} d^4x Homework...
  29. K

    The variation of a scalar field (from Ryder's QFT book)

    Hello! Im currently reading Ryder's QFT book and am confused with the variation of a scalarfield. He writes that the variation can be done in two ways, \phi(x) \rightarrow \phi'(x) = \phi(x) + \delta \phi(x) and x^\mu \rightarrow x'^\mu = x^\mu + \delta x^\mu. This seems...
  30. S

    Early Universe scalar field, inflaton and analogies in electric field

    I have been trying to get my head around this topic for a while. As I go through the description of scalar fields, the inflation and the potential inflaton, (in description as in ned.ipac.caltech.edu), I constantly miss a concept. There must be a fundamental difference between the type of...
  31. L

    Transformation question; first order shift of a scalar field

    Hi to all! I have the following transformation \tau \to \tau' = f(\tau) = t - \xi(\tau). Also I have the action S = \frac{1}{2} \int d\tau ( e^{-1} \dot{X}^2 - m^2e) where e = e(\tau) . Then in the BBS String book it says that $$ {X^{\mu}}' ({\tau}') = X^{\mu}(\tau)$$ and...
  32. P

    Why is the Higgs field a scalar field?

    as i understand it the higgs field is a spin-0 scalar field that gives mass to elementry particles. How is it a scalar field? I thought it was homogenous.
  33. P

    Is This the Correct Method for Quantizing the Scalar Field?

    Hi can I just check that i haven't done anyhting foolish here whe quantising the scalar field; \ddot{\phi} - \frac{1}{a^2}\nabla \phi + 3H\dot{\phi} - 3\frac{H}{a^2}\nabla \phi + m^2 \phi with \phi = \int \frac{d^3 K}{(2\pi)^{\frac{3}{2}}}(\chi \exp(+ikx) +\chi \dagger \exp(-ikx))...
  34. D

    How can the gradient of a scalar field be covarient?

    According to Carroll, \nabla \phi is covariant under rotations. This really confuses me. For example, how could equations like \vec{F}=-\nabla V be rotationally covariant if force is a contravariant vector? I know this is strictly speaking more of a mathy question, but I still figured this...
  35. snoopies622

    Seeking derivation of real scalar field Lagrangian

    Here and there I come across the following formula for the Lagrangian density of a real scalar field, but not a deriviation. \mathcal{L} = \frac {1}{2} [ \dot \phi ^2 - ( \nabla \phi ) ^2 - (m \phi )^2 ] Could someone show me where this comes from? The m squared term in particular...
  36. O

    How to Express Contractions of Complex Scalar Field Operators via Propagators?

    i am trying to understand how to express contractions of field operators via propagators. we are talking about an interacting theory of 2 complex scalar fields, lets call them ψ1 and ψ2. the interaction term is: Lint=λ(ψ2)^3(ψ1) i have found the free propagator defined as...
  37. soothsayer

    Quantum gravity - Planck's constant as a scalar field?

    "Quantum" gravity -- Planck's constant as a scalar field? I was just reading about a fascinating new theory on the solution to the quantum gravity problem: http://arxiv.org/pdf/1212.0454.pdf I really like it, but I have one big problem with it: The author states that G = \frac{\hbar...
  38. R

    Line Integral of Scalar Field Along a Curve

    Homework Statement For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as \int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b)...
  39. T

    Lagrangian density for a complex scalar field (classical)

    Hi. Let's say we have a complex scalar field \varphi and we separate it into the real and the imaginary parts: \varphi = (\varphi1 + i\varphi2) It's Lagrangian density L is given by: L = L(\varphi1) + L(\varphi1) Can you tell the argument behind the idea that in summing the densities of...
  40. A

    Deriving charge for Noether current in free complex scalar field QFT

    Homework Statement Hi I a attempting to derive the expression for the conserved Noether charge for a free complex scalar field. The question I have to complete is: " show, by using the mode expansions for the free complex scalar field, that the conserved Noether charge (corresponding to complex...
  41. M

    Wick rotation, scalar field and invariants

    One point about Wick rotation is puzzling me and I can not find explanations in books. It concerns the invariants formed from scalar product and solutions to equation. So I will expose my way of reasoning to let you see if it is correct and at the end ask more specific questions. Let's start...
  42. Y

    Divergence Theorem: Multiplied by Scalar Field

    Homework Statement Homework Equations Definitely related to the divergence theorem (we're working on it): The Attempt at a Solution I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...
  43. I

    Transformation properties of derivative of a scalar field

    Hi all, I'm a part III student and taking the QFT course. The following seems "trivial" but when I went and asked the lecturer, the comment was that they too hate such nitty gritty details! The problem is page 12 of Tong's notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf All...
  44. A

    Mass dimension of a scalar field in two dimensions?

    Which is the mass dimension of a scalar filed in 2 dimensions? In 4 dim I know that a scalar field has mass dimension 1, by imposing that the action has dim 0: S=\int d^4 x \partial_{\mu} A \partial^{\mu} A where \left[S\right]=0 \left[d^4 x \right] =-4 \left[ \partial_{\mu} \right]=1...
  45. A

    Is Vacuum Energy of a Free Scalar Field Zero?

    Homework Statement I have the following task: In quantum free scalar field theory find commutators of creation and anihilation operators with total four-momentum operator, starting with commutators for fields and canonical momenta. Show that vacuum energy is zero. Homework Equations...
  46. E

    Noether current for SO(N) invariant scalar field theory

    Homework Statement I understand the premise of Noether's theorem, and I've read over it in as many online lectures as I can find as well as in An Introduction to Quantum Field Theory; Peskin, Schroeder but I can't seem to figure out how to actually calculate it. I feel like I'm missing a...
  47. M

    Poisson Equation for a Scalar Field

    We all know that for the gravitational field we can write the Poisson Equation: \nabla^2\phi=-4\pi G\rho But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as r^{-2}. Here is the thing. When you deal with gravity, the Poisson equation is...
  48. C

    Is \(\nabla \times (\phi \nabla \phi) = 0\) for a Differentiable Scalar Field?

    How to prove that \nabla x (\phi\nabla\phi) = 0? (\phi is a differentiable scalar field) I'm a bit confused by this "differentiable scalar field" thing...
  49. A

    How Does a Line Integral of a Scalar Field Differ from a Regular Integral?

    Okay this might be a nooby question, but it bothers me. What is the difference between the line integral of a scalar field and just a regular integral over the scalar field? For a function of one variable i certainly can't see the difference. But then I thought they might be identical in...
  50. H

    Is My Gradient Solution for a Scalar Field Correct?

    Homework Statement Consider the scalar field V = r^n , n ≠ 0 expressed in spherical coordinates. Find it's gradient \nabla V in a.) cartesian coordinates b.) spherical coordinates Homework Equations cartesian version: \nabla V = \frac{\partial V}{\partial x}\hat{x} +...
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