Scalar field Definition and 207 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. 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...
  2. Ken Gallock

    I Lorentz transformation and its Noether current

    Hi. I'd like to ask about the calculation of Noether current. On page16 of David Tong's lecture note(http://www.damtp.cam.ac.uk/user/tong/qft.html), there is a topic about Noether current and Lorentz transformation. I want to derive ##\delta \mathcal{L}##, but during my calculation, I...
  3. 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...
  4. 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 -...
  5. 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...
  6. F

    I How to compute second-order variation of an action?

    Starting with the action for a free scalar field $$S[\phi]=\frac{1}{2}\int\;d^{4}x\left(\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-m^{2}\phi^{2}(x)\right)=\int\;d^{4}x\mathcal{L}$$ Naively, if I expand this to second-order, I get $$S[\phi+\delta\phi]=S[\phi]+\int\;d^{4}x\frac{\delta...
  7. Ken Gallock

    I What does it mean: "up to total derivatives"

    Hi. I don't understand the meaning of "up to total derivatives". It was used during a lecture on superfluid. It says as follows: --------------------------------------------------------------------- Lagrangian for complex scalar field ##\phi## is $$ \mathcal{L}=\frac12 (\partial_\mu \phi)^*...
  8. 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...
  9. L

    Hamiltonian in terms of creation/annihilation operators

    Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
  10. 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...
  11. F

    I A question about assumptions made in derivation of LSZ formula

    I've been reading through a derivation of the LSZ reduction formula and I'm slightly confused about the arguments made about the assumptions: $$\langle\Omega\vert\phi(x)\vert\Omega\rangle =0\\ \langle\mathbf{k}\vert\phi(x)\vert\Omega\rangle =e^{ik\cdot x}$$ For both assumptions the author first...
  12. 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...
  13. 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...
  14. 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...
  15. 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 ##...
  16. 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...
  17. 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?
  18. 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...
  19. 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...
  20. D

    A Exact solutions of quintessence models of dark energy

    Hi everyone, I got the basic ideas quintessence (minimally coupled) and derived the KG equation for scalar field: $$ \ddot{\phi} + 3 H \dot{\phi} + \frac{\partial V(\phi)}{\partial \phi} = 0 $$ where $$H=\frac{\dot{a}}{a}$$ and $\phi$ is the scalar field. There are various models depending...
  21. 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...
  22. 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...
  23. 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...
  24. I

    Why Don't First-Order Terms Disappear in the Taylor Expansion for Scalar Fields?

    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'}...
  25. 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?
  26. D

    A Time dependence of field operators

    In field theory we most of deal with theories whose Lagrangian densities are of the form (sticking to scalar fields for simplicity) $$\mathcal{L}= -\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m_{\phi}^{2}\phi^{2} + \cdots$$ where ##\partial := \frac{\partial}{\partial x^{\mu}}##...
  27. 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)...
  28. 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)...
  29. 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...
  30. 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?
  31. S

    Can Scalar Field Exert Torque on Particle?

    Can a scalar field exert a torque on a particle?
  32. 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)...
  33. 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...
  34. C

    How Do You Analyze Two-Body Scattering in Scalar Field Theory?

    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...
  35. 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) +...
  36. 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...
  37. 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...
  38. 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(...
  39. 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...
  40. 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...
  41. 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})...
  42. 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)]...
  43. P

    Ladder operators for real scalar field

    Puting a minus in front of the momentum in the field expansion gives ##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde...
  44. S

    Lorentz transformation of a scalar field

    Hello, I'm reading Tong's lecture notes on QFT and I'm stuck on the following problem, found on p.11-12. A scalar field \phi , under a Lorentz transformation, x \to \Lambda x , transforms as \phi(x) \to \phi'(x) = \phi(\Lambda^{-1} x) and the derivative of the scalar field transforms...
  45. D

    Lorentz scalars - transformation of a scalar field

    When one considers a Lorentz transformation between two frames ##S## and ##S'##, such that the coordinates in ##S## are given by ##x^{\mu}## and the coordinates in ##S'## are given by ##x'^{\mu}##, with the two related by x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu} then a scalar field ##\phi (x)##...
  46. L

    In the interacting scalar field theory, I have a question.

    First of all, I copy the text in my lecture note. - - - - - - - - - - - - - - - - - - - In general, $$e^{-iTH}$$ cannot be written exactly in a useful way in terms of creation and annihilation operators. However, we can do it perturbatively, order by order in the coupling $$ \lambda $$. For...
  47. Spinnor

    Give mass to a massless scalar field in 1+1, Higgs like?

    Is it possible to have a free massless scalar field in 1+1 spacetime and then add another field of the right type which interacts with some adjustable strength with the massless field to give mass to the massless field? Is there a Higgs-like mechanism in 1+1 spacetime? Thanks for any help!
  48. K

    Solution of equation for decaying real scalar field

    Suppose there is a real scalar field ##\phi## with some decay width ##\Gamma## to some fermion. The quantum equation of motion after one-loop correction takes the form ##\ddot{\phi}+(m^2+im\Gamma)\phi=0## where ##m## is the renormalized mass. The solution can be obtained as ##\phi=\phi_0...
  49. K

    Scalar field energy density and pressure in hot universe

    Kolb&Turner in "the early universe" mentioned that for a scalar field ##\phi## at finite temperature, ##p=-V_T(\phi)## and ##\rho=-p+T\frac{d p(T)}{d T}## where ##V_T## is potential energy including temperature correction. My question is: when we consider the evolution of the universe using...
  50. I

    Expectation value of a real scalar field in p state

    Hello, I've been trying to find <p'|φ(x)|p> for a free scalar field. and integral of <p'|φ(x)φ(x)|p> over 3d in doing the space In writing φ(x) as In doing the first, I get the creation and annihilation operators acting on |p> giving |p+1> and |p-1> which are different from the bra state |p>...
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