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

    I A directional, partial derivative of a scalar product?

    Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...
  2. ubergewehr273

    B Confusion between ##\theta## and ##d\theta##

    Why is ##\theta## a scalar whereas ##d\theta## a vector ?
  3. H

    A scalar on a semi-infinite domain with source and sink

    Hi everyone, I've been looking at a problem that seems simple at first, but appears to be deceptively difficult (unless I'm missing something). 1. Homework Statement I've been looking at a problem that involves the diffusion of a scalar quantity, ##q(x)##, on the semi-infinite domain, ##\leq...
  4. M

    A Why are open strings vectors or scalars, or massive?

    In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...
  5. P

    B Is Center of Mass a vector or scalar quantity?

    I am a little bit confused on Center of Mass and Center of Gravity about what are they vectors or scalars . As I may think they are neither because they are simply two points . Am I saying right or wrong .
  6. T

    I Scalar quantities and complex numbers

    I was taught a scalar is a quantity that consists of a number (positive or negative) and it might include a measuring unit, e.g. 6, 5 kg, -900 J, etc. I was wondering if complex numbers like 3 + 7j (where j is the square root of minus 1) can be considered scalar quantities too, or is it that...
  7. S

    A Possible decay process for a cubic scalar self-interaction

    Consider the Lagrangian $$\mathcal{L}=\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}m^{2}h^{2}-\frac{\lambda}{3!}h^{3}$$ for a real scalar field ##h##. This is the Klein-Gordon Lagrangian with a cubic self-interaction term. Does this model allow the decay process $$h \rightarrow h +...
  8. rezkyputra

    Covariant Derivatives (1st, 2nd) of a Scalar Field

    Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...
  9. P

    Induced Magnetic Moment (vector) vs. Induced EMF (scalar)

    When I induce magnetic flux through a closed loop, I should expect the lines of flux produced by current in that loop to oppose the change of flux through that loop. But what happens when that loop, say a rectangular loop, is curved into the shape of the letter J (like a candy cane) and my flux...
  10. S

    A Factors in the theory of a complex scalar field

    The theory of a complex scalar field ##\chi## is given by $$\mathcal{L}=\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi.$$ Why is it not common to include a factor of ##\frac{1}{2}## in front of the complex ##\chi## kinetic term? What is the effect on the propagator of...
  11. binbagsss

    Real Scalar Field, Hamiltonian, Conjugate Momentum

    ## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z## In order to compute the Hamiltonian I first of all need to compute...
  12. binbagsss

    Real scalar field , Action, variation, deriving EoM

    ## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density. ## S= \int d^{4}(x) L (x) ##, the action. ## \phi -> \phi + \delta \phi ## (just shortened the notation and dropped the x dependence) I have ##...
  13. R

    MHB Proving Positive Definite Scalar Product for $n \times n$ Matrices

    Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product: $$\langle X, Y \rangle = tr(X Y^T),$$ and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are (1) $\langle X, X\rangle \geq 0$ and (2)...
  14. Y

    I Is angle a vector or a scalar quantity?

    Is angle a vector or a scalar quantity? (eg- θ in l=rθ)
  15. donaldparida

    Why are pressure and current scalar?

    I know that a vector quantity or simply a vector is a physical quantity which has a magnitude and is associated with some definite direction. According to this definition should not pressure and current be vectors since both are associated with some definite direction?In some places they are...
  16. K

    B Can Newton's G and the "Einstein Scalar" Be Related?

    Is it possible to take the "Eistein Scalar" from the Einstein Tensor, like one can take the Ricci Scalar from the Ricci Tensor? If so, is the G of Newton's law the same as this "Einstein Scalar" or is it just the same symbol used in very different things. (Sorry for my bad English. I think...
  17. ATY

    I Complex conjugation in scalar product?

    Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...
  18. C

    Grad of a Scalar Field: Computing ∇T in Spherical Coordinates

    Homework Statement Let T(r) be a scalar field. Show that, in spherical coordinates ∇T = (∂T/∂r) rˆ + (1/r)(∂T/∂θ) θˆ + (1/(r*sin(θ)))(∂T/∂φ) φˆ Hint. Compute T(r+dl)−T(r) = T(r+dr, θ+dθ, φ+dφ)−T(r, θ, φ) in two different ways for the infinitesimal displacement dl = dr rˆ + rdθ θˆ + r*sin(θ)dφ...
  19. S

    A Scalar mass and quantum corrections

    What exactly are short-distance quantum corrections? Why is the mass term of a fundamental scalar field highly sensitive to short-term quantum corrections?
  20. S

    A Phi-nth quantum scalar field theories where n is not integer

    Consider a quantum scalar field theory with interaction terms of the form ##\phi^{n}##, where ##n## is not an integer. Where are some examples of physical theories which involve such interaction terms?
  21. S

    Conservation of Noether charge for complex scalar field

    Homework Statement Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time. Homework Equations ##\pi=\dot{\phi}^{*}## The Attempt at a Solution...
  22. S

    A Generalising the Euler-Lagrange equation for scalar fields

    The Euler-Lagrange equation obtained from the action ##S=\int\ d^{4}x\ \mathcal{L}(\phi,\partial_{\mu}\phi)## is ##\frac{\partial\mathcal{L}}{\partial\phi}-\partial_{\mu}\big(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\big)=0##. My goal is to generalise the Euler-Lagrange equation...
  23. S

    A Complex scalar field - commutation relations

    I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...
  24. F

    B How do we know if something is vector or scalar quantity?

    I am well-versed with the definition of scalar and vector quantities.The confusion I mainly have is at many points, my textbook makes ambiguous statements like "Because force is vector quantity it follows that field strength is also a vector quantity." What relationship should arbitrary...
  25. O

    I Vector to scalar potential, transformation of fields

    Hey guys. So, as i was going through Griffith's Electrodynamics, and i came across this problem: In the solutions: How to they actually get to that expression for V = (V(bar)+vAx(bar) )Ɣ? I understand everything after that, but this just made me very confused. How do they get this from the...
  26. T

    Linear Algebra, subset of R2 not closed under scalar multipl

    Homework Statement Construct a subset of the x-y plane R2 that is (a) closed under vector addition and subtraction, but not scalar multiplication. Hint: Starting with u and v, add and subtract for (a). Try cu and cv Homework Equations vector addition, subtraction and multiplication The...
  27. V

    I Why are scalar fields Lorentz invariant?

    Hi. This question most probably shows my lack of understanding on the topic: why are scalar fields Lorentz invariant? Imagine a field T(x) [x is a vector; I just don't know how to write it, sorry] that tells us the temperature in each point of a room. We make a rotation in the room and now...
  28. S

    I Questions about gradient and scalar product

    I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...
  29. Mr-R

    A Landau Lifshitz Gravitational field equation

    Book: Landau Lifshitz, The Classical Theorey of Fields, chapter 11, section 95. I have gone through the derivation of Einstein field equations but not without holes to fill and fix in my understanding. Let's start with the action for the grtavitational field ##S_g## which after some explanation...
  30. D

    A A question about the mode expansion of a free scalar field

    In the canonical quantisation of a free scalar field ##\phi## one typical constructs a mode expansion of the corresponding field operator ##\hat{\phi}## as a solution to the Klein-Gordon equation...
  31. D

    I Exploring the Properties of Scalar Product & Law of Cosines

    Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##. With the construct of a triangle, the Law of Cosines is proved. ##\mathbf A## points to the tail of ##\mathbf B##. Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...
  32. F

    B Explaining vector & scalar quantities to a layman

    I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts: A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence...
  33. C

    Line integral of scalar field ( piecewise curve)

    Homework Statement for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ? I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ? Homework EquationsThe Attempt at a Solution is...
  34. H

    I Is Force Truly a Scalar in Rope Friction Problems?

    Consider a rope wraps an angle ##\theta## around a pole with a coefficient of static friction ##\mu##. You pull one end with a tension ##T_0##. The force that the other end can support is given by ##T=T_0e^{\mu\theta}##(derivation below). For ##\mu=1## and ##\theta=2\pi##, ##T=530T_0##. That...
  35. C

    Time derivative of a time-dependent vector and scalar

    Homework Statement ## \frac{d}{dt}\gamma(t)\vec{u(t)} ## Homework Equations See above The Attempt at a Solution This comes from trying to verify a claim in Chapter 12 of Griffiths Electrodynamics, 4th. edition (specifically Eq. 12.62 -> Eq. 12.63, if anyone has it on hand). I would have...
  36. I

    Taylor Expansion For Scalar Field

    Homework Statement Page 35 of Jackson's Electrodynamics (3rd ed), it gives the following equation (basically trying to prove your standard 1/r potential is a solution to Poisson equation): \nabla^2 \Phi_a = \frac{ -1 }{ \epsilon_0 } \int \frac{ a^2 }{( r^2 + a^2)^{5/2} } \rho( \boldsymbol{x'}...
  37. R

    Magnetic Flux Vectors: How Are Flux Scalar?

    how is flux scalar if the magnetic field and area both are vector quantities?
  38. carllacan

    I Why can't the real scalar field and the EM be coupled?

