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. C

    I Spontaneous Symmetry breaking of multiplet of scalar fields

    Consider a theory with two multiplets of real scalar fields ##\phi_i## and ##\epsilon_i##, where ##i### runs from 1 to N. The Lagrangian is given by: $$\mathcal L = \frac{1}{2} (\partial_{\mu} \phi_i) (\partial^{\mu} \phi_i) + \frac{1}{2} (\partial_{\mu} \epsilon_i) (\partial^{\mu} \epsilon_i)...
  2. V

    A Evolution of Scalar Field: Equation Demonstration

    I'm looking for a demonstration of the equation governing the evolution of the scalar field: ## \Box \phi = \frac{1}{\sqrt{g}} \frac{ \partial}{\partial x^{\mu}} \sqrt(g)g^{(\mu)(\nu)} \frac{\partial}{\partial x^{\nu}} \phi=0## I used the lagrangian for a scalar field: ## L = \nabla_{\mu}\phi...
  3. Han_Cholo

    I Computing the normal scalar component of acceleration

    Hi, I have this math problem where I need to find the scalar component of acceleration at a given time under certain conditions. Usually these problems aren't bad for me, but this one has left me scratching my head. Its giving me ||a|| = 4 and (aT)(T) = 5i +5j -k I have the formula aN =...
  4. H

    A QED vs Scalar QED: Proving Divergence in P&S 10.1

    In Peskin and Schroeder problem 10.1 is about showing that superficially divergent diagrams that would destroy gauge invariance converge or vanish. We are supposed to prove it for the 1-photon, 3-photon, and 4-photon vertex diagrams. Does this change for scalar QED?
  5. L

    I Wave function: vector or scalar?

    Meant as element of Hilbert space of L^2 functions... etc., the wave function is a vector. In the abstract description with kets and operators on these, the wave function is the scalar product between a ket |Psi> and the "eigenkets" |x> of the position operator: psi(x) = <x|Psi>. So: psi is a...
  6. S

    A Can a Scalar Field Exist Without a Net Source?

    The magnetic field has no net source or sinks i.e. number of sources are equal to the number of sinks. Can a scalar field also have no net source? Or a source is required for a scalar field?
  7. S

    Can Scalar Field Exert Torque on Particle?

    Can a scalar field exert a torque on a particle?
  8. K

    I Function of angle between vector field and scalar function

    I was curious, if you were given a vector field F(x,y,z) = <Fx(x,y,z), Fy(x,y,z), Fz(x,y,z)>, and then some scalar function f(x,y,z), how would you define a function θ(x,y,z) of the angle θ between the scalar function and the vector field at any given point. I know how I would find this at a...
  9. darida

    Derivative of Mean Curvature and Scalar field

    Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...
  10. H

    A Canonical quantization of scalar fields

    In the srednicki notes he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int d^{3}p\frac{1}{2m}P^{2}\tilde{a}^{\dagger}(p)\tilde{a}(p) $$ Where $$\tilde{a}(p) = \int \frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}e^{-ipx}a(x)$$ Is this as simple as...
  11. C

    Computation of non scalar loop integrals

    Consider the following integral that comes out of a loop calculation along with some fermionic propagators (e.g virtual one loop correction to a ##p \gamma^* \rightarrow p'## process such as in DIS): $$ \int \frac{\text{d}^d l}{l^2 (l-p)^2 (p+q-l)^2} \text{Tr}(\not p \gamma^{\nu} (\not p + \not...
  12. Lamia

    SU(3) octet scalar quartic interactions

    Hi. General question: Is there a fixed way to find all invariant tensor for a generic representation? Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field ## \phi^{a} ## in the adjoint representation of SU(3). They will be terms like ##...
  13. W

    Pressure tensor reduces to scalar pressure for isotropic dis

    1. Does anyone know why for an isotropic distribution function, pressure tensor reduces to a scalar pressure? For instance, for a Maxwellian distribution P=A ∫ vx vy exp-(vx2 + vy2 + vz2) dvx dvy dvz is not zero. I think everybody should realize how bogus some of the authors are. Google...
  14. F

    Physical motivation for integrals over scalar field?

