Scalar Definition and 777 Threads

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...

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...

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...

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...

I'm learning time-dependent Maxwell's Equations and having difficulty understanding the following derivative: Given f(\textbf{r}, \textbf{r}', t) = \frac{[\rho(\textbf{r}, t)]}{|\textbf{r} - \textbf{r}'|} where \textbf{r} = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}, in...

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^*...

I have managed to get to the end of Chapter 6 and have done almost all of the exercises (I didn't get anywhere with exercise 5.6 (d) and have seen the Cesarth/TSny exchange and still don't feel I have a satisfactory solution...) but I have hit a bit of a wall with Exercise 7.1 (a).First question...

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...

Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...

Dear all! I think the main difference between scalar and vector fields is that vectorial fields are composed of vector elements that varies among them. Scalar fields are fields that have large regions of equal magnitude, variations are just presented in different regions. Please bring me help...

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...

Hi, Are ALL scalar products of four-vectors Lorentz-invariant (as opposed to just the scalar product of a four-vector with itself)? And, if yes, what is the proof?

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...

For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum: P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2, where: c = G = ħ = 1, k is the...

Is Average Speed a Scalar quantity?

Apparently, f \nabla^2 f = \nabla \cdot f \nabla f - \nabla f \cdot \nabla f where f is a scalar function. Can someone please show me why this is step by step. Feel free to use suffix notation. Thanks in advance.

Why is a charge from say an electron a scalar. It has a constant magnitude, and it has a direction.

Hello, On p.573 of Jackson 2nd Ed. (section 12.1), he says, "From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extremum condition, \delta A=0." I agree that if the action is a Lorentz scalar, then...

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...