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. E

    B Understanding the active/passive transformation of a scalar field

    ##\mathcal{P}## is a point in Minkowski spacetime ##M##, and ##\varphi_1: U \in M \mapsto \mathbb{R}^4## and ##\varphi_2: U \in M \mapsto \mathbb{R}^4## are two coordinate systems on the spacetime. A scalar field is a function ##\Phi(\mathcal{P}): M \mapsto \mathbb{R}##, and we can define...
  2. W

    I Renormalization of scalar field theory

    I was reading about the renormalization of ##\phi^4## theory and it was mentioned that in order to renormalize the 2-point function ##\Gamma^{(2)}(p)## we add the counterterm : \delta \mathcal{L}_1 = -\dfrac{gm^2}{32\pi \epsilon^2}\phi^2 to the Lagrangian, which should give rise to a...
  3. Hiero

    I Scalar powers of a matrix exponential

    Starting from the definition of a matrix exponential as a power series, how would we show that ##(e^A)^n=e^{nA}##? I know how to show that if A and B commute then ##e^Ae^B = e^{A+B}## and from this we can show that the first identity is true for integer values of n, but how can we show it’s...
  4. R

    Finding Scalar Curl and Divergence from a Picture of Vector Field

    For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...
  5. E

    B Gradient of scalar field is zero everywhere given boundary conditions

    I'm struggling with a few steps of this argument. It's given that we have a surface ##S## bounding a volume ##V##, and a scalar field ##\phi## such that ##\nabla^2 \phi = 0## everywhere inside ##S##, and that ##\nabla \phi## is orthogonal to ##S## at all points on the surface. They say this is...
  6. G

    Why is the velocity (vector) the same value as the speed (scalar)?

  7. P

    B Derivative of a constant scalar field at a point

    Wikipedia defines the derivative of a scalar field, at a point, as the cotangent vector of the field at that point. In particular; The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each...
  8. E

    Nonlinear scalar equation in field theory

    Thank you very much
  9. twilder

    I Scalar product of biharmonic friction with velocity components

    I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is $$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2 $$ where ##\underline u = (u,v)## and ##A## is a...
  10. G

    A Validity of Scalar Field Lagrangian with Linear and Quadratic Terms

    Hi, if I want to construct the most general Lagrangian of a single scalar field up to two fields and two derivatives, I usually see that is $$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ i.e. the Klein-Gordon Lagrangian. My question is, would be valid the Lagrangian $$\mathscr{L} = \phi...
  11. JD_PM

    A Commutation relations between HO operators | QFT; free scalar field

    I am getting started in applying the quantization of the harmonic oscillator to the free scalar field. After studying section 2.2. of Tong Lecture notes (I attach the PDF, which comes from 2.Canonical quantization here https://www.damtp.cam.ac.uk/user/tong/qft.html), I went through my notes...
  12. Eclair_de_XII

    How to prove that a scalar multiple of a continuous function is continuous

    Suppose ##\alpha=0##. Then ##\alpha f=0##, the zero map. Hence, the distance between the images of any two ##x_1,x_2 \in D## through ##f##, that is to say, the absolute difference of ##(\alpha f)(x_1)=0## and ##(\alpha f)(x_2)=0##, is less than any ##\epsilon>0## regardless of the choice of...
  13. K

    I Scalar Hamiltonian and electromagnetic transitions

    Hello! This is probably a silly question (I am sure I am missing something basic), but I am not sure I understand how a Hamiltonian can be a scalar and allow transitions between states with different angular momentum at the same time. Electromagnetic induced transitions are usually represented...
  14. W

    I Propagator of a Scalar Field via Path Integrals

    I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...
  15. andylatham82

    B What does the scalar product of two displacements represent?

    Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...
  16. PeroK

    I What do we do with the massive scalar quantum field in QFT?

    I'm learning some QFT from QFT for the Gifted Amateur. Chapter 11 develops the massive scalar quantum field but they don't seem subsequently to do anything with it. I've looked ahead at the next few chapters, which move on to other stuff, which leaves me wondering what we we actually do with...
  17. M

    Wick contraction in scalar QED

    While writing out the Dyson series due to the time ordering above I encountered the two expressions $$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$ I was able to write out the first term in terms of contractions using Wick's theorem and then finally...
  18. Athenian

    [Special Relativity] Scalar Invariant under a Lorentz-transformation

    "My" Attempted Solution To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation. With that said, below is my...
  19. T

    A Numerically Solving Scalar Propagation in Curved Spacetime

    Hey everybody, Background: I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric. The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0): $$ \partial_t\phi =...
  20. P

    A Einstein Tensor and Stress Energy Tensor of Scalar Field

    Hi All. Given that we may write And that the Stress-Energy Tensor of a Scalar Field may be written as; These two Equations seem to have a similar form. Is this what would be expected or is it just coincidence? Thanks in advance
  21. M

