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. N

    Scalar triple product coplanarity

    Homework Statement Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0. Homework Equations The Attempt at a Solution if u · (v × w) = 0, then u is orthogonal to vxw, and vxw is orthogonal to v and w. therefore, u must lie in the same plane...
  2. T

    Determine Scalar Potential Function

    Hi, I wonder if someone could help me. I'm trying to find the potential function,\phi, of the field: v = y2z3i + 2xyz3j + 3xy2 z2k So using v = \nabla\phi, I have found: \frac{\partial \phi}{\partial x} = y2z3x + F(y,z) \frac{\partial \phi}{\partial y} = y2z3x + G(x,z)...
  3. R

    Units of Scalar Field \phi & Lagrangian Density

    What are the dimensions of a scalar field \phi ? The Lagrangian density is: \mathcal L= \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi So in order to make all the terms have the same units, you can try either: \mathcal L=\frac{\hbar^2}{c^2} \partial_\mu \phi \partial^\mu \phi -...
  4. T

    How to Determine Time Ordering in Phi-3 Theory for a 2-to-3 Particle Process?

    i have been given a problem for writing s matrix in second order perturbative theory for an interaction hamiltonian with phi 4 and phi 3 contributions. it is also given that our initial state is of 2 particles and final state is of three particles. now in solving that i have to take time...
  5. P

    Scalar fields: why symmetric ener-mom. tensor?

    I'm studying the properties of the energy momentum tensor for a scalar field (linked to the electromagnetic field and corresponding energy-momentum tensor) and now I'm facing the statement: "for a theory involving only scalar fields, the energy-momentum tensor is always symmetric". But I've...
  6. G

    Why is wavefunction a lorentz scalar? So confused

    I hope someone can explain this to me: In multiple textbooks I've seen it said that a single particle wave function (no spin) transforms as a Lorentz scalar. I.e. if we have a Lorentz transformation from an old frame to a new frame \overline{x}=\Lambda x (x is short for (t,x,y,z)) then...
  7. B

    D dimension scalar potential for point charge

    Homework Statement Show that with d spatial dimensions the potential \phi due to a point charge q is given by \phi (r) = \frac{\Gamma(\frac{d}{2}-1)}{4\pi^{d/2}}\frac{q}{r^{d-2}} Homework Equations The Attempt at a Solution The electric field strength is known to be: E(r) =...
  8. G

    Scalar product of many-particle states?

    How do you find the scalar product of two non-orthogonal many particle states? For example <\leftarrow,\rightarrow|\uparrow,\downarrow> I wanted to express both as a 4-vector in the up/down basis, but this seems weird, since then a state |\uparrow\downarrow+\downarrow\uparrow> is like...
  9. M

    Gradient of a scalar field in a given direction

    I have to find the gradient of a scalar field, h, at a certain point in a direction given by a vector. I know, \vec{\nabla}h will give me the direction of maximum slope, and its magnitude is the magnitude of the slope, but i don't know where to start in finding the slope in any other...
  10. B

    Is a scalar field incompatible with the Principle of Equivalence?

    As I know, Einstein initially tried describe the gravitational interaction as mediated by a scalar field, but he later gave up this idea because it is incompatible with the Principle of Equivalence.I don't know how this idea is incompatible with the Principle of Equivalence. Please help me. Thanks
  11. O

    How Do You Derive Feynman Rules for Scalar QED Using Functional Methods?

    Homework Statement Hello all, thanks for reading... I was assigned to calculate feynman rules for the scalar QED theory via functional methods. The fields are a scalar complex field \phi, \phi^* and gauge field A_\mu, and the lagrangean is \mathfrak{L} = (D_\mu \phi)^* (D^\mu \phi) - m^2...
  12. D

    Understanding Scalar Fields and the Laplace Equation: How Do They Relate?

