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. P

    A How the mass term of the Hamiltonian for a scalar fields transform?

    The Hamiltonian for a scalar field contains the term $$\int d^3x m^2 \phi(x) \phi(x)$$, does it changs to the following form? $$\int d^3x' {m'}^2 \phi'(x') \phi'(x')=\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$? As it is well known for a scalar field: $$\phi'(x')=\phi(x)$$ .
  2. 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?
  3. Hill

    Second order diagram for the "scalar graviton"

    It has been shown in the text that ##h_0 = \frac 1 {\Box} J## with the diagram and that ##h_1 = \lambda \frac 1 {\Box} (h_0 h_0) = \lambda \frac 1 {\Box} [( \frac 1 {\Box} J)( \frac 1 {\Box}J)]## with the diagram I was not sure if the next order diagram is or rather Thus, I substitute...

    A Scalar Field Dynamics in Inflation

    I am facing a problem while wanting ##\phi## dynamics in a cubic potential; ##g\phi^{3}##. The equation of motion I get for my case is(this follows from the usual Euler-Lagrange equations for ##\phi## in cosmology--Briefly discussed in Carol's Spacetime Geometry, inflation chapter)...
  5. T

    I Prefactor in Canonical Quantization of Scalar Field

    Hey all, I am encountering an issue reconciling the choice of prefactors in the canonical quantization of the scalar field between Srednicki and Peskin's books. In Peskin's book (see equation (2.47)), there is a prefactor of ##\frac{1}{\sqrt{2E_{p}}}## whereas in Srednicki's book (see equation...
  6. Tertius

    I Can a scalar field model account for the cosmic redshift?

    A minimally coupled scalar field can model a cosmological fluid model where And where the equation of state can be the standard ## \omega = \frac {p} {\rho}## I can see how this does a fine job modeling matter, because as the scale factor increases, the density will go as ##\frac {1} {a^3}##...
  7. S

    I Noether currents for a complex scalar field and a Fermion field

    For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...
  8. G

    A Scalar decay to one-loop in Yukawa interaction

    I am trying to calculate the amplitude for a decay ##\phi \to e^+e^-## under a Yukawa interaction ##\mathcal{L}_I = -g\phi \bar{\psi}\psi## to one-loop order (with massless fermions for simplicity). If I'm not wrong, there are 4 diagrams that contribute to 1 loop, three diagrams involving...
  9. P

    I Question about implication from scalar product

    Hi, Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have...
  10. heroslayer99

    Suvat vector versus the scalar form

    Hi I was just wondering about the suvat formulae and a question popped into my head, which I'd like someone to try and explain the reason as to why please. So I know that when we have a formula such as F=ma or v = u + at, you can evaluate the magnitude of both sides and arrive at a scalar...
  11. C

    Difference between scalar and cross product

    Hi! For example, how do you tell whether to use the scalar or cross product for an problem such as, However, I do know that instantaneous angular momentum = cross product of the instantaneous position vector and instantaneous momentum. However, what about if I didn't know whether I'm meant to...
  12. H

    I Plane wave decomposition method in scalar optics

    Suppose an optical scalar wave traveling in Z direction. Using the diffraction theory of Fourier Optics, we can predict its new distribution after a distance Z. The core idea of Fourier Optics is to decompose a scalar wave into plane waves traveling in different directions. But this...
  13. guyvsdcsniper

    Evaluating scalar products of two functions

    I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx## The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) =...
  14. Onyx

    B Sign of Expansion Scalar in Expanding FLRW Universe

    Considering the FLWR metric in cartesian coordinates: ##ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)## With ##a(t)=t##, the trace of the extrinsic curvature tensor is ##-3t##. But why is it negative if it's describing an expanding universe, not a contracting one?
  15. StenEdeback

    I Best book for Lagrangian of classical, scalar, relativistic field?

    Hi all experts! I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck
  16. L

    A Vector analysis question. Laplacian of scalar and vector field

    If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
  17. T

    I What is the relationship between force and potential in particle interactions?

