Scalar Definition and 829 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. 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...
  2. M

    Feynman Diagrams for Interacting Scalar Fields

    Homework Statement Consider four real massive scalar fields, \phi_1,\phi_2,\phi_3, and \phi_4, with masses M_1,M_2,M_3,M_4. Let these fields be coupled by the interaction lagrangian \mathcal{L}_{int}=\frac{-M_3}{2}\phi_1\phi_{3}^{2}-\frac{M_4}{2}\phi_2\phi_{4}^{2}. Find the scattering amplitude...
  3. G

    I Magnitude of the gradient of a constant scalar field

    Hey! Short definition: A gradient always shows to the highest value of the scalar field. That's why a gradient field is a vector field. But let's assume a constant scalar field f(\vec r) The gradient of f is perpendicular to this given scalar field f. My Questions: 1. Why does the gradient...
  4. J

    Generalized coordinates- scalar product

    Homework Statement a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself. b: What would be the r, θ components of the unit vector in the θ direction? Homework Equations Scalar product of 2 vectors = AαgαβBβ The Attempt at a Solution For part a, I used the...
  5. D

    I Can You Add a Scalar to a Matrix Directly?

    So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...
  6. J

    I How does the Higgs scalar potential evolve with temperature?

    How does the Higgs scalar potential evolve with temperature? Is there possibility they are independent? Besides temperature. What else can theoretically affect the higgs scalar potential?
  7. S

    A Transformation of a scalar field

    I read somewhere that, suppose a scalar field Σ transforms as doublet under both SU(2)L and SU(2)R, its general rotation is δΣ = iεaRTaΣ - iεaLΣTa. where εaR and εaL are infinitesimal parameters, and Ta are SU(2) generators. I don't quite understand this. First, why does the first term have...
  8. e2m2a

    I Vector and Scalar Tensor Invariance

    I am confused about tensor invariance as it applies to velocity and energy. My understanding is a tensor is a mathematical quantity that has the same value for all coordinate systems. I also understand that a vector is a first order tensor and energy is a zero order tensor. Thus, they should...
  9. Spinnor

    I Vector potential A_mu from scalar function theta(x_mu)?

    Suppose we have a scalar function θ(x,y,z,t) of space and time where theta is some angle (0≤θ≤2π) that represents the compact coordinate of a 3 dimensional space (x,y,z) filling membrane at the space time point (x,y,z,t) in a compact space dimension w. Suppose that charge density "pushes" on the...
  10. Y

    Scalar in terms of multiple variables, Nyquist & Bode Plot

    Homework Statement A scalar is given by It is controlled by With step time h = 0.2 s 1. Find the discrete equivalent model 2. Check the stability of closed loop (K = +1) 3. Obtain the via the Bode plot Homework EquationsThe Attempt at a Solution So for question 1. This is where I'm...
  11. Physics345

    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 the point P(9, 2, –5). Homework Equations Ax+By+Cz+D=0 (x,y,z)=(x0,y0,z0)+s(a1,a2,a3)+t(b1,b2,b3) x=x0+sa1+tb1 y=y0+sa2+tb2 z=z0+sa3+tb3 The Attempt at a...
  12. Aleberto69

    Electrodynamics: questions about vector and scalar potentials....

    and Lorentz Gauge. Manipulating Maxwell equations and introducing ##\vec A## vector and ##Φ## scalar potentials the following equations are obtained: ## \nabla^2 \vec A+k^2 \vec A=-μ\vec J+\nabla(\nabla⋅\vec A+jωεμΦ) ~~~~~~~~~~(1)## ## \nabla^2 Φ+k^2 Φ=- \frac ρ ε -jω(\nabla⋅\vec...
  13. binbagsss

    A Why Scalar Quantities Matter in Singularity Tests

    Apologies if this is a stupid question, so for e.g, in a Schwarzschild space-time we look at ##R^{abcd}R_{abcd} ## (we seek some scalar quantity that blows-up and can not use ##R##as we are looking at vacuum solutions so we know this is zero) The reason we seek a scalar is because it is the...
  14. E

    B Current values for Friedman's scalar and its derivative?

    What are the current values for Friedman's scalar and its first derivative with respect to time? Thanks.
  15. A

    Understanding the Relationship Between Potential Energy and Direction in Fields

    Why do we not need to consider direction when determining the change in potential energy? Why do we need to consider it in case of force? Or am I interpreting the question correctly?
  16. F

    Euler's equation pressure difference

    Homework Statement I am after PC - PA However I must do so without breaking into components. My problem has different values L=3 H=4 SG=1.2 downward a = 1.5g horizontal a = 0.9g and my coordinate is conventional positive y up and positive x to the right cos##\theta## = 3/5 sin##\theta## =...
  17. C

    A Variation of Scalar Field Action: Polchinski's AdS/CFT Review

    I am reading Polchinski's review on AdS/CFT https://arxiv.org/abs/1010.6134. I have a very simple question, and please help me out. Thanks in advanced. The question abou formula (3.19) The scalar effective bulk action is given by $$ S_0=\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl}...
  18. T

    Potential of particle - why is there a scalar product here?