    I've recently read about Null Identities of vector analysis. I'm having a problem in understanding what is it by "taking the curl of the grad of any scalar field is equal to zero." What is by definition of scalar field then? How would it looks like? Is position vector a scalar field? If No...
  13. E

    Statics - Moment using scalar analysis

    Homework Statement Homework Equations I think that I need to find the moment of the force F about the line BC first. Then, using that moment, find the "projection" of M_{BC} onto the X axis to find the answer. M_{BC}\sin \left(45^{o}\right) However, I am not getting anything...
  14. nicksauce

    Total Momentum Operator for Free Scalar Field

    Homework Statement I want to show that \mathbf{P} = -\int d^{3}x}\pi(x)\nabla\phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\mathbf{p}a_{p}^{\dagger}a_p for the KG field.Homework Equations \phi(x) = \int{\frac{d^{3}p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p + a_{-p}^{\dagger})e^{ipx} \pi(x) =...
  15. maverick280857

    Functional Quantization of Scalar Fields

    Hi everyone, I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is \frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re...
  16. D

    Comparing Tensor Double Dot Scalar Product Definitions

    Ok I have seen the tensor double dot scalar product defined two ways and it all boils down to how the multiplication is defined. Does anyone know which is correct? I believe the first one is correct but I keep seeing the second one in various books on finite element methods. 1. \nabla \vec{u}...
  17. H

    Partial differential: partial scalar partial vector

    Hi, I have a problem to find the meaning of a special partial differential: partial scalar partial vector. i.e. dF/dn where F is a scalar and n is a i.e. normal vector. This is a partial diff. n could be a vector consisting of partial differentials, (dT/dx,dT/dy) I have looked in...
  18. M

    Finding the Scalar Potential of a Sphere with Non-Uniform Charge Density

    Homework Statement A sphere of radius a has a charge density which varies with distance r from the center according to http://img14.imageshack.us/img14/9577/wangsnessproblem594.gif where A is a constant and http://img35.imageshack.us/img35/555/wangsnessproblem593.gif .[/URL] Find the...
  19. E

    Why Is My Calculation of the Scalar Product Incorrect?

    Homework Statement Find the scalar product of the 2 vectors. Vector A is north of east at 70 degrees with a magnitude of 3.60m Vector B is south of west at 30 degrees with a magnitude of 2.40mHomework Equations ABcosxThe Attempt at a Solution I did dot product using the formula...
  20. R

    Are Current, Potential, and Potential Difference Scalar or Vector Quantities?

    Hi Just wondering if someone can tell me if the following are scalar or vector quantities and why Current Potential Potential Difference Also, I'm wondering if we include plus/minus signs in calculations depending on the charge. Ex. would current be negative if it was a negative...
  21. C

    Sketching the Gradient of a Scalar Field: How to Implement and Interpret?

    Homework Statement Calculate the gradient of the scalar field f(x,y) = x^{2} - y^{2} . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch. Homework Equations The Attempt at a Solution So I worked...
  22. S

    Is work a scalar or a vector quantity and why?

    Is work a scalar or a vector quantity and why?
  23. T

    MATLAB Matlab code for plotting magnetic scalar potential of a sextupole

    First let me introduce myself. I'm and electronics engineer with 10 years of experience in the instrumentation of particle beams. I left the field for 10 years and have been fortunate to be granted a second lease of life on particle accelerators. Slowly over time the job has come to involve more...
  24. C

    Kretschmann Scalar: Flat Spacetime & Singularities

    The Kretschmann scalar (the full contraction of the Reimann tensor K = R_abcd R^abcd) is often used to identify singularities - i.e. for a Schwarzschild black hole, K \propto 1/r^6, so we have a singularity at r=0, but not at the Schwarzschild horizon). Clearly, as r->\infinity, K->0. Is K=0...
  25. C

    Product of a scalar and a vector

    Homework Statement I am trying to prove that: \nabla \cdot (\psi\mathbf{A}) = \mathbf{A} \cdot\nabla\psi + \psi\nabla \cdot \mathbf{A} Where nabla is a scalar function and A is a vector field The Attempt at a Solution I tried expanding both the LHS and RHS, but I think I am getting...
  26. C

    How Is the Scalar Product of 4-Vectors Defined and Proven Lorentz Invariant?