    Momentum Operator for the real scalar field

    I think the solution to this problem is a straightforward calculation and I think I was able to make reasonable progress, but I'm not sure how to finish this... $$\begin{align*} \vec{P}&=-\int dx^3 \pi \nabla \phi\\ &= -\int\int\int dx^3\frac{dp^3}{(2\pi)^3 2e(p)} \frac{du^3}{(2\pi)^3}...
  22. F

    I Scattering of a scalar particle and a Fermion

    Hello everyone, I am working on the following problem: I would like to determine the invariant Matrix element of the process ##\psi\left(p,s\right)+\phi\left(k\right)\rightarrow\psi\left(p',s'\right)+\phi\left(k'\right)## within Yukawa theory, where ##\psi\left(p,s\right)## denotes a fermion...
  23. N

    Vector and scalar potentials for an EM plane wave in a vacuum

    Lorentz gauge: ∇⋅A = -μ0ε0∂V/∂t Gauss's law: -∇2V + μ0ε0∂2V/∂t2 = ρ/ε0 Ampere-Maxwell equation: -∇2A + μ0ε0∂2A/∂t2 = μ0J I started with the hint, E = -∇V - ∂A/∂t and set V = 0, and ended up with E0 ei(kz-ωt) x_hat = - ∂A/∂t mult. both sides by ∂t then integrate to get A = -i(E0/ω)ei(kz-ωt)...
  24. E

    I Textbook gives the gradient of a scalar as a scalar

    Background: I am currently reading up on homogenization theory. I have a simple conductivity model (image attached). u is a scalar function (such as potential or temperature). The textbook proceeds by giving a series expansion for the gradient of u (image attached). the problem is that the...
  25. R

    B Energy as a non relativistic scalar and Galilean invariance

    Summary: Why is there no contradiction between energy as a non relativistic scalar and Galilean invariance? If energy is a non relativistic scalar, doesn't it mean that there is a contradiction with Galilean invariance? What i mean is that if i try to accelerate an object within the Galilean...
  26. Milsomonk

    Did I correctly set up the two loop scalar integral?

  27. J

    MHB Planes 1 & 2 Intersect: Find Scalar Equations

    Two planes, plane 1 and plane 2, intersect in the line with symmetric equation (x-1)/2 = (y-2)/3 = (z+4)/1. Plane 1 contains the point A(2,1,1) and plane 2 contains the point B(1,2,-1). Find the scalar equations of planes plane 1 and plane 2. I have no idea how to do it, all help will be...
  28. K

    I Is the Delta Function a Scalar?

    I read that ##\delta^4(x-y)## is invariant under Lorentz transformations. I was trying to show myself this, so I procceded as follows. The following integrals are both equal to 1 ##\int \delta^4(x-y) d^4 x## and ##\int \delta^4 (x'-y') d^4x'## so I assume they are equal to one another, as long...
  29. O

    A Verifying the Relation in Yang-Mills Theory with a Scalar Field

    I'm trying yo verify the relation \begin{equation} [D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi, \end{equation} where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here, \begin{equation} D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi], \end{equation} and \begin{equation}...
  30. nineteen

    Why is Kinetic Energy a scalar quantity?

    Why is Kinetic energy a scalar quantity? I read in an article, it said, when the velocity is squared, it is not a vector quantity anymore. Can someone fill in the gaps for me? I can't quite get what that article said. And I would be pleased if you provide some other examples other than kinetic...
  31. A

    What is Current? I know it is a scalar but I found something weird....

    While I was going through "Introduction to Electrodynamics" by David J. Griffith I see the line "Current is a vector quantity". But we know it doesn't obey the vector algebra (addition ). Then how it can be a vector?... Please help me
  32. F

    A Classical scalar field as Dark Matter

    The pressure of a scalar field is: Φ˙2−V(Φ) so to have zero or negligeable pressure it needs to have equipartition of its energy in potential and kinetic form ==> the potential must be positive. In particular a mass term m2Φ2 ... could be all right: the field should tend to roll down this...
  33. E

    Altitude - Why is it a Scalar?

    Homework Statement How come altitude of a mountain is a scalar? Homework Equations Scalars = only magnitude Vectors = have magnitude & direction The Attempt at a Solution - Doesn't altitude of a mountain have both magnitude and direction (direction being measured straight up 90 degrees to the...
  34. E

    Why does Celsius temperature in degrees have +/- signs, since it's scalar?

    Why does Celsius degrees have +/- signs, since it's scalar?
  35. B

    Can Scalar Fields Be Decomposed Similar to Vector Fields?

