Scalar Definition and 777 Threads

##\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...

Starting from the definition of a matrix exponential as a power series, how would we show that ##(e^A)^n=e^{nA}##? I know how to show that if A and B commute then ##e^Ae^B = e^{A+B}## and from this we can show that the first identity is true for integer values of n, but how can we show it’s...

For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...

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...

Hi, if I want to construct the most general Lagrangian of a single scalar field up to two fields and two derivatives, I usually see that is $$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ i.e. the Klein-Gordon Lagrangian. My question is, would be valid the Lagrangian $$\mathscr{L} = \phi...

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...

Suppose ##\alpha=0##. Then ##\alpha f=0##, the zero map. Hence, the distance between the images of any two ##x_1,x_2 \in D## through ##f##, that is to say, the absolute difference of ##(\alpha f)(x_1)=0## and ##(\alpha f)(x_2)=0##, is less than any ##\epsilon>0## regardless of the choice of...

Hello! This is probably a silly question (I am sure I am missing something basic), but I am not sure I understand how a Hamiltonian can be a scalar and allow transitions between states with different angular momentum at the same time. Electromagnetic induced transitions are usually represented...

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...

Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...

I'm learning some QFT from QFT for the Gifted Amateur. Chapter 11 develops the massive scalar quantum field but they don't seem subsequently to do anything with it. I've looked ahead at the next few chapters, which move on to other stuff, which leaves me wondering what we we actually do with...

"My" Attempted Solution To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation. With that said, below is my...

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}...

Hello everyone, I am working on the following problem: I would like to determine the invariant Matrix element of the process ##\psi\left(p,s\right)+\phi\left(k\right)\rightarrow\psi\left(p',s'\right)+\phi\left(k'\right)## within Yukawa theory, where ##\psi\left(p,s\right)## denotes a fermion...

Lorentz gauge: ∇⋅A = -μ0ε0∂V/∂t Gauss's law: -∇2V + μ0ε0∂2V/∂t2 = ρ/ε0 Ampere-Maxwell equation: -∇2A + μ0ε0∂2A/∂t2 = μ0J I started with the hint, E = -∇V - ∂A/∂t and set V = 0, and ended up with E0 ei(kz-ωt) x_hat = - ∂A/∂t mult. both sides by ∂t then integrate to get A = -i(E0/ω)ei(kz-ωt)...

Background: I am currently reading up on homogenization theory. I have a simple conductivity model (image attached). u is a scalar function (such as potential or temperature). The textbook proceeds by giving a series expansion for the gradient of u (image attached). the problem is that the...

Summary: Why is there no contradiction between energy as a non relativistic scalar and Galilean invariance? If energy is a non relativistic scalar, doesn't it mean that there is a contradiction with Galilean invariance? What i mean is that if i try to accelerate an object within the Galilean...

Two planes, plane 1 and plane 2, intersect in the line with symmetric equation (x-1)/2 = (y-2)/3 = (z+4)/1. Plane 1 contains the point A(2,1,1) and plane 2 contains the point B(1,2,-1). Find the scalar equations of planes plane 1 and plane 2. I have no idea how to do it, all help will be...

I read that ##\delta^4(x-y)## is invariant under Lorentz transformations. I was trying to show myself this, so I procceded as follows. The following integrals are both equal to 1 ##\int \delta^4(x-y) d^4 x## and ##\int \delta^4 (x'-y') d^4x'## so I assume they are equal to one another, as long...

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}...

Why is Kinetic energy a scalar quantity? I read in an article, it said, when the velocity is squared, it is not a vector quantity anymore. Can someone fill in the gaps for me? I can't quite get what that article said. And I would be pleased if you provide some other examples other than kinetic...

While I was going through "Introduction to Electrodynamics" by David J. Griffith I see the line "Current is a vector quantity". But we know it doesn't obey the vector algebra (addition ). Then how it can be a vector?... Please help me

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...

Homework Statement How come altitude of a mountain is a scalar? Homework Equations Scalars = only magnitude Vectors = have magnitude & direction The Attempt at a Solution - Doesn't altitude of a mountain have both magnitude and direction (direction being measured straight up 90 degrees to the...

Why does Celsius degrees have +/- signs, since it's scalar?

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?

Homework Statement Let a and b be non-zero vectors in space. Determine comp a (a × b). Homework Equations comp a (b) = (a ⋅ b)/|a| The Attempt at a Solution [/B] comp a (a × b) = a ⋅ (a × b)/|a| = (a × a) ⋅b/|a| = 0 ⋅ b/|a| = 0 Is this the answer? Or is there more to it?

Given metric ds2=dr2-r2dθ2 Gamma comes as Γ122=r,Γ212=Γ221=1/r The Reimann tensor comes as R11=R2121=∂1Γ212-Γm12Γ21m=0,only non zero terms . Similary R22=R1212=∂1Γ122-Γm21Γ1m2=0,only non zero terms. Therefore R(ricci scaler)=0 Is the space flat??

I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms." Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1)...

Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5) Homework EquationsThe Attempt at a Solution Finding the scalar equation: Ax+By+Cz+D=0 3x-4y+6z+D=0 3(9)-4(2)+6(-5)+D=0 -11+D=0 D=11...

Hello, I have this potential: ## V(\chi) = \frac { F ’’ (\chi) [ 2 F(\chi) - \chi F’ (\chi) ]}{ (F’(\chi))^3} ## How to get ## \frac{ d V(\chi)}{d \chi} = \frac{ \chi F’’ + F’ - F’ }{ F’^2} - 2 \frac{ \chi F’ -F }{ F’^3} F’’ ~~~~~~(*)## My trail, ## V( \chi) = 2 F F’’ F’^{-3} - \chi F’’...

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...

I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...

Hello! Can someone direct me towards a reading where the Feynman rules for scalar QED are derived? Thank you!

1 Is Equilibrium a Scalar or Vector quantity? 2 What is the unit of Equilibrium? Thanks & Regards, Prashant S Akerkar

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...

Knowing that a scalar quantity doesn't change under rotation of a coordinate system. Do we consider a point in a Cartesian coordinate system (i.e. A (4,5)) a scalar quantity? If yes, why do the components of point A change under rotation of the coordinate system? According to my understanding...

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}...

Homework Statement Two charged balls are placed in point A and B and the distance between them is 9,54cm. Each of the balls are charged with 8,0 x 10^-8 C. Find the scalar value and direction of the electric field in point C placed 5 cm from A and 6 cm from B. Homework Equations Cosine Rule...

I was going through vector and scalar quantities (the way they are taught in high school), and this is how I think students are supposed to understand it: Scalar quantities are quantities that add like numbers. For e.g. Mass. If I add 100 g of water to a bucket and then add a further 100 g, I...

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...

Homework Statement a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself. b: What would be the r, θ components of the unit vector in the θ direction? Homework Equations Scalar product of 2 vectors = AαgαβBβ The Attempt at a Solution For part a, I used the...

So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...