# What is Scalar field: Definition and 205 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. ### A Lagrangian density , for scalar field , vector field and Spinor field

hi, I have go through many books - they derive Dirac equation from Dirac Lagrangian, KG equation from scalar Lagrangian - but my question is how do we get Dirac or scalar Lagrangian at first place as our starting point - kindly help in this regard or refer some book - which clearly elaborate...

17. ### Varying an action with respect to a scalar field

Let us work with ##(-+++)## signature Where the metric ##g_{\mu \nu}## is the flat version (i.e. ##K=0##) of the Robertson–Walker metric (I personally liked how Weinberg derived it in his Cosmology book, section 1.1) \begin{equation*} (ds)^2 = -(dt)^2 + a^2(t) (d \vec x)^2 \end{equation*}...
18. ### A Renormalizability conditions for a real scalar field in d dimensions

I am studying the real scalar field theory in ##d## spacetime dimensions as beautifully presented by M. Srednicki QFT's draft book, chapter 18 (actually, for the sake of simplicity, let us include polynomial interactions of degree less than or equal to 6 only) \begin{equation*} \mathcal{L}...
19. ### I Classical equivalent of scalar free field in QFT

Hi there, In QFT, a free scalar field can be represented by the lagrangian density $$\mathcal{L} = \frac{1}{2}\left(\partial\phi\right)^2 - \frac{1}{2}m^2\phi^2$$ I would like to find a classical system that has the same lagrangian. If we consider the transversal motion of an elastic string...
20. ### Lagrangian for the electromagnetic field coupled to a scalar field

It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...
21. ### I EDE - Solving the Klein - Gordon Equation for a scalar field

Let us suppose we have a scalar field ##\phi##. The Klein-Gordon equations for the field can be written as $$\ddot{\phi} + 3H \dot{\phi} + \frac{dV(\phi)}{d\phi} = 0$$ The other two are the Friedmann equations written in terms of the ##\phi## H^2 =...
22. E

### B Understanding the active/passive transformation of a scalar field

##\mathcal{P}## is a point in Minkowski spacetime ##M##, and ##\varphi_1: U \in M \mapsto \mathbb{R}^4## and ##\varphi_2: U \in M \mapsto \mathbb{R}^4## are two coordinate systems on the spacetime. A scalar field is a function ##\Phi(\mathcal{P}): M \mapsto \mathbb{R}##, and we can define...
23. ### I Renormalization of scalar field theory

I was reading about the renormalization of ##\phi^4## theory and it was mentioned that in order to renormalize the 2-point function ##\Gamma^{(2)}(p)## we add the counterterm : \delta \mathcal{L}_1 = -\dfrac{gm^2}{32\pi \epsilon^2}\phi^2 to the Lagrangian, which should give rise to a...
24. E

### B Gradient of scalar field is zero everywhere given boundary conditions

I'm struggling with a few steps of this argument. It's given that we have a surface ##S## bounding a volume ##V##, and a scalar field ##\phi## such that ##\nabla^2 \phi = 0## everywhere inside ##S##, and that ##\nabla \phi## is orthogonal to ##S## at all points on the surface. They say this is...
25. ### B Derivative of a constant scalar field at a point

Wikipedia defines the derivative of a scalar field, at a point, as the cotangent vector of the field at that point. In particular; The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each...
26. ### A Commutation relations between HO operators | QFT; free scalar field

I am getting started in applying the quantization of the harmonic oscillator to the free scalar field. After studying section 2.2. of Tong Lecture notes (I attach the PDF, which comes from 2.Canonical quantization here https://www.damtp.cam.ac.uk/user/tong/qft.html), I went through my notes...
27. ### I Propagator of a Scalar Field via Path Integrals

I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...

31. ### A Verifying the Relation in Yang-Mills Theory with a Scalar Field

I'm trying yo verify the relation $$[D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi,$$ where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here, $$D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi],$$ and ...
32. ### A Classical scalar field as Dark Matter

The pressure of a scalar field is: Φ˙2−V(Φ) so to have zero or negligeable pressure it needs to have equipartition of its energy in potential and kinetic form ==> the potential must be positive. In particular a mass term m2Φ2 ... could be all right: the field should tend to roll down this...
33. ### Scalar field decomposition

If a vector field can be decomposed into a curl field and a gradient field, is there a similar decomposition for scalar fields, say into a divergence field plus some other scalar field?
34. ### I Lorentz invariance and equation of motion for a scalar field

Hi there, I just saw some lectures where they claim that the Klein Gordon equation is the lowest order equation which is Lorentz invariant for a scalar field. But I could easily come up with a Lorentz invariant equation that is first order, e.g. $$(M^\mu\partial_\mu + m^2)\phi=0$$ where M is...
35. ### SE tensor for scalar field

Homework Statement Show that if the Lagrangian only depends on scalar fields ##\phi##, the energy momentum tensor is always symmetric: ##T_{\mu\nu}=T_{\nu\mu}## Homework Equations ##T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L## The Attempt at a...
36. ### I Nonrelativistic limit of scalar field theory

The Klein-Gordon equation has the Schrodinger equation as a nonrelativistic limit, in the following sense: Start with the Klein-Gordon equation (for a complex function ##\phi##) ## \partial_\mu \partial^\mu \phi + m^2 \phi = 0## Now, define a new function ##\psi## via: ##\psi = e^{i m t}...
37. ### A Restrictions Placed on a Scalar Field by the Vacuum

Hi Everyone! I have been told that even for an entirely LOCAL scalar field φ with Lagrangian density say of the form, L = ∂/∂xμ∂/∂xμφ ± φ4 + Aφ3 + Bφ2. +. Cφ + D, that it is really bad, bad, bad because the coefficient (C) of φ is not zero! That is, ∂/∂φ(L) ≠ 0 when φ...
38. ### 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...
39. ### 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...
40. ### 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...

49. ### 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...
50. ### 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...