Scalar Definition and 829 Threads

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)]...

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...

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...

From a textbook. proof that the scalar product ##A\centerdot B## is a scalar: Vectors A' and B' are formed by rotating vectors A and B: $$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$ $$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...

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)##...

Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS: ##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'## I posted this question long ago and i was told that this is the scalar product...

I've started f(T) theory but I have a simple question like something that i couldn't see straightforwardly. In Teleparallel theories one has the torsion scalar: And if you take the product you should obtain But there seems to be the terms like . How does this one vanish? because we know...

What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value

Homework Statement How does the scalar product of displacement four vector with itself give the square of the distance between them? Homework Equations (Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector) or how ds2=ηαβdxαdxβ The Attempt at a Solution Clearly I am completely new to the...

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...

Hey, all. I have a question concerning the treatment and use of vectors when solving problems (or in general, really). I know that vectors have both magnitude and direction, while scalars only have magnitude. However, in solving problems and looking at how others have solved them, I've noticed...

we know work is F.s which is a scalar quantity. but why it is defined this way? work actually has something to do with direction. Negative work means reduction in velocity and velocity has direction. What was the problem of defining work as vector that lies in the direction of displacement??

Question is regarding Scalars and Vectors Article. Q: One of the two forces is double the other and their resultant is equal to the greater force . The angle between them is ? Ans : Its answer is cos^-1 (-1/4) My solution : The formula for Cosine law is R= √A²+B² +2ABcos theta Our teacher...

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!

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...

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...

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>...

What does it mean if a scalar field φ is said to be constant on a surface S? Does φ then have a particular mathematical relationship with S?

Homework Statement Prove that for any three vectors ##\hat a, \hat b ## and ## \hat c##, ##\hat a \cdot (\hat b \times \hat c)## = ##(\hat a \times \hat b) \cdot \hat c ## Homework Equations [/B] ## \hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = (1)(1)\cos(0) = 1 ## ##...

Homework Statement (a) Find the christoffel symbols (Done). (b) Show that ##\phi## is a solution and find the relation between A and B.[/B] Homework EquationsThe Attempt at a Solution Part(b) \nabla_\mu \nabla^\mu \phi = 0 I suppose for a scalar field, this is simply the normal derivative...

Does this integration of Ricci scalar over surface apply in general or just for compact surfaces? ∫RdS = χ(g) where χ(g) is Euler characteristic. And could anybody give me some good references to prove the formula?

Hello, Thanks to all of you for the help that you provide so that we can move forward. I need help about the computation of the Kretschmann scalar without using software, I face to some difficulties on its computation for instance in the case of Schwarzschild metric. Could some one help me with...

Hi, This is overwhelmingly more of a maths problem than a physics problem, because it's all theoretical. I'll give some background to modle it incase the math's isn't enough. Say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0...

Hi to all the readers of the forum. I cannot figure out the following thing. I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V). I know that a scalar (in Galileian Physics) is something that is invariant under rotation. How can I reconcile this...

Homework Statement Fact: Being the dot product of force and distance, work is a scalar. Fragment from my textbook: The work done on the spring is ##\frac{1}{2}kx^2##, and so the work done by the spring is ##-\frac{1}{2}kx^2##. Homework Equations ##W = f \cdot d ## The Attempt at a Solution I...

I have that ∇2∅ = 0 everywhere. ∅ is a scalar potential and must be finite everywhere. Why is it that ∅ must be a constant? I'm trying to understand magnetic field B in terms of the Debye potentials: B = Lψ+Lχ+∇∅. I get this from C.G.Gray, Am. J. Phys. 46 (1978) page 169. Here they found that...

In examining the work energy theorem on vector fields, I have concluded that friction must be a scalar field with a negative value. This is because one must integrate the line integral with respect to ds instead of the function dotted with dr. Am I correct in my understanding or am I missing...

I am reading some of "Planck 2013 results. XXII. Constraints on inflation." The paper is full of values for various inflationary parameters under various models, with their confidence intervals. For instance, in Table 5 on page 13, the authors report that — for a model including both running of...

Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?

I have a question that is very basic and could not seem to find it online or I have not searched the right way. What is the propagator of a scalar boson? I found that of a fermion line and that of a vector boson but could not find that of a scalar boson.

Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...

Homework Statement How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation? Homework EquationsThe Attempt at a Solution

Can u describe VECTOR and SCALAR in detail with example

so for surface integral for scalar quantities. Why do we use cross product not dot product in the integral? but can we just add an unit normal vector n to make the direction the same? My question seems really stupid too a lot people, but this is really my confusion to surface integral. please...

Hi guys, So I've got a real scalar field which is the sum of the positive frequency part and negative frequency part: \phi(x)=\phi^{(+)}(x)+\phi^{(-)}(y) and I'm looking at the time-ordered product: T(\phi(x)\phi(y))=\theta(x^{0}-y^{0})\phi(x)\phi(y)+\theta(y^{0}-x^{0})\phi(y)\phi(x) for...

Homework Statement Find the equation of the plane that goes through points P, Q and R. P = (3, -1, 2), Q = (8, 2, 4) and R = (-1, -2, -3) Homework Equations Eq of plane 0 = a(x - x0) + b(y - y0) + c(z - z0) The Attempt at a Solution In order to find vector normal to the plane, my teacher...

Let's see if I think correctly first: I think a vector is a group of numbers independent of each other. What we say 3D vector means "it takes three numbers to specify a position and these numbers are not (explicitly) dependent on each other. The so called 'direction' of a vector is a...

Dear all, Can anyone please explain how the linear combination of non-coplanar and non-orthogonal coordinate axes representing a point x as shown below is derived. Please use the reference text attached in this post to explain to me as i will find it a bit relevant. I want to...

Dear all, My question is from the text of Alan F. Beardon, Algebra and Geometry concerning the scalar triple product. I have attached the text in this post. In order for the STP to be non-zero. The 3 vectors must be distinct and they are not coplanar. 2 vectors can be coplanar...

Homework Statement I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a...

Homework Statement I am a teacher currently teaching very introductory physics. I just came across a test question asking the students to choose whether acceleration is vector or scalar. Homework EquationsThe Attempt at a Solution I have always thought that acceleration can be either vector or...

Homework Statement Consider the following real scalar field in two dimensions: S = \int d^2 x ( \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2 - g \phi^3) What are the Feynman rules for calculating < \Omega | T(\phi_1 ... \phi_n ) | \Omega > 2. Homework...

Homework Statement [/B] I am trying to compute ##R## from the 3-d metric: ##ds^{2}=d\chi^{2}+f^{2}\chi(d\theta^{2}+sin^{2}\theta d\phi^{2})##Homework Equations [/B] The space also satisfies the below relationships: ##R=3k## ## R_{abcd}=\frac{1}{6}R(g_{ac}g_{db}-g_{ad}g_{bc})## [1] The Attempt...

If the physical quantity angle doesn't follow the vector addition property (only infinitesimal angles follow this), why is it even considered to be a vector? Because i thought electrical current isn't considered to be a vector because it doesn't follow this rule. Why isn't it enough to rule out...

Would you please explain in details , why pressure is a scalar, though, pressure = \frac {force}{area} and force is a vector ?

The volume of a triangular prism is given by: v = ½ |a • b x c| Where b and c are two of the sides of the triangular face of the prism, and a is the length of the prism. The volume of a rectangular/parallelogram-based pyramid is given by: V = ⅓ |a • b x c| My question is, what are a, b...

[A,B] = AB-BA, so the commutator should be a matrix in general, but yet [x,p]=i*hbar...which is just a scalar. Unless by this commutator, we mean i*hbar*(identity matrix) ? I am asking because I see in a paper the following: tr[A,B] Which I interpret to mean the trace of the commutator...

Homework Statement Demonstrate the equivalence between the gauge fields A1=(0,bx,0) and A2=)-yB/2,xB/2,0) and find the scalar field Φ for which A1= A2 + ∇ΦHomework Equations B = ∇XA The Attempt at a Solution The first part is fine, you just plug it into the above relevant equation and you get...

Im currently reading these lectures notes on yukawa scattering (charged scalars and real scalars). http://www.damtp.cam.ac.uk/user/tong/qft.html In the interaction part he focuses strictly on 2 particle to 2 particle scattering, is there a reason other types are not discussed? For example a 2...

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...