    If a vector field can be decomposed into a curl field and a gradient field, is there a similar decomposition for scalar fields, say into a divergence field plus some other scalar field?
  36. hnnhcmmngs

    Vectors and scalar projections

    Homework Statement Let a and b be non-zero vectors in space. Determine comp a (a × b). Homework Equations comp a (b) = (a ⋅ b)/|a| The Attempt at a Solution [/B] comp a (a × b) = a ⋅ (a × b)/|a| = (a × a) ⋅b/|a| = 0 ⋅ b/|a| = 0 Is this the answer? Or is there more to it?
  37. A

    B Is the Space with Ricci Scalar Zero Flat?

    Given metric ds2=dr2-r2dθ2 Gamma comes as Γ122=r,Γ212=Γ221=1/r The Reimann tensor comes as R11=R2121=∂1Γ212-Γm12Γ21m=0,only non zero terms . Similary R22=R1212=∂1Γ122-Γm21Γ1m2=0,only non zero terms. Therefore R(ricci scaler)=0 Is the space flat??
  38. George Keeling

    I Why are scalars and dual vectors 0- and 1-forms?

    I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms." Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1)...
  39. Specter

    Find the scalar, vector, and parametric equations of a plane

    Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5) Homework EquationsThe Attempt at a Solution Finding the scalar equation: Ax+By+Cz+D=0 3x-4y+6z+D=0 3(9)-4(2)+6(-5)+D=0 -11+D=0 D=11...
  40. Safinaz

    Differentiating a scalar potential

    Hello, I have this potential: ## V(\chi) = \frac { F ’’ (\chi) [ 2 F(\chi) - \chi F’ (\chi) ]}{ (F’(\chi))^3} ## How to get ## \frac{ d V(\chi)}{d \chi} = \frac{ \chi F’’ + F’ - F’ }{ F’^2} - 2 \frac{ \chi F’ -F }{ F’^3} F’’ ~~~~~~(*)## My trail, ## V( \chi) = 2 F F’’ F’^{-3} - \chi F’’...
  41. E

    I Lorentz invariance and equation of motion for a scalar field

    Hi there, I just saw some lectures where they claim that the Klein Gordon equation is the lowest order equation which is Lorentz invariant for a scalar field. But I could easily come up with a Lorentz invariant equation that is first order, e.g. $$ (M^\mu\partial_\mu + m^2)\phi=0 $$ where M is...
  42. archaic

    B Dot product scalar distributivity

    I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...
  43. S

    I Feynman Rules for Scalar QED: LRZ Reading

    Hello! Can someone direct me towards a reading where the Feynman rules for scalar QED are derived? Thank you!
  44. prashantakerkar

    B Equilibrium - Scalar or Vector?

    1 Is Equilibrium a Scalar or Vector quantity? 2 What is the unit of Equilibrium? Thanks & Regards, Prashant S Akerkar
  45. B

    Is the Energy Momentum Tensor for Scalar Fields Always Symmetric?

    Homework Statement Show that if the Lagrangian only depends on scalar fields ##\phi##, the energy momentum tensor is always symmetric: ##T_{\mu\nu}=T_{\nu\mu}## Homework Equations ##T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L## The Attempt at a...
  46. sams

    I Do we consider a point in a coordinate system to be a scalar?

    Knowing that a scalar quantity doesn't change under rotation of a coordinate system. Do we consider a point in a Cartesian coordinate system (i.e. A (4,5)) a scalar quantity? If yes, why do the components of point A change under rotation of the coordinate system? According to my understanding...
  47. stevendaryl

    I Nonrelativistic limit of scalar field theory

    The Klein-Gordon equation has the Schrodinger equation as a nonrelativistic limit, in the following sense: Start with the Klein-Gordon equation (for a complex function ##\phi##) ## \partial_\mu \partial^\mu \phi + m^2 \phi = 0## Now, define a new function ##\psi## via: ##\psi = e^{i m t}...
  48. H

    Find the scalar value and direction of the electric field

    Homework Statement Two charged balls are placed in point A and B and the distance between them is 9,54cm. Each of the balls are charged with 8,0 x 10^-8 C. Find the scalar value and direction of the electric field in point C placed 5 cm from A and 6 cm from B. Homework Equations Cosine Rule...
  49. Q

    A Restrictions Placed on a Scalar Field by the Vacuum

    Hi Everyone! I have been told that even for an entirely LOCAL scalar field φ with Lagrangian density say of the form, L = ∂/∂xμ∂/∂xμφ ± φ4 + Aφ3 + Bφ2. +. Cφ + D, that it is really bad, bad, bad because the coefficient (C) of φ is not zero! That is, ∂/∂φ(L) ≠ 0 when φ...
  50. T

    Is temperature considered a scalar quantity?

    I was going through vector and scalar quantities (the way they are taught in high school), and this is how I think students are supposed to understand it: Scalar quantities are quantities that add like numbers. For e.g. Mass. If I add 100 g of water to a bucket and then add a further 100 g, I...
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