    Suppose I have some interaction potential, u(r), between two repelling particles. We will name them particles 1 and 2. I want to find the force vectors F_12 and F_21. Would I be correct in saying that the x-component of F_12 would be given by -du/dx, y-component -du/dy etc? And to find the...
  18. Tertius

    A Local phase invariance of complex scalar field in curved spacetime

    I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...
  19. E

    I Schwartz derivation of the Feynman rules for scalar fields

    Hi everyone, In his book "Quantum field theory and the standard model", Schwartz derives the position-space Feynman rules starting from the Schwinger-Dyson formula (section 7.1.1). I have two questions about his derivation. 1) As a first step, he rewrites the correlation function as $$...
  20. R

    I Why is momentum considered a vector and kinetic energy a scalar?

    I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...
  21. E

    I Commutation relations for an interacting scalar field

    Hi there, In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states: I do not understand this explanation. Can you please...
  22. H

    I Is the scalar magnetic Potential the sum of #V_{in}# and ##V_{out}##

    Hi, I'm wondering if I have an expression for the scalar magnetic potential (V_in) and (V_out) inside and outside a magnetic cylinder and the potential is continue everywhere, which mean ##V^1 - V^2 = 0## at the boundary. Does it means that ##V^1 - V^2 = V_{in} - V_{out} = 0## ?
  23. Soony143

    A Numerically solving Scalar field coupled to Friedman equation

    I am a research student of MS PHYSICS. I have to numerically solve Friedman equation coupled to scalar field(phi). It is given in research paper of Sean Carroll, Mark Trodden and Hoffman entitled as ""can the dark energy equation of state parameter w be less than-1?""...
  24. A

    Green's theorem with a scalar function

    Greetings! My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!
  25. A

    Showing that the gradient of a scalar field is a covariant vector

    In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...
  26. T

    I Unruh Effect (1+1)D: Understanding Equation 5.68

    Hi all, I am trying to work through the Unruh Effect for the (1+1)-dimensional massive scalar field case and came across the paper I attached. However, I am trying to derive equation 5.68, but am greatly struggling with the prefactor on the left hand side. When comparing the left hand side to...
  27. T

    A Covariant Derivative of Stress Energy Tensor of Scalar Field on Shell

    Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.
  28. U

    Engineering Path integrals in scalar fields when the path is not provided

    I cannot seem to start answering the question as a result of the path not being provided. How do I solve this when the path is not provided? See picture below
  29. brotherbobby

    Line integral of a scalar function about a quadrant

    Problem : We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using (plane) polar coordinates. (1) In (plane)...
  30. M

    Feynman rules and the tree level cross section of two scalar fields

    Hi there. I'm trying to solve the problem mentioned above, the thing is I'm truly lost and I don't know how to start solving this problem. Sorry if I don't have a concrete attempt at a solution. How do I derive the Feynman rules for this Lagrangian? What I think happens is that in momentum...
  31. Tertius

    A Difference Between Scalar Field Solutions in Curved Spacetime

    A general free field Lagrangian in curved spacetime (- + + +), is given by: L = -1/2 ∇cΦ ∇cΦ - V(Φ) when the derivative index is lowered, we obtain: L = -1/2 gdc∇dΦ ∇cΦ - V(Φ) then we can choose to replace V(Φ) with something like 1/2 b2 Φ2 so: L = -1/2 gdc∇dΦ ∇cΦ - 1/2 b2 Φ2 ** I will...
  32. D

    I Scalar product and generalised coordinates

    Hi If i have 2 general vectors written in Cartesian coordinates then the scalar product a.b can be written as aibi because the basis vectors are an orthonormal basis. In Hamiltonian mechanics i have seen the Hamiltonian written as H = pivi - L where L is the lagrangian and v is the time...
  33. fenyutanchan

    How Does Scalar QCD Beta Function Calculation Differ in Various Sources?

    I have calculated $Z$s as $$ \begin{aligned} Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\ Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\ Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} +...
  34. S

    Constants in scalar and vector potentials

    We have a scalar potential $$\Phi(\vec{r})=\frac{q}{4\pi\epsilon_0} \left( \frac{1}{r} - \frac{a^2\gamma e^{-\gamma t}\cos\theta}{r^3}\right)$$ and a vector potential $$\vec{A}(\vec{r})=\frac{a^2qe^{-\gamma t}}{4\pi\epsilon_0r^4}\left(3\cos\theta\hat{r} + \sin\theta\hat{\theta} \right) .$$ how...
  35. Arman777

    Python Using np.einsum to calculate Ricci scalar

    I was trying to calculate $$R = g^{ij}R_{ij}$$ bu using einsum but I couldn't not work it out. Anyone can help me ? Here are some of the resources https://stackoverflow.com/questions/26089893/understanding-numpys-einsum...
  36. U

    I Why is Scalar Massless Wave Equation Conformally Invariant?