    I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found: If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral It appears to me that...
  15. W

    Prove the transformation is scalar

    Homework Statement 1.) Prove that the infinitesimal volume element d3x is a scalar 2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar. Homework EquationsThe Attempt at a Solution [/B] 1.) Rkh = ∂x'h/∂xk By coordinate transformation, x'h = Rkh xk dx'h =...
  16. FrancescoS

    Performing Wick Rotation to get Euclidean action of scalar f

    I'm working with the signature ##(+,-,-,-)## and with a Minkowski space-stime Lagrangian ## \mathcal{L}_M = \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi ## The Minkowski action is ## S_M = \int dt d^3x \mathcal{L}_M ## I should obtain the Euclidean action by Wick rotation. My...
  17. S

    Taylor expansion of a scalar potential field

    Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...
  18. H

    ##\overline{MS}## in scalar theory references

    Does anyone know any good references for discussion of ##\overline{MS}## theory in phi^4 theory?
  19. RJLiberator

    Inner product propety with Scalar Matrix (Proof)

    Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...
  20. C

    Self interacting scalar field

    Homework Statement A self-interacting real scalar field ##\psi(x)## is described by the Lagrangian density ##\mathcal L = \mathcal L_o + \mathcal L_I = \frac{1}{2} (\partial_{\mu}\psi)(\partial^{\mu}\psi) − \frac{1}{2}m^2\psi^2 − \frac{g}{4!}\psi^4 ## where g is a real coupling constant, and...
  21. L

    Confusion regarding the scalar potential

    Homework Statement Consider the following in cylindrical coordinates \rho,\varphi,z. An electric current flows in an infinitely long straight cylindrical wire with the radius R. The magnetic field \mathbf{B} outside of the thread is...
  22. H

    Order of scalar interaction impact Feynman diagrams

    On page 60 of srednicki (72 for online version) for the $$\phi^{3}$$ interaction for scalar fields he defines $$Z_{1}(J) \propto exp\left[\frac{i}{6}Z_{g}g\int d^{4}x(\frac{1}{i}\frac{\delta}{\delta J})^{3}\right]Z_0(J)$$ Where does this come from? I.e for the quartic interaction does this...
  23. C

    Derivation of momentum for the complex scalar field

    The conserved 4-momentum operator for the complex scalar field ##\psi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2)## is given in terms of the mode operators in ##\psi## and ##\psi^{\dagger}## as $$P^{\nu} = \int \frac{d^3 p}{(2\pi)^3 }\frac{1}{2 \omega(p)} p^{\nu} (a^{\dagger}(p) a(p) +...
  24. S

    Lorentz transformation of a scalar field

    Hi, the following is taken from Peskin and Schroeder page 36: ##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)## It describes the transformation law for a scalar field ##\phi(x)## for an active...
  25. B

    Triangle inequality implies nonnegative scalar multiple

    I'm not really sure if this is true, which is why I want your opinion. I have been trying to prove it, but it will help me a lot if someone can confirm this. Let ## v_{1}, v_{2} ... v_{n} ## be vectors in a complex inner product space ##V##. Suppose that ## | v_{1} + v_{2} +...+ v_{n}| =...
  26. J

    Example of curvature scalar diverging at infinity?

    Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity. Can someone give an example of space-time whose scalar invariant diverges...
  27. F

    Non conservative electric field and scalar potential V

    Hello forum. The electric field generated by a changing magnetic field is not conservative. A conservative field is a field with the following features: the closed line integral is zero the line integral from point A to point B is the same no matter the path followed to go from A to B it is...
  28. bcrowell

    Curvature singularity with well-behaved Kretschmann scalar

    Does anyone know of an example, preferably a simple one, that can be used to demonstrate that we can have a curvature singularity without a singularity in the Kretschmann scalar?
  29. S

    Negative scale factor RW metric with scalar field

    Homework Statement The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier. Homework Equations I have this action S=-\frac{1}{2}\int...
  30. M

    Grassmann Integral into Lagrange for scalar superfields

    I have a more philosophical question about the interpretation of a mathematical process. We have a chiral superscalarfield shown as partiell Grassmann Integral and transform it into a lagrange. where S and P are real components of a complex scalarfield and D and G are real componentfields of...
  31. loops496

    Scalar Field Solution to Einsteins Equations

    Homework Statement Compute $$T_{\mu\nu} T^{\mu\nu} - \frac{T^2}{4}$$ For a massless scalar field and then specify the computation to a spherically symmetric static metric $$ds^2=-f(r)dt^2 + f^{-1}(r)dr^2 + r^2 d\Omega^2$$Homework Equations $$4R_{\mu\nu} R^{\mu\nu} - R^2 = 16\pi^2 \left(...
  32. K

    Scalar interactions amd chirality

    why do scalar interactions(for example the higgs vev or its components) reverse the chirality of the interacting particle?? i think this is the key for understanding the mass generation of fermions, but i can't think of a logical reason of the reversed chirality.
  33. N

    Does the scalar of Weight (W) = mg all the time?