    Homework Statement If a and b are 4-vectors give the definition of the scalar product a.b and demonstrate its Lorentz invariance Homework Equations The Attempt at a Solution So (with 4-vectors double underlined!) a.b = a0b0-a1b1-a2b2-a3b3 a' = (a0*gamma - beta*gamma a1 ...
  27. S

    How can I solve this problem?

    Find the scalar equation of the plane which passes through the line intersection of planes x+y+z-4=0 and y+z-2= 0 that goes through (2,4,7) and satisfies the conditions a) it is 2 units from the orgin b) it is 3 units from the point a(5,-3,7) I would really apprecaite if some toke me...
  28. T

    Proving trace of (1,1) Tensor is a scalar

    Homework Statement Given a tensor Mab, Prove that its trace is a scalar. Homework Equations The Attempt at a Solution To prove the trace is a scalar, I know I have to prove it doesn't transform under coordinate transformations. Now, we can transform M^a_b as follows...
  29. S

    Weyl invariant scalar field theory

    I'm not sure if this is the right place for this question, so feel free to move it. Anyway, my question is, is there any good reason why the following field theory should be Weyl invariant in an arbitrary dimension d>1: S = \int d^d x \sqrt{g} \left( g^{\mu \nu} \partial_\mu \phi \partial_\nu...
  30. R

    Understanding the Relationship Between Force, Displacement, and Distance

    If force is always parallel to displacement and displacement is always in a straight line and doesn't change direction. Then the displacement is equal to distance right? Distance is a scalar quantity. Does this mean that force parallel to displacement or distance as it would be is a scalar...
  31. M

    Proof of the existence of a scalar potential

    Homework Statement Hi there - I'm wondering about how you can actually show the existence of a scalar potential for an irrotational vector field E - if \nabla \times E = 0 everywhere, then how does one show there exists a scalar potential \phi(x) such that E=- \nabla \phi ? The Attempt at...
  32. C

    Find the formula of the level curve (scalar) for the graph T(x,y) = (2x+y)/(x^2 -y^2)

    Hi this is the question that I am unsure about... Find and then sketch the level curves of the scalar field T(x,y) = (2x+y)/(x^2 -y^2) for; T = -1 T = -0.5 T = 0 T = 0.5 T = 1 I am unsure about these answers which I got by subbing each of the values into T(x,y) 1) T = -1 After...
  33. M

    Finding the Scalar Potential for a Complex 3D Conservative Force

    I have a conservative force given in three vector components as normal. Each component is a function of x, y and z. I need to find the scalar potential. I am a bit confused about this because of the force's complexity. I know the V = (integral) F.dx, for a 1 dimensional problem, but i...
  34. C

    Question About Complex Scalar Field: Advantages/Disadvantages?

    I wanted to ask a quick question about the complex scalar field. My question is that does the scalar field need to be complex in order to include the part for anti-particles or do you regards the scalar field for particles and anti-particles seperate. I saw this specifically when you second...
  35. T

    Decay rate of a scalar particle under scalar/pseudoscalar lagrangian

    Hi, I'm trying to solve problem 48.4 of Srednickis QFT-Book. It goes something like this: Homework Statement We have a scalar field with mass M and a Dirac field with mass m (M>2m). The interaction part of the lagrangian is L_a = g \varphi \bar{\Psi}\Psi L_b = g \varphi...
  36. N

    Scalar product of position vectors

    http://img520.imageshack.us/img520/9580/56788025.th.jpg See the problem above. I can do all of the problem, barring the last part. I have found r and r.r: http://img520.imageshack.us/img520/1590/29349935.jpg How does this allow me to find the minimum and maximum distance...
  37. B

    Vanishing Ricci scalar always implies vacuum?

