# 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.

View More On Wikipedia.org
1. ### 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. ### 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. ### 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...
4. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

35. ### 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...
36. ### 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...
37. ### 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?
38. ### 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...
39. E