    I have a question in my textbook where I'm given weight of a "penguin in a sled" as 80N but the object is on a 40 degree angle. Is it telling me that on a normal flat surface the weight is 80N so that way to figure out the force of gravity on the x-axis I must divide the W by 9.8 then plug in my...
  34. S

    Why work done by a force is a scalar product

    Why work done by a force was taken as dot product between force applied and displacement caused?
  35. P

    Double covariant derivative of function of scalar

    If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.
  36. D

    SR & Lorentz Scalar Fields: Covariant Diff. & Wave Amplitude

    Hi. In GR , covariant differentiation is used because the basis vectors are not constant. But , what about in SR ? If the basis vectors are not Cartesian then they are not constant. Does covariant differentiation exist in SR ? And are for example spherical polar basis vectors which are not...
  37. A

    Statics - Moment using both vector and scalar approaches

    Homework Statement Homework Equations Mo=Fd Mo=r x F The Attempt at a Solution Alright guys, I did the whole process but I'm pretty sure I just made a little bump somewhere in my calculations which screwed up my answers. First I found everything I could find OA = 350j, so the unit vector...
  38. Korbid

    How to Solve the Scalar Density Integral in Spherical Coordinates?

    hi! i need to solve this integral: \rho_s=\int (m/\omega)e^{-\omega/T}d{\vec k} where \omega=\sqrt{m^2+{\vec k}} is the dispersion relation, T is the temperature of the system and m the mass of a particle Thank you!
  39. m4r35n357

    Sign of Kretschmann Scalar in Kerr Metric

    This question is motivated by one on stack exchange, and on this paper (which comes across a bit student-y but it claims to have been reviewed, and in any case I have reproduced its results in ctensor and gnuplot). So: the KS (abbreviation!) conveys an overview of curvature at a given point in...
  40. H

    Index Notation, multiplying scalar, vector and tensor.

    I am confused at why ##V_{i,j}V_{j,k}A_{km,i}## the result will end up being a vector (V is a vector and A is a tensor) What are some general rules when you are multiplying a scalar, vector and tensor?
  41. G

    Scalar product using right hand rule ?

    Homework Statement Refer to solution II , the author used the scalar analysis( dot product) to get the direction of moment ...IMO , this is incorrect ... Only cross product can be determined this way . correct me if I'm wrong . Homework EquationsThe Attempt at a Solution
  42. H

    Scalar fields and the Higgs boson

    This is more of a QFT question, so the moderator may want to move it to another forum. The simplest example of a QFT that I learned was the scalar field; in Sakurai's 1967 textbook. I know the Higgs is a J=0 particle. Is it described by the simple scalar field discussed in Sakurai's text? I ask...
  43. E

    Determinant of 3x3 matrix equal to scalar triple product?

    The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...
  44. jk22

    Does Dirac notation apply to tensor product in tensor analysis?

    Just a question : do we have in Dirac notation $$\langle u|A|u\rangle\langle u|B|u\rangle=\langle u|\langle u|A\otimes B|u\rangle |u\rangle$$ ?
  45. loops496

    Directional Derivative of Ricci Scalar: Lev-Civita Connection?

    I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...
  46. T

    Prove Determinant Using the Triple Scalar Product

    Homework Statement I'm supposed to prove det A = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{kr} using the triple scalar product. Homework Equations \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{ kr} (\vec u \times \vec v) \cdot \vec w = u_i v_j w_k...
  47. auditt241

    Unit Tangent Vector in a Scalar Field

    Hello, I am attempting to calculate unit normal and tangent vectors for a scalar field I have, Φ(x,y). For my unit normal, I simply used: \hat{n}=\frac{\nabla \phi}{|\nabla \phi|} However, I'm struggling with using this approach to calculate the unit tangent. I need to express it in terms of the...
  48. S

    Normalization of free scalar field states

    Hi, if we adopt the convention, a^{\dagger}_\textbf{p} |0\rangle = |\textbf{p}\rangle then we get a normalization that is not Lorentz invariant, i.e. \langle \textbf{p} | \textbf{q} \rangle = (2\pi)^3 \delta^{(3)}(\textbf{p} - \textbf{q}) . How do I explicitly show that this delta...
  49. S

    Self-adjointness of the real scalar field

    Hello, This problem is in reference to the QFT lecture notes (p.18-19) by Timo Weigand (Heidelberg University). He writes: For the real scalar fields, we get self-adjoint operators \phi(\textbf{x}) = \phi^{\dagger}(\textbf{x}) with the commutation relations [\phi(\textbf{x})...
  50. N

    Curl of Gradient of a Scalar Field

    Hello, new to this website, but one question that's been killing me is how can curl of a gradient of a scalar field be null vector when mixed partial derivatives are not always equal?? consider Φ(x,y,z) a scalar function consider the determinant [(i,j,k),(∂/∂x,∂/∂y,∂/∂z),(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)]...
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