    I'm reading up on the Lagrangian equation, but what I'm asking is to do with electromagnetism. In the first equation here: http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf L equals the kinetic minus the potential energy. For the potential energy term, I just don't...
  19. binbagsss

    Complex scalar field, conserved current, expanding functional

    Homework Statement [/B] Hi I am looking at this action: Under the transformation ## \phi \to \phi e^{i \epsilon} ## Homework Equations [/B] So a conserved current is found by, promoting the parameter describing the transformation- ##\epsilon## say- to depend on ##x## since we know that...
  20. S

    Lagrangian for a single scalar field

    Homework Statement Hello all ! Over the past few day's, I've been trying to understand how Sean Carroll comes to the conclusion that he does on equation 1.153. I've tried to look for various resources online but I still have trouble understanding how he is able to add both partial derivatives...
  21. Milsomonk

    How to Approach Scalar Electron-Muon Scattering?

    Homework Statement Consider the scattering of a scalar electron and a spin half muon, draw and label the feynman diagram for this process. write down the invariant amplitude M and the spin average |M|^2, use the trace notation.Homework Equations Photon propagator $$\dfrac...
  22. Milsomonk

    Spontaneous symmetry breaking scalar field masses

    Homework Statement Determine the mass of the scalars and show that one remains zero in accordance with goldstones theorem.Homework Equations $$L=\dfrac {1}{2} (\partial_\mu \phi_a)(\partial^\mu \phi_a)-\dfrac{1}{2} \mu^2 (\phi_a \phi_a) - \dfrac{1}{4} \lambda (\phi_a \phi_a)^2+ i\bar{\psi}...
  23. R

    Contraction of a tensor to produce scalar

    Homework Statement Explain how it is possible to perform a contraction of the tensor ##T^{\beta \gamma}_{\delta \epsilon}## in order to produce a scalar T Homework EquationsThe Attempt at a Solution $$T^{\beta \gamma}_{\delta \epsilon}T_{\beta \gamma}^{\delta \epsilon}=T$$ Not sure if that is...
  24. F

    I Covariant derivative of Ricci scalar causing me grief

    Hi all I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity: \begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla...
  25. Milsomonk

    Why Are the EOMs for a Complex Scalar Field Not Independent?

    Homework Statement Find the equations of motion for the Lagrangian below: $$ L=\partial_\mu \phi^* \partial^\mu \phi - V( \phi,\phi^* ) $$ Where : $$ V( \phi,\phi^* )= m^2 \phi^* \phi + \lambda (\phi^* \phi)^2 $$Homework Equations Euler Lagrange equation: $$ \partial_\mu \dfrac {\partial L}...
  26. D

    Peskin complex scalar field current

    Homework Statement i'm trying to calculate the charge operator for a complex scalar field. I've got the overal problem right but I'm confused about this: On page 18 of Peskin, it is written that the conserved current of a complex scalar field, associated with the transformation ##\phi...
  27. Math Amateur

    MHB The Directional Derivative .... in Scalar Fields and Vector Fields ....

    I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...
  28. Philosophaie

    Scalar Potentials and Electromagnetic Current

    For a Dipole and a Torad (or a Solenoid) I need to find the scalar Potential,phi, Charge Density,rho, and then 4-Electromagnetic Current,J(rho*c,j) where A and J are 4-vectors and a and j are 3-vectors. -grad^2(phi) + 1/c^2*d/dt(phi) = rho/epsillon0 where grad(A(phi/c,a)) = -1/c^2*d/dt(phi)...
  29. P

    B Minkowski metric, scalar product, why the minus sign?

    In Schutz's A First Course in General Relativity (second edition, page 45, in the context of special relativity) he gives the scalar product of four basis vectors in a frame as follows: $$\vec{e}_{0}\cdot\vec{e}_{0}=-1,$$...
  30. P

    Show this integral defines a scalar product.

    Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
  31. romsofia

    How Do You Vary the Action of a Lagrangian for a Scalar Field?

    Homework Statement Vary the action of a Lagrangian for a scalar field. I kind of just need someone to read over this, I'm not sure if my steps are actually logical (especially the one before we do integration by parts). Since this isn't actually homework, we can move it to the classical...
  32. TeethWhitener

    I Complex scalar field commutation relations

    I'm trying to derive the commutation relations of the raising and lowering operators for a complex scalar field and I had a question. Let's start with the commutation relations: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...
  33. hugo_faurand

    B What is the reality of the scalar product?