    According to David Tong's notes the real scalar field can't be coupled to the electromagnetic field because it doesn't have any "suitable" conserved currents. What does "suitable" mean? The real field does have conserved currents, why aren't those suitable?
  39. P

    Short Aerial: Find the scalar potential (retarded sources)

    Homework Statement Determine the vector potential due to a short wire running from (−L/2, 0, 0) to (L/2, 0, 0) carrying a current I = I0cos(ωt) (consider only points at distance r >> L from the origin). (You may neglect the effect of the return circuit.) Now determine the corresponding scalar...
  40. K

    I Why Does the Gradient Point Towards the Greatest Increase?

    Hi, I am looking for a proof that explains why gradient is a vector that points to the greatest increase of a scalar function at a given point p. http://math.stackexchange.com/questions/221968/why-must-the-gradient-vector-always-be-directed-in-an-increasing-direction I understand the proof...
  41. I

    What Is the Issue with Scalar Loop Corrections in Non-Abelian SU(N) Theories?

    Hello all, I hope you can give me a hand with a QFT homework I'm working on. We are to compute the beta equation of a Non-abelian SU(N) theory with: Complex scalars (massless), bosons, ghosts. My question is referring to the Boson self-energy scalar loop correction. 1. Homework Statement We...
  42. Calaver

    Scalar Multiple of Vector - Vector & Scalar = 0?

    Note: I am not in the course where this problem is being offered; it was simply an interesting linear algebra "thought question" that I found online to which I believe I have found a solution. However, there is one step in my solution that I am unsure about, so thank you to anyone who spares the...
  43. S

    B Understanding Scalar and Vector Products in Geometric Algebra

    (Scalar)·(Scalar) = Scalar (Scalar)·(Vector) = Scalar (Vector)·(Vector) = Scalar (Scalar)x(Scalar) = Not valid (Scalar)x(Vector) = Vector (Vector)x(Vector) = VectorDid I get them right, if not why? Thanks
  44. Z

    MHB Closure under Scalar multiplication

    Hey guys, I really need help in revising my Axiom 6 for my Linear Algebra course. My professor said, "You need to refine your statement. You want to show rx1 and rx2 are real numbers. You should not state they are real numbers." Here is my work: Proof of Axiom 6: rX is in R2 for X in R2...
  45. C

    A Q: Scalar Boundary Condition & U(1) Isometry - Lewkowycz & Maldacena

    I have a simple question about Lewkowycz and Maldacena's paper http://arxiv.org/abs/1304.4926v2'][/PLAIN] http://arxiv.org/abs/1304.4926v2 In section 2, they consider the scalar field in BTZ background ground and require boundary condition of the scalar field, $\phi \sim e^{i\tau}$ . This...
  46. C

    I Superficial degree of divergence for scalar theories

    I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##) I am trying to understand what ##[g_E]## is here? Since in this set up we are summing over...
  47. prashant singh

    I Why does A.A = ||A||^2 in the scalar product formula?

    Why A.A = ||A||^2 , I know that from product rule we can prove this where theta =0 , I am asking this because I have seen many proves for A.B = ||A||||B||cos(theta) and to prove this they have used A.A = ||A||^2, how can they use this , this is the result of dot product formula. I havee seen...
  48. prashant singh

    I Scalar product and vector product

    why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove...
  49. H

    Finding Equation of Motion Using Scalar Field Lagrangian

    Homework Statement I must find the following equation of motion: φ'' + 3Hφ' + dV/dφ = 0 Using the scalar field Lagrangian: (replace the -1/2m^2φ^2 term with a generic V(φ) term though) with the Euler-Lagrange Equation I know that I must assume φ = φ(t) and the scale factor a = a(t)...
  50. H

    I Deriving Equation of Motion for Quintessence Scalar Field?

    Hello, I am having trouble deriving the equation of motion for the quintessence field. The equation of motion which I am meant to get at the end point is: (with ' denoting derivative w.r.t time) φ'' + 3Hφ' + dV/dφ = 0 Using the inflaton lagrangian: (although with a generic potential V(φ)...
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