Scalar field Definition and 190 Threads

Homework Statement My question is just about a small mathematical detail, but I'll give some context anyways. (From Rubakov Sec. 2.2) An expression for energy is given by E= \int{}d^3x\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}\dot{\phi}(\vec{x}) - L, where L is the Lagrangian...

Hi, f(X)=\frac{xy^2}{x^2+y^4} is the function in question, this is the value of the function at ##X=(x,y)## when ##x\neq0##, and ##f(X)=0## when ##X=(0,y)## for any ##y## even ##y=0##. Now, along any vector or line from the origin the directional derivative ##f'(Y,0)## (where ##Y=(a,b)## is...

I was trying to derive current for Complex Scalar Field and I ran into the following:So we know that the Lagrangian is: $$L = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^* \phi$$ The Lagrangian is invariant under the transformation: $$\phi \rightarrow e^{-i\Lambda} \phi $$ and $$\phi^*...

Hello everybody. I have a free scalar in two dimensions. I know that its propagator will diverge for lightlike separations, that is t= ±x. I have to find the prefactor for this delta function, and I don't know how to do this. How do I see from, for example, \int \frac{dk}{\sqrt{k^2+m^2}} e^{i k...

If a question asks for the direction of the maximum gradient of a scalar field, is it acceptable to just use del(x) as the answer or is the question asking for a unit vector? Thanks

Homework Statement Consider the Lagrangian, L, given by L = \partial_{\mu}\phi^{*}(x)\partial^{\mu}\phi(x) - m^2\phi^{*}(x)\phi(x) . The conjugate momenta to \phi(x) and \phi^{*}(x) are denoted, respectively, by \pi(x) and \pi^{*}(x) . Thus, \pi(x) = \frac{\partial...

Homework Statement Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field \phi(x) is written as \phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik...

I am having some problem with this attached question. I also attached my answer... My problem is the appearence of the term: 2 e (A \cdot \partial C) |\phi|^2 which shouldn't appear...but comes from cross terms of the: A \cdot A \rightarrow ( A + \partial C) \cdot (A + \partial C) In my...

I would like to ask something. How is the solution of EOM for the action (for FRW metric): S= \int d^{4}x \sqrt{-g} [ (\partial _{\mu} \phi)^{2} - V(\phi) ] give solution of: \ddot{\phi} + 3H \dot{\phi} + V'(\phi) =0 I don't in fact understand how the 2nd term appears... it...

To every scalar field s(x,y) there corresponds a 'constant' vector field x = A s(x,y) and y = B s(x,y), where A,B are direction cosines. The vector field is only partially constant since only the directions, and not the magnitudes, which are equal to |f(x,y)|, of the field vectors are constant...

Question: For the scalar field \Phi = x^{2} + y^{2} - z^{2} -1, sketch the level surface \Phi = 0 . (It's advised that in order to sketch the surface, \Phi should be written in cylindrical polar coordinates, and then to use \Phi = 0 to find z as a function of the radial coordinate \rho)...

A scalar field \psi is dependent only on the distance r = \sqrt{x^{2} + y^{2} + z^{2}} from the origin. Show: \partial_{x}^{2}\psi = \left(\frac{1}{r} - \frac{x^{2}}{r^{3}}\right)\frac{d\psi}{dr} + \frac{x^{2}}{r^{2}}\frac{d^{2}\psi}{dr^{2}} I've used the chain and product rules so...

If a vector field can be decomposed how a sum of a conservative + solenoidal + harmonic field... so, BTW, a scalar field can be decomposed in anothers scalar fields too?

Hello All, In Carroll's there is a brief introduction to a dynamical dark energy in which the equation of motion for slowly rolling scalar field is discussed. Then to give an idea about the mass scale of this field it is compared to the Hubble constant, saying that it has an energy of...

Homework Statement Evaluate the scalar field ##f(r, \theta, \phi)= \mid 2\hat{r}+3\hat{\phi} \mid## in spherical coords. Homework Equations Law of Cosines? ##\mid \vec{A} + \vec{B} \mid = \sqrt{A^2+B^2+2ABCos(\theta)}## The Attempt at a Solution I'm not sure the law of cosines...

Good afternoon fellow scientists,i have a small problem in evaluating the propagator for the complex Klein-Gordon field. Although the procedure is the one followed for the computation of the propagator of the real K-G field, a problem comes up: As known: <0|T\varphi^{+}(x)\varphi(y)|0> =...