    Hi everyone ! I would like to know the real meaning of scalar product. So, I know scalar product is defined as : ||a||.||b||.cos(a;b) = k But what k is ? (Sorry for my english, I am french). Regards :)
  34. E

    Line Integral Notation wrt Scalar Value function

    I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...
  35. hilbert2

    A Constant Solutions of Real Scalar Field

    Suppose I have a self interacting real scalar field ##\phi## with equation of motion ##\partial^i \partial_i \phi + m^2 \phi = -A \phi^2 - B\phi^3##, and I attempt to find constant solutions ##\phi (x,t) = C## for it. The trivial solution is the zero solution ##\phi (x,t) = 0##, but there can...
  36. Ken Gallock

    Non-relativistic complex scalar field

    This is spontaneous symmetry breaking problem. 1. Homework Statement Temperature is ##T=0##. For one component complex scalar field ##\phi##, non-relativistic Lagrangian can be written as $$ \mathcal{L}_{NR}=\varphi^* \Big( i\partial_t + \dfrac{\nabla^2}{2m} \Big)\varphi -...
  37. hilbert2

    A Scalar Fields with the Same Mass

    In the Peskin&Schröder's QFT book there's an exercise that's about a pair of scalar fields, ##\phi_1## and ##\phi_2##, having the field equations ##\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_1 = 0## ##\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_2 = 0## where the mass parameter...
  38. M

    Scalar Moment around a point due to a force

    Homework Statement In two dimensions, the moment of a force can be calculated using the scalar method, MO=Fd, where F is the magnitude of the force and d is the perpendicular distance from the line of force to the point where the moment is being considered. Using the scalar method, calculate...
  39. G

    I Solve Scalar Product Problem in Set R of Functions [0,1]

    Alright, so we ran into a peculiarity in answering this question. Let R be the set of all functions f defined on the interval [0,1] such that - (1) f(t) is nonzero at no more than countably many points t1, t2, . . . (2) Σi = 1 to ∞ f2(ti) < ∞ . Define addition of elements and multiplication...
  40. S

    I Experimental bound on the scalar spectral index

    Based on the paper by Visinelli (https://arxiv.org/abs/1605.06449), He stated in page 6 that the scalar spectral index as given by the Planck 2013 data (https://arxiv.org/abs/1303.5076) is, ##n_s = 0.9655 \pm 0.0062~~## (##68\%## C.L.) but when I looked into the Planck 2013 paper, I did not...
  41. L

    Ward-Takahashi identities at tree level in scalar QED

    Homework Statement Let \Gamma^\mu be the three-point vertex in scalar QED and \Gamma^{\mu\nu} be the four-point vertex. Use Feynman's rule at tree level and verify that the Ward-Takahashi identities are satisfied: q^\mu \Gamma_\mu(p_1,p_2)=e[D_F^{-1}(p_1)-D_F^{-1}(p_2)],\\...
  42. S

    I Understanding Lorentz Transformation on Scalar Fields

    Hello! Can someone explain to me how does a scalar field changes under a Lorentz transformation? I found different notations in different places and I am a bit confused. Thank you!
  43. S

    A Calculating the power spectra of scalar perturbation

    I'd like to numerically calculate the power spectra of the scalar perturbation at the Hubble crossing in warm inflation, my problem is that I don't know how to do it. As I know, the Hubble crossing happens at the onset of warm inflation where the different modes become larger than the Hubble...
  44. L

    A Why Is the Gradient of a Scalar Product Best Evaluated in Cartesian Coordinates?

    It is very well known result that ##grad[e^{i\vec{k}\cdot \vec{r}}]=i\vec{k}e^{i\vec{k}\cdot \vec{r}}##. Also ##\vec{k}\cdot \vec{r}=kr\cos \theta## and ##gradf(r)=\frac{df}{dr} grad r##. Then I can write grad e^{ikr\cos \theta}=ik\cos \theta e^{i \vec{k}\cdot \vec{r}}...
  45. binbagsss

    Complex scalar field -- Quantum Field Theory -- Ladder operators

    Homework Statement STATEMENT ##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})## where ##w_k=\sqrt{{k}.{k}+m^2}## The only non vanishing commutation relations of the creation and annihilation operators are: ## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3...
  46. Eclair_de_XII

    Bijective function between an ordered pair and a scalar?

    Homework Statement "Prove that ##S = \{(a, b) : a, b ∈ ℕ## and ##b ≥ 2a\}## is denumerable." Homework Equations Basically, my aim is to find a bijection ##f: ℕ→S##. The Attempt at a Solution Define ##f: ℕ→S## by ##f(x)≥2x##. Then suppose that there exist ##x_1∈ℕ## and ##x_2∈ℕ## such that...
  47. binbagsss

    I Does dP_x dP_y = d^2\vec P : Integration scalar / vector var

    Excuse me if this is a bad question but: Does ##d P_x d P_y = d^2 \vec P ##? I thought not because ##P_x ## is a scalar , a component of the vector, whereas ##\vec P ## is a vector? Thanks in advance
  48. S

    Differential operator acting on scalar fields

    Homework Statement I really cannot seem to be able to follow the logic of how you would use the product rule when using 4 vector differential operator. ∂μ is the differential operator, Aμ is a scalar field and φ and φ* is it's complex conjugate scalar field. I have the answer, I'd just really...
  49. andylatham82

    I Angle between vectors via scalar product vs vector product

    Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...
  50. L

    I Understanding the scalar field quantization

    I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field. The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
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