Scalar field Definition and 207 Threads

I was wondering how the notion of a time-independent field translates into the context of General Relativity. In order to specify my confusion, consider a scalar field ##\phi## in Schwarzschild spacetime with usual coordinates ##(t,r,\theta,\phi)##. Its metric is $$g = - f(r) \, dt^2 + f(r)^{-1}...

Varying ##\partial_\lambda\phi\,\partial^\lambda\phi## wrt the metric tensor ##g_{\mu\nu}## in two different ways gives me different results. Obviously I'm doing something wrong. Where am I going wrong? Method 1: \begin{equation} (\delta g_{\mu\nu})\,\partial^\mu\phi\,\partial^\nu\phi...

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...

For the solution of the equation of motion, we take a plane wave ##\phi(x) = e^{ik_\mu x^\mu}##. Plugged in, we obtain $$ -(k_0)^2 + (\vec{k})^2 = 2c_2 \Rightarrow k_\mu k^\mu = 2c_2 $$ One can then find the group velocity (using ##(k_0)^2 = \omega^2##) to be $$ \vec{v}_g =...

I am facing a problem while wanting ##\phi## dynamics in a cubic potential; ##g\phi^{3}##. The equation of motion I get for my case is(this follows from the usual Euler-Lagrange equations for ##\phi## in cosmology--Briefly discussed in Carol's Spacetime Geometry, inflation chapter)...

Hey all, I am encountering an issue reconciling the choice of prefactors in the canonical quantization of the scalar field between Srednicki and Peskin's books. In Peskin's book (see equation (2.47)), there is a prefactor of ##\frac{1}{\sqrt{2E_{p}}}## whereas in Srednicki's book (see equation...

A minimally coupled scalar field can model a cosmological fluid model where And where the equation of state can be the standard ## \omega = \frac {p} {\rho}## I can see how this does a fine job modeling matter, because as the scale factor increases, the density will go as ##\frac {1} {a^3}##...

I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the ak operators are ladder operators and fk is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...

For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...

I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...

Hi there, In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states: I do not understand this explanation. Can you please...

I am a research student of MS PHYSICS. I have to numerically solve Friedman equation coupled to scalar field(phi). It is given in research paper of Sean Carroll, Mark Trodden and Hoffman entitled as ""can the dark energy equation of state parameter w be less than-1?""...

In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...

Hi all, I am trying to work through the Unruh Effect for the (1+1)-dimensional massive scalar field case and came across the paper I attached. However, I am trying to derive equation 5.68, but am greatly struggling with the prefactor on the left hand side. When comparing the left hand side to...

Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.

Problem statement : As a part of the problem, the diagram shows the contour ##C##above on the left. The contour ##C## is divided into three parts, ##C_1, C_2, C_3## which make up the sides of the right triangle. Required to prove : ##\boxed{\oint_C x^2 y \mathrm{d} s = -\frac{\sqrt{2}}{12}}##...

A general free field Lagrangian in curved spacetime (- + + +), is given by: L = -1/2 ∇cΦ ∇cΦ - V(Φ) when the derivative index is lowered, we obtain: L = -1/2 gdc∇dΦ ∇cΦ - V(Φ) then we can choose to replace V(Φ) with something like 1/2 b2 Φ2 so: L = -1/2 gdc∇dΦ ∇cΦ - 1/2 b2 Φ2 ** I will...

Homework Statement:: The solution to the KG equation is assumed to take the form$$\Phi = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{1}{r} \phi_{lm}(t,r) Y_{lm}(\theta, \phi)$$ Relevant Equations:: N/A To first show that $$\left[ \frac{\partial^2}{\partial t^2} - \frac{d^2}{dr_*^2} +...

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*}...

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}...

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...

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. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...

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

##\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...

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...

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...

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...

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...

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...

Hey everybody, Background: I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric. The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0): $$ \partial_t\phi =...

Hi All. Given that we may write And that the Stress-Energy Tensor of a Scalar Field may be written as; These two Equations seem to have a similar form. Is this what would be expected or is it just coincidence? Thanks in advance

I think the solution to this problem is a straightforward calculation and I think I was able to make reasonable progress, but I'm not sure how to finish this... $$\begin{align*} \vec{P}&=-\int dx^3 \pi \nabla \phi\\ &= -\int\int\int dx^3\frac{dp^3}{(2\pi)^3 2e(p)} \frac{du^3}{(2\pi)^3}...

I'm trying yo verify the relation \begin{equation} [D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi, \end{equation} where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here, \begin{equation} D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi], \end{equation} and \begin{equation}...

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...

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?

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...

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...

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}...

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...

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...

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...

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...

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}...

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...

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...

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}...

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}...

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...

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...

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: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...