Homework Statement Show that [\hat{\phi}(x_1),\hat{\phi}^\dagger(x_2)] = 0 for (x_1 - x_2)^2 < 0 where \phi is a complex scalar field Homework Equations \hat{\phi}=\int\frac{d^3 \mathbf{k}}{(2\pi)^3 \sqrt{2\omega}}[\hat{a}(k)e^{-ik\cdot x} + b^\dagger(k)e^{ik\cdot x}]...

Sorry for reopening a closed thread. But I have exactly the same doubt as this guy: https://www.physicsforums.com/showthread.php?t=346730 And the answer doesn't actually answer his question. I do get delta(p+p'), but they just help me in getting a_{p}a_{-p} and a_{p}^{\dagger}a_{-p}^{\dagger}...

Homework Statement I am studying inflation theory for a scalar field \phi in curved spacetime. I want to obtain Euler-Lagrange equations for the action: I\left[\phi\right] = \int \left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + V\left(\phi\right) \right]\sqrt{-g} d^4x Homework...

Hello! Im currently reading Ryder's QFT book and am confused with the variation of a scalarfield. He writes that the variation can be done in two ways, \phi(x) \rightarrow \phi'(x) = \phi(x) + \delta \phi(x) and x^\mu \rightarrow x'^\mu = x^\mu + \delta x^\mu. This seems...

I have been trying to get my head around this topic for a while. As I go through the description of scalar fields, the inflation and the potential inflaton, (in description as in ned.ipac.caltech.edu), I constantly miss a concept. There must be a fundamental difference between the type of...

as i understand it the higgs field is a spin-0 scalar field that gives mass to elementry particles. How is it a scalar field? I thought it was homogenous.

Hi can I just check that i haven't done anyhting foolish here whe quantising the scalar field; \ddot{\phi} - \frac{1}{a^2}\nabla \phi + 3H\dot{\phi} - 3\frac{H}{a^2}\nabla \phi + m^2 \phi with \phi = \int \frac{d^3 K}{(2\pi)^{\frac{3}{2}}}(\chi \exp(+ikx) +\chi \dagger \exp(-ikx))...

According to Carroll, \nabla \phi is covariant under rotations. This really confuses me. For example, how could equations like \vec{F}=-\nabla V be rotationally covariant if force is a contravariant vector? I know this is strictly speaking more of a mathy question, but I still figured this...

Here and there I come across the following formula for the Lagrangian density of a real scalar field, but not a deriviation. \mathcal{L} = \frac {1}{2} [ \dot \phi ^2 - ( \nabla \phi ) ^2 - (m \phi )^2 ] Could someone show me where this comes from? The m squared term in particular...

"Quantum" gravity -- Planck's constant as a scalar field? I was just reading about a fascinating new theory on the solution to the quantum gravity problem: http://arxiv.org/pdf/1212.0454.pdf I really like it, but I have one big problem with it: The author states that G = \frac{\hbar...

Homework Statement For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as \int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b)...

Hi. Let's say we have a complex scalar field \varphi and we separate it into the real and the imaginary parts: \varphi = (\varphi1 + i\varphi2) It's Lagrangian density L is given by: L = L(\varphi1) + L(\varphi1) Can you tell the argument behind the idea that in summing the densities of...

Homework Statement Hi I a attempting to derive the expression for the conserved Noether charge for a free complex scalar field. The question I have to complete is: " show, by using the mode expansions for the free complex scalar field, that the conserved Noether charge (corresponding to complex...

Homework Statement Homework Equations Definitely related to the divergence theorem (we're working on it): The Attempt at a Solution I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...

Hi all, I'm a part III student and taking the QFT course. The following seems "trivial" but when I went and asked the lecturer, the comment was that they too hate such nitty gritty details! The problem is page 12 of Tong's notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf All...

Which is the mass dimension of a scalar filed in 2 dimensions? In 4 dim I know that a scalar field has mass dimension 1, by imposing that the action has dim 0: S=\int d^4 x \partial_{\mu} A \partial^{\mu} A where \left[S\right]=0 \left[d^4 x \right] =-4 \left[ \partial_{\mu} \right]=1...

Homework Statement I have the following task: In quantum free scalar field theory find commutators of creation and anihilation operators with total four-momentum operator, starting with commutators for fields and canonical momenta. Show that vacuum energy is zero. Homework Equations...