    I have been searching through the literature and popular textbooks for this simple answer. I know that in the absence of soures, i.e. matter fields the Ricci scalar is zero. This is synonymous with saying that the Ricci scalar vanishes in vacuum and that the resulting space is flat. However...
  38. M

    Why is the volume element a scalar density of weight -1?

    In Ray d'Inverno's book on general relativity, he defines the volume element in a way which makes it a scalar density of weight -1, meaning it transforms with the inverse of the Jacobian. Every other source I have looked at seems to say it should transform with the Jacobian, making it a scalar...
  39. B

    Gradient of Scalar - Find Direction for Mosquito (1,1,2)

    The temperature in an auditorium is given by T = x2 + y2 - z. A mosquito located at (1,1,2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly? I know that the gradient of T will point to the direction where the...
  40. B

    Vectors, magitude, scalar components

    Homework Statement the displacement vector A has scalr omponents of Ax= 80.0 m and Ay= 60.0 m. the displacement vector B has a scalr component Bx= 60.0 m and a magnitude of B=75.0m. The displacement vecto C has a magnitude of C= 100.0 m and is directed at an angle of 36.9 degrees above the +...
  41. A

    A Visual Representation of the Vector Scalar Product?

    To any teachers or students, either instructing or taking, a Calculus-based Physics I course: I tutor a calculus-based general physics course in kinematics, and similar topics, and, I recently had a student approach me about his inability to grasp the scalar/dot product, in vector operations...
  42. R

    Exploring Lorentz Scalar of Chern-Simons Lagrangian

    Homework Statement Imagine a spatially 2d world. The electromagnetic field could be richer here, because you could add to the Lagrangian L an additional term (known as the Chern-Simons Lagrangian) L_{CS} = \epsilon_{0}\frac{\kappa}{2}\epsilon^{\alpha \beta...
  43. H

    Scalar & Vector: Differences & Direction

    what are the differences between scalar and vector quantities? why do we use */-in specifying direction of any scalar quantity
  44. F

    Scalar component and the vector projection of F

    A force F of 6 units acts in the direction 30 degrees west of north. An object is constrained to move north-westerly, that is, 45 degrees west of north. (a) Sketch the force vector roughly to scale on a set of axes that has the positive y axis pointing north, and write F using exact values...
  45. I

    [QFT] Feynman rules for self-interacting scalar field with source terms

    I'm not sure if this is the right place to post a graduate level course material, but I have a question about perturbative expansion of the 2n-point function of a scalar field theory. Homework Statement First, the question: In which space (position or momentum) is the topological distinctness...
  46. M

    Defining scalar product from norm

    Euclidean norm is defined usually as|v|2= g(v,v), where g is a nondegenerate, positive definite, symmetric bilinear form. But how can make it backwards? What properties must norm have that g(v,w) = (|v+w|2 - |v|2 - |w|2)/2 be a positive definite, symmetric bilinear form?
  47. I

    Second Hermitian scalar product

    Homework Statement Let (u,v)1 be a second Hermitian scalar product on a vector space V. Claim: There exists a positive transformation T with respect to the given scalar product (u,v) such that (u,v)1 = (Tu,v) for all u,v in V. Homework Equations A transformation T is positive if...
  48. M

    Question about quantization of scalar field

    Why the quantization of scalar field resolves the energy negative problem that exist in the klein-gordon equation?
  49. J

    Maxwell equations + scalar and vector potentials

    Im doing some study on scalar and vector potentials in the area of electromagnetics, and the author of the book derived this equation \vec{E} = -j\omega\vec{A} - \nabla\phi where \vec{A} = vector potential and \phi = scalar potential and \vec{E} = time harmonic form of electric field...
  50. P

    Dot, Scalar, Inner Product Question

    I have been searching for a way to relate known concepts (known to me) to the computation of the dot product in an effort to understand why it takes the form it does. I ran into a little snippet in a classical dynamics book that seems like it just may be the ticket. Here is what it says...
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