    It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?
  37. G

    I Feynman diagram for scalar - vector interaction

    The term for the electromagnetic interaction of a Fermion is ##g \bar{\Psi} \gamma_\mu \Psi A^\mu##, where ##g## is a dimensionless coupling constant, ##\Psi## is the wave function of the Fermion, ##\gamma## are the gamma matrices and ##A## is the electromagnetic field. One can quite simply see...
  38. E

    A Prove that the Kruskal solution is stable to scalar field pertubations

    Homework Statement:: The solution to the KG equation is assumed to take the form$$\Phi = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{1}{r} \phi_{lm}(t,r) Y_{lm}(\theta, \phi)$$ Relevant Equations:: N/A To first show that $$\left[ \frac{\partial^2}{\partial t^2} - \frac{d^2}{dr_*^2} +...
  39. JD_PM

    Varying an action with respect to a scalar field

    Let us work with ##(-+++)## signature Where the metric ##g_{\mu \nu}## is the flat version (i.e. ##K=0##) of the Robertson–Walker metric (I personally liked how Weinberg derived it in his Cosmology book, section 1.1) \begin{equation*} (ds)^2 = -(dt)^2 + a^2(t) (d \vec x)^2 \end{equation*}...
  40. JD_PM

    A Renormalizability conditions for a real scalar field in d dimensions

    I am studying the real scalar field theory in ##d## spacetime dimensions as beautifully presented by M. Srednicki QFT's draft book, chapter 18 (actually, for the sake of simplicity, let us include polynomial interactions of degree less than or equal to 6 only) \begin{equation*} \mathcal{L}...
  41. J

    I Ricci scalar for FRW metric with lapse function

    I need the Ricci scalar for the FRW metric with a general lapse function ##N##: $$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$ Could someone put this into Mathematica as I don't have it?
  42. George Keeling

    I Is Scale Factor a Scalar? Sean Carroll Invitation

    Is the scale factor a scalar? I think that the answer is no but I want to check because god (or the universe) has been playing tricks on me... At Sean Carroll's invitation I wanted to check that the tensor$$ K_{\mu\nu}=a^2\left(g_{\mu\nu}+U_\mu U_\nu\right) $$was a Killing tensor...
  43. E

    I Classical equivalent of scalar free field in QFT

    Hi there, In QFT, a free scalar field can be represented by the lagrangian density $$\mathcal{L} = \frac{1}{2}\left(\partial\phi\right)^2 - \frac{1}{2}m^2\phi^2$$ I would like to find a classical system that has the same lagrangian. If we consider the transversal motion of an elastic string...
  44. F

    Lagrangian for the electromagnetic field coupled to a scalar field

    It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...
  45. Ntip

    Vector Fields vs Scalar Fields

    I am looking at antenna theory and just came upon scalar fields. I found an site giving an example of a scalar field as measuring the temperature in a pan on a stove with a small layer of water. The temperature away from the heat source will be cooler than near it but it doesn't have a...
  46. Delta2

    The forgotten magnetic scalar potential

    I wonder if there is any book that discusses the possibility of existence of a magnetic scalar potential. That is a scalar potential ##\chi## such that $$\vec{B}=\nabla\chi+\nabla\times\vec{A}$$. From Gauss's law for the magnetic field B we can conclude that it will always satisfy laplace's...
  47. Arman777

    I EDE - Solving the Klein - Gordon Equation for a scalar field

    Let us suppose we have a scalar field ##\phi##. The Klein-Gordon equations for the field can be written as \begin{equation} \ddot{\phi} + 3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0 \end{equation} The other two are the Friedmann equations written in terms of the ##\phi## \begin{equation} H^2 =...
  48. J

    I Variation of Ricci scalar wrt derivative of metric

    I understand from the wiki entry on the Einstein-Hilbert action that: $$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$ What is the following? $$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$ Is there a place I could look up such GR expressions on the internet? Thanks