Homework Statement I understand the premise of Noether's theorem, and I've read over it in as many online lectures as I can find as well as in An Introduction to Quantum Field Theory; Peskin, Schroeder but I can't seem to figure out how to actually calculate it. I feel like I'm missing a...

How to prove that \nabla x (\phi\nabla\phi) = 0? (\phi is a differentiable scalar field) I'm a bit confused by this "differentiable scalar field" thing...

Okay this might be a nooby question, but it bothers me. What is the difference between the line integral of a scalar field and just a regular integral over the scalar field? For a function of one variable i certainly can't see the difference. But then I thought they might be identical in...

Homework Statement Consider the scalar field V = r^n , n ≠ 0 expressed in spherical coordinates. Find it's gradient \nabla V in a.) cartesian coordinates b.) spherical coordinates Homework Equations cartesian version: \nabla V = \frac{\partial V}{\partial x}\hat{x} +...

Homework Statement A scalar field is given by the function: ∅ = 3x2y + 4y2 a) Find del ∅ at the point (3,5) b) Find the component of del ∅ that makes a -60o angle with the axis at the point (3,5) Homework Equations del ∅ = d∅/dx + d∅/dy The Attempt at a Solution I completed part a: del ∅ =...

Homework Statement I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf. Homework Equations Everything is attached, but I...

Homework Statement For a real scalar field \phi, the propagator is \frac{i}{(k^2-m_\phi^2)}. If we instead assume a complex scalar field, \phi = \frac{1}{\sqrt{2}} (\phi_1 + i \phi_2), where \phi_1,\phi_2 are real fields with masses m_{\phi 1},m_{\phi 2}, what is the propagator...

Hallo, I was wondering what is the physical significance of scalar field \Phi (x) as an quantum operator. \Phi (x) have canonical commutation relation such as [ \Phi (x) , \pi (x) ] so it must be an opertor, thus what are his eigenstates? Thanks, Omri

Homework Statement We wish to find, in 2+1 dimensions, the analogue of E = - \frac{1}{4\pi r} e^{-mr} found in 3+1 dimensions. Here r is the spatial distance between two stationary disturbances in the field. Homework Equations In 3+1 we start from E = - \int \frac{ d^3 k }{(2\pi)^3}...

Hi, I was looking at the lagrangian and conserved currents for the free complex scalar field and it looks like it has a striking similarity to the conserved current for probability: \frac{\partial \rho}{\partial t}=\nabla\cdot \vec{j} where j_i =-i(\psi^{\ast}\partial_i \psi -...

Hi guys, If I use the definition of the scalar complex field as the combination of two scalar real fields, I can get \phi (x) = \int \frac{d^3 p}{(2\pi )^3} \frac{1}{\sqrt{2p_0}} [ \hat a _{\vec{p}} e^{-ip.x} + \hat b _{\vec{p}}^{\dagger } e^{ip.x}] which I can rewrite in terms of...

In the Kallen-Lehmann spectral representation (http://en.wikipedia.org/wiki/K%C3%A4ll%C3%A9n%E2%80%93Lehmann_spectral_representation) the interacting propagator is given as a weighted sum over free propagators. The pole of the integracting propagator is, of course, given by p^2=m^2, m being the...

Suppose I have the scalar field f in the xy-plane and that it is smooth. Its total derivative is given the normal way, i.e. df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy and the gradient of f is given the normal way as well. I read in a paper that, due to the...

For the following scalar field: \psi(x,y,z) = (y-1)z2 Find grad \psi Here is my attempt at: Multiplying out brackets: yz2 - z2 Therefore grad \psi = 0+Z2 J -2ZK Is this correct??

Hi all :) can anybody help me out in understanding scalar function and vector function? the difference between them

The effective action Γ[ϕ] for a scalar field theory is a functional of an auxiliary field ϕ(x). Both Γ and ϕ are defined in terms of the generating functional for connected graphs W[J] as W[J] + \Gamma[\phi] = \int d^dx J \phi , \quad \frac{\delta}{\delta J(x)} W[J] = \phi(x) Show - \int...

Basic question on scalar filed theory that is getting on my nerves. Say that we have the langrangian density of the free scalar (not hermitian i.e. "complex") field L=-1/2 (\partial_{\mu} \phi \partial^{\mu} \phi^* + m^2 \phi \phi^*) Thus the equations of motion are (\partial